![List of Figures
Figure 1-1. Diagram showing various applications of different interconnect technologies.
Optical links are currently only used at longer distances (>100m) [2].................................... 4
Figure 1-2. Timeline of information carrying capacity for links in various electrical and
optical links. Those in optical systems have data rates roughly 2 orders of magnitude higher
than those in electrical systems [2].......................................................................................... 5
Figure 1-3. Relationship between processor and front side bus (FSB) clock speeds for
various models over the past 15 years. Note the significant improvement in processor
speeds over FSB speeds in recent years [2]............................................................................. 6
Figure 1-4. Diagram showing individual components of an optical transmitter-receiver
system. The transmitter is the top diagram and the receiver is the bottom [4]........................ 7
Figure 1-5. Wafer cost per unit area for various materials versus wafer size. Notice that
InP is currently only available in one wafer size and is prohibitively expensive compared to
a Si wafer of the same size [2]................................................................................................. 9
Figure 1-6. Hypothetical timeline for the transition between electrical and optical based
communication systems for various distances [2]. .................................................................. 9
Figure 1-7. Energy band diagrams for silicon (above) and gallium arsenide (below).
Notice that Si has an indirect bandgap while GaAs has a direct bandgap. This explains
why GaAs is a much better light emitter than Si [2].............................................................. 10
Figure 1-8. Cross section of the device designed by the Intel researchers. It is a ridge Si
waveguide surrounded by SiO2. The silicon on insulator (SOI) technology used allows for a
large mode confinement within the Si waveguide so that high enough Raman amplification
is reached. The p and n type Si on either side help to draw electrons generated by two
photon absorption out of the device and reduce parasitic losses such as free carrier
absorption (FCA) [11]............................................................................................................ 12
Figure 1-9. Scanning electron microscope (SEM) image of a p-i-n diode waveguide used
for Raman amplification and lasing experiments [12]........................................................... 12
Figure 1- 10. Outline of the actual structure (s) to be studied in this thesis. ........................ 17
Figure 1-11. (a) Complete diagram of the device. The upper left shows a top view of the
ring resonator which is used to accomplish frequency selective feedback and amplification
as well as output coupling. The bottom shows a cross section of the cavity. The upper right
shows a zoomed in picture of the gain medium and the surrounding layers. In this work, the
gain medium is actually composed of alternating layers of nc-Si and SiO2:Er as depicted in
vii](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-7-320.jpg)
![both Figure 1-10 and in (b). This particular diagram shows the horizontal slot ridge design.
Electrons and holes are injected by tunneling through the p and n contacts into the nc-Si
layers where they will form excitons. Also visible is the ring resonator structure used to
achieve frequency selective feedback and amplification as well as output coupling [13]. .. 18
Figure 2-1. A monochromatic plane wave incident upon a dielectric interface. The subscripts
‘i’, ‘r’, and ‘t’ denote the incident, reflected and transmitted portions of the wave. The
diagram shows the case of TE polarization meaning that the E field is directed out of the
page in all regions. 27
Figure 2-2. Representation of three possible scenarios for light incident upon a dielectric
interface for the case when n1>n2. At left is the case for θi<θc. There will be a transmitted
ray in this case. The figure in the center shows the case for θi=θc in which there will be a
transmitted ray along the interface itself. Finally TIR is shown at right when θi>θc. There
will be no transmission. This is the physical basis for waveguides....................................... 29
Figure 2-3. A basic asymmetrical planar slab waveguide. Note the orientation of the axes.30
Figure 2-4. A diagram illustrating the relationship between wavevector k, transverse
wavevector κ, and longitudinal wavevector β. ...................................................................... 32
Figure 2-5. Graphical solution of the transcendental equation (2.24) for an asymmetrical
waveguide. Each intersection of the red and blue graphs (aside from the vertical lines
where the value of tangent is infinite) represents a possible mode........................................ 34
Figure 2-6. Graphs (a) – (d) show pictures of the cross section of the E field for modes 0–3
respectively. The value of β for each mode is also given on the graphs. The evanescent
decay into the substrate and cladding is shown by the blue and green sections of the plots
respectively. It can be observed that higher order modes penetrate further into the substrate
and cladding........................................................................................................................... 36
Figure 2-7. Picture of the two dimensional slot confined waveguide proposed in [32]. It is
essentially two planar slab waveguides placed in close proximity in a low index SiO2
background [32, 33]............................................................................................................... 38
Figure 2-8. Graph of the x component of the transverse E field of the fundamental TM
mode across the regions of the slot confinement waveguide. The blue sections show the
evanescent decay into the cladding regions, the black sections show the distribution in the
waveguides and the red section shows the behavior within the slot. In this case the slot
width is 144nm while the guiding layers are each 180nm..................................................... 40
Figure 2-9. Figures (a) – (d) show graphs of the E field distribution of the transverse
fundamental TM mode for various slot widths and slot to guide layer thickness ratios.
Notice how as the slot thickness increases and consequently lowers the overall effective
index, a larger evanescent tail leaking into the cladding materials results. Thus the
confinement factor begins to decrease as the total amount of the low index material present
in the structure increases beyond a certain point. .................................................................. 45
viii](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-8-320.jpg)
![Figure 2-10. Graph showing the relationship between confinement factor and ratio of slot
width to waveguide width. The width of the guiding layers is kept constant at 180nm while
the slot width is chosen to fit the desired ratio. The CF increases up to a ratio of about 0.80
because the added low index material allows a larger amount of light in the desired region.
The values saturate between 0.80 and about 1.1.................................................................... 45
Figure 2-11. (a) Simplified representation of absorption, (b) spontaneous emission, and (c)
stimulated emission................................................................................................................ 48
Figure 2-12. A four level system. L3 and L1 are both unstable states so electrons rapidly
decay to L2 and L0, usually in a non-radiative transition...................................................... 50
Figure 2-13. (a) Simple Fabry-Perot cavity with mirrors on either end. Also shown is the
pumping (energy) source [9]. (b) Ring resonator structure which is the type of structure
used for the device in this work............................................................................................. 52
Figure 2-14. A simple energy band diagram for a bulk material (left) and a quantum-
confined nanocrystal (right)................................................................................................... 56
Figure 2-15. Diagram of the field-effect structure for achieving electrical injection [14].
The array of silicon nanocrystals exists in the light gray area directly beneath the gate....... 57
Figure 2-16. (a) Schematic energy level scheme of Er3+
showing the different types of
transitions that may occur. The energy levels are very sharp for the free ions. Doped into a
SiO2 lattice, the levels split into many discrete levels.(b) A more detailed view of the
4
I13/2
4
I15/2 transition. There are many discrete energy states in which electrons can exist
within each energy band. The electrons will obey Boltzmann statistics. The graphs on the
right show how the number of electrons (N(E)) changes with increasing energy (E). It can
be observed that absorption takes place from the most populated states in the lower band to
the least populated states in the upper band and vice versa for emission. This difference
accounts for the slightly different wavelength and frequency of photons involved in these
two processes. ........................................................................................................................ 59
Figure 2-17. Solid lines show the absorption spectrum of an Er-doped optical fiber for
wavelengths between 450-850nm. This shows the principal absorption bands within the
range of wavelengths. [36]..................................................................................................... 60
Figure 2-18. Photoluminescence (PL) emission spectrum for Er -doped (~5 x 1015
cm-2
)
silica film on a Si substrate when pumped with 488nm light. The wavelengths at which the
two peak intensities occur are labeled [37]............................................................................ 61
Figure 2-19. Diagram showing pumping of the 1.55µm transition in Er via Förster energy
transfer between nc-Si and Er. Non-radiative decay occurs from the I11/2 level to the I13/2
level. This is followed by radiative recombination from I13/2 I15/2 at 1.55µm..................... 62
Figure 3-1. Cross section of structure to be analyzed. T_C, T_Si and T_SiO2 denote the
thicknesses of the cladding, silicon and silicon dioxide respectively. The multilayer region
consists of layers between 15 and 30nm thick. The layers alternate between low index SiO2
ix](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-9-320.jpg)
![and higher index Si with Si forming both the top and bottom layers. It is flanked on top by
a cladding of SiO2 and is deposited on a substrate of the same material. The dimensions of
the cladding are set for all simulations while the slab height varies depending on layer
thicknesses and the SiO2 to Si ratio. Note the orientation of the x, y, and z axes…..67
Figure 3-2. Complete cross sectional diagram of the structure. The active gain medium is
actually composed of alternating layers of nc-Si and SiO2:Er. This particular diagram
shows a horizontal slot ridge waveguide design. Electrons and holes are injected through
the p and n contacts into the p and n type device layers. They then form excitons by
tunneling into the nc-Si layers. Also visible is the ring resonator structure used to achieve
frequency selective feedback and amplification............................................................... 68
Figure 3-3. (a) Slot confinement structure proposed by MIT [32]. The structure consists of
upper and lower high index cladding layers with alternating thin low index slot layers with
high index thicker guide layers in between. In this case, w is the width, tc, tH , and tl are the
thickness of the cladding, low index slots and high index layers respectively. The
refractive indices of the high and low index layers are nH and nl respectively while n0 is the
refractive index of the surrounding air. It operates on the principle of confinement within
low index slot layers but is much more difficult to fabricate owing to the extremely high
precision required to etch the sidewalls. (b) Cross section profile showing the quasi TM
modal distribution of the y component of the E field in the structure proposed in [32]. Also
shown is the distribution of the y component of the E field along the structure (on the right
side)................................................................................................................................... 69
Figure 3-4. Outline of the structure and picture of the major (y) and minor (x) components
of the E field of the fundamental TM eigenmode looking along the z direction found using
the BeamProp mode solver. This is a structure with N = 9 periods, T_Si = 20nm and a ratio
of 1.00. The slab height is 0.380µm. The high concentration of the field in the low index
layers is clearly visible...................................................................................................... 72
Figure 3-5. 3D picture of the power distribution of the TM mode in a structure with N = 9
periods, T_Si = 20nm and a ratio of 1.00. The high power confinement in the low index
layers is readily apparent here. Also noted is the relatively large tail showing a significant
amount of leakage of the mode into the low index substrate............................................ 74
Figure 3-6. 2D cross section at x=0 of the power distribution of the TM mode of the same
structure as in Figure 3-5. The SiO2 layers are labeled along with the other regions of the
structure. The long tail extending into the substrate is more discernable in this figure. .. 75
Figure 3-7. Cross section of the z component of power distribution of the TM mode across
the layers in the structure shown in Figure 3-1. Again the high confinement in the low
index layers is evident. Note also the oblong shape of the mode as it propagates down the
structure. The evanescent tail in the substrate is visible here as in the other two figures. 76
Figure 3-8. Confinement factor versus SiO2/Si thickness ratio of the TE and TM modes for
both the FDTD and TMM methods. The FDTD plots are shown as crosses and circles and
x](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-10-320.jpg)
![3
It is appropriate to begin with an overview of the major advantages afforded by
optical solutions over their respective electrical counterparts. Two of the most obvious
improvements in an optical system are the reduction or complete elimination of
electromagnetic interference (EMI) and noise as well as immunity from short circuits.
These are inherent problems that have plagued electrical systems for years. Examples of
EMI include cross talk between adjacent cables in a bundle, stray wires or electromagnetic
radiation in the atmosphere. It is impossible to completely remove either of these from such
systems. Metallic shielding is one approach but it adds extra weight, cost and parasitic
capacitance which limits the bandwidth – all significant enough drawbacks to warrant
consideration as to whether or not it is necessary in the first place [1]. Another minor yet
critical benefit of optical fibers is that they are much more difficult to tap and therefore
more secure.
The most significant advantages offered by optical solutions are: 1) lower loss
transmission at higher frequencies, 2) larger bandwidth (multiplexing capabilities), and 3)
smaller size, weight and overall power consumption (at smaller interconnection distances).
As it has been previously noted, electrical interconnects are currently the better choice for
systems at the chip level but at higher data rates and larger distances optical links are far
superior. The hope is that this will be able to scale to the chip level as optical circuits based
in Si become more of a reality. There are a few other major factors hindering a full
transition. These are the need for: 1) wavelength independence over the entire
communication spectrum, 2) polarization independence and 3) higher optical power levels
– all of which could be addressed utilizing nonlinear effects in Si [1]. In order for this to
take place, many of the current problems confronting optoelectronic integrated circuits
(OIC) will require a variety of innovative solutions to meet the ever increasing data and
speed requirements of the rapidly evolving global communications network.
As the lowest level of integration, chip-to-chip optical circuitry will realize the
greatest benefit and also presents the greatest challenge due to the extreme conditions
inherent at smaller dimensional sizes such as high temperatures which create the problems
with resistor-capacitor coupling, bandwidth limitations and durability. Optical interconnects
become increasingly competitive at higher levels and longer transmission distances where
their electrical counterparts experience greater losses due to parasitic effects. For data rates](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-14-320.jpg)
![4
in excess of 1Gb/s and distances of greater than 100m, the use of optical interconnects is
currently standard practice with the slightly higher manufacturing cost easily justified by the
large increase in performance. At more intermediate distances of 1 – 100m there are
solutions in both forms. A diagram showing the applications of different interconnect
technologies is presented in Figure 1-1 [2]. Figure 1-2 [2] is a timeline of the information
carrying capacity of various electrical and optical links. It is clear that optical systems afford
much higher data rates than even the best electrical systems.
Figure 1-1. Diagram showing various applications of different interconnect technologies. Optical links are
currently only used at longer distances (>100m) [2].](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-15-320.jpg)
![5
Figure 1-2. Timeline of information carrying capacity for links in various electrical and optical links. Those
in optical systems have data rates roughly 2 orders of magnitude higher than those in electrical systems [2].
The need for higher data rates and bandwidths places increasing strain on current
electrical interconnects. The degradation of performance, particularly at the chip-to-chip
level, due to the trend towards smaller dimensions and higher temperatures has proven to
be a hindrance to the continued progress of the microelectronics. A diagram showing the
relationship between processor and front side bus (FSB) clock speeds for different models
within the past 15 years is shown in Figure 1-3 [2]. Recently, the potential of optical
integrated circuits (OICs) has been realized due to an increased need for higher
performance, especially on smaller scales. Despite their aforementioned drawbacks
however, electrical ICs remain the most cost effective and reliable solution particularly at
the chip-to-chip level largely due to highly developed fabrication methods and technologies
along with the readily accessible infrastructure that has been firmly established over the
years.](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-16-320.jpg)
![6
Figure 1-3. Relationship between processor and front side bus (FSB) clock speeds for various models over
the past 15 years. Note the significant improvement in processor speeds over FSB speeds in recent years [2].
Until recently, most optical devices have been made from what are known as exotic
materials such as gallium arsenide (GaAs) or indium phosphide (InP) owing to their
excellent optical properties. These materials are both expensive and complicated to process,
however. Before integrated optics became a viable option, most devices were discrete
components which were hand-assembled and required extreme attention to detail as far as
aligning them to fiber connectors to maximize light coupling. Thus there is often a
significant cost disadvantage when working with optical systems and devices. Complete
integration of optical components fabricated in silicon would eliminate the need for such
precision by automation of this process with a subsequent reduction in cost. The hope is
that specific components can be fabricated to perform all the necessary functions in the
transmission and reception of optical data with the end result being a completely optical-
based system on a single chip. Each of the necessary functions in this system can be
thought of as an optical analog to a similar electrical function accomplished with transistors.
These functions include:
1) light generation
2) light guiding](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-17-320.jpg)
![7
3) light detection
4) signal multiplexing
5) high speed (≥ 1 GHz) modulation
6) low cost assembly
7) high coupling efficiency to optical fibers.
A block diagram for a typical transmitter-receiver system is shown in Figure 1-4 [3, 4].
Figure 1-4. Diagram showing individual components of an optical transmitter-receiver system. The
transmitter is the top diagram and the receiver is the bottom [4].](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-18-320.jpg)
![8
Si has been shown to be adept at all of these functions except for light generation – the
major barrier that must be conquered. Recent results by Intel have shown devices capable
of high speed modulation [5]. Si based components are great for light guiding solutions
because Si is transparent (its bandwidth is larger than the energy of the photons) at the
infrared wavelengths used in communication systems, namely 1.55µm. However, Si is
notorious for high free carrier absorption losses and Auger processes – problems which,
along with poor light emission, have plagued Si-based systems for years. This is part of the
tradeoff when utilizing a material with such excellent electrical properties. In order for
photonic components to take hold at the lowest level of integration, performance demands
must outweigh cost drawbacks as is the case with long haul telecommunications links. If a
transition to exclusively optical-based communication systems is to take place, it will most
likely take place through slow developments and advances in fabrication techniques and
device technologies as performance demands increase and costs decrease.
Demand for superior performance will continue to increase as it has in the past and
will continue to do so at a faster rate in the future. Global communication networks are
expanding rapidly every day and the need for large quantities of information in short
periods of time has never been greater. The burden thus falls on innovation of new
fabrication techniques and design technologies to help precipitate the transition,
particularly in terms of the information processing level (as opposed to information
transmission which already employs optical solutions). With the microelectronics industry
firmly centered on silicon and baseline CMOS fabrication processes, these innovations will
almost certainly involve Si-based components. Although hybrid integration involving III-
V-based semiconductors or other types of materials is possible, they are highly unlikely to
firmly take hold since Si is by far more plentiful, inexpensive and well understood [1, 2].
Evidence of how much cheaper fabrication in Si is than in III-V materials is shown in
Figure 1-5 [2]. At best they will provide alternative or customized solutions. A hypothetical
timeline showing the transition between electrical and optical-based systems for
communication over various distances is visualized in Figure 1-6 [2]. Some experts in the
area of silicon photonics instead foresee a merging of the two technologies and believe that
future systems will rely on parallel solutions using smart electronic-photonic partitioning.](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-19-320.jpg)
![9
Figure 1-5. Wafer cost per unit area for various materials versus wafer size. Notice that InP is currently only
available in one wafer size and is prohibitively expensive compared to a Si wafer of the same size [2].
Figure 1-6. Hypothetical timeline for the transition between electrical and optical based communication
systems for various distances [2].](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-20-320.jpg)
![10
The inability of Si to behave as an efficient light emitter is due to the indirect nature
of its band structure. The energy band diagrams for both Si and GaAs are shown in Figure
1-7 [2]. Notice that GaAs has a direct bandgap which helps explain why it is a very
efficient light emitter, while Si does not. Numerous attempts have been made over the
years to circumvent this problem, most with only varying or irreproducible success.
Bandgap engineering (controlling the band structure with alloys and Brillouin zone folding
in superlattices are two notable methods) and enhanced quantum confinement via quantum
wells, wires and dots are among the more successful and notable approaches in recent years.
See Chapter 1 in [2] and references therein for a complete overview of these techniques.
Figure 1-7. Energy band diagrams for silicon (above) and gallium arsenide (below). Notice that Si has an
indirect bandgap while GaAs has a direct bandgap. This explains why GaAs is a much better light emitter
than Si [2].](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-21-320.jpg)
![11
It should be mentioned that while getting Si to emit light as efficiently as a direct
gap semiconductor is somewhat difficult and may never be achieved, light amplification
and lasing have in fact been achieved in Si, mostly as a result of the phenomenon of
stimulated Raman scattering (SRS). This phenomenon arises from the Raman Effect
whereby photons from a pump beam transfer their energy to longer wavelength photons
through interaction with vibrating atoms within a material. Spontaneous Raman scattering
and emission at 1540nm in Si waveguides was first observed and reported in 2002 by a
group at UCLA [6]. This was followed by a report of SRS and amplification in a 2003
publication [7]. Finally the first successful demonstration of lasing at 1675nm in a Si
waveguide [8] was given in 2004. One of the major drawbacks of the initial Raman laser
was that at the high pumping powers required for Raman amplification to achieve lasing,
the two photon absorption problem becomes more significant. Two photon absorption
(TPA) occurs when two incoming photons come together at the same exact spot in the
crystal lattice and provide enough energy to raise an electron from the valence band to the
conduction band. This is not possible for single photons at the infrared wavelengths
normally used in silicon photonic devices because they do not provide enough energy for
the electrons to cross the bandgap. Once in the conduction bands, the electrons then
become free carriers and contribute to the problem of free carrier absorption (FCA) at
higher power levels by absorbing light propagating through the waveguide and weakening
the overall signal strength in the device. At high pump intensities, TPA is enough to either
cancel out the Raman gain or prevent SRS altogether. Because of this, only pulsed-pump
operation was possible with the UCLA design. In 2006 a group of Intel researchers got
around this problem by surrounding the silicon waveguide with p and n type silicon to
effectively form a PIN diode [9-12]. When the diode is reverse biased, the free carriers
(electrons) generated by TPA are ‘swept’ away by the electric field. This reduces the free
carrier absorption losses enough to allow continuous operation. Diagrams are shown in
Figure 1-8 and Figure 1-9 [11, 12].](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-22-320.jpg)
![12
Figure 1-8. Cross section of the device designed by the Intel researchers. It is a ridge Si waveguide
surrounded by SiO2. The silicon on insulator (SOI) technology used allows for a large mode confinement
within the Si waveguide so that high enough Raman amplification is reached. The p and n type Si on either
side help to draw electrons generated by two photon absorption out of the device and reduce parasitic losses
such as free carrier absorption (FCA) [11].
Figure 1-9. Scanning electron microscope (SEM) image of a p-i-n diode waveguide used for Raman
amplification and lasing experiments [12].](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-23-320.jpg)
![13
The important thing to realize about these results however is that SRS is a non-
linear process that involves the emission or absorption of a phonon. This means that the
overall efficiency for this type of lasing will be much lower than that which relies primarily
on first order stimulated emission processes. Also, it requires high powered pumping from
an additional source of laser light (usually a mode-locked fiber laser) which complicates
the fabrication process and means that the device will not be exclusively based in Si, both
of which will increase cost. It also exacerbates problems like two photon absorption and
free carrier absorption as noted. Devices based on SRS are still being actively studied
however, and in all likelihood will play a role in the future of silicon photonics.
There is one other recent breakthrough that warrants mention. In 2006 a
collaborative team from both Intel and the University of California at Santa Barbara
designed and fabricated an electrically pumped hybrid Si laser. Being the first electrically
pumped Si laser, this was a major development. Instead of relying on SRS which
necessitates optical pumping, it achieves lasing by electrical pumping – injecting electrons
and holes into InP which is a direct bandgap material and an efficient light emitter. Light
generated in the InP is coupled to the Si waveguide layer which sits underneath. The device
is hybrid because it is actually InP that directly emits and amplifies the photons of light [3].
One of the main advantages of this technique is that it eliminates the need to align
external light sources with the Si chip by either pre-fabricated lasers on chip or optical
fibers off chip, both of which would not be practical for low-cost, high-volume production.
The novelty of the design is based on a new way of bonding together the InP and Si
substrates. The main problem with growing III-V materials like InP onto Si is that they
have a large lattice mismatch and dislocations develop which negatively impact device
performance and reliability. The new bonding technique uses a low temperature plasma
enhanced oxidation process that acts as a ‘glass glue’ between the two materials. It allows
InP substrates to be bonded directly to pre-patterned Si photonic chips without the need for
alignment. This technique may prove to be very successful as a solution for large scale
optical integration onto a Si platform by simplifying the manufacturing process and
reducing cost. It is a hybrid approach however, so it remains to be seen whether or not it
will be able to realize the low cost – high volume potential that a monolithic approach](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-24-320.jpg)
![14
could simply because both InP and other III-V materials are more expensive and not as
abundant or as well understood as is Si [1].
The inability to achieve efficient light emission in Si based components and devices
– aside from the fact that a transition to technology based completely on hybrid materials
would not be practical in terms of cost – remains the major impeding factor to the
successful implementation of low cost-high volume optical integration at all system levels.
It will be seen that the use of nanocrystalline silicon and its novel energy transfer properties
with erbium ions in close proximity poses a very viable solution to this problem.
1.2 MURI Silicon Laser Project
A Multi University Research Initiative (MURI) is a large scientific project
composed of a collection of university research teams and possibly partners from national
or industrial labs. For this particular MURI project, the primary task at hand is the design
and development of practical and useful Si-based photonic components – particularly a
laser system. The institutions involved are listed in alphabetical order as follows:
Boston University
California Institute of Technology
Cornell University
Lehigh University
Massachusetts Institute of Technology
Stanford University
University of Delaware
University of Rochester
Also taking part are researchers from the McMaster University, University of
Toronto and the Army, Navy, and Air Force research laboratories.
The technical approach as stated in the executive summary reads:
“Our program will deliver a nanophotonic, silicon-based laser for monolithic, CMOS-
compatible integration with electronic and photonic chip-level circuits. The silicon-
based laser must be electrically pumped; it must have a high wallplug efficiency and
low heat dissipation; it must have a long operating lifetime; and it must emit at a carrier
frequency and output power that meet the expectation of a > 10Gbit/s optical](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-25-320.jpg)
![15
communications link, operating at a Bit-Error Rate tolerance < 10-12
. Two principal
challenges frame the science and engineering research context of this program: high
lasing efficiency cavities and high gain efficiency media [13]”
The program is organized into three major tasks, each with supporting subtasks. The
first task is an extrinsic gain laser with the subtasks including a slot waveguide structure
with Er-doped oxide and a traditional index guided structure with Er-doped silicon nitride
and quantum dot doped oxides. The second task is an intrinsic gain laser with the subtasks
of a conventional edge emitter structure with bulk and quantum-confined germanium (Ge),
and a conventional edge emitter structure with strained and alloyed Ge. Finally, the third
task is the design of the application drivers. The subtasks are a CMOS compatible Er-
doped optical micro-amplifier for 1550 nm light, a CMOS-compatible optical signal
processor link and a CMOS-compatible, multi-wavelength optical power supply.
In particular this thesis deals with the design and optimization of a structure falling
under the first subtask of the first major task. A table summarizing the four laser design
approaches for this initiative with their compositional materials, device parameters,
projected performance and risk assessment is presented in Table 1-1 [13]:](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-26-320.jpg)
![16
Table 1-1. The four design approaches for the MURI initiative with their specific parameter values.
1.3 Project Overview – Objectives, Approaches, Tools Used
This project is concerned with the design, modeling and optimization of a slot-
confined, multilayer waveguide structure to be used as the optical cavity for a silicon-based
laser system. It consists of alternating nanometer-thin layers of amorphous Si (a-Si) and
erbium (Er)-doped silicon dioxide (SiO2:Er) with a cap layer and substrate both composed
of SiO2. A carefully chosen thermal budget during fabrication will result in the formation
of silicon nanocrystals in the Si layers producing nanocrystalline Si (nc-Si). An outline of
the actual structure analyzed can be seen in Figure 1- 10. A more complete picture of the
overall system of which it is to be a part is shown in Figure 1-11[13]. There is one minor
clarification with this figure however: instead of a random array of nc-Si and Er3+
ions in a
SiO2 matrix, the gain medium (the central layer in the figure) consists of distinct multiple
layers of SiO2:Er and nc-Si [13]. This is also shown in Figure 1-11.](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-27-320.jpg)
![18
(b)
SiO2:Er (low index)
nc-Si (high index)
Figure 1-11. (a) Complete diagram of the device. The upper left shows a top view of the ring resonator
which is used to accomplish frequency selective feedback and amplification as well as output coupling. The
bottom shows a cross section of the cavity. The upper right shows a zoomed in picture of the gain medium
and the surrounding layers. In this work, the gain medium is actually composed of alternating layers of nc-Si
and SiO2:Er as depicted in both Figure 1-10 and in (b). This particular diagram shows the horizontal slot
ridge design. Electrons and holes are injected by tunneling through the p and n contacts into the nc-Si layers
where they will form excitons. Also visible is the ring resonator structure used to achieve frequency selective
feedback and amplification as well as output coupling [13].
The high index contrast at each of the Si/ SiO2 interfaces allows for a large
concentration of light within the low index (SiO2) layers. The idea is to use field effect
injection to bring electrons and holes into the surrounding p and n type device layers and
then rely on tunneling to allow them to travel into the nc-Si layers and form excitonic pairs
[14]. The nc-Si will allow efficient excitation and energy transfer to the Er atoms through a
dipole-dipole energy coupling interaction. This dipole-dipole energy coupling phenomenon
is known as Förster energy transfer. Essentially the nanocrystals provide a convenient
means of exciting the Er. This gives rise to light emission from Er at 1550nm – the
operating wavelength for most currently active optical communications links in the world.
