Microstructural design of 2D and 3D photonic …...II In the next step, 2D symmetric and asymmetric...

Microstructural Design of 2D and 3D Photonic Crystals via Topology Optimization

A thesis submitted in fulfilment of the requirements for the degree of

Doctor of Philosophy

Fei Meng

Faculty of Science, Engineering and Technology

Swinburne University of Technology

June 2018

ACKNOWLEDGEMENT

I would first like to thank my coordinating supervisor, Professor Xiaodong Huang for his

guidance throughout my studies in these years. Completing this project would be impossible without his suggestions and patience. I would also like to express the gratitude to my associate supervisor, Professor Baohua Jia. I have received many valuable advices from her.

I would like to thank Professor Mike Xie, Dr Shiwei Zhou, Dr Shanqing Xu, Dr Jianhu

Shen from RMIT University and Professor Guoxing Lu, Professor Dong Ruan, Dr Han Lin from Swinburne University of Technology for the beneficial discussions we have had.

I would also like to thank my fellow PhD students, Yang Fan Li, Shuo Li, Dingjie Lu,

Qiming Liu, Wei Chen, Zhaoxuan Zhang, Weibai Li for their friendship and help. I would like to thank my family for their support and encouragement in these years. Finally, I would like to thank the financial support from the Australian Research Council

(FT130101094 and DE120100291) and the China Scholarship Council (Student NO. 201306370093).

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PUBLICATION LIST

Journal Papers:

1. Meng Fei, Jia Baohua, Huang Xiaodong, Topology-Optimized 3D Photonic Structures

with Maximal Omnidirectional Bandgaps, Advanced Theory and Simulations, 2018,

1800122.

2. Li Shuo, Lin Han, Meng Fei, Moss David, Huang Xiaodong, Jia Baohua, On-Demand

Design of Tunable Complete Photonic Band Gaps based on Bloch Mode Analysis,

Scientific Reports, 2018, 8 (1), 14283.

3. Li Yang Fan, Meng Fei, Li Shuo, Jiao Baohua, Zhou Shiwei, Huang Xiaodong, Designing

broad phononic band gaps for in-plane modes. Physics Letters A, 2018, 382 (10), 679-684.

4. Meng Fei, Li Shuo, Li Yang Fan, Jia Baohua, Huang Xiaodong, Microstructural design

for 2D photonic crystals with large polarization-independent band gaps. Materials Letters,

2017, 207, 176-178.

5. Meng Fei, Li Yang fan, Li Shuo, Lin Han, Jia Baohua, Huang Xiaodong, Achieving large

band gaps in 2D symmetric and asymmetric photonic crystals, Journal of Lightwave

Technology, 2017, 35 (9), 1670-1676.

6. Chen Yafeng, Meng Fei, Sun Guangyong, Li Guangyao, Huang Xiaodong. Topological

design of phononic crystals for unidirectional acoustic transmission, Journal of Sound and

Vibration, 2017, 410, 103-123.

7. Li Shuo, Meng Fei, Lin Han, Huang Xiaodong, Jia Baohua, All-angle negative refraction

flatlens with a broad bandwidth, Photonics and Nanostructures - Fundamentals and

Applications, 2017, 27, 11-16.

8. Li Yang Fan, Meng Fei, Zhou Shiwei, Lu Minghui, Huang Xiaodong, Broadband All-

angle Negative Refraction by Optimized Phononic Crystals. Scientific Reports, 2017, 7,

7445.

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9. Zhang Zhaoxuan, Li Yang Fan, Meng Fei, Huang Xiaodong, Topological design of

phononic band gap crystals with sixfold symmetric hexagonal lattice. Computational

Materials Science, 2017, 139, 97-105.

10. Meng Fei, Li Shuo, Lin Han, Jia Baohua, Huang Xiaodong, Topology optimization of

photonic structures for all-angle negative refraction, Finite Elements in Analysis and

Design, 2016, 117–118, 46-56.

11. Li Yang fan, Huang Xiaodong, Meng Fei, Zhou Shiwei, Evolutionary topological design

for phononic band gap crystals, Structural and Multidisciplinary Optimization, 2016, 54

(3), 595–617.

12. Meng Fei, Huang Xiaodong, Jia Baohua, Bi-directional evolutionary optimization for

photonic band gap structures, Journal of Computational Physics, 2015, 302, 393-404.

Conference Papers:

1. Meng Fei, Li Shuo, Jia Baohua, Huang Xiaodong, Designing photonic crystals with

complete band gaps. in: The 7th International Conference on Computational Methods,

Berkeley, United States, 2016.

2. Meng Fei, Huang Xiaodong, Jia Baohua, A new topology optimization algorithm for

photonic band gap structures, in: The 11th World Congress on Structural and

Multidisciplinary Optimization, Sydney, Australia, 2015.

3. Li Yang fan, Huang Xiaodong, Meng Fei, Zhou Shiwei, Topology optimization of 2D

phononic band gap crystals based on BESO methods, in: The 11th World Congress on

Structural and Multidisciplinary Optimization, Sydney, Australia, 2015.

DECLARATION

I hereby declare that this thesis contains no material that has been submitted previously, in whole or in part, to qualify for any other award or degree. Moreover, any other material taken from other people’s work included in this thesis, published or otherwise, are fully acknowledged in accordance with standard referencing practices.

Fei Meng 2018

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ABSTRACT

Photonic crystals are periodic microstructures composed of two or more materials with

different electromagnetic properties. They can modulate electromagnetic waves with specific

wavelengths and generate unusual optical phenomena. Photonic crystals have shown a

promising future in medical treatment, telecommunication, national defense, and many other

applications. The property of photonic crystals relies on the space distribution of material. This

study develops a new approach to design the microstructure of photonic crystals based on finite

element analysis and the bi-directional evolutionary structural optimization (BESO) method.

Innovative 2D and 3D photonic crystal designs with maximized photonic band gaps, complete

band gaps and all angle negative refraction (AANR) ranges are obtained.

In the first part of this research, the finite element method for calculating the photonic band

diagram is developed. It is the foundation for the analysis and design of the 2D and 3D photonic

crystals. The master equations and transversality constraints are first presented. Solving the

master equations is an eigenvalue problem. After finite element discretization, the master

equations are rewritten in matrix format and solved for given wave vectors. The programs are

developed in MATLAB and verified by comparing the results to that of the plane wave

expansion method.

In the second stage, a new algorithm based on BESO is proposed to open and enlarge the

band gaps in the 2D photonic crystals for both the transverse electric (TE) and transverse

magnetic (TM) modes. The algorithm starts from a simple initial design without a band gap.

Through iterative finite element analysis and sensitivity analysis, BESO gradually re-distributes

dielectric materials within the unit cell so that the resulting photonic crystal possesses a

maximum gap between two specified adjacent bands. The proposed optimization algorithm can

successfully open band gaps from between bands 1~2 to bands 10~11 for both the TM and TE

modes. Some optimized structures exhibit novel patterns that are remarkably different from

traditional designs. The proposed algorithm is computationally efficient as the solution usually

converges within 100 iterations.

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In the next step, 2D symmetric and asymmetric photonic crystals are optimized using

BESO to study the influence of lattice symmetry on the size of maximum band gaps. For both

the TM and TE modes, square and hexagonal lattices are considered. The results show that the

band gaps of the asymmetric designs are larger than those of the symmetric ones. Moreover,

the largest TM band gap for photonic crystals with a square lattice achieves a comparable size

to photonic crystals with a hexagonal lattice, extending current state-of-the-art understanding.

The origin of wide band gaps and the unusual geometric characteristics of asymmetric photonic

crystals are discussed.

Complete band gaps in 2D photonic crystals are polarization-independent, but they are

challenging to achieve because photonic crystals for TE and TM polarizations have opposite

geometric characteristics. In this study, an efficient approach to design 2D photonic crystals

with large complete band gaps is proposed. An initial structure is first created by superposing a

TE band gap structure and a TM band gap structure so that the band number of the gap can be

determined. Then, BESO is utilized to enlarge the objective complete band gap. Some

innovative designs are obtained for both the square and hexagonal lattices—one of them

achieves the largest complete band gap ever reported. The potential application of these

optimization results in hollow-core photonic crystal fibers is discussed.

Negative refraction, especially AANR is essential for applications like flat lens. In this

stage, BESO is employed to design 2D photonic crystals with a maximized AANR frequency

range. An objective function is offered based on the definition of the upper and lower limit

frequency of AANR. Then, a BESO algorithm is proposed to generate AANR property in 2D

photonic crystals and to maximize its frequency range for both the TM and TE modes. The

numerical results show that the novel patterns have a much larger AANR frequency range than

the intuitive designs. The minimum permittivity contrast for the AANR in 2D photonic crystals

with a square lattice is 8.6 for the TE modes and 13.4 for the TM modes. Adopting a sensitivity

filter improves the manufacturability of the optimized designs.

In the final part of this study, the 3D photonic crystals with omnidirectional band gaps are

designed. Symmetric and asymmetric cubic lattices are considered. Band gaps between bands

2~3 to 16~17 are explored for silicon-air material. In contrast to previous research, this method

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acquires band gaps between any two neighbor bands for asymmetric cubic lattices and obtains

novel 3D photonic structures. Compared to the symmetric designs, the asymmetric results tend

to have larger band gaps. Among the 15 maximized band gaps, 13 are larger than 15 percent

and three are larger than 24 percent. The topology optimized designs have a uniform

bicontinuous characteristic that originates from the orthogonal distribution of electric fields at

the bounds of the band gaps. The distribution of the dielectric material follows the directions of

the wave vectors of the maximum eigenfrequencies at the lower bounds. Consequently, the

macroscopic effective property such as permittivity exhibits corresponding directionality. The

isotropy of the effective permittivity indicates the lower bound flatness, and photonic crystals

with large band gaps tend to have isotropic effective permittivity.

Keywords: structural topology optimization; bi-directional evolutionary structural

optimization; photonic crystal; microstructure; photonic band gap; negative refraction

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CONTENTS

ABSTRACT ........................................................................................................................... I

NOTATIONS ......................................................................................................................... I

CHAPTER 1 Introduction and Literature Review ...................................................................1

1.1 Research Background................................................................................................1

1.2 Structural Topology Optimization .............................................................................1

1.2.1 Origin of topology optimization......................................................................1

1.2.2 Notions in structural topology optimization ....................................................4

1.2.3 Structural topology optimization methods .......................................................5

1.3 Photonic Crystal ...................................................................................................... 12

1.3.1 Development of photonic crystal .................................................................. 12

1.3.2 Properties and applications of photonic crystal ............................................. 14

1.4 Topology Optimization of Photonic Crystals ........................................................... 19

1.4.1 2D photonic band gap crystal........................................................................ 20

1.4.2 3D Photonic band gap crystal ....................................................................... 21

1.4.3 Photonic crystals with other properties ......................................................... 22

1.4.4 Asymmetric photonic crystals ....................................................................... 22

1.4.5 Numerical methods for photonic crystal research .......................................... 23

1.5 Content of the Thesis .............................................................................................. 24

1.5.1 Gap in knowledge ......................................................................................... 24

1.5.2 Project Objectives......................................................................................... 25

1.5.3 Organization of contents ............................................................................... 25

CHAPTER 2 Finite Element Method for Photonic Crystals .................................................. 27

2.1 Maxwell Equations and the Master Equations ......................................................... 27

2.2 FEM for 2D Photonic Crystals ................................................................................ 29

2.2.1 2D FEM formulation .................................................................................... 29

2.2.2 Verification of 2D FEM program.................................................................. 32

2.3 FEM for 3D Photonic Crystals ................................................................................ 33

2.3.1 3D FEM formulation .................................................................................... 33

2.3.2 Verification of 3D FEM program.................................................................. 37

2.4 Conclusions ............................................................................................................ 38

CHAPTER 3 Bi-Directional Evolutionary Optimization for 2D Photonic Band Gap Crystals ............................................................................................................................................. 40

II

3.1 BESO Formulation .................................................................................................. 40

3.1.1 Objective function ........................................................................................ 40

3.1.2 Design variable............................................................................................. 41

3.1.3 Sensitivity analysis ....................................................................................... 43

3.1.4 Mesh-independency filter ............................................................................. 44

3.1.5 Implementation of BESO.............................................................................. 45

3.2 Numerical Results and Discussion........................................................................... 47

3.3 Conclusions ............................................................................................................ 55

CHAPTER 4 Achieving Large Band Gaps in 2D Asymmetric Photonic Crystals .................. 56

4.1 Numerical Analysis and Topology Optimization ..................................................... 56


4.2.1 Topology optimization results ...................................................................... 58

4.2.2 Origin of large band gaps.............................................................................. 66

4.3 Conclusions ............................................................................................................ 69

CHAPTER 5 Topological Design for 2D Photonic Crystals with Large Complete Band Gaps ............................................................................................................................................. 71

5.1 Construction of Initial Design ................................................................................. 71



5.2.2 Potential application in the photonic crystal fiber .......................................... 78

5.3 Conclusions ............................................................................................................ 80

CHAPTER 6 Topology Optimization of 2D Photonic Crystals for All-Angle Negative Refraction ............................................................................................................................. 81

6.1 AANR Frequency Range......................................................................................... 81

6.2 Topology Optimization ........................................................................................... 85

6.2.1 Modified objective function .......................................................................... 85

6.2.2 Sensitivity analysis ....................................................................................... 86

6.2.3 Numerical implement of BESO .................................................................... 88


6.3.1 Optimized designs ........................................................................................ 89

6.3.2 Influence of the filter .................................................................................... 93

6.4 Conclusions ............................................................................................................ 94

CHAPTER 7 Topology Optimization of 3D Photonic Crystals with Large Omnidirectional Band Gaps ............................................................................................................................ 95

7.1 Topology Optimization Algorithm .......................................................................... 95


III


7.2.2 Geometric characteristics of 3D photonic crystals ....................................... 104

7.3 Conclusions .......................................................................................................... 107

CHAPTER 8 Conclusions .................................................................................................. 108

REFERENCES ................................................................................................................... 111

I

NOTATIONS

e elemental sensitivity

e modified elemental sensitivity ke~ historical averaged elemental sensitivity kth threshold of the sensitivities

ε relative permittivity

ε0 permittivity of vacuum

Θ solid area in the design domain,

∂Θ boundary of structure

λ lagrange multipliers

µ relative permeability

µ0 permeability of vacuum

v test function

ρ free charge

ρ(r) design variable/density function

t,r level-set function

ω frequency

ωl, ωu lower and upper limit of the AANR frequency range

ωbot, ωtop lower and upper of the band gap

Ω design domain/region of the unit cell (3D cases)

a lattice constant

A region of the unit cell (2D cases)

B magnetic induction fields

c light speed in vacuum

V volume fraction of material

D electric displacement fields

e nodal solution for the electric field

II

E elastic modulus

E electric field

E0 elastic modulus of solid material

ER evolution rate

f(X), F objective function

xf , yf partial derivatives of ω to k

xxf , yyf , xyf second-order partial derivatives of ω to k

fsk generalized gradient vectors

G constraint

h nodal solution for the magnetic field

H magnetic field

J current densities

k iteration number

k wave vector

K overall stiffness matrix

Kr reduced stiffness matrix

Ke elemental stiffness matrix

eexpK expanded elemental stiffness matrix

m band number

M overall mass matrix

Mr reduced mass matrix

Me elemental mass matrix

eexpM expanded elemental mass matrix

n total number of nodes

N shape function matrix

p penalty factor

r coordinates

rmin radius of sensitivity filter

R radius of EFC curvature

III

t time

u nodal solution for magnetic or electric field

v velocity

wi, wij, ijw weight factor

vg group velocity

xe design variable of one element

X design variable vector

Chapter 1 Introduction and Literature Review

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CHAPTER 1

Introduction and Literature Review

1.1 Research Background

Light is electromagnetic waves that convey energy and information. Controlling and using

light has been the dream of humanity since ancient times. Many optical devices have been

invented and developed to extract information or harvest energy, including the traditional

mirror, magnifying lens, modern radio telescope, and solar panel. These devices were designed

and fabricated from a variety of optical materials, ranging from glass, metal, and semiconductor,

to superconductor, metamaterial, and photonic crystal.

Photonic crystals are a unique kind of material with periodic microstructures. Their

property relies on the constitutive materials and their layout. Photonic crystals have been used

in devices, such as cavity, optical fiber, and waveguide. The unique structures of photonic

crystals give rise to these extraordinary applications, but also raises the question: How can the

distribution of materials be designed to achieve the desired performance? In this research, the

topology optimization method — a state-of-the-art structural design approach — will be used

to create the microstructure of photonic crystals.

In this chapter, the notion of structural topology optimization is first introduced. Several

popular topology optimization methods for continuum structures are reviewed. Then the origin

and development of photonic crystal and current research topics are introduced. The application

of structural topology optimization methods in the design of photonic crystals and other optical

devices are reviewed. Last, the objective, methodology, and contents of this research are

presented.

1.2 Structural Topology Optimization

1.2.1 Origin of topology optimization

Seeking the optimal design of a structure is an eternal question in structural engineering.

The word “optimal” may have multiple implications here, including best performance, lightest


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weight, or highest efficiency, and the design should satisfy some specific constraints. To

achieve the set goals, the design process usually starts from a simple or rough structure. After

computation and analysis of the structure, the initial design is modified and improved to achieve

a better design. Structural optimization can be generally classified into three categories: size,

shape, and topology. Size optimization searches for the best combination of some size

parameters, like the cross-section area of bars or thickness of shells, while shape optimization

modifies a shape and boundary of a design. The topology of an initial structure (e.g., the

connection of bars or the number of structural components) remains unchanged. Instead,

topology optimization is not confined by the initial structure and can change the connection

between bars or introduce new holes into a continuum structure. The result can be any topology,

shape, and size as long as it is within the design domain. Topology optimization provides the

most freedom in structural design. It can answer the basic question: How should material be

placed in a given domain to obtain the best performance of a structure that fulfills several

prescribed requirements?

(a) (b)

(c) (d) Fig. 1-1 Comparison of size optimization, shape optimization and topology optimization for continuum structure. (a) Initial structure. (b) Size optimization. (c) Shape optimization. (d) Topology optimization. A vertical load is exerted on the middle of the left end. The optimal structure takes up 50% of the design domain (dashed box).


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The concept of structural topology optimization originated from the design of truss

structures. In 1854, Maxwell 1 studied the computation of the internal force of truss structures

and optimized them accordingly. In 1904, Michell 2 conducted research to minimize the weight

of trusses with a single load under a stress constraint. Several rules, called the Michell criteria,

were suggested that optimal trusses should satisfy. The optimal trusses were named Michell

trusses and were composed of infinitely fine and intensive rods that had little practical value.

However, this research is still a milestone in the history of structural topology optimization.

From then on, new topology optimization methods are usually inspected by comparing them to

the Michell trusses 3-4. Many other scholars have also studied the optimization of trusses based

on Michell’s research 5-9.

Compared to trusses, continuum structures are difficult to describe by finite parameters.

