COMPARATIVE STUDY ON BENDING LOSS BETWEEN DIFFERENT S-SHAPED WAVEGUIDE BENDS USING MATRIX METHOD

Rupak Bhattacharyya et al. (Eds) : ACER 2013,
pp. 317–327, 2013. © CS & IT-CSCP 2013 DOI : 10.5121/csit.2013.3230
COMPARATIVE STUDY ON BENDING LOSS
BETWEEN DIFFERENT S-SHAPED
WAVEGUIDE BENDS USING MATRIX METHOD
Koushik Bhattacharya1
and Nilanjan Mukhopadhyay2
1,2
Department of Electronics and Communication Engineering, GIMT,
Krishnagar, India
{k.bhattacharyya10, nilu.opt}@gmail.com
ABSTRACT
Bending loss in the waveguide as well as the leakage losses and absorption losses along with a
comparative study among different types of S-shaped bend structures has been computed with
the help of a simple matrix method.This method needs simple 2×2 matrix multiplication. The
effective-index profile of the bended waveguide is then transformed to an equivalent straight
waveguide with the help of a suitable mapping technique and is partitioned into large number of
thin sections of different refractive indices. The transfer matrix of the two adjacent layers will
be a 2×2 matrix relating the field components in adjacent layers. The total transfer matrix is
obtained through multiplication of all these transfer matrices. The excitation efficiency of the
wave in the guiding layer shows a Lorentzian profile. The power attenuation coefficient of the
bent waveguide is the full-width-half-maximum (FWHM) of this peak .Now the transition losses
and pure bending losses can be computed from these FWHM datas.The computation technique
is quite fast and it is applicable for any waveguide having different parameters and wavelength
of light for both polarizations(TE and TM).
KEYWORDS
Excitation Efficiency, S-shape Bend, Transition Loss, Leakage Loss, FWHM, Attenuation
Coefficient.
1. INTRODUCTION
Optical integrated circuits (OICs) has gained importance both in terms of performance and cost
effectiveness. The size of the OICs is reducing by bending the waveguide channels. The optical
loss introduced by these bends is a key factor which determines the overall performance and the
achievable density of components of an OIC fabricated on a single substrate[1].There are various
bend configurations such as two-corner bends, S-shaped bends, coherently coupled multiple
section bends etc. S-shaped bends[1] are widely used in integrated optical circuits because they
provide low-loss transitions between parallel waveguides with a lateral offset and are relatively
easy to design and fabricate[1].BPM is the most powerful technique to analyze devices with
structural variations along the propagation direction, and it provides detailed information about
the optical field[1]. However, even 2D-BPM is a highly computation-intensive program requiring
huge computer run-time and memory. 3D-BPM applicable to channel waveguides is even more
computation-intensive [1]. An approach for analyzing the propagation of electromagnetic waves
through a stratified medium is the matrix method [2]. Ghatak et al. [3] presented an analysis of
2D optical waveguides which is computationally fast involving only straightforward

318 Computer Science & Information Technology (CS & IT)
multiplication of 2x2 transfer matrices of layers. Since this method uses rarely any iteration, it is
extremely fast in computation using a standard PC [1].
2. BENDING LOSS AND TRANSITION LOSS
There are different types of losses occurred in the channel waveguide due to bending, they are
mainly bending and transition losses shown in Fig. 1[1].The major two types of losses are: [a]
pure bending loss due to any curvature in waveguide and [b] transition loss arising from the
discontinuity in waveguidecurvature.Total bending loss is the addition of these two losses [1].
Fig 1. (a)S-type bending structure with transition length L and lateral offset h[1] (b) The transition
and pure bending loss region[1]
The scope of this work is to investigate the light propagation within a channel waveguide and to
calculate these losses for mainly three types of S-shaped bends(Double arc, cosine generated and
sine generated) and to find an optimized solution.
3. MATHEMATICAL BACKGROUND: MATRIX METHOD
A stratified dielectric medium is considered as shown in Fig. (2).The electric field in each
medium for a plane wave incident at angle θ1may be written in the form [4]
exp[ ( cos )] exp[ ( cos )]i ii i
i i i i i i i i ie E e i t k x z e E e i t k x zε ω θ β ω θ β∆ − ∆+ + − −
= + − − + + − …… (1)
(a)
(b)

