EXHIBIT C

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ELSEVIER Optics Communications 233 (2004) 57-65

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Design of multilayer polarizing beam splitters using

genetic algorithm

M. Shokooh-Saremi a"*, M. Nourian b, M.M. Mirsalehi a, S.H. Keshmiri a

a Department ofElectrical Engineering, Faculty ofEngineering, Ferdowsi Vniversity ofMashhad, P.O. Box 9I77S-I111, Mashhad, Iran

b

Faculty ofElectrical Engineering, K.N. Toosi University of Technology. Tehran, Iran

Received 30 August 2003; received in revised form 6 January 2004; accepted 9 January 2004

Abstract

Polarizing beam splitters (PBSs) are key elements of optics and optical communication systems. In this paper, cubic

and plate multilayer PBSs have been designed using genetic algorithm (GA) method which is a powerful optimization

tool. In this work, after an introduction to genetic algorithm, a plate PBS has been designed using this method to show

the ability of GA technique to design polarizing beam splitters with predefined target. Then, a GA-designed cubic PBS

has been compared with a MacNeille one on the spectral characteristics and index profile. Finally, a C-band cubic PBS

has been designed with GA and the results have been compared with ones obtained from needle, flip-flop and combined

GA and gradient design methods with similar design parameters. The results show that the genetic algorithm method

has the ability to design the thin film polarizing beam splitters and in comparison to other techniques has reasonable

results when the time is not a critical parameter in the design process.

© 2004 Elsevier B.V, All rights reserved.

Keywords: Polarizing beam splitter; Thin film filters; Genetic algorithm

1. Introduction

Polarizing beam splitters (PBSs) are optical filters that separate two orthogonal polarization

components (S and P) of light into different directions, In these filters the S and P polarizations

have equal importance. Wavelength band, extinc-

" Corresponding author. Tel.: +985118615100; fax:

+985118629541,

E-mail address: saremi@wali.um.ac.ir (M. Shokooh-Saremi).

tion ratio, reflectance or transmittance of desired

polarization and angular field are main cliaracteristics of a PBS. Multilayer polarizing beam

splitters are a kind of polarization separators that

are based on optical interference thin films, Plate

and cube are the main two configurations of thin

film PBSs. In the plate type, the layers are deposited on a plane substrate while in the cubic form,

the layers are deposited on the hypotenuse of two

prisms and then these prisms arc cemented together with an optical adhesive to produce a cube

[1], Fig. I shows a schematic view of a cubic PBS.

There are good reviews on thin film PBSs in [2-4],

0030-401R/S - see front matter © 2004 Elsevier B.V. All rights reserved,

doi; 10.1016/j,opteora.2004,01.008

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5S M. Shokooh-Saremi e! a!. I Optics Communications 233 (2004) 57-65

Input

Fig, 1. Schematic view of a cubic PBS.

PBS is a main component in coherent optical

communication systems based on the polarization

diversity, optical instruments and other optical

elements such as add/drop multiplexers, tunable

acousto-optical filters and optical switching [4—10],

There are reports on the design of thin film polarizing beam splitters using different techniques

[4,11,12], From a general point ofview, thedesign of

a multilayer thin film filter (TFF) with an arbitrary

spectral profile, is an optimization problem. Genetic

algorithm (GA) is a powerful and attractive optimization tool and has been applied to several electromagnetic problems, including the multilayer

opticalfilters since 1992 [13-20], This method has an

evolutionary nature and roots in genetic relations.

We have used GA to design multilayer PBSs. In this

paper, first, genetic algorithm is described as a design tool for multilayer filters. Then, thin film PBSs

are designed using the GA and the results are presented and discussed and comparisons with other

design techniques are given.

2. Genetic algorithm

Several optimization techniques have been used

to design multilayer filters. From the optimization

point of view, these techniques can be categorized

into two types. First, methods that require an

initial design on which the optimization is performed to obtain the desired specifications. Second, methods that start with a random design and

then apply an optimization procedure on it. Genetic algorithm is a suitable and attractive tool for

optimization and can be utilized as a design or

optimization method to the thin film filters.

In an optimization problem, GA modifies the

initial design parameters in a way that a better

filter would be obtained through steps. In a design

problem, this algorithm produces a set of initial

designs named initial generation. Then, the genetic

operators arc applied to produce new generations

with better specifications. Using this procedure, a

suitable filter is finally obtained.