Therefore, topology optimization for continuum structures did not develop until numerical

methods, especially the finite element method (FEM), emerged. In 1981, Cheng and Olhoff 10

minimized the compliance of a thin elastic plate using its thickness as design variables. They

found the phenomenon of mesh dependence in their optimization results and introduced the

notion of the microstructure. Bendsøe and Kikuchi 11 proposed the homogenization method in

1988. By introducing a model of a unit cell with holes, effective macroscopic material

parameters were obtained and the topology optimization problems were transferred to size

optimization problems. After that, many sophisticated topology optimization methods for

continuum structures have been developed.

Topology optimization initially sought the best structure with the maximum stiffness or

minimum compliance. However, today, the objective of structural topology optimization is not

solely restricted to optimizing the mechanical property of structures. It has expanded to study

thermal, fluid-flow, electromagnetic, optic and multi-physical problems. As a method that

combines physical analysis and mathematical programming, topology optimization is a rapidly

expanding research area in computer-aided engineering. With the development of computer

technology, topology optimization has become a powerful technique for engineers to find

optimal designs for both macro and microstructures.


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1.2.2 Notions in structural topology optimization

A typical structural optimization problem with a single material can be described as:

,...n,iρG

Vd

F F

i 10 1or 0

:Subject to

:Minimize*

Ω

rr

r

r

(1-1)

where Ω is the design domain. r is the coordinates, and the design variable ρ(r) represents the

material at r. This problem means searching the optimal distribution of material in design

domain Ω to minimize objective function F. The optimization result should satisfy several

requirements: (1) the volume of material equals to V*, (2) ρ for every point in the design domain

equals either 0 or 1, and (3) other constraints Gi ≤ 0.

The design domain refers to the space in which materials distribute in the optimization

process, such as the dashed box in Fig. 1-1. In practice, it is usually a two-dimensional (2D) or

three-dimensional (3D) region. The optimization algorithm cannot modify the structure outside

the design domain. The design domain is contained in the computation area and discretized

using the FEM or another numerical method. The design domain should be chosen according

to the practical need, rather than as large a domain as possible, it will result in the use of too

many computational resources or a loss in the accuracy.

The objective function is a function of the distribution of materials and is the aim of

topology optimization and the basis to evaluate optimization results. It is noteworthy that

topology optimization does not “create” an optimal design but searches the optimal design in a

solution space. Therefore, when the objective function is irrational or conflicts with the

constraints, topology optimization will yield unwanted results. The selection of the objective

function is a critical issue in topology optimization.

Design variables represent the distribution of materials in the design domain. They update

iteratively in the optimization process. After FEM discretization, for a single material

optimization problem, the material of an element can be reflected by a binary variable. ρ(r) can

be substituted by a vector composed of all the design variables. In an ideal case, design variables


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can only be 0 or 1 and the resulting structure is called a 0/1 design. However, intermediate

design variables have been employed in many optimization algorithms.

Constraints are the requirements that the optimization results must satisfy. They can be the

volume of material, displacement or stress at some points, or the overall performance of the

structure.

1.2.3 Structural topology optimization methods

Although many researchers have pursued different directions of topology optimization,

the difference between most methods is inconspicuous 12. In this thesis, several popular methods

including the solid isotropic microstructure with penalty (SIMP) method, genetic algorithm,

level-set method, and evolutionary methods are introduced.

(1) SIMP method

In 1989, Bendsøe 13 introduced intermediate design variables based on the homogenization

method. It was assumed that there is a nonlinear relationship between material property and

design variables, also referred to as “density.” and that virtual “intermediate material” has

property between the two constituent materials. The problem of searching for the best structure

became searching for a required density field. Afterward, Rozvany 14 proposed the notion of

SIMP because the nonlinear interpolation function will exert a penalty on the intermediate

design variables between 0 and 1. Consequently, the efficiency of the intermediate elements

will be suppressed and the final result tends to be a 0/1 solution.

Taking the elastic modulus as an example, the SIMP method uses the following

interpolation function:

0ExxE pee (1-2)

where E0 is the elastic modulus of the solid material, p is the penalty factor, and xe is the design

variable with an intermediate value. When p = 1, the optimization problem corresponds to

designing a thin plate with a variable thickness 15. When p > 1, Eqn. 1-2 exerts a penalty for the

intermediate elements and reduces their contribution to structural performance. As a result,


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these intermediate elements are pushed to void or solid elements in the subsequent optimization

process. There will be too many intermediate elements in the result if p is too small. Conversely,

if p is too large, the evolution process may rapidly converge to a local optimal 12.

In 1999, Bendsøe and Sigmund 16 combined the Hashin-Strikhman upper bound with the

homogenization theory and explained the physical meaning of the intermediate elements, which

had been a confusing problem in the SIMP method. Stolpe and Svanberg 17 suggested the

rational approximation for material properties method, which uses different interpolation

function to impel the optimization result to be a 0/1 solution. Bruns 18 proposed SINH method

which employs an interpolation function opposite to the SIMP method, to impose the penalty

on the intermediate elements. Rozvany 19 and Sigmund 20 put forward the continuation method

to increase the probability of convergence to a global optimal. In this method, the penalty factor

and sensitivity filter parameters are gradually adjusted during the optimization process. Further,

to solve the numerical instabilities in topology optimization, such as checkerboard patterns and

mesh-dependency 21, different sensitivity filters or density filters have been proposed 22-23.

These studies significantly prompt the study of the SIMP method. There are many free open

source programs available 24-25 for SIMP, and the method has been adopted in commercial

software like Optistruct and Tosca.

Fig. 1-2 The interpolation function in the SIMP method when penalty factor p equals to 1, 2, 3 and the Hashin-Strikhman upper bound 26. The latter one is the theoretical limit of elastic modulus when the volume fraction of solid material is xe. The elastic modulus is assumed to be 1 for the solid material and 0 for the void.


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(2) Genetic algorithm

The genetic algorithm was developed based on the “survival of the fittest” strategy from

the Darwinian evolution theory. It is a stochastic searching approach in which the core idea

seeks an optimal solution by simulating heredity and mutation in natural organisms. The genetic

algorithm was first suggested by John Holland 27 and applied in artificial intelligence systems 28. Today, many scholars use the genetic algorithm in structural topology optimization 29-32.

The genetic algorithm deals with the topology optimization problem as the optimization

of a series of discretized (usually binary) design variables. The combination of design variables

resembles the DNA of creatures and the “fitness” of the designs are evaluated by a holistic

analysis of the whole structure. At the beginning of the optimization, the initial designs are

randomly generated, the large number of which are necessary. Then, the algorithm evaluates

the objective function and assesses the fitness of the different designs. After the first round of

analysis, a couple of the best solutions are chosen to be the “parent” designs. The parent design

variables are then broken into segments and exchanged with each other. This stage is called

“crossover.” A series of children designs are obtained.

To ensure that the searching procedure explores the entire design space — in other words,

to ensure that the optimization result is the global optimal — a mutation procedure is used in

the genetic algorithm. Several design variables occasionally switch from 0 to 1 or 1 to 0, a

mutation procedure that brings different features from parent designs into children designs. The

degree of mutation is controlled by assigning a mutation probability, after which, the new

children designs are assessed. The break-crossover-mutation procedure will repeat iteratively

until the optimization process converges or reaches the prescribed maximum iteration number.

In gradient-based optimization methods like SIMP, the evolution of the design is directed

by elemental sensitivities. The structure changes from one design to another after an iteration.

However, for the genetic algorithm, several different designs are simultaneously analyzed in a

single step and more than one point in the design space is explored. Obviously, this feature

increases the probability to find the global optimal 33, making this method suitable for non-

convex problems. Moreover, in the optimization process, only the objective function needs to


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be evaluated, while other information like elemental sensitivities is unnecessary. Therefore, the

objective function can be discontinuous or non-differentiable.

Conversely, the drawback of the genetic algorithm is evident. Compared to the gradient-

based optimization methods, more than one design should be analyzed in one step and the

numerical analysis requires many computation resources 34. Further, the genetic algorithm

cannot guarantee the continuity and integrity of the resulting structure since the addition and

removal of materials are stochastic. Consequently, the numerical analysis may be non-

convergent or incorrect and the optimization procedure may fail 35. Some techniques have been

introduced to manage this problem, such as changing the isolated materials into a void 36 or

penalizing the unconnected elements 32.

(3) Level-set method

The mathematical concept of the level-set method was put forward by Osher and Sethian 37 in 1988. It was initially used to compute the merge and split of moving interfaces and was

later developed into a structural topology optimization method 38-39. The level-set method is

capable of explicitly describing the structural boundary, without any intermediate area or

checkerboard patterns. Today, the level-set method has been applied to solve various structural

optimization problems 40-44.

The name “level-set” comes from the function describing the boundary of structure that is

usually implicit and scalar without an analytic expression:

rrr

r 000

, t (1-3)

where Θ is the solid area in the design domain and ∂Θ is the boundary of the structure. For 2D

problems, the interface is an isoline defined by the value of t,r in the design domain. When

modeling the structure by FEM analysis, the discretized level-set definition is usually used. The

value of t,r is defined based on the position of the center of elements; it is positive when

the element is solid and negative when the element is void.


9

(a)

(b)

Fig. 1-3 (a) Typical level-set function 45. (b) The corresponding region with solid material.

In the optimization process, the structural boundary varies with the change of the level-set

function. The evolution of the interface is determined by appointing a series of velocity vectors

at several points on the interfaces. Taking the derivative of the objective function with pseudo-

time, using the chain rule, there is:

vdtd

t

r (1-4)

Eqn. 1-4 builds a relationship between the objective function and the velocity vector v. The

boundary of the optimal structure is the numerical solution of this differential equation.

Compared to the density method, which uses continuous design variables, the most

significant advantage of the level-set method is its clear imitation of the boundaries that makes

the recognition and fabrication of the structure easier. In the optimization process of the level-

set method, the interface can merge to evolve the topology. However, new holes cannot be

introduced into the existing structures. The most commonly used solution is creating an initial


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design with plenty of holes. In the following optimization process, these holes will merge and

evolve the structure. Some modified level-set methods have been developed to import new

holes 46-47. Generally speaking, the level-set method is mathematically more complicated and

is more difficult to apply.

(4) ESO/BESO method

The evolutionary structural optimization (ESO) method was proposed by Xie and Steven 48 in 1992. ESO is based on a simple idea that gradually removes low-efficiency materials step

by step, allowing the structure to evolve and be more efficient. The efficiency of the materials

is evaluated by mechanical data of structure such as the von-Mises stress. Later, the modified

ESO — bi-directional evolutionary structural optimization (BESO) method was proposed 49-50.

In BESO, materials can be added to the existing structure even after they are removed. In

ESO/BESO, the methods are easy to understand and apply but the early versions have many

defects, such as the optimization process is unstable and hard to converge. More importantly,

they are heuristic methods and lack a mathematical basis 21, 34, 51.

In the last decade, Huang and Xie 52 made great efforts to improve BESO such as through

the introduction of material interpolation functions and sensitivity analysis based on objective

functions and the so-called soft-kill BESO 53. By comparing BESO with other methods, some

of its controversies like the Zhou-Rozvany problem and optimization with a displacement

constraint were clarified 54. These works greatly improved the BESO method. BESO can obtain

mesh-independent results, and the stability and convergence of the optimization process have

been significantly enhanced, making the modern BESO completely different from its early

version while keeping it easy to understand and use.

BESO is based on the FEM analysis of structures. After the discretization of the design

domain, the binary design variables are employed to represent the material of each element. For

hard-kill BESO, the design variable can be either 0 or 1, while for soft-kill BESO, the design

variables can be a small value (like 10-3) or 1. BESO evaluates the efficiency of each element

by computing the elemental sensitivities that are the derivatives of the objective function with

regards to each design variable. BESO then updates the design variables according to the


11

volume constraint of the structures. The design variables of the elements with high sensitivities

are raised while the design variables of the elements with low sensitivities are reduced. In this

way, the new design will approach an optimal structure. BESO uses a sensitivity filter to

eliminate the checkerboard patterns and ensure the optimization results are mesh-independent

and averages the elemental sensitivity with historical values to increase the stability of the

optimization process.

(a)

(b) (c)

Fig. 1-4 (a) Illustration of sensitivity filter 55. Sensitivities of all the elements in the circle area are weighted averaged. The value is taken as the sensitivity for the element at the center of the circle. rmin is the radius of the filter. (b) Topology optimization result for the question in Fig. 1-1 using BESO without sensitivity filter. There are many checkerboard structures. (c) Topology optimization result for the same structure using BESO with sensitivity filter. Checkerboard patterns and the other fine structures have been successfully eliminated.

BESO has proven to be effective and reliable for various structural optimization problems.

The current BESO method is capable of generating practical structures for optimization

problems such as natural frequency 56, compliant mechanisms 53, 57 and nonlinear structures 58.

In recent years, the BESO algorithm has been extended to design the microstructures of

materials with extreme mechanical properties 59-61 and electromagnetic properties 62.


12

1.3 Photonic Crystal

1.3.1 Development of photonic crystal

Periodic structures affect the propagation of electromagnetic waves (photons) in a similar

way that the periodic potential in a semiconductor affects electrons. Wave scattering from

periodic structures has been known for a long time. One-dimensional (1D) photonic crystals

are the simplest form of photonic crystals that consist of periodic multi-layer films with

different dielectric constants. In 1887, Lord Rayleigh analyzed the optical property of 1D

photonic crystals and found that such structures have a frequency range of high reflectivity.

This phenomenon is now known as the 1D photonic band gap.

Later, in 1914, Darwin studied the dynamical theory of X-ray diffraction 63. Bloch 64

investigated the motion of an electron in a crystal lattice by employing a periodic force field.

Abelès 65 studied the propagation of electromagnetic waves in periodic media. Flouquet-Bloch

waves were then used to investigate the propagation of electromagnetic waves in stratified

media 66-69. The notion of the photonic band diagram (or band structure) was introduced into

these analyses.

In 1987, Yablnovitch 70 proposed that spontaneous emissions can be inhibited

omnidirectionally using 3D periodic dielectric structures, while John 71 found that the

phenomenon that electrons in semiconductors tend to localize at small energies also appeared

for photons in 3D periodic structures with a certain degree of disorder. Thus, the band gap

concept was extended to 3D cases and the term “photonic crystal” was first used.

However, designing and fabricating 3D photonic crystals is more difficult than first

thought by the pioneers. Yablnovitch and Gmitter 72 proposed a 3D face-centered-cubic

structure to realize the band gap, but this design was later proved through full vector wave

calculation to have no true band gaps despite having a deep density of states 73-74. It seemed that

the photonic crystal had come to an end. At the close of 1990, it was claimed in Nature that the

photonic crystal had “bite the dust” 75. However, at almost the same time, Ho et al. 76 proved

the existence of the photonic band gap in a diamond structure that consisted of dielectric spheres

for refractive-index contrasts as low as two.


13

After verification of the 3D photonic crystals, the concept was extended to 2D cases by

Plihal et al. 77-78 in 1991 and then experimentally measured 79-80. In the same year, Yablonovitch

et al. 81 reported that local electromagnetic modes will occur within the band gap when the

perfect periodicity of photonic crystals is broken by a local defect. These pioneering works

prompted many subsequent studies in the following decades about the mechanism, optical

property, fabrication, and application of photonic crystals.

(a) (b)

(c)

Fig. 1-5 Artificial photonic crystals. (a) 1D photonic crystal (Bragg mirror) 82. (b) 2D photonic crystal 83. The pillars are created by nanoimprint lithography. (c) 3D photonic crystal 84.

Although photonic crystal research originated from artificial structures, it has been

recognized that periodic nanostructures are abundant in the natural world. For example, opal (a

national gemstone of Australia) consists of a structure of closely packed spheres of silicon

dioxide (SiO2) 85-86 that diffracts light and shows many colors 87. In the animal world, photonic

crystal structures are also found in various species, such as weevil 88-89, beetles 90-91, moth 92,

butterfly 93-95, and peacock 96-97. The abnormal structures in their feathers, shells, villus, or skins


14

give these animals bright iridescent colors. Some other species like jellyfish 98 and chameleon 99 can even actively change their color. These systems that have evolved for biological purposes

provide us with numerous inspiration and design protocols.

(a) (b)

(c)

Fig. 1-6 Photonic crystals in the natural world. (a) Opal. (b) Feather of a peacock. (c) Wings of butterfly 100.

Now, photonic crystals are defined as microstructures that are periodic in one, two or three

dimensions, although the 1D form is usually called Bragg mirror. In contrast to metamaterials,

photonic crystals can only affect electromagnetic waves that have wavelengths comparable to

their lattice constant. Photonic crystals consist of two or more materials that can be metal,

dielectric material and, air or vacuum. A high enough contrast between dielectric indices of

components is necessary.

1.3.2 Properties and applications of photonic crystal


15

Due to their unique structures and optical properties because of their different constituent

materials, photonic crystals are capable of manipulating electromagnetic waves and

demonstrate unusual optical properties.

(1) Photonic band gap

The behavior of the electromagnetic waves in photonic crystals depends on their

frequencies. Frequencies that can propagate form modes, from which the groups of allowed

modes form photonic bands. The range of disallowed frequencies is named photonic band gaps.

Within the frequency range of the photonic band gaps, light waves are entirely prohibited 101-

103. The unique photonic band structures and band gaps are the foundations of many other

optical properties and give rise to many applications.

The most straightforward form of photonic crystal, the 1D photonic band gap crystal, also

has surprising properties. The layered structure can be designed to reflect light incidents from

any angle with any polarization, forming an omnidirectional reflector, but in general, the

reflection is only for near-normal incidence. They are widely used in modern optic instruments,

such as highly reflective mirrors in vertical-cavity surface-emitting lasers 104, Bragg grating in

optical fibers 105, and the confining layers in LEDs 106.

Compared to 1D photonic crystals, 2D and 3D photonic band gap crystals are more

complicated, resulting in more diverse applications like cavities, photonic crystal fibers, and

waveguides. For 2D photonic crystals, there are two polarization modes: transverse magnetic

(TM) and transverse electric (TE). In the TM modes, the electric field is perpendicular to the

crystal cross-section, while in the TE modes the magnetic field is perpendicular to the crystal

cross-section. Band gaps that work for both polarizations are named complete band gaps.

Optical cavities are the major components of laser devices. They amplify a light source by

confining electromagnetic energy in a small space. Traditionally, these cavities consist of two

opposing flat mirrors. For 2D or 3D photonic crystals, a point defect can be created and a

localized mode around the defect can be induced by removing one or several components in the

periodic structures. The photonic crystal around the defect reflects light with frequencies in the

band gap and confines them to a small region where they cannot escape. Photonic crystal


16

cavities can be designed to have an ultra-high quality factor 102, 107-108. These cavities can be

used to control and extract spontaneous emission 109-111, concentrate light 112, or as nanoscale

filters 113 and sensors 114-115.

For 2D cavities, emissions in the third direction are prevented by using two metallic plates

to construct a sandwich structure — if the transmission of light in the third direction is not

confined, it will transport along the point defect. Photonic crystal fibers are designed based on

this mechanism. A traditional optical fiber consists of a central core surrounded by the cladding

with a lower dielectric constant and the light is confined by an index-guiding mechanism.