Erbium is chosen for this reason as well as the aforementioned energy coupling capabilities
with silicon nanocrystals, and because it is generally well understood based on its use in
erbium doped fiber amplifiers (EDFAs) since the advent of the fiber optic section of the
telecom industry. The ability to confine a high percentage of light within the SiO2 layers
containing Er ions enables the device to more easily generate the gain necessary to achieve](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-29-320.jpg)
![19
lasing. In this way, the electronic properties of nanocrystalline silicon and the light emitting
capabilities of Er are each leveraged in such a way as to produce laser light emission in a
silicon based structure.
Due to the complex nature of the proposed structure, the use of numerical
simulation software becomes necessary in obtaining an accurate picture of the propagating
modes. These simulations are done using the RSoft Design Group, Inc. Photonics Suite and
in particular the BeamProp and FullWave tools. Both BeamProp and FullWave are
software packages designed to compute and model the propagation of light waves in
arbitrary geometries. Both are inherently based on finite difference computational methods.
BeamProp uses the Beam Propagation Method while FullWave uses another well-known
technique called the finite-difference time-domain (FDTD) method. Both techniques are
considered very robust and efficient at giving accurate pictures of how light propagates in
even the most complicated photonic devices. A full explanation with mathematical details
is not within the scope of this document but interested readers are referred to the numerous
publications on such methods [15-28]. An understanding of these numerical methods is not
critical to the full comprehension of the concepts presented but references are given for the
sake of completeness. It should be noted that the BeamProp package was used only for its
mode-solving capabilities while FullWave was used for actual simulations. Matlab was
used for data analysis and visualization purposes.
Simulations of the proposed structure are performed using different periods of
repeating Si/SiO2 layers and different thickness ratios between the Si and SiO2 layers. The
raw data from these simulations is then read into Matlab where it is used to calculate the
confinement factor (CF) within the SiO2 layers for each structure. By examining simpler
2D slot confined waveguide structures and something known as the effective index method,
it is possible to predict that the thickness ratio between the Si and SiO2 layers is the critical
parameter affecting the CF. This hypothesis is confirmed by analysis of simulation results.
Based on these results, a few optimum structures are chosen and a gain/loss
performance study is undertaken. This entails simulations of each structure, with loss
inserted into the Si layers to model the free carrier absorption effects in silicon, and gain
within the SiO2 layers to model the light emission from erbium. The ratio of the gain in the
SiO2 to loss in the Si is then varied over a range of values beginning with zero. For each](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-30-320.jpg)
![Chapter 2
Theory
Any theoretical description of propagating light waves begins with a discussion
of Maxwell’s equations and the solutions to the electromagnetic wave equation in an
isotropic, linear and source free medium. This is followed by explanations of the
boundary conditions at a dielectric interface and total internal reflection (TIR), and a
discussion of wave propagation in guiding media. Two simplified 2D waveguide
examples are then presented. First is a three layer slab waveguide with graphical
depictions of the solutions for the eigenmodes of propagation. Second is a slightly more
complicated slotted waveguide, also with graphical results. The idea of optical
confinement within the guiding layers is established alongside these examples. The
graphs are intended to give an indication as to both the advantages and limitations
inherent in the slotted structure. A very well-written and thorough introduction to this
area of electromagnetism can be found in chapter 9 of [29].
2.1 Maxwell’s Equations
The packaging of four previously derived relationships (Gauss’ Law, Gauss’
Law for magnetism, Faraday’s Law and Ampere’s Law) into a compact and consistent
form and the slight correction provided for one of them (Ampere’s Law) by James
Maxwell is without question among the crowning scientific achievements of the 19th
century. Maxwell’s equations are presented below in differential form. The integral
21](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-32-320.jpg)
![22
forms can be found in almost any text on electromagnetism and can be easily derived
from the differential forms using either Gauss’ divergence theorem or Stokes’ theorem
[30].
t∂
∂−
=×∇
B
E (2.01a)
J
D
H +
∂
∂
=×∇
t
(2.01b)
0=⋅∇ B (2.01c)
ρ=⋅∇ D (2.01d)
In order from top to bottom they are: (a) Faraday’s Law, (b) Ampere’s Law with
Maxwell’s correction term, (c) Gauss’ Law for magnetism, and (d) Gauss’ Law. A table
summarizing the various quantities and their units in MKS form is given in Table 2-1:
Quantity Description Units (MKS)
E Electric field amplitude Volts/meter (V/m)
H Magnetic field amplitude Amps/meter (A/m)
D Electric flux density Coulombs/meter2
(C/m2
)
B Magnetic flux density Webers/meter2
(W/m2
)
J Current density Amps/meter2
(A/m2
)
ρ Charge density Coulombs/meter3
(C/m3
)
Q Charge Coulombs (C)
Table 2-1. Various quantities used in electromagnetics and their units expressed in MKS units [30].
The constitutive relationships between the flux densities (D and B) and the field
amplitudes (E and H) are determined by the properties of the medium through which the
waves are propagating. In simple linear and isotropic media1
, these are written as,
HB µ= and ED ε= (2.02)
1
This is only true for monochromatic fields.](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-33-320.jpg)
![24
By further analysis the first term on the left hand side of the middle equation can be
shown to be effectively negligible [30]. This produces the standard form for the wave
equation:
02
=
∂
∂
−∇
t
E
E
2
µε . (2.07)
This same equation can be derived for H by starting with equation (2.04b) instead of
(2.04a).
It is also important to note that the wave equation reproduced here is actually a
vector equation which involves the vector Laplacian. It can be easily simplified to a
scalar equation for a rectangular coordinate system however. In a non-rectangular
coordinate system it is usually more difficult to simplify the vector equation because of
the need to break it into orthogonal components. Every structure presented in this work
lends itself readily to a rectangular coordinate system, so (2.07) is sufficient. A more
detailed derivation of the wave equation can be found in reference [30].
Solving the vector wave equation involves first choosing an appropriate
coordinate system, as mentioned above. For a simple rectangular system the scalar wave
equation can easily be found for each vector component and then solved by separation
of variables to get
)exp(),( tjj(t)(t) 0 ωψφψψ −⋅== rkrr , (2.08)
where ψ0 is the amplitude, ω is the temporal frequency and k is the wavevector in
radians/second. The magnitude of k is written as
µεω=k . (2.09)
This is known as the dispersion relationship for a light wave. The wavevector is defined
as k = 2π/λ, where λ is the wavelength of the light. It can be thought of as a ‘spatial’
frequency. The solutions to the scalar wave equation in a rectangular coordinate system
represent a sinusoidal wave in both space and time. Since this same equation can be
solved for H just as easily, it confirms the origin of the term electromagnetic radiation.
Maxwell’s equations can be decoupled to produce two distinct forms of the wave
equation – one each describing the magnetic and electric fields, each of which oscillates](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-35-320.jpg)
![25
through space and time. It can further be proven that E and H are orthogonal to each
other2
[29].
Through examination of the wave equation, it can be shown that the velocity of
the wave in a medium with unique values for µ and ε is
µε
1
=v , (2.10)
where v is the phase velocity.
Using the values of permittivity and permeability for vacuum which are given in
(2.03), the value for the speed of light (2.99 x 108
m/s) can be calculated. This leads to a
definition for the index of refraction of a material as
000 ε
ε
εµ
µε
==n , (2.11)
when µ = µ0. Thus the index of refraction is exactly one in vacuum and is very close to
one in air. This quantity determines the factor by which the speed of light is decreased
when traveling through a particular medium. It will be seen in the next section that it
also determines the manner in which light bends or refracts when passing from one
medium into another.
2.3 Boundary Conditions at Dielectric Interfaces
Before taking on a discussion of light propagation in a guiding medium such as a
waveguide, it is essential that the behavior of light waves at a dielectric interface be first
understood. A dielectric interface occurs at the boundary between materials with two
different indices of refraction (n1 ≠ n2). The general rule for the wave solutions in the
two regions is that they must be continuous across the boundary in some way. Specific
boundary conditions, presented below, explain the exact nature of this continuity for
each type of field or flux quantity.
When electromagnetic radiation is incident upon an interface of any type, it will
undergo one of the following phenomena or some combination thereof:
Transmission – no interaction of light with the new material
2
This is only true when the fields are in a plane wave and does not apply in general.](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-36-320.jpg)
![26
Absorption – energy lost as transmitted light interacts with impurities in the new
material
Reflection – light is re-directed back into the original material
Refraction – light passes into the new material but with an altered path
Technically there is no such thing as pure transmission. There is some degree of
absorption in almost any type of material and at different wavelengths of light.
Absorption becomes important when modeling the loss mechanisms within a waveguide
and is covered in a later section of this chapter. The two remaining phenomena will be
of primary concern in discussing the behavior of light at a dielectric interface.
The diagram in Figure 2-1 shows a simple case of a plane wave of light incident
on a dielectric interface with electric field (E), magnetic field (H), and wavevector (k)
all labeled. The subscripts ‘i’, ‘r’, and ‘t’ stand for incident, reflected and transmitted
portions of each field quantity respectively. The direction of the wavevector denotes the
direction of propagation for each ray and can be determined through calculation of the
cross product between E and H or more simply by the right hand rule [29]. In this
diagram, the E field points directly out of the page for all rays and the H field is oriented
perpendicular to this, as required by Maxwell’s equations. This is known as the
transverse electric (TE) polarization. Polarization is discussed in the next section.
Using the integral form of Maxwell’s Equations, the aforementioned boundary
conditions can be derived [29]. They are listed below for a source-free medium.
.0)(ˆ
,0)(ˆ
,0)(ˆ
,0)(ˆ
12
12
12
12
=−⋅
=−⋅
=−×
=−×
DD
BB
HH
EE
s
s
s
s
(2.12)
In this case, s is a unit vector that is normal to the surface. A simple way of
summarizing these rules is: “D-B normal, E-H tangential”, meaning that the normal
components of D and B are continuous across the boundary, as are the tangential
components of E and H.
ˆ
Using these continuity rules for the fields, three additional relationships can be
derived [29]. The first of these simply states that the incident, reflected, and transmitted
wave vectors form a ‘plane of incidence’ which also includes the normal to the surface](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-37-320.jpg)
![27
(in this case, the z axis). The other two are also known as the law of reflection and the
law of refraction (Snell’s Law). These are given below and also visualized in Figure 2-1.
k
n n
E
k
E
θ θ
Ht
θ z
k
E
H
y
Figure 2-1. A monochromatic plane wave incident upon a dielectric interface. The subscripts ‘i’, ‘r’, and
‘t’ denote the incident, reflected and transmitted portions of the wave. The diagram shows the case of TE
polarization meaning that the E field is directed out of the page in all regions.
ri θθ = , (2.13a)
ti nn θθ sinsin 21 = . (2.13b)
These relationships, along with the original boundary conditions can further be
extended to derive general equations relating the reflected and transmitted field
amplitudes to the incident field amplitude for both E and H as a function of θi [30].
These equations are not given since they are not as pertinent to the nature of this work,
however they can be found in chapter 1 of [30]. The physical implications of these four
relatively simple boundary conditions on the electric and magnetic fields and flux
densities are astonishing – they essentially lead to the two most fundamental laws of
optics. They also have other important consequences as the remainder of this chapter
will address.](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-38-320.jpg)
![28
2.4 Field Polarization
Another key point to hit on is the orientation of either the electric or magnetic
field with respect to the interface. If the electric field is perpendicular to the interface the
polarization is transverse electric (TE). If the magnetic field is perpendicular to the
interface the polarization is transverse magnetic (TM). The diagram used in the last
section showed a case of TE polarization. The four boundary conditions, the law of
reflection and Snell’s Law all remain the same regardless of the polarization, however
the behavior of the equations relating reflected and transmitted fields to the incident
ones is different [30].
For more complicated 3D geometries – specifically rectangular and circular
waveguides – these characterizations are meaningless so instead the terms quasi-TE and
quasi-TM are used to describe the polarization, depending on whether the particular
mode is primarily TE or TM respectively. There are some additional quirks in the
description of the polarization with regard to the simulation software and the particular
structure being analyzed. These will be referenced in Chapter 3.
2.5 Poynting Vector
An important quantity in determining the power incident on a surface from an
electromagnetic wave is called the Poynting vector. It is defined as
HES ×= , (2.14)
where S has units of Watts/meter2
(intensity). To get the total incident power it is
necessary to integrate S over the entire surface of interest.
Since S in general points in the direction of propagation, which in this case is
perpendicular to both E and H, it is often of interest to obtain the time average intensity
along this direction (in this case, z). This is shown below for propagation along the z
direction.
]ˆRe[
2
1 *
zSz ⋅×= HE . (2.15)
The factor of one half and the reason for the Re and complex conjugate operators
arises from taking the time average intensity for a sinusoidal waveform. This is derived](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-39-320.jpg)
![29
fully in [31]. Again, the important point is that this quantity represents the power per
unit cross sectional area or instantaneous intensity for a wave and can be integrated over
the cross section of an interface to determine the total power flow [30].
2.6 Total Internal Reflection (TIR)
The critical process in any guided wave approach is that of total internal
reflection (TIR). A pictographic representation of TIR is seen in Figure 2-2 .
Examination of Snell’s Law and Figure 2-2 shows that when n1 > n2 and θi (referred to
as θ1 here) is greater than a certain critical angle θc, the incident ray is completely
reflected back into the first medium. The critical angle is a function of both n1 and n2
and can be written as
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
= −
1
21
sin
n
n
cθ (2.16)
using Snell’s Law (2.13b) and substituting 90° for θ2.
Based on this it can easily be seen how waveguiding becomes a possibility by
surrounding a high index medium with layers of a lower index medium. Light must be
coupled into the waveguide in such a way that its angle of incidence at the boundary
with the lower index layer is always greater than the critical angle. This forms the basis
for waveguides which will be presented in the next section.
Figure 2-2. Representation of three possible scenarios for light incident upon a dielectric interface for the
case when n1>n2. At left is the case for θi<θc. There will be a transmitted ray in this case. The figure in the
center shows the case for θi=θc in which there will be a transmitted ray along the interface itself. Finally
TIR is shown at right when θi>θc. There will be no transmission. This is the physical basis for waveguides.
θ
nn
n n
θ
θ
n
θ
n
θ](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-40-320.jpg)
![32
where 22
0 )( βκα −±== ink and is known as the transverse wavevector. When β >
k0ni the term under the radical is negative and the solution becomes exponential:
)exp()( 0 xExEy γ−= (2.22)
where 2
0
2
)( ink−== βγα m and is known as the attenuation coefficient.
The diagram in Figure 2-4 helps to clarify the relationship between wavevector k,
transverse wavevector κ and longitudinal wavevector β. The only physical situation
occurs when β satisfies the condition:
fs nknk 00 << β . (2.23)
In this case, the wave is oscillatory in the guiding layer (nf) and decays
exponentially in the substrate (ns) and cladding (nc) layers. The above condition is
necessary to produce a guided wave in a structure where nf > ns >nc.
k
k2
= κ2
+ β2
x
κ
β
z
Figure 2-4. A diagram illustrating the relationship between wavevector k, transverse wavevector κ, and
longitudinal wavevector β.
To find the exact values for β which describe the allowed modes for the waveguide, the
generalized solutions in each layer must be connected via the boundary conditions. Each
solution has a transverse portion that describes the distribution in x and a longitudinal
portion describing the distribution along the z direction. As mentioned previously, the
longitudinal portions of all solutions have the same form so they can be left out for
simplicity. The transverse parts of the electric field amplitude are [30]:
⎪
⎩
⎪
⎨
⎧
+
+
−
=
))(exp(
)sin()cos(
)exp(
)(
hxD
xCxB
xA
xE
s
ff
c
y
γ
κκ
γ
(2.22)](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-43-320.jpg)
![33
where A, B, C and D are the amplitude coefficients which will be determined by
application of the boundary conditions. γc and γs are the attenuation coefficients in the
cover and substrate respectively and κf is the transverse component of the wavevector k
within the guiding layer. In this case only two of the four boundary conditions from
(2.12) will be used. To reiterate, the tangential components of both E and H are
continuous across each boundary. The other two conditions will not be needed. This
results in an underspecified homogeneous system of four linear equations (2 boundaries
with 2 boundary conditions for each boundary) with five unknowns (A,B,C,D and the
value of κf for each mode) which, when fully solved, yields a transcendental equation in
the remaining unknown variable. Using Faraday’s law (2.01a) along with (2.02) to put
the tangential component of H in terms of E (the curl must be expanded into individual
components on the left hand side of the equation), three of the four boundary conditions
can be applied to find A in terms of B,C and D. The equation describing the amplitude of
the transverse E field across the structure is [30]:
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎨
⎧
+
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
−
=
))(exp()sin()cos(
)sin()cos(
)exp(
)(
hxxxA
xxA
xA
xE
sf
f
c
f
f
f
c
f
c
y
γκ
κ
γ
κ
κ
κ
γ
κ
γ
(2.23)
The final boundary condition produces a transcendental equation in κf:
( )
scf
scf
f h
γγκ
γγκ
κ
−
+
= 2
)tan( (2.24)
Known as the characteristic equation, this equation can only be solved using numerical
or graphical methods. The result will show the discrete values for κf which can be used
in turn to solve for the values of β, the propagation constant for each mode. This type of
analysis can be performed for the TM polarized eigenmodes as well and will yield
similar equations [30]. Theoretically, any type of waveguide can be analyzed and
completely solved using this approach. However, in more complicated geometries or
multiple layer structures the analysis – particularly finding and solving the characteristic](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-44-320.jpg)
![37
In communications it is common to design waveguides and fibers to only
support one eigenmode of propagation due to the fact that multiple modes will distort a
signal as a result of modal dispersion. A full explanation of different kinds of dispersion
in waveguides can be found in [30]. The formula describing the approximate number of
modes that will propagate in a waveguide is:
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡ −
=
π
22
Int
sf nnhk
m . (2.25)
In this case ‘Int’ means to round towards 0 (for example, if the number in parenthesis
came out to be 5.68395, m = 5). The number of modes is therefore dependent on three
parameters: the wavelength λ, the width h and the difference of the squares of the
indices of refraction of the guiding layer and the substrate or cladding layer (whichever
has the larger index of refraction). Although it is 1µm in the previous example, λ is
usually taken to be 1.55µm because at this particular wavelength the lowest possible
transmission losses are attained in most silica waveguides and optical fibers [30]. For
this reason it has been dubbed the communication wavelength. Unless otherwise
mentioned all simulations, calculations and results will assume this value for λ.
An interesting phenomenon to take note of is the evanescent decay of the mode
in both the cladding and substrate layers. It can be clearly observed that a finite portion
of each mode actually extends into these layers. This can be likened to a similar
occurrence in the energy levels of the finite potential well of quantum mechanics and
forms the basis for coupling between waveguides and other optical components – a very
important issue in the field of integrated optics. One other thing of note is the slight yet
noticeable asymmetry in the mode distributions which is not surprisingly a result of the
fact that the waveguide itself is asymmetric. Similar analysis can be performed on a
symmetric waveguide with similar results. A more complete discussion and analysis of
these topics can be found in [30, 31].](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-48-320.jpg)
![38
2.7.2 Slot Confinement Waveguide
As it was noted before, the ideas from the earlier parts of this section and in
particular those of the previous section can be extended to more complicated structures.
Recently, some research has focused on guiding light in low-index materials by using
high index contrast waveguides. One of these structures is a slot confined waveguide or
simply the slot waveguide. This structure was proposed by a group at Cornell [32, 33]
and used as the basis for further analysis in another recent publication [34]. It is
essentially two planar Si slab waveguides placed in close proximity in a low index SiO2
background. A picture of this is seen in Figure 2-7 [32]. In this case nc is still the index
of refraction for the upper cladding but ns is now the index of refraction of the slot
material and nH is that of each guiding layer. It can be seen that the width of each guide
is |b – a| and the width of the slot is 2a.
Figure 2-7. Picture of the two dimensional slot confined waveguide proposed in [32]. It is essentially two
planar slab waveguides placed in close proximity in a low index SiO2 background [32, 33].
The idea is that by situating the two waveguides next to each other to form a slot
made of SiO2 with a lower index than the Si waveguides, the evanescent decay from
each waveguide will result in a highly concentrated flux density (D) in the slot as a
result of the boundary condition for D – namely, the continuity of its normal component.
The advantage of this over past methods of guiding and confining light in low index
materials is that the light is guided by TIR and not external reflections from interference
effects which are both lossy and wavelength dependent. The guided modes will
therefore be eigenmodes of the structure that are fundamentally lossless and not as
heavily dependent on wavelength as previous structures.](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-49-320.jpg)
![39
The constitutive relations (2) show that the high index contrast at the interface
between the slot and the guiding layer is responsible for the large discontinuity in the E
field that results. Physically the best way to understand what is going on is to think of
the large discontinuity in E as a redistribution of photons resulting from the boundary
condition on D. This only occurs for the TM polarized modes since for TE modes the E
field (and hence the D field3
) has no component perpendicular to the interface. The x
(normal) component of the transverse E field of the fundamental TM eigenmode of the
slot waveguide can be found through analysis similar to that in the previous section to
be [32]
[ ] [ ]
[ ] [ ] [ ]⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎨
⎧
−−−⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+−⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
)(exp)(sin)sinh()(cos)cosh(
1
)(sin)sinh()(cos)cosh(
1
)cosh(
1
)(
2
2
2
22
2
bxaxa
n
n
aba
n
axa
n
axa
n
x
n
AxE
cHs
Hs
sH
Hs
c
Hs
Hs
s
Hs
H
s
s
y
γκγ
κ
γ
κγ
κγ
κ
γ
κγ
γ
(2.26)
where κH is the transverse wavevector, γc and γs are the attenuation coefficients in the
cladding and slot respectively. The value of A is written as [32]
( )
0
22
0
0
k
nk
AA HH κ−
= , (2.27)
where A0 is an arbitrary constant and k0 = 2π/λ0 is the wave number in vacuum. This
eigenmode can be thought of as an interaction between the fundamental TM eigenmodes
of each individual waveguide. The values of κH, γc, and γs can all be related to the
propagation constant (β) for each mode in a similar fashion to the simple planar
waveguide:
22
0
2
0
2
2
0
2
)(
)(
)(
βκ
βγ
βγ
−=
−=
−=
HH
ss
cc
nk
nk
nk
(2.28a-c)
3
This is not true in the most general case. Here, D is in fact parallel to E because it is assumed that this is
an isotropic and linear material. For further discussion see [27].](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-50-320.jpg)
![40
The values of β for each eigenmode are calculated by solving the characteristic equation
for this structure [32]:
( )[ ] ( a
n
n
ab s
Hs
sH
H γ
κ
γ
κ tanhtan 2
2
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=Φ−− ). (2.29)
The continuity condition on D causes the E field to be enhanced by a factor of
(nH / ns)2
on the slot side of the boundary between the slot and the higher index
waveguides. This creates a higher overall power confinement within the thinner low
index slot layers. Power confinement will be discussed in the next section. If the two
waveguides are close enough together the enhancement can be quite significant but
deteriorates rapidly as the separation increases. A graph of the x component of the
transverse E field distribution of the fundamental TM mode is shown in Figure 2-8. In
this case the values used were: λ0=1.55µm, nH=3.48, ns= nc=1.44, a=72 nm, and b=152
nm and the slot width is 144nm [32]. Additional graphs and discussion of this type of
waveguide can be found in the next section on power confinement.
−6 −4 −2 0 2 4 6
x 10
−7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Transverse E−field distribution of fundamental TM eigenmode
x (m)
Ex(x)(a.u.)
Figure 2-8. Graph of the x component of the transverse E field of the fundamental TM mode across the
regions of the slot confinement waveguide. The blue sections show the evanescent decay into the cladding
regions, the black sections show the distribution in the waveguides and the red section shows the behavior
within the slot. In this case the slot width is 144nm while the guiding layers are each 180nm.](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-51-320.jpg)
![41
2.7.3 Power Confinement in Waveguides
One of the most important figures of merit for a waveguide is the power
confinement or confinement factor (CF) of each of its eigenmodes. It has already been
stated that the total power incident on a surface or cross sectional area of a waveguide
can be calculated in general by integrating the time averaged Poynting vector along the
direction of propagation (2.14) over the area comprising the surface. In the case of the
2D planar slab waveguide, the integral is instead performed across just the x direction
since it is considered to be essentially flat. The power confinement is obtained by
calculating the ratio of power flowing within the core or guiding layer to the total power
in all regions of the structure. For the planar slab waveguide this is written as
∫
∫
∫
∫
∞
∞−
−
==
dxxHxE
dxxHxE
dxxHxE
dxxHxE
P
P
xy
h
xy
regionsall
xy
core
xy
total
core
)()(
)()(
)()(
)()(
*
0
*
*
*
. (2.30)
Usually this quantity is expressed as a percentage by multiplying it by one
hundred. It is usually the case that higher order modes are less confined, mostly due to
the fact that they have lower attenuation factors and thus extend further into the low
index cladding and substrate layers.
The importance of the confinement factor is due to the fact that poorly confined
modes often exhibit higher power loss due to bending in the guiding structure (as in ring
resonators which will be discussed in a later section) and unintentional evanescent
coupling to nearby components. These phenomena are not directly relevant to this
discussion but can be explored further in [1, 30, 31].
The major advantage of the aforementioned slot waveguide is that it provides a
very high concentration of light in a relatively thin, low index layer (the slot) [32]. The
high power confinement of light that results is especially favorable in integrated optics
where power efficiency and dissipation are extremely important in device design. The
fact that light is concentrated in thin, nanometer-size layers could be considered even
more important however, since the diffraction limit has imposed limits on the size of
optical components and devices. The diffraction limit states that the lower limit of these](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-52-320.jpg)
![47
2.8.2 Three Processes: Absorption, Spontaneous Emission, Stimulated Emission
Stimulated absorption or simply absorption is the process by which an electron is
raised from a lower energy level (usually the ground level) to a higher one through
interaction with a photon. A diagram of this process for a simple two level system is
seen in Figure 2-11a. Absorption will only occur when the energy of the photon, given
by the quantity hν where h is Planck’s constant, is equal to or greater than the difference
in energy between the two levels.
A second critical process in a laser system is emission. Emission is just the exact
opposite of absorption – an electron makes a transition from a higher energy state to a
lower one and gives off a photon. There are two different types of emission:
spontaneous and stimulated. Spontaneous emission is the more common type of optical
emission and occurs when an excited electron randomly relaxes to a lower state emitting
a photon in the process. A diagram is shown in Figure 2-11b. By its nature, spontaneous
emission is completely unpredictable. It is impossible to know exactly where the emitted
photon will appear or which direction it will head in. It is possible however, to
determine a statistical average for the length of time between emissions for particular
transitions within a given material system. Often, spontaneous emission is treated as a
source of noise within a system since it creates additional light sources which are neither
predictable nor controllable [30].
Stimulated emission on the other hand is much more predictable and controllable.
It takes place in the same fashion as absorption except that the incoming photon now
interacts with an electron already in a higher energy state. The interaction of the photon
with the electron induces the transition and yields the release of another photon identical
in phase, direction, polarization and frequency to the original. A diagram of the process
is shown in Figure 2-11c. Thus, stimulated emission produces amplification of light
incident upon a material with electrons in excited states. It is one of the crucial elements
in producing laser light.](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-58-320.jpg)
![49
2.8.3 Population Inversion and Pumping Schemes
Since stimulated emission acts as an amplification process and produces a
photon identical to one incident onto an electron in an excited energy state, it is really
the fundamental process in the production of laser light. In order to maximize the
number of identical photons and increase the intensity of the laser light, there must first
be a sufficient number of electrons in excited states with which incoming photons can
interact. For most systems in equilibrium, electrons will tend to relax to the lowest
energy state. Therefore energy must be introduced into the system in some way to raise
electrons from lower energy levels to higher ones by the process of absorption. This
results in the non-equilibrium condition of population inversion. The method of
achieving population inversion is called the pumping scheme.
The most common pumping schemes that are used in almost all laser systems are:
1) optical sources 2) electrical charge injection 3) chemical reactions 4) ion beams and 5)
electron discharges or beams [30]. Optical pumping involves the use of either a flash
lamp or another source of laser light to raise electrons to a higher energy state. Since it
requires an external source of light this type of pumping is not conducive to the laser
system that is the focus of this work. Instead the laser system studied here employs
electrical injection pumping which is commonly used in semiconductor lasers. These are
discussed further in a forthcoming section. The other types of pumping are not as
common and are not discussed in detail.
The simplest optical pumping scheme is a two level system. The idea is to raise
electrons from lower energy states to a specific higher energy state by providing them
the amount of energy unique to that specific absorption wavelength or frequency. In a
three level system the electrons are first excited to a higher energy level (L3) that is
unstable after which they undergo spontaneous emission or non-radiative decay and
relax to a lower, more stable level (L2) after a characteristic lifetime in the higher state.