The basic block that GA works with is called

"chromosome". In a TFF design problem, each

chromosome is equivalent to a filter. A chromosome consists of "genes" that represent the refractive indices and thicknesses of the layers of a

TFF. A "generation" is a set of chromosomes

(filters) with various layers and jVp0(, is the number

offilters in each generation. The numerical quality

measure of a PBS filter is its "fitness function"

which we have chosen as

F- 1

n^s-ii + i/jp (i)

where is the wavelength ofthe /th sample and

and are the reflectances for the S and P polarizations. Referring to Fq, (1), F would increase as

R% and R? approach 1 and 0, respectively. A filter

will be "stronger" if its F is greater.

The gene-based structure of chromosomes is

shown in Fig, 2(a). Each chromosome (filter)

consists of two gene strings, one string for refractive index of layers and one for layers thicknesses

and these strings are interrelated for each filter.

Here, refractive indices are selected from a predetermined set such as {«(!), nfl),..., n{M)} and the

genetic operations are performed on the indices

1,2,instead of the refractive index values

(indexed variable). Layer thicknesses are integer

variables measured in Angstroms. The GA based

design procedure can be summarized as follows;

(1) Creation ofthe initial generation. In this step,

jV1x>p filters with layer numbers between 1 to Nmm.

the refractive indices of layers chosen from the

{«(!), nfl),...,«(iVf)} set and layer thicknesses hi

the [r/miii^max] interval are created randomly. The

values of N^, n(I) to n{M), dmj[1 and are

defined by user.

(2) Calculation of the fitness function (F) for

each filter. The fitness function is calculated for

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M. Shokooh-Saremi el at. / Optics Communications 233 (2004) 57-65 59

thickness gene

Filter (Chrornosorne)

(a) f

Refractive index gene '

11 tj . • •

t k

A A

n i n 2 • • n k

I

Cross-over

r- n

z "3 nj n, "6 ll2' m" "4

h

L- Is h *4 t5 Ify tr h' w

r" n

i 112 it "5 IT, "3 nq "5*

fic

(b) U h h

1

tq' t5 Ifi v t5' -J

fic

to)

111 "2 nq H5 Mutation nr nj* nj' nq' nsT

(V

t| tv h U ts h "tT T7 U U ts k

Fig, 2. (a) Chromosome (filler) structure of GA, (b) cross-over operator (c) mulation operator.

eacli filter using Eq. (1) and relations for thin film

filter analysis [1], Then, the filters are sorted according to their fitness function values (the strongest filter is at the top). Let be a desired fitness

function and Fs be the fitness function of the

strongest filter in the current generation. Thus, if

Fs F Fi, then the strongest filter is chosen as the

desired filter and the operation is terminated.

Otherwise, procedure is transferred to step 3.

(3) Applying the genetic operators. The genetic

operators (selection, cross-over and mutation) are

applied to the chromosomes according to the following steps. The number of chromosomes (filters)

after applying each operator remains constant

equal to In these steps, G

k

and G

t+I represent

the current and the new generations, respectively.

(3.1) Applying the selection (S) operator. The

selection operator consists of the natural selection

and reproduction steps. In the natural selection

step, the average of fitness functions is computed

(Fa). Then, filters with fitness functions greater or

equal to Fs are selected (Ah). After that, the remaining filters ({Ap0p — Ni) filters) are chosen

through reproduction in which filters are selected

randomly and proportional to their fitness function values. It means that the filters with higher

fitness function value have more chance to select

than the ones with lower fitness function. Also it is

possible for weak filters to be selected according to

random nature of this reproduction. Reproduction

increases the density of strong filters in the selection stage. Thus we have

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60 M. Shokooh-Saremi e! a!. I Optics Communications 233 (2004) 57-65

Gs=S{&}. (2)

(3.2) Applying the cross-over (C) operator. In

this step, slices of refractive index genes and layer

thickness genes arc transferred between random

seleeted pairs of filters. For this purpose, pairs of

filters are seleeted randomly, then a slice with

random length that is smaller than the length of

the shorter filter, is defined over the two filters in

each pair. Finally, the genes in this slice are

transferred between two filters (Fig, 2(b)). Crossover, basically, transfers genetic properties between chromosomes and after applying it we have

Gc = C{GS}. (3)

(3.3) Applying the mutation (M) operator. In

this step, a few filters are chosen randomly and

then their layer refractive indices and thicknesses

are altered. The alteration procedure is to replace

the current refractive index with one from the

predetermined set. Also, the current layer thickness is subtracted from a predetermined value to

produce a new layer thickness (Fig, 2(c)). Mutation is basically used for escaping from the local

optima.