Conversely, photonic band gap fibers confine light using band gaps that transport the light

within a hollow core, minimizing the loss and unwanted interference that can be induced by

solid materials. Photonic crystal fibers can also be designed with cladding composed of circular,

concentric 1D photonic crystal layers 116, known as Bragg fibers. The endless single-mode

property of photonic crystal fibers is critical for long-distance optical communication 117-120. By

filling the air void in cladding with special materials such as liquid crystal, the fiber properties

can be tuned through electricity or temperature 121-123. Photonic crystal fiber currently plays an

important role in many areas like sensors 124-126, high-power laser transmission 127-130 and non-

linear devices 131-133. For example, hollow-core photonic crystal fibers have already been used

in surgery to guide surgical lasers and remove tumors 134.

A waveguide is a structure that can guide the flow of electromagnetic waves. Conventional

waveguides are based on the total internal reflection mechanism, meaning the light is confined

to the high dielectric materials and has a strong interaction with materials. Light in photonic

crystals can be trapped by a point defect. If a linear defect is introduced, through the removal

of several adjacent units then, similar to the hollow-core photonic crystal fibers, the light within

the frequency range of the band gap is trapped and guided along the air defect. Waveguides can

be realized in both 2D and 3D photonic crystals. The most significant difference between

photonic crystal waveguides and conventional index-guiding waveguides is that the radiation

losses at sharp corners or imperfections can be effectively eliminated 135-136. This property is

key in many optical or electronic devices, such as on-chip photonic integrated circuits 137-138,


17

3D cavity waveguide systems 139, frequency splitters 140-142, polarization splitters 143-144, optical

switches 145 and optical sensors 146.

(2) Negative refraction

Negative refraction is a phenomenon in which a light beam travels to an interface of two

materials and is refracted to the direction at the same side of the surface as the incident beam.

This is opposite to what is normally expected. Theoretical analysis of negative refraction

originated from the investigation of “left-handed material” that had both a negative permittivity

and negative permeability, such as that predicted by Veselago in the 1960s 147. Its unique

property gives rise to some interesting applications, such as superlens 148. Although materials

with negative refractive indices do not exist in nature, Pendry et al. 149, Smith et al. 150 and

Shelby et al. 151 have successfully constructed negative refractive metamaterials constituted by

periodic metallic resonant structures. These composites are capable of working at the

microwave region, which has a wavelength much larger than the periodicity of the structure. In

recent years, the available frequency of metamaterials has expanded to visible or higher

frequency regions 152-155. However, the high loss caused by resonance and intrinsic fabrication

limitations are still significant difficulties in the design of metamaterials.

For photonic crystals, negative refraction can be realized using their unique dispersion

properties. Luo et al. 156 described the all-angle negative refraction (AANR) effect in 2D

photonic crystals as that the effective refractive index remains positive. The diffraction in

photonic crystals may result in one of the components of the group velocity propagating the

opposite direction of the wave vector k, inducing negative refraction. Thus, negative refraction

generates in photonic crystals even without negative indices. Many researchers have

investigated negative refraction in photonic crystals 157-161. Compared to metamaterials,

photonic crystals can interfere with light with wavelengths comparable to its lattice constant,

meaning they can work at higher frequencies such as optical or infrared regions. Further, typical

photonic crystals composed of dielectric materials have a lower absorption in high-frequency

regions 103. Negative refraction in photonic crystals can be utilized to achieve a superprism

effect 162-164 and perfect imaging 160-161, 165-166.


18

(3) Other properties and applications

Besides the properties mentioned above, photonic crystals can have other functions such

as slow-light and self-collimation. To achieve maximized confinement, the frequency of the

light should be set at the middle frequency of the band gap. Conversely, the interaction between

the light and material can be enhanced for frequencies at the edges of photonic band gaps, in

which the bands tend to be flat and the band slope approaches zero. The slope can be interpreted

as the group velocity at which the light energy transmits in the waveguide, a phenomenon called

slow-light 167. Light can be delayed or even temporarily stored in all-optical memories 168-170.

This phenomenon can be utilized to study nonlinear optics 171-173, and design highly efficient

optical devices 174-175.

The way light propagation in photonic crystals is being modulated is determined by the

dispersion properties of the crystals. Similar to the mechanism of negative refraction, when the

dispersion surface is flat, incident light from different angles will be refracted to the same

direction and collimated. In 1999, Kosaka et al. 176 observed collimated light propagation by a

3D photonic crystal, and self-collimating effects together with negative refraction have been

subsequently studied in various aspects 163, 177-179.

In practice, 1D, 2D and 3D photonic crystals are considered key components to enhance

the performance of optical devices like LEDs 180-182 and solar cells 183-185. Photonic crystals

have also been used in the research of fundamental physics. For example, topological photonic

crystals — an analog of the electronic topological insulator — have attracted much attention in

recent years 186-191. The realization of macroscopic quantum coherence at room temperature is

one of the holy grails in quantum physics 192, and Bose-Einstein condensation and superfluidity

have been achieved recently with the help of 1D and 3D photonic band gap crystals for excitons

and polaritons 193-195.


19

(a) (b)

(c)

(d)

Fig. 1-7 Some properties and applications of photonic crystals. (a) Waveguide bend 196. (b) Hollow-core photonic crystal fiber 130. (c) Slow-light waveguide with kagome-lattice 197. (d) Negative refraction flat lens 198.

1.4 Topology Optimization of Photonic Crystals

The traditional design approach of photonic crystals is a trial-and-error process based on

physical intuitions and parametric studies 199-205. This process is inefficient and time-consuming,

and the resulting design may deviate from the optimum. The property of photonic crystals stems

from their constituent materials and spatial distribution 103. For the given materials, the design

of photonic crystals becomes a typical topology optimization question. The systematic way to


20

find the optimal design of photonic crystals is to formulate the problem with appropriate

objective functions and constraints and then solving it by topology optimization methods.

1.4.1 2D photonic band gap crystal

In the particular frequency range of photonic band gaps, the propagation of

electromagnetic waves is entirely prohibited. In practice, a broader band gap range means a

broader available bandwidth of signals and applications that are of great significance in the

design of photonic crystals with large band gaps. Many researchers have investigated the design

of band gap structures, especially for 2D photonic crystals, using various topology optimization

methods.

Cox and Dobson used a gradient-based topology optimization algorithm to maximize the

band gaps in 2D photonic crystals for both the TM 206 and TE modes 207. Kao et al. 208 employed

the level-set method to design 2D photonic crystals for both TM and TE modes. He et al. 209

used the level-set method to discover the optimal shapes for the maximal band gaps in photonic

crystals. Sigmund and Jensen 210-211 proposed a two-stage optimization in which the first stage

used coarse grids to generate the five best topologically different candidates for each band. In

the second stage, the SIMP method was employed using the candidate topologies with an

opening band gap to find the optimal band gap structures. Men et al. 212-213 systematically

reported the design of 2D photonic crystals using a semidefinite programming and subspace

method. The optimization starts from a series of randomly generated initial designs to obtain

the band gaps but without a guarantee of the occurrence. Takezawa and Kitamura 214 developed

the phase field method for shape optimization of 2D photonic crystal and metamaterial.

For 2D photonic crystals, there are two possible light wave polarizations. A photonic band

gap is usually called a complete band gap if both the TM and TE waves are prohibited. A

complete band gap is more practical because it is independent of the polarization of light.

However, the formation of a TM and TE band gap has different mechanisms 215. For the TM

band gap, the Mie resonance is the dominating mechanism, while for the TE band gap, the

Bragg scattering is more significant. Consequently, photonic band gap crystals for both the TM

and TE modes show different structural characters 103, 210, 212, 216 and are more difficult to design.


21

Wen et al. 217, Wang et al. 202, and Shi et al. 203 designed and optimized the complex

structures of photonic crystals with complete band gaps through parameter analysis. Zhang et

al. 218 realized complete photonic band gaps and tunable self-collimation in 2D plasma photonic

crystals. Shen et al. 219 used the genetic algorithm to design a large complete band gap in 2D

photonic crystals. Goh et al. 220 designed both 1D and 2D photonic crystals by genetic algorithm,

including 2D photonic crystals with dual TE and TM modes band gap. Li et al. 221 designed 2D

photonic crystals with complete band gaps that consisted of circular rods with different radii

arranged asymmetrically. Jia et al. 222 optimized a 2D photonic crystal slab with a complete

band gap using the genetic algorithm. Men et al. 213 obtained 2D photonic crystals with multiple

and complete band gaps using the subspace method. Jia and Thomas 223 obtained large complete

photonic band gaps by superposing two substructures of 2D photonic crystals. Cheng and Yang 224 maximized complete band gaps for 2D photonic crystals with square lattices using the level-

set method

1.4.2 3D Photonic band gap crystal

Following the verification of photonic band gaps in a 3D photonic crystal consisting of

dielectric spheres 76, previous researchers have devised many interesting designs, such as

woodpile 225-228, layered structures 229-230, spiral structures 231-232, continuous framework 233-235

(including the gyroid structure), level-set models 236-237, and inverse opal 238-239. These “hand-

design” 3D photonic crystals are mainly based on physical intuitions together with parameter

analysis.

Compared to their 2D counterparts, 3D photonic crystals have more potential in practical

application because they can control electromagnetic waves propagating in all directions.

However, applying the topology optimization methods above in the design of 3D photonic

crystals is not an easy task —3D photonic crystals have more degrees of freedom and the

Brillouin zone is more complex. Although the topology optimization of 2D photonic crystals is

mature, their application in 3D photonic crystals is rare. James et al. 240 developed a genetic

algorithm to design photonic crystals and obtained the classic diamond structure. Men et al. 241

combined the subspace method with the MIT Photonic Bands (MPB) software to search the


22

optimal band gap in simple cubic (SC), body-centered cubic (BCC) and face-centered cubic

(FCC) lattices. The results show some improvement upon traditional designs. Lu et al. 242

investigated the mechanic counterpart of photonic crystal — phononic crystal — by using a

gradient based topology optimization method. 3D phononic crystals with omnidirectional band

gaps for the SC, BCC and FCC lattices were designed.

1.4.3 Photonic crystals with other properties

Except for the band gap phenomenon, topology optimization methods have been applied

to achieve other functions. Borel et al. 196, Jensen and Sigmund 243, and Wang et al. 244 designed

a photonic crystal waveguide bend using the SIMP method. Matzen et al. 245 and Saucer and

Sih 246 designed photonic crystal cavities using topology optimization. Matzen et al. 247

designed slow-light photonic waveguides. Yang et al. 248 enhanced the slow-light coupling

efficiency in photonic crystal waveguides via topology optimization. Florescu et al. 249 designed

disordered 2D photonic crystal with complete band gaps. Wang et al. 250 achieved a highly

efficient light trapping photonic crystal structure using the genetic algorithm. Piggott et al. 251

and Louise et al. 252 designed a broadband on-chip wavelength multiplexer and demultiplexer.

Shen et al. 253 designed an integrated-nanophotonics polarization beamsplitter. Lin et al. 111

formed third-order Dirac points to enhance spontaneous emissions in photonic crystals. Zhou

et al. 254-255, Huang et al. 62, and Yamasaki et al. 256 designed metallic metamaterials with

negative refraction properties and maximized electromagnetic parameters (i.e., permittivity and

permeability).

1.4.4 Asymmetric photonic crystals

In the current research into photonic crystal optimization, the lattices are considered

symmetric so that the band diagram can be calculated by only evaluating the wave vectors along

the boundary of the irreducible Brillouin zone 103. This reduces the total number of design

variables and minimizes the computational costs, However, from an optimization point of view,

the symmetric requirement exerts an extra geometric constraint on the optimization problem

and limits the designable space to achieve the targeted properties.


23

To study the influence of symmetry, Preble and Lipson 257 proposed an asymmetric

photonic crystal for TE polarization using an evolutionary algorithm and demonstrated a larger

band gap with asymmetric designs. Similar designs were also reported in 258-260. Wen et al. 217

designed photonic crystal slabs with complete band gaps by reducing their symmetry. Ohlinger

et al. 261 maximized complete band gaps in the asymmetric rectangular lattice. Jia and Thomas 223 put forward a set of 2D aperiodic and asymmetric photonic structures by superposing two

substructures with different lattices. Jiang et al. 262 designed a photonic crystal slab comprised

of circular silicon rods within an asymmetric square lattice. Giden et al. 179 designed a star-

shaped photonic crystal to realize super-collimation over a broad bandwidth. Dong et al. 263-264

found some phononic structures for square and hexagonal unit cells with a reduced symmetry.

1.4.5 Numerical methods for photonic crystal research

To design photonic crystals, a proper method is necessary to compute its dispersion

properties. Due to the complex structures in the topology optimization process, numerical

methods are the only choice. The computation problem in photonic crystal research can be

classified into three kinds: frequency-domain eigenvalue problems, frequency-domain

responses and time-domain simulations 103. Several numerical methods have been developed to

solve these problems, including the FEM 265-268, plane wave expansion (PWE) method 269-271

and finite-difference time-domain method 272-273. Some commercial software like COMSOL 274,

HFSS, and MPB have been developed to analyze photonic crystals and other electromagnetic

problems.

The band diagram is an important tool to reveal the dispersion properties of photonic

crystals. In this research, the band diagram should be obtained first to design the photonic

crystal microstructures of photonic crystals. The Maxwell equation should be transformed into

an eigenvalue problem and solved using the frequency-domain eigensolver. The PWE method

and corresponding MPB software are popular in calculating the band diagram of photonic

crystals. However, for the PWE method, the Fourier transform converges slowly in photonic

crystals with discontinuous dielectric structures. Interpolation schemes are usually introduced

to smooth the sharp dielectric interfaces. Further, the PWE method cannot manage metallic


24

materials and aperiodic structures 275. The FEM is more flexible compared to the PWE method

and is not sensitive to the continuities of the material distributions, which is required for

simulating the diverse abnormal structures in the topology optimization process. With coarse

discretization, the FEM usually obtains a better accuracy than the PWE method. The main

drawback of the FEM is the high demand for computing resources 275.

1.5 Content of the Thesis

1.5.1 Gap in knowledge

Much research has demonstrated that topology optimization methods possess a powerful

capability to design innovative photonic crystal structures and optical devices with extreme or

exotic properties. These methods are attractive and have successfully obtained the optimized

solutions of different optical materials and devices. However, sometimes the computational

cost for the optimization is expensive because hundreds — even thousands — of iterations are

inevitable. Therefore, it is important to attempt new and different optimization algorithms such

as BESO to explore the topological designs of photonic crystals.

For 2D photonic band gap crystals, the previous research has indicated that the

optimization results depend highly on the parameters and algorithms used in the optimization

process. In most cases, there is no unique solution for the design of photonic crystals. A

systemic topology optimization method for photonic structures with band gap is still a challenge

because of the complicated nature of the solution space. The topology optimization for complete

band gaps is even harder because of the opposite structural characteristics between the TE and

TM mode designs. Since a simple or randomly generated structure is usually taken as the initial

design 213, 216, many iterations are inevitable, and the computational cost is consequently

significant. Only a few complete band gap designs have been so far reported. Further, although

there are several reports on the influence of symmetry in photonic crystals, no systematic study

has been done. Both symmetric and asymmetric photonic crystals should be considered in this

research.

For 3D photonic band gap crystals, the topology optimization research is relatively rare.

The optimization results show improvements over traditional designs, but many of the designs


25

in the previous research are similar or variants of each other. 3D photonic crystal designs with

large omnidirectional band gaps are limited. Provided that omnidirectional band gaps could

occur at any neighbor bands, innovative designs of 3D photonic crystals are still required for

the further theoretical study and various applications.

In terms of the AANR property of photonic crystals, most research has so far focused on

the mechanisms and experiments of AANR in photonic crystals. It is of significant importance

to design low-absorption photonic crystals with a broad AANR frequency range that can result

in a large available bandwidth or can lower the monochromaticity requirement of light sources.

The AANR property possibly occurs by rationally designing the photonic crystal structures.

However, the topology optimization of AANR in photonic crystals has not been reported yet.

1.5.2 Project Objectives

Photonic crystals have shown a promising future in medical treatment, telecommunication,

national defense, and many other applications. The property of photonic crystals relies on the

spatial distribution of materials. The main aim of this research is to propose an efficient,

effective and systematic method to design 2D and 3D photonic crystals for various optical

properties.

In this research, the mathematical formulation and program for analyzing photonic crystals

are developed. The BESO method is modified to optimize the photonic crystal on the basis of

FEM analysis of the periodic structures. Focusing on band gap and negative refraction

properties, the microstructure of 2D and 3D photonic crystals are designed. 2D photonic

crystals with maximized TE band gaps, TM band gaps, and complete band gaps, 2D photonic

crystals with AANR property, 3D photonic crystals with large band gaps are designed.

Furthermore, the symmetry of the photonic crystal structures will be discussed. The physical

origin of the complicated topology optimized designs is then revealed.

1.5.3 Organization of contents

This thesis is organized as follows:

Chapter 1: Introduce structural topology optimization methods. Illustrate the notion,

history, properties and applications of photonic crystals. Point out the aims of this research.


26

Chapter 2: Present the FEM formulation for 2D and 3D photonic crystals. Compile the

corresponding program using MATLAB.

Chapter 3: Put forward a new optimization algorithm to design 2D photonic band gap

crystals for both TM and TE modes.

Chapter 4: Study the influence of symmetry on photonic band gaps. Analyze the unusual

geometric characteristics of asymmetric photonic crystals.

Chapter 5: Propose a new approach to design 2D photonic crystals with large complete

band gaps. Explore their potential application in photonic crystal fibers.

Chapter 6: Propose an algorithm to generate negative refraction in 2D photonic crystal and

then maximize the AANR frequency range.

Chapter 7: Design 3D photonic crystals with omnidirectional band gaps. Conclude the

geometric characteristics.

Chapter 8: Conclude the main achievements of this research, and discover the future of

topology optimization for photonic crystals.

Chapter 2 Finite Element Method for Photonic Crystals

27

CHAPTER 2

Finite Element Method for Photonic Crystals

In this chapter, the FEM formulations for analyzing 2D and 3D photonic crystals are

deduced. The corresponding programs are compiled using MATLAB and then verified by

comparing the results with the PWE method. These works will lay the foundation for the

optimization of photonic crystals.

2.1 Maxwell Equations and the Master Equations

All the macroscopic electromagnetism, including the propagation of electromagnetic

waves in photonic crystals, is generally governed by the macroscopic Maxwell’s equations 103:

0 B (2-1a)

D (2-1b)

0

tBrE (2-1c)

JDrH

t (2-1d)

where D and B are the displacements and magnetic induction fields. E and H are the electric

and magnetic fields respectively. ρ and J are the free charges and current densities.

To simplify Eqn. 2-1, several assumptions are made in this research103: (1) The structure

of photonic crystal does not change with time. (2) There are no sources of light in the photonic

crystal, therefore ρ = 0 and J = 0. (3) The photonic crystals are periodic structures consist of

linear, lossless, homogeneous and isotropic dielectric materials and air. The relative

permittivity of material ε is real and positive. (4) The material dispersion (the property that ε

changes with light frequency) is neglected. The value of dielectric constants is chosen according

to the considered frequency range. (5) The field strengths are small enough so that D(r) =

ε0ε(r)E(r), B(r) = µ0µ(r)H(r). ε0 and µ0 are the permittivity and permeability of vacuum. The


28

relative magnetic permeability µ ≈ 1 for most dielectric materials, therefore B = µ0H. ε(r) is a

scalar dielectric function.