This lower yet more stable level is still higher than the initial ground level (L1) of the
electrons. The lasing phase is between L2 and L1, where stimulated emission induces
transitions to lower energy states and the release of photons. The most efficient scheme
is a four level system, however. A diagram is shown in Figure 2-12. In this system the](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-60-320.jpg)
![50
ground level is now L0 and there is an added unstable state above the ground state, L1.
The advantage of this over a three level system is that population inversion can be
achieved with much less effort since L1 will not be as heavily populated since it is a
more unstable state than L0. The reason that a simple two level system is highly
impractical is because the light used for pumping is just as likely to decrease the
population as to enhance it. Incoming photons stimulate emission when interacting with
electrons in the higher energy state and also cause absorption of energy and a transition
to the higher energy state for electrons in the lower state. Overall this means that even
with an infinite rate of pumping, the populations in both levels would at best equalize.
Four level systems provide the opportunity for the highest possible population inversion
and hence more efficient lasing [30].
L3
Relaxation (Fast)
L2
Pumping Lasing
L1
Relaxation (Fast)
L0
Figure 2-12. A four level system. L3 and L1 are both unstable states so electrons rapidly decay to L2 and
L0, usually in a non-radiative transition.
2.84 Optical Resonators and Amplification
Some of the excited electrons will then begin to decay back to their lower energy
states by spontaneous emission. The photons emitted interact with other electrons still in
the higher energy state giving rise to more photons as they traverse the optical cavity.
An external optical cavity is formed by surrounding the source of the laser light with
two highly reflective surfaces. An internal cavity would be formed by fabricating the
two end surfaces of the gain medium with specific reflectivity. This causes most of the
photons to continue bouncing back and forth between the two surfaces producing
amplification by further stimulated emission provided there are no losses within the](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-61-320.jpg)
![51
cavity. The situation when there are losses within the cavity will be discussed in the next
section. The intensity of light grows exponentially with each trip through the cavity until
it begins to saturate. This situation will also be discussed in the next section. Gain is a
measure of how much the intensity is amplified from the initial value put into the gain
medium. The expression for gain is [30]
))(exp(
0
z
I
I
G νγ== , (2.33)
where I and I0 are the intensity and initial intensity, γ(ν) is the gain coefficient at a given
frequency ν, and z is the distance traversed within the cavity. The gain coefficient can in
turn be expressed as [30]
Nv ∆= )()( σνγ , (2.32)
where σ(ν) is called the gain cross section at frequency ν and ∆N is a measure of the
population inversion for the particular type of gain medium. The gain cross section
varies depending on frequency, index of refraction of the material, and the spontaneous
emission coefficient. It is an important measure of how much gain per cross sectional
area will be ‘seen’ by an incoming beam of light and is an important performance metric
when discussing potential light emitting materials for use in the particular laser system
in this work.
Pictures of two optical cavities exhibiting amplification are shown in Figure 2-
13a [9] and Figure 2-13b. The cavity in (a) is designed so that some of the light at one
end is transmitted or coupled out of the cavity on each round trip. This particular type of
cavity is called a Fabry-Perot cavity. The ring structure is seen in (b). It can also be seen
as part of the device in Figure 1-5b. It is often more useful when electrical injection is
used to pump the gain medium.
The length and reflectivity of the mirrors on either end can be custom-designed
to establish longitudinal mode selection and frequency-selective feedback. This means
that the resonator essentially acts as a filter allowing only photons of a certain frequency
or wavelength to be amplified while all other modes will slowly die off because they
aren’t being reinforced with each round trip through the cavity.](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-62-320.jpg)
![52
(a)
(b)
Figure 2-13. (a) Simple Fabry-Perot cavity with mirrors on either end. Also shown is the pumping
(energy) source [9]. (b) Ring resonator structure which is the type of structure used for the device in this
work.](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-63-320.jpg)
![54
sustain oscillation and quickly die out [30]. This is very important since there are
generally a large number of possible modes into which light can couple and the goal is
to couple into only one of them. With proper design, the resonance frequency can be
tuned to a desired value by selective feedback within the cavity.
2.8.6 Electrical Charge Injection
It is now appropriate to discuss electrical injection since it is the pumping
scheme utilized in the MURI Si laser project and is necessary for any practical, cost-
effective light emission system since it eliminates the need for an external source of
light. Electrical injection is used to achieve population inversion within a material by
introducing carriers (electrons, holes, or both) into excited states which result in light
emission or electroluminescence (EL) upon radiative recombination. There are various
ways of electrically pumping a medium, but for the extrinsic, slot confined cavity design
task of the MURI project the ideal method is carrier tunneling. Band injection is used
primarily for the intrinsic laser design. To accurately understand how electrical injection
works it is first necessary to review the nature of energy bands and carrier states within
both bulk and quantum confined semiconductor structures.
In bulk semiconductor materials the discrete energy levels in atomic systems are
replaced by large energy bands with nearly continuous (yet technically still discrete)
distributions of energy states within them. These energy bands describe the relationship
between the energy E and the crystal momentum or wavevector k. The particular
arrangement and value of the energy levels varies depending on the type of material.
The energy band structures for Si and GaAs were shown in Figure 1-7. The valence
bands are those below E = 0eV and the conduction bands are those above 0eV. The
fundamental emission process is the recombination of an electron from the conduction
band and a hole from the valence band to generate a photon. Both the conservation of
energy and momentum apply to this process and are stated as follows [30]:
νhEE ba =− (Conservation of energy) (2.33)
valenceconductionphoton hkhkhk −= (Conservation of momentum) (2.34)](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-65-320.jpg)
![56
Figure 2-14. A simple energy band diagram for a bulk material (left) and a quantum-confined
nanocrystal (right).
The development of an efficient method for electrical injection into silicon
nanocrystal based devices presents a critical challenge that is crucial to overall device
performance. There have been a number of schemes proposed for electrical injection
into nc-Si based devices, the details of which are quite involved and beyond the scope of
this project. One of the more interesting methods – which is also the currently proposed
method for the MURI project – is provided by the group at Stanford [14]. This particular
scheme is known as field effect injection and is carried out using a floating-gate
transistor structure. It shows that excitons can be efficiently produced in Si nanocrystals.
Basically holes and electrons tunnel into the floating gate array of silicon nanocrystals
where they form excitonic pairs and undergo radiative recombination. The bandgap of
the nanocrystals is specifically tailored to enable efficient energy coupling to the Er ions.
This type of structure enables light emission in CMOS-compatible Si structures and is
also suitable for the device proposed in this work. A diagram is seen in Figure 2-15.](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-67-320.jpg)
![57
Figure 2-15. Diagram of the field-effect structure for achieving electrical injection [14]. The array of
silicon nanocrystals exists in the light gray area directly beneath the gate.
An important figure of merit for any electrically-pumped laser system is the
threshold current density. The threshold current density is the minimum amount of
current per unit area required to achieve lasing. Generally it is desirable to minimize this
parameter since it dictates the power consumption of the device. It depends primarily on
the internal radiative efficiency of the excitons, the gain cross section of the light
emitting material, mode confinement, and losses within the optical cavity or gain
medium. All of these are important considerations and tradeoffs that must be adequately
addressed in the design of cost-effective, efficient light emitting devices with low power
consumption.
2.8.7 MURI Si Laser Project – Er3+
as an Extrinsic Light Emitter
One of the solutions to the unfavorable characteristics of Si as a light-emitting
material is to add a Si-compatible extrinsic material into the laser cavity’s host material
to act as a light emitter. In the extrinsic gain laser task of the MURI project, the two
extrinsic dopant candidates are Er3+
ions and Lead Sulfide/Lead Selenide quantum dots.
This work in particular involves the use of the Er3+
ion. It is incorporated into
nanostructured layers of SiO2 to promote population inversion. The two major reasons
for this, according to the MURI executive summary, are that it “leverages an extensive
library of glass materials knowledge from the fiber optic telecommunications industry
and exploits the novel energy transfer properties of Si-nanocrystals and Si-nanowires”
[13].](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-68-320.jpg)
![58
The multilayer waveguide structure consists of alternating layers of nanometer-
thin layers of Si nanocrystals and erbium-doped SiO2 embedded in an active horizontal
slot region and can easily be made into a resonant cavity. The Si nanocrystals are
formed by first depositing amorphous Si and then appropriately tuning the thermal
budget. In particular, it is found that a ring-shaped resonator provides the best
configuration for efficient electrical injection and low optical loss. It is important to
ensure maximum confinement and minimal loss of the optical modes within the cavity,
especially since erbium has a relatively low emission cross section at wavelengths near
1.55µm. The device is pumped utilizing the field effect injection method described
previously. The thin nc-Si layers allow for excellent carrier transport (tunneling). In this
way, nc-Si provides a way to avoid the problems inherent in bulk Si as well as the
excessive degradation of dielectric materials that would occur as a result of hot carrier
injection [13]. Results show that the internal radiative quantum efficiency for excitons
in Si nanocrystals is sufficiently high – between 60 and 100% [35]. This is important for
maximizing the efficiency of the energy transfer to the Er3+
light emitters. Pictures of
the device structure are seen in Figure 1- 10 and Figure 1-11.
Since Er is the light emitting material, it is important to discuss its absorption
and emission properties. An energy level diagram for Er3+
ions in SiO2 is shown in
Figure 2-16a. This shows the many possible energy level transitions that may occur. In
this case there is an optical pumping source which raises electrons to higher energy
states. A more detailed view of the 4
I15/2
4
I13/2 transition in particular is shown in
Figure 2-16b. This is the transition that produces emission at 1.55µm. The large number
of transitions from lower to higher energy states between these two bands can clearly be
seen here. The electrons and holes both obey Boltzmann statistics and as a result the
strongest absorption will occur from highly populated levels in the lower energy band to
lesser populated levels in the conduction band. Because the energy transfer process will
effectively act as the pumping source for the Er, it is useful to look at its absorption
spectrum. An absorption spectrum for Er3+
ions in silica is shown in Figure 2-17 [36].
In this case, the ions are excited by an optical source.](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-69-320.jpg)
![60
Figure 2-17. Solid lines show the absorption spectrum of an Er-doped optical fiber for wavelengths
between 450-850nm. This shows the principal absorption bands within the range of wavelengths. [36].
The radiative emission in Er3+
that will be useful in this project occurs in the
4
I15/2
4
I13/2 transition. The strongest emission takes place between the most highly
populated levels in the upper band and the lesser populated levels in the lower band, as
shown in Figure 2-16b. A photoluminescence (PL) emission spectrum for Er3+
is shown
in Figure 2-18 [37]. A very favorable aspect of Er-doped silica is that two of its primary
emission peaks are around 1.55µm (actually about 1.535 and 1.551µm) which happens
to be the minimum attenuation wavelength for silica waveguides and fibers. This is the
reason that it is the most commonly used wavelength for optical fiber-based
communication links. Using erbium as the extrinsic light-emitting material is therefore a
very viable alternative to devices that use intrinsic silicon simply because of the many
problems encountered in obtaining direct light emission from silicon.](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-71-320.jpg)
![61
Figure 2-18. Photoluminescence (PL) emission spectrum for Er -doped (~5 x 1015
cm-2
) silica film on a Si
substrate when pumped with 488nm light. The wavelengths at which the two peak intensities occur are
labeled [37]
After the creation of an exciton (either electrically or optically) in the nc-Si, it
will couple its energy to the Er3+
ions in a process known as Förster energy transfer,
shown in Figure 2-19. The bandgap of the nc-Si is tuned so that it closely matches the
resonant absorption levels of Er – this helps ensure maximum energy transfer. The Er
ions are excited to higher energy states where they undergo non-radiative decay to the
4
I15/2 level. Finally, radiative recombination and the emission of photons will occur from
4
I15/2 to 4
I13/2. Extensive investigation has shown that the energy transfer to Er is faster
than both the exciton recombination in Si nanocrystals and the radiative emission rate in
Er. This enables Er ions to be easily inverted – another practical benefit of using Er. It
has also been shown that Er can be incorporated at fairly high concentrations within
SiO2 which allows the nc-Si:Er gain medium to compensate for the notoriously small
emission cross section of Er. One additional advantage in using Si nanocrystals is that
they will not cause significant scattering losses within the cavity [13].](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-72-320.jpg)
![Chapter 3
Simulation & Results
3.1 Introduction
In order to analyze and design realistic photonic components such as waveguides
it becomes necessary to move beyond two-dimensional slab-based structures and to now
consider those of three dimensions. As noted, this complicates the analytical techniques
presented in the previous chapter. A mathematical solution for even the simplest 3D
structures can be quite involved even with a number of limiting approximations [1, 38].
Therefore the use of a commercial software package designed for numerical simulation
of photonic components is needed to provide more rapid and tractable analysis.
For all simulations in this study the RSoft Design Group’s Photonics CAD Suite
is used. Analysis and graphing is performed with Mathworks’ Matlab. For a better
understanding of the RSoft tools FullWave and BeamProp please refer to [15, 16] and
the references therein. The results of these simulations are presented beside results from
a technique known as the transfer matrix method (TMM) which serves as a basis for
comparison to the more realistic FDTD simulations. An application of the TMM is
developed specifically for the structure presented in this work by a fellow colleague [39].
One minor thing to note is that for 3D structures there are no purely transverse magnetic
(TM) or transverse electric (TE) modes as discussed in chapter 2. Instead there are only
hybrid modes which are either ‘TM like’ or ‘TE like’. These are usually denoted quasi-
TM and quasi-TE but in order to avoid confusion will continue to be referred to simply
63](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-74-320.jpg)
![64
as TM and TE throughout the remainder of the chapter where it is implied that these are
hybrid modes.
There are a few geometrical differences between the 3D and 2D structures as
well. In the 2D model, the structure simply consisted of two waveguides situated side by
side with a low index slot between them. The 3D version presented here is slightly more
complicated. Instead of only two waveguides and one slot, it will consist of repeating
periods of high and low index layers or slots. To visualize this, think of multiple
waveguides lined up next to each other. The guiding layers of each waveguide are the
high index layers and the slots between the waveguides are the low index layers.
Additionally, because it is easier to render alternating material layers by horizontal
deposition than by vertical side wall fabrication, the structure will have its layers
oriented horizontally – a 90° rotation from the configuration of the 2D model shown
previously. The structure is referred to as a multilayer waveguide. A 3D model of this
type with only a few repeating periods of high index guiding layers and low index slots
is possible and has been studied [34], though it will be shown that the confinement
factor can be further enhanced by fabricating structures with additional thinner layers,
and a higher SiO2 to Si layer thickness ratio. When the total height of the structure is
large enough and the layers are thin enough, the actual number of layers or periods does
not affect the confinement factor as much as the ratio between the low index (SiO2) slot
layer and high index (Si) guiding layer thicknesses – from this point forth referred to
simply as ratio or thickness ratio. In fact, it was observed in the discussion of chapter 2
that the confinement factor (CF) is maximized only for a range of ratio values. This
result can be extrapolated to the 3D structure where it can be assumed that a similar
situation exists.
A more tangible explanation of why the ratio is the critical parameter is provided
by the effective index method or effective index approximation [30]. This is a way of
treating the entire multilayer region as a single layer of the same thickness with an
effective index of refraction determined by a weighted average of the Si and SiO2 layers
based on their thicknesses and indices of refraction. Because the weighted average
depends on the two thicknesses, their ratio becomes a critical parameter. The initial](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-75-320.jpg)
![66
structures can be accurately modeled in this way. This is the reason that only a few ideal
structures are chosen for the gain/loss analysis. The benefits of the TMM are realized
through parameter scanning since it is able to produce accurate results for a continuous
range of structures as well as for layer thicknesses as low as 3nm in much less time than
the FDTD method. It provides a very practical model for approximating performance in
3D rib waveguide structures to which the results of FDTD simulations can be compared.
It is found that these methods are very consistent with each other. It should be noted that
most of the results presented here, along with other details beyond the context of this
work are also presented in the TMM article [39].
3.2 Structure
A cross-section of the proposed device structure is shown in Figure 3-1 with the
x and y axes as shown. A more complete picture of the entire laser system in which the
waveguide is used is shown for reference in Figure 3-2. This picture shows a gain
medium with only one layer of nc-Si and SiO2:Er when in fact, the proposed device
structure will have multiple layers of these materials in the gain medium as shown in
Figure 3-1 and Figure 1-11. The z axis is directed into the page and is the direction of
propagation. This particular orientation of the axes will be used consistently throughout
the remainder of the chapter. The structure consists of alternating nanometer-thin layers
of Si and SiO2 with a SiO2 cap layer and a SiO2 substrate. Notice that starting with the
bottom layer, there are N repeating periods of Si/SiO2 with a final layer of Si underneath
the SiO2 cap layer. Thus there is always one additional layer of Si. The refractive index
of Si and SiO2 are taken to be 3.458 and 1.440 for all simulations and the height of the
cap layer is set at 50nm. The height of the multilayer region and the width of the cap –
from this point forth referred to simply as the slab height and width – are each
constrained by the requirement of single mode operation in both the horizontal and
vertical directions. The width, w, is set at 1µm while the slab height is dependent on the
number of layers and thickness of each layer but is restricted to values less than about
0.4µm. These values were determined using both TMM calculations and the
aforementioned effective index method [30, 39]. The limitation on the slab height is
especially important since it determines the maximum number of repeating Si/ SiO2](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-77-320.jpg)
![68
Figure 3-2. Complete cross sectional diagram of the structure. The active gain medium is actually
composed of alternating layers of nc-Si and SiO2:Er. This particular diagram shows a horizontal slot ridge
waveguide design. Electrons and holes are injected through the p and n contacts into the p and n type
device layers. They then form excitons by tunneling into the nc-Si layers. Also visible is the ring
resonator structure used to achieve frequency selective feedback and amplification [13].
The logic behind such a design can be explained in a fairly straightforward
manner. The primary advantage of this type of waveguide is that it is relatively easy to
fabricate. Multiple layers of Si and SiO2 can first be deposited using a conventional
deposition process with a very high level of precision. The cap layer is formed by
selective chemical etching of previously deposited layers [13, 39]. The cap is used to
confine the propagating light in the region of the multiple layer waveguide which sits
directly underneath it. The design proposed recently by MIT is shown in Figure 3-3a
and Figure 3-3b and is based upon the same theoretical concepts presented towards the
end of Chapter 2 but is not nearly as easy to make since it would require incredibly high
precision etching along the side walls [34]. Therefore it is a somewhat impractical
design. By using a ridge structure with a cap layer, the device enables propagating
modes to be confined to one area of the waveguide (which is effectively infinite in the
horizontal dimensions) without requiring expensive and difficult sidewall etching.](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-79-320.jpg)
![69
(a)
(b)
Figure 3-3. (a) Slot confinement structure proposed by MIT [34]. The structure consists of upper and
lower high index cladding layers with alternating thin low index slot layers with high index thicker guide
layers in between. In this case, w is the width, tc, tH , and tl are the thickness of the cladding, low index
slots and high index layers respectively. The refractive indices of the high and low index layers are nH
and nl respectively while n0 is the refractive index of the surrounding air. It operates on the principle of
confinement within low index slot layers but is much more difficult to fabricate owing to the extremely
high precision required to etch the sidewalls. (b) Contour picture showing the quasi TM mode distribution
of the y component of the E field of the structure proposed in [34]. Also shown is the distribution of the y
component of the E field along the structure (on the right side) [34].](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-80-320.jpg)
![76
0.00100161
0.167571
Power (in z’ direction) @ cT=36.0448µm [Mon#2 (0,0,9.4)]
X’ (µm)
1- 0 1
Y’(µm)
0.4-
0.3-
0.2-
0.1-
0.0
0.1
0.2
0.3
0.4
0.5
Figure 3-7. Contour picture of the z component of power distribution of the TM mode across the layers in
the structure shown in Figure 3-1. Again the high confinement in the low index layers is evident. Note
also the oblong shape of the mode as it propagates through the structure. The evanescent tail in the
substrate is visible here as in the two previous figures.
Another point that needs to be addressed is that of the effective thickness ratio
between the SiO2 and Si layers. It was noted that there is always one extra layer of Si
within the multilayer region of each structure. Therefore even though the actual ratio
between the SiO2 and Si layers is 1.25 for example, the overall effective ratio will be the
total thickness of all the SiO2 layers divided by the total thickness of all the Si layers.
Each original ratio value is then adjusted to an appropriate effective thickness ratio.
From this point forth, the term ratio will refer to the effective thickness ratio unless
otherwise stated. Table 3-1 shows the original ratio values alongside their
corresponding effective ratio values.
Figure 3-8 shows plots of CF versus effective SiO2/Si thickness ratio using both
the FDTD simulation as well as the TMM method for both TE and TM polarizations.
The reason for the additional simulations using TE polarization is to serve as a
comparison to the TM modes. The FDTD results are given for slab heights of 0.370 –
0.405µm and are shown as discrete data points. Each point represents the simulation of a
single structure. It was mentioned previously that this range of heights guarantees single
mode operation for both TE and TM modes at a wavelength of 1.55µm. The maximum](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-87-320.jpg)
![77
CF is about 55% and occurs for a ratio of about 0.80. The TMM plot is given for a slab
height of exactly 0.380µm. This enables the most accurate comparison to the FDTD data
points.
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
Thickness ratio between SiO2
and Si layer
Confinementfactor
TMM and FDTD result for multilayer thickness of 0.38 µm
TE mode confinement
TM mode confinement
FDTD data points for TM
FDTD data points for TE
Figure 3-8. Confinement factor versus SiO2/Si thickness ratio of the TE and TM modes for both the
FDTD and TMM methods. The FDTD plots are shown as crosses and circles and are for slab heights
t of
or reasons mentioned earlier, it is impossible to achieve reasonable, non-
skewed ith
t
between 0.370µm and 0.405µm. The TMM plots are in dashed and solid lines and are for a slab heigh
exactly 0.380µm [39].
F
simulation results with the slab height set at a specific value. It has to do w
the fact that varying the thickness ratio continuously necessitates layer thicknesses that
are not whole numbers. Because of the lower limit on the resolution and layer thickness
as well as the need for these values to be whole numbers, there cannot be one set slab
height. For each ratio value, the combination of layer thickness and number of Si/SiO2
periods which results in the slab height closest to 0.400µm is used. A listing of the exac
parameters for each structure is shown in Table 3-1.](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-88-320.jpg)
![78
N Ratio Effective
Ratio
T_Si
(nm)
T_SiO2
(nm)
Total Height
(nm)
CF
(%)
6 2.00 1.71 20 40 380 42.57
7 1.50 1.31 20 30 370 48.56
7 1.75 1.53 20 35 405 52.03
8 0.80 0.71 25 20 385 61.56
8 1.25 1.11 20 25 380 58.46
9 1.00 0.90 20 20 380 60.41
Table 3-1. A listing of parameters for various structures spanning the presumed range of ideal ratios. The
value of 20nm for Si layer thickness was chosen since it is most amenable to a resolution of 5nm per grid
point. A height of 0.380µm is chosen for the corresponding TMM CF values in the graph simply because
it is the most common height of the structures presented here.
There are three structures with a slab height of 0.380µm and another that is
0.385µm. It is for this reason that the slab height for the TMM plot is set at 0.380µm.
The agreement between these two models is quite evident from Figure 3-8 even though
the FDTD values are slightly higher. There is an obvious source of error stemming from
the fact that not every FDTD point has a slab height of exactly 0.380µm. This explains
the slight jump in the value of the CF for the structure that is 0.405µm. Another possible
source of error is the fact that the propagation length is a mere 10µm owing to limited
time and memory resources. Such a relatively short propagation length means the
eigenmode is not able to completely form. Overall a general trend is clear for both plots.
The behavior of this graph is not surprising given the results shown in Figure 2-
10 from the last chapter as well as the effective index method. There is clearly a point at
which the CF begins to saturate followed by a range of optimum ratio values and finally
a slowly decreasing tail for ratios beyond the optimum range. As expected, this range of
ratios is roughly 0.71 – 1.11 (0.80 – 1.25 for the original ratio values). This is also the
range for which the difference between the CF in the TM modes and that in the TE
modes is highest.
Figure 3-9 shows an additional graph for a similar structure with a slab height of
0.520µm for reference [39]. This plot shows only TMM results but exhibits a trend](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-89-320.jpg)
![79
similar to Figure 3-8. The structure exceeds the cutoff height for single mode operation,
but only for TE polarized modes. There will still be only one TM mode. It is observed
that a CF as high as about 75% can be obtained for a ratio of 1.00. This could be very
useful for devices in which single mode operation is only required only for the TM
modes – a distinct possibility for potential designs based on this study.
0.5 1 1.5 2 2.5
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
Thickness ratio between Si and SiO2
layer
Confinementfactor
TMM result for multilayer thickness of 0.52 µm
TE mode confinement
TM mode confinement
Figure 3-9. Confinement factor versus SiO2/Si thickness ratio of the TE and TM modes for the TMM
method. It exhibits a similar trend to Figure 3-8. The ideal ratios fall within the same range as for the
0.380µm structure with the maximum being about 75% occurring at a ratio of about 1.1 [39].
With an ideal range of ratios confirmed through the agreement between the
TMM and FDTD methods, three structures with ratios distributed within this ideal range
are chosen and an analysis of the gain and loss behavior is commenced.
Before the next section is presented however, it should be noted that the
structures proposed here are not physically viable since the Si and SiO2 layers are both
too thick to support the necessary dipole interaction (Förster energy transfer) between
the Er dopants in SiO2 and the Si nanocrystals in the Si layers. Of course the main idea
presented in this section is the fact that thickness ratio (along with the material
properties of each layer) and not necessarily layer thickness, is the critical parameter.](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-90-320.jpg)
![82
meaningless and that only relative power is important. The losses in the TM and TE
modes are computed and denoted as γTM and γTE. In all three structures the TE mode loss
was approximately four times greater than the TM mode loss. The worst case scenario
occurred for the first structure where γTE/γTM = 3.89. These values are consistent with
TMM calculations and show that a significantly larger loss is incurred by TE modes
than by TM.
The next step in the procedure is to perform repeated simulations with the same
amount of loss in the Si layers as before while adding gain to the SiO2 layers in small
increments starting with zero. Doing so will gradually increase the gain to loss
coefficient ratio as well as the peak power observed at the time monitor, Pmon. The gain
is programmed into the structure in the same way as the loss except that the sign of the
imaginary component of the refractive index is negative instead of positive. The
objective is to determine the ratio between the gain coefficient in SiO2 and the loss
coefficient in the Si (gain to loss coefficient ratio) that yields net gain in the cavity. This
will be denoted Rng where ‘ng’ denotes net gain. The net gain within the cavity, G is
computed as
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
0
ln
P
P
G mon
, (3.03)
where Pmon is the peak power at the time monitor and P0 is the peak power at the time
monitor with no loss or gain in the structure.
If the light distribution were equal among all layers, the value of Rng would be
exactly one. However, since this is not the case for the TM modes, it will be less than
one. Using Matlab, a quick calculation shows that the power confined in the SiO2 layers
is about four times that in the Si layers. This means that the loss in Si can be roughly
four times greater than the gain in SiO2 and net gain will still be observed. Thus the
expected value of Rng for TM modes is around 0.25. Figure 3-10 and Figure 3-11 show
plots of the overall gain G versus the gain/loss coefficient ratio for each polarization for
the TMM and FDTD methods respectively for the first structure from Table 3-2 [39].
Each point on the FDTD graph is a separate simulation. Rng is simply the x intercept of
each plot. The slopes of the corresponding polarization plots for the two methods are
noticeably different due to the fact that different gain and loss coefficients were used but](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-93-320.jpg)
![83
this is not relevant (the gain and loss coefficients for the FDTD simulations were
magnified by a factor of about ten to account for the shorter propagation distance). The
important thing to notice is that the value of Rng is approximately the same for
corresponding polarizations of each method. Both TM plots cross the x axis at a value of
roughly 0.27 while the TE plots cross at around 2.5. The behavior of the TM plot is as
expected but that of the TE plot is somewhat odd. There is no discontinuity of the E
field for this polarization so an Rng value of about one is expected. However, the fact that
the value is much greater than one seems to suggest that these modes see more of the
high index Si and experience more losses as a result. One possible explanation is that
the large discrepancy between the gain profile (high contrast between layers) and the
actual mode distribution (approximately a smooth Gaussian) leads to a coupling effect
which causes a leakage of photons from the outer SiO2 layers where they experience
amplification [39]. This area requires further investigation in order to find a more
sufficient explanation.
0 0.5 1 1.5 2 2.5 3
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Modal gain for TE and TM polarization by TMM
gain/loss coefficient ratio
modeloss/gain(1/cm)
TE mode gain
TM mode gain
Figure 3-10. Plot of modal gain G (in cm-1
) versus gain to loss coefficient ratio for the TMM method for
both the TE and TM modes. The zero gain value for TE modes is about 2.45 and about 0.25 for TM
modes [39].](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-94-320.jpg)
![84
0 0.5 1 1.5 2 2.5 3
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
Gain/Loss coefficient ratio
ModalGaincoefficientA.U.
