Optimal design of photonic crystals 1 4

Chapter 4Bandgap Optimization of Photonic Crystal Structures Having solved the infinite-dimensional eigenvalue problem, which is ized by the wave vector and dielectric function, we are r

Chapter 4

Bandgap Optimization of

Photonic Crystal Structures

Having solved the infinite-dimensional eigenvalue problem, which is ized by the wave vector and dielectric function, we are ready to describe the

with the simple single band gap optimization in the two-dimensional TE case

as a model problem, and formulate a linear fractional SDP using the subspacemethod and mesh adaptivity Then we describe the extension to TM polarizationand combined TE/TM polarizations, as well as multiple band gaps optimization.Results on both square and hexagonal lattices are then presented and discussed

The objective in photonic crystal design is to maximize the band gap betweentwo consecutive frequency modes Due to the lack of a fundamental length scale

in Maxwell equations, it can be shown that the magnitude of the band gap scales

by a factor of s when the crystal is expanded by a factor of 1/s Therefore, it

is more meaningful to consider the gap-midgap ratio instead of the absolute

The optimization problem is to look for an optimal configuration ε∗(r) which isthe solution of

sup

ε(r)

After discretizing this infinite optimization problem by procedures described insection 3.1.2, we obtain the following finite-dimensional optimization problem:

k∈Phλmh(ε, k) ,

(4.1.3)

In this problem a subtle difference between TE and TM polarizations lies in the

from (3.1.10)

Unfortunately, this optimization problem is non-convex; furthermore, it fers from a lack of regularity at the optimum The reason for this is that the

multiplicity, and multiple eigenvalues at the optimum are typical of structureswith symmetry As a consequence, the gradient of the objective function J (ε)with respect to ε is not well-defined at points of eigenvalue multiplicity, and thusgradient-based descent methods can run into serious numerical difficulties andconvergence problems

In this section we describe our approach in solving the band gap optimizationproblem based on a subspace method and SDP In our approach, we first re-formulate the original problem as an optimization problem in which we aim tomaximize the band gap obtained by restricting the operator to two orthogonalsubspaces The first subspace consists of eigenfunctions associated to eigenvaluesbelow the band gap, and the second subspace consists of eigenfunctions whoseeigenvalues are above the band gap In this way, the eigenvalues are no longer

explicitly present in the formulation, and eigenvalue multiplicity no longer leads

to a lack of regularity The reformulated optimization problem is exact butnon-convex and large-scale To reduce the problem size, we truncate the high-dimensional subspaces to only a few eigenfunctions below and above the bandgap [17, 47], thereby obtaining a new small-scale yet non-convex optimizationproblem Finally, we keep the subspaces fixed at a given decision parameter vec-tor to obtain a convex semidefinite optimization problem for which SDP solutionmethods can be efficiently applied

where Φ(ε, k) and Ψ(ε, k) consist of the first m eigenvectors and the remaining

N − m eigenvectors column-wise, respectively, of the eigenvalue problem (3.1.9)

We will also denote the subspaces spanned by the vectors of Φ(ε, k) and Ψ(ε, k)

maximizing the gap-midgap ratio between the two subspaces sp(Φ(ε, k)) andsp(Ψ(ε, k)) The latter viewpoint allows us to develop an efficient subspace ap-proximation method for solving the band gap optimization problem as discussedbelow.

associated matrices

Θ(ˆε, k) := [Φ(ˆε, k) | Ψ(ˆε, k)] = [u1h(ˆε, k) umh(ˆε, k) | um+1h (ˆε, k) uNh (ˆε, k)] ,

where Ψ(ˆε, k) and Ψ(ˆε, k) consist of the first m eigenvectors and the remaining

