Chapter Bandgap Optimization of Photonic Crystal Structures Having solved the infinite-dimensional eigenvalue problem, which is parameterized by the wave vector and dielectric function, we are ready to describe the optimal design problem, and propose our approach for solving it. We start 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 subspace method and mesh adaptivity. Then we describe the extension to TM polarization and combined TE/TM polarizations, as well as multiple band gaps optimization. Results on both square and hexagonal lattices are then presented and discussed 4.1 The Band Gap Optimization Problem The objective in photonic crystal design is to maximize the band gap between two 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 band gap [29]. The gap-midgap ratio between λm and λm+1 is defined as inf λm+1 (ε(r), k) − sup λm (ε(r), k) J(ε(r)) = k∈∂B k∈∂B inf λm+1 (ε(r), k) + sup λm (ε(r), k) k∈∂B . (4.1.1) k∈∂B 65 The optimization problem is to look for an optimal configuration ε∗ (r) which is the solution of sup J(ε(r)), (4.1.2) ε(r) with J(ε(r)) defined in (4.1.1), and the eigenvalues λm , λm+1 satisfying (3.1.3). After discretizing this infinite optimization problem by procedures described in section 3.1.2, we obtain the following finite-dimensional optimization problem: (ε, k) − max λm λm+1 h (ε, k) h max Jh (ε) = ε∈Qh s.t. k∈Ph max λm+1 (ε, k) k∈Ph h Ah (ε, k)ujh = λjh Mh (ε)ujh , k∈Ph + max λm h (ε, k) , k∈Ph (4.1.3) j = m, m + 1, k ∈ Ph . In this problem a subtle difference between TE and TM polarizations lies in the operators of the eigenvalue problem: Ah and Mh take the corresponding forms from (3.1.10). Unfortunately, this optimization problem is non-convex; furthermore, it suffers from a lack of regularity at the optimum. The reason for this is that the m+1 are typically not smooth functions of ε at points of eigenvalues λm h and λh multiplicity, and multiple eigenvalues at the optimum are typical of structures with 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 thus gradient-based descent methods can run into serious numerical difficulties and convergence problems. 4.2 Band Structure Optimization In this section we describe our approach in solving the band gap optimization problem based on a subspace method and SDP. In our approach, we first reformulate the original problem as an optimization problem in which we aim to maximize the band gap obtained by restricting the operator to two orthogonal subspaces. The first subspace consists of eigenfunctions associated to eigenvalues below the band gap, and the second subspace consists of eigenfunctions whose eigenvalues are above the band gap. In this way, the eigenvalues are no longer 66 explicitly present in the formulation, and eigenvalue multiplicity no longer leads to a lack of regularity. The reformulated optimization problem is exact but non-convex and large-scale. To reduce the problem size, we truncate the highdimensional subspaces to only a few eigenfunctions below and above the band gap [17, 47], thereby obtaining a new small-scale yet non-convex optimization problem. Finally, we keep the subspaces fixed at a given decision parameter vector to obtain a convex semidefinite optimization problem for which SDP solution methods can be efficiently applied. 