Take these assumptions into account, the Maxwell equations become

0, trH (2-2a)

0, trEr (2-2b)

0,, 0

ttt rHrE (2-2c)

0,, 0

ttt rErrH (2-2d)

E and H are complicated functions of both time and space. Using Fourier analysis, a

solution of E and H can be built by the combination of a set of harmonic modes as a spatial

pattern multiplies a complex exponential. For mathematical convenience, a complex-valued

field is employed and then its real part is used to recover the physical fields.

tiet rErE , (2-3a)

tiet rHrH , (2-3b)

Substitute Eqns. 2-3 into Eqns. 2-2, the four Maxwell equations become two divergence

equations and two curl equations 103:

0 rH (2-4a)

0 rEr (2-4b)

00 rHrE i (2-4c)

00 rErrH i (2-4d)

where r = (x, y, z) denotes the space coordinates. ω is the frequency of a harmonic mode.

Combining 2-4(c) and 2-4(d), the master equations entirely in H(r) or E(r) are

rHrHr

21

c

(2-5a)

rErrE

2

c (2-5b)


29

where c is the light speed in vacuum, 1/c2 = ε0μ0.

Solving Eqs. 2-5(a) and 2-5(b) as eigenvalue problems, the eigenfrequencies and

corresponding electric or magnetic field can be obtained. In the calculation, the transversality

requirements 2-4(a) and 2-4(b) must be enforced. Noticing that the divergence of a curl always

equals to 0, there is actually only one requirement for each master equation (Eq. 2-4(a) for

solving H, Eq. 2-4(b) for solving E).

In 2D cases, light waves are decomposed to two polarizations. One of the electric and

magnetic fields is set perpendicular to the plane of wave propagation while the other one is

parallel to this plane. Since the electromagnetic field and ε(r) are unchanged on the vertical

direction, the transversality requirements are naturally satisfied. However, for 3D photonic

crystals, this convenience doesn’t exist anymore because the structure of photonic crystals may

vary in three directions. As a result, the transversality requirements should be implemented as

extra constraints.

2.2 FEM for 2D Photonic Crystals

2.2.1 2D FEM formulation

There are two possible polarizations of the magnetic and electric fields for 2D cases. In

the TM modes, the magnetic field is confined to the plane of wave propagation and the electric

field E = (0, 0, E(k, r)) is perpendicular to this plane, where the vector r=(x, y) denotes the

coordinates in the plane and k is the wave vector. In contrast, the electric field of the TE modes

is confined to the plane of wave propagation and the magnetic field H = (0, 0, H(k, r)) is

perpendicular to this plane. Since there are no point sources or sinks of electric displacement

and magnetic fields in the photonic crystal, the time-harmonic Maxwell equations can be

reduced to two decoupled equations as

),()(),(2

rkrrk Ec

E

for TM mode (2-6a)

),(),()(

1 2

rkrkr

Hc

H

for TE mode (2-6b)


30

Due to the periodicity of the photonic crystal, the dielectric function satisfies ε(r) = ε(r+R), and

E(k, r) = E(k, r+R), H(k, r) = E(k, r+R). R is the lattice translation vector.

According to the Bloch-Floquet theory 276, magnetic and electric fields can be represented

as the product of a periodic function and an exponential factor as

rkrrk ieEE , for TM modes (2-7a)

rkrrk ieHH , for TE modes (2-7b)

Thus, the governing equations can be converted to eigenvalue problems within a unit cell

as

rrkk Ec

Eii2

for TM modes (2-8a)

rrkk Hc

Hii21

for TE modes (2-8b)

For a given wave vector k = (kx, ky), Eqns. 2-8(a) and 2-8(b) are typical eigenvalue

problems and can be solved by the finite element method. Multiplying an arbitrary test function

on both side and taking the integral over the unit cell lead to their weak forms as

0

22

2

2

2

2

dAEc

Eky

Eikx

Eik

Eiky

Eikxy

ExEv

yx

A yx

rrrr

rrrr

(2-9a)

0111

1111

22

dAHc

Hky

Hikx

Hik

Hiky

Hikxy

Hyx

Hx

v

yx

A yx

rrrr

rrrr

(2-9b)

where v is the test function, A is the region of the unit cell.

To conduct finite element analysis, the unit cell is discretized into finite elements. The

components E(r) and H(r) are expressed as 277

e

ei

n

ii ENE

1)(r (2-10a)

e

ei

n

ii HNH

1(r) (2-10b)


31

where e denotes element and n is the total number of nodes for each element. 4-node linear

square elements are used here. n = 4. Shape function N = {N1, N2, N3, N4}. Following the

standard formulation of finite element method, Eqns. 2-9(a) and 2-9(b) can be written in matrix

form as

eMeK

e

ee

e

e

c

2

for TM mode (2-11a)

hMhK

e

e

e

e

e c

21

for TE mode (2-11b)

where h and e are the nodal eigenvectors for the magnetic and electric fields, respectively. Eqns.

2-11(a) and 2-11(b) can be written in the matrix format as

02

uMK

c

(2-12)

where u is the magnetic or electric field corresponds to the eigenfrequency ω. K and M are the

overall stiffness mass matrices, which depend on the spatial distribution of dielectric materials

within the primitive unit cell. K is a complex matrix, while M is a real matrix. Both K and M

are symmetric, singular, and sparse.

eKK , eeMM , u = e for TM modes (2-13a)

e

e

KK

1, eMM , u = h for TE modes (2-13b)

Ke and Me are customarily referred to as elemental stiffness and mass matrices. Ke and Me can

be expressed by

4321 KKKKK e (2-14)

dAA

e NNM T (2-15)

Where

dAkkAyx NNK T22

1

dAxx

ikAx

NNNNK TT

2


32

dAyy

ikAy

NNNNK TT

3

dAyyxxA

NNNNKTT

4

Field u is a periodic function. The periodicity constraint is exerted by coupling the degree

of freedoms on the boundary of the unit cell. Solve Eqn. 2-12 for a series of wave vector k, the

band diagram and corresponding electric and magnetic field can be obtained.

2.2.2 Verification of 2D FEM program

Based on the FEM formulations above, the corresponding program is developed using

MATLAB. The results are compared to those obtained from the PWE method. A photonic

structure with highly symmetric square lattice is employed here. The 2D photonic crystal is

composed of air (relative permittivity 1, no color) and GaAs (relative permittivity 11.4, blue

color). It consists of circular rods equidistantly arranged with a periodicity length of a, as

illustrated in Fig. 2-1(a). Fig. 2-1(b) shows the cross section of a unit cell, corresponding

reciprocal lattice, the first Brillouin zone, and the irreducible Brillouin zone. Thus, the band

diagram can be calculated by sweeping the wave vector in the first Brillouin zone. However,

due to the rotational/mirror symmetry of the photonic crystal, only the irreducible Brillouin

zone should be considered, which is 1/8 of the first Brillouin zone. Furthermore, the local

maximum/minimum value of eigenfrequencies are on the vertices and boundaries of the IBZ

with rare exceptions. In the symmetric cases, the wave vectors on the boundary of irreducible

Brillouin zone are calculated to identify a band gap 241, 276, 278. Hence, to plot the band structure

of the photonic crystal, the wave vector k varies from point Γ (0,0) to Χ(π/a,0), then Μ (π/a,

π/a), and finally comes back point Γ. The 2D unit cell is uniformly meshed by 64×64 bilinear

square elements. The number of plane waves is 32×32. It can be seen from Fig. 2-1(c) that the

FEM results agree well with the results obtained from the PWE method.


33

(a)

(b)

(c)

Fig. 2-1. (a) Structure of the 2D photonic crystal. (b) Cross section of the unit cell, FEM grid, and corresponding first Brillouin zone (the square box), irreducible Brillouin zone (the shaded triangular). (c) Band diagram and the comparison with PWE results.

2.3 FEM for 3D Photonic Crystals

2.3.1 3D FEM formulation

As mentioned above, there is a transversality requirement for each master equation (Eqn.

2-4(a) for solving H, Eqn. 2-4(b) for solving E). Eqn. 2-4(b) depends on ε(r), which makes it


34

hard to handle in numerical analysis. Therefore, the finite element formulation will base on

solving the magnetic field. The electric field is obtained using the magnetic field.

For 3D photonic crystals, due to the periodicity of the photonic crystal, the magnetic field

H can be expressed as a periodic function h times an exponential factor according to the Bloch-

Floquet theory 276.

rkrhrkH ie, (2-16)

where wave vector k = (kx, ky, kz). Substitute Eqn. 2-16 into Eqn. 2-5(a):

hhkr

k21

cii

(2-17)

For a given k, Eqn. 2-17 is an eigenvalue problem and can be solved by the finite element

method. Multiplying an arbitrary test function on both sides and taking the integral over the

unit cell will lead to its weak form as

01

1

2

2

dc

iiv

hh

hkhkhkkhk

(2-18)

where v is the test function, Ω is the volume of the unit cell. Unlike 2D cases, expand Eqn. 2-

18, there are three equations.

011

11

11

11

1

1

2

22

dhc

xh

zxh

y

zh

zyh

y

hz

khy

k

hz

khy

ki

xhk

xh

kzhk

yhki

hkkhkkhkkv

xzy

xx

zxyx

xzxy

zz

yy

xz

xy

zzxyyxxzy

rr

rr

rr

rr

r

r

(2-19a)


35

011

11

11

11

1

1

2

22

dhc

yh

zzh

z

xh

xyh

x

hz

khz

k

hx

khx

ki

yhk

zh

kxh

kyhki

hkkhkkhkkv

y

zy

yx

zyyz

yxxy

zz

yz

yx

xx

zzyyzxxyx

rr

rr

rr

rr

r

r

(2-19b)

011

11

11

11

1

1

2

22

dhc

yh

yxh

x

zh

yzh

x

hy

khx

k

hy

khx

ki

yhk

xhk

zh

kzhki

hkkhkkhkkv

z

zz

yx

zyzx

yzxz

zy

zx

yy

xx

zyxyzyxzx

rr

rr

rr

rr

r

r

(2-19c)

After finite element discretization of the unit cell, for an element e, h can be represented

by interpolating its value at nodes, he, with a shape function N. Using the Galerkin method, the

matrix format of Eqn. 2-19 can be written as

01 2

eee

e chMK

(2-20)

Substitute Eqn. 2-16 into Eqn. 2-4(a), the transversality requirement for magnetic field is

0 hki (2-21)

For an element e, its matrix format can be expressed as


36

0 ehA (2-22)

This constraint can be implemented using Lagrange multiplier method 279. An

underdetermined coefficient λe is introduced as an extra degree of freedom, and Eqn. 2-20 is

expanded to

00

1 T2

e

eee

e c

h

A

AMK (2-23)

Take the expanded elemental stiffness and mass matrices as

0

T

exp AAK

Ke

e ,

000

exp

ee M

M (2-24)

Assign a multiplier λ for each element, then assemble the expanded elemental stiffness and

mass matrices of all elements by the sequence of degree of freedoms, the general finite element

format of this eigenvalue problem can be obtained:

0*2

λh

MKc

(2-25)

where K and M are the overall stiffness and mass matrices. h* is the value of field h at all nodes.

λ is the Lagrange multipliers of all the elements. After solving h, the magnetic field can be

recovered using Eqn. 2-16 and the electric field can be recovered by Eqn. 2-4(d).

Noted that the value of λ does not have real meaning for us, the overall stiffness and mass

matrices are

e

eexp

1 KK

, eexpMM (2-26)

Similarly, in the expanded matrices, Ke and Me are expressed as

4321 KKKKK e (2-27)

de NNM T (2-28)

Where

dkkkkkkkkkkkkkkkkkk

yxzyzx

zyzxyx

zxyxzy

NNK22

22

22

T1


37

dz

kkk

k

yk

kkk

xk

kkk

i

yx

z

z

y

zx

y

x

x

zy NNNNNNK00000

000

00

0000

0TTT

2

dkk

kk

zkk

kk

ykk

kkx

i yz

xz

yz

xy

xz

xy NNNNNNK000

00

00000

00000

TTT

3

dzzyzxz

zyyyxy

zxyxxx

NNNNNN

NNNNNN

NNNNNNK

000010001

000100000

000000100

010000000

100000001

100010000

001000000

000001000

100010000

TTT

TTT

TTT

4

2.3.2 Verification of 3D FEM program

Similar to the 2D case, the FEM program is developed in MATLAB. The results are

compared to the widely used MPB software which is on the basis of the PWE method. A 3D

photonic crystal with simple cubic lattice is employed here. It is composed of air (relative

permittivity 1, no color) and silicon (relative permittivity 12.96, blue color). The photonic

crystal structure is cubic dielectric material blocks equidistantly arranged in air background, as

illustrated in Fig. 2-2(a). The band structure is also only for wave vectors on the boundary of

the irreducible Brillouin zone, the wave vector k varies from the point Χ (π/a, 0, 0) to Μ (π/a,

π/a, 0), Γ (0, 0, 0), R (π/a, π/a, π/a) and then back to X (0, 0, 0). The unit cell is uniformly

meshed by 24×24×24 8-node linear cubic elements. The number of plane waves is 32×32×32.

It can be seen from Fig. 2-2(c) that the FEM results agree well with the PWE results, which

verifies the correctness of the FEM program.


38

(a)

(b)

(c)

Fig. 2-2. (a) Structure of the 3D photonic crystal. (b) Unit cell and corresponding first Brillouin zone (the cubic box), irreducible Brillouin zone (the shaded triangular cone). (c) Band diagram and the comparison with PWE results.

2.4 Conclusions

This chapter presents the FEM formulation for 2D and 3D photonic crystals. Using a series

of assumptions and approximations, the master equations and corresponding transversality

requirements are derived from Maxwell equations. Solving the master equations for given wave

vectors is an eigenvalue problem. After finite element discretization, this problem is rewritten

in the matrix format. The corresponding program is compiled in MATLAB and verified by


39

comparing its results with the PWE method. In this research, the lattice of 2D and 3D photonic

crystals are meshed by uniform square or cubic elements. The complex boundaries of 2D and

3D photonic crystals can be approximately simulated by zig-zag edges together with “grey”

elements with intermediate properties without introducing obvious errors. MATLAB solve the

eigenvalue problem using the standard package ARPACK. The memory needed is proportional

to the square of the number of degree of freedoms. For the 3D problem, the computing resource

it consumes increase rapidly with the increase of resolution. Other efficient methods that can

be combined with topology optimization should be investigated in the future.

Chapter 3 Bi-Directional Evolutionary Optimization for 2D Photonic Band Gap Crystals

40

CHAPTER 3

Bi-Directional Evolutionary Optimization for 2D Photonic Band

Gap Crystals

Toward an efficient and easy-implement optimization for photonic band gap structures,

this chapter extends the BESO method for maximizing band gaps, including the objective

function, sensitivity analysis, sensitivity filter and the implementation of BESO. Then the

optimization results are obtained for both TM and TE modes at different frequency levels. The

characteristics of optimization results and the advantages of this method are discussed.

3.1 BESO Formulation

3.1.1 Objective function

Topology optimization of photonic crystals aims to find an optimal structure of the

primitive unit cell with the maximum band gap between two adjacent bands. Due to the lack of

fundamental length scale in Maxwell’s equation, the ratio between the size and the central value

of the band gap, which is independent of the lattice constant of the photonic crystal, is more

meaningful than the absolute value of the band gap. Therefore, the objective function for

designing photonic crystals can be changed to maximize the band gap-midgap ratio between

two adjacent bands (referred to as band m and band m+1) as

nmm

mm xxxf ..., ,maxminmaxmin2 :Max 21

1

1

XX

(3-1)

where X is the design variables which will be explained later. It should be noted that the

frequency is the function of the wave vector. The above optimization problem is hardly solved

directly because the locations of max(ωm) and min(ωm+1) are changing with the change of the

topology during the optimization process. To overcome this difficulty, it is assumed that any

point from the first to the mth whose frequency larger than C1 = 0.9 × max(ωm(k)) may become

the location of max(ωm) in the next iteration. Similarly, any point from the (m+1)th to the highest


41

band whose frequency is less than C2 = 1.1 × min(ωm+1(k)) may become the location of

min(ωm+1) in the next iteration. The coefficient 0.9 and 1.1 are arbitrarily chosen based on our

numerical experience. Thus, a new objective function is established as

max

1 11 1

1 11 1

ˆ

ˆ2

i

m

jijij

i mjijij

i

m

jijij

i mjijij

e

ww

wwxf

kk

kk

(3-2)

where i is the number of wave vectors k. j is the number of bands. ijw and ijw are weight

factors. At first, the following two parameters are defined:

0

)(2 ijij

CA

k otherwise

)(when 2Cij k ( ,,1mj ) (3-3a)

0)(ˆ 1C

A ijij

k otherwise

)(when 1Ckij ( mj ,,1 ) (3-3b)

Then, the weight factors are given as

1 1i mjij

ijij

A

Aw

(3-4a)

1 1

ˆ

ˆˆ

i

m

jij

ijij

A

Aw

(3-4b)

As a result,

1 1

1i mj

ijw and

1 1

1ˆi

m

jijw .

3.1.2 Design variable

Since the primitive unit cell is discretized with finite elements, the structure of a photonic

crystal can be represented by assigning artificial design variables to all elements. The design

variable xe is the artificial variable of an element e which will be used to represent the material

of the element. It is assumed that a photonic crystal is composed of two materials with

permittivity ε1 and ε2 (where ε1 < ε2). The elemental design variable xe can be constructed by

following that the element is composed of material 1 with permittivity ε1 when xe = 0 and


42

material 2 permittivity ε2 when xe = 1. Thus, the design variables can represent the structure of

the unit cell, and the optimization is to seek the optimal combination of design variables, X, so

that the resulting photonic crystal has a maximal gap between the specified bands.

The traditional BESO method 52-53, 55 used the discrete design variable xe = 0 or 1 only.

However, numerical experience shows that the design of photonic crystals is very sensitive to

the change of the design variable so that a very fine mesh should be used. Instead of using an

extremely fine mesh, the design variable of the BESO method in this chapter can be assigned

with discrete intermediate design values. xe can be 0, 0.1, 0.2, … 1. In other words, the variation

of the design variable in each iteration is limited to be Δxe = 0.1.

For the compliance minimization of structures, the well-known SIMP model 15 makes

elements with intermediate design variables density uneconomical in the optimization process

and thus the solution naturally tends to be 0/1. In the design of photonic crystals, a 0/1 solution

results in a larger band gap since intermediate design variables reduce the contrast between the

materials. Therefore, the linear material interpolation law has been applied successfully to the

design of photonic crystals for TM modes 206-207 as

eee xxx 21 1)( for TM mode (3-5)

In the case of TE modes, the above linear material interpolation has also been applied in

the literature 196. However, the numerical experience indicated that this material interpolation

scheme worked only for maximizing some TE band gaps. Here, the following inverse linear

material interpolation is adopted for TE modes

eee

xxx 21

111)(

1

for TE mode (3-6)

The above material interpolation scheme has also been used in references 211. It is noted

that this inverse linear material interpolation implies the penalization of permittivity for

intermediate elements as shown in Fig. 3-1.