N − m eigenvectors, respectively, of the eigenvalue problem

Ah(ˆε, k)ujh= λjhMh(ˆε)ujh, 1 ≤ j ≤ N

of the subspaces sp(Φ(ε, k)) and sp(Ψ(ε, k)) and are no longer functions of thedecision variable vector ε

typi-cally be quite large In order to reduce the size of the inclusions, we reduce thedimensions of the subspaces by considering only the “important” eigenvectorsamong u1h(ε, k) umh(ε, k), um+1h (ε, k) uNh (ε, k), namely those ak eigenvec-

the formulation P2ˆε yields the following reduced optimization formulation:

eigenvalue constraints, in exact extension of active-set methods in nonlinear

small-est integers that satisfy

well-spanned by including all relevant eigenvectors corresponding to those eigenvalueswith multiplicity at or near the current min/max values

to reduce the nonlinearity of the underlying problem Furthermore, we show

as a linear fractional semidefinite program, and hence is solvable using moderninterior-point methods

4.2.3 Fractional SDP formulations for TE and TM polarizations

We now show that by a simple change of variables for each of the TE and TM

program and hence can be further converted to a linear semidefinite program

for convenience:

y := (y1, y2, , yny) := (1/ε1, , 1/εnε, L, U )

We also amend our notation to write various functional dependencies on y

We note that the objective function is a linear fractional expression and

is a linear fractional SDP Using a standard homogenization [18, 21], a linearfractional SDP can be converted to a linear SDP, as discussed in section 2.4 Weintroduce new decision variable notation:

Y := (Y1, Y2, , YnY) := (θy1, , θyny, θ)

Amending the notation to write the previous functional dependencies on Y

in-stead of y, we finally re-write PTEyˆ as,

for convenience:

z := (z1, z2, , znz) := (ε1, , εnε, 1/L, 1/U )

Similar to the TE case, we amend our notation to write various functional

U − L

znz−1− znz

utilizing the parameter dependence introduced in (3.1.4), and multiplying the

for the TM polarization as

a linear SDP

We introduce new decision variable notation:

Z := (Z1, Z2, , ZnZ) := (θz1, , θznz, θ)

Amending the notation to write the previous functional dependencies on Z

very efficiently by using modern interior point methods Here we use the SDPT3

software [56] for this task

4.2.4 Multiple band gaps optimization formulation

In this section, we derive the most general optimization formulation – multiplecomplete band gaps optimization, i.e., multiple prohibitive frequency ranges that

a set of J TM-polarized frequency bands for which we seek to achieve completegaps We define the discrete eigenvalue gap-midgap ratio for the jth gap of TEand TM cases respectively as

JhT E,j(ε) = 2

min

k∈PhλT E,mj+1h (ε, k) − max

k∈PhλT E,mjh (ε, k)max

band gaps for only TE or TM polarization, one can simply omit the non-relevant

then formulation (4.2.12) is generalized to treat more generic cases in which thenumber and location of TE band gaps are allowed to differ from those of TM

band gaps Some auxiliary variables are defined as

If ˆε is a given parameter vector, we can similarly derive a ˆγ = (1/ˆε1, , 1/ˆεnε).One can easily see that the stiffness matrix of the eigensystem in TE case

i=1γiAT E

4.2.2, the following approximate and reduced matrices and the associated spaces are constructed:

determined similarly as described in (4.2.5) The resulting optimization

U T E

j +L T E j

,

T M

j −U T M j

L T M

j +U T M j

x = (γ, ε, LT E1 , , LT EJ , U1T E, , UJT E, 1/LT M1 , , 1/LT MJ , 1/U1T M, , 1/UJT M, F )

(4.2.16)The bilinear constraints (the fifth through the seventh set of constraints

in (4.2.15)) can be linearized around the previous solution to obtain a linear

ˆiˆi+ ˆεi(γi− ˆγi) + ˆγi(εi− ˆεi) = 1 The resulting linearized semidefinite program

can be efficiently solved using modern interior point methods such as the SDPT3

software [56]

Our computational procedure incorporating mesh adaptivity in conjunction with

0 start with a coarse mesh, e.g., 8 × 8;

an error tolerance εtol;

program (4.2.15);

and go to 1 ;

Computation procedures for (4.2.7) and (4.2.9) follow the above in a similarfashion

We consider a two-dimensional photonic crystal confined in the computationaldomain of a unit cell of either square lattice or hexagonal lattice shown in Figure3.1, with lattice constant a = 2 At the beginning of the simulation, the domain