GM=M{Gc}. (4)

After applying all these operators, a new generation (G*"1"

1

) is produced (G*"1"

1

= A/{C{S{Gt}}})

and the procedure continues by returning to step 2.

The above mentioned algorithm is implemented

thorough a software in C++. Its inputs are: the

refractive indices of materials, substrate and incident medium, the angle of incidence, the maximum

and minimum allowable layer thicknesses and the

maximum allowable number of layers. Also, the

desired wavelengths and the corresponding desired

jTs are provided by the user. After completion of

the design procedure, the structure of the filter is

provided as a file.

3. Design results

In this section, we provide sample designs of

multilayer PBSs, using the genetic algorithm. The

first filter is a plate PBS. The target is to design a

C-band plate PBS over 1528-1560 nm wavelength

band. The angle of incidence is 60°, the incidence

medium and substrate are air (« = 1) and BK.7

glass (« = 1.5), respectively. Two materials with

refractive indices of 1,465 (SiCfi) and 2.20 (TiCb)

have been used. Fig. 3 shows the profile of the

designed filter which has 20 layers. Figs. 4 and 5

represent the percentage of reflectance versus

wavelength and angle of incidence, respectively.

The fitness function for this design has been equal

to 5.25 for 10 wavelength points and 13,500 generations. The design took approximately 35 min

on a Pii/700 MHz computer.

To compare the results from GA with a previously designed classic PBS, a wide-band MacNeille

PBS which is mentioned in [12], has been considered. This filter has 22 layers. The angle of incidence is 45° in prism, two materials with refractive

2.4

2.2

1,B

1.6

1.4

1.2

2 3 4 5

Thickness (pm)

Fig, 3, Refractive index profile of tlie designed plate PBS,

100'

BO

60

—— Rs

Up

1.525 1,53 1.535 1.54 1,545 1,55 1.565 1.56 1.565

Wavelength (m) x 10

Fig. 4. Reflectance vs. the wavelength spectrum ofthe designed

plate PBS.

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M. Shokooh-Saremi el at. / Optics Communications 233 (2004) 57-65 &1

100-

„ 60

Hp

20 30 40 SO 60 70 00

Angle ol Incidence (degrees)

Fig. 5, Reflectance vs. the angle of incidence of the designed

plate PBS at 1550 nra.

indices of 2.2 and 1.45 are is utilized and the refractive index of prisms is 1.7. Figs. 6(a, c) and 7(a)

represent the calculated results and index profile of

this PBS. For GA-designed PBS, it has been considered that angle of incidence is equal to 45° and

the refractive indices of materials and prisms are

the same as MacNeille one. Figs. 6(b, d) and 7(b)

represent calculated results and index profile of

this GA-designed PBS.

2.2-

2

1.8-

1.6-

1.4-

(«)

0.5 1 1.5 2

ThickncKM tjum)

2.5

I B

1.6

0.5 10

Thickness (pntl

15 2.0

Fig, 7. Index profile of (a) MacNeille and (b) GA-designed

PBSs.

As it is seen, the spectral characteristics of GAbased PBS is considerably better than MacNeille

one and the total physical thickness of the former

is less than the later PBS. Eq. (5), represents

quality measure (P) for PBSs called degree of polarization [1]

too

80

60

40

20

0

i

i

\i\

it '<

•i i

!i 5 i

5 i

• j

(a)

0.4 0.5 0.6 0.7

Wavelength (pm)

0.6

100

,X 60

40

20

0.4

(b)

0.5 0.6 0.7

Wavclcnglh (fim)

0.8

100

80

so a^550 nm

40

20

30 40 50 60

Angle of Incidence (*)

100

n

80

5 60 Nl ^.=550 nm 1 \ f

« 40

20

n

(d)

30 40 50 60

Angle of Incidence (')

70

Fig. 6. Spectral profiles of MacNeille and GA-designed PBSs; (a, b) reflectance vs. wavelength and (e, d) reflectance vs angle of incidence in 550 nm for MacNeille and GA-designed PBSs, respectively (R*: —, ftp; —).

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62 M. Shokooh-Saremi et al. / Optics Communications 233 (2004) 57-65

P =

R* — Rp

Rk + Rv (5)

Considering P for comparing the quality of two

PBSs for the desired wavelength band. Table 1

shows some parameters such as obtained wavelength band, angular sensitivity, total physical

thickness as well as average and standard deviation of P. As it is seen, the GA-designed PBS has

better performance based on the mentioned parameters. However, it should be noted that because of the random nature of GA, in a

predetermined execution time, it is required to

design some PBSs and then select the best one.