43

Fig. 3-1 Material interpolation schemes for TM and TE modes.

3.1.3 Sensitivity analysis

For the given wave vector, k, the corresponding frequencies, ω and eigenvectors, u can be

extracted by solving Eqn. 2-8(a) or Eqn. 2-8(b). The derivation of ω with regard to xe can be

easily formulated as

uMkKu

kk

ee

T

e xxc

x22

21

(3-7)

Using the material interpolation scheme, Eqn. 3-5, for TM modes, the derivations of matrix

K and M are

0

exK

(3-8a)

e

exMM

12

(3-8b)

where the dimensions of Me is the same to that of M but all components in M which are not

related to element e are zero. Similarly, using the material interpolation scheme, Eqn. 3-6, for

TE modes, the derivations of matrix K and M are

e

exKK

12

11

(3-9a)

0

exM

(3-9b)


44

Substituting the above equations into Eqn. 3-7, the sensitivity of all eigenfrequencies can

be easily obtained since their corresponding eigenvectors can be outputted from FEA. Therefore,

no additional computational cost is needed for sensitivity analysis, excluding storage

consideration.

It should be noted that the above sensitivity analysis is only suitable for the case of a single

eigenvalue. Multiple eigenvalues widely exist in the design of photonic crystals. Investigation

of sensitivity analysis of multiple eigenvalues is available in many papers 280-282. Take the case

of a double eigenvalue ω with two corresponding eigenvectors u1 and u2 as an example, but the

extension of a higher number of multiplicities is straightforward. The multiplicity of the

eigenvalue implies that any linear combination of the eigenvectors u1 and u2, (e.g. c1u1 + c2u2)

corresponding to ω, will also satisfy the original eigenvalue problem. The sensitivities of double

eigenvalues are eigenvalues of a 2D algebraic subeigenvalue problem as

** uuf tsk (3-10)

where t is the eigenvalue, its two solutions are the sensitivity corresponding to the two multiple

eigenvalues. fsk (s,k = 1,2) denote generalized gradient vectors as

kee

Tssk xx

c uMKuf

22

21

(3-11)

With the sensitivities for multiple eigenvalues, it is natural to assign the lowest sensitivity

to the lowest order eigenvalue and the highest sensitivity to the highest order eigenvalue.

In the practical implementation, the existence of multiple eigenvalues for any given wave

vectors should be carefully checked in each optimization step. When there are multiple

eigenvalues, the sensitivities should be calculated according to Eqn. 3-10. Thus, the sensitivity

number defined with the derivation of the modified objective function can be calculated

accordingly

e

ee x

xf

)( (3-12)

3.1.4 Mesh-independency filter


45

In the topology optimization of mechanical structures, the filtering scheme can effectively

alleviate the numerical instabilities of the checkerboard pattern and mesh-dependency 52-53, 55.

For the design of photonic crystals, the usage of the filter scheme is arguable 211. The numerical

experience indicated that the checkerboard pattern does not exist in the design of photonic

crystals, but a different mesh may result in a different optimized design. This mesh-dependency

problem is different from that of mechanical structures, where a fine mesh results in a more

detail design of the structures. To avoid this mesh-dependency problem, the filter scheme is

still used for the design of photonic crystals in this chapter. The filtering scheme modifies the

sensitivity number as

n

ii

n

iei

e

w

w

1

1ˆ

(3-13)

where the weight factor wi is defined by

ei

ei

ei

i rrrrrr

wmin

minmin

if ,0 if ,

(3-14)

where eir denotes the distance between the center of element e and i. rmin is the radius of the

filter, defined to identify the neighboring elements that affect the sensitivity number of element

e. rmin = 0.0283a (1/50 of the diagonal line of the square unit cell) is used for all numerical

examples in this chapter.

In order to improve the stability and convergence of solution, elemental sensitivity

numbers can be further averaged with their corresponding values in the previous iteration as 52

ke

ke

ke ˆ~

21~ 1 (3-15)

where k is the current iteration number.

3.1.5 Implementation of BESO

Different from most topology optimization problems, there is no volume constraint for the

optimal design of photonic crystals. It is understandable as there is no band gap when the unit

cell is full of either material 1 or material 2. Therefore, the optimal volume fraction of material


46

2 must be between 0% and 100%. For simplicity, the volume fraction, Vf, is for material 2 in

the thereafter sections. BESO starts from an initial design which is nearly full of material 2 and

there is no band gap for the initial design. The optimization for the variation of the volume

fraction has three stages. At the first stage, the volume fraction gradually decreases when it is

larger than a prescribed volume, *fV , as

*11 when 1 fkf

kf

kf VVERVV (3-16)

where ER is the evolution rate. ER = 2% and *fV = 40% are used throughout this chapter. Due

to the complexity of the optimization of photonic band gap crystals, many locally optimal

solutions may exist. The setting of *fV means searching the optimal solution with a volume

fraction around *fV .

The volume fraction at the second stage is determined according to the variation history

of the band gap as

ERVVxfxfVVxfxf

VVkf

kf

ke

ke

kf

kf

ke

kek

fkf 11

111 1 (3-17)

The second stage is ended until the volume fraction is vibrantly convergent to a constant

value, **fV . Then the volume fraction keeps a constant at the third stage until the topology of

the unit cell and objective function are stably convergent.

Similar to the conventional BESO method 52, 55, the design variables are updated according

to the relative values of sensitivity numbers and volume fraction. Based on the relative ranking

of the calculated sensitivity numbers ke~ , a threshold of the sensitivity number, k

th , is

determined by using the bi-section method so that the volume fraction in the next iteration is

equal to 1kfV as described in the above section. The design variable of each element is

modified by comparing its sensitivity number with the threshold as

kth

ke

ke

kth

ke

kek

e xxxx

x

~ if ,)0 ,max(

~ if ,)1 ,min(1 (3-18)

where Δx = 0.1 throughout the chapter, which means BESO uses discrete design variables.

Although BESO may take some discrete intermediate design variables during the optimization


47

process, the final design is naturally convergent to an almost 0/1 design due to the adoption of

the material interpolation schemes.

Figure 3-2 depicts the proposed BESO procedure for the design of photonic crystals with

the desired band gap. At first, the FE analysis for the unit cell of a photonic crystal is conducted

by MATLAB, to obtain the eigenfrequencies, corresponding eigenvectors, and the band

diagram. Then, the sensitivity numbers are calculated and the topology of the unit cell is

updated accordingly. The thereafter iterative process evolves the topology of the unit cell

towards its optimum until both the topology and objective function are convergent.

Fig. 3-2 Flow chart of the BESO procedure.

3.2 Numerical Results and Discussion

In this section, numerical examples will be presented to illustrate the effectiveness of the

proposed optimization algorithm. The photonic crystals to be designed have a square lattice

with a = 1. However, the proposed method can be equally applied for photonic crystals with

other lattices. It is assumed that the photonic crystals are composed of two materials: air (ε1 =

1) and gallium arsenide (GaAs) (ε2 = 11.4). The unit cell represented the structure of photonic

crystals is meshed by 64×64 bilinear square elements. BESO starts from the initial design which

is full of material 2 (GaAs) except for four elements in the center of the unit cell with material


48

1 (air). To illustrate microstructures in figures, white and black colors denote air and GaAs,

respectively.

Figure 3-3 gives the optimized 3×3 unit cells of photonic crystals for TM modes with

maximum band gaps from the first to the tenth band (where the dashed box represents the unit

cell). Their corresponding band diagrams are also given in Fig. 3-3. Generally, GaAs with the

high permittivity is isolated by air with low permittivity which is consistent with the previous

reports 210, 212. The distribution shapes of GaAs can be solid circles or circle rings. The obtained

maximum band gap-midgap ratio for TM mode is 42.89% which is the gap between the 7th and

8th bands. Although the lattice constant a = 1 is used for all examples, it is interesting to find

that some optimized topologies are similar but alike with a half of the lattice constant, e.g. the

optimized topologies between bands 1~2 and 4~5 or between bands 2~3 and 9~10.

It should be noted that the resulting topology is not a unique solution for the design of

photonic crystals. For instance, the equivalent solution can also be found by shifting the

optimized unit cell with half the lattice along x and y directions simultaneously. It well indicates

that the optimized design of photonic crystals highly depends on the initial design.

(a)

(b)


49

(c)

(d)

(e)

(f)

(g)


50

(h)

(i)

(j) Fig. 3-3 Optimized 3×3 unit cells and their band diagrams for TM band gaps. (a) The first band gap. (b) The second band gap. (c) The third band gap. (d) The fourth band gap. (e) The fifth band gap. (f) The sixth band gap. (g) The seventh band gap. (h) The eighth band gap. (i) The ninth band gap. (j) The tenth band gap.

Fig. 3-4 shows the optimized 3×3 unit cells of photonic crystals for TE modes with

maximum band gaps from the first to the tenth band and their corresponding band diagrams. It

should be pointed out that the third band gap doesn’t open using the given initial design. Instead,

a new initial design which is full of material 2 (GaAs) except for two elements at the middle of

four edges with material 1 (air) is used. Opposite to the optimized designs for TM modes, GaAs

tends to be connected together for the optimized designs for TE modes except for those of the

second, fourth and eighth band gaps. The obtained maximum band gap-midgap ratio for TE

mode is 44.27% which is the gap between the 7th and 8th bands. It seems that maximizing band

gaps at high frequency level usually generate more complex topologies which are hardly figured

out by scientists.


51

(a)

(b)

(c)

(d)

(e)


52

(f)

(g)

(h)

(i)

(j) Fig. 3-4 Optimized 3×3 unit cells and their band diagrams for TE band gaps. (a) The first band gap. (b) The second band gap. (c) The third band gap. (d) The fourth band gap. (e) The fifth band gap. (f) The sixth band gap. (g) The seventh band gap. (h) The eighth band gap. (i) The ninth band gap. (j) The tenth band gap.


53

In order to illustrate the whole optimization process, Fig. 3-5 shows the evolution histories

of the objective function and volume fraction for maximizing the seventh band gap of TE mode.

It can be seen that the band gap-midgap ratio for the initial design is -10.47%, which means no

band gap. When the volume fraction of GaAs gradually decreases, the band gap-midgap ratio

gradually increases from a negative value to a positive one which means the band gap turns up.

This band gap continues to increase and then converges to the maximum value, 44.27%,

meanwhile the volume fraction is also convergent to a stable value, 39.82%. The whole

optimization process only needs 60 iterations. Fig. 3-6 gives the evolution history of the unit

cell. It can be seen that the optimized topology is very close to a 0/1 design although there are

many grey areas during the optimization process.

Fig. 3-5 Evolution histories of the objective function and volume fraction for maximizing the seventh band gap of TE mode.


54

Fig. 3-6 Evolution history of the unit cell for maximizing the seventh band gap of TE mode. (a) Initial design. (b) Iteration 10. (c) Iteration 20, (d) Iteration 30. (e) Iteration 40. (f) Final optimized design.

It can be seen that all optimized solutions in Figs. 3-3 and 3-4 have clear topologies without

any blur areas although some discrete intermediate design variables are used. The obtained band

gaps demonstrate the effectiveness of the proposed optimization method. Some optimized

topologies are similar to those in the literature 207-208, 210, 212, but some topologies such as Figs.

3-3(f) and (h) for TM mode and Figs. 3-4(d), (e), (h), (i), (j) for TE modes are first reported. It

should be noted that the computational burden for the optimization of photonic crystals mainly

comes from a large number of FEM analysis. Compared to other optimization methods with

hundreds, even thousands of iterations, all of the present optimized designs are obtained with

around 100 iterations and the computational efficiency of the proposed optimization method is

therefore obvious.

It should be noted that the optimization problem for photonic band gap structures may

have many local optima and the optimized solution may highly depend on the initial guess as

mentioned before. To successfully obtain the desirable band gap, the optimization algorithms

developed by Sigmund and Jensen 210-211 were only applied to the initial design with band gaps.

Men et al. 212-213 examined two different types of initial configurations: photonic crystals

exhibiting band gaps and random distribution. Starting from an initial random design without

any band gap, the optimization algorithm often failed to find a proper design with band gaps,

for example, the success rates for the ninth TM and TE band gaps were only 16.7% and 13.3%

respectively. In this chapter, a simple initial design without any band gap is used and BESO

can successfully found all band gaps except for the 3rd TE band gap. By slightly changing the

initial design, the 3rd TE band gap can still be successfully obtained. Certainly, other solutions


55

could also be found if BESO starts from another initial design. However, further investigation

into seeking the global optima of photonic band gap crystals is still needed.

3.3 Conclusions

This chapter systematically investigates the topology optimization of 2D photonic crystals

for both TM and TE modes. A new optimization algorithm based on BESO is proposed for the

design of band gap structures. Numerical results indicate that the algorithm proposed in this

chapter is effective and applicable to any frequency level. Moreover, the proposed algorithm is

computationally efficient as the solution usually converges within about 100 iterations. The

optimization of 2D symmetric photonic crystal with square lattice is relatively simple, but this

efficient and reliable algorithm will lay the ground for more complicated research in following

chapters.

Chapter 4 Achieving Large Band Gaps in 2D Asymmetric Photonic Crystals

56

CHAPTER 4

Achieving Large Band Gaps in 2D Asymmetric Photonic Crystals

To promote the theoretical analysis for photonic crystal, and provide multiple alternative

solutions for practical applications, new systematic design method with and without the

symmetric constraint is necessary. This chapter systematically studies the influence of

symmetry on photonic band gaps of 2D photonic crystals using topology optimization method.

Optimal designs of symmetric and asymmetric photonic crystals of square and hexagonal

lattices with band gaps of both transverse electric (TE) and transverse magnetic (TM) modes

are designed and analyzed. The formation of large band gaps and the unusual geometric

characteristics of asymmetric photonic crystals are discussed.

4.1 Numerical Analysis and Topology Optimization

Two types of lattices, square and hexagonal, are considered in this chapter. The symmetric

unit cells are illustrated in Fig. 4-1(a) and 4-1(b). The square lattice has 4-fold reflection

symmetry and 4-fold rotational symmetry (C4v). Meanwhile, the hexagonal lattice has 6-fold

reflection symmetry and 6-fold rotational symmetry (C6v). Their irreducible Brillouin zones are

1/8 and 1/12 of the first Brillouin zone, respectively. To calculate the band diagram for

symmetric photonic crystals, only wave vectors along the boundary of the irreducible Brillouin

zone are evaluated. In contrast, for asymmetric photonic crystals, the entire lattices have a 1-

fold rotational symmetry (C1). The irreducible Brillouin zone is the entire first Brillouin zone,

as shown in Fig. 4-1(c) and 4-1(d). With the time-reversal symmetry ω(k) = ω(-k) (k = (kx, ky)

is the wave vector) 217, the wave vectors in half of the first Brillouin zone should be calculated.


57

(a)

(b)

(c)

(d)

Fig. 4-1 (a) C4v square lattice; (b) C6v hexagonal lattice; (c) C1 square lattice; (d) C1 hexagonal lattice. Left: symmetry of the primitive unit cell. Right: the first Brillouin zone and irreducible Brillouin zone (grey area) in the reciprocal space.

The FEM analysis and BESO algorithm are the same with that in Chapter 3, except that

the optimization process starts from a randomly generated structure of the primitive unit cell,


58

which has no existing band gap. Then, the finite element analysis is conducted to obtain the

band diagram and the corresponding eigenvectors. According to the calculated sensitivities,

BESO increases design variables for elements with high sensitivities (from 0 to 1) and decreases

design variables for elements with low sensitivities (from 1 to 0) simultaneously. As a result, a

new design is formed with the updated design variables, and thereafter the finite element

analysis and BESO update scheme are repeated. Such an iterative process continues and evolves

the topology of the primitive unit cell towards its optimum. Because of the randomness of the

initial design, the same optimization process is repeated for 5 times, and 5 designs are obtained.

The one with largest band gap will then be signed out as the final result.

(a)

(b)

Fig. 4-2 (a) Randomly generated initial design for square lattice. (b) Randomly generated initial design for the hexagonal lattice. It should be noted the grid of structure is much coarser than the grid of finite element discretization to increase the randomness.


4.2.1 Topology optimization results

Photonic crystals are assumed to be composed of two materials: low permittivity material

1, air (ε1 = 1); high permittivity material 2, GaAs (ε2 = 11.4). To systematically investigate the

band gap properties, both TM and TE modes from between bands 1~2 to bands 10~11 are

optimized, in which square and hexagonal lattices with the highest symmetry (C4v or C6v) and


59

lowest symmetry (C1) are considered. The resulting designs are shown in Fig. 4-3 ~ 4-6 with

3×3 unit cells. In these figures, the red dashed box represents the unit cell, and white and blue

colors denote materials 1 and 2, respectively.

(a) C4v 1st (b) C4v 2nd

(c) C4v 3rd (d) C4v 4th

(e) C4v 5th (f) C4v 6th

(g) C4v 7th (h) C4v 8th

(i) C4v 9th (j) C4v 10th


60

(k) C1 1st (l) C1 2nd

(m) C1 3rd (n) C1 4th

(o) C1 5th (p) C1 6th

(q) C1 7th (r) C1 8th

(s) C1 9th (t) C1 10th

Fig. 4-3 TM mode designs for C4v and C1 square lattice.



61





(k) C1 1st (l) C1 2nd

(m) C1 3rd (n) C1 4th


62

(o) C1 5th (p) C1 6th

(q) C1 7th (r) C1 8th

(s) C1 9th (t) C1 10th

Fig. 4-4 TE mode designs for C4v and C1 square lattice.





63



(k) C1 1st (l) C1 2nd

(m) C1 3rd (n) C1 4th

(o) C1 5th (p) C1 6th

(q) C1 7th (r) C1 8th

(s) C1 9th (t) C1 10th Fig. 4-5 TM mode designs for C6v and C1 hexagonal lattice.


64






(k) C1 1st (l) C1 2nd

(m) C1 3rd (n) C1 4th


65

(o) C1 5th (p) C1 6th

(q) C1 7th (r) C1 8th

(s) C1 9th (t) C1 10th Fig. 4-6 TE mode designs for C6v and C1 hexagonal lattice.

It can be seen from Figs. 4-3 ~ 4-6 that the designs of TM modes are composed of isolated

dielectric rods of various shapes and sizes. In contrast, dielectric materials tend to be connected

and form walls or rings in the designs of TE modes. Designs for band gaps at higher frequencies

usually have more complex and subtle topologies. For symmetric optimal designs, the number

of rods in the primitive unit cell of TM modes or subpartitions in TE modes exactly equals the

band order210. The results show that such conclusions are also applicable to asymmetric

photonic crystals. It is also noticed that some optimized topologies share the same topological

patterns with the symmetric designs. In some cases, the primitive unit cell of the higher band is

formed by multiplying the one for the lower band, e.g. the 1st and 4th TM designs with C4v

square lattice. In some other cases, the structure of the higher band is formed by the rotation

and multiplication of the one for the lower band, e.g. the 4th and 8th TM designs for C1 square

lattice and the 1st and 3rd TE designs for C1 hexagonal lattice. Although the primitive unit cells

of the corresponding asymmetric and symmetric designs are always different, some of them

should be classified to the same topology through the observation of the 3×3 unit cells, e.g. the

3rd and 7th TE designs of the square lattice.