Propagation Gain from FDTD simulation
TM propagation gain
Reference
TE propagation gain
Reference
Figure 3-11. Plot of propagation gain in the optical cavity, G (in arbitrary units) versus gain to loss
coefficient ratio for the FDTD method for both the TE and TM modes. The net gain value for the TE
modes is about 2.5 and about 0.25 for TM modes [39].
In conclusion it has been shown that by virtue of the large refractive index
contrast between Si and SiO2 layers, there is a redistribution of the E field that leads to
high confinement within lower index SiO2 layers. This allows for a much lower gain to
loss coefficient ratio to be realized since more photons of light see amplification in the
SiO2 layers than absorption and loss in the Si layers. Thus, the same amount of overall
gain can be achieved with a lower threshold current density and less overall power input
in the device design. This represents a significant breakthrough with respect to the
design and optimization of optical systems and devices based on Si as low power
consumption and dissipation are always a high priority in these endeavors.](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-95-320.jpg)
![Bibliography
[1] R. G. Hunsperger, Integrated Optics: Theory and Technology, 5th ed. Berlin:
Springer-Verlag, 2002. pp. 1-71
[2] L. Pavesi and D. J. Lockwood, "Silicon Photonics," in Topics in Applied
Physics. vol. 94 Berlin: Springer-Verlag, 2004, pp. 1-90.
[3] "Silicon Photonics Research at Intel." vol. 2007 Santa Clara, CA, 2006.
Overview of Silicon Photonics research at Intel, Inc. with diagrams,
explanations and links to further resources and documents. Available:
http://www.intel.com/research/platform/sp/
[4] M. Paniccia, V. Krutul, and S. Koehl, "Introducing Intel's Advances in
Silicon Photonics," pp. 1, Feb 2004.
Available: http://www.intel.com/technology/silicon/sp/doc.htm
[5] M. Paniccia, V. Krutul, and S. Koehl, "Intel Unveils Silicon Photonics
Breakthrough: High-Speed Silicon Modulation," in Technology @ Intel
Magazine, 2004, pp. 1-6.
Available: http://www.intel.com/technology/silicon/sp/doc.htm
[6] R. Claps, D. Dimitropoulos, Y. Han, and B. Jalali, "Observation of Raman
emission in silicon waveguides at 1.54 microns," Optics Express, vol. 10, no.
22, pp. 1305, Nov 2002.
[7] R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali,
"Observation of stimulated Raman amplification in silicon waveguides,"
Optics Express, vol. 11, no. 15, pp. 1731, Jul 2003.
[8] O. Boyraz and B. Jalali, "Demonstration of a silicon Raman laser," Optics
Express, vol. 12, no. 21, pp. 5269, Oct 2004.
[9] M. Paniccia, S. Koehl, and V. Krutul, "Continuous Silicon Laser," pp. 1,
Feb 2004.
Available: http://www.intel.com/technology/silicon/sp/doc.htm
[10] H. Rong, R. Jones, A. Liu, et al., "A continuous-wave Raman silicon laser,"
Nature, vol. 433, no. 7027, pp. 725, Feb 2005.
[11] J. Faist, "Silicon shines on," Nature, vol. 433, pp. 691, Feb 2005.
87](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-98-320.jpg)
![88
[12] A. Liu, H. Rong, R. Jones, O. Cohen, D. Hak, and M. Paniccia, "Optical
Amplification and Lasing by Stimulated Raman Scattering in Silicon
Waveguides," Journal of Lightwave Technology, vol. 24, no. 03, pp. 1440,
Mar 2006.
[13] P. M. Fauchet, "MURI: Si-based Laser Grant Award," University of
Rochester: AFOSR/MIT/University of Rochester, 2006.
[14] R. J. Walters, G. I. Bourianoff, and H. A. Atwater, "Field-effect
electroluminescence in silicon nanocrystals," Nature, vol. 04, pp. 143, Feb
2005.
[15] RSoft, "BeamPROP 7.0 User Guide," Ossining, NY: RSoft Design Group,
Inc., 1993-2006, pp. 13-23.
[16] RSoft, "FullWAVE 5.0 User Guide," Ossining, NY: RSoft Design Group,
Inc., 1999-2006, pp. 15-19.
[17] R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, "Numerical
Techniques for Modeling Guided-Wave Photonic Devices," IEEE Journal of
Selected Topics in Quantum Electronics, vol. 06, no. 01, pp. 150, Jan/Feb
2000.
[18] R. Scarmozzino and R. M. Osgood Jr., "Comparison of finite-difference
and Fourier-transform solutions of the parabolic wave equation with
emphasis on integrated-optics applications," Journal of the Optical Society
of America A, vol. 08, no. 05, pp. 724, May 1991.
[19] A. Tavlove, Computational Electrodynamics: The Finite-Difference Time-
Domain Method. Norwood, MA: Artech House, 1995.
[20] A. Tavlove and M. E. Browdin, "Numerical Solution of Steady-State
Electromagnetic Scattering Problems Using the Time-Dependent Maxwell's
Equations," IEEE Transactions on Microwave Theory and Techniques, vol.
MTT-23, no. 08, pp. 623, Aug 1975.
[21] K. S. Yee, "Numerical solution of initial boundary value problems involving
Maxwell's equations in isotropic media," IEEE Transactions on Antennas
and Propagation, vol. AP-14, no. 03, pp. 302, May 1966.
[22] D. Yevick, "A guide to electric field propagation techniques for guided-
wave optics," Optical and Quantum Electronics, vol. 26, p. S185, 1994.
[23] D. Yevick and B. Hermansson, "New formulations of the matrix beam
propagation method: Application to rib waveguides," Journal of Quantum
Electronics, vol. 25, p. 221, 1989.](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-99-320.jpg)
![89
[24] D. Yevick and B. Hermansson, "Efficient beam propagation techniques,"
Journal of Quantum Electronics, vol. 26, p. 109, 1990.
[25] J. P. Berenger, "A Perfectly Matched Layer for the Absorption of
Electromagnetic Waves," Journal of Computational Physics, vol. 114, no. 02,
pp. 185, 1994.
[26] M. D. Feit and J. A. Fleck, "Light propagation in graded-index optical
fibers," Applied Optics, vol. 17, p. 3990, 1978.
[27] M. D. Feit and J. A. Fleck, "Computation of mode properties in optical
fiber waveguides by a propagating beam method," Applied Optics, vol. 19, p.
1154, 1980.
[28] G. R. Hadley and R. E. Smith, "Full-vector waveguide modeling using an
iterative finite-difference method with transparent boundary conditions,"
Journal of Quantum Electronics, 1995.
[29] D. J. Griffiths, Introduction to Electrodynamics, 3rd ed. Upper Saddle River,
NJ: Prentice Hall, 1999. pp. 321-91
[30] C. Pollock, Fundamentals of Optoelectronics, 1st ed. Chicago: Irwin, 1995.
pp. 1-71,399-489
[31] D. L. Lee, Electromagnetic Principles of Integrated Optics. New York, NY:
John Wiley & Sons, 1986. pp. 1-50
[32] V. R. Almeida, Q. Xu, C. Barrios, and M. Lipson, "Guiding and confining
light in void nanostructure," Optics Letters, vol. 29, no. 11, pp. 1209, Jun
2004.
[33] Q. Xu, V. R. Almeida, R. R. Panepucci, and M. Lipson, "Experimental
demonstration of guiding and confining light in nanometer-size low-
refractive-index material," Optics Letters, vol. 29, no. 14, pp. 1626, Jul 2004.
[34] N. N. Feng, J. Michel, and L. C. Kimerling, "Optical Field Concentration in
Low-Index Waveguides," IEEE Journal of Quantum Electronics, vol. 42, no.
09, pp. 885, Sept 2006.
[35] R. J. Walters, J. Kalkman, A. Polman, H. A. Atwater, and M. J. A. de Dood,
"Photoluminescence quantum efficiency of dense silicon nanocrystal
ensembles in SiO2," Physical Review B, vol. 73, pp. 132302, 2006.
[36] A. J. Kenyon, C. Chryssou, and C. W. Pitt, "Indirect excitation of 1.5
micron emission from Er3+ in silicon-rich silica," Applied Physics Letters,
vol. 76, no. 06, pp. 688, Feb 2000.](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-100-320.jpg)
![90
[37] A. Polman, "Erbium implanted thin film photonic materials," Journal of
Applied Physics, vol. 82, no. 01, pp. 1, Jul 1997.
[38] S. E. Miller, "Integrated Optics: An Introduction," Bell System Technical
Journal, vol. 48, no. 07, pp. 2059, Sept 1969.
[39] Y. Fu, D. Riley, and P. M. Fauchet, "Photon Confinement in multi-nano-
layer waveguides," Rochester, NY: University of Rochester, p. 3, Aug 2007.](https://image.slidesharecdn.com/9aa29467-3551-4010-9002-ee7310939ca2-160209203316/85/Toward-an-Electrically-Pumped-Silicon-Laser-Modeling-and-Optimization-101-320.jpg)
Toward an Electrically-Pumped Silicon Laser Modeling and Optimization
- 1. Toward an Electrically-Pumped Silicon Laser: Modeling and Optimization by Daniel B. Riley Submitted in Partial Fulfillment of the Requirements for the Degree Master of Science Supervised by Professor Philippe M. Fauchet Department of Electrical and Computer Engineering The College School of Engineering and Applied Sciences University of Rochester Rochester, New York
- 2. Curriculum Vitae Daniel B. Riley was born on October 15th , 1982 in Rochester, NY. He attended the State University of New York at Geneseo where he graduated with the Bachelor of Science degree in Applied Physics and the Bachelor of Arts degree in Mathematics in May of 2005. While in attendance, he was inducted into the Geneseo chapters of the Phi Beta Kappa, Sigma Pi Sigma, and Phi Eta Sigma honor societies. He attended University of Rochester from August 2005 – August 2007 where he pursued the Master of Science degree in Electrical and Computer Engineering with a concentration in optoelectronics. He worked in the research group of Dr. Philippe Fauchet and also served as a graduate teaching assistant during his first year in attendance. ii
- 3. Acknowledgements I would like to first thank my advisor, Dr. Philippe Fauchet who never failed to enlighten me with his insight and incredible knack for explaining even the most difficult concepts in a matter of minutes. I also thank all members of the Fauchet research group over the last year. You have all provided me a constant source of support, knowledge and encouragement. In particular I thank Yjing Fu and Jidong Zhang for valuable suggestions and vital discussion. I am also deeply indebted to Vicki Heberling for her tireless efforts in support of my work – especially with the color printer and all the adventures it presented. I must also thank the members of all participating institutions in the MURI silicon laser project for the continuous insight and enlightenment they have provided, as well as all sources of funding for the project. Finally, I thank the members of my Masters Thesis examination committee: Dr. Miguel Alonso from the Institute of Optics and Dr. Thomas Hsiang from the Department of Electrical and Computer Engineering. iii
- 4. Abstract A slot-confined, multilayer waveguide structure to be used as the resonant optical cavity in a silicon (Si)-based laser intended for use in chip-scale, nano-photonic systems has been modeled and optimized using both numerical and theoretical methods. The structure is a rib waveguide and consists of alternating nanometer-thin layers of amorphous Si (a-Si) and erbium (Er)-doped silicon dioxide (SiO2) surrounded by a cap layer and substrate each composed of SiO2. A carefully chosen thermal budget during fabrication will result in the formation of silicon nanocrystals in the Si layers producing nanocrystalline Si (nc-Si). This creates a series of high-index-contrast interfaces which enables high optical mode confinement and higher gain within the low index SiO2:Er layers for quasi-TM modes. A series of full-vectorial, finite-difference time domain (FDTD) simulations are performed on a range of optimized structures yielding confinement factors equal to or higher than those currently reported for similar structures. Additional simulations are performed with gain in the SiO2 layers and loss in the Si layers to model a realistic cavity. Varying the ratio of gain coefficient to loss coefficient between simulations and normalizing the output power yields a value for the minimum gain to loss ratio required to achieve net gain within the cavity. This in turn leads to a reasonable approximation for the lower limit on the threshold current density for electrical injection – a critical parameter in the design of devices with low power dissipation and consumption. All results are reported along with those of a different model known as the transfer matrix method (TMM) with analysis and discussion of the consistencies and discrepancies. Overall considerable agreement is found between the two approaches. This enhances the understanding of performance expectations for such device structures. iv
- 5. Table of Contents Curriculum Vitae .............................................................................ii Acknowledgements..........................................................................iii Abstract............................................................................................ iv Table of Contents ............................................................................. v List of Figures.................................................................................vii Foreword........................................................................................... 1 1 Introduction................................................................................... 2 Motivation.......................................................................................................2 MURI Silicon Laser Project........................................................................14 Project Overview..........................................................................................16 2 Theory .......................................................................................... 21 Maxwell’s Equations....................................................................................21 The Wave Equation......................................................................................23 Boundary Conditions at Dielectric Interfaces...........................................25 Field Polarization .........................................................................................28 Poynting Vector............................................................................................28 v
- 6. Total Internal Reflection .............................................................................29 Waveguides ...................................................................................................30 Planar Slab Waveguide ............................................................................................... 30 Slot Confinement Waveguide...................................................................................... 38 Power Confinement in Waveguides ........................................................................... 41 Lasers.............................................................................................................46 Introduction.................................................................................................................. 46 Three Processes: Absorption, Spontaneous Emission, Stimulated Emission......... 47 Population Inversion and Pumping Schemes............................................................ 49 Optical Resonators and Amplification....................................................................... 50 Gain Saturation and Lasing Threshold ..................................................................... 53 Electrical Charge Injection Pumping and Semiconductor Lasers.......................... 54 MURI Silicon Laser Project and Erbium Doped Fiber Amplifiers (EDFAs)........ 57 3 Simulation & Results .................................................................. 63 Introduction ..................................................................................................63 Structure .......................................................................................................66 Investigation of Eigenmodes Using BeamProp .........................................70 FDTD Simulations........................................................................................72 Gain and Loss Analysis................................................................................80 4 Conclusion.................................................................................... 85 Bibliography ................................................................................... 87 vi
- 7. List of Figures Figure 1-1. Diagram showing various applications of different interconnect technologies. Optical links are currently only used at longer distances (>100m) [2].................................... 4 Figure 1-2. Timeline of information carrying capacity for links in various electrical and optical links. Those in optical systems have data rates roughly 2 orders of magnitude higher than those in electrical systems [2].......................................................................................... 5 Figure 1-3. Relationship between processor and front side bus (FSB) clock speeds for various models over the past 15 years. Note the significant improvement in processor speeds over FSB speeds in recent years [2]............................................................................. 6 Figure 1-4. Diagram showing individual components of an optical transmitter-receiver system. The transmitter is the top diagram and the receiver is the bottom [4]........................ 7 Figure 1-5. Wafer cost per unit area for various materials versus wafer size. Notice that InP is currently only available in one wafer size and is prohibitively expensive compared to a Si wafer of the same size [2]................................................................................................. 9 Figure 1-6. Hypothetical timeline for the transition between electrical and optical based communication systems for various distances [2]. .................................................................. 9 Figure 1-7. Energy band diagrams for silicon (above) and gallium arsenide (below). Notice that Si has an indirect bandgap while GaAs has a direct bandgap. This explains why GaAs is a much better light emitter than Si [2].............................................................. 10 Figure 1-8. Cross section of the device designed by the Intel researchers. It is a ridge Si waveguide surrounded by SiO2. The silicon on insulator (SOI) technology used allows for a large mode confinement within the Si waveguide so that high enough Raman amplification is reached. The p and n type Si on either side help to draw electrons generated by two photon absorption out of the device and reduce parasitic losses such as free carrier absorption (FCA) [11]............................................................................................................ 12 Figure 1-9. Scanning electron microscope (SEM) image of a p-i-n diode waveguide used for Raman amplification and lasing experiments [12]........................................................... 12 Figure 1- 10. Outline of the actual structure (s) to be studied in this thesis. ........................ 17 Figure 1-11. (a) Complete diagram of the device. The upper left shows a top view of the ring resonator which is used to accomplish frequency selective feedback and amplification as well as output coupling. The bottom shows a cross section of the cavity. The upper right shows a zoomed in picture of the gain medium and the surrounding layers. In this work, the gain medium is actually composed of alternating layers of nc-Si and SiO2:Er as depicted in vii
- 8. both Figure 1-10 and in (b). This particular diagram shows the horizontal slot ridge design. Electrons and holes are injected by tunneling through the p and n contacts into the nc-Si layers where they will form excitons. Also visible is the ring resonator structure used to achieve frequency selective feedback and amplification as well as output coupling [13]. .. 18 Figure 2-1. A monochromatic plane wave incident upon a dielectric interface. The subscripts ‘i’, ‘r’, and ‘t’ denote the incident, reflected and transmitted portions of the wave. The diagram shows the case of TE polarization meaning that the E field is directed out of the page in all regions. 27 Figure 2-2. Representation of three possible scenarios for light incident upon a dielectric interface for the case when n1>n2. At left is the case for θi<θc. There will be a transmitted ray in this case. The figure in the center shows the case for θi=θc in which there will be a transmitted ray along the interface itself. Finally TIR is shown at right when θi>θc. There will be no transmission. This is the physical basis for waveguides....................................... 29 Figure 2-3. A basic asymmetrical planar slab waveguide. Note the orientation of the axes.30 Figure 2-4. A diagram illustrating the relationship between wavevector k, transverse wavevector κ, and longitudinal wavevector β. ...................................................................... 32 Figure 2-5. Graphical solution of the transcendental equation (2.24) for an asymmetrical waveguide. Each intersection of the red and blue graphs (aside from the vertical lines where the value of tangent is infinite) represents a possible mode........................................ 34 Figure 2-6. Graphs (a) – (d) show pictures of the cross section of the E field for modes 0–3 respectively. The value of β for each mode is also given on the graphs. The evanescent decay into the substrate and cladding is shown by the blue and green sections of the plots respectively. It can be observed that higher order modes penetrate further into the substrate and cladding........................................................................................................................... 36 Figure 2-7. Picture of the two dimensional slot confined waveguide proposed in [32]. It is essentially two planar slab waveguides placed in close proximity in a low index SiO2 background [32, 33]............................................................................................................... 38 Figure 2-8. Graph of the x component of the transverse E field of the fundamental TM mode across the regions of the slot confinement waveguide. The blue sections show the evanescent decay into the cladding regions, the black sections show the distribution in the waveguides and the red section shows the behavior within the slot. In this case the slot width is 144nm while the guiding layers are each 180nm..................................................... 40 Figure 2-9. Figures (a) – (d) show graphs of the E field distribution of the transverse fundamental TM mode for various slot widths and slot to guide layer thickness ratios. Notice how as the slot thickness increases and consequently lowers the overall effective index, a larger evanescent tail leaking into the cladding materials results. Thus the confinement factor begins to decrease as the total amount of the low index material present in the structure increases beyond a certain point. .................................................................. 45 viii
- 9. Figure 2-10. Graph showing the relationship between confinement factor and ratio of slot width to waveguide width. The width of the guiding layers is kept constant at 180nm while the slot width is chosen to fit the desired ratio. The CF increases up to a ratio of about 0.80 because the added low index material allows a larger amount of light in the desired region. The values saturate between 0.80 and about 1.1.................................................................... 45 Figure 2-11. (a) Simplified representation of absorption, (b) spontaneous emission, and (c) stimulated emission................................................................................................................ 48 Figure 2-12. A four level system. L3 and L1 are both unstable states so electrons rapidly decay to L2 and L0, usually in a non-radiative transition...................................................... 50 Figure 2-13. (a) Simple Fabry-Perot cavity with mirrors on either end. Also shown is the pumping (energy) source [9]. (b) Ring resonator structure which is the type of structure used for the device in this work............................................................................................. 52 Figure 2-14. A simple energy band diagram for a bulk material (left) and a quantum- confined nanocrystal (right)................................................................................................... 56 Figure 2-15. Diagram of the field-effect structure for achieving electrical injection [14]. The array of silicon nanocrystals exists in the light gray area directly beneath the gate....... 57 Figure 2-16. (a) Schematic energy level scheme of Er3+ showing the different types of transitions that may occur. The energy levels are very sharp for the free ions. Doped into a SiO2 lattice, the levels split into many discrete levels.(b) A more detailed view of the 4 I13/2 4 I15/2 transition. There are many discrete energy states in which electrons can exist within each energy band. The electrons will obey Boltzmann statistics. The graphs on the right show how the number of electrons (N(E)) changes with increasing energy (E). It can be observed that absorption takes place from the most populated states in the lower band to the least populated states in the upper band and vice versa for emission. This difference accounts for the slightly different wavelength and frequency of photons involved in these two processes. ........................................................................................................................ 59 Figure 2-17. Solid lines show the absorption spectrum of an Er-doped optical fiber for wavelengths between 450-850nm. This shows the principal absorption bands within the range of wavelengths. [36]..................................................................................................... 60 Figure 2-18. Photoluminescence (PL) emission spectrum for Er -doped (~5 x 1015 cm-2 ) silica film on a Si substrate when pumped with 488nm light. The wavelengths at which the two peak intensities occur are labeled [37]............................................................................ 61 Figure 2-19. Diagram showing pumping of the 1.55µm transition in Er via Förster energy transfer between nc-Si and Er. Non-radiative decay occurs from the I11/2 level to the I13/2 level. This is followed by radiative recombination from I13/2 I15/2 at 1.55µm..................... 62 Figure 3-1. Cross section of structure to be analyzed. T_C, T_Si and T_SiO2 denote the thicknesses of the cladding, silicon and silicon dioxide respectively. The multilayer region consists of layers between 15 and 30nm thick. The layers alternate between low index SiO2 ix
- 10. and higher index Si with Si forming both the top and bottom layers. It is flanked on top by a cladding of SiO2 and is deposited on a substrate of the same material. The dimensions of the cladding are set for all simulations while the slab height varies depending on layer thicknesses and the SiO2 to Si ratio. Note the orientation of the x, y, and z axes…..67 Figure 3-2. Complete cross sectional diagram of the structure. The active gain medium is actually composed of alternating layers of nc-Si and SiO2:Er. This particular diagram shows a horizontal slot ridge waveguide design. Electrons and holes are injected through the p and n contacts into the p and n type device layers. They then form excitons by tunneling into the nc-Si layers. Also visible is the ring resonator structure used to achieve frequency selective feedback and amplification............................................................... 68 Figure 3-3. (a) Slot confinement structure proposed by MIT [32]. The structure consists of upper and lower high index cladding layers with alternating thin low index slot layers with high index thicker guide layers in between. In this case, w is the width, tc, tH , and tl are the thickness of the cladding, low index slots and high index layers respectively. The refractive indices of the high and low index layers are nH and nl respectively while n0 is the refractive index of the surrounding air. It operates on the principle of confinement within low index slot layers but is much more difficult to fabricate owing to the extremely high precision required to etch the sidewalls. (b) Cross section profile showing the quasi TM modal distribution of the y component of the E field in the structure proposed in [32]. Also shown is the distribution of the y component of the E field along the structure (on the right side)................................................................................................................................... 69 Figure 3-4. Outline of the structure and picture of the major (y) and minor (x) components of the E field of the fundamental TM eigenmode looking along the z direction found using the BeamProp mode solver. This is a structure with N = 9 periods, T_Si = 20nm and a ratio of 1.00. The slab height is 0.380µm. The high concentration of the field in the low index layers is clearly visible...................................................................................................... 72 Figure 3-5. 3D picture of the power distribution of the TM mode in a structure with N = 9 periods, T_Si = 20nm and a ratio of 1.00. The high power confinement in the low index layers is readily apparent here. Also noted is the relatively large tail showing a significant amount of leakage of the mode into the low index substrate............................................ 74 Figure 3-6. 2D cross section at x=0 of the power distribution of the TM mode of the same structure as in Figure 3-5. The SiO2 layers are labeled along with the other regions of the structure. The long tail extending into the substrate is more discernable in this figure. .. 75 Figure 3-7. Cross section of the z component of power distribution of the TM mode across the layers in the structure shown in Figure 3-1. Again the high confinement in the low index layers is evident. Note also the oblong shape of the mode as it propagates down the structure. The evanescent tail in the substrate is visible here as in the other two figures. 76 Figure 3-8. Confinement factor versus SiO2/Si thickness ratio of the TE and TM modes for both the FDTD and TMM methods. The FDTD plots are shown as crosses and circles and x
- 11. are for slab heights between 0.370µm and 0.405µm. The TMM plots are in dashed and solid lines and are for a slab height of exactly 0.380µm................................................... 77 Figure 3-9. Confinement factor versus SiO2/Si thickness ratio of the TE and TM modes for the TMM method. It exhibits a similar trend to Figure 3-8. The ideal ratios fall within the same range as for the 0.380µm structure with the maximum being about 75% occurring at a ratio of about 1.1............................................................................................................... 79 Figure 3-10. Plot of overall gain G (in cm-1 ) versus gain to loss coefficient ratio for the TMM method for both the TE and TM modes. The zero gain value for TE modes is about 2.45 and about 0.25 for TM modes................................................................................... 83 Figure 3-11. Plot of overall gain in the optical cavity, G (in arbitrary units) versus gain to loss coefficient ratio for the FDTD method for both the TE and TM modes. The net gain value for the TE modes is about 2.5 and about 0.25 for TM modes................................. 84 xi
- 12. Foreword This thesis was undertaken as part of a multi university research initiative (MURI). The project involves all aspects in the design and development of a complete electrically- pumped silicon based laser for chip-scale nanophotonic systems. My research was primarily concerned with the design of a waveguide for the optical cavity that would provide high mode confinement, minimize losses and realize lower threshold current densities. I worked closely with a fellow member of my research group, Yjing Fu. He helped me with many of the design aspects with which I was unfamiliar. He also developed the TMM algorithm and provided the data for comparison in this work. There is currently an unpublished article based in part on this research. More complete details regarding this article can be found in the Bibliography. 1
- 13. Chapter 1 Introduction 1.1 Motivation The communications industry has in recent years seen a rapid rise in the demand for high speed, high bandwidth and low loss connections at all levels of system integration. This increased demand necessitates better performance, particularly at the chip-to-chip and datacom levels which almost exclusively use electronic circuitry. Many experts have predicted that Moore’s law will break down as device dimensions continue to shrink. Resistor-capacitor coupling, noise, and other parasitic effects contribute to the bandwidth bottleneck for interconnects. These effects are further agitated on smaller scales. As data rates and performance demands continue to increase, this poses a serious problem. Generating, transmitting, receiving, and processing data in the form of optical signals provides a viable solution to this problem. The advantages of optical systems are abundant and their continued development and implementation is held back by only a few remaining obstacles – chief among them being cost. The ability to leverage the extensive knowledge and manufacturing capabilities of a well developed and well established microelectronics industry centered on silicon provides the most plausible path to success in this regard. The major objective remains the development of systems based primarily on optical signals and fabricated in silicon materials in such a way that a gradual transition or partitioning between the two technologies can occur. 2
- 14. 3 It is appropriate to begin with an overview of the major advantages afforded by optical solutions over their respective electrical counterparts. Two of the most obvious improvements in an optical system are the reduction or complete elimination of electromagnetic interference (EMI) and noise as well as immunity from short circuits. These are inherent problems that have plagued electrical systems for years. Examples of EMI include cross talk between adjacent cables in a bundle, stray wires or electromagnetic radiation in the atmosphere. It is impossible to completely remove either of these from such systems. Metallic shielding is one approach but it adds extra weight, cost and parasitic capacitance which limits the bandwidth – all significant enough drawbacks to warrant consideration as to whether or not it is necessary in the first place [1]. Another minor yet critical benefit of optical fibers is that they are much more difficult to tap and therefore more secure. The most significant advantages offered by optical solutions are: 1) lower loss transmission at higher frequencies, 2) larger bandwidth (multiplexing capabilities), and 3) smaller size, weight and overall power consumption (at smaller interconnection distances). As it has been previously noted, electrical interconnects are currently the better choice for systems at the chip level but at higher data rates and larger distances optical links are far superior. The hope is that this will be able to scale to the chip level as optical circuits based in Si become more of a reality. There are a few other major factors hindering a full transition. These are the need for: 1) wavelength independence over the entire communication spectrum, 2) polarization independence and 3) higher optical power levels – all of which could be addressed utilizing nonlinear effects in Si [1]. In order for this to take place, many of the current problems confronting optoelectronic integrated circuits (OIC) will require a variety of innovative solutions to meet the ever increasing data and speed requirements of the rapidly evolving global communications network. As the lowest level of integration, chip-to-chip optical circuitry will realize the greatest benefit and also presents the greatest challenge due to the extreme conditions inherent at smaller dimensional sizes such as high temperatures which create the problems with resistor-capacitor coupling, bandwidth limitations and durability. Optical interconnects become increasingly competitive at higher levels and longer transmission distances where their electrical counterparts experience greater losses due to parasitic effects. For data rates
- 15. 4 in excess of 1Gb/s and distances of greater than 100m, the use of optical interconnects is currently standard practice with the slightly higher manufacturing cost easily justified by the large increase in performance. At more intermediate distances of 1 – 100m there are solutions in both forms. A diagram showing the applications of different interconnect technologies is presented in Figure 1-1 [2]. Figure 1-2 [2] is a timeline of the information carrying capacity of various electrical and optical links. It is clear that optical systems afford much higher data rates than even the best electrical systems. Figure 1-1. Diagram showing various applications of different interconnect technologies. Optical links are currently only used at longer distances (>100m) [2].
