Lecture22 non linear fiber optics

Module 22 : Non-linear fiber optics I
Lecture : Non-linear fiber optics I
Objectives
In this lecture you will learn the following
Basics of Non-Linear Fiber Optics
Kerr Non-linearity
Mixing of frequencies
Wave Equation for Non-linear Medium
Non-Linear Schrodinger Equation
Qualitative picture of wave propagation in a Non-linear medium
What is Non-linear fiber optics?
When the optical intensity inside an optical fiber increases, the refractive index of the fiber
gets modified. The wave propagation characteristics the become a function of optical
power. Unlike linear fiber optics, where the propagation constant is a function of fiber and
the wavelength only, the propagation constant becomes a function of optical power in
addition to the other parameters.
Inside a single mode optical fiber, an optical power of few tens of mW may drive the
medium into non-linearity.
Why should we Study Non-linear Fiber optics?
When the optical fiber becomes non-linear the pulse propagation gets significantly
modified.
Also new frequencies get generated inside the optical fiber. The spectrum of the output

signal is not same as that of the input signal.
To observe the non-linearity in the bulk medium generally a large optical power is
required. Whereas, optical non-linearity can be easily observed inside an optical fiber with
very low optical power of few tens of mW. This can be explained as follows:
The efficiency of non-linear process is proportional to the optical intensity (Optical power
per unit area) and the interaction length.
For a bulk medium the efficiency is
Where is the optical power and is the optical wavelength.
On the other hand the efficiency for the optical fiber is
Where is the core radius of the fiber and is the attenuation constant of the fiber. Note
that is the power density in the fiber core, and is the effective interaction length
on the fiber. Since is very small for the optical fiber, the interaction length is few Km of a
typical single mode optical fiber. Also since the core radius is very small, few micrometer,
the optical intensity inside the core is very high.
The ratio of the fiber to bulk efficiency (assuming
) is
That means the non-linear efficiency inside a fiber is about billion times more than that in
the bulk medium.
It is therefore easy to get non-linearity inside an optical fiber than in a bulk medium.
Since the non-linearity affects the signal propagation as a whole, there may be two
approaches to handle it.
• Avoid non-linearity in the system by keeping low optical power levels.
• Understand the non-linear signal propagation and make intelligent use of it for
increasing the transmission capability of the fiber.
The first option does not seem very feasible since the optical power has be increased for
long distance communication. Also in a multi-channel system like WDM, even if each
channel has low power, the total power of all channels together is large enough to drive
the optical fiber into non-linear regime.
The second option therefore is more appropriate and desirable. This option is interesting
and at the same time very challenging because the non-linear pulse propagation is a very
complex phenomenon. It requires far more in-depth understanding of the wave

propagation. Nevertheless we will investigate some of the aspects of the non-linear pulse
propagation on an optical fiber. The saving grace is, the non-linear effects are week and
therefore certain approximations can be rightly made in the analysis.
Kerr Non-linearity
In the presence of intense optical field, the induced electric polarization inside the material
can be generally written as
Where is the free-space permittivity, and are the first, second, and third
order susceptibilities of the medium. The susceptibilities are in general tensors.
Quantity essentially is the dielectric constant of the medium. The first order
susceptibility represents linear property of the medium.
The higher order susceptibilities give non-linear effects. These susceptibilities depend upon
the molecular structure of the material. For Pure silica glass, the is very small.
The lowest order non-linear susceptibility which affects the optical fibers is the third order
susceptibility . This susceptibility can provide many effects like, third harmonic
generation, Four Wave mixing (FWM), and non-linear refraction.
Let investigate here the phenomenon of non-linear refraction.
Due to third order susceptibility, the refractive index of the medium can be written as
For an electromagnetic wave the Poynting vector (power density) is proportional to the
square of the magnitude of the electric field. The refractive index of the medium has two
components, , the linear refractive index which could be a function of frequency, and
, the non-linear refractive index which is a function of optical power density.
This non-linearity is called the Kerr Non-linearity and the effect is called the Kerr-
effect.
The non-linear index coefficient is related to the third order susceptibility through a
relation
For glass .

One of the important feature of medium non-linearity is generation of new frequencies.
This can be explained as follows.
Let there are three electric fields with three frequencies and photon
momenta respectively incident on a medium with high third order susceptibility.
The non-linear polarization in presence of these fields is
The interaction of three photons takes place inside the material and a new photon in
created such that the total energy and momentum is conserved. If the fourth photon has a
frequency and momentum , then we must have
This process is called Four Wave Mixing (FWM) or Four Photon Mixing .
The conservation of momentum requires what is called the phase matching of the signals
inside the medium.
One of the combinations of the frequencies which give the new frequency close to the
original frequencies is
There are 9 combinations possible in general for the new frequencies plus three
combinations which would produce new frequencies same as the original frequencies.
They are
However, if the two input frequencies are same then we get limited frequencies.
If , then the new frequency is
And if , then there is no new frequency generated since

In a WDM system when the channels are equi-spaced,
And the new frequency is
This then means that the two neighboring channels of a WDM system exchange power
with the middle channel. Or in other words there is a cross talk introduced between the
WDM channels in the presence of four wave mixing.
The efficiency of the FWM depends upon the phase matching between the waves. Since
due to dispersion on an optical fiber, different frequencies travel with different speeds,
there is phase mismatch between the channels.
The FWM efficiency is given as
Where is the difference in the propagation constants of the various waves due to
dispersion.
And are the phase constants and are the frequencies of the four
waves inside the optical fiber. D is the dispersion at the wavelength of interest.
For equi-spaced channels , and we get the phase mismatch as
For generation of FWM components, the dispersion and its slope should be as small as
possible. In this situation a dispersion flattened fiber is to be used.

