Modeling and Design of Mixed-coherence Optical Stacks

by William Beck

ARL-TR-6137 September 2012

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Army Research Laboratory

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ARL-TR-6137 September 2012

Modeling and Design of Mixed-coherence Optical Stacks

William Beck

Sensors and Electron Devices Directorate, ARL

Approved for public release; distribution unlimited. 
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Modeling and Design of Mixed-coherence Optical Stacks

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William Beck

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14. ABSTRACT

A Mathematica-based program for designing layered optical structures is described that can compute the optical properties at 

an arbitrary angle of incidence for stacks containing anisotropic and dispersive materials as well as stacks that contain a 

mixture of coherent and incoherent sections. The program can be used to calculate the properties of a fully specified optical 

stack or can be used to optimize some property of the stack by varying the thickness or optical properties of the constituent

layers. Examples are provided for optimization of a solar cell that has coherent optical layers deposited on an essentially 

incoherent substrate.

15. SUBJECT TERMS

Optical stack multi-layer coherence transfer-matrix

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William Beck

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iii

Contents

List of Figures iv

Acknowledgments v

1. Introduction 1

2. Design and Modeling Method 3

2.1 4x4 Matrix Method..........................................................................................................3

2.2 Mixed Coherence ............................................................................................................5

2.2.1 Overview .............................................................................................................5

2.2.2 Mixed Coherence ................................................................................................5

2.3 Dispersive Index Functions.............................................................................................6

2.4 Determining Absorption in each Layer...........................................................................7

2.5 Structure Optimization ....................................................................................................7

2.6 Efficient Computation .....................................................................................................8

3. Example: Optimization of a GaAs Solar Cell Design 8

3.1 Overview .........................................................................................................................8

3.2 Optimization of Antireflection Coating ..........................................................................9

3.3 Enhancement of QD Absorption by Bragg Reflector ...................................................11

4. Conclusion 12

5. References 14

List of Symbols, Abbreviations, and Acronyms 15

Distribution List 16
iv

List of Figures

Figure 1. Calculated transmission for a 200-nm CaF2 layer on a 100-μm quartz substrate 

assuming coherent propagation in all layers..............................................................................1

Figure 2. Measured transmission for the structure modeled in figure 1. ........................................2

Figure 3. Imaginary part of refractive index (corresponding to absorption) for GaAs (blue) 

and QD-GaAs (purple)...............................................................................................................9

Figure 4. Basic structure of GaAs solar cell with 2-layer AR coating. ..........................................9

Figure 5. Solar power spectrum. The purple shaded area is the largest component that can 

be absorbed in GaAs. ...............................................................................................................10

Figure 6. Calculated reflection for an AR-coated structure with 2-m-thick GaAs on top of a 

10-period Bragg reflector.........................................................................................................11

Figure 7. Absorption efficiency with Bragg reflector under plain GaAs and QD-enhanced 

GaAs layer. ..............................................................................................................................12
v

Acknowledgments

I offer my thanks to Kimberly Sablon-Ramsey and John Little for the design of the gallium 

arsenide (GaAs) solar cells and for many helpful discussions about their operation. Thanks also 

go to Kimberly Olver for all of the measured data.
vi

INTENTIONALLY LEFT BLANK.
1

1. Introduction

Multilayer optical structures are used in a wide variety of devices including photodetectors, 

lasers, and antireflection (AR) coatings on lenses. The optical properties are typically calculated 

using a transfer matrix method, which matches the tangential components of the electric and 

magnetic field at the interfaces between the layers. A method using 2x2 matrices provides a full

vector solution for isotropic materials, while Yeh’s (1) 4x4 matrix method can be used for

birefringent materials.

However, most optical transfer matrix methods assume that light propagates coherently through 

the structures, i.e., that the phase of the light changes in a fully predictable manner as the light

propagates through the layers and across the layer interfaces. This coherent propagation leads to 

interference oscillations in the reflection and transmission of the stack. However, some of the 

interference oscillations predicted by the coherent model are not observed in experiments. For 

example, figure 1 shows the calculated transmission for a 200-nm calcium fluoride (CaF2) layer 

on a 100-μm-thick quartz substrate. The low-frequency oscillations are associated with 

interference in the CaF2 layer and the high-frequency oscillations are associated with the 

substrate, but the measured transmission (figure 2) does not contain the high-frequency 

oscillations.

Figure 1. Calculated transmission for a 200-nm CaF2 layer on a 100-μm quartz substrate

assuming coherent propagation in all layers.