Computer Science & Information Technology (CS & IT) 319
Where
1 2 3 3 2 3 4 4 4 2 4 0
1 1 2 2 4 4
0; cos ; cos ( );
and sin sin ......= sin
i i ik d k d d k k n n
c
k k k
ω
θ θ
β θ θ θ
∆ = ∆ = ∆ = ∆ = + = =
= = =
βIs an invariant of the system; ie +
and ie −
represent the unit vectors along the direction of the
electric field, and iE+
and iE−
represent the electric field amplitudes of waves propagating in the
downward and upward directions in ith
medium (Fig. 2), respectively [4]. On applying the
appropriate boundary conditions at the interfaces obtain
൬
‫ܧ‬ଵ
ା
‫ܧ‬ଵ
ି൰ = ܵଵ ൬
‫ܧ‬ଶ
ା
‫ܧ‬ଶ
ି൰ = ⋯ = ܵଵܵଶܵଷ ൬
‫ܧ‬ସ
ା
‫ܧ‬ସ
ି൰……………………… (2)
Where,
ܵ௜ = 1/‫ݐ‬௜ ቆ
݁௜ఋ೔ ‫ݎ‬௜݁௜ఋ೔
‫ݎ‬௜݁௜ఋ೔ ݁௜ఋ೔
ቇ…………………………… (3)
‫ݎ‬࢏And‫ݐ‬࢏represent, respectively, the amplitude reflection and transmission coefficients at the ith
interface; ߜ௜ = ݇௜݀௜ܿ‫ߠݏ݋‬௜[4]. For TE polarization
1 1
1 1
1 1
cos cos
cos cos
2 cos
and
cos cos
i i i i
i
i i i i
i i
i
i i i i
n n
r
n n
n
t
n n
θ θ
θ θ
θ
θ θ
+ +
+ +
+ +
−
=
+
=
+
………………… (4)
Fig 2. The incidence of a plane wave at an angle θ1[4]
and for TM polarization
1 1 1
1 1 1
1 1 1
cos cos
cos cos
2 cos
and
cos cos
i i i i
i
i i i i
i i
i
i i i i
n n
r
n n
n
t
n n
θ θ
θ θ
θ
θ θ
+ + +
+ + +
+ + +
−
=
+
=
+
…….……………… (5)
Obviously, for a wave incident from the first medium, there would be no upward propagating
wave in the last medium, and thus for the structure shown, in (Fig.2), ‫ܧ‬ସ
ି
= 0. Using this
condition, and using (2) and (3) the fields are calculated the fields in terms of‫ܧ‬ଵ
ି
.In order to use
the above matrix method in determining the propagation characteristics of planar waveguides, the
second, third, and fourth media (in Fig. 2) may correspond to the superstrate, the wave guiding
film, and the substrate, respectively [4]. If one now evaluates the excitation efficiency of the wave
݊ଵ
݊ଶ
݊ଷ
݊ସ
‫ܧ‬ଵ
ା
‫ܧ‬ଶ
ା
‫ܧ‬ଵ
ି
‫ܧ‬ଶ
ି
‫ܧ‬ଷ
ି
‫ܧ‬ଷ
ା
‫ܧ‬ସ
ି ‫ܧ‬ସ
ା
݀ଶ
݀ଷ
Z
X