- 16. 5 Figure 1-2. Timeline of information carrying capacity for links in various electrical and optical links. Those in optical systems have data rates roughly 2 orders of magnitude higher than those in electrical systems [2]. The need for higher data rates and bandwidths places increasing strain on current electrical interconnects. The degradation of performance, particularly at the chip-to-chip level, due to the trend towards smaller dimensions and higher temperatures has proven to be a hindrance to the continued progress of the microelectronics. A diagram showing the relationship between processor and front side bus (FSB) clock speeds for different models within the past 15 years is shown in Figure 1-3 [2]. Recently, the potential of optical integrated circuits (OICs) has been realized due to an increased need for higher performance, especially on smaller scales. Despite their aforementioned drawbacks however, electrical ICs remain the most cost effective and reliable solution particularly at the chip-to-chip level largely due to highly developed fabrication methods and technologies along with the readily accessible infrastructure that has been firmly established over the years.
- 17. 6 Figure 1-3. Relationship between processor and front side bus (FSB) clock speeds for various models over the past 15 years. Note the significant improvement in processor speeds over FSB speeds in recent years [2]. Until recently, most optical devices have been made from what are known as exotic materials such as gallium arsenide (GaAs) or indium phosphide (InP) owing to their excellent optical properties. These materials are both expensive and complicated to process, however. Before integrated optics became a viable option, most devices were discrete components which were hand-assembled and required extreme attention to detail as far as aligning them to fiber connectors to maximize light coupling. Thus there is often a significant cost disadvantage when working with optical systems and devices. Complete integration of optical components fabricated in silicon would eliminate the need for such precision by automation of this process with a subsequent reduction in cost. The hope is that specific components can be fabricated to perform all the necessary functions in the transmission and reception of optical data with the end result being a completely optical- based system on a single chip. Each of the necessary functions in this system can be thought of as an optical analog to a similar electrical function accomplished with transistors. These functions include: 1) light generation 2) light guiding
- 18. 7 3) light detection 4) signal multiplexing 5) high speed (≥ 1 GHz) modulation 6) low cost assembly 7) high coupling efficiency to optical fibers. A block diagram for a typical transmitter-receiver system is shown in Figure 1-4 [3, 4]. Figure 1-4. Diagram showing individual components of an optical transmitter-receiver system. The transmitter is the top diagram and the receiver is the bottom [4].
- 19. 8 Si has been shown to be adept at all of these functions except for light generation – the major barrier that must be conquered. Recent results by Intel have shown devices capable of high speed modulation [5]. Si based components are great for light guiding solutions because Si is transparent (its bandwidth is larger than the energy of the photons) at the infrared wavelengths used in communication systems, namely 1.55µm. However, Si is notorious for high free carrier absorption losses and Auger processes – problems which, along with poor light emission, have plagued Si-based systems for years. This is part of the tradeoff when utilizing a material with such excellent electrical properties. In order for photonic components to take hold at the lowest level of integration, performance demands must outweigh cost drawbacks as is the case with long haul telecommunications links. If a transition to exclusively optical-based communication systems is to take place, it will most likely take place through slow developments and advances in fabrication techniques and device technologies as performance demands increase and costs decrease. Demand for superior performance will continue to increase as it has in the past and will continue to do so at a faster rate in the future. Global communication networks are expanding rapidly every day and the need for large quantities of information in short periods of time has never been greater. The burden thus falls on innovation of new fabrication techniques and design technologies to help precipitate the transition, particularly in terms of the information processing level (as opposed to information transmission which already employs optical solutions). With the microelectronics industry firmly centered on silicon and baseline CMOS fabrication processes, these innovations will almost certainly involve Si-based components. Although hybrid integration involving III- V-based semiconductors or other types of materials is possible, they are highly unlikely to firmly take hold since Si is by far more plentiful, inexpensive and well understood [1, 2]. Evidence of how much cheaper fabrication in Si is than in III-V materials is shown in Figure 1-5 [2]. At best they will provide alternative or customized solutions. A hypothetical timeline showing the transition between electrical and optical-based systems for communication over various distances is visualized in Figure 1-6 [2]. Some experts in the area of silicon photonics instead foresee a merging of the two technologies and believe that future systems will rely on parallel solutions using smart electronic-photonic partitioning.
- 20. 9 Figure 1-5. Wafer cost per unit area for various materials versus wafer size. Notice that InP is currently only available in one wafer size and is prohibitively expensive compared to a Si wafer of the same size [2]. Figure 1-6. Hypothetical timeline for the transition between electrical and optical based communication systems for various distances [2].
- 21. 10 The inability of Si to behave as an efficient light emitter is due to the indirect nature of its band structure. The energy band diagrams for both Si and GaAs are shown in Figure 1-7 [2]. Notice that GaAs has a direct bandgap which helps explain why it is a very efficient light emitter, while Si does not. Numerous attempts have been made over the years to circumvent this problem, most with only varying or irreproducible success. Bandgap engineering (controlling the band structure with alloys and Brillouin zone folding in superlattices are two notable methods) and enhanced quantum confinement via quantum wells, wires and dots are among the more successful and notable approaches in recent years. See Chapter 1 in [2] and references therein for a complete overview of these techniques. Figure 1-7. Energy band diagrams for silicon (above) and gallium arsenide (below). Notice that Si has an indirect bandgap while GaAs has a direct bandgap. This explains why GaAs is a much better light emitter than Si [2].
- 22. 11 It should be mentioned that while getting Si to emit light as efficiently as a direct gap semiconductor is somewhat difficult and may never be achieved, light amplification and lasing have in fact been achieved in Si, mostly as a result of the phenomenon of stimulated Raman scattering (SRS). This phenomenon arises from the Raman Effect whereby photons from a pump beam transfer their energy to longer wavelength photons through interaction with vibrating atoms within a material. Spontaneous Raman scattering and emission at 1540nm in Si waveguides was first observed and reported in 2002 by a group at UCLA [6]. This was followed by a report of SRS and amplification in a 2003 publication [7]. Finally the first successful demonstration of lasing at 1675nm in a Si waveguide [8] was given in 2004. One of the major drawbacks of the initial Raman laser was that at the high pumping powers required for Raman amplification to achieve lasing, the two photon absorption problem becomes more significant. Two photon absorption (TPA) occurs when two incoming photons come together at the same exact spot in the crystal lattice and provide enough energy to raise an electron from the valence band to the conduction band. This is not possible for single photons at the infrared wavelengths normally used in silicon photonic devices because they do not provide enough energy for the electrons to cross the bandgap. Once in the conduction bands, the electrons then become free carriers and contribute to the problem of free carrier absorption (FCA) at higher power levels by absorbing light propagating through the waveguide and weakening the overall signal strength in the device. At high pump intensities, TPA is enough to either cancel out the Raman gain or prevent SRS altogether. Because of this, only pulsed-pump operation was possible with the UCLA design. In 2006 a group of Intel researchers got around this problem by surrounding the silicon waveguide with p and n type silicon to effectively form a PIN diode [9-12]. When the diode is reverse biased, the free carriers (electrons) generated by TPA are ‘swept’ away by the electric field. This reduces the free carrier absorption losses enough to allow continuous operation. Diagrams are shown in Figure 1-8 and Figure 1-9 [11, 12].
- 23. 12 Figure 1-8. Cross section of the device designed by the Intel researchers. It is a ridge Si waveguide surrounded by SiO2. The silicon on insulator (SOI) technology used allows for a large mode confinement within the Si waveguide so that high enough Raman amplification is reached. The p and n type Si on either side help to draw electrons generated by two photon absorption out of the device and reduce parasitic losses such as free carrier absorption (FCA) [11]. Figure 1-9. Scanning electron microscope (SEM) image of a p-i-n diode waveguide used for Raman amplification and lasing experiments [12].
- 24. 13 The important thing to realize about these results however is that SRS is a non- linear process that involves the emission or absorption of a phonon. This means that the overall efficiency for this type of lasing will be much lower than that which relies primarily on first order stimulated emission processes. Also, it requires high powered pumping from an additional source of laser light (usually a mode-locked fiber laser) which complicates the fabrication process and means that the device will not be exclusively based in Si, both of which will increase cost. It also exacerbates problems like two photon absorption and free carrier absorption as noted. Devices based on SRS are still being actively studied however, and in all likelihood will play a role in the future of silicon photonics. There is one other recent breakthrough that warrants mention. In 2006 a collaborative team from both Intel and the University of California at Santa Barbara designed and fabricated an electrically pumped hybrid Si laser. Being the first electrically pumped Si laser, this was a major development. Instead of relying on SRS which necessitates optical pumping, it achieves lasing by electrical pumping – injecting electrons and holes into InP which is a direct bandgap material and an efficient light emitter. Light generated in the InP is coupled to the Si waveguide layer which sits underneath. The device is hybrid because it is actually InP that directly emits and amplifies the photons of light [3]. One of the main advantages of this technique is that it eliminates the need to align external light sources with the Si chip by either pre-fabricated lasers on chip or optical fibers off chip, both of which would not be practical for low-cost, high-volume production. The novelty of the design is based on a new way of bonding together the InP and Si substrates. The main problem with growing III-V materials like InP onto Si is that they have a large lattice mismatch and dislocations develop which negatively impact device performance and reliability. The new bonding technique uses a low temperature plasma enhanced oxidation process that acts as a ‘glass glue’ between the two materials. It allows InP substrates to be bonded directly to pre-patterned Si photonic chips without the need for alignment. This technique may prove to be very successful as a solution for large scale optical integration onto a Si platform by simplifying the manufacturing process and reducing cost. It is a hybrid approach however, so it remains to be seen whether or not it will be able to realize the low cost – high volume potential that a monolithic approach
- 25. 14 could simply because both InP and other III-V materials are more expensive and not as abundant or as well understood as is Si [1]. The inability to achieve efficient light emission in Si based components and devices – aside from the fact that a transition to technology based completely on hybrid materials would not be practical in terms of cost – remains the major impeding factor to the successful implementation of low cost-high volume optical integration at all system levels. It will be seen that the use of nanocrystalline silicon and its novel energy transfer properties with erbium ions in close proximity poses a very viable solution to this problem. 1.2 MURI Silicon Laser Project A Multi University Research Initiative (MURI) is a large scientific project composed of a collection of university research teams and possibly partners from national or industrial labs. For this particular MURI project, the primary task at hand is the design and development of practical and useful Si-based photonic components – particularly a laser system. The institutions involved are listed in alphabetical order as follows: Boston University California Institute of Technology Cornell University Lehigh University Massachusetts Institute of Technology Stanford University University of Delaware University of Rochester Also taking part are researchers from the McMaster University, University of Toronto and the Army, Navy, and Air Force research laboratories. The technical approach as stated in the executive summary reads: “Our program will deliver a nanophotonic, silicon-based laser for monolithic, CMOS- compatible integration with electronic and photonic chip-level circuits. The silicon- based laser must be electrically pumped; it must have a high wallplug efficiency and low heat dissipation; it must have a long operating lifetime; and it must emit at a carrier frequency and output power that meet the expectation of a > 10Gbit/s optical
- 26. 15 communications link, operating at a Bit-Error Rate tolerance < 10-12 . Two principal challenges frame the science and engineering research context of this program: high lasing efficiency cavities and high gain efficiency media [13]” The program is organized into three major tasks, each with supporting subtasks. The first task is an extrinsic gain laser with the subtasks including a slot waveguide structure with Er-doped oxide and a traditional index guided structure with Er-doped silicon nitride and quantum dot doped oxides. The second task is an intrinsic gain laser with the subtasks of a conventional edge emitter structure with bulk and quantum-confined germanium (Ge), and a conventional edge emitter structure with strained and alloyed Ge. Finally, the third task is the design of the application drivers. The subtasks are a CMOS compatible Er- doped optical micro-amplifier for 1550 nm light, a CMOS-compatible optical signal processor link and a CMOS-compatible, multi-wavelength optical power supply. In particular this thesis deals with the design and optimization of a structure falling under the first subtask of the first major task. A table summarizing the four laser design approaches for this initiative with their compositional materials, device parameters, projected performance and risk assessment is presented in Table 1-1 [13]:
- 27. 16 Table 1-1. The four design approaches for the MURI initiative with their specific parameter values. 1.3 Project Overview – Objectives, Approaches, Tools Used This project is concerned with the design, modeling and optimization of a slot- confined, multilayer waveguide structure to be used as the optical cavity for a silicon-based laser system. It consists of alternating nanometer-thin layers of amorphous Si (a-Si) and erbium (Er)-doped silicon dioxide (SiO2:Er) with a cap layer and substrate both composed of SiO2. A carefully chosen thermal budget during fabrication will result in the formation of silicon nanocrystals in the Si layers producing nanocrystalline Si (nc-Si). An outline of the actual structure analyzed can be seen in Figure 1- 10. A more complete picture of the overall system of which it is to be a part is shown in Figure 1-11[13]. There is one minor clarification with this figure however: instead of a random array of nc-Si and Er3+ ions in a SiO2 matrix, the gain medium (the central layer in the figure) consists of distinct multiple layers of SiO2:Er and nc-Si [13]. This is also shown in Figure 1-11.
- 28. 17 1µm Cap Layer50nm T_C(SiO2) Figure 1- 10. Outline of the actual structure (s) to be studied in this thesis. (a) Substrate (SiO2) nc-Si SiO2:Er T_SiO T_Si Multilayer Region y Slab Height ~370 – 400nm z x SiO2:Er nc-Si
- 29. 18 (b) SiO2:Er (low index) nc-Si (high index) Figure 1-11. (a) Complete diagram of the device. The upper left shows a top view of the ring resonator which is used to accomplish frequency selective feedback and amplification as well as output coupling. The bottom shows a cross section of the cavity. The upper right shows a zoomed in picture of the gain medium and the surrounding layers. In this work, the gain medium is actually composed of alternating layers of nc-Si and SiO2:Er as depicted in both Figure 1-10 and in (b). This particular diagram shows the horizontal slot ridge design. Electrons and holes are injected by tunneling through the p and n contacts into the nc-Si layers where they will form excitons. Also visible is the ring resonator structure used to achieve frequency selective feedback and amplification as well as output coupling [13]. The high index contrast at each of the Si/ SiO2 interfaces allows for a large concentration of light within the low index (SiO2) layers. The idea is to use field effect injection to bring electrons and holes into the surrounding p and n type device layers and then rely on tunneling to allow them to travel into the nc-Si layers and form excitonic pairs [14]. The nc-Si will allow efficient excitation and energy transfer to the Er atoms through a dipole-dipole energy coupling interaction. This dipole-dipole energy coupling phenomenon is known as Förster energy transfer. Essentially the nanocrystals provide a convenient means of exciting the Er. This gives rise to light emission from Er at 1550nm – the operating wavelength for most currently active optical communications links in the world. Erbium is chosen for this reason as well as the aforementioned energy coupling capabilities with silicon nanocrystals, and because it is generally well understood based on its use in erbium doped fiber amplifiers (EDFAs) since the advent of the fiber optic section of the telecom industry. The ability to confine a high percentage of light within the SiO2 layers containing Er ions enables the device to more easily generate the gain necessary to achieve
- 30. 19 lasing. In this way, the electronic properties of nanocrystalline silicon and the light emitting capabilities of Er are each leveraged in such a way as to produce laser light emission in a silicon based structure. Due to the complex nature of the proposed structure, the use of numerical simulation software becomes necessary in obtaining an accurate picture of the propagating modes. These simulations are done using the RSoft Design Group, Inc. Photonics Suite and in particular the BeamProp and FullWave tools. Both BeamProp and FullWave are software packages designed to compute and model the propagation of light waves in arbitrary geometries. Both are inherently based on finite difference computational methods. BeamProp uses the Beam Propagation Method while FullWave uses another well-known technique called the finite-difference time-domain (FDTD) method. Both techniques are considered very robust and efficient at giving accurate pictures of how light propagates in even the most complicated photonic devices. A full explanation with mathematical details is not within the scope of this document but interested readers are referred to the numerous publications on such methods [15-28]. An understanding of these numerical methods is not critical to the full comprehension of the concepts presented but references are given for the sake of completeness. It should be noted that the BeamProp package was used only for its mode-solving capabilities while FullWave was used for actual simulations. Matlab was used for data analysis and visualization purposes. Simulations of the proposed structure are performed using different periods of repeating Si/SiO2 layers and different thickness ratios between the Si and SiO2 layers. The raw data from these simulations is then read into Matlab where it is used to calculate the confinement factor (CF) within the SiO2 layers for each structure. By examining simpler 2D slot confined waveguide structures and something known as the effective index method, it is possible to predict that the thickness ratio between the Si and SiO2 layers is the critical parameter affecting the CF. This hypothesis is confirmed by analysis of simulation results. Based on these results, a few optimum structures are chosen and a gain/loss performance study is undertaken. This entails simulations of each structure, with loss inserted into the Si layers to model the free carrier absorption effects in silicon, and gain within the SiO2 layers to model the light emission from erbium. The ratio of the gain in the SiO2 to loss in the Si is then varied over a range of values beginning with zero. For each
- 31. 20 simulation, light waves are propagated along the entire length of the structure (~10µm). The maximum power at a point just shy of the end (~9.5µm) is recorded for simulations with no gain or loss (P0) and then compared to those with both loss and gain in varying ratios (Pmon). The net gain throughout the propagation (G) can be calculated from these values. Repeating these steps for both transverse magnetic (TM) and transverse electric (TE) polarizations and then plotting a graph of net gain (G) versus the gain/loss coefficient ratio reveals a linear relationship. The value of the ratio for which G = 0 is the cutoff ratio of gain in SiO2 to loss in Si for which net gain is observed. This in turn can be used to realize a reasonable approximation for the minimum injection current density in the system – a useful parameter in terms of minimizing the overall power consumption of the device and improving its overall efficiency.
- 32. Chapter 2 Theory Any theoretical description of propagating light waves begins with a discussion of Maxwell’s equations and the solutions to the electromagnetic wave equation in an isotropic, linear and source free medium. This is followed by explanations of the boundary conditions at a dielectric interface and total internal reflection (TIR), and a discussion of wave propagation in guiding media. Two simplified 2D waveguide examples are then presented. First is a three layer slab waveguide with graphical depictions of the solutions for the eigenmodes of propagation. Second is a slightly more complicated slotted waveguide, also with graphical results. The idea of optical confinement within the guiding layers is established alongside these examples. The graphs are intended to give an indication as to both the advantages and limitations inherent in the slotted structure. A very well-written and thorough introduction to this area of electromagnetism can be found in chapter 9 of [29]. 2.1 Maxwell’s Equations The packaging of four previously derived relationships (Gauss’ Law, Gauss’ Law for magnetism, Faraday’s Law and Ampere’s Law) into a compact and consistent form and the slight correction provided for one of them (Ampere’s Law) by James Maxwell is without question among the crowning scientific achievements of the 19th century. Maxwell’s equations are presented below in differential form. The integral 21
- 33. 22 forms can be found in almost any text on electromagnetism and can be easily derived from the differential forms using either Gauss’ divergence theorem or Stokes’ theorem [30]. t∂ ∂− =×∇ B E (2.01a) J D H + ∂ ∂ =×∇ t (2.01b) 0=⋅∇ B (2.01c) ρ=⋅∇ D (2.01d) In order from top to bottom they are: (a) Faraday’s Law, (b) Ampere’s Law with Maxwell’s correction term, (c) Gauss’ Law for magnetism, and (d) Gauss’ Law. A table summarizing the various quantities and their units in MKS form is given in Table 2-1: Quantity Description Units (MKS) E Electric field amplitude Volts/meter (V/m) H Magnetic field amplitude Amps/meter (A/m) D Electric flux density Coulombs/meter2 (C/m2 ) B Magnetic flux density Webers/meter2 (W/m2 ) J Current density Amps/meter2 (A/m2 ) ρ Charge density Coulombs/meter3 (C/m3 ) Q Charge Coulombs (C) Table 2-1. Various quantities used in electromagnetics and their units expressed in MKS units [30]. The constitutive relationships between the flux densities (D and B) and the field amplitudes (E and H) are determined by the properties of the medium through which the waves are propagating. In simple linear and isotropic media1 , these are written as, HB µ= and ED ε= (2.02) 1 This is only true for monochromatic fields.
- 34. 23 where ε is the electric permittivity of the medium in Farads/meter and µ is the magnetic permeability of the medium in Henrys/meter. For reference, the accepted values for permittivity and permeability in vacuum are given as and (2.03)Farads/m10854.8 12 0 − ×=ε Henrys/m104 7 0 − ×= πµ For the sake of simplicity and relevance, this section deals only with linear and isotropic media – that is, materials in which the index of refraction does not depend on the strength of the fields or on the direction of propagation within the material. Maxwell’s equations in a linear, isotropic and source free medium (ρ = 0, J = 0) are given below. t∂ ∂− =×∇ B E (2.04a) t∂ ∂ =×∇ D H (2.04b) 0=⋅∇ B (2.04c) 0=⋅∇ D (2.04d) Note that equations (2.04a) and (2.04c) are unchanged. Also note the nice symmetry that results in this case. These equations are now much easier to work with and put into a more useful and insightful form. 2.2 The Wave Equation Using equations (2.04) the wave equation can be derived in a fairly straightforward manner. The approach involves decoupling the two curl equations for B and D by first taking the curl of both sides of (2.04a). Substituting µH for B and further realizing that the time derivative of H can come outside of the curl operation yields )()( HE ×∇ ∂ ∂ −=×∇×∇ t µ , (2.05) assuming that µ and ε are both time- and position-invariant. Applying a vector identity to the left hand side of this result and using (2.04b) and (2.02) gives 2 2 2 )()( t∂ ∂ −=∇−⋅∇∇=×∇×∇ E EEE µε . (2.06)
- 35. 24 By further analysis the first term on the left hand side of the middle equation can be shown to be effectively negligible [30]. This produces the standard form for the wave equation: 02 = ∂ ∂ −∇ t E E 2 µε . (2.07) This same equation can be derived for H by starting with equation (2.04b) instead of (2.04a). It is also important to note that the wave equation reproduced here is actually a vector equation which involves the vector Laplacian. It can be easily simplified to a scalar equation for a rectangular coordinate system however. In a non-rectangular coordinate system it is usually more difficult to simplify the vector equation because of the need to break it into orthogonal components. Every structure presented in this work lends itself readily to a rectangular coordinate system, so (2.07) is sufficient. A more detailed derivation of the wave equation can be found in reference [30]. Solving the vector wave equation involves first choosing an appropriate coordinate system, as mentioned above. For a simple rectangular system the scalar wave equation can easily be found for each vector component and then solved by separation of variables to get )exp(),( tjj(t)(t) 0 ωψφψψ −⋅== rkrr , (2.08) where ψ0 is the amplitude, ω is the temporal frequency and k is the wavevector in radians/second. The magnitude of k is written as µεω=k . (2.09) This is known as the dispersion relationship for a light wave. The wavevector is defined as k = 2π/λ, where λ is the wavelength of the light. It can be thought of as a ‘spatial’ frequency. The solutions to the scalar wave equation in a rectangular coordinate system represent a sinusoidal wave in both space and time. Since this same equation can be solved for H just as easily, it confirms the origin of the term electromagnetic radiation. Maxwell’s equations can be decoupled to produce two distinct forms of the wave equation – one each describing the magnetic and electric fields, each of which oscillates
- 36. 25 through space and time. It can further be proven that E and H are orthogonal to each other2 [29]. Through examination of the wave equation, it can be shown that the velocity of the wave in a medium with unique values for µ and ε is µε 1 =v , (2.10) where v is the phase velocity. Using the values of permittivity and permeability for vacuum which are given in (2.03), the value for the speed of light (2.99 x 108 m/s) can be calculated. This leads to a definition for the index of refraction of a material as 000 ε ε εµ µε ==n , (2.11) when µ = µ0. Thus the index of refraction is exactly one in vacuum and is very close to one in air. This quantity determines the factor by which the speed of light is decreased when traveling through a particular medium. It will be seen in the next section that it also determines the manner in which light bends or refracts when passing from one medium into another. 2.3 Boundary Conditions at Dielectric Interfaces Before taking on a discussion of light propagation in a guiding medium such as a waveguide, it is essential that the behavior of light waves at a dielectric interface be first understood. A dielectric interface occurs at the boundary between materials with two different indices of refraction (n1 ≠ n2). The general rule for the wave solutions in the two regions is that they must be continuous across the boundary in some way. Specific boundary conditions, presented below, explain the exact nature of this continuity for each type of field or flux quantity. When electromagnetic radiation is incident upon an interface of any type, it will undergo one of the following phenomena or some combination thereof: Transmission – no interaction of light with the new material 2 This is only true when the fields are in a plane wave and does not apply in general.
- 37. 26 Absorption – energy lost as transmitted light interacts with impurities in the new material Reflection – light is re-directed back into the original material Refraction – light passes into the new material but with an altered path Technically there is no such thing as pure transmission. There is some degree of absorption in almost any type of material and at different wavelengths of light. Absorption becomes important when modeling the loss mechanisms within a waveguide and is covered in a later section of this chapter. The two remaining phenomena will be of primary concern in discussing the behavior of light at a dielectric interface. The diagram in Figure 2-1 shows a simple case of a plane wave of light incident on a dielectric interface with electric field (E), magnetic field (H), and wavevector (k) all labeled. The subscripts ‘i’, ‘r’, and ‘t’ stand for incident, reflected and transmitted portions of each field quantity respectively. The direction of the wavevector denotes the direction of propagation for each ray and can be determined through calculation of the cross product between E and H or more simply by the right hand rule [29]. In this diagram, the E field points directly out of the page for all rays and the H field is oriented perpendicular to this, as required by Maxwell’s equations. This is known as the transverse electric (TE) polarization. Polarization is discussed in the next section. Using the integral form of Maxwell’s Equations, the aforementioned boundary conditions can be derived [29]. They are listed below for a source-free medium. .0)(ˆ ,0)(ˆ ,0)(ˆ ,0)(ˆ 12 12 12 12 =−⋅ =−⋅ =−× =−× DD BB HH EE s s s s (2.12) In this case, s is a unit vector that is normal to the surface. A simple way of summarizing these rules is: “D-B normal, E-H tangential”, meaning that the normal components of D and B are continuous across the boundary, as are the tangential components of E and H. ˆ Using these continuity rules for the fields, three additional relationships can be derived [29]. The first of these simply states that the incident, reflected, and transmitted wave vectors form a ‘plane of incidence’ which also includes the normal to the surface
- 38. 27 (in this case, the z axis). The other two are also known as the law of reflection and the law of refraction (Snell’s Law). These are given below and also visualized in Figure 2-1. k n n E k E θ θ Ht θ z k E H y Figure 2-1. A monochromatic plane wave incident upon a dielectric interface. The subscripts ‘i’, ‘r’, and ‘t’ denote the incident, reflected and transmitted portions of the wave. The diagram shows the case of TE polarization meaning that the E field is directed out of the page in all regions. ri θθ = , (2.13a) ti nn θθ sinsin 21 = . (2.13b) These relationships, along with the original boundary conditions can further be extended to derive general equations relating the reflected and transmitted field amplitudes to the incident field amplitude for both E and H as a function of θi [30]. These equations are not given since they are not as pertinent to the nature of this work, however they can be found in chapter 1 of [30]. The physical implications of these four relatively simple boundary conditions on the electric and magnetic fields and flux densities are astonishing – they essentially lead to the two most fundamental laws of optics. They also have other important consequences as the remainder of this chapter will address.