1000 1500 2000

n m

0 .6

0 .7

0 .8

0 .9

1 .0

Tp p

2
2

Figure 2. Measured transmission for the structure modeled in figure 1.

The absence of interference oscillations from thicker substrates is typically due to variations in 

the thickness of the substrate that are larger than the wavelength of the light. Interference can 

also be suppressed by a rough surface on the back of the substrate. In both cases, the light that is 

reflected from the back surface arrives at the front surface with a randomized phase, so that the 

interference predicted by the purely coherent model does not occur. Therefore, it is desirable to 

have a calculation that properly describes the incoherence and yields results that more closely 

match experiments. In an ideal case, the calculation would include the ability to describe partial

coherence, which could treat cases in which the phase of the light is only partially randomized by 

transmission through a layer or interface. However, partial coherence is more complicated to 

describe and usually requires Monte Carlo calculations (2) that average over a large set of 

random phase changes. This makes them slow and generally unsuitable for use in structure 

optimization, which requires many runs. The model reported here assumes that layers and 

interfaces are either fully coherent or fully incoherent.

Many of the structures that we would like to model include materials that have a large variation 

of their refractive indices over the wavelengths of interest. This is particularly true for 

semiconductor layers, whose absorption drops to nearly zero at wavelengths longer than their 

cutoff wavelength. Any realistic model should include this variation and provide a variety of 

tools for specifying the wavelength-dependent complex refractive index.

While it is useful to be able to compute the optical properties of a fully specified structure, it is 

even more useful to be able to determine the dimensions or materials that optimize some 

property of the structure. For example, when designing an AR coating for a photodetector that is 

detecting a laser, the designer would like to determine the AR layer thicknesses that yield the 

minimum reflection at the laser wavelength. But when designing a broad-band solar cell, the 

program should maximize the overall solar cell efficiency.

1000 1500 2000

0 .1

0 .2

0 .3

0 .4

0 .5
3

2. Design and Modeling Method

The model that has been implemented uses the Yeh 4x4 matrix method, which provides a full 

vector electromagnetic solution at any angle of incidence for coherent stacks of isotropic or 

birefringent materials, but it generalizes that method to accommodate stacks that contain an 

arbitrary mixture of coherent and incoherent layers and interfaces. The complex refractive 

indices of all materials are specified by user-definable functions of wavelength, so that arbitrary 

dispersive properties can be included. Further, the thickness and optical properties of all layers 

can be specified using variables that are then varied to maximize a user-specified merit function.

2.1 4x4 Matrix Method

The details of the 4x4 matrix method are provided in section 9.7 of Yeh’s book (1) and are not 

repeated here, but some of the essential features are described in this section.

In isotropic layered media, the electromagnetic radiation can be divided into two uncoupled 

modes: s modes with an electric field vector perpendicular to the plane of incidence and p modes 

with an electric field parallel to the plane of incidence. In the case of birefringent layered media,

the s and p modes are mixed, but the radiation can be represented by four partial waves, each 

with a wavevector kz, electric field polarization vector p, and magnetic field polarization vector 

q that can be calculated from the birefringent material properties. If the four kz are all real, then 

two modes have group velocities to the left and the other two have group velocities to the right.

Therefore, the optical field at any point in a layer i can be represented by a vector whose 

elements are the amplitudes of the four modes. The components of the tangential electric (E) and 

magnetic (H) fields are related to the amplitudes (A) in the four-component mode vector by a Dmatrix such that

1

2

3

4

x

y

i

y

x i

E A

H A

D

E A

H A

       

    

   

       

(1)

Therefore, the tangential fields in layer (i-1) can be matched to the fields in layer i by

1 1

2 2 1

1

3 3

4 4 1

i i

i i

A A

A A

D D

A A

A A







       

    

   

       

, (2)
4

where 

1 D Di i 1



is called the dynamical matrix that relates the mode amplitudes in layer i to the 

mode amplitudes in the adjacent layer (i-1).

Within a layer, the mode amplitudes change exponentially so that the amplitudes at the left side 

of a layer are related to those at the right side by

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4 , , ,

exp[ ] 0 0 0

0 exp[ ] 0 0

0 0 exp[ ] 0

0 0 0 exp[ ]

z

z

i

z

z i left i i right i right

A ik t A A

A ik t A A

P

A ik t A A

A ik t A A

               

                 

               

, (3)

where 

Pi

is the propagation matrix for layer i. 