4.2.1 Reformulation of the band gap optimization problem using subspaces We first define two additional decision variables: U := λm+1 (ε, k) , h k∈Ph L := max λm h (ε, k), k∈Ph and then rewrite the original problem (4.1.3) as P0 : max ε,U,L s.t. U −L , U +L m+1 (ε, k), λm h (ε, k) ≤ L , U ≤ λh ∀k ∈ Ph , m m Ah (ε, k)um h = λh Mh (ε)uh , ∀k ∈ Ph , Ah (ε, k)um+1 = λm+1 Mh (ε)um+1 , h h h ∀k ∈ Ph , εL ≤ εi ≤ εH , i = 1, . . . , nε , (4.2.1) U , L > 0. Next, we introduce the following matrices: m+1 Θ(ε, k) := [Φ(ε, k) | Ψ(ε, k)] := [u1h (ε, k) . . . um (ε, k) . . . uN h (ε, k) | uh h (ε, k)], 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) 67 as sp(Φ(ε, k)) and sp(Ψ(ε, k)), respectively. The first three sets of constraints in (4.2.1) can be represented exactly as Φ∗ (ε, k)[Ah (ε, k) − LMh (ε)]Φ(ε, k) 0, ∀k ∈ Ph , Ψ∗ (ε, k)[Ah (ε, k) − U Mh (ε)]Ψ(ε, k) 0, ∀k ∈ Ph , where “ ” is the generalized ordering on symmetric matrices, i.e., A B if and only if A − B is positive semidefinite, as introduced in section 2.4. We therefore obtain the following equivalent optimization problem: P1 : max ε,U,L s.t. U −L , U +L Φ∗ (ε, k)[Ah (ε, k) − LMh (ε)]Φ(ε, k) 0, ∀k ∈ Ph , Ψ∗ (ε, k)[Ah (ε, k) − U Mh (ε)]Ψ(ε, k) 0, ∀k ∈ Ph , εL ≤ εi ≤ εH , i = 1, . . . , nε , U , L > 0. (4.2.2) Although the reformulation P1 is exact, there is however a subtle difference in the interpretation of P0 and P1 : P0 can be viewed as maximizing the gap-midgap m+1 ; whereas P1 can be viewed as ratio between the two eigenvalues λm h and λh maximizing the gap-midgap ratio between the two subspaces sp(Φ(ε, k)) and sp(Ψ(ε, k)). The latter viewpoint allows us to develop an efficient subspace approximation method for solving the band gap optimization problem as discussed below. 4.2.2 Subspace approximation and reduction ˆ. We then introduce the Let us assume that we are given a parameter vector ε associated matrices Θ(ˆ ε, k) := [Φ(ˆ ε, k) | Ψ(ˆ ε, k)] = [u1h (ˆ ε, k) . . . um ε, k) | um+1 (ˆ ε, k) . . . uN ε, k)] , h (ˆ h (ˆ h 68 where Ψ(ˆ ε, k) and Ψ(ˆ ε, k) consist of the first m eigenvectors and the remaining N − m eigenvectors, respectively, of the eigenvalue problem Ah (ˆ ε, k)ujh = λjh Mh (ˆ ε)ujh , ≤ j ≤ N. Under the presumption that sp(Φ(ˆ ε, k)) and sp(Ψ(ˆ ε, k)) are reasonable approxˆ, we replace Φ(ε, k) with imations of sp(Φ(ε, k)) and sp(Ψ(ε, k)) for ε near ε Φ(ˆ ε, k) and Ψ(ε, k) with Ψ(ˆ ε, k) to obtain P2εˆ : max ε,U,L s.t. U −L , U +L Φ∗ (ˆ ε, k)[Ah (ε, k) − LMh (ε)]Φ(ˆ ε, k) 0, ∀k ∈ Ph , Ψ∗ (ˆ ε, k)[Ah (ε, k) − U Mh (ε)]Ψ(ˆ ε, k) 0, ∀k ∈ Ph , εL ≤ εi ≤ ε H , i = 1, . . . , nε , U , L > 0. (4.2.3) Note in P2εˆ that the subspaces sp(Φ(ˆ ε, k)) and sp(Ψ(ˆ ε, k)) are approximations of the subspaces sp(Φ(ε, k)) and sp(Ψ(ε, k)) and are no longer functions of the decision variable vector ε. The semidefinite inclusions in P2εˆ are large-scale, i.e., the rank of either the first or second inclusion is at least N /2, for each k ∈ Ph , and N will typically be quite large. In order to reduce the size of the inclusions, we reduce the dimensions of the subspaces by considering only the “important” eigenvectors m+1 (ε, k) . . . uN among u1h (ε, k) . . . um