Ω is decomposed into a uniform grid of dimension 8 × 8, which yields a mesh

the adaptively refined mesh is able to capture details as finely as a 128 × 128uniform mesh

The dielectric function ε is composed of two materials with dielectric

exploited to further reduce the dielectric function to be defined in only 1/8 ofthe computational domain in a square lattice, and 1/12 in a hexagonal lattice

At the finest refinement level (128 × 128), the maximum number of decision ables associated with the dielectric constants is (1 + 128/2) × 128/4 = 2080 forboth square and hexagonal lattices

vari-In all the results presented herein, the eigenvalues are plotted in the

results, we also describe our results in terms of the frequency gap-midgap ratio

mj h

min

k∈Phωmjh (ε, k)min

k∈Phωmjh (ε, k).

The frequency gap-midgap ratio is used as the objective function in somepublished research (see, e.g., [33, 42], while the eigenvalue gap-midgap ratio isused in other published research (e.g., [51, 50, 44]) Despite the obvious differencebetween the eigenvalue and frequency (the former being the square of the latterdivided by the speed of light), the optimal crystal structures have been observed

to be astonishingly consistent when either is used in the gap objective function insingle band gap optimization problems [33, 50, 44] While it is intuitive that thefrequency relative gap-midgap ratio should be monotone in the eigenvalue gap-midgap ratio, one can create pathological counterexamples Herein we choose tooptimize the eigenvalue gap-midgap ratio because the first four sets of constraints

of our optimization model (4.2.15) are linear in the eigenvalues and so require

no extra linearization themselves, and the fifth and sixth constraints are onlymodestly nonlinear in the eigenvalues Of course, should one wish to optimizethe frequency gap-midgap ratio, the resulting nonlinear constraints (the fifth andsixth set of constraints in (4.2.15)) could be linearized as discussed earlier whenconstructing the linear semidefinite program

4.3.2 Choices of parameters

Initial configuration

Because the underlying optimization problem may have many local optima, theperformance of the proposed algorithm can be sensitive to the choice of the

optima as shown in Figure 4.1 for the second TE band gap and in Figure 4.2for the fourth TM band gap Therefore, the choice of the initial configuration isimportant We examine here two different types of initial configurations: pho-tonic crystals exhibiting band gaps at the low frequency spectrum and randomdistribution

The well-known photonic crystals (e.g., dielectric rods in air – Figure 4.2(a),air holes in dielectric material, orthogonal dielectric veins – Figure 4.1(d)) exhibitband-gap structures at the low frequency spectrum Such a distribution seems

to be a sensible choice for the initial configuration as it resembles various knownoptimal structures [16] When these well-known photonic crystals are used as theinitial configuration, our method easily produces the band-gap structures at thelow frequency mode (typically, the first three TE and TM modes) On the otherhand, maximizing the band gap at the high frequency mode (typically, above thefirst three TE and TM modes) tends to produce more complicated structureswhich are very different from the known photonic crystals mentioned above As

a result, when these photonic crystals are used as the initial configurations formaximizing the band gap at the high frequency mode, the obtained results areless satisfactory

Random initial configurations such as Figures 4.1(a) and 4.2(d) obtainedfrom a 64 × 64 uniform mesh have very high spatial variation and may thus besuitable for maximizing the band gap at the high frequency mode Indeed, weobserve that random distributions often yield larger band gaps (better results)than the known photonic crystals for the high frequency modes Of course, therandom initialization does not eliminate the possibility of multiple local optimaintrinsic to the physical problem In view of this effect, we use multiple random

distributions to initialize our method In particular, we start our main algorithmwith a number of uniformly random distributions as initial configurations toobtain the optimal structures in our numerical results discussed below.