The transfer matrix to propagate the mode vector through a stack of layers can be obtained by 

stringing together the dynamical and propagation matrices for each layer

1 1 1 1

0 1 1 1 2 2 2 3 3 3 ( )( )( )... M D D PD D P D D P D Dn

   



(4)

If the stack is sandwiched between two isotropic media, the modes in the end sections are leftand right-moving s- and p-waves. Assume that the light is incident from the left and let 

, , , A A B B

s p s p

and 

, C Cs p

be the incident, reflected, and transmitted waves, respectively. Then 

11 12 13 14

21 22 23 24

31 32 33 34

41 42 43 44

0

0

s s

s

p p

p i

A M M M M C

B M M M M

A M M M M C

B M M M M

                  

                   

, (5)

and the reflection and transmission coefficients can be expressed in terms of the transfer matrix 

elements by eight relations, including these four examples

21 33 23 31

0 11 33 13 31

p

s

ss

s A

B M M M M

r

A M M M M 

  

      

(6)

41 33 43 31

0 11 33 13 31

p

p

sp

s A

B M M M M

r

A M M M M 

  

      

(7)

33

0 11 33 13 31

p

s

ss

s A

C M

t

A M M M M 

 

      

(8)

31

0 11 33 13 31

p

p

sp

s A

C M

t

A M M M M 

  

      

, (9)
5

where the subscripts or r and t indicate the modes of the incident and reflected/transmitted 

waves.

2.2 Mixed Coherence

2.2.1 Overview

As described in section 1, the essential property of a coherent layer or interface is that light 

propagating through the layer or interface has a predictable phase change based on the thickness 

and refractive index of the layer or by the change in refractive index across the interface. 

However, light that propagates through a fully incoherent layer or interface is assumed to emerge 

with a random phase, which is the result of an average over paths with many different phases. In 

the case of a thick substrate, the random phases are generated by thickness variation within the 

optical beam area. In the case of an interface, the random phases are typically associated with 

interface roughness.

An important property of two waves that are incoherent is that the total intensity is equal to the 

sum of the two constituent intensities. This contrasts with two coherent waves, where the total 

amplitude is equal to the sum of the amplitudes. The transfer matrices described in section 2.1

are amplitude transfer matrices that express the amplitudes on one side of the layer, interface, or 

stack as a linear combination of the amplitudes on the other side. For example, with

1 1

2 2

3 3

4 4 L R

A A

A A

M

A A

A A

       

    

   

       

(10)

the amplitude of the first component on the left is

A M A M A M A M A 1 11 1 12 2 13 3 14 4 L R R R R    

(11)

But if the layer, interface, or stack represented by M is incoherent, the result should express the 

intensity on the left as an incoherent sum of the individual intensities on the right:

2 2 2 2 2 2 2 2 2 A M A M A M A M A 1 11 1 12 2 13 3 14 4 L R R R R    

(12)

It is apparent that the appropriate intensity transfer matrix for an incoherent section is just the 

amplitude transfer matrix with each element replaced by the square of its absolute amplitude.

2.2.2 Mixed Coherence

A stack comprised of all coherent elements can be modeled using amplitude transfer matrices, 

and a stack comprised of all incoherent elements can be modeled using intensity transfer 

matrices. Yet, many real structures contain some coherent and some incoherent elements. For 

example, when a series of thin layers are deposited onto a thick substrate, the thin layers usually 
6

transmit coherently while the substrate transmits incoherently. A purely coherent analysis of 

such a structure using amplitude transfer matrices would yield the spurious interference 

oscillations from the substrate, while a purely incoherent analysis using intensity transfer 

matrices would eliminate the interference effects that do occur in the thin coherent layers. In this 

case, a mixed analysis must be used in which the coherent sections are first modeled using 

amplitude transfer matrices and those coherent sections are then incorporated along with the 

incoherent sections into an analysis by intensity transfer matrices.

The model described in this report first identifies all coherent sub-stacks within the full stack. 

The amplitude transfer matrix of each sub-stack is calculated using equation 4. Then, those substack amplitude transfer matrices are converted to intensity transfer matrices and combined with 

the intensity transfer matrices for the incoherent sections to form an overall stack intensity 

transfer matrix yielding a structure that looks like

IntensityTransfer Matrix forCoherentSection

2 IncoherentSections

2 2

1 1 1 1 1

Intensity 1 1 1 2 2 2 3 3 3

AmplitudeTransfer Matrix for CoherentSection

M D D PD D P D D P D D P D D air s s s air

    

 , (13)

where the numbered subscripts indicate thin coherent layers, and the s subscript indicates an 

incoherent substrate. The structure could have an arbitrary number of coherent and incoherent 

sections.