h (ε, k), namely those ak eigenvech (ε, k), uh tors whose eigenvalues lie below but nearest to λm h (ε, k) and those bk eigenvectors whose eigenvalues lie above but nearest to λm+1 (ε, k), for small values of ak , bk , h typically chosen in the range between and 5, for each k ∈ Ph . This yields reduced matrices m−ak +1 Θak +bk (ˆ ε, k) := [Φak (ˆ ε, k) | Ψbk (ˆ ε, k)] = [uh m+bk (ˆ ε, k) . . . um ε, k) | um+1 (ˆ ε, k) . . . uh h (ˆ h (ˆ ε, k)]. Substituting Φak (ˆ ε, k) in place of Φ(ˆ ε, k) and Ψbk (ˆ ε, k) in place of Ψ(ˆ ε, k) in 69 the formulation P2εˆ yields the following reduced optimization formulation: P3εˆ : max ε,U,L s.t. U −L , U +L ε, k) Φ∗ak (ˆ ε, k)[Ah (ε, k) − LMh (ε)]Φak (ˆ 0, ∀k ∈ Ph , Ψ∗bk (ˆ ε, k)[Ah (ε, k) − U Mh (ε)]Φbk (ˆ ε, k) 0, ∀k ∈ Ph , εL ≤ εi ≤ εH , i = 1, . . . , nε , U , L > 0. (4.2.4) In this way the formulation P3εˆ seeks to model only the anticipated “active” eigenvalue constraints, in exact extension of active-set methods in nonlinear optimization. The integers ak , bk are determined indirectly through user-defined parameters rl > 0, and ru > 0, where we only retain those eigenvectors whose ε, k) or whose eigenvalues are within eigenvalues are within 100rl % beneath λm h (ˆ 100ru % above λm+1 (ˆ ε, k). This translates to choosing ak , bk ∈ N+ as the smallh est integers that satisfy k +1 k ε, k) − λm−a (ˆ ε, k) λm ε, k) − λm−a (ˆ ε, k) λm h (ˆ h (ˆ h h ≤ rl ≤ , m m λh (ˆ ε, k) λh (ˆ ε, k) (4.2.5a) k +1 k (ˆ ε, k) (ˆ ε, k) (ˆ ε, k) − λm+1 (ˆ ε, k) − λm+1 λm+b λm+b h h h h ≤ r ≤ . u m+1 m+1 ε, k) ε, k) λh (ˆ λh (ˆ (4.2.5b) The dimensions of the resulting subspaces sp(Φak (ˆ ε, k)) and sp(Φbk (ˆ ε, k)) are typically very small (ak , bk ∼ 2, . . . , 5). Furthermore, the subspaces are wellspanned by including all relevant eigenvectors corresponding to those eigenvalues with multiplicity at or near the current min/max values. We observe that P3εˆ has significantly smaller semidefinite inclusions than if ˆ in order the full subspaces were used. Also, the subspaces are kept fixed at ε to reduce the nonlinearity of the underlying problem. Furthermore, we show below that for the TE and TM polarizations that P3εˆ can be easily re-formulated as a linear fractional semidefinite program, and hence is solvable using modern interior-point methods. 70 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 polarizations, problem P3εˆ can be converted to a linear fractional semidefinite program and hence can be further converted to a linear semidefinite program. TE polarization. We introduce the following new decision variable notation for convenience: y := (y1 , y2 , . . . , yny ) := (1/ε1 , . . . , 1/εnε , L, U ) . We also amend our notation to write various functional dependencies on y instead of ε such as Φ(ˆ y , k), Ψ(ˆ y , k). Utilizing the parameter dependence introduced in (3.1.4), we re-write P3εˆ for the TE polarization as ˆ y : max PTE y s.t. yny − yny −1 , yny + yny −1 Φ∗ak (ˆ y , k) ny −2 TE i=1 yi Ah,i (k) − yny −1 MhTE Φak (ˆ y , k) y , k) Ψ∗bk (ˆ ny −2 TE i=1 yi Ah,i (k) − yny MhTE Ψbk (ˆ y , k) 1/εH ≤ yi ≤ 1/εL , 0, 0, ∀k ∈ Ph , ∀k ∈ Ph , i = 1, . . . , ny − 2, yny −1 , yny > 0. (4.2.6) We note that the objective function is a linear fractional