The third PBS is a cubic one. This PBS has been

designed as a C-band (1528-1560 nm) polarization

splitter which has applications in WDM optical

communication systems. The angle of incidence is

45° in prism, the refractive indices of incident and

substrate mediums are 1.5 (BK7 glass) and a

maximum of 20 layers is allowed. For manufacturing purposes only two materials with refractive

indices of 1.465 (SiCF) and 2.065 (TaiOs) have

been used. The designed filter has 19 layers. The

fitness function for this design has been equal to

28.65 for 10 wavelength points and 13,400 generations. The design took approximately 25 min on a

PII/700 MHz computer. Also for comparing the

designs with GA and other synthesis techniques,

the third example (C-band cube PBS) has been

considered and compared with the results obtained

from major synthesis techniques such as needle

[21], flip-flop [22] and combined GA and gradient

methods. For needle and flip-flop designs, the

FilmWizard from Scientific Computing International (SCI) has been utilized in which the mentioned synthesis methods exist. Table 2 represents

the results obtained from these methods. In all

designs, the angle ofincidence, refractive indices of

materials, incidence medium and substrate are the

same as GA designed filter.

In needle design, initial total thickness, maximum layer numbers and minimum layer thickness

are set to be 6000 nm, 20 and 10 nm, respectively.

The result for 50 iterations is a 20 layer PBS. This

design took 1 min on the above mentioned computer. In flip-flop design, the initial total thickness,

maximum layer numbers and minimum layer

thickness are set to be the same as needle design

and the mesh size is 10 nm. The result for 25 iterations is a 19 layer PBS. This design took 30 s on

the above mentioned computer. The last design is

based on the combination of GA and gradient

methods to lower the design time of the basic GA

method. For this purpose, the result of first 200

generations ofthe GA program, that had 17 layers

and took 2 min, has been optimized with the global gradient method. The optimization process

had 100 iterations and took 3 min. Figs. 8-11

represent the dependency of spectral characteristics of the above PBSs on wavelength and angle of

incidence.

For considering the effect of manufacturing issues on the designed filters, the layer thicknesses of

each filter has been perturbed 100 times with

random changes of ±5 A. Then, for each perturbation, the average of P over desired bandwidth

has been compared with the average of P for unTable 1

A comparison between some parameters of MacNeille and GA-designed PBSs

Angle of PaVe

incidence P)

(in glass)

(%) A. (%) Effective

wavelength band

(nm) (P > 95%)

Angular

sensitivity P)

Total physical

thickness (pm)

GA-designed PBS 44 96.39 2.438 ~400

44.5 98.07 1.346 ~400

45 99.06 0.661 ~400 2 1.8

45.5 99.26 0.521 ~400

46 98.6 0.938 ~400

MacNeille PBS 45 98.25 3.447 ~375

45.5 97.61 8.074 ~375 1 2.7

46 96.83 7.87 ~375

'Standard deviation of P over desired wavelength band.

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M. Shokooh-Saremi el at. / Optics Communications 233 (2004) 57-65 63