- 39. 28 2.4 Field Polarization Another key point to hit on is the orientation of either the electric or magnetic field with respect to the interface. If the electric field is perpendicular to the interface the polarization is transverse electric (TE). If the magnetic field is perpendicular to the interface the polarization is transverse magnetic (TM). The diagram used in the last section showed a case of TE polarization. The four boundary conditions, the law of reflection and Snell’s Law all remain the same regardless of the polarization, however the behavior of the equations relating reflected and transmitted fields to the incident ones is different [30]. For more complicated 3D geometries – specifically rectangular and circular waveguides – these characterizations are meaningless so instead the terms quasi-TE and quasi-TM are used to describe the polarization, depending on whether the particular mode is primarily TE or TM respectively. There are some additional quirks in the description of the polarization with regard to the simulation software and the particular structure being analyzed. These will be referenced in Chapter 3. 2.5 Poynting Vector An important quantity in determining the power incident on a surface from an electromagnetic wave is called the Poynting vector. It is defined as HES ×= , (2.14) where S has units of Watts/meter2 (intensity). To get the total incident power it is necessary to integrate S over the entire surface of interest. Since S in general points in the direction of propagation, which in this case is perpendicular to both E and H, it is often of interest to obtain the time average intensity along this direction (in this case, z). This is shown below for propagation along the z direction. ]ˆRe[ 2 1 * zSz ⋅×= HE . (2.15) The factor of one half and the reason for the Re and complex conjugate operators arises from taking the time average intensity for a sinusoidal waveform. This is derived
- 40. 29 fully in [31]. Again, the important point is that this quantity represents the power per unit cross sectional area or instantaneous intensity for a wave and can be integrated over the cross section of an interface to determine the total power flow [30]. 2.6 Total Internal Reflection (TIR) The critical process in any guided wave approach is that of total internal reflection (TIR). A pictographic representation of TIR is seen in Figure 2-2 . Examination of Snell’s Law and Figure 2-2 shows that when n1 > n2 and θi (referred to as θ1 here) is greater than a certain critical angle θc, the incident ray is completely reflected back into the first medium. The critical angle is a function of both n1 and n2 and can be written as ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = − 1 21 sin n n cθ (2.16) using Snell’s Law (2.13b) and substituting 90° for θ2. Based on this it can easily be seen how waveguiding becomes a possibility by surrounding a high index medium with layers of a lower index medium. Light must be coupled into the waveguide in such a way that its angle of incidence at the boundary with the lower index layer is always greater than the critical angle. This forms the basis for waveguides which will be presented in the next section. Figure 2-2. Representation of three possible scenarios for light incident upon a dielectric interface for the case when n1>n2. At left is the case for θi<θc. There will be a transmitted ray in this case. The figure in the center shows the case for θi=θc in which there will be a transmitted ray along the interface itself. Finally TIR is shown at right when θi>θc. There will be no transmission. This is the physical basis for waveguides. θ nn n n θ θ n θ n θ
- 41. 30 2.7 Waveguides 2.7.1 Planar Slab Waveguide Often the simplest way to introduce the concept of a waveguide is with the example of a planar slab waveguide. A diagram of this structure is seen in Figure 2-3. It can be seen from this that the structure is infinite in the y direction and therefore not very realistic. The axis of propagation here, as in most cases, is taken to be the z direction. It will be seen that due to the confined nature of a waveguide as well as the physical principles of TIR and the boundary conditions on the electric and magnetic fields, that light propagation in such structures takes the form of discrete eigenmodes. The value and number of these modes depends on both the physical parameters and geometry of the waveguide as well as the operating wavelength. For optical communications, the wavelength is usually set at around 1550nm – the transparency window for most optical fibers – and the remaining quantities are left to be specified by the designer. For this problem however, all values will be specified and the primary objective will be to solve for the modes of propagation. It is also important to note that there are two mutually exclusive sets of modes comprised by the TE and TM polarizations. The TE polarization will be used in this particular analysis. The distribution of values for the index of refraction of each layer in the waveguide must be known in most cases. The waveguide in Figure 2-3 has nf =1.50, ns=1.45 and nc=1.40 where nf is the guiding layer, ns is the substrate layer and nc is the cap layer. This type of structure is asymmetric since nf > ns >nc. In a symmetric structure the substrate and cap layers have equal indices of refraction. The width of the guiding layer h is 5µm and the operating wavelength λ is 1µm. The x=0 position is arbitrarily chosen as the top interface since it is an asymmetric waveguide and this point cannot be in the direct center. nc = 1.40 x λ = 1 µm h = 5 µmnf = 1.50 ns = 1.45 z Figure 2-3. A basic asymmetrical planar slab waveguide. Note the orientation of the axes.
- 42. 31 An important first step in this kind of problem is the choice of a proper coordinate system. Because the individual components of each field will remain independent of each other (no component coupling) throughout the structure during propagation, the best coordinate system to use is a rectangular (Cartesian) one. To completely specify the eigenmodes, the wave equation must be solved in each dielectric layer and each solution must be connected at the layer boundaries using the established boundary conditions. Using (2.07) and (2.09) the wave equation can be rewritten as , (2.17)0)( 2 0 2 =+∇ yiy EnkE where ni = nf ,ns, or nc depending on the layer. The electric field points strictly in the y direction (out of the page) because the polarization is TE as mentioned previously (refer to Figure 2-2). The y dependence of the electric field can be ignored since each layer is infinite in that direction. This means Ey will only be a function of x and z. Knowledge of differential equations enables a simple trial solution for Ey: )exp()(),( zjxEzxE yy β+= . (2.18) In this equation β is the propagation coefficient in the z direction (aka the longitudinal portion of the wave vector). The trial solution is plugged into (2.17) to yield 0))(( 22 02 2 =−+ ∂ ∂ yi y Enk x E β . (2.19) This equation completely describes the behavior of the E field along the x direction (the transverse portion of the wave). The solution has the general form )exp()( 0 xjExEy α±= , (2.20) where E0 is the field amplitude at x=0 and 22 0 )( βα −= ink . The longitudinal part has been left out since it is simply sinusoidal in z for all regions. There are two possible scenarios here. When β < k0ni , the term under the radical is positive so the solution takes an oscillatory form: )exp()( 0 xjExEy κ= (2.21)
- 43. 32 where 22 0 )( βκα −±== ink and is known as the transverse wavevector. When β > k0ni the term under the radical is negative and the solution becomes exponential: )exp()( 0 xExEy γ−= (2.22) where 2 0 2 )( ink−== βγα m and is known as the attenuation coefficient. The diagram in Figure 2-4 helps to clarify the relationship between wavevector k, transverse wavevector κ and longitudinal wavevector β. The only physical situation occurs when β satisfies the condition: fs nknk 00 << β . (2.23) In this case, the wave is oscillatory in the guiding layer (nf) and decays exponentially in the substrate (ns) and cladding (nc) layers. The above condition is necessary to produce a guided wave in a structure where nf > ns >nc. k k2 = κ2 + β2 x κ β z Figure 2-4. A diagram illustrating the relationship between wavevector k, transverse wavevector κ, and longitudinal wavevector β. To find the exact values for β which describe the allowed modes for the waveguide, the generalized solutions in each layer must be connected via the boundary conditions. Each solution has a transverse portion that describes the distribution in x and a longitudinal portion describing the distribution along the z direction. As mentioned previously, the longitudinal portions of all solutions have the same form so they can be left out for simplicity. The transverse parts of the electric field amplitude are [30]: ⎪ ⎩ ⎪ ⎨ ⎧ + + − = ))(exp( )sin()cos( )exp( )( hxD xCxB xA xE s ff c y γ κκ γ (2.22)
- 44. 33 where A, B, C and D are the amplitude coefficients which will be determined by application of the boundary conditions. γc and γs are the attenuation coefficients in the cover and substrate respectively and κf is the transverse component of the wavevector k within the guiding layer. In this case only two of the four boundary conditions from (2.12) will be used. To reiterate, the tangential components of both E and H are continuous across each boundary. The other two conditions will not be needed. This results in an underspecified homogeneous system of four linear equations (2 boundaries with 2 boundary conditions for each boundary) with five unknowns (A,B,C,D and the value of κf for each mode) which, when fully solved, yields a transcendental equation in the remaining unknown variable. Using Faraday’s law (2.01a) along with (2.02) to put the tangential component of H in terms of E (the curl must be expanded into individual components on the left hand side of the equation), three of the four boundary conditions can be applied to find A in terms of B,C and D. The equation describing the amplitude of the transverse E field across the structure is [30]: ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − − = ))(exp()sin()cos( )sin()cos( )exp( )( hxxxA xxA xA xE sf f c f f f c f c y γκ κ γ κ κ κ γ κ γ (2.23) The final boundary condition produces a transcendental equation in κf: ( ) scf scf f h γγκ γγκ κ − + = 2 )tan( (2.24) Known as the characteristic equation, this equation can only be solved using numerical or graphical methods. The result will show the discrete values for κf which can be used in turn to solve for the values of β, the propagation constant for each mode. This type of analysis can be performed for the TM polarized eigenmodes as well and will yield similar equations [30]. Theoretically, any type of waveguide can be analyzed and completely solved using this approach. However, in more complicated geometries or multiple layer structures the analysis – particularly finding and solving the characteristic
- 45. 34 equation – becomes intractable. This necessitates the use of more powerful computational packages such as RSoft. Applying this approach to the example waveguide in Figure 2-3 and using the graphical approach to solve for the modes of propagation produces nice pictorial representations. A graph of each side of (2.24) is seen in Figure 2-5 while Figure 2-6 shows simple cross sectional pictures of the E field for each mode. Each intersection point in Figure 2-5 represents a propagating mode. The total power in each mode is normalized to be one. 0 0.5 1 1.5 2 x 10 4 −10 −8 −6 −4 −2 0 2 4 6 8 10 Graphical Solution for Modes Propagating in a Simple 3−Layer Waveguide Kappa RHSandLHSofEquation2.24 Figure 2-5. Graphical solution of the transcendental equation (2.24) for an asymmetrical waveguide. Each intersection of the red and blue graphs (aside from the vertical lines where the value of tangent is infinite) represents a possible mode.
- 46. 35 (a) −10 −8 −6 −4 −2 0 2 4 x 10 −4 0 1 2 3 4 5 6 7 Mode 0 β = 94087 x (cm) Ey(x)(a.u.) (b) −10 −8 −6 −4 −2 0 2 4 x 10 −4 −4 −3 −2 −1 0 1 2 3 4 Mode 1 β = 93608 x (cm) Ey(x)(a.u.)
- 47. 36 (c) −10 −8 −6 −4 −2 0 2 4 x 10 −4 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 Mode 2 β = 92819 x (cm) Ey(x)(a.u.) (d) −10 −8 −6 −4 −2 0 2 4 x 10 −4 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Mode 3 β = 91752 x(cm) Ey(x)(a.u.) Figure 2-6. Graphs (a) – (d) show pictures of the cross section of the E field for modes 0–3 respectively. The value of β for each mode is also given on the graphs. The evanescent decay into the substrate and cladding is shown by the blue and green sections of the plots respectively. It can be observed that higher order modes penetrate further into the substrate and cladding.
- 48. 37 In communications it is common to design waveguides and fibers to only support one eigenmode of propagation due to the fact that multiple modes will distort a signal as a result of modal dispersion. A full explanation of different kinds of dispersion in waveguides can be found in [30]. The formula describing the approximate number of modes that will propagate in a waveguide is: ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − = π 22 Int sf nnhk m . (2.25) In this case ‘Int’ means to round towards 0 (for example, if the number in parenthesis came out to be 5.68395, m = 5). The number of modes is therefore dependent on three parameters: the wavelength λ, the width h and the difference of the squares of the indices of refraction of the guiding layer and the substrate or cladding layer (whichever has the larger index of refraction). Although it is 1µm in the previous example, λ is usually taken to be 1.55µm because at this particular wavelength the lowest possible transmission losses are attained in most silica waveguides and optical fibers [30]. For this reason it has been dubbed the communication wavelength. Unless otherwise mentioned all simulations, calculations and results will assume this value for λ. An interesting phenomenon to take note of is the evanescent decay of the mode in both the cladding and substrate layers. It can be clearly observed that a finite portion of each mode actually extends into these layers. This can be likened to a similar occurrence in the energy levels of the finite potential well of quantum mechanics and forms the basis for coupling between waveguides and other optical components – a very important issue in the field of integrated optics. One other thing of note is the slight yet noticeable asymmetry in the mode distributions which is not surprisingly a result of the fact that the waveguide itself is asymmetric. Similar analysis can be performed on a symmetric waveguide with similar results. A more complete discussion and analysis of these topics can be found in [30, 31].
- 49. 38 2.7.2 Slot Confinement Waveguide As it was noted before, the ideas from the earlier parts of this section and in particular those of the previous section can be extended to more complicated structures. Recently, some research has focused on guiding light in low-index materials by using high index contrast waveguides. One of these structures is a slot confined waveguide or simply the slot waveguide. This structure was proposed by a group at Cornell [32, 33] and used as the basis for further analysis in another recent publication [34]. It is essentially two planar Si slab waveguides placed in close proximity in a low index SiO2 background. A picture of this is seen in Figure 2-7 [32]. In this case nc is still the index of refraction for the upper cladding but ns is now the index of refraction of the slot material and nH is that of each guiding layer. It can be seen that the width of each guide is |b – a| and the width of the slot is 2a. Figure 2-7. Picture of the two dimensional slot confined waveguide proposed in [32]. It is essentially two planar slab waveguides placed in close proximity in a low index SiO2 background [32, 33]. The idea is that by situating the two waveguides next to each other to form a slot made of SiO2 with a lower index than the Si waveguides, the evanescent decay from each waveguide will result in a highly concentrated flux density (D) in the slot as a result of the boundary condition for D – namely, the continuity of its normal component. The advantage of this over past methods of guiding and confining light in low index materials is that the light is guided by TIR and not external reflections from interference effects which are both lossy and wavelength dependent. The guided modes will therefore be eigenmodes of the structure that are fundamentally lossless and not as heavily dependent on wavelength as previous structures.
- 50. 39 The constitutive relations (2) show that the high index contrast at the interface between the slot and the guiding layer is responsible for the large discontinuity in the E field that results. Physically the best way to understand what is going on is to think of the large discontinuity in E as a redistribution of photons resulting from the boundary condition on D. This only occurs for the TM polarized modes since for TE modes the E field (and hence the D field3 ) has no component perpendicular to the interface. The x (normal) component of the transverse E field of the fundamental TM eigenmode of the slot waveguide can be found through analysis similar to that in the previous section to be [32] [ ] [ ] [ ] [ ] [ ]⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎨ ⎧ −−−⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ +−⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ +−⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = )(exp)(sin)sinh()(cos)cosh( 1 )(sin)sinh()(cos)cosh( 1 )cosh( 1 )( 2 2 2 22 2 bxaxa n n aba n axa n axa n x n AxE cHs Hs sH Hs c Hs Hs s Hs H s s y γκγ κ γ κγ κγ κ γ κγ γ (2.26) where κH is the transverse wavevector, γc and γs are the attenuation coefficients in the cladding and slot respectively. The value of A is written as [32] ( ) 0 22 0 0 k nk AA HH κ− = , (2.27) where A0 is an arbitrary constant and k0 = 2π/λ0 is the wave number in vacuum. This eigenmode can be thought of as an interaction between the fundamental TM eigenmodes of each individual waveguide. The values of κH, γc, and γs can all be related to the propagation constant (β) for each mode in a similar fashion to the simple planar waveguide: 22 0 2 0 2 2 0 2 )( )( )( βκ βγ βγ −= −= −= HH ss cc nk nk nk (2.28a-c) 3 This is not true in the most general case. Here, D is in fact parallel to E because it is assumed that this is an isotropic and linear material. For further discussion see [27].
- 51. 40 The values of β for each eigenmode are calculated by solving the characteristic equation for this structure [32]: ( )[ ] ( a n n ab s Hs sH H γ κ γ κ tanhtan 2 2 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =Φ−− ). (2.29) The continuity condition on D causes the E field to be enhanced by a factor of (nH / ns)2 on the slot side of the boundary between the slot and the higher index waveguides. This creates a higher overall power confinement within the thinner low index slot layers. Power confinement will be discussed in the next section. If the two waveguides are close enough together the enhancement can be quite significant but deteriorates rapidly as the separation increases. A graph of the x component of the transverse E field distribution of the fundamental TM mode is shown in Figure 2-8. In this case the values used were: λ0=1.55µm, nH=3.48, ns= nc=1.44, a=72 nm, and b=152 nm and the slot width is 144nm [32]. Additional graphs and discussion of this type of waveguide can be found in the next section on power confinement. −6 −4 −2 0 2 4 6 x 10 −7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Transverse E−field distribution of fundamental TM eigenmode x (m) Ex(x)(a.u.) Figure 2-8. Graph of the x component of the transverse E field of the fundamental TM mode across the regions of the slot confinement waveguide. The blue sections show the evanescent decay into the cladding regions, the black sections show the distribution in the waveguides and the red section shows the behavior within the slot. In this case the slot width is 144nm while the guiding layers are each 180nm.
- 52. 41 2.7.3 Power Confinement in Waveguides One of the most important figures of merit for a waveguide is the power confinement or confinement factor (CF) of each of its eigenmodes. It has already been stated that the total power incident on a surface or cross sectional area of a waveguide can be calculated in general by integrating the time averaged Poynting vector along the direction of propagation (2.14) over the area comprising the surface. In the case of the 2D planar slab waveguide, the integral is instead performed across just the x direction since it is considered to be essentially flat. The power confinement is obtained by calculating the ratio of power flowing within the core or guiding layer to the total power in all regions of the structure. For the planar slab waveguide this is written as ∫ ∫ ∫ ∫ ∞ ∞− − == dxxHxE dxxHxE dxxHxE dxxHxE P P xy h xy regionsall xy core xy total core )()( )()( )()( )()( * 0 * * * . (2.30) Usually this quantity is expressed as a percentage by multiplying it by one hundred. It is usually the case that higher order modes are less confined, mostly due to the fact that they have lower attenuation factors and thus extend further into the low index cladding and substrate layers. The importance of the confinement factor is due to the fact that poorly confined modes often exhibit higher power loss due to bending in the guiding structure (as in ring resonators which will be discussed in a later section) and unintentional evanescent coupling to nearby components. These phenomena are not directly relevant to this discussion but can be explored further in [1, 30, 31]. The major advantage of the aforementioned slot waveguide is that it provides a very high concentration of light in a relatively thin, low index layer (the slot) [32]. The high power confinement of light that results is especially favorable in integrated optics where power efficiency and dissipation are extremely important in device design. The fact that light is concentrated in thin, nanometer-size layers could be considered even more important however, since the diffraction limit has imposed limits on the size of optical components and devices. The diffraction limit states that the lower limit of these
- 53. 42 sizes is on the order of the wavelength of the light within a particular material. Thus the value of light intensity (power per unit area) becomes an important design parameter. One of the primary objectives of this research is to determine optimum device dimensions in terms of maximizing the power confinement within relatively small layers. Through analysis it is found that the most crucial parameter in this endeavor is the ratio of the slot width to that of the guiding layers. To provide a visualization of this, a few graphs of the fundamental TM eigenmode of the slot waveguide are presented with different values for the slot width (Figure 2-9) along with a plot of CF versus ratio (Figure 2-10). In each plot the width of the surrounding guides is kept constant at a specific value while the width of the slot is chosen to fit the chosen ratio. The first thing to notice is that the advantage of confinement in a low index slot is only realized when the slot is relatively thin since the evanescent decay becomes quite substantial at larger thicknesses. It can also be seen that the relationship between slot to guide ratio and CF is not linear and that there is instead a range of optimum values for the ratio. One might be tempted to think a higher ratio and hence larger amount of low index material would yield a higher confinement and that the best approach is to incorporate as much low index material as possible. This trend is indeed true for ratios up to about 0.80. After this point however, there is a steady tailing off of the confinement values. This is due to the nature of the interaction between the eigenmodes of the two waveguides. The width of the slot affects the ratio of the slot to guide width which in turn influences the combined effective index of the two guiding layers and the slot layer, causing more light to be redistributed into the outermost cladding and substrate layers. This results in a balancing act between slot-to-guide ratio and CF, and is perhaps the most important idea to retain from this section, since it provides valuable insight into the nature of light propagation in such a device and also forms the basis of the major design concepts utilized in the next chapter. A more enlightening explanation is given in the next chapter with the presentation of simulation results and graphics for a more realistic device based on this structure. The slot waveguide provides a workable theoretical basis for the design of a three dimensional, multilayer, waveguide to be used as the resonant cavity in a Si-based laser system which can achieve high light enhancement in low index Er-doped SiO2
- 54. 43 layers in order to maximize gain, minimize losses and potentially yield lower threshold current densities. (a) −6 −4 −2 0 2 4 6 x 10 −7 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Transverse E−field distribution of fundamental TM eigenmode Slot Width = 50nm; R = 0.28 x (m) Ex(x)(a.u.) (b) −6 −4 −2 0 2 4 6 x 10 −7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Transverse E−field distribution of fundamental TM eigenmode Slot Width = 144nm; R = 0.80 x (m) Ex(x)(a.u.)
- 55. 44 (c) −6 −4 −2 0 2 4 6 x 10 −7 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 Transverse E−field distribution of fundamental TM eigenmode Slot Width = 180nm; R = 1.00 x (m) Ex(x)(a.u.)
- 56. 45 (d) −6 −4 −2 0 2 4 6 x 10 −7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Transverse E−field distribution of fundamental TM eigenmode Slot Width = 270nm; R = 1.50 x (m) Ex(x)(a.u.) Figure 2-9. Figures (a) – (d) show graphs of the E field distribution of the transverse fundamental TM mode for various slot widths and slot to guide layer thickness ratios. Notice how as the slot thickness increases and consequently lowers the overall effective index, a larger evanescent tail leaking into the cladding materials results. Thus the confinement factor begins to decrease as the total amount of the low index material present in the structure increases beyond a certain point. Confinement Factor vs Ratio of Slot Width to Waveguide Width 55 56 57 58 59 60 61 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Ratio of Slot Width to Waveguide Width ConfinementFactor(%) Figure 2-10. Graph showing the relationship between confinement factor and ratio of slot width to waveguide width. The width of the guiding layers is kept constant at 180nm while the slot width is chosen to fit the desired ratio. The CF increases up to a ratio of about 0.80 because the added low index material allows a larger amount of light in the desired region. The values saturate between 0.80 and about 1.1.
- 57. 46 2.8 Lasers 2.8.8 Introduction Because this project deals with the design and optimization of a waveguide to be used as the resonant cavity of a laser system, it is necessary to give an introduction to the basic operating principles of lasers. Laser is an acronym that stands for Light Amplification by Stimulated Emission of Radiation. All lasers are based on the fundamental principles of absorption and emission of electromagnetic radiation. In order to best understand the functionality of lasers it is critical to first understand these processes. The next topic to cover is pumping and the different methods that are used to achieve population inversion which is a necessity for lasing to occur. Next the topic of amplification via resonant optical cavities will be covered. This is the most critical aspect in any type of laser system since it is what distinguishes laser light from that of a light emitting diode (LED) or any other arbitrary source for that matter. The three distinguishing characteristics of laser light are: 1) directionality, 2) coherence, and 3) it is monochromatic – that is, the range of wavelengths or bandwidth is relatively small compared to arbitrary sources. It will be seen that particularly the last property of laser light is the result of the resonant optical cavity in which the light emitting medium is placed. This is the main reason that light from a laser is preferable over other light sources in many systems and experiments. The cavity can be tuned to produce light of very specific frequencies and wavelengths depending on what is needed in the particular system or experiment. The next two ideas discussed are gain saturation and lasing threshold which play a role in limiting the output power of a lasing cavity. The minimum injection current needed to achieve lasing is called the threshold current and is a very important parameter in the design of practical semiconductor lasers. Electrical injection pumping, particularly with respect to silicon nanocrystals, is covered in the next section. This is an important concept since the MURI project of which this study is a part considers an electrically-pumped device. The concluding section deals with the MURI laser design proposed by the principle investigators at each university and their research groups. It also discusses the use of erbium as the extrinsic material which provides light amplification.
- 58. 47 2.8.2 Three Processes: Absorption, Spontaneous Emission, Stimulated Emission Stimulated absorption or simply absorption is the process by which an electron is raised from a lower energy level (usually the ground level) to a higher one through interaction with a photon. A diagram of this process for a simple two level system is seen in Figure 2-11a. Absorption will only occur when the energy of the photon, given by the quantity hν where h is Planck’s constant, is equal to or greater than the difference in energy between the two levels. A second critical process in a laser system is emission. Emission is just the exact opposite of absorption – an electron makes a transition from a higher energy state to a lower one and gives off a photon. There are two different types of emission: spontaneous and stimulated. Spontaneous emission is the more common type of optical emission and occurs when an excited electron randomly relaxes to a lower state emitting a photon in the process. A diagram is shown in Figure 2-11b. By its nature, spontaneous emission is completely unpredictable. It is impossible to know exactly where the emitted photon will appear or which direction it will head in. It is possible however, to determine a statistical average for the length of time between emissions for particular transitions within a given material system. Often, spontaneous emission is treated as a source of noise within a system since it creates additional light sources which are neither predictable nor controllable [30]. Stimulated emission on the other hand is much more predictable and controllable. It takes place in the same fashion as absorption except that the incoming photon now interacts with an electron already in a higher energy state. The interaction of the photon with the electron induces the transition and yields the release of another photon identical in phase, direction, polarization and frequency to the original. A diagram of the process is shown in Figure 2-11c. Thus, stimulated emission produces amplification of light incident upon a material with electrons in excited states. It is one of the crucial elements in producing laser light.
- 59. 48 (a) Photon (Spontaneous) Absorption After: Before: Electron in excited state after absorbing the photon’s energy.Electron in ground state (b) Spontaneous Emission Photon After: Before: Electron in ground state after emitting a photon.Electron in excited state. (c) Stimulated Emission 2 Photons Photon Before: After: Electron in excited state. Electron in ground state after emitting photon.2 identical photons.Incoming photon interacts with it. Figure 2-11. (a) Simplified representation of absorption, (b) spontaneous emission, and (c) stimulated emission.
- 60. 49 2.8.3 Population Inversion and Pumping Schemes Since stimulated emission acts as an amplification process and produces a photon identical to one incident onto an electron in an excited energy state, it is really the fundamental process in the production of laser light. In order to maximize the number of identical photons and increase the intensity of the laser light, there must first be a sufficient number of electrons in excited states with which incoming photons can interact. For most systems in equilibrium, electrons will tend to relax to the lowest energy state. Therefore energy must be introduced into the system in some way to raise electrons from lower energy levels to higher ones by the process of absorption. This results in the non-equilibrium condition of population inversion. The method of achieving population inversion is called the pumping scheme. The most common pumping schemes that are used in almost all laser systems are: 1) optical sources 2) electrical charge injection 3) chemical reactions 4) ion beams and 5) electron discharges or beams [30]. Optical pumping involves the use of either a flash lamp or another source of laser light to raise electrons to a higher energy state. Since it requires an external source of light this type of pumping is not conducive to the laser system that is the focus of this work. Instead the laser system studied here employs electrical injection pumping which is commonly used in semiconductor lasers. These are discussed further in a forthcoming section. The other types of pumping are not as common and are not discussed in detail. The simplest optical pumping scheme is a two level system. The idea is to raise electrons from lower energy states to a specific higher energy state by providing them the amount of energy unique to that specific absorption wavelength or frequency. In a three level system the electrons are first excited to a higher energy level (L3) that is unstable after which they undergo spontaneous emission or non-radiative decay and relax to a lower, more stable level (L2) after a characteristic lifetime in the higher state. This lower yet more stable level is still higher than the initial ground level (L1) of the electrons. The lasing phase is between L2 and L1, where stimulated emission induces transitions to lower energy states and the release of photons. The most efficient scheme is a four level system, however. A diagram is shown in Figure 2-12. In this system the
- 61. 50 ground level is now L0 and there is an added unstable state above the ground state, L1. The advantage of this over a three level system is that population inversion can be achieved with much less effort since L1 will not be as heavily populated since it is a more unstable state than L0. The reason that a simple two level system is highly impractical is because the light used for pumping is just as likely to decrease the population as to enhance it. Incoming photons stimulate emission when interacting with electrons in the higher energy state and also cause absorption of energy and a transition to the higher energy state for electrons in the lower state. Overall this means that even with an infinite rate of pumping, the populations in both levels would at best equalize. Four level systems provide the opportunity for the highest possible population inversion and hence more efficient lasing [30]. L3 Relaxation (Fast) L2 Pumping Lasing L1 Relaxation (Fast) L0 Figure 2-12. A four level system. L3 and L1 are both unstable states so electrons rapidly decay to L2 and L0, usually in a non-radiative transition. 2.84 Optical Resonators and Amplification Some of the excited electrons will then begin to decay back to their lower energy states by spontaneous emission. The photons emitted interact with other electrons still in the higher energy state giving rise to more photons as they traverse the optical cavity. An external optical cavity is formed by surrounding the source of the laser light with two highly reflective surfaces. An internal cavity would be formed by fabricating the two end surfaces of the gain medium with specific reflectivity. This causes most of the photons to continue bouncing back and forth between the two surfaces producing amplification by further stimulated emission provided there are no losses within the
- 62. 51 cavity. The situation when there are losses within the cavity will be discussed in the next section. The intensity of light grows exponentially with each trip through the cavity until it begins to saturate. This situation will also be discussed in the next section. Gain is a measure of how much the intensity is amplified from the initial value put into the gain medium. The expression for gain is [30] ))(exp( 0 z I I G νγ== , (2.33) where I and I0 are the intensity and initial intensity, γ(ν) is the gain coefficient at a given frequency ν, and z is the distance traversed within the cavity. The gain coefficient can in turn be expressed as [30] Nv ∆= )()( σνγ , (2.32) where σ(ν) is called the gain cross section at frequency ν and ∆N is a measure of the population inversion for the particular type of gain medium. The gain cross section varies depending on frequency, index of refraction of the material, and the spontaneous emission coefficient. It is an important measure of how much gain per cross sectional area will be ‘seen’ by an incoming beam of light and is an important performance metric when discussing potential light emitting materials for use in the particular laser system in this work. Pictures of two optical cavities exhibiting amplification are shown in Figure 2- 13a [9] and Figure 2-13b. The cavity in (a) is designed so that some of the light at one end is transmitted or coupled out of the cavity on each round trip. This particular type of cavity is called a Fabry-Perot cavity. The ring structure is seen in (b). It can also be seen as part of the device in Figure 1-5b. It is often more useful when electrical injection is used to pump the gain medium. The length and reflectivity of the mirrors on either end can be custom-designed to establish longitudinal mode selection and frequency-selective feedback. This means that the resonator essentially acts as a filter allowing only photons of a certain frequency or wavelength to be amplified while all other modes will slowly die off because they aren’t being reinforced with each round trip through the cavity.