66

The size of the band gaps for different lattices and symmetries are compared in Fig. 4-7.

In all cases, asymmetric designs have larger TM and TE band gaps than those of the symmetric

designs. In some cases, the band gap size increases significantly, e.g. the 5th TM band gap and

the 2nd TE band gap of the hexagonal lattice are enlarged by more than 200%. On average, the

size of band gaps of the square lattice increases by 6.5% for the TM modes and 13.7% for the

TE modes. As for the hexagonal lattice, it increases by 25.7% for the TM modes and 44.9% for

the TE modes. Evidently, reducing symmetry leads to significantly larger band gaps as expected.

(a)

(b)

Fig. 4-7 Comparison between the size of band gaps for symmetric and asymmetric optimal designs: (a) Square lattice; (b) Hexagonal lattice.

4.2.2 Origin of large band gaps


67

For the hexagonal lattice, the largest band gap sizes for TM and TE modes are 0.476 and

0.517, respectively, for both designs at the 1st band as shown in Fig. 4-7(b). Meanwhile, both

symmetric and asymmetric designs have the same gap sizes and share the same topologies as

shown in Fig. 4-5(a), 4-5(k), Fig. 4-6(a) and 4-6(k). Circular dielectric rods are distributed as

equilateral triangles for the TM mode, and the dielectric material forms the hexagonal

honeycomb pattern for the TE mode. This confirms that the asymmetric optimization is able to

include symmetric designs if the optimal design is symmetric. From the perspective of

optimization, the designable space is enlarged by reducing the symmetry of the photonic

crystals. These two designs are similar to the traditional designs where the hexagonal holes in

the TE mode design were replaced by the circular holes in the application 210.

For the square lattice, Fig. 4-7(a) shows that the largest TM band gap is the asymmetric

design of the 4th band gap and its size is 0.478; the largest TE band gap is the asymmetric design

at the 9th band with a size of 0.487. To the author’s best knowledge, these two structures have

the largest band gaps ever found in the square lattice and both designs are reported for the first

time. In terms of symmetric photonic crystals, it is well-known that the hexagonal lattice tends

to have larger band gaps than that of the square lattice 210, 283-284 for both the TM and TE modes.

But in this study, the largest TM band gap in the square lattice is comparable to the largest one

in the hexagonal lattice, 0.476. After parallel shifting, the primitive unit cell of the 4th TM design

consists of 4 rods and there is an implied 4-fold rotational symmetry. The centroid of each rod

coincides with the centroid of the isosceles triangle. Similarly, the primitive unit cell of the 9 th

TE design has implied 2-fold mirror symmetry about its 2 diagonals. Their primitive unit cells

and corresponding band diagrams over the first Brillouin zone are given in Fig. 4-8 and 4-9.

For the 4th TM band gap design, the electric fields at the Γ point of bands 4 and 5 are

illustrated in Figs. 4-8(c) and 4-8(d). For band 4, the electric field has a high amplitude in the

dielectric rods; while for band 5, the electric field concentrates in the air regions and the

amplitude in the rods is nearly zero. Such orthogonal patterns down-shift the lower band and

up-shift the upper band, and a larger band gap appears. Similarly, the magnetic fields at the Γ

point of bands 9 and 10 are illustrated in Figs. 4-9(c) and 4-9(d) for the 9th TE band gap design.

In contrast to the TM design, the energy of magnetic field for the lower band 9 concentrates in


68

the air region, while that for the upper band 10 concentrates in the dielectric material. They also

form a pair of orthogonal patterns to enlarge the corresponding band gap. This phenomenon is

consistent with the analysis in 103. As a result, obtaining orthogonal patterns of electric or

magnetic fields at higher bands naturally leads to complex and subtle designs.

(a) (b)

(c) (d)

Fig. 4-8 The 4th TM band gap design for C1 square lattice. (a) Topology of the primitive unit cell; (b) Band diagram (blue – odd bands; red – even bands); (c) Electric field distribution at the Γ point for band 4 (3×3 unit cells); (d) Electric field distribution at the Γ point for band 5 (3×3 unit cells).

(a) (b)


69

(c) (d)

Fig. 4-9 The 9th TE band gap design for C1 square lattice. (a) Topology of the primitive unit cell; (b) Band diagram (blue – odd bands; red – even bands); (c) Magnetic field distribution at the Γ point for band 9 (3×3 unit cells); (d) Magnetic field distribution at the Γ point for band 10 (3×3 unit cells).

The formation of band gaps can be qualitatively explained by Mie resonance and Bragg

scattering theory 103, 215, but it is still impossible to obtain analytical results for 2D problems so

far. Generally, it is considered that dielectric rods with uniformly spatial distribution will result

in large TM band gaps 210, 285. Figs. 4-3 and 4-5 also indicate that most asymmetric TM designs

tend to have more even distributed dielectric rods since the dielectric materials can distribute

freely in the design space. However, for the 4th TM band gap of square lattice, the uniformity

of asymmetric design decreases obviously compared to the symmetric one, while the band gap

size increases significantly from 0.379 to 0.478. Physically, forming orthogonal patterns of

electric or magnetic fields is critical for maximizing TE or TM band gaps as discussed above.

However, for a 2D problem, to find an appropriate structure satisfying this requirement is still

a challenge especially for higher bands since the underlying relationship between the topology

of photonic structure and the band gap is too complicated. This also provides the research

opportunity and significance for topology optimization, which seeks the optimal designs based

on numerical analysis.

4.3 Conclusions

This chapter systematically investigated the designs of 2D symmetric and asymmetric

photonic crystals based on the finite element analysis and topology optimization. Designs of

both TM and TE modes, square and hexagonal lattices are obtained for band gaps at from the

first to 10th band. The results show that the band gap sizes of asymmetric designs are larger than


70

those of the symmetric ones. The largest TM and TE band gaps for the square and hexagonal

lattices are identified. Moreover, the largest TM band gap of the square lattice photonic crystals

achieves a comparable size to that of the hexagonal lattice. This asymmetric design method

opens up a new possibility in photonic crystal designs towards larger band gaps due to the break

of the symmetric rules in the conventional design approach.

Chapter 5 Topological Design for 2D Photonic Crystals with Large Complete Band Gaps

71

CHAPTER 5

Topological Design for 2D Photonic Crystals with Large Complete

Band Gaps

This chapter presents a systematic approach to design photonic crystals with large

complete band gaps. A heuristic design is first generated by direct superposition. Afterward, it

is used as the initial design and evolved to an optimal structure with a large complete band gap

by topology optimization. The geometric characteristics of photonic crystals with complete

band gaps and their potential applications in photonic crystal fibers are discussed.

5.1 Construction of Initial Design

A complete band gap works for both TM and TE polarizations, so the photonic crystal

should have the characteristics of both TM and TE band gap structures. Therefore superposing

a TE band gap structure with a TM band gap structure may create a complete band gap.

However, TE and TM band gaps resulting from such structural superposition may not overlap

or even be entirely eliminated. Nevertheless, the negative mutual influence can be mitigated, if

the TE and TM band gap structures coincide well with each other or share a similar pattern.

The topology optimization results in previous chapters for TE or TM band gap 208, 210, 212, 216

provide adequate designs, from which some can be manually selected as the initial structures.

An example is illustrated in Fig. 5-1. The unit cells of TE and TM photonic band gap

crystals are composed of air (relative permittivity = 1, denoted by the white area) and GaAs

(relative permittivity = 11.4, denoted by the red or yellow area). Both photonic crystals have

the square lattice with a 4-fold reflection and 4-fold rotation symmetry. The first photonic

crystal has a TE gap between the 5th and 6th bands and the second has a TM gap between the 9th

and 10th bands. The normalized frequency ranges (ωa/2πc, a is the lattice constant, ω and c are

frequency and speed of light) of the band gaps are 0.613~0.898 and 0.865~1.285, respectively.

It can be seen that the TE and TM structures match well, and a new design is obtained by


72

superposing these two structures. The band diagram of the new design is given in Fig. 5-1(d),

and there is a narrow complete band gap formed by the overlapping of the 5th TE gap and 9th

TM gap, the frequency range is 0.689~0.718. The size of the complete band gap is measured

by the band gap-midgap ratio, which is 4.20%.

(d)

Fig. 5-1 (a) The unit cell of the 5th TE band gap structure. (b) The unit cell of the 9th TM band gap structure. (c) Superposition of the TE and TM designs. (d) Corresponding band diagram of the new structure.

The resulting complete band gap illustrated in Fig. 5-1 is too small in practice. It should

be also noted that the superposition of TE and TM designs may not directly result in a complete

band gap in some cases. But such a superposition method provide a clue to form a complete

band gap. Thereafter, topology optimization is applied to open and enlarge the band gap in the

superposition structure. Unlike the conventional topology optimization method starting from a

randomly generated initial design 212, the superposition method eliminates the randomness in

(a)

(b)

(c)


73

the optimization process and thus improve the computational efficiency and ensure a high

success rate.

In this chapter, BESO is utilized to achieve large complete band gaps. Assuming a

complete band gap is constructed by the overlapping of the ith TE band gap between adjacent

bands i and i+1, and the jth TM band gap between bands j and j+1. The upper bound of the band

gap is ωtop = min( )(TE1 ki , )(TM

1 kj ) and the lower bound is ωbot = max( )(TE ki , )(TM kj ),

where k = (kx, ky) is the wave vectors within the irreducible Brillouin zone. To maximize the

complete band gap, the objective function f(X) can be expressed as

bottop

bottopf

2 :Max X (5-1)

where X = [x1, x2, …, xn] denotes design variable and n is the total number of finite elements.

The design variable, xe = 0 means that element e is air, while xe = 1 means element e is GaAs.

The structure of the unit cell can be represented by X. Elemental sensitivity is then calculated

to evaluate the variation of the objective function caused by the change of design variables

2)(

4bottop

e

bottop

e

topbot

ee

xxx

f

X (5-2)

BESO increases design variables for elements with high sensitivities and decreases design

variables for elements with low sensitivities. The optimization will iteratively update the design

variable X, which represents the structure of the unit cell of a photonic crystal, so that the band

gap is gradually enlarged to a maximum.



The evolution history of the band gap-midgap ratio and the primitive unit cell is illustrated

in Fig. 5-2(a) when BESO starts from the superposition design shown in Fig. 5-1. The structure

of the unit cell gradually evolves to its optima and the size of complete band gap increases

continuously. The entire evolution process stably converges after 73 iterations, within 57

minutes on a workstation (Dual Intel Xeon E5 CPU, 192 GB RAM). The resulting band gap


74

has a size of 20.79% at normalized frequency range 0.726~0.894, as shown in the band diagram

Fig. 5-2(b), which is approximately 4 times larger than the initial superposition design. The

obtained structure given in Fig. 5-2(b) is the same to the one extensively discussed in Refs. 202,

213, 219, which is usually assumed to be the largest complete band gap in silicon or GaAs based

square lattice photonic crystals.

(a)

(b) Fig. 5-2 (a) Evolution history of the band gap-midgap ratio and the structure of unit cell. The primitive unit cells are uniformly discretized into 96×96 elements. (b) Optimization result and the band diagram.

Using the same approach, more structures with complete band gaps at different frequency

levels can be obtained for both square and hexagonal lattices, as shown in Fig. 5-3 and Fig. 5-

4 respectively. Note the hexagonal lattice has 6-fold reflection and 6-fold rotation symmetry.


75

For each group in Figs. 5-3 and 5-4, the left picture shows the superposition of TE and TM band

gap structures and the right one gives the corresponding optimized structure. The approach

works well for band gaps at the low normalized frequency, e.g. Fig. 5-3(a) with a complete

band gap at 0.407~0.487, and high normalized frequency, e.g. Fig. 5-3(f) with a complete band

gap at 1.043~1.330. In this study, the possible superposition between the 1st ~ 10th TE band gap

and the 1st ~ 20th TM band gap in the square lattice and hexagonal lattice are considered. For a

photonic crystal with a complete band gap, the band number of TM band gap is generally larger

than that of the TE band gap. In terms of the band gaps for a single polarization, hexagonal

lattices tend to have larger band gaps 210, 284. However, the largest complete band gap, 24.12%,

is formed between the 10th TE band gap and 17th TM band gap for the square lattice as shown

in Fig. 5-3(f). To the author’s best knowledge, this is the largest complete band gap ever

reported for GaAs/air photonic crystals.

(a)

(b)


76

(c)

(d)

(e)

(f)

Fig. 5-3 Square photonic crystal structures with large complete band gaps (Left - initial superposition structure; Right - optimized structure). (a) Square lattice, 2nd TE + 3th TM, band gap size 18.02%. (b) Square lattice, 4th TE + 6th TM, band gap size 17.76%. (c) Square lattice, 4th TE + 8th TM, band gap size 17.32%. (d) Square lattice, 5th TE + 8th TM, band gap size 18.26%. (e) Square lattice, 7th TE + 12th TM, band gap size 18.75%. (f) Square lattice, 10th TE + 17th TM, band gap size 24.12%.


77

(a)

(b)

(c)

(d)

(e)


78

(f)

Fig. 5-4 Hexagonal photonic crystal structures with large complete band gaps (Left - initial superposition structure; Right - optimized structure). (a) Hexagonal lattice, 1st TE + 2nd TM, band gap size 20.99%. (b) Hexagonal lattice, 3rd TE + 6th TM, band gap size 18.06%. (c) Hexagonal lattice, 3rd TE + 7th TM, band gap size 17.63%. (d) Hexagonal lattice, 4th TE + 7th TM, band gap size 19.51%. (e) Hexagonal lattice, 7th TE + 11th TM, band gap size 20.75%. (f) Hexagonal lattice, 9th TE + 15th TM, band gap size 21.38%.

Compared to the TE and TM band gap structures, the structures in Figs. 5-3 and 5-4 are

more complex and show different characteristics from each other. Generally, these structures

can be decomposed and the components can be classified into “rods”, “rings” (hollow rods) and

thin “walls”. The traditional complete band gap structures were inspired by connecting the rods

or rings structures with thin walls 203, 218. The similar pattern can be found for the structures in

Figs. 5-2, 5-3(a), (b), (e), (f) and 5-4(a), (b), (e), (f). However, isolated dielectric rods (Fig. 5-

3(d), Fig. 5-4(c) and (d)) and the combination of rods and rings (Fig. 5-3(c)) may also generate

a large complete band gap. It well demonstrates that the thin “wall” is not a prerequisite for the

formation of complete band gaps, which is different from the traditional understanding.

The band gap structures of TM modes usually have a much lower filling ratio than that of

the TE modes. For the 13 examples in this chapter, the average filling ratio is 0.154 for the TM

modes and 0.388 for the TE modes. The average filling ratio of the complete band gap structures

is 0.277, which approximately equals the mean filling ratio of TM and TE band gap structures

(0.271). Obviously, the resulting structure reflects the compromise between the corresponding

TM and TE band gap structures. The large complete band gap emerges because the resulting

designs have the characteristics of both TE and TM structures, and hence both Mie resonance

and Bragg scattering are supported.

5.2.2 Potential application in the photonic crystal fiber


79

As one of the most fundamental properties of photonic crystals, a wide complete band gap

can lead to enormous potential applications, e.g. band-gap photonic crystal fibers (PCFs), in

which light propagates perpendicularly to the periodic plane and are concentrated in the air core

by the band gap effect. The potential application of complete band gap structures in PCFs is

investigated. The relative permittivity of the dielectric material is 8.4, corresponding to the

index of chalcogenide glasses which have been successfully applied in the fabrication of

infrared PCFs 286. Other optical properties they possess, such as high refractive indices and high

nonlinearity, make them ideal for fiber lasers and other active devices. The complete band gap

results discussed above correspond to one case that the out-of-plane wave vectors kz = 0. To

make the complete band gap structures useful in a fiber, the gaps must extend over a range of

nonzero kz. Fig. 5-5 shows the complete band gap structure in Fig. 5-4(e) which can be

employed as the cladding of a PCF, and its corresponding band diagram for various kz. Since

the optimization result depends on the selected material properties, the optimized structure is

with a slightly different size. It can be observed that a large complete band gap presents in a

wide kz range. When kz = 0, the size of complete band gap is 21.92%. For comparison, the

traditional triangular arranged holes design has a complete band gap of only 6.81%. Within this

large complete band gap, highly localized states can be introduced by generating a defect in the

crystal, acting as the core to guide light in a hollow-core PCF. The yellow region above the air

line ω = ckz shows the possible guided modes in the air core, which indicates the incident light

with a broad range of wavelengths can be confined in the core area by the band-gap effect. It

means the complete band gap structures have great potential in optical communications and

high power laser delivery.

(a)


80

(b)

Fig. 5-5 (a) Cross section of the PCF. (b) Band diagram as a function of the out-of-plane wave vector kz. The blue area is the light cone of the photonic crystal, and the main band gap is illustrated by the yellow region. The red line shows the light line.

5.3 Conclusions

In this chapter, a new approach is proposed to efficiently design 2D photonic crystals with

large complete band gaps. The numerical examples demonstrate the effectiveness of the method

and some innovative designs with both square lattice and hexagonal lattice are obtained. One

of them achieves the largest complete band gap ever reported for GaAs/air photonic crystals,

with a relative size of more than 24%. Large complete band gaps emerge because the resulting

designs have the characteristics of both TE and TM structures, and hence both Mie resonance

and Bragg scattering are supported. The optimization result can be used in hollow core photonic

crystal fibers to enhance its performance.

Chapter 6 Topology Optimization of 2D Photonic Crystals for All-Angle Negative Refraction

81

CHAPTER 6

Topology Optimization of 2D Photonic Crystals for All-Angle

Negative Refraction

In order to create novel photonic crystals with broad AANR frequency range, a new

topology optimization algorithm based on BESO is proposed. The mechanism of AANR in 2D

dielectric photonic crystals and the calculation of AANR frequency range are described. Then

the objective function of the optimization problem, sensitivity analysis and the implement of

BESO method are presented. At last numerical results are demonstrated, and the effects of

dielectric permittivity contrast and filter are discussed.

6.1 AANR Frequency Range

Negative refraction of electromagnetic waves for metamaterials requires its effective

permittivity and permeability to be negative simultaneously. However, the mechanism of

negative refraction for photonic crystals is totally different. It arises due to the dispersion

characteristics of waves in a periodic medium. The band diagram along the boundary of the

irreducible Brillouin zone and the equi-frequency contours (EFC) in k-space can be employed

to identify the possibility of AANR. AANR in a photonic crystal may happen in different

frequency ranges, but AANR at lower frequencies is more meaningful in superlensing, because

high order diffractions can be suppressed and all the optical energy is negatively refracted 158,

287. Therefore, this chapter focuses on the AANR on the first photonic band.

To illustrate the mechanism of AANR in the first band, an example from Ref. 158 is

demonstrated in Fig. 6-1. The 2D square-lattice photonic crystal is composed of air holes in

dielectric materials with a relative permittivity of 12. As shown in Fig. 6-1(a), the lattice

constant is a, and the radius of air holes is 0.35a. The normal of the air-photonic crystal surface

is arranged to be parallel to the Γ-M direction. The EFC of the first TE band is shown in Fig.