Table 2

Comparison among various design techniques for a cubic C-band PBS

GA Needle Flip-Hop GA +gradient

rf<A) n rf(A) n d (A) n d (A) n

Structure A 1,5 A 1.5 A 1.5 A 1.5

2237 2.065 3178.2 1.465 13,600 1.465 2096.8 2.065

2730 1.465 2364.7 2,065 1500 2,065 2708.3 1.465

2207 2,065 4464.7 1,465 3500 1.465 2031.5 2,065

2313 1.465 2605.6 2.065 2000 2.065 3154 1.465

2144 2,065 4796.1 1,465 4600 1.465 1840.5 2.065

2587 1.465 2482.2 2,065 1400 2.065 3322.8 1,465

2275 2.065 11997 1.465 2600 1.465 2003 2.065

2731 1.465 2361.3 2.065 1500 2.065 3303.9 1.465

2343 2.065 4163.6 1.465 6100 1.465 1835.4 2.065

3933 1.465 2334.7 2.065 800 2.065 4130.5 1.465

2017 2.065 4089.3 1.465 5000 1.465 1706.8 2,065

2946 1,465 2417.6 2.065 1900 2.065 3493.8 1.465

2322 2.065 3976.4 1.465 2000 1.465 1803.3 2,065

2347 1.465 2518 2.065 2700 2.065 3142.9 1.465

2177 2.065 4343.4 1.465 2700 1.465 1988 2.065

3084 1,465 2864.7 2,065 2000 2.065 2846,5 1.465

1625 2,065 3553.6 1.465 3300 1.465 2062.8 2,065

3033 1.465 2892.9 2.065 1800 2.065 S 1,5

2321 2.055 4285 1.465 600 1.465

S 1,5 2392 2,065 S 1.5

S 1,5

4,7372 7.4081 5,96 4.3471

(tun) 8,2401 12.367 9,6674 7,6104

Max. (f) 1.0 1.0 0.9995 1.0

Min. (P) 0.9954 0.9930 0.9710 0.9898

Mean (P) 0.9986 0.9980 0.9927 0.9968

Std, (P) 0,0012 0.0016 0,0077 0,0028

Layer number 19 20 19 17

Yield {%) 100 100 2 9

Design time 25 min 1 min 1/2 min 5 min

A, incident medium; S, substrate; P. degree of polarization; Std., standard deviation.

perturbed filter. If the absolute change is less than

0.005, the filter is selected and otherwise rejected.

These calculations have been brought in Table 2

under "Yield" item. Considering the results in

'fable 2 and Figs. 8 11, the best results belong to

GA and needle methods. Although the GA is a

powerful optimization tool and searches for the

global optimum point, when its search space becomes very large and complicated, the design lime

increases considerably and in comparison with

other methods may bring a drawback to GA. The

design time is highly dependent on the number of

wavelength points, maximum number of layers

and the maximum layer thickness. In simpler designs such as antireflection filters, GA is able to

100

W) CC

HH i-I

MM

99.2

GA

Need e

Flip flop

GA-bGrad ent

Wavelength (m)

Fig, 8, S-polarization reflectance (/?s) for various design

methods for the C-band cubic PBS.

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64 M. Shokooh-Saremi et ai / Optics Communications 233 (2004) 57-65

OA

Needle

Flip flop

GA+Gradlent

2.5

1.52

100

1.53 1.54 1.55

Wavelength (m)

1.56

x 10

Fig. 9. P-polarization reflectance for various design

methods for the C-band cubic PBS.

v

90

GA

Needle

Flip flop

GA+Gradient

43 43.5 44 44.5 45 45^ 46 46.5

Angle of Incidence (degrees)

Fig. 11. Degree of polarization (P) for C-band cubic PBS versus angle of incidence in prism.

^ 97

—- GA

Needle

—Flip flop -B- GA+Gradient

1.52 1.53 1.54 1.55 1.56

Wavelength (m) x1q

1.57

-6

Fig. 10. Degree of polarization (F) for C-band cubic PBS vs.

wavelength for various design methods.

produce an optimum design, similar to oilier wellknown melliods, in a few seconds.

4. Conclusions

In this paper, the genetic algorithm optimization technique has been used to design multilayer

PBSs. For design, only the optimization of reflectances of S and P polarizations over a predetermined wavelength-band have been considered and

the materials are considered to be perfect dielectrics. First, for evaluating GA to design PBSs, a

plate PBS has been designed over a wavelengthband which is suitable for WDM optical communication systems. The plate PBS has 20 layers with

wavelength baud of 1528-1560 nm. This design,

shows the ability of GA in providing results with

suitable agreements with desired specifications.

For comparison with classic designs, a cube PBS

has been designed and compared with similar

MacNeille one. Based on the results in Table 1 and

considering degree of polarization as a measure of

comparison, the wavelength band, angular sensitivity and tbe total physical thickness of the GAdesigned PBS are better than the MacNeille one.

Table 1 represents a summery of some of these

parameters based on P, angle of incidence and

spectral band of the two mentioned PBSs, As it is

seen, the designed PBS has better spectral profile

as well as less total physical thickness with respect

to MacNeille one. Considering other synthesis

techniques, a C-band cube PBS has been designed

using GA, needle, flip-flop and combined GA and

gradient optimization methods with similar design

parameters and targets. As it is seen in Figs. 8-11

and Table 2 and the information in the previous

section, GA and Needle methods have represented

the best results. From the reported results and

other designs with the GA, it is observed that for

filters that have complicated profile or do not have

a classical or straight method of design, GA is a

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M. Shokooh-Saremi et al. / Optics Communications 233 (2004) 57-65 65

suitable design tool when the design time is not a

critical parameter. The choice of some design parameters such as layer thickness range, maximum

number of layers and wavelength points have direct effect on the optimization process. So, the user

of the GA should take into account the effect of

the mentioned parameters.

Acknowledgements

The authors would like to thank Dr. Kambiz

Farnaam from Scientific Computing International

(SCI) for the FilmWizard package and his helpful

suggestions and comments.

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