expression and ˆ y the constraint functions are linear functions of the variables y. Therefore PTE is a linear fractional SDP. Using a standard homogenization [18, 21], a linear fractional SDP can be converted to a linear SDP, as discussed in section 2.4. We introduce new decision variable notation: Y := (Y1 , Y2 , . . . , YnY ) := (θy1 , . . . , θyny , θ) . Amending the notation to write the previous functional dependencies on Y in- 71 ˆ y stead of y, we finally re-write PTE as, ˆ Y : max PTE (YnY −1 − YnY −2 ) , Y s.t. Φ∗ak (Yˆ , k) nY −3 TE i=1 Yi Ah,i (k) − YnY −2 MhTE Φak (Yˆ , k) 0, ∀k ∈ Ph , Ψ∗bk (Yˆ , k) nY −3 TE i=1 Yi Ah,i (k) − YnY −1 MhTE Ψbk (Yˆ , k) 0, ∀k ∈ Ph , YnY −2 + YnY −1 = 1, YnY /εH ≤ Yi ≤ YnY /εL , i = 1, . . . , nY − 3, YnY −2 , YnY −1 , YnY > 0. (4.2.7) ˆ Y is the final linear SDP form that can be solved using various SDP solvers. PTE TM polarization. We introduce slightly different decision variable notation 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 dependencies on z instead of ε such as Φ(ˆ z , k), Ψ(ˆ z , k). Noting that zn −1 − znz U −L = z , U +L znz −1 + znz utilizing the parameter dependence introduced in (3.1.4), and multiplying the semidefinite inclusions of (4.2.4) by znz −1 and znz , respectively, we re-write P3εˆ 72 for the TM polarization as ˆ z PTM : max z znz −1 − znz , znz −1 + znz nz −2 TM i=1 zi Mh,i Φ∗ak (ˆ z , k) znz −1 ATM h (k) − s.t. Ψ∗ak (ˆ z , k) znz ATM h (k) − ε L ≤ z i ≤ εH , nz −2 TM i=1 zi Mh,i z , k) Φak (ˆ z , k) Ψak (ˆ ∀k ∈ Ph , 0, ∀k ∈ Ph , 0, i = 1, . . . , nz − 2, znz −1 , znz > 0. (4.2.8) Here again the objective function is a linear fractional form and the constraint ˆ z functions are linear functions of the variables z. Therefore PTM is a linear ˆ y , and can be homogenized to fractional SDP with format similar to that of PTE 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 inˆ z stead of z, we finally re-write PTM as, ˆ Z : max PTM (ZnZ −2 − ZnZ −1 ) , Z s.t. ˆ k) Zn −2 ATM (k) − Φ∗ak (Z, Z h nZ −3 TM i=1 Zi Mh,i ˆ k) Φak (Z, 0, ∀k ∈ Ph , ˆ k) Zn −1 ATM (k) − Ψ∗bk (Z, Z h nZ −3 TM i=1 Zi Mh,i ˆ k) Ψbk (Z, 0, ∀k ∈ Ph , ZnZ −2 + ZnZ −1 = 1, εL ZnZ ≤ Zi ≤ εH ZnZ , i = 1, . . . , nZ − 3, ZnZ −2 , ZnZ −1 , ZnZ > 0. (4.2.9) ˆ ˆ Y and P Z are linear semidefinite programs, they can be solved Since both PTE TM very efficiently by using modern interior point methods. Here we use the SDPT3 software [56] for this task. 73 4.2.4 Multiple band gaps optimization formulation In this section, we derive the most general optimization formulation – multiple complete band gaps optimization, i.e., multiple prohibitive frequency ranges that are independent of the polarization of the waves. Let J T E = {mj | ≤ j ≤ J} denotes a set of J TE-polarized frequency bands, and J T M = {nj | ≤ j ≤ J} a set of J TM-polarized frequency bands for which we seek to achieve complete gaps. We define the discrete eigenvalue gap-midgap ratio for the jth gap of TE and TM cases respectively as T E,mj +1 JhT E,j (ε) λh =2 T E,mj +1 max λh k∈Ph JhT M,j (ε) = T E,mj (ε, k) − max λh (ε, k) k∈Ph k∈Ph T E,mj (ε, k) + max λh k∈Ph T M,nj +1 λh (ε, k) k∈Ph T M,nj +1 max λh (ε, k) k∈P h T M,nj − max λh k∈Ph T M,nj + max λh k∈Ph , (4.2.10a) . (4.2.10b) (ε, k) (ε, k) (ε, k) The jth complete band gap is hence defined as Jhj (ε) = min(JhT E,j (ε), JhT M,j (ε)). (4.2.11) To design the photonic crystal structure that supports multiple complete band gaps, the following optimization problem is proposed: maxε∈Qh min1≤j≤J αj Jhj (ε) , s.t. ATh E (ε, k)uh T E,mj T M,nj ATh M (ε, k)uh T E,mj T E,mj MhT E (ε)uh T M,nj MhT M (ε)uh = λh = λh , T M,nj (4.2.12) , k ∈ Ph , j = 1, . . . , J + 1. Here αj are prescribed weights of each band gap. To model multiple complete band gaps for only TE or TM polarization, one can simply omit the non-relevant eigen equations in the above formulation. Note further that if J T E = J T M , then formulation (4.2.12) is generalized to treat more generic cases in which the number and location of TE band gaps are allowed to differ from those of TM 74 (ωa/2πc)2 0.4 J 5h = 0.037, Q5h = 0.018 0.3 0.2 2 J h = 0.043, Qh = 0.022 0.1 M Γ K Γ K Γ K Γ K Γ K Γ 0.9 (ωa/2πc)2 5 J h = 0.268, Qh = 0.135 0.6 J = 0.268, Q2 = 0.134 0.3 h (ωa/2πc)2 Γ 0.7 h M 5 J h = 0.371, Qh = 0.187 0.5 0.3 J h = 0.371, Qh2 = 0.187 0.1 Γ M (ωa/2πc)2 0.7 5 J h = 0.360, Qh = 0.182 0.5 0.3 2 J h = 0.360, Qh = 0.182 0.1 Γ M (ωa/2πc)2 0.7 5 J h = 0.349, Qh = 0.176 0.5 0.3 J h = 0.349, Qh = 0.176 0.1 Γ M Figure 4.8: Mesh adaptivity results show the grids (left), optimal structures (middle), and bands (right) for the second and fifth TM band gaps in the hexagonal lattice: Grid resolution varies from h = a/8 (top panel) to hmin = a/128 (bottom panel). 89 table, the execution time is averaged over 10 runs corresponding to 10 randomly chosen starting point configurations. The adaptive procedure starts with the coarsest uniform mesh of resolution h = a/8, and is refined up to the resolution of hmin = a/64. The results are compared with a fixed uniform mesh of resolution h = a/64. It can be seen that the computational saving using mesh adaptivity is as high as 80%, and is more prominent in the TM problems than in TE problems. Execution time (min) Uniform mesh Adaptive mesh Computational saving (%) JhT E,1 1.3 0.46 64.6 JhT E,2 1.4 0.65 53.5 JhT E,3 2.4 0.98 59.1 JhT E,4 1.7 0.43 74.5 JhT E,5 2.9 1.4 51.7 JhT E,6 3.2 2.0 37.5 JhT E,7 3.0 2.2 26.7 JhT E,8 3.4 1.4 58.8 Execution time (min) Uniform mesh Adaptive mesh Computational saving (%) JhT M,1 0.42 0.12 71.4 JhT M,2 0.72 0.19 73.6 JhT M,3 0.81 0.22 72.8 JhT M,4 1.6 0.31 80.6 JhT M,5 1.7 0.47 72.3 JhT M,6 2.2 0.51 76.8 JhT M,7 2.3 0.81 64.8 JhT M,8 2.2 0.58 73.6 Table 4.3: Comparison of average executive time for optimizing various band gaps using uniform mesh (h = a/64) and adaptive mesh (hmax = a/8, hmin = a/64) in a square lattice. Here JhT E,i denotes the gap-midgap ratio between the ith and (i + 1)th eigenvalue for the TE polarization. A ”success” run is defined as one that results in a gap-midgap ratio of at least 20%. 