- 63. 52 (a) (b) Figure 2-13. (a) Simple Fabry-Perot cavity with mirrors on either end. Also shown is the pumping (energy) source [9]. (b) Ring resonator structure which is the type of structure used for the device in this work.
- 64. 53 2.8.5 Gain Saturation and Lasing Threshold The number of electrons in the excited state from which stimulated emission and lasing occurs decreases with each photon emitted. This in turn lowers the overall population inversion by two since the higher level has one less electron and the lower level has one additional electron. Depending on the rate at which electrons are pumped into the higher level as well as their average lifetime there, this can cause a decrease in the amplification factor of the particular medium. When the intensity of the initial beam of light incident upon the material is small (lower number of stimulated events) compared to the population inversion, this effect is negligible. However, as intensity increases, more stimulated events will occur and this will lead to a more significant decrease in the inversion. Even when no stimulated emission takes place, the amount of inversion will saturate at a value that is the product of the pumping rate and the average lifetime for electrons in the higher energy level. This is a good approximation for cases of low initial intensity within the lasing medium. The saturation intensity is the value of the intensity for which the population inversion is half of its unsaturated value (when there are no stimulated emission events). It provides a point of reference for determining when saturation will play a significant role. For the most part it is desirable to operate at intensities above this value. As light propagates back and forth through a cavity it will encounter loss mechanisms. One source of loss is the output coupling to the mirrors at either end – or to the waveguide in the ring structure. Usually the mirrors reflect most of the light back into the cavity but the mirror at one end is designed to allow some light through to be put to use in a particular application. Other sources of loss are excited-state or free carrier absorption (which presents a major design challenge with Si based materials), scattering, and defects within the gain medium. In order for the intensity in a given mode to grow exponentially, the round-trip gain seen by the mode must be at least sufficient to offset the round-trip losses it encounters. This is known as the threshold gain condition for lasing. For modes that meet this condition, intensity continues to amplify until it saturates at exactly the threshold value. At this point, the gain is just enough to sustain lasing by balancing off the losses within the medium. Modes not supported by the specific cavity design will find the saturated gain level too low to
- 65. 54 sustain oscillation and quickly die out [30]. This is very important since there are generally a large number of possible modes into which light can couple and the goal is to couple into only one of them. With proper design, the resonance frequency can be tuned to a desired value by selective feedback within the cavity. 2.8.6 Electrical Charge Injection It is now appropriate to discuss electrical injection since it is the pumping scheme utilized in the MURI Si laser project and is necessary for any practical, cost- effective light emission system since it eliminates the need for an external source of light. Electrical injection is used to achieve population inversion within a material by introducing carriers (electrons, holes, or both) into excited states which result in light emission or electroluminescence (EL) upon radiative recombination. There are various ways of electrically pumping a medium, but for the extrinsic, slot confined cavity design task of the MURI project the ideal method is carrier tunneling. Band injection is used primarily for the intrinsic laser design. To accurately understand how electrical injection works it is first necessary to review the nature of energy bands and carrier states within both bulk and quantum confined semiconductor structures. In bulk semiconductor materials the discrete energy levels in atomic systems are replaced by large energy bands with nearly continuous (yet technically still discrete) distributions of energy states within them. These energy bands describe the relationship between the energy E and the crystal momentum or wavevector k. The particular arrangement and value of the energy levels varies depending on the type of material. The energy band structures for Si and GaAs were shown in Figure 1-7. The valence bands are those below E = 0eV and the conduction bands are those above 0eV. The fundamental emission process is the recombination of an electron from the conduction band and a hole from the valence band to generate a photon. Both the conservation of energy and momentum apply to this process and are stated as follows [30]: νhEE ba =− (Conservation of energy) (2.33) valenceconductionphoton hkhkhk −= (Conservation of momentum) (2.34)
- 66. 55 where Ea and Eb are the energy of the electron in the conduction band and valence band respectively and the terms in (2.32) represent the momentum of a photon, an electron in the conduction band, and an electron in the valence band respectively. In the case of momentum conservation, it turns out that the momentum of a photon is negligible compared to that of an electron in either band so the initial and final momentums of the electron must be approximately equal. This is exactly the reason why direct bandgap semiconductors such as GaAs are much more efficient light emitters than indirect bandgap semiconductors like Si. In order for conservation of momentum to be satisfied in an indirect material the electron transition must be accompanied by the creation of a phonon at the exact same time. The phonon provides the difference in momentum. This is a two-particle process and has a much lower probability of occurring; the number of light emission events is not as high as it would be in a direct bandgap material. This is part of the problem in using Si as the base material for a light emission system. It was pointed out in the introduction that silicon nanocrystals pose a very practical solution to many of the inherent problems with Si as a light emitter. Nanocrystals differ from bulk materials in that they are quantum confined in all three dimensions to a smaller space and therefore exhibit completely different band structures. Silicon is no exception to this rule. A simple band diagram for a nanocrystal is seen in Figure 2-14 along with the energy diagram for a bulk material. As the size of the nanocrystals decreases, the bandgap widens and the energy states spread out and become more discrete. A more complete energy band diagram for bulk Si was shown in Figure 1-7. The beauty of a nanocrystal lies in the ability to tune its bandgap within a certain range. This means that emission over a much wider range of wavelengths can be realized. The benefits of this will be explained in the next section which also describes the ability of nc-Si to couple energy to Er3+ ions.
- 67. 56 Figure 2-14. A simple energy band diagram for a bulk material (left) and a quantum-confined nanocrystal (right). The development of an efficient method for electrical injection into silicon nanocrystal based devices presents a critical challenge that is crucial to overall device performance. There have been a number of schemes proposed for electrical injection into nc-Si based devices, the details of which are quite involved and beyond the scope of this project. One of the more interesting methods – which is also the currently proposed method for the MURI project – is provided by the group at Stanford [14]. This particular scheme is known as field effect injection and is carried out using a floating-gate transistor structure. It shows that excitons can be efficiently produced in Si nanocrystals. Basically holes and electrons tunnel into the floating gate array of silicon nanocrystals where they form excitonic pairs and undergo radiative recombination. The bandgap of the nanocrystals is specifically tailored to enable efficient energy coupling to the Er ions. This type of structure enables light emission in CMOS-compatible Si structures and is also suitable for the device proposed in this work. A diagram is seen in Figure 2-15.
- 68. 57 Figure 2-15. Diagram of the field-effect structure for achieving electrical injection [14]. The array of silicon nanocrystals exists in the light gray area directly beneath the gate. An important figure of merit for any electrically-pumped laser system is the threshold current density. The threshold current density is the minimum amount of current per unit area required to achieve lasing. Generally it is desirable to minimize this parameter since it dictates the power consumption of the device. It depends primarily on the internal radiative efficiency of the excitons, the gain cross section of the light emitting material, mode confinement, and losses within the optical cavity or gain medium. All of these are important considerations and tradeoffs that must be adequately addressed in the design of cost-effective, efficient light emitting devices with low power consumption. 2.8.7 MURI Si Laser Project – Er3+ as an Extrinsic Light Emitter One of the solutions to the unfavorable characteristics of Si as a light-emitting material is to add a Si-compatible extrinsic material into the laser cavity’s host material to act as a light emitter. In the extrinsic gain laser task of the MURI project, the two extrinsic dopant candidates are Er3+ ions and Lead Sulfide/Lead Selenide quantum dots. This work in particular involves the use of the Er3+ ion. It is incorporated into nanostructured layers of SiO2 to promote population inversion. The two major reasons for this, according to the MURI executive summary, are that it “leverages an extensive library of glass materials knowledge from the fiber optic telecommunications industry and exploits the novel energy transfer properties of Si-nanocrystals and Si-nanowires” [13].
- 69. 58 The multilayer waveguide structure consists of alternating layers of nanometer- thin layers of Si nanocrystals and erbium-doped SiO2 embedded in an active horizontal slot region and can easily be made into a resonant cavity. The Si nanocrystals are formed by first depositing amorphous Si and then appropriately tuning the thermal budget. In particular, it is found that a ring-shaped resonator provides the best configuration for efficient electrical injection and low optical loss. It is important to ensure maximum confinement and minimal loss of the optical modes within the cavity, especially since erbium has a relatively low emission cross section at wavelengths near 1.55µm. The device is pumped utilizing the field effect injection method described previously. The thin nc-Si layers allow for excellent carrier transport (tunneling). In this way, nc-Si provides a way to avoid the problems inherent in bulk Si as well as the excessive degradation of dielectric materials that would occur as a result of hot carrier injection [13]. Results show that the internal radiative quantum efficiency for excitons in Si nanocrystals is sufficiently high – between 60 and 100% [35]. This is important for maximizing the efficiency of the energy transfer to the Er3+ light emitters. Pictures of the device structure are seen in Figure 1- 10 and Figure 1-11. Since Er is the light emitting material, it is important to discuss its absorption and emission properties. An energy level diagram for Er3+ ions in SiO2 is shown in Figure 2-16a. This shows the many possible energy level transitions that may occur. In this case there is an optical pumping source which raises electrons to higher energy states. A more detailed view of the 4 I15/2 4 I13/2 transition in particular is shown in Figure 2-16b. This is the transition that produces emission at 1.55µm. The large number of transitions from lower to higher energy states between these two bands can clearly be seen here. The electrons and holes both obey Boltzmann statistics and as a result the strongest absorption will occur from highly populated levels in the lower energy band to lesser populated levels in the conduction band. Because the energy transfer process will effectively act as the pumping source for the Er, it is useful to look at its absorption spectrum. An absorption spectrum for Er3+ ions in silica is shown in Figure 2-17 [36]. In this case, the ions are excited by an optical source.
- 70. 59 (a) 4 H11/22000 Energy(cm-1 ) 4 F9/2 Relaxation 4 I9/2 4 I11/21000 Pumping Excited State Absorption 4 I13/2 (b) Figure 2-16. (a) Schematic energy level scheme of Er3+ showing the different types of transitions that may occur. The energy levels are very sharp for the free ions. Doped into a SiO2 lattice, the levels split into many discrete levels. (b) A more detailed view of the 4 I13/2 4 I15/2 transition. There are many discrete energy states in which electrons can exist within each energy band. The electrons will obey Boltzmann statistics. The graphs on the right show how the number of electrons (N(E)) changes with increasing energy (E). It can be observed that absorption takes place from the most populated states in the lower band to the least populated states in the upper band and vice versa for emission. This difference accounts for the slightly different wavelength and frequency of photons involved in these two processes. 4 I15/2 Emission at ~1536nm 4 I13/2 E N(E) 4 I15/2 Emission at 1.55µm E N(E)
- 71. 60 Figure 2-17. Solid lines show the absorption spectrum of an Er-doped optical fiber for wavelengths between 450-850nm. This shows the principal absorption bands within the range of wavelengths. [36]. The radiative emission in Er3+ that will be useful in this project occurs in the 4 I15/2 4 I13/2 transition. The strongest emission takes place between the most highly populated levels in the upper band and the lesser populated levels in the lower band, as shown in Figure 2-16b. A photoluminescence (PL) emission spectrum for Er3+ is shown in Figure 2-18 [37]. A very favorable aspect of Er-doped silica is that two of its primary emission peaks are around 1.55µm (actually about 1.535 and 1.551µm) which happens to be the minimum attenuation wavelength for silica waveguides and fibers. This is the reason that it is the most commonly used wavelength for optical fiber-based communication links. Using erbium as the extrinsic light-emitting material is therefore a very viable alternative to devices that use intrinsic silicon simply because of the many problems encountered in obtaining direct light emission from silicon.
- 72. 61 Figure 2-18. Photoluminescence (PL) emission spectrum for Er -doped (~5 x 1015 cm-2 ) silica film on a Si substrate when pumped with 488nm light. The wavelengths at which the two peak intensities occur are labeled [37] After the creation of an exciton (either electrically or optically) in the nc-Si, it will couple its energy to the Er3+ ions in a process known as Förster energy transfer, shown in Figure 2-19. The bandgap of the nc-Si is tuned so that it closely matches the resonant absorption levels of Er – this helps ensure maximum energy transfer. The Er ions are excited to higher energy states where they undergo non-radiative decay to the 4 I15/2 level. Finally, radiative recombination and the emission of photons will occur from 4 I15/2 to 4 I13/2. Extensive investigation has shown that the energy transfer to Er is faster than both the exciton recombination in Si nanocrystals and the radiative emission rate in Er. This enables Er ions to be easily inverted – another practical benefit of using Er. It has also been shown that Er can be incorporated at fairly high concentrations within SiO2 which allows the nc-Si:Er gain medium to compensate for the notoriously small emission cross section of Er. One additional advantage in using Si nanocrystals is that they will not cause significant scattering losses within the cavity [13].
- 73. 62 Er3+ Si-nc Energy Transfer Figure 2-19. Diagram showing pumping of the 1.55µm transition in Er via Förster energy transfer between nc-Si and Er. Non-radiative decay occurs from the I11/2 level to the I13/2 level. This is followed by radiative recombination from I13/2 I15/2 at 1.55µm. 4 I11/2 4 I13/2 4 I15/2 Relaxation Exciton Creation + Emission at ~1550nm -
- 74. Chapter 3 Simulation & Results 3.1 Introduction In order to analyze and design realistic photonic components such as waveguides it becomes necessary to move beyond two-dimensional slab-based structures and to now consider those of three dimensions. As noted, this complicates the analytical techniques presented in the previous chapter. A mathematical solution for even the simplest 3D structures can be quite involved even with a number of limiting approximations [1, 38]. Therefore the use of a commercial software package designed for numerical simulation of photonic components is needed to provide more rapid and tractable analysis. For all simulations in this study the RSoft Design Group’s Photonics CAD Suite is used. Analysis and graphing is performed with Mathworks’ Matlab. For a better understanding of the RSoft tools FullWave and BeamProp please refer to [15, 16] and the references therein. The results of these simulations are presented beside results from a technique known as the transfer matrix method (TMM) which serves as a basis for comparison to the more realistic FDTD simulations. An application of the TMM is developed specifically for the structure presented in this work by a fellow colleague [39]. One minor thing to note is that for 3D structures there are no purely transverse magnetic (TM) or transverse electric (TE) modes as discussed in chapter 2. Instead there are only hybrid modes which are either ‘TM like’ or ‘TE like’. These are usually denoted quasi- TM and quasi-TE but in order to avoid confusion will continue to be referred to simply 63
- 75. 64 as TM and TE throughout the remainder of the chapter where it is implied that these are hybrid modes. There are a few geometrical differences between the 3D and 2D structures as well. In the 2D model, the structure simply consisted of two waveguides situated side by side with a low index slot between them. The 3D version presented here is slightly more complicated. Instead of only two waveguides and one slot, it will consist of repeating periods of high and low index layers or slots. To visualize this, think of multiple waveguides lined up next to each other. The guiding layers of each waveguide are the high index layers and the slots between the waveguides are the low index layers. Additionally, because it is easier to render alternating material layers by horizontal deposition than by vertical side wall fabrication, the structure will have its layers oriented horizontally – a 90° rotation from the configuration of the 2D model shown previously. The structure is referred to as a multilayer waveguide. A 3D model of this type with only a few repeating periods of high index guiding layers and low index slots is possible and has been studied [34], though it will be shown that the confinement factor can be further enhanced by fabricating structures with additional thinner layers, and a higher SiO2 to Si layer thickness ratio. When the total height of the structure is large enough and the layers are thin enough, the actual number of layers or periods does not affect the confinement factor as much as the ratio between the low index (SiO2) slot layer and high index (Si) guiding layer thicknesses – from this point forth referred to simply as ratio or thickness ratio. In fact, it was observed in the discussion of chapter 2 that the confinement factor (CF) is maximized only for a range of ratio values. This result can be extrapolated to the 3D structure where it can be assumed that a similar situation exists. A more tangible explanation of why the ratio is the critical parameter is provided by the effective index method or effective index approximation [30]. This is a way of treating the entire multilayer region as a single layer of the same thickness with an effective index of refraction determined by a weighted average of the Si and SiO2 layers based on their thicknesses and indices of refraction. Because the weighted average depends on the two thicknesses, their ratio becomes a critical parameter. The initial
- 76. 65 objective is therefore to determine an optimum range of ratios for which the CF is maximized. The next step is to then select a few structures with ratios in the ideal range and perform gain/loss by adding a gain mechanism to the SiO2 layers and a loss mechanism to the Si layers to reproduce conditions in the actual device. The main objective is to establish a lower limit on the ratio of gain in SiO2 to loss in Si which in turn provides an estimate of the threshold current density from the electron injection. The major tradeoffs involved with FDTD simulation are time, computing power, and limitations within the software itself. Specifically, it is impossible to use resolutions more precise than about 5nm per grid point without exceeding the virtual memory limit on the machine being used (~ 2 GB). Exceeding the virtual memory limit causes simulations to run for days at a time as opposed to hours at a time and in some cases causes the machine to crash. Because of this it was decided early on that some accuracy would be sacrificed in the interest of saving time. Additionally, since 5nm is the lowest possible resolution for reasonable simulation times, the smallest layer thickness that can be used while still achieving workable results is about 15nm (3 grid points). Thus, all of the structures presented will have layer thicknesses of at least 20nm. This is somewhat unrealistic since the ideal device will have layers between 3-5nm; the key point, however, is validation of the theory provided by simulations of the proposed design. A second problem – possibly a result of the limits on resolution and virtual memory – is that it is difficult to obtain accurate results from simulations unless all physical proportions of the structure (layer thicknesses, height, width, length) as well as the resolution for each dimension have whole number values. For example, a structure with layers 33.585µm thick and a resolution of 7.7µm per grid point yield unrealistic or bizarre results whereas values of 30µm and 5µm for layer thickness and resolution give realistic, physical results. Furthermore, these values must be evenly divisible by the resolutions in each corresponding dimension. For example, a layer thickness of 24nm and resolution of 5nm per grid point would produce somewhat unreliable results whereas a layer thickness of 25nm at this resolution would not. This places severe restrictions on the structure’s physical measurements and eliminates the possibility of using scripts for parameter scanning. Therefore, only a discrete number of unique
- 77. 66 structures can be accurately modeled in this way. This is the reason that only a few ideal structures are chosen for the gain/loss analysis. The benefits of the TMM are realized through parameter scanning since it is able to produce accurate results for a continuous range of structures as well as for layer thicknesses as low as 3nm in much less time than the FDTD method. It provides a very practical model for approximating performance in 3D rib waveguide structures to which the results of FDTD simulations can be compared. It is found that these methods are very consistent with each other. It should be noted that most of the results presented here, along with other details beyond the context of this work are also presented in the TMM article [39]. 3.2 Structure A cross-section of the proposed device structure is shown in Figure 3-1 with the x and y axes as shown. A more complete picture of the entire laser system in which the waveguide is used is shown for reference in Figure 3-2. This picture shows a gain medium with only one layer of nc-Si and SiO2:Er when in fact, the proposed device structure will have multiple layers of these materials in the gain medium as shown in Figure 3-1 and Figure 1-11. The z axis is directed into the page and is the direction of propagation. This particular orientation of the axes will be used consistently throughout the remainder of the chapter. The structure consists of alternating nanometer-thin layers of Si and SiO2 with a SiO2 cap layer and a SiO2 substrate. Notice that starting with the bottom layer, there are N repeating periods of Si/SiO2 with a final layer of Si underneath the SiO2 cap layer. Thus there is always one additional layer of Si. The refractive index of Si and SiO2 are taken to be 3.458 and 1.440 for all simulations and the height of the cap layer is set at 50nm. The height of the multilayer region and the width of the cap – from this point forth referred to simply as the slab height and width – are each constrained by the requirement of single mode operation in both the horizontal and vertical directions. The width, w, is set at 1µm while the slab height is dependent on the number of layers and thickness of each layer but is restricted to values less than about 0.4µm. These values were determined using both TMM calculations and the aforementioned effective index method [30, 39]. The limitation on the slab height is especially important since it determines the maximum number of repeating Si/ SiO2
- 78. 67 periods when the layer thicknesses and ratio are set. Based on Figure 3-1, the equation for determining the slab height is: T_SiT_Si)r(T_SiN +⋅+⋅=HeightSlab . (3.01) where N is the number of repeating Si/SiO2 periods, T_Si is the thickness of the Si layer, and r is the thickness ratio between the SiO2 layers and the Si layers. Also, T_SiO2 will be used to denote thickness of the SiO2 layer from this point forth. 1µm Cap Layer (SiO2) 50nm T_C Figure 3-1. Cross section of structure to be analyzed. T_C, T_Si and T_SiO2 denote the thicknesses of the cladding, silicon and silicon dioxide respectively. The multilayer region consists of layers between 15 and 30nm thick. The layers alternate between low index SiO2 and higher index Si with Si forming both the top and bottom layers. It is flanked on top by a cladding of SiO2 and is deposited on a substrate of the same material. The dimensions of the cladding are set for all simulations while the slab height varies depending on layer thicknesses and the SiO2 to Si ratio. Note the orientation of the x, y, and z axes. Substrate (SiO2) nc-Si SiO2:Er y Slab Height ~370 – 400nm Multilayer Region z x SiO2:Er T_SiO nc-Si T_Si
- 79. 68 Figure 3-2. Complete cross sectional diagram of the structure. The active gain medium is actually composed of alternating layers of nc-Si and SiO2:Er. This particular diagram shows a horizontal slot ridge waveguide design. Electrons and holes are injected through the p and n contacts into the p and n type device layers. They then form excitons by tunneling into the nc-Si layers. Also visible is the ring resonator structure used to achieve frequency selective feedback and amplification [13]. The logic behind such a design can be explained in a fairly straightforward manner. The primary advantage of this type of waveguide is that it is relatively easy to fabricate. Multiple layers of Si and SiO2 can first be deposited using a conventional deposition process with a very high level of precision. The cap layer is formed by selective chemical etching of previously deposited layers [13, 39]. The cap is used to confine the propagating light in the region of the multiple layer waveguide which sits directly underneath it. The design proposed recently by MIT is shown in Figure 3-3a and Figure 3-3b and is based upon the same theoretical concepts presented towards the end of Chapter 2 but is not nearly as easy to make since it would require incredibly high precision etching along the side walls [34]. Therefore it is a somewhat impractical design. By using a ridge structure with a cap layer, the device enables propagating modes to be confined to one area of the waveguide (which is effectively infinite in the horizontal dimensions) without requiring expensive and difficult sidewall etching.
- 80. 69 (a) (b) Figure 3-3. (a) Slot confinement structure proposed by MIT [34]. The structure consists of upper and lower high index cladding layers with alternating thin low index slot layers with high index thicker guide layers in between. In this case, w is the width, tc, tH , and tl are the thickness of the cladding, low index slots and high index layers respectively. The refractive indices of the high and low index layers are nH and nl respectively while n0 is the refractive index of the surrounding air. It operates on the principle of confinement within low index slot layers but is much more difficult to fabricate owing to the extremely high precision required to etch the sidewalls. (b) Contour picture showing the quasi TM mode distribution of the y component of the E field of the structure proposed in [34]. Also shown is the distribution of the y component of the E field along the structure (on the right side) [34].
- 81. 70 3.3 Investigation of Eigenmodes Using BeamProp Due to the time consuming nature of the FDTD simulation method (even when the virtual memory limit is not exceeded), the BeamProp mode solver built into the RSoft toolset is used to perform quick analysis on a large number of structures and establish a rough idea as to which of these might exhibit ideal characteristics. For both BeamProp and FDTD simulations a generic structure is first drawn in the CAD layout window. The orientation of the axes is as noted in Figure 3-1 with z denoting the propagation axis. The wavelength (λ) is set to 1.55µm and the substrate index is set to 1.440. Based on the diagram in Figure 3-1, the RSoft convention defines TM modes to have the H field parallel to the structure layers and the E field orthogonal to the layers and vice-versa for the TE modes. Thus, both the BeamProp and FDTD simulations will primarily use TM polarization since only these modes exhibit the large E field discontinuity and hence high confinement within the low index SiO2 layers. TE modes will be discussed later on in the gain/loss analysis section, but for the most part it is desirable to couple light into only the TM modes. The appropriate values for width (the width of the cap layer) and Si and SiO2 layer thickness are also entered into the appropriate dialogue box while the number of layers and their indices of refraction are entered into the layer table editor. In actuality only the value of the Si layer thickness is entered along with a particular ratio value. The value of the SiO2 layer thickness is then expressed using an equation within the symbol table editor. Due to the software limitations mentioned previously, only a discrete range of ratios in combination with Si layer thicknesses are possible. The range of ratio values used is based on the information from the graph of CF versus ratio for the 2D model (Figure 2-10) and the effective index method. The particular values are chosen so as to provide whole number values for all layer thicknesses. The ratios used are 0.50, 0.80, 1.00, 1.25, 1.50, 1.75, 2.00, and 4.00. The values for Si layer thickness are restricted to be 15, 20, 25 or 30nm depending on the particular ratio (for example, the ratios of 1.25 and 1.75 only give whole number values for Si layer thicknesses that are divisible by four). For each ratio the number of periods is varied starting with 1 and increasing until
- 82. 71 the slab height, as determined by (3.1), crosses the single mode cutoff height of approximately 400nm to produce an array of unique, discrete structures. The x domain is chosen to extend 0.50µm on either side of the edge of the 1µm cap layer and has a resolution of 0.05µm per grid point. The z domain is the propagation length and is set to 10µm with the same resolution as the x domain. The y range starts 0.4µm within the substrate (-0.4µm) and extends up to between 0.10 and 0.15µm above the top of the structure (which is 0.15 – 0.20µm above the top of the multilayer region). In the case of either simulation tool, the resolution in the y direction is much more crucial than that in either the x or z directions, mainly because the electric field undergoes the biggest changes in this direction. Therefore the y resolution is set to 0.005µm per grip point – the highest possible resolution that will not cause skewed results. Upon setting up a generic layout for each ratio, each structure is simulated in BeamProp by a batch processing technique in which the Si layer thickness is varied over the appropriate range of values. Because the SiO2 layer thickness is computed by an equation based on the specific ratio value of the structure, it will also vary over a range of values. The excitation field is a Gaussian beam launched into the TM modes of the structure. Figure 3-4 shows an outline of a structure along with a picture of the mode profile for the major and minor components of the E field looking along the z direction. The major component is actually the y component in this case. The high concentration in the alternating layers of SiO2 is readily apparent. The output for each structure with a given ratio is a data file expressing the value of the field of the fundamental eigenmode at each grid point. Importing each of these files into Matlab, the confinement factor is then calculated for each structure using a simple program. The end result is a collection of plots or data sets for each structure with a specific ratio. It is very important to realize that the CF values found in this undertaking are not physically achievable. BeamProp merely uses numerical techniques to solve for the eigenmodes of the structure based purely on its geometry, the index of refraction profile and the operating wavelength. What this means it that the fundamental eigenmode for which it solves will only take form after relatively long propagation lengths when the more abundant and continuous array of leaky and radiation modes
- 83. 72 have settled out. The propagation lengths necessary to allow for the settling out of non- guided leaky and radiation modes is far too large to be considered for use in any kind of integration scheme nor for simulations that could be done within a reasonable time frame. Thus the BeamProp mode solver is used only as a simplified preliminary analysis to provide a rough idea as to what the optimal structures might be. 0.0 1.0 E-Major Mode Profile (m=0,neff=2.06248) Horizontal Direction (µm) 1- 0 1 VerticalDirection(µm) 0.4- 0.3- 0.2- 0.1- 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.0 1.0 E-Minor Mode Profile (Mag=5) Horizontal Direction (µm) 1- 0 1 VerticalDirection(µm) 0.4- 0.3- 0.2- 0.1- 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Figure 3-4. Outline of the structure and contour picture of the major (y) and minor (x) components of the E field of the fundamental TM eigenmode looking along the z direction found using the BeamProp mode solver. This is a structure with N = 9 periods, T_Si = 20nm and a ratio of 1.00. The slab height is 0.380µm. The high concentration of the field in the low index layers is clearly visible. 3.4 FDTD Simulations Upon choosing a group of ideal structure parameters based on both the BeamProp analysis as well as the results from the 2D model presented in chapter 2, FDTD simulations are performed using the RSoft FullWave tool. The cutoff slab height (0.4µm), the lower limit on layer thickness (0.015µm) and equation (3.1) determine the number of periods (N) in each structure. N varies between 5 and 12. The ratios range from 0.80 – 1.25 and are chosen using the graph of CF versus ratio for the 2D model (Figure 2-10) as well as the effective index method.