82

6-1(b). For wave vectors k on the Γ-M edge, the eigenfrequency increases monotonically, while

the curvature of the equal-frequency contours gradually changes from concave to convex. As

demonstrated in Fig. 6-1(c), for an incident beam from the air with frequency ω, wave vector

k1 and group velocity vg1, if there is a convex EFC on the first band of the photonic crystal and

the contour intersects with the surface normal, a negative refracted beam with wave vector k2

and group velocity vg2 will arise.

(a)

(b)

(c)

Fig. 6-1 (a) Structure of the photonic crystal; (b) EFC of the first TE photonic band (unit of frequency is 2πc/a); (c) Negative refraction of a light beam. k1 and vg1 are the wave vector and group velocity of the incident beam, while k2 and vg2 are the wave vector and group velocity of the refracted beam.


83

The AANR frequency range is illustrated in Fig. 6-2. The black curve is the first photonic

band in Γ-M direction and the red line is the light line ω = ck which is shifted to M point 103.

The modes at the first photonic band below the light line can exist and propagate inside photonic

crystals to generate negative refraction. Light above the light line cannot be confined in

photonic crystals thus the upper limit of AANR ωu can be obtained by finding the intersection

of the light line and the first band. The corresponding frequency of the intersection is the upper

limit, ωu, of the AANR frequency range. The relationship between the radius of EFC curvature

and wave number |k| along Γ-M is the blue curve depicted in Fig. 6-2. As illustrated in [10],

negative refraction can only be generated when EFC curvature becomes convex. The lower

limit, ωl, of AANR is therefore determined by the point where the concave EFC changes to a

convex one. In other words, the radius of EFC curvature changes from a positive infinity to a

negative infinity at the lower limit ωl of AANR frequency range. An incident light from any

angle within the range between ωu and ωl will result in a negative refraction beam.

Fig. 6-2 Upper limit and lower limit of the AANR frequency range and the reference point used in optimization.

To identify the lower and upper limit of AANR frequency range, the first photonic band

and the radius of EFC curvature along Γ-M should be obtained. In this chapter, they are

calculated by using the finite element method. TM modes and TE modes are both considered.


84

In order to determine a proper finite element mesh size, the photonic band and radius of

EFC curvature of a simple structure at different mesh size are calculated and compared. As

shown in Fig. 6-3(a) and Fig. 6-4(a), the microstructure within a unit cell is a dielectric square

column in the air for TM modes and a square frame for TE modes. The relative permittivity of

the dielectric material is 18. The unit cell is meshed uniformly by 16×16, 32×32, 64×64 and

128×128 square elements, respectively. Following the proposed FEA method, the

corresponding photonic band and radius of EFC curvature are calculated and depicted in Fig.

6-3(b) and Fig. 6-4(b). The results indicate that, apart from the coarse mesh with 16×16, other

meshes give the nearly identical results for both TM and TE modes. Therefore, the 64×64 mesh

has sufficient accuracy and will be adopted in the following optimization.

(a) (b)

Fig. 6-3 Grid convergence analysis for TM modes, (a) Microstructure of the unit cell; (b) Corresponding photonic band and radius of EFC curvature at different mesh size.

(a) (b)


85

Fig. 6-4 Grid convergence analysis for TE modes, (a) Microstructure of the unit cell; (b) Corresponding photonic band and radius of EFC curvature at different mesh size.

6.2 Topology Optimization

6.2.1 Modified objective function

The aim of this chapter is to design dielectric photonic crystals with a broad AANR

frequency range. Due to the lack of fundamental length scale in Maxwell’s equation, the AANR

frequency range can be measured by the non-dimensional ratio between the difference of upper

limit ωu and lower limit ωl and the mean value of them. Thus, the objective function of the

present optimization problem can be expressed as

Maximize 2)(

lu

luf

X (6-1)

Although the above objective function has a clear physical meaning, it is hard to formulate

ωu or ωl mathematically. An alternative way can be developed to overcome this issue. It is noted

that an initial guess design may not have AANR, which means ωl might be larger than ωu. Even

so, both ωu and ωl can be easily determined following the procedure described in Section 2. A

reference point on the first band just left to the lower limit ωl as shown in Fig. 6-2 is selected,

and then the radius of EFC curvature at the reference point is maximized. Consequently, the

lower limit ωl will be moved toward the reference point. As this process is repeated step by step

and the lower limit ωl will be gradually moved toward to the Γ point. As ωu is determined by

the constant light line, the decrease of ωl is equivalent to enlarge the AANR frequency range.

In this chapter, the normalized wave vector |k|a/2π of the reference point is set to be 20/2

smaller than that of the lower limit. During the optimization process, the reference point updates

iteratively with the change of lower limit.

To calculate the EFC, it is necessary to calculate the first eigenfrequency, ω, for various

wave vectors k = (kx, ky) within the irreducible Brillouin zone. In this chapter, 1/4 of the first

Brillouin zone is divide by 100×100 square grids and the first eigenfrequencies at grid points

are calculated numerically. This is illustrated by Fig. 6-5 with a coarse grid. The radius of EFC

curvature can be defined with


86

xyyxxyyyxx

yx

fffffffff

R

222

2/322

(6-2)

where x

x kf

and y

y kf

. xxf , yyf and xyf are the second-order partial derivatives.

Those derivatives will be calculated by using the finite difference method.

Due to the symmetry of EFC, only the first eigenfrequencies at the points on Γ-M boundary

and two adjacent parallel rows as shown in Fig. 6-5 will be used to calculate the first- and

second-order partial derivatives. Considering Δkx = Δky , the first partial derivatives of an

eigenfrequency, e.g. at point 1, can be expressed by

x

yx kff

254

(6-3)

Similarly, the second order partial derivatives are

2

1816

4 xyyxx k

ff

and

27372

4 xxy k

f

(6-4)

In summary, the present objective of the optimization problem is modified by maximizing

the radius of EFC curvature at the reference point as

Maximize xyxx

x

fffR

2 at the reference point (6-5)

Fig. 6-5 Illustration of the square grid within the irreducible Brillouin zone.

6.2.2 Sensitivity analysis


87

After the finite element discretization, the topology of the unit cell is represented by

assigning each element a design variable which is related to the property of this element. The

photonic crystals in this chapter are composed of two materials with permittivity ε1 and ε2

(where ε1 < ε2), respectively. A design variable xe are constructed by assuming its corresponding

element e is composed of material 1 with low permittivity ε1 when xe = 0, and material 2 with

high permittivity ε2 when xe = 1. In the optimization process, the evolution of topology is

reflected by the change of design variables, which is updated iteratively based on the elemental

sensitivity numbers. Sensitivity numbers denote the relative ranking of elemental sensitivities

which is the derivative of the objective function with regard to a design variable. The present

objective is to maximize the radius of EFC curvature at the reference point as described before.

Therefore, the sensitivity number αe for element e can be defined according to the chain rule as

8,4,27,5,16,3 i e

i

i

xy

xyi e

i

i

xx

xxi e

i

i

x

x

ee

xf

fR

xf

fR

xf

fRxR

(6-6)

For any grid point, i, within the irreducible Brillouin zone, it corresponds to the wave

vector ki, the first eigenfrequency ωi and eigenvector ui. The derivative of ωi with regard to xe

can be easily calculated by

ie

ie

Ti

ie

i

xxc

xuMKu

22

21

(6-7)

In this chapter, as the same as the band gap optimization of photonic crystals 216, a linear

interpolation scheme is used for TM modes and an inverse linear material interpolation is

adopted For TE modes:

eee xxx 21 1)( TM modes (6-8a)

eee

xxx 21

111)(

1

TE modes (6-8b)

Based on the material interpolation functions, the derivatives of matrices K and M to xe

For TM modes and TE modes can be obtained:

0

exK

and e

exMM

12

TM modes (6-9a)


88

e

exKK

12

11

and 0

exM

TE modes (6-9b)

Afterward, in order to improve the stability and convergence of the optimization process,

elemental sensitivity numbers except for that of the first iteration are averaged with their

historical information as

ke

ke

ke 1ˆ

21ˆ (6-10)

where k is the current iteration number.

6.2.3 Numerical implement of BESO

After obtaining the sensitivity number of every element, BESO will increase design

variables for elements with highest sensitivity numbers and decrease design variables for

elements with lowest sensitivity numbers simultaneously to maximize the radius of the EFC

curvature at the reference point. Different from other topology optimization problems, e.g.

maximizing stiffness of a structure with a given volume constraint, the optimization of photonic

crystals for AANR has no volume constraint. Numerical analysis indicates that the optimal

volume fraction of photonic crystals for AANR is normally around 50%. In order to accelerate

the search process, BESO starts from an initial design which is almost full of material 2 and

gradually decreases the volume fraction until reaches a prescribed target. Thereafter, the

evolution of the volume fraction will be determined by the variation history of the fractional

AANR frequency range. When the AANR frequency range achieves its maximum value, the

volume fraction would fluctuate around a constant. In the final stage, the volume fraction will

keep a constant until the topology is stably convergent.

BESO updates the design variables according to the relative ranking of sensitivity numbers

and the volume fraction of the next iteration. However, numerical experience indicates that the

design of photonic band gap crystals is very sensitive to the change of the design variables 216.

For the sake of caution, the design variable is assigned with discrete intermediate design values

between 0 and 1 with a constant step size Δxe. Δxe = 0.1 throughout the chapter, which means


89

the possible variation of the design variable in each iteration is limited to 0.1. Figure 6-6

illustrate the proposed BESO procedure for the design of AANR photonic crystals.

Fig. 6-6 Flow chart of the BESO procedure.


6.3.1 Optimized designs

In this chapter, photonic crystals for TM and TE modes are analyzed and optimized for

AANR property independently. A square lattice with lattice constant a = 1 is considered. The

photonic crystals are composed of two materials: low permittivity material 1, ε1 = 1 (air) and

high permittivity material 2, ε2 = 18 (Ge at 1.55 µm). The unit cell is meshed by 64×64 bilinear

square elements and plotted as greyscale images to illustrate the topologies of photonic crystals.

White and black areas denote materials 1 and 2, respectively, while grey areas denote the

transitional elements with permittivity between ε1 and ε2 in the optimization process. The


90

optimization process starts from an initial design which fully consists of material 2 except that

a circle with a radius of 0.1 in the center is material 1, as shown in the inset of Fig. 6-7.

Figure 6-7 (a) and (b) present the evolution histories of AANR frequency range, volume

fraction and topology of the unit cell during the optimization process of photonic crystals for

TM and TE modes, respectively. It can be seen that the fractional frequency ranges of the initial

guess design are -13.78% for TM mode and -17.89% for TE mode, which means the initial

design has no AANR property. BESO redistributes the constituent materials within the unit cell

step by step so that the AANR frequency range gradually increases. Meanwhile, the volume

fraction of material 2 reduces to the target volume fraction and then fluctuates up and down till

an optimal one around the target volume fraction is obtained. Both the AANR frequency range

and topologies are stably converged after 199 and 229 iterations for TM and TE modes,

respectively.

(a)


91

(b)

Fig. 6-7 Evolution histories of AANR frequency range, the volume fraction of material 2 and topology during the optimization for (a) TM mode; (b) TE mode.

The optimized designs with 3×3 unit cells for TM and TE modes and their corresponding

band diagrams with indicated AANR frequency range are shown in Fig. 6-8. The optimized

topology for TM modes shows that the high permittivity materials are separated by the air. The

maximized AANR range is 2.56% and the optimal volume fraction of material 2 is 40.80%.

Conversely, the optimized topology for TE modes shows that the high permittivity material is

connected and forms the hollow square with inner “comb” corners as shown in Fig. 6-8(b). The

corresponding AANR frequency range is 14.08% and the optimal volume fraction of material

2 is 50.98%.

(a)


92

(b) Fig. 6-8 Optimized designs with 3×3 unit cells and band diagrams for (a) TM mode; (b) TE mode. The corresponding AANR frequency range is indicated with green area.

Surprisingly, the distribution characteristics of constituent materials for AANR of TM and

TE modes are similar to those for the formation of the first TM and TE band gap 216 although

the band gap has not been considered in the proposed optimization algorithm. Luo et. al. 158 has

demonstrated photonic band gap crystals with AANR property, and their intuitive designs with

3×3 unit cells are shown in Fig. 6-9 for comparison. It shows the current optimized designs

have larger AANR frequency ranges for both TM and TE modes. Numerical calculations

indicate that the current AANR frequency ranges are 115% larger than that of the intuitive

design for TE mode ε2 = 12, and 24% larger for TM mode when ε2 = 18.

The numerical experiments also indicate that the optimized topology is independent of the

permittivity contrast of the constitutive materials except for the magnitude of the AANR

frequency range. Fig. 6-9 also shows the variation of fractional AANR frequency range against

the permittivity contrast of the constitutive materials. Similar to the conclusion of Luo et. al. 158, optimized AANR frequency range reduce monotonically with the decrease of permittivity

contrast. The smallest permittivity contrast for a 2D square-lattice photonic crystal having

AANR is about 8.6 for TE mode, and 13.4 for TM mode.


93

Fig. 6-9 Relationship between fractional AANR frequency range and permittivity contrast of two constitutive materials.

6.3.2 Influence of the filter

It is observed that the optimized topology for TE mode as shown in Fig. 6-8(b) contains

some inside “comb” teeth which might be difficultly manufactured. To overcome this problem,

similar to the topology optimization of elastic structures, the heuristic filter scheme with little

computational effort can be integrated into the optimization algorithm. The filtering scheme for

the BESO method can refer to Chapter 3. Fig. 6-10 shows the resulting topologies by using

various filter radii. It can be observed that the number of “comb” teeth reduces but do not totally

disappear when rmin = 0.025a, 0.05a and 0.075a. When the filter radius increases to 0.1a, the

“comb” teeth are successfully eliminated as shown in Fig. 6-10(d). Its fractional AANR

frequency range is 13.81% which is slightly smaller than that of the optimized design in Fig. 6-

8(b) without using the filter. However, the optimized design is shown in Fig. 6-10(d) may be

more practical and easily manufactured compared to the complicated topology shown in Fig.

8(b).


94

(a) (b)

(c) (d)

Fig. 6-10 Optimized topologies (3×3 unit cells) for TE modes using various filter radii, (a) rmin=0.025a, AANR frequency range 14.00%; (b) rmin=0.05a, AANR frequency range 14.01%; (c) rmin=0.075a, AANR frequency range 13.93%; (d) rmin=0.1a, AANR frequency range 13.81%.

6.4 Conclusions

This chapter develops a new topology optimization algorithm based on the BESO method

to design photonic crystals with broad AANR frequency ranges for both TM and TE modes.

The photonic crystals are assumed to be 2D and made of dielectric materials and air. An

objective function is put forward based on the definition of the upper limit and lower limit of

the AANR range. Numerical simulations show that the BESO algorithm is effective and

applicable. The resulting novel patterns have larger AANR frequency range than the intuitive

designs. The results indicate that the smallest permittivity contrast for a 2D square-lattice

photonic crystal having AANR is about 8.6 for TE modes, and 13.4 for TM modes. Furthermore,

the filter scheme can be used to improve the manufacturability of the optimized design for TE

mode. Since the principle of AANR for 3D photonic crystals and their equal-frequency contours

become extremely complicated, future work on topology optimization of 3D photonic crystals

with AANR is recommended.

Chapter 7 Topology Optimization of 3D Photonic Crystals with Large Omnidirectional Band Gaps

95

CHAPTER 7

Topology Optimization of 3D Photonic Crystals with Large Omnidirectional Band Gaps

In this chapter, a topology optimization algorithm is proposed based on BESO method. 3D

dielectric photonic crystals with asymmetric cubic lattices are systematically investigated.

Some novel structures of 3D photonic crystals with large omnidirectional band gaps are

obtained. Then the geometric characteristics of 3D photonic crystals are analyzed based on the

optimization results.

7.1 Topology Optimization Algorithm

The aim for designing a 3D photonic crystal is to generate and maximize a band gap

between two adjacent photonic bands (referred to as band m and band m+1). The dimensionless

band gap-midgap ratio is used to measure the size of the band gap, and the objective function

of this topology optimization problem can be expressed as

bottop

bottopf

2 :Max X (7-1)

where X is the design variables representing the structure of the unit cell. X = [x1, x2, …, xn], n

is the total number of finite elements. The upper bound of the photonic band gap is ωtop =

min(ωm+1(k)) and the lower bound is ωbot = max(ωm(k)) for all the wave vectors in the

irreducible Brillouin zone.

A design variable xe is a value associated to element e according to its constitutive material.

The photonic crystals in this chapter are assumed to consist of air (ε1 = 1) and silicon (ε2 =

12.96). The value is assigned by appointing xe = 0 when element e is composed of air, while xe

= 1 when element e is composed of silicon. In the design of photonic crystals, the size of band

gap is very sensitive to the change of the design variable. To ensure a stable optimization

process, the material of an element changes gradually from one to another by using a discrete

design variable. The relative permittivity of element e will be interpolated using Eqn. 6-2.


96

eee

xxx 21

111)(

1

(7-2)

Although this method can be applied for photonic crystals with any lattice types, only the

cubic lattice is explored in this chapter. For the highly symmetric structures with SC lattice, the

irreducible Brillouin zone is 1/48 of the first Brillouin zone (Fig. 7-1(a)), and only wave vectors

on the path connecting 4 high symmetry points are evaluated to obtain the band diagram of the

photonic crystal. Cubic lattice with no symmetry constraints includes both asymmetric

structures and symmetric structures (SC, BCC or FCC). However, the corresponding

irreducible Brillouin zone will expand to the entire first Brillouin zone, and the computational

load to solve the band diagram increases significantly. Considering time reversal symmetry

(ω(k) = ω(-k)), the wave vectors in half of the first Brillouin zone should be taken into account

(Fig. 7-1(b)).

(a) (b)

Fig. 7-1 The first Brillouin zone of (a) symmetric simple cubic lattice and (b) asymmetric cubic lattice. The shaded zone denotes in which wave vectors need to be calculated.

Sensitivity analysis is conducted to evaluate the variation of the objective function caused

by the variation of a design variable. The method is the same with that for 2D photonic crystals

in Chapter 2. An elemental sensitivity αe is the derivative of the objective function regarding

the design variable xe.

2)(

4bottop

e

bottop

e

topbot

ee

xxx

f

X (7-3)


97

For a wave vector k and its corresponding frequency ω and magnetic field h, the derivative

of ω with regard to xe can be calculated by

h

Mk

Kh

kk

e

r

e

rT

e xxc

x22

21

(7-4)

where Kr and Mr are the reduced stiffness and mass matrices without the transversality

requirement part. Using the material interpolation scheme Eqn. 7-2, the derivations of Kr and

Mr to xe are

e

e

r

xK

K

12

11

and 0

e

r

xM

(7-5)

Thus, the sensitivity of the objective function can be calculated for every element. In order

to improve the stability and convergence of the algorithm, the elemental sensitivities are further

averaged with their historical information and processed by a heuristic filter scheme. The

optimal volume fraction is determined according to the increase and decrease history of band

gap size. The detailed procedures are the same with Chapter 2.