Once the intensity transfer matrix for the whole stack is obtained, we can use intensity versions 

of equations 5–9 to determine the reflection and transmission coefficients.

2.3 Dispersive Index Functions

The model specifies the complex refractive index in each layer by a user-defined function of 

wavelength so that arbitrary dispersive properties can be included. Three commonly used 

functions are the following:

1. A constant function (no dispersion)

2. Interpolation of tabulated experimental data

3. Sellmeier coefficients

The Sellmeier equation (3) is an empirical relationship between refractive index and wavelength 

for a transparent material that typically provides precise fits to experimental data with a small 

number of coefficients. A typical form is

2 2 2 2

2 1 2 3

2 2 2 2

1 2 3

( ) 1 ... n

n

B B B B

n

C C C C

    

   

     

   

, (14)

where the series is carried out to as few as n=1 or as many as n=5.
7

In most cases, I have used published Sellmeier coefficients for transparent optical materials such 

as silicon dioxide (SiO2), CaF2, and titanium oxide (TiO2), and interpolated experimental data for 

semiconductors such as silicon (Si) and gallium arsenide (GaAs).

2.4 Determining Absorption in each Layer

It is easy to determine the total absorption A in a stack from the reflection R and transmission T 

coefficients using

A R T    1 , but in some cases, we need to know the absorption in the 

individual constituent layers of the stack. Again using a solar cell as an example, the cell may 

contain a junction area in which absorption leads to photocurrent, but also absorbing contact 

layers that generate no photocurrent. Knowing the absorption in each individual layer allows us 

to design the contact layer thickness so that it doesn’t absorb a significant fraction of the incident 

light. We can then choose to optimize the AR coating thicknesses to maximize the absorption in 

the useful junction region.

The model uses two passes to determine the layer-by-layer absorption. The first pass determines 

the reflection and transmission coefficients for the full stack. Using these coefficients with 

equation 5, we see that the right-side intensity vector that corresponds to an incident s-wave from 

the left side with unit intensity is

T T

ss sp 0 0

. The right-side intensity vector corresponding 

to an incident p-wave from the left side is 

T T

ps pp 0 0 .

In the second pass, the right-side vector determined in the first pass is propagated through each 

layer and interface of the stack from right to left, thereby determining the vector at every location

in the stack. This then determines the transverse electric and magnetic fields at every location 

through equation 1. The power flow can then be computed from the z-component of the Poynting 

vector

1 * * Re

2

z x y y x

S E H E H      

(15)

Knowing the power flow at every interface allows us to compute the power lost in each layer, 

which is the absorption. Note that equation 15 is valid even in absorbing regions with complex kvector, whereas the more commonly used expression

2

2

S E





k

(16)

is only valid in regions with real k. 

2.5 Structure Optimization

When defining the stack, any of the stack parameters can be left as a variable. The user can then 

define an arbitrary merit function that depends on the optical coefficients of the stack and use the 

powerful nonlinear optimization routines that are built into Mathematica to find the values of the 
8

variable parameters that maximize the merit function. The variables can be real numbers, such as 

the thickness of a layer, but they can also be integers that specify which material to use out of a 

list of candidate materials. For example, the optimization for an AR coating can determine the 

best combination of thicknesses and material to use. Furthermore, the goal function can include 

any of the built-in Mathematica functions such as integration and differentiation, so that quite 

complex optimization problems can be set up.

2.6 Efficient Computation

Execution speed is important for doing structure optimization. For example, if we want to 

optimize the thicknesses in a multilayer AR coating on a solar cell, we need to maximize the 

overall solar cell efficiency, which is an integral of the absorption over the full solar wavelength 

band. So, computing the efficiency for a single AR coating requires computing the transfer 

matrices and optical coefficients at hundreds of wavelengths. Then, optimizing the AR coating 

thicknesses requires computing the full spectrum over hundreds, or even thousands, of different 

combinations of thicknesses. 

The most direct way to evaluate the stack transfer matrix shown in equation 4 or 13 is to simply 

compute the transfer matrix for each layer and interface, in turn, and then multiply them together. 

However, that method leads to lots of repeated calculation of the same matrices. For example, 

the D matrices are only a function of the material and the wavelength, so all layers and interfaces 

that use a particular material have the same D matrix. To maximize execution efficiency, the 

model first looks through the stack to identify all of the materials that are used. At each 

wavelength it pre-computes the D matrices and other parameters that only depend on material.