in the film (i.e. ɳሺߚሻ = ฬ
‫ܧ‬ଷ
ି
‫ܧ‬ଵ
ାฬ
ଶ
‫ݎ݋‬ ฬ
‫ܧ‬ଷ
ା
‫ܧ‬ଵ
ାฬ
ଶ
) is evaluated as a function of β , then one would obtain
resonance peaks appeared [5] which are Lorentzianin shape; the value of βat which the peaks
appear gives the real part of the propagation constant and the full widthat half maximum
(FWHM) represents the power attenuation coefficient which is just twice the imaginary part of
the propagation constant. If 2G represents the FWHM of the resonance peak, then pure bending
loss for length L’of a constant curvature bent waveguide is given by [1]
B=4.34(2G) L’ (dB)………………………… (6)
Once the propagation constant has been determined, the fields throughout the system can be
calculated by using the appropriate matrices at the interfaces. This method involves only
straightforward multiplicationof 2 X 2 matrices, and no iteration needs to be carried out to
determine the real and imaginary parts of the propagation constant, and those accuracies better
than one part in a million can readily be obtained [4].
Representing the waveguide bend of (Fig. 1) by an equivalent straight waveguide of length equal
to the arc length of the central line of radius of curvature R, the effective refractive index of the
equivalent waveguide along the transverse axis x may be expressed as [6],
( ) ( )(1 )eq eff
x
n x n x
R
= + ………………………. (7)
݊௘௤ሺ‫ݔ‬ሻ Is the transverse effective index profile of the original waveguide without any
curvature.The fundamental mode of the waveguide can be well approximated by a Gaussian
distribution of the form [1]
2
0 2
( ) exp( )
2 x
x
E x E
a
= − ………………………… (8)
The modal offset between two waveguides of radii of curvature Rଵ andܴଶ is then given by [7],
2 4
1
2 2 2
1 2
1 1
( )( )xV a
R R
δ
ρ
= −
∆
………………………… (9)
It is evident from Eq. (9) that the modal offset increases with increasing difference in curvature.
In Eq. (9), ρ is the waveguide half-width and
2 2
max
2
max2
subn n
n
−
∆ = ,…………………………………. (10)
1 max
2
2V n
π
ρ
λ
= ∆ …….……................................. (11)
Where݊௠௔௫is the maximum refractive index value on the surface of the waveguide and݊௦௨௕ is the
refractive index of the substrate at the wavelength, λ[1]. The spot sizeܽ௫appearing in Eq.(8). And
(9) may be calculated from an Eigen value equation of the form [8]
Exp (1
ܽ௫
ଶൗ ) =2‫ܣ‬௫
௏భ
మ
√గ
,‫ܣ‬௫ = ሺ
௔ೣ
ఘ
ሻ…………………. (12)

Since in almost all practical cases, this modal offset is a small quantity, the transition loss can be
expressed as [6]
T= − 4.34ln (1-
గమఋమ
ସௐమ ) dB, ………………………..(13)
Where W is the waveguide width.
Next, the model is used to calculate the bending losses of more complicated S-shaped channel
waveguide bends (Double arc, cosine generated and sine generated).The corresponding curvature
variations along the three different S-curves are [1],
ଵ
ோభሺ௬ሻ
=
ଶగ௛
௅మ sin ቀ
ଶగ௬
௅
ቁ(For sine generated)……………….. (14)
ଵ
ோమሺ௬ሻ
=
గమ௛
ଶ௅మ cos ቀ
గ௬
௅
ቁ(For cosine generated)…………….. (15)
ܴଷሺ‫ݕ‬ሻ = േ
௅మ
ସ௛
ሺ1 ൅
௛మ
௅మሻ(For double arc)………………...... (16)
Where h is the lateral offset between the two parallel sections and L is the transition length in the
longitudinal direction (fig.1).
5. COMPUTATIONAL RESULT
When light incident on the first medium at the guiding medium is excited by the portion of that
light wave. So to study the excitation efficiency as a function of incident angle is the key thing for
every computation. The nature of the plot is same for TE and TM mode.
5.1. Computation of Excitation Efficiency for TE mode
(c)
Fig: 4. Variation of excitation efficiency with angle of incidence at first medium for TE mode for resolution of incident
angle (a) 10-4
(b) 10-5
and (c) 10-6
with refractive indices for first medium
݊ଵ ൌ 1.5, second medium ݊ଶ ൌ 1.4, third medium ݊ଷ ൌ 1.5, forth medium ݊ସ ൌ 1.4 and the thickness of
the layers as ݀ଵ ൌ 10µ݉, ݀ଶ ൌ 0.2µ݉, ݀ଷ ൌ 2µ݉.
1.42 1.44 1.46 1.48 1.5 1.52 1.54 1.56
0
5
10
15
20
Incident angle
Excitationefficiency
1.42 1.44 1.46 1.48 1.5 1.52 1.54 1.56
0
10
20
30
Incident angle
1.42 1.44 1.46 1.48 1.5 1.52 1.54
0
2
4
6
8
10
12
14
16
18
x 10
4
Incident angle
(a)
(b)