(a) Initial crystal

lattice

Subspace dimensions

eigenvectors to enhance convergence to an optimum In our numerical

(a) Initial crystal

configura-tion #1

(b) Optimized crystal ture #1

struc-0 0.1 0.2 0.3 0.4 Jh4 = 0.659, Qh4 = 0.339

the dimensions of the subspaces are determined through (4.2.5), the upper

as-sociated with the other band gaps, which causes a conflict of interest Since

are chosen to be identical and are spanned by only the eigenfunctions in tween: sp(Φbk,`ˆ (k)) = sp(Φˆak,m(k)) ≡ [u`+1h (ˆε, k), , umh(ˆε, k)] Effectively, theoptimization would try to flatten the sandwiched subspace

All the computations have been carried out using MATLAB and run on a Linux

PC with an Intel Xeon E5550, 2.67GHz processor One run of the algorithmtypically takes 0.5 − 10.0 minutes, which includes 4–30 outer iterations; eachiteration consists of one pass of Steps 2–6 of the main algorithm in section 4.2.5.Due to the presence of local maxima and our use of randomly chosen startingpoint configurations, the times as well as the quality of the resulting solution

can vary For this reason, we make 10 runs of our algorithm corresponding to 10

randomly chosen starting point configurations, and report our aggregate results

in Table 4.1 To eliminate other contributing factors to the computational coat

such as mesh refinement, all the simulations are performed on uniform 64 × 64

grid The table shows average execution times and average number of outer

iterations for each case Since our method and the choice of initial configurations

do not guarantee that the solutions will have a positive band gap, we also report

the number of successful runs, where a run is judged to be successful if the

resulting gap-midgap ratio is at least 20%

Table 4.1 shows that in general TM problems usually solve faster and require

fewer outer iterations than TE problems This observation is consistent with the

result reported in [33], and is possibly caused by higher non-convexity of the

orig-inal TE optimization problem Table 4.1 also shows that lower eigenvalue band

gap optimization problems tend to require less computational time, and yields

wider band gaps more consistently To explore this further, we also performed 30

runs with random initial conditions for two representative band gap cases (the

second and the ninth bands), and report histograms of the gap-midgap ratios

for these runs in Figure 4.3 For the second band gap, the histograms show that

53.3% and 90% of the cases are successful for the TE and the TM problems,

re-spectively For the ninth band gap the numbers are substantially lower, namely

13.3% and 16.7%, respectively

JhT E,1 JhT E,2 JhT E,3 JT E,4h JhT E,5 JhT E,6 JhT E,7 JhT E,8 JhT E,9

Average Execution time (min) 0.42 0.72 0.81 1.6 1.7 2.2 2.3 2.2 4.6

Table 4.1: Average computational time, average number of outer iterations, and

total number of successes of 10 runs for optimizing various band gaps, for both

Table 4.2 shows the progressive increase in computational cost as the mesh

computational cost is decomposed into solving the eigenvalue problem, solving

the SDP problem, and the remaining cost We observe that the computational

−0.1 00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

5 10 15 20 25 30

JTM,9 h

Figure 4.3: Histograms of gap-midgap ratios for 30 runs with random initialconfigurations, for the second and ninth bands

time increases rapidly finer meshes are used and that the computation is inated by the time to solve the SDP problem This is because the number ofdecision variables increases by a factor of four after each mesh refinement, due

dom-to the fact that uniform mesh refinement increases the number of elements by afactor of four as the mesh size is halved

The optimized crystal and band structures for four different meshes are shown

in Figures 4.4 and 4.5 We see that the optimized crystal structures of coarsermeshes topologically resemble those of finer meshes and that their gap-midgapratios are quite close to each other This seems to indicate that adaptive meshrefinement strategies may be effective for computing these structures

Before ending this subsection, we discuss possible ways to improve the putational cost of our procedure, while capturing the micro features of the op-timized photonic crystals The above insight on mesh size refinement points topotentially large saving from using mesh adaptivity and incorporating a non-uniform grid for the representation of the dielectric function, as well as the

for all cell elements i) Moreover, the shapes (circular or connected rods) of theinclusions are much more visible and defined on the finer grids.

A comparison of computational cost in Table 4.3 shows the efficiency ofmesh adaptivity Due to the presence of local maxima and the randomly choseninitial configurations, the computational cost is likely to vary Hence, in this