By using this approach, as well as similar strategies to reduce duplicated calculations, the 

execution time was reduced by a factor of 7, reducing the time for solar cell optimizations from 

more than an hour to 5–10 min (on a Dell Latitude E4300 laptop).

3. Example: Optimization of a GaAs Solar Cell Design

3.1 Overview

This example came out of a project within the Electro-optic (EO)/Infrared (IR) Materials and 

Devices Branch at the U.S. Army Research Laboratory (ARL) to develop high-performance 

GaAs-based solar cells for Army applications. GaAs absorbs strongly at wavelengths below its 

cutoff wavelength of 870 nm, but the solar cell efficiency could be improved by extending the 

absorption to longer wavelengths by adding layers of quantum dots (QDs), which extend the 

absorption to roughly 1000 nm. However, as shown in figure 3, the extended absorption yielded 

by the QDs is relatively weak, so an optical structure was desired to enhance the absorption in 

the region between 870 and 1000 nm. In addition, we wanted to design an AR coating that would 
9

maximize the solar cell efficiency for plain GaAs structures and also for the wavelengthextended QD-GaAs structures.

Figure 3. Imaginary part of refractive index (corresponding to absorption) for GaAs (blue) 

and QD-GaAs (purple).

3.2 Optimization of Antireflection Coating

The basic structure of the plain GaAs solar cell is shown in figure 4. The goal was to choose 

materials and thicknesses for the AR coating to maximize the solar cell efficiency. The full solar 

cell efficiency is determined by the efficiency with which the cell converts incident photons into 

photoelectrons and then the efficiency with which the device extracts those photoelectrons and 

sends them to an external circuit. This modeling addressed only the first step so the efficiency 

was defined as the ratio of the power in photo excited carriers divided by the total incident solar 

power.

Figure 4. Basic structure of GaAs solar cell with 2-layer AR coating.

850 900 950 1000 1050 1100

nm

0.02

0.04

0.06

0.08

Im n

Substrate

p-GaAs

AR Coating

Top Contact

Bottom Contact

n-GaAs
10

The solar power spectrum is shown in figure 5 along with shading indicating the maximum 

fraction of that power that can be absorbed in GaAs with its cutoff of 870 nm. The actual 

efficiency will be

 

cutoff cutoff

0 0 cutoff

Eff S A S [ ] d [ ]d

 



    



  

(17)

where 

S[ ] 

is the solar power spectrum, 

A

is the fraction of incident light absorbed at 

wavelength 



, and 

cutoff  /

reflects the fact that when a photon with energy 

E E  

gap

is 

absorbed, it generates an excited carrier with energy 

E

that rapidly decays to a carrier with 

energy 

Egap

, with the excess energy 

E E   gap

lost as heat. With this factor included, the 

maximum efficiency for a GaAs solar cell is 40%. The goal for the optical design was to choose 

an AR coating that maximized

Eff

to as close as possible to that theoretical maximum.

Figure 5. Solar power spectrum. The purple shaded area is the largest component that can 

be absorbed in GaAs.

An example of one run to find an optimum two-layer AR coating illustrates the process. The 

calculation used interpolated numerical data for the solar spectrum and the complex refractive 

index of GaAs. A “menu” of possible transparent materials to use in the AR coating was 

constructed including yttrium oxide (

YO2 3

), tantalum pentoxide (

Ta O2 5

), CaF2, magnesium 

fluoride (

MgF2

), 

2 SiO , silicon nitride (

3 4 Si N

), and 

TiO2

with the refractive index of each 

specified by published Sellmeier coefficients The programmed specification for each layer of the 

AR coating included an integer variable to specify which material from the “menu” to use and a 

real variable to specify the thickness of the layer. The GaAs absorbing layer was 2 μm thick. All 

layers were assumed to be coherent except the substrate, which was incoherent. The 
11

Mathematica optimization routine was then used to find the combination of materials and 

thicknesses that yielded the largest value of 

Eff

in equation 17. 

After sorting through all of the possible material combinations in the AR coating, the model 

found that the best result was obtained with 46.33 nm of

TiO2

followed by 89.78 nm of

CaF2

, 

which yielded an efficiency of 38.6%.