For TE mode the resonance occurring at the guiding medium at a particular incident angle. At this
incident angle the excitation efficiency becomes large. Resolution of incident angle is taken to
be10-5
. All the results are obtained with the help of MATLAB (version 7.5) software.
Similarly for TM mode the same results for the resonance occurring at the guiding medium for a
particular value of the incident angle can be obtained.
5.2. Effect of increasing the thickness of superstrate (d2)
The thickness of the superstrate i.e. d2has an effect on the excitation efficiency. If the thickness is
increased from 0.3 to 0.6 then the amplitude of the excitation efficiency for the leakage mode
goes on decreasing and the guided mode becomes more confined as shown below.
Fig: 5 Variation of excitation efficiency with thickness of superstrate for (a) ݀ଶ ൌ 0.3µ݉(b) ݀ଶ ൌ 0.4µ݉
(c) ݀ଶ ൌ 0.5µ݉and (d)݀ଶ ൌ 0.6µ݉
1.42 1.44 1.46 1.48 1.5 1.52 1.54 1.56
0
5
10
15
Incident angle
1.42 1.44 1.46 1.48 1.5 1.52 1.54 1.56
0
5
10
15
20
25
Incident angle
1.42 1.44 1.46 1.48 1.5 1.52 1.54 1.56
0
5
10
15
20
25
30
Incident angle
1.42 1.44 1.46 1.48 1.5 1.52 1.54 1.56
0
5
10
15
20
25
30
Incident angle
(a) (b)
(c) (d)

5.3. Effect of bending the waveguide with a bending radius(R)
In a bended waveguide if the thickness of the superstrate is increased then the amplitude of the
excitation efficiency reduced and the excitation efficiency as a function of incident angle yields a
Lorentzian shaped resonance curve. The value of βat which the peak appears gives the real part of
the propagation constant of the guided waves, and the full width at half maximum FWHM
represents the power attenuation coefficient which is twice the imaginary part of the propagation
constant.
5.3.1. Calculation of pure bending loss fora double arc and cosine-generated S-
shaped bend waveguide
The pure bending loss for double arc and cosine-generated S-shaped bend structure has been
computed here. For the double arc is a structure having two radius of curvature with one value
positive and other is negative. Here the value of the radius of curvature has been kept same. The
radius of curvature for the double arc and cosine-generated structure follows Eqn.16 and Eqn.15
respectively. The bending loss can be calculated using Eqn.6.Depending on the variation of
bending radius (R) and the waveguide length (L’) the pure bending loss also changes.
Fig: 6 Excitation efficiency as a function of incident angle for the computation of pure bending loss as (a)
R=7.2mm, L’=1.2mm, calculated FWHM=0.006980 radian (for positive arc) and (b) R=8.5mm,
L’=1.3mm, calculated FWHM=0.006916 radian (for positive arc).
1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49 1.5 1.51 1.52
0
1
2
3
4
5
6
7
8
Incident angle in radian
Peak value=7.29328
1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49 1.5 1.51 1.52
0
1
2
3
4
5
6
7
8
Incident angle in radian
Peak value=7.29229
(a) (b)

Fig: 7 Variation of pure bending loss for cosine-generated bend waveguide as a function of transition
length and bending radius for different transition length (L) as (a) L=500µm (b) L=1000µm.Values of the
parameters are taken as h (lateral offset) =100µm and λ=1µm.Calculated FWHM are given with the figure
itself.
From the above results it is cleared that the amplitude of the excitation efficiency decreases as the
bending radius increases.The FWHM is calculated from each profile and then with the help of
Eqn.6 the pure bending loss can be calculated. It may be noted that there is a rapid increase of
bending loss if the radius of curvature decreases below 10 mm.
5.3.2. Transition loss and total bending loss as a function of bending radius for an S-curve
made of two circular arcs
Variation of the radius of curvature has the effect on transition loss as well as total bending loss
as observed from the following results. The transition loss is calculated with the help of Eq. (13).
Fig: 8 Variation of (a) transition loss and (b) total bending loss as a function of bending radius with the
required parameters values W (waveguide width)=2µm,ρ (waveguide half width)=1µm and λ=1µm.
(a) (b)
(a) (b)

Total bending loss is calculated by adding the transition loss and the pure bending loss. From the
above two plot it can be seen that the transition loss is less than the pure bending loss. Also the
transition loss increases slightly below R=10mmcompared to total bending loss. For sine and
cosine generated structures the same variation can be obtained.
5.3.3. Comparison of total bending loss between different S-shaped bends waveguide
as a function of transition length
Now with the help of the transition loss calculated from the Eqn.13, the total bending loss can be
obtained by adding that with the calculated pure bending loss. This total bending loss changes
with the variation of transition length of the waveguide as found from the below results.
Fig:9 Variation of total bending loss as a function of transition length for (a) double arc (b) sine- generated
(c) cosine-generated S-shaped bend waveguide with the required parameter values taken as h(lateral
offset)=100µm and λ=1µm.
It can be observed that all the loss profiles are of same nature. The results shows that the total
bending loss decreases with the transition length but amount of total bending loss is much less in
case of cosine-generated S-shaped bend waveguide than the other two bend structures. So in case
of OIC design the use of cosine-generated bend structure is very handful.
(a)
(b)
(b)
(c)