- 84. 73 The approach is exactly the same as that used for the BeamProp simulations with a few exceptions, most of which stem from the fact that time, accuracy, and memory usage warrant much more consideration here. First of all, setting up the domain for FDTD simulations is slightly more challenging in that there are many parameters that influence the tradeoff between simulation time, accuracy and memory usage. The x, y and z domains must be balanced carefully so that the upper limit of virtual memory is not exceeded. It has been noted that the resolution in the y direction must be allotted the highest priority so it remains 0.005µm per grid point. Also the resolution in the z direction is lowered to 0.03µm per grid point in order to free up additional memory. Since the propagation length is important in obtaining accurate results - it must be large enough to allow some of the leaky and radiation modes to die out and leave a reasonable picture of the eigenmode – it is kept at 10µm which is roughly one tenth of the propagation length for typical integrated photonic components. These settings provide the best balance between the accuracy of the mode picture and the length of the simulation time, which turns out to be between 3 – 5 hours for each structure. Secondly, a cross-sectional time monitor is placed at a point just shy of the end of the device – at about 9.4µm or so. It is set up to report the normal component of the power incident upon its surface at all points in the cross section of the structure. It works by using numerical methods to compute the Poynting vector at each grid point and then integrates it over the area immediately surrounding the grid point. Finally, the launch field is set to be Gaussian. It is later found that by instead using a file with the launch field output from the BeamProp mode simulation for the same layout, a slightly more accurate result is obtained. However, for the purpose of determining CF, the difference is negligible. The launch field will be used only for the gain/loss analysis section. These steps are then repeated for each structure using TE polarization. The output for each simulation is again a data file which can be imported into Matlab for analysis and graphing. The file contains the values of power for each grid point on the cross section reported by the time monitor at the end of the simulation time. The CF for each input data file can be easily computed using a simple Matlab program. Also both 3D pictures and 2D cross sections (taken at x=0) can be produced as well as a cross sectional picture of the distribution of the z component of the power.
- 85. 74 Figure 3-5 and Figure 3-6 show the 3D picture and 2D cross section at x=0 of the power distribution in a structure with 9 Si/SiO2 periods of 20nm thick layers in a thickness ratio of 1.00. This gives evidence of the large E field discontinuity between the layers and shows that indeed most of the light is confined in the lower index SiO2 layers as expected. Figure 3-7 shows a cross section of the distribution of the z component of the power across the layers. As with Figure 3-4 the high concentration in the SiO2 layers is very apparent. −400 −300 −200 −100 0 100 200 300 400 500 −1.5 −1 −0.5 0 0.5 1 1.5 0.2 0.4 0.6 0.8 1 Y (nanometers) Power Distribution X (microns) Power(arbitraryunits) Figure 3-5. 3D picture of the power distribution of the TM mode in a structure with N = 9 periods, T_Si = 20nm and a ratio of 1.00. The high power confinement in the low index layers is readily apparent here. Also noted is the relatively large tail showing a significant amount of leakage of the mode into the low index substrate.
- 86. 75 −400 −300 −200 −100 0 100 200 300 400 500 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 X−Cut Cross Section of Power Distribution in Mode Y(nanometers) Power(arbitraryunits) Substrate Guiding Region Cladding and Air SiO2 layers Figure 3-6. 2D cross section at x=0 of the power distribution of the TM mode of the same structure as in Figure 3-5. The SiO2 layers are labeled along with the other regions of the structure. The long tail extending into the substrate is more discernable in this figure.
- 87. 76 0.00100161 0.167571 Power (in z’ direction) @ cT=36.0448µm [Mon#2 (0,0,9.4)] X’ (µm) 1- 0 1 Y’(µm) 0.4- 0.3- 0.2- 0.1- 0.0 0.1 0.2 0.3 0.4 0.5 Figure 3-7. Contour picture of the z component of power distribution of the TM mode across the layers in the structure shown in Figure 3-1. Again the high confinement in the low index layers is evident. Note also the oblong shape of the mode as it propagates through the structure. The evanescent tail in the substrate is visible here as in the two previous figures. Another point that needs to be addressed is that of the effective thickness ratio between the SiO2 and Si layers. It was noted that there is always one extra layer of Si within the multilayer region of each structure. Therefore even though the actual ratio between the SiO2 and Si layers is 1.25 for example, the overall effective ratio will be the total thickness of all the SiO2 layers divided by the total thickness of all the Si layers. Each original ratio value is then adjusted to an appropriate effective thickness ratio. From this point forth, the term ratio will refer to the effective thickness ratio unless otherwise stated. Table 3-1 shows the original ratio values alongside their corresponding effective ratio values. Figure 3-8 shows plots of CF versus effective SiO2/Si thickness ratio using both the FDTD simulation as well as the TMM method for both TE and TM polarizations. The reason for the additional simulations using TE polarization is to serve as a comparison to the TM modes. The FDTD results are given for slab heights of 0.370 – 0.405µm and are shown as discrete data points. Each point represents the simulation of a single structure. It was mentioned previously that this range of heights guarantees single mode operation for both TE and TM modes at a wavelength of 1.55µm. The maximum
- 88. 77 CF is about 55% and occurs for a ratio of about 0.80. The TMM plot is given for a slab height of exactly 0.380µm. This enables the most accurate comparison to the FDTD data points. 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 Thickness ratio between SiO2 and Si layer Confinementfactor TMM and FDTD result for multilayer thickness of 0.38 µm TE mode confinement TM mode confinement FDTD data points for TM FDTD data points for TE Figure 3-8. Confinement factor versus SiO2/Si thickness ratio of the TE and TM modes for both the FDTD and TMM methods. The FDTD plots are shown as crosses and circles and are for slab heights t of or reasons mentioned earlier, it is impossible to achieve reasonable, non- skewed ith t between 0.370µm and 0.405µm. The TMM plots are in dashed and solid lines and are for a slab heigh exactly 0.380µm [39]. F simulation results with the slab height set at a specific value. It has to do w the fact that varying the thickness ratio continuously necessitates layer thicknesses that are not whole numbers. Because of the lower limit on the resolution and layer thickness as well as the need for these values to be whole numbers, there cannot be one set slab height. For each ratio value, the combination of layer thickness and number of Si/SiO2 periods which results in the slab height closest to 0.400µm is used. A listing of the exac parameters for each structure is shown in Table 3-1.
- 89. 78 N Ratio Effective Ratio T_Si (nm) T_SiO2 (nm) Total Height (nm) CF (%) 6 2.00 1.71 20 40 380 42.57 7 1.50 1.31 20 30 370 48.56 7 1.75 1.53 20 35 405 52.03 8 0.80 0.71 25 20 385 61.56 8 1.25 1.11 20 25 380 58.46 9 1.00 0.90 20 20 380 60.41 Table 3-1. A listing of parameters for various structures spanning the presumed range of ideal ratios. The value of 20nm for Si layer thickness was chosen since it is most amenable to a resolution of 5nm per grid point. A height of 0.380µm is chosen for the corresponding TMM CF values in the graph simply because it is the most common height of the structures presented here. There are three structures with a slab height of 0.380µm and another that is 0.385µm. It is for this reason that the slab height for the TMM plot is set at 0.380µm. The agreement between these two models is quite evident from Figure 3-8 even though the FDTD values are slightly higher. There is an obvious source of error stemming from the fact that not every FDTD point has a slab height of exactly 0.380µm. This explains the slight jump in the value of the CF for the structure that is 0.405µm. Another possible source of error is the fact that the propagation length is a mere 10µm owing to limited time and memory resources. Such a relatively short propagation length means the eigenmode is not able to completely form. Overall a general trend is clear for both plots. The behavior of this graph is not surprising given the results shown in Figure 2- 10 from the last chapter as well as the effective index method. There is clearly a point at which the CF begins to saturate followed by a range of optimum ratio values and finally a slowly decreasing tail for ratios beyond the optimum range. As expected, this range of ratios is roughly 0.71 – 1.11 (0.80 – 1.25 for the original ratio values). This is also the range for which the difference between the CF in the TM modes and that in the TE modes is highest. Figure 3-9 shows an additional graph for a similar structure with a slab height of 0.520µm for reference [39]. This plot shows only TMM results but exhibits a trend
- 90. 79 similar to Figure 3-8. The structure exceeds the cutoff height for single mode operation, but only for TE polarized modes. There will still be only one TM mode. It is observed that a CF as high as about 75% can be obtained for a ratio of 1.00. This could be very useful for devices in which single mode operation is only required only for the TM modes – a distinct possibility for potential designs based on this study. 0.5 1 1.5 2 2.5 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 Thickness ratio between Si and SiO2 layer Confinementfactor TMM result for multilayer thickness of 0.52 µm TE mode confinement TM mode confinement Figure 3-9. Confinement factor versus SiO2/Si thickness ratio of the TE and TM modes for the TMM method. It exhibits a similar trend to Figure 3-8. The ideal ratios fall within the same range as for the 0.380µm structure with the maximum being about 75% occurring at a ratio of about 1.1 [39]. With an ideal range of ratios confirmed through the agreement between the TMM and FDTD methods, three structures with ratios distributed within this ideal range are chosen and an analysis of the gain and loss behavior is commenced. Before the next section is presented however, it should be noted that the structures proposed here are not physically viable since the Si and SiO2 layers are both too thick to support the necessary dipole interaction (Förster energy transfer) between the Er dopants in SiO2 and the Si nanocrystals in the Si layers. Of course the main idea presented in this section is the fact that thickness ratio (along with the material properties of each layer) and not necessarily layer thickness, is the critical parameter.
- 91. 80 Therefore the results give testament to the overall agreement between the TMM and FDTD methods for modeling device performance but also give important guidelines particularly with respect to designing more feasible structures that might find application in future Silicon based photonic components and systems. 3.5 Gain and Loss Analysis With knowledge of which range of ratios maximize the confinement of light within the low index SiO2 layers, it is now desirable to perform simulations with gain in the SiO2 layers and loss in the Si layers. These provide a way to model the Er light emitters doped into the SiO2 layers which provide gain in the optical cavity, and the high free carrier absorption in Si which decreases electron-hole pair recombination respectively. Since relatively large amounts of power are confined within the SiO2, the idea is to figure out how much loss in the Si layers can be tolerated while still achieving net gain purely through the large discontinuity of the E field. Before undertaking this analysis however, it must first be confirmed that the loss experienced by the TE modes is sufficiently larger than that experienced by the TM modes. It is preferable to maximize the amount of light coupled into the TM modes and minimize the amount coupled into the TE modes since only the TM modes will produce the large E field discontinuity. Simply due to the fact that more light is distributed within the SiO2 than within the lossy Si layers, it is expected that light in the TM modes will experience less loss than the TE modes anyways. The TE modes effectively see more loss than do the TM modes.
- 92. 81 To verify this, further FDTD simulation is required. Three optimized structures are chosen to be studied throughout the remainder of this work. They are listed in Table 3-2: Ratio Effective Ratio N T_Si (nm) T_SiO2 (nm) 1.00 0.89 9 20 20 0.80 0.71 8 25 20 1.25 1.11 8 20 25 Table 3-2. A list of three optimal structures which best cover the optimum range of ratios and structure heights. In this case, T_Si and T_SiO2 denote thickness of the Si layers and SiO2 layers respectively. The ratios are chosen so as to sufficiently cover the ideal range of ratios observed in Figure 3-8. Initially simulations are performed exactly as before, with no loss or gain. One small adjustment is made in that instead of using a Gaussian distribution as the launch field, the data file output from the BeamProp simulation of the structure is used. The reason is that this represents the fundamental eigenmode for each structure and will therefore yield slightly more accurate results for simulations with loss and/or gain inserted into the layers. These are repeated for each structure and polarization (TE and TM). For each case, the maximum power seen at the time monitor is recorded. This is denoted as P0 since it signifies the power with no loss. Next, the loss coefficient in the Si is set at a somewhat arbitrary value. Since the propagation length is only about one tenth of that in realistic structures, it must be higher than what would be expected in order to effectively mimic actual device performance. The loss is programmed into each structure by adding an imaginary component to the refractive index value in the Si layers. Simulations are again performed for each structure and each polarization and the maximum power at the time monitor recorded and denoted as Ploss. The overall loss is computed using the following formula: ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −= 0 ln 1 P P L loss γ , (3.02) where γ is loss and L is propagation length (which is actually the position of the time monitor). This shows that the value of actual power fluxing through the device is
- 93. 82 meaningless and that only relative power is important. The losses in the TM and TE modes are computed and denoted as γTM and γTE. In all three structures the TE mode loss was approximately four times greater than the TM mode loss. The worst case scenario occurred for the first structure where γTE/γTM = 3.89. These values are consistent with TMM calculations and show that a significantly larger loss is incurred by TE modes than by TM. The next step in the procedure is to perform repeated simulations with the same amount of loss in the Si layers as before while adding gain to the SiO2 layers in small increments starting with zero. Doing so will gradually increase the gain to loss coefficient ratio as well as the peak power observed at the time monitor, Pmon. The gain is programmed into the structure in the same way as the loss except that the sign of the imaginary component of the refractive index is negative instead of positive. The objective is to determine the ratio between the gain coefficient in SiO2 and the loss coefficient in the Si (gain to loss coefficient ratio) that yields net gain in the cavity. This will be denoted Rng where ‘ng’ denotes net gain. The net gain within the cavity, G is computed as ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = 0 ln P P G mon , (3.03) where Pmon is the peak power at the time monitor and P0 is the peak power at the time monitor with no loss or gain in the structure. If the light distribution were equal among all layers, the value of Rng would be exactly one. However, since this is not the case for the TM modes, it will be less than one. Using Matlab, a quick calculation shows that the power confined in the SiO2 layers is about four times that in the Si layers. This means that the loss in Si can be roughly four times greater than the gain in SiO2 and net gain will still be observed. Thus the expected value of Rng for TM modes is around 0.25. Figure 3-10 and Figure 3-11 show plots of the overall gain G versus the gain/loss coefficient ratio for each polarization for the TMM and FDTD methods respectively for the first structure from Table 3-2 [39]. Each point on the FDTD graph is a separate simulation. Rng is simply the x intercept of each plot. The slopes of the corresponding polarization plots for the two methods are noticeably different due to the fact that different gain and loss coefficients were used but
- 94. 83 this is not relevant (the gain and loss coefficients for the FDTD simulations were magnified by a factor of about ten to account for the shorter propagation distance). The important thing to notice is that the value of Rng is approximately the same for corresponding polarizations of each method. Both TM plots cross the x axis at a value of roughly 0.27 while the TE plots cross at around 2.5. The behavior of the TM plot is as expected but that of the TE plot is somewhat odd. There is no discontinuity of the E field for this polarization so an Rng value of about one is expected. However, the fact that the value is much greater than one seems to suggest that these modes see more of the high index Si and experience more losses as a result. One possible explanation is that the large discrepancy between the gain profile (high contrast between layers) and the actual mode distribution (approximately a smooth Gaussian) leads to a coupling effect which causes a leakage of photons from the outer SiO2 layers where they experience amplification [39]. This area requires further investigation in order to find a more sufficient explanation. 0 0.5 1 1.5 2 2.5 3 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Modal gain for TE and TM polarization by TMM gain/loss coefficient ratio modeloss/gain(1/cm) TE mode gain TM mode gain Figure 3-10. Plot of modal gain G (in cm-1 ) versus gain to loss coefficient ratio for the TMM method for both the TE and TM modes. The zero gain value for TE modes is about 2.45 and about 0.25 for TM modes [39].
- 95. 84 0 0.5 1 1.5 2 2.5 3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 Gain/Loss coefficient ratio ModalGaincoefficientA.U. Propagation Gain from FDTD simulation TM propagation gain Reference TE propagation gain Reference Figure 3-11. Plot of propagation gain in the optical cavity, G (in arbitrary units) versus gain to loss coefficient ratio for the FDTD method for both the TE and TM modes. The net gain value for the TE modes is about 2.5 and about 0.25 for TM modes [39]. In conclusion it has been shown that by virtue of the large refractive index contrast between Si and SiO2 layers, there is a redistribution of the E field that leads to high confinement within lower index SiO2 layers. This allows for a much lower gain to loss coefficient ratio to be realized since more photons of light see amplification in the SiO2 layers than absorption and loss in the Si layers. Thus, the same amount of overall gain can be achieved with a lower threshold current density and less overall power input in the device design. This represents a significant breakthrough with respect to the design and optimization of optical systems and devices based on Si as low power consumption and dissipation are always a high priority in these endeavors.
- 96. Chapter 4 Conclusion A multilayer waveguide with alternating nanometer-thin layers of nc-Si and SiO2:Er to be used as the resonant optical cavity in a Si based laser system has been designed and analyzed using theoretical and numerical methods. The 2D slot confinement slab structure provides a good theoretical model on which many assumptions regarding more realistic 3D structures can be based. The high index contrast between the layers of alternating material composition results in high discontinuity of the electric displacement as a result of Maxwell’s equations. High mode confinement factors within a low index medium can be achieved by taking advantage of this result. Using the effective index method it is understood that thickness ratio is the critical design parameter in obtaining the best possible confinement of the quasi-TM mode within the low index SiO2:Er layers. High confinement helps to realize high light emission efficiency from the erbium dopants in these layers and also minimizes loss. FDTD and TMM simulations both confirm the assumptions based on the 2D model, show consistent results when applied to this structure and provide a very reliable indication of overall device performance. It is found that the optimum range of thickness ratios is between about 0.70 and 1.11 for total slab heights of 0.380µm and 0.520µm. Maximum confinement factors of roughly 55% and 75% are found for the 0.380µm and 0.520µm slab heights respectively. 85
- 97. 86 Additional investigation as to the modal gain and loss is also performed in order to accurately model realistic devices. Since the TM modes exhibit such high confinement within the SiO2 layers containing the light emitting Er ions, loss in the Si layers (mostly free carrier absorption) can be higher than the gain provided by the Er ions and net gain can still be realized. Both FDTD and TMM methods are used to determine the minimum gain to loss coefficient ratio at which net gain is still seen in the cavity. Both methods confirm that this ratio is about 0.27 for TM modes and 2.5 for TE modes. It is therefore preferable to couple light only into the TM modes. The fact that net gain can still be achieved even when loss is more substantial than gain within the cavity is quite remarkable and results in lower threshold current density and overall power consumption of the device. This approach takes advantage of the superior electronic properties of Si and the desirable optical properties of Er doped SiO2 (most notably emission at 1.55µm) while minimizing the impact of the poor optical qualities of Si and circumventing the inability to electrically pump Er ions by making use of the dipole-dipole energy transfer mechanism between the Si nanocrystals and the Er ions. The device should find widespread application in chip-scale nanophotonic systems as well as other solutions at all levels of integration. Such systems will surely become more prevalent and useful in the future with more parallel approaches and the probability of electronic-photonic partitioning of systems. There is still much research to be done however. Among other things, the exact nature of the energy transfer between the silicon nanocrystals and erbium ions is not yet fully understood. Further investigation of structures with layer thicknesses of ~3-5nm is also needed. The somewhat bizarre behavior of the TE mode with respect to minimum gain/loss coefficient ratio should also be addressed. There are a few other issues – in both design and fabrication – that must be resolved. Finally, further simulation with perhaps more powerful computing capabilities (and hence more grid points and longer propagation distances) will enable even more accurate modeling results and allow for better optimization of this particular structure.
- 98. Bibliography [1] R. G. Hunsperger, Integrated Optics: Theory and Technology, 5th ed. Berlin: Springer-Verlag, 2002. pp. 1-71 [2] L. Pavesi and D. J. Lockwood, "Silicon Photonics," in Topics in Applied Physics. vol. 94 Berlin: Springer-Verlag, 2004, pp. 1-90. [3] "Silicon Photonics Research at Intel." vol. 2007 Santa Clara, CA, 2006. Overview of Silicon Photonics research at Intel, Inc. with diagrams, explanations and links to further resources and documents. Available: http://www.intel.com/research/platform/sp/ [4] M. Paniccia, V. Krutul, and S. Koehl, "Introducing Intel's Advances in Silicon Photonics," pp. 1, Feb 2004. Available: http://www.intel.com/technology/silicon/sp/doc.htm [5] M. Paniccia, V. Krutul, and S. Koehl, "Intel Unveils Silicon Photonics Breakthrough: High-Speed Silicon Modulation," in Technology @ Intel Magazine, 2004, pp. 1-6. Available: http://www.intel.com/technology/silicon/sp/doc.htm [6] R. Claps, D. Dimitropoulos, Y. Han, and B. Jalali, "Observation of Raman emission in silicon waveguides at 1.54 microns," Optics Express, vol. 10, no. 22, pp. 1305, Nov 2002. [7] R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, "Observation of stimulated Raman amplification in silicon waveguides," Optics Express, vol. 11, no. 15, pp. 1731, Jul 2003. [8] O. Boyraz and B. Jalali, "Demonstration of a silicon Raman laser," Optics Express, vol. 12, no. 21, pp. 5269, Oct 2004. [9] M. Paniccia, S. Koehl, and V. Krutul, "Continuous Silicon Laser," pp. 1, Feb 2004. Available: http://www.intel.com/technology/silicon/sp/doc.htm [10] H. Rong, R. Jones, A. Liu, et al., "A continuous-wave Raman silicon laser," Nature, vol. 433, no. 7027, pp. 725, Feb 2005. [11] J. Faist, "Silicon shines on," Nature, vol. 433, pp. 691, Feb 2005. 87
- 99. 88 [12] A. Liu, H. Rong, R. Jones, O. Cohen, D. Hak, and M. Paniccia, "Optical Amplification and Lasing by Stimulated Raman Scattering in Silicon Waveguides," Journal of Lightwave Technology, vol. 24, no. 03, pp. 1440, Mar 2006. [13] P. M. Fauchet, "MURI: Si-based Laser Grant Award," University of Rochester: AFOSR/MIT/University of Rochester, 2006. [14] R. J. Walters, G. I. Bourianoff, and H. A. Atwater, "Field-effect electroluminescence in silicon nanocrystals," Nature, vol. 04, pp. 143, Feb 2005. [15] RSoft, "BeamPROP 7.0 User Guide," Ossining, NY: RSoft Design Group, Inc., 1993-2006, pp. 13-23. [16] RSoft, "FullWAVE 5.0 User Guide," Ossining, NY: RSoft Design Group, Inc., 1999-2006, pp. 15-19. [17] R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, "Numerical Techniques for Modeling Guided-Wave Photonic Devices," IEEE Journal of Selected Topics in Quantum Electronics, vol. 06, no. 01, pp. 150, Jan/Feb 2000. [18] R. Scarmozzino and R. M. Osgood Jr., "Comparison of finite-difference and Fourier-transform solutions of the parabolic wave equation with emphasis on integrated-optics applications," Journal of the Optical Society of America A, vol. 08, no. 05, pp. 724, May 1991. [19] A. Tavlove, Computational Electrodynamics: The Finite-Difference Time- Domain Method. Norwood, MA: Artech House, 1995. [20] A. Tavlove and M. E. Browdin, "Numerical Solution of Steady-State Electromagnetic Scattering Problems Using the Time-Dependent Maxwell's Equations," IEEE Transactions on Microwave Theory and Techniques, vol. MTT-23, no. 08, pp. 623, Aug 1975. [21] K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Transactions on Antennas and Propagation, vol. AP-14, no. 03, pp. 302, May 1966. [22] D. Yevick, "A guide to electric field propagation techniques for guided- wave optics," Optical and Quantum Electronics, vol. 26, p. S185, 1994. [23] D. Yevick and B. Hermansson, "New formulations of the matrix beam propagation method: Application to rib waveguides," Journal of Quantum Electronics, vol. 25, p. 221, 1989.
- 100. 89 [24] D. Yevick and B. Hermansson, "Efficient beam propagation techniques," Journal of Quantum Electronics, vol. 26, p. 109, 1990. [25] J. P. Berenger, "A Perfectly Matched Layer for the Absorption of Electromagnetic Waves," Journal of Computational Physics, vol. 114, no. 02, pp. 185, 1994. [26] M. D. Feit and J. A. Fleck, "Light propagation in graded-index optical fibers," Applied Optics, vol. 17, p. 3990, 1978. [27] M. D. Feit and J. A. Fleck, "Computation of mode properties in optical fiber waveguides by a propagating beam method," Applied Optics, vol. 19, p. 1154, 1980. [28] G. R. Hadley and R. E. Smith, "Full-vector waveguide modeling using an iterative finite-difference method with transparent boundary conditions," Journal of Quantum Electronics, 1995. [29] D. J. Griffiths, Introduction to Electrodynamics, 3rd ed. Upper Saddle River, NJ: Prentice Hall, 1999. pp. 321-91 [30] C. Pollock, Fundamentals of Optoelectronics, 1st ed. Chicago: Irwin, 1995. pp. 1-71,399-489 [31] D. L. Lee, Electromagnetic Principles of Integrated Optics. New York, NY: John Wiley & Sons, 1986. pp. 1-50 [32] V. R. Almeida, Q. Xu, C. Barrios, and M. Lipson, "Guiding and confining light in void nanostructure," Optics Letters, vol. 29, no. 11, pp. 1209, Jun 2004. [33] Q. Xu, V. R. Almeida, R. R. Panepucci, and M. Lipson, "Experimental demonstration of guiding and confining light in nanometer-size low- refractive-index material," Optics Letters, vol. 29, no. 14, pp. 1626, Jul 2004. [34] N. N. Feng, J. Michel, and L. C. Kimerling, "Optical Field Concentration in Low-Index Waveguides," IEEE Journal of Quantum Electronics, vol. 42, no. 09, pp. 885, Sept 2006. [35] R. J. Walters, J. Kalkman, A. Polman, H. A. Atwater, and M. J. A. de Dood, "Photoluminescence quantum efficiency of dense silicon nanocrystal ensembles in SiO2," Physical Review B, vol. 73, pp. 132302, 2006. [36] A. J. Kenyon, C. Chryssou, and C. W. Pitt, "Indirect excitation of 1.5 micron emission from Er3+ in silicon-rich silica," Applied Physics Letters, vol. 76, no. 06, pp. 688, Feb 2000.
- 101. 90 [37] A. Polman, "Erbium implanted thin film photonic materials," Journal of Applied Physics, vol. 82, no. 01, pp. 1, Jul 1997. [38] S. E. Miller, "Integrated Optics: An Introduction," Bell System Technical Journal, vol. 48, no. 07, pp. 2059, Sept 1969. [39] Y. Fu, D. Riley, and P. M. Fauchet, "Photon Confinement in multi-nano- layer waveguides," Rochester, NY: University of Rochester, p. 3, Aug 2007.