However, in a DWDM system where the FWM is undesirable (since FWM introduces cross-
talk between different channels), the fiber should have some dispersion. A low but non-
zero dispersion fiber is necessary for WDM transmission for suppressing FWM.
In presence of non-linearity the refractive index is intensity dependent. So if the dielectric
constant is appropriately replaced with non-linear term we get the wave equation for the
non-linear medium.
The wave equation in general then is given as
The term is the dielectric constant of the medium. The linear refractive index of
the medium is . The last two terms in the equation are the non-linear terms.
For silica glass since is negligibly small, the wave equation reduces to
For time harmonic field with angular frequency , the equation reduces to
Where the non-linear coefficient has been defined above.
The equation is not very to solve. However, under the assumption that the electric field
evolves slowly due to non-linear effects, the equation can be linearized.
The non-linear Schrodinger equation (NSE) has special importance in the propagation of
light in the optical fibers in presence of non-linearity.
Although the equation has been derived from the classical Maxwell's equations, the name
has been assigned to the equation due to the fact that in presence of non-linearity, the
energy packets show particle kind of behavior (this aspect will become clear when we
discuss solitonic propagation).

To derive the NSE from the wave equation some assumptions have to be made which are
following.
• The fiber is a single mode fiber.
• All scattering effects in the fiber are neglected.
• The amplitude envelop of the wave changes slowly with respect to the carrier. That is,
the electric field is quasi-monochromatic meaning fractional bandwidth of the wave is very
small. This assumption is quite valid for most of the communication signals, since the
carrier frequency is of the order of Hz (optical frequency) and the modulation
frequency (same as data rate) is of the order of Hz.
• The non-linearity is a small perturbation to the linear term.
• Polarization of the wave remains same throughout the propagation.
• The transverse distribution of the modal fields is unaffected by the non-linearity.
The electric field can be written as
Where the cylindrical coordinate system is assumed and is the modal field
distribution and is a slowly varying envelope function of z.
The modal field distribution satisfies the linear wave equation giving
Where is the phase constant of the carrier in the optical fiber and is the modal
propagation constant. This equation is identical to the one solved in the module on wave
propagation inside an optical fiber.
Assuming that the envelop function varies slowly as a function of distance, the terms like
are neglected and then the envelop function satisfies the equation
The propagation constant gets contributions from two effects, one due to dispersion,
that is its dependence on frequency, and other due to loss and non-linear effects. Let us
therefore write
Where

And
is the attenuation constant of the fiber.
Now since , we can approximate, and the equation of
the envelop function becomes
Let us take the Fourier transform of this equation over time to get
Where is the Fourier transform of defined as
And the inverse Fourier transform is defined as
Now let us expand in Taylor series around as
Where,
Substituting for in the envelop equation and retaining only up to the second
derivative terms of , we get

Taking inverse Fourier transform the equation of the envelop is
Substituting for in the equation the Non-Linear Schrodinger Equation (NSE) as
Where we have defined the non-linearity coefficient as
The parameter is called the effective core area and is given by
We can identify the various terms of the NSE as follows:
• The first term gives the rate of change of the wave envelop as a function of distance.
• The second term is related to the group velocity since
• The third term gives the group velocity dispersion (GVD) of the envelop.
• The fourth term is due to the loss (attenuation) on the optical fiber.
• The fifth term is due to the fiber Kerr non-linearity.
In subsequent module we will solve the NSE to see the effect of each term on pulse
propagation in an optical fiber.
By putting in the NSE we can get the pulse propagation equation for the linear
medium.
We have seen earlier that the light is guide in the core of the optical fiber, because

refractive index of the core is higher compared to its surrounding medium, the cladding.
In other words the energy has tendency to concentrate in the region of higher refractive
index.
Let us now consider an optical pulse (say of Gaussian shape ). Due to Kerr non-linearity
the entire pulse does not see the same refractive index, but the refractive index in
maximum at the peak of the pulse and minimum at the edges of the pulse as shown in Fig.
The refractive index profile along the length of the fiber is no more constant now but has a
shape same as the pulse power profile. Since the refractive index profile is created by the
pulse itself, it slides along with the pulse.
The pulse energy has now preference to stay near its peak, as refractive index is higher
here. The non-linearity has tendency to compress the pulse.
If there is a dispersion in the medium, the pulse energy has tendency to spread way at it
travels along the fiber. In a linear medium therefore the pulse broadens.
The non-linearity has effect opposite of the dispersion. It may then be used to counteract
the dispersion on the fiber.
If the compressing effect due to non-linearity is completely balanced by the spreading
effect due to the dispersion, then the pulse can transmit indefinitely without either
expansion or contraction. The pulse will therefore travel undistorted on the optical fiber for
infinite length.
From communication view point this aspect is very interesting, since a high data rate can
be sent on the long length of the fiber with out worrying about the dispersion.
It can very easily be noted that at any point in time, the non-linear effect should not
become weaker compared to the dispersion. Because, if dispersion overcomes the non-
linearity, the pulse spreads bring the peak of the pulse down. This in tern reduces the non-
linear effect, making the dispersion to further dominate and broaden the pulse further. The
process is regenerative and the non-linearity can never catch up with the dispersion if it
loses once. Every effort therefore has to be made to maintain the power level of the pulse.
In the presence of fiber loss, the pulse power decreases with distance making non-linearity
weaker. Appropriate amplification mechanisms have to worked out for maintaining the
optical power in the pulse.
Recap

In this lecture you have learnt the following
Kerr Non-linearity
Congratulations, you have finished Lecture 22. To view the next lecture select it from the
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Lecture22 non linear fiber optics

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