6. APPLICATIONS
This method has been vastly used in OIC design. In case of designing a directional coupler the
bending losses can be computed and optimized by following the efficient bend structure through
this method. With the help of this method it is possible to design couplers for any level of
coupling. Other applications involve designing of distributed Bragg reflector (DBR) laser,
distributed feedback (DFB) lasers and other passive and active optical devices [12].
7. CONCLUSIONS
To calculate the losses due to waveguide bending, a simple method including matrix
multiplication is shown here. Solution of any transcendental or differential equation is not
required here. Also a comparative study among three different S-shaped waveguide bend
structures (double arc, sine generated, cosine generated) on the basis of total bending loss is
shown here and it has been found that the suitable method to minimize the loss is to use the
cosine generated S-shaped bend structure. The transition loss has to be taken into account for this
computation. For any wavelength and polarization of input light, this technique is applicable.
Hence this method is very handful in OIC design. Amplifier can be used to compensate the losses
due to bending; but it would be expensive. If the refractive index of the guiding medium can be
modified so that it will compensate the losses then it can be a very useful method. This method
can be implemented to minimize the losses in channel waveguide by index matching. It does not
involve any approximations and thus gives exact numerical results for TE and TM modes. Also
the computation using the present model is very fast.
ACKNOWLEDGEMENTS
The authors would like to thank Dr. Rajib Chakraborty, Department of Applied Optics and
Photonics, Calcutta University for his proper guidance and valuable suggestions towards the
completion of this paper.
REFERENCES
[1] P. Ganguly, J.C. Biswas & S.K. Lahiri, (1998) “Modeling of titanium in diffused lithium niobate
channel waveguide bends: a matrix approach”, Optics Communication, No. 155, pp 125-134.
[2] M. Born & E. Wolf, (1980) Principle of Optics, Pergamum, LondoN.
[3] A.K. Ghatak, K. Thyagrajan, M.R. Shiny, (1987) J. Light wave Technol.Vol. LT-J, pp 660.
[4] Ajoy K. Ghatak, K. Thyagarajan & M.R. Shenoy, (1987)”Numerical analysis of planar optical
waveguides using matrix approach ”, Journal of light wave technology,Vol. LT-5,No. 5,pp 660-667.
[5] R. Ulrich, (1970) “Theory of prism-film coupler by plane-wave analysis,”J. Opt. Soc. Amer., vol. 60,
pp. 1337-1350.
[6] M. Heiblum, J.H. Harris, (1980) IEEE J. Quantum Electron. Vol QE-11. pp 1154.
[7] F. Ladouceur, P. Lab eye, (1995) J. Light wave Technology, LT-13. 481.
[8] F. Ladouceur, J.D. Love, I.M. Skinuer, (1991) IEEE Proc. J. Opto-electron. pp 138. 253.
[9] A.K. Ghatak, (1985) “Leaky modes in optical waveguides,” Opt. Quantum Electron, vol. 17, pp.
311-321.
[10] Yariv and P. Yeh, (1984) Optical Waves in Crystals. New York: Wiley, p. 420.
[11] E. M. Conwell, (1973) “Modes in optical waveguides formed by diffusion,” Appl. Phys. Lett., vol.
23, pp. 328-329.
[12] T. Makino, (1995) “Transfer matrix method with applications to distributed feedback optical
devices,” PIER 10, pp. 271-319.

AUTHORS
Koushik Bhattacharyya is a 4th year student of Electronics and Communication
Engineering. He is pursuing his M.Tech from Global Institute of Management &
Technology (GIMT), Krishnagar, Nadia.
Nilanjan Mukhopadhyay has received the B. Sc. degree in Physics from Calcutta
University, India in 2006. Then the B.Tech & M.Tech degrees were received from Calcutta
University in 2009 & 2011 respectively. From 2011 to 2012 he was working in Tata
Consultancy Services, Bangalore, India. Since Aug, 2012 he has been working as an
Assistant Professor in the Department of Electronics & Communication Engineering,
Global Institute of Management & Technology, Krishnagar, India.

COMPARATIVE STUDY ON BENDING LOSS BETWEEN DIFFERENT S-SHAPED WAVEGUIDE BENDS USING MATRIX METHOD

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