3.3 Enhancement of QD Absorption by Bragg Reflector

The absorption strength of GaAs well below its cutoff is very strong so that nearly all light in 

that wavelength range is absorbed in a 2-μm-thick layer. But as shown in figure 3, the extended 

absorption provided by QDs between 870 and 1000 nm is much weaker, and incident light in that 

wavelength range is not fully absorbed in a 2-μm-thick layer. Therefore, the efficiency at those 

wavelengths should be improved by a back reflector that sends light back through the layer on a 

second pass. This reflector should be placed between the substrate and the device layers to avoid 

parasitic absorption of light going through the substrate. The simplest back reflector would be a 

layer of gold, but gold does not grow epitaxially on GaAs so it cannot be placed between the 

substrate and other epitaxial layers of the structure. Thus, the structure modeled here used a 

Bragg reflector consisting of alternating layers of GaAs and Al0.8Ga0,2As. The best results were 

obtained with a reflector consisting of 10-periods of 63-nm GaAs/75-nm Al0.8Ga0,2As, which 

yields a reflection band centered at 900 nm. The strong effect of the reflector can be seen in 

figure 6, which shows the calculated overall reflection from an AR-coated structure containing a 

2-μm-thick layer of plain GaAs (which does not absorb in the 870–1100 nm region) on top of a 

10-period Bragg reflector. 

Figure 6. Calculated reflection for an AR-coated structure with 2-mthick GaAs on top of a 10-period Bragg reflector.
12

When the Bragg reflector is used under a QD-enhanced GaAs layer, it increases the absorption in 

the QD enhancement region (870–1000 nm) as well as at wavelengths just below the GaAs 

cutoff where the GaAs absorption is relatively weak. Figure 7 shows the calculated absorption 

efficiency (

A

in equation 17) for a simple cell using plain GaAs and no Bragg reflector, QDenhanced GaAs with no Bragg reflector, and QD-enhanced GaAs with a Bragg reflector. The 

Bragg reflector nearly doubles the absorption in the QD-enhanced wavelength range. The 

corresponding solar cell efficiencies were 38.6% for the simple cell, 40.5% for QD-enhanced 

without Bragg reflector, and 42.1% for QD-enhanced with Bragg reflector.

Figure 7. Absorption efficiency with Bragg reflector under plain GaAs and QD-enhanced 

GaAs layer.

4. Conclusion

The Mathematica model described in this report is a powerful tool for designing layered optical 

structures. It includes capability for anisotropic (birefringent) materials whose complex 

refractive index can be defined by arbitrarily complex functions of wavelength. Standard 

functions are provided for materials defined by Sellmeier coefficients or by interpolating 

experimental data. Optical properties of the structures can be computed at an arbitrary angle of 

incidence.

0

0.2

0.4

0.6

0.8

1

300 400 500 600 700 800 900 1000 1100

GaAs - No Bragg

QD - No Bragg

QD - 10-period Bragg (900 nm)

Absorption Efficiency

Wavelength (nm)

89.8 nm CaF

2

46.2 nm TiO2

2 m GaAs or QD-GaAs

Bragg Reflector

Infinite Substrate

Absorption Efficiency
13

The model can treat structures containing arbitrary combinations of coherent and incoherent 

layers and interfaces. This eliminates the spurious, high-frequency oscillations in the optical 

properties of thick incoherent layers that are predicted by purely coherent models while 

preserving interference effects within the coherent regions.

The model uses the powerful nonlinear optimization capabilities of Mathematica to find the 

structure dimensions and materials that maximize a user-defined merit function. This capability 

has been used to design optimized AR coatings and Bragg reflectors for QD-enhanced GaAs 

solar cells.
14

5. References

1. Yeh, P. Optical Waves in Layered Media; John Wiley & Sons, New York, 1988.

2. Troparevsky, M. C.; Sabau, A. S.; Lupini, A. R.; Zhang, Z. Optics Express 2010, 18 (24), 

24715-24721.

3. Wikipedia. Sellmeier equation. http://en.wikipedia.org/wiki/Sellmeier_equation (accessed

August 26, 2012).
15

List of Symbols, Abbreviations, and Acronyms

AR antireflection 

ARL U.S. Army Research Laboratory 

CaF2 calcium fluoride 

EO electro-optic 

GaAs gallium arsenide 

IR infrared 

MgF2 magnesium fluoride 

QDs quantum dots 

Si silicon 

Si3N4 silicon nitride

SiO2 silicon dioxide 

Ta2O5 tantalum pentoxide 

TiO2 titanium oxide 

Y2O3 yttrium oxide 
16

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