The optimization algorithm starts from a randomly generated structure. Then, FEM

analysis is conducted to obtain the photonic bands and the corresponding magnetic fields.

According to the calculated sensitivities, BESO increases design variables for elements with

high sensitivities (from 0 to 1) and decreases design variables for elements with low sensitivities

(from 1 to 0) simultaneously. A new design is formed with the updated design variables, and

thereafter the FEM analysis and BESO scheme are repeated. Such an iterative process continues

and evolves the topology of the primitive unit cell towards its optimum. The optimization

process is divided into two stages. In stage 1, the unit cell is meshed by coarse 16×16×16 cubic

elements, and an intermediate design is obtained. Then, this intermediate design is refined by a

24×24×24 grid, and a more elaborate design is achieved. The volume fraction of the dielectric

material keeps a constant in stage 2. The flow chart of this BESO process is shown in Fig. 7-2.


98

Fig. 7-2 Flow chart of BESO algorithm



Denote the band gap between the mth and (m+1)th photonic bands as “G-m”, m is the band

number. A typical optimization process for maximizing band gap G-4 is illustrated in Fig. 7-3.

The insets demonstrate the evolution history of the unit cell structure. Blue denotes silicon,

white denotes air, and the colors in between indicate elements with intermediate design

variables. For the initial design, the relative size of band gap is -23.8%, which means there is

no band gap at all. The volume fraction of silicon at the commencing point is 0.591. It gradually

decreases while the band gap size stably increases from negative to positive. A band gap arises

and converges after 76 iterations in stage 1 and 16 iterations in stage 2. The maximized band

gap is between normalized frequency 0.4066~0.5352, with a relative size of 27.4%. Although

many intermediate elements emerge in the optimization process, the unit cell converges to a

clear 0/1 design with few intermediate elements at the end of stage 2. The lattice of the


99

optimized G-4 structure is BCC. The final design and corresponding band diagram are shown

in Fig. 7-4(a) and 7-4(c). To clearly demonstrate the photonic structure, the FEM model is

smoothed by drawing the isosurface of threshold sensitivity αth in the last iteration.

Fig. 7-3 Evolution history for maximizing G-4.

(a) (b)

(c) Fig. 7-4 (a) Unit cell structure with maximized G-4. (b) Prototype of the optimized photonic crystal fabricated by 3D printing ABS. (c) Corresponding band diagram. The wave vectors are taken on the nodes of a uniform 5×10×10 grid covering the shaded zone in Fig. 7-1(b).


100

It should be noted that the topology optimization of 3D photonic crystals is a non-convex

problem, and there are many local optimums that may possess large band gaps. For example,

the Gyroid structure (shown in Fig. 7-5) is another solution for G-4, with a slightly narrower

band gap. The optimized design of photonic crystal depends highly on the initial design. In this

chapter, the optimization process is repeated for 10 times for every potential position of band

gap. 10 different initial designs are generated and then the optimized results with the largest

band gap are singled out. The 3D photonic crystals with maximized G-2 to G-10 are obtained

and illustrated in Fig. 7-6 in 2×2×2 unit cells. Apparently, G-1 does not exist because the two

fundamental bands cannot separate.

Fig. 7-5 Another optimization result for G-4 (single Gyroid structure).

It can be seen from Fig. 7-6 and Table 7-1 that, compared to the limited photonic crystal

structures in previous research, the proposed topology optimization approach is able to find the

band gap at any specified bands, which provides the possibility for systematically exploring the

geometric rules for the design of 3D photonic bandgap crystals. Among these 15 topology

optimization results, G-2 and G-3 are relatively small, but all the other PBGs are larger than

15%. With the band number of PBGs increasing, the lower boundary of band gap stably elevates,

and the PBGs work for electromagnetic waves with higher frequencies. The geometry of

photonic crystal structures for PBG at the higher bands has more subtle branches and

connections to modulate electromagnetic wave with a shorter wavelength. Such a complicated

geometry is hardly identified without the help of the developed topology optimization technique.


101

In previous research, almost all the band gaps are found between bands 2~3, 4~5, 5~6 and

8~9 88, 225-226, 228-230, 234, 236-238, 241, 288-290. However, this method successfully realizes band gaps

between any two adjacent bands from 2~3 to 16~17. In Fig. 7-6, G-8 is identical with the

diamond structure that has been widely reported, but all other optimized structures in this paper

are first reported. Several representative traditional designs made from dielectric materials with

cubic lattices are compared to the optimization results in Fig. 7-7, denoted by red squares.

Notice that the band number for photonic crystals in BCC and FCC lattices are adjusted. For

example, the band gap of the diamond structure is between bands 8~9 in the asymmetric SC

lattice instead of 2~3 in FCC lattice. This is because in the real space the unit cell of an SC

lattice is 3 times larger than that of an FCC lattice, and the number of scattering and resonant

structures in one unit cell quadruples.

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)


102

(j) (k) (l)

(m) (n) (o)

Fig. 7-6 Optimization results with maximized band gaps between different photonic bands. (a) G-2. (b) G-3. (c) G-4. (d) G-5. (e) G-6. (f) G-7. (g) G-8. (h) G-9. (i) G-10. (j) G-11. (k) G-12. (l) G-13. (m) G-14. (n) G-15. (o) G-16.

Table 7-1 Maximized band gaps at different frequency levels

Volume fraction

Frequency range ∆ω (ωa/2πc)

Central frequency ω0 (ωa/2πc)

Band gap size (%)

G-2 0.242 0.3395 ~ 0.3794 0.3595 11.1 G-3 0.279 0.3639 ~ 0.3869 0.3754 6.1 G-4 0.234 0.4010 ~ 0.5281 0.4645 27.4 G-5 0.249 0.4377 ~ 0.5231 0.4804 17.8 G-6 0.271 0.4544 ~ 0.5313 0.4929 15.6 G-7 0.275 0.4746 ~ 0.5527 0.5136 15.2 G-8 0.233 0.5015 ~ 0.6744 0.5880 29.4 G-9 0.261 0.5183 ~ 0.6298 0.5741 19.4 G-10 0.273 0.5239 ~ 0.6252 0.5746 17.6 G-11 0.265 0.5525 ~ 0.6628 0.6077 18.2 G-12 0.269 0.5865 ~ 0.6949 0.6173 18.4 G-13 0.256 0.5829 ~ 0.7110 0.6470 19.8 G-14 0.269 0.5861 ~ 0.7105 0.6483 19.2 G-15 0.254 0.6188 ~ 0.7451 0.6810 18.8 G-16 0.257 0.6288 ~ 0.8032 0.7160 24.4


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Fig. 7-7 Size of PBGs for optimization results (black line) and traditional designs (red squares). Ⅰ - Orthogonally crossing rods, ε = 12.96, f = 6.6% 234. Ⅱ - Single Gyroid structure, ε = 13, f = 26.6% 291. Ⅲ - BCC Spiral structure, ε = 12.25, f = 20% 231. Ⅳ - Hollow spheres connected by rods, ε = 12.96, f = 16.26% 241. Ⅴ - Diamond structure, ε = 13, f = 29.6% 103. Ⅵ - FCC woodpile structure, ε = 13, f = 19.5% 103, 226.

The traditional designs mainly focused on symmetric photonic crystals since the symmetry

offers a convenient way to categorize the electromagnetic modes. For example, the designs a

and d in Fig. 7-7 have symmetric SC lattice (Oh). The former one is constituted by orthogonally

crossing rods (Fig. 7-8(a)), and the PBG appears between bands 2~3. The latter one has a similar

topology but with hollow spheres at the joints (Fig. 7-8(b)), and its PBG appears between bands

5~6. These two structures and their variants have been widely reported in 233-234, 241, 290. Their

sizes of band gap are respectively 6.6% and 16.5%, which are smaller than their asymmetric

counterparts, indicating removing symmetry constraints leads to larger band gaps. This is

because the symmetry constraint limits the subsets of optimization solution. Therefore the

symmetric solutions are much less and the band gaps are narrower, which is consistent with the

findings in 2D photonic crystals 292. In some other cases such as G-2, G-4, the structures with

certain rotational or mirror symmetries lead to a true maximal PBGs. In all, the optimized

structures reported in this paper provide more options in devising practical optical components

for the selection of the frequency range, member size, and structural manufacturability.


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(a) (b)

Fig. 7-8 Optimization results for symmetric SC lattice.

7.2.2 Geometric characteristics of 3D photonic crystals

The topology optimization results provide the possibility for systematically exploring the

geometric rules for the design of 3D photonic bandgap crystals. The geometries of 3D photonic

crystals are very complicated as shown in Fig. 7-6 but they share some common characteristics.

Generally, the dielectric material forms a continuous framework similar to open cell foams,

which allows air to freely flow through. Such bicontinuous open cell photonic crystals provide

excellent structural stiffness, as well as fluid permeability, which benefits the fabrication of 3D

photonic crystals. The results further confirm that 3D bicontinuous structures are the ideal

layouts of a dielectric material for 3D omnidirectional band gaps.

The bicontinuous characteristic originates from the formation of PBG, which is

determined by the photonic modes at the edge of the band gap. The photonic modes at bands 3

and 4 for G-3, and bands 16 and 17 for G-16 are illustrated in Fig. 7-9. The wave vectors

correspond to those at the peak of the lower bound of the PBG. To demonstrate where the

electric field energy concentrates, an intensity zone is defined, where the electric field intensity

is 20% of the highest ones in the unit cell, as shown in Fig. 7-9 by the red region.

It can be seen from Fig. 7-9 (a) and (c) that, on the lower boundary of the band gap (the

4th band), the electric field concentrates most energy in silicon to lower the frequency 103. The

connected silicon branches provide a channel for the electric field to transmit, forming a

continuous dielectric structure. This is similar to the 2D TE photonic band gap crystals 210, 212,

216. On the upper boundary (the 5th band), the distribution of electric field is orthogonal to that

on the lower boundary. In order to promote the upper bound and enlarge the band gap, the


105

electric field shuns the dielectric materials and concentrates in the air void as much as possible.

In 2D TE photonic crystals, the connected dielectric structures will lead to isolated air voids

inevitably. Hence, the high-intensity electric fields at the upper boundary are compelled to cut

through dielectric materials, which prejudice the expansion of band gap. Conversely, for 3D

photonic crystals, connected dielectric framework and connected air voids are compatible.

Silicon and air will interpenetrate into each other to imitate the orthogonal electric fields and

enlarge the band gap, generating a bicontinuous photonic structure. This geometric property is

also found in many traditional designs 76, 103, 235-236, 239-241, 289, 293-294. But of course, it does not

rule out the possibility that other types of designs (structures with separated dielectric materials

or isolated air voids) possess large band gaps, like the spiral structure 231, inverse opal structure 238, and the one shown in Fig. 7-8(b).

Fig. 7-9 (a) (b) Distribution of high electric intensity zone for photonic modes on band 3 and band 4, G-3 structure, k = (0, 0, 0.4π). (c) (d) Distribution of high electric intensity zone for photonic modes on band 16 and band 17, G-16 structure, k = (π, 0, 0). (e) Effective permittivity for G-3 structure on different directions, which shows obvious anisotropy. Maximum value = 3.91 is on the z direction while minimum value = 2.12, A = 0.543. (f) Effective permittivity for G-16 structure on different directions, indicating it tends to be isotropic. Maximum value = 2.82, minimum value = 2.73, A = 0.967. (g) Colorbar.


106

As shown in Fig. 7-9(c), for G-3 structure, the direction of wave vector at the lower edge

of the band gap (0, 0, 0.4π) coincides with the direction of maximum effective permittivity (z

direction). When the band number arises, the density of states around PBG increase. Photonic

bands below the band gap tend to be flat and wave vectors in many different directions should

be minimized to enlarge the gap, and hence the effective permittivity tends to be isotropic, such

as the G-16 structure in Fig. 7-9(f). This phenomenon can be employed to design novel optical

devices with simultaneously band gaps (complete or partial) and objective anisotropic/isotropic

properties like effective permittivity or permeability.

To obtain the maximum band gap, the eigenfrequencies for the wave vector at max(ωm(k))

are minimized. Therefore, during the evolution of optimized structures, the dielectric material

gradually gathers and forms continuous paths along the directions of the specific wave vector.

As a result, the macroscopic effective property such as permittivity, εeff, exhibits its

directionality. , which can be calculated by the homogenization method 62. As shown in Fig.

7-9(e) for G-3 structure, the direction of wave vector at the lower edge of the band gap (0, 0,

0.4π) coincides with the direction of maximum effective permittivity (z direction). When the

density of states at the edge of PBG increases, the lower bound of PBGs tends to be flat and

wave vectors in many directions should be minimized in order to enlarge the gap. Thus, the

effective permittivity tends to be isotropic, such as the G-16 structure in Fig. 7-9(f).

An isotropy factor is defined as A = min(εeff)/max(εeff). For a given structure, a smaller A

indicates higher anisotropy, and A = 1 means that it has isotropic effective permittivity. The

isotropy factors for G-2 ~ G-16 are calculated and plotted in Fig. 7-7. It can be seen that there

is an apparent correlation between the isotropy of εeff and the size of PBG. Generally, photonic

crystals with large band gaps tend to be isotropic with a higher A. However, it should be noted

that the structural isotropy does not necessarily cause the increase of a certain band gap. To

some extent, the geometry symmetry in the past design guarantees the isotropy of photonic

crystals. Overemphasis on the structural isotropy during the design process, e.g., the adoption

of the symmetry as discussed above leads to a less optimal solution with a small band gap.


107

7.3 Conclusions

In this chapter, 3D photonic crystals with large omnidirectional band gaps are

systematically designed. Symmetric and asymmetric cubic lattice are considered in this chapter,

and the potential band gaps below the 17th band are explored. Different from previous research,

this method successfully acquires band gaps at all the investigated positions for the asymmetric

cubic lattice. Among the 15 topology optimization designs, 13 of them possess band gaps larger

than 15% and 3 of them are larger than 24%. Compared to the designs with symmetric simple

cubic lattice, asymmetric designs tend to have larger band gaps. The optimization results have

a uniform bicontinuous characteristic. The underlying reason is that, for the modes at the lower

boundary of the band gap, the dielectric material needs to provide a path for the electric field

to bring down the frequency, while for the upper boundary, the corresponding orthogonal

electric field concentrates in the air to promote the frequency. The unconstrained optimization

process built a connection between photonic bands and the macroscopic material property. This

property can be employed to guide the design of 3D photonic crystals. Further work is

recommended for exploring other materials and lattice types, quantitatively analysis of the

geometric property, and optimizing 3D photonic crystals for other optical phenomena.

Chapter 8 Conclusions

108

CHAPTER 8

Conclusions

Photonic crystals have many intriguing applications. For certain materials, the property of

photonic crystals relies on their structures. The objective of this study is to design the novel 2D

and 3D photonic crystal structures to realize and improve their performance on band gap and

AANR by using BESO method.

In the first stage, a new optimization algorithm based on BESO is proposed to open and

enlarge band gaps in the 2D photonic crystals for both the TE and TM modes. The proposed

optimization algorithm can successfully open band gaps from between bands 1~2 to 10~11 for

both the TM and TE modes. Some optimized structures exhibit novel patterns that are

remarkably different from traditional designs. The proposed algorithm is computationally

efficient as the solutions usually converge within 100 iterations.

To systematically study the influence of symmetry on the size of band gaps, 2D symmetric

and asymmetric photonic crystals are optimized using BESO. Designs for both the TM and TE

modes, square and hexagonal lattices are obtained for photonic band gaps from between bands

1~2 to 10~11. The results show that the band gaps of the asymmetric designs are larger than

those of the symmetric ones. Moreover, the largest TM band gap for photonic crystals with a

square lattice achieves a comparable size to photonic crystals with a hexagonal lattice,

extending current state-of-the-art understanding. This asymmetric design method opens up a

new possibility in photonic crystal designs towards larger band gaps due to the break of the

symmetric rules in the conventional design approach.

Next, an efficient approach is proposed to design 2D photonic crystals with large complete

band gaps. The numerical examples demonstrate the effectiveness of the method and some

innovative designs with both the square lattice and hexagonal lattices are obtained, one with a

relative size of 24.12% achieves the largest complete band gap ever reported. The large

complete band gap emerges because the resulting designs have the characteristics of both TE


109

and TM structures and both Mie resonance and Bragg scattering are supported. The

optimization result can be used in hollow core photonic crystal fibers.

Further, a BESO algorithm is developed to generate AANR property in 2D photonic crystal

and maximize its frequency range. Numerical simulations show the novel patterns have a much

larger AANR frequency range than the intuitive designs. The results indicate that the minimal

permittivity contrast for a 2D square lattice photonic crystal having AANR is about 8.6 for TE

modes, and 13.4 for TM modes. The sensitivity filter can be used to improve the

manufacturability of the optimized design for TE mode. Future work on topology optimization

of 3D photonic crystals with AANR is recommended.

Finally, the topology optimization method is applied to design the 3D photonic crystals

with omnidirectional band gaps. Band gaps from between bands 2~3 to 16~17 are explored for

both symmetric and asymmetric cubic lattices. In contrast to previous research, this method

acquires band gaps between any two neighbor bands for asymmetric cubic lattice and obtains

novel 3D photonic structures. Compared to the symmetric designs, the asymmetric results tend

to have larger band gaps. Among the 15 maximized band gaps, 13 are larger than 15 percent

and three are larger than 24 percent. The topology-optimized designs have a uniform

bicontinuous characteristic that originates from the orthogonal distribution of electric fields at

the bounds of the band gaps. The distribution of the dielectric material follows the directions of

the wave vectors of the maximum eigenfrequencies at the lower bounds. Consequently, the

macroscopic effective property such as permittivity exhibits corresponding directionality. The

isotropy of the effective permittivity indicates the lower bound flatness. Photonic crystals with

large band gaps tend to have isotropic effective permittivity.

Although this research has obtained these interesting results and innovative designs, there

are some limitations. First, the topology optimized designs have not been verified by

experiments. The manufacture of photonic crystals with arbitrary structures and high

permittivity materials are still difficult. Limited by the complicated fabrication techniques of

2D and 3D photonic crystals, the experimental verification will be carried out in the future as

an independent study. Another limitation is that this research only considers the ideal photonic


110

crystals. Topological design of photonic crystal slabs, waveguides, cavities, and other

applications and devices are not considered in this thesis.

In the future, several promising topics shown below can be put effort into: (1) Improve the

optimization algorithm, enhance its efficiency and convergence. Increase the clearness of the

structural boundary. (2) Employ other numerical method or commercial software to increase

the speed of FEM analysis. (3) Topology optimization for 3D photonic crystals with AANR

properties. (4) Take the nonlinear property of materials into account in topology optimization.

(5) Fabricate and test the topology optimization designs. (6) Explore the other optical property

such as topological photonics.

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