4.3.5 Optimal structures with single band gap We start the optimization procedure with a random distribution of the dielectric on a coarse uniform mesh of resolution h = a/8. In Figures 4.9 through 4.12, we present plots of the final optimized crystal structures and the corresponding band structures for first 10 band gaps of either TE or TM polarizations in square and hexagonal lattices. We see that the optimized TM band gaps are exhibited in isolated high-ε structures, while the optimized TE band gaps appear mostly in connected high-ε structures. This observation has also been pointed out in [29] (p75) “the TM band gaps are favored in a lattice of isolated high-ε regions, and TE band gaps are favored in a connected lattice”, and observed in [33] previously. For both TE and TM polarizations, the crystal structures become more and more complicated as we progress to higher bands. It would be very difficult to create such structures using physical intuition alone. The largest gap-midgap ratio for the TM case is 90.7% between the seventh and eighth frequency bands in a hexagonal lattice, and it corresponds to a frequency gap-midgap ratio of 47.9% ; while the largest ratio for the TE case is 97.8%, between the third and forth 90 bands in the hexagonal lattice, and the frequency gap-midgap ratio is evaluated to be 52.3%. The results presented here are no way guaranteed to be globally optimal, as pointed out in Section 4.2.1. While most crystal structures in the TM cases appear similar to those presented in [33], we have shown quite different TE structures. A qualitative comparison between the two results in the background indicates larger band gaps (both in absolute value and in the gap-midgap ratio) in our results. 0.4 (ωa/2πc)2 (ωa/2πc)2 0.3 0.2 J = 0.558, Q1 = 0.284 h h 0.1 Γ X M 0.2 Γ (b) (ωa/2πc)2 (ωa/2πc)2 J = 0.541, Q 3= 0.276 h h 0.2 0.1 0.5 X M Γ JhT E,3 (ωa/2πc)2 (ωa/2πc)2 J5 = 0.643, Q5 = 0.330 h h 0.3 Γ X M 0.6 M Γ J = 0.695, Q = 0.359 h h 0.3 Γ JhT E,5 Γ X M Γ (f) JhT E,6 1.4 (ωa/2πc)2 1.2 (ωa/2πc)2 X 0.9 0.6 J = 0.885, Q = 0.466 h h 0.8 0.4 Γ X M 0.4 Γ (h) X M Γ JhT E,8 1.5 1.2 J = 0.642, Q = 0.330 h h (ωa/2πc)2 0.8 J8 h = 0.685, Q h = 0.353 Γ JhT E,7 (ωa/2πc)2 Γ (d) JhT E,4 0.9 Γ JhT E,9 J10 = 0.564, Q10= 0.288 h h 0.5 0.4 (i) J = 0.643, Q4 = 0.330 h h 0.1 Γ (g) Γ 0.3 (e) M 0.7 0.3 X JhT E,2 0.4 (c) J = 0.655, Q = 0.337 h h 0.1 Γ JhT E,1 (a) 0.3 X M Γ (j) Γ X M Γ JhT E,10 Figure 4.9: Optimal structures with the first ten single TE band gap in square lattice. 91 0.5 (ωa/2πc)2 (ωa/2πc)2 0.3 0.2 0.1 (a) J1= 0.748, Q = 0.388 h h Γ X M JhT M,1 0.1 Γ X M Γ J = 0.733, Q h= 0.380 h Γ (d) X M Γ JhT M,4 0.9 0.6 (ωa/2πc)2 0.8 J = 0.806, Q 5= 0.421 h h 0.6 0.4 Γ X M Γ Γ (f) M Γ 1.5 J = 0.880, Q = 0.463 h h J = 0.789, Q = 0.411 h h 0.5 0.4 Γ X M Γ JhT M,7 (h) Γ X M Γ JhT M,8 1.6 1.6 (ωa/2πc)2 1.2 X JhT M,6 (ωa/2πc)2 (ωa/2πc)2 0.8 J = 0.698, Q 6= 0.360 h h 0.3 0.2 JhT M,5 (ωa/2πc)2 M 0.3 1.2 J = 0.762, Q 9= 0.396 h h 1.2 0.8 Γ JhT M,9 J10= 0.767, Q10 = 0.399 h h 0.8 0.4 (i) X (g) Γ 0.6 Γ JhT M,3 (ωa/2πc)2 0.1 JhT M,2 (ωa/2πc)2 (ωa/2πc)2 0.2 J = 0.747, Q = 0.387 h h 0.9 J = 0.728, Q 3= 0.377 h h 0.3 (e) 0.2 (b) 0.4 0.3 Γ 0.5 (c) 0.4 0.4 X M Γ Γ (j) X M Γ JhT M,10 Figure 4.10: Optimal structures with the first ten single TM band gap in square lattice. 92 0.4 (ωa/2πc)2 (ωa/2πc)2 0.4 0.3 0.2 0.1 J = 0.968, Q 1= 0.516 h h Γ M K 0.2 J = 0.978, Q = 0.523 h h 0.2 Γ M M K (ωa/2πc)2 (ωa/2πc)2 Γ 0.6 0.4 0.2 Γ M K Γ JhT E,6 1.2 J = 0.513, Q = 0.261 h h (ωa/2πc)2 (ωa/2πc)2 K J = 0.223, Q = 0.112 h h (f) J 8= 0.521, Q 8= 0.265 h h 0.8 0.6 0.6 0.4 0.4 0.2 0.2 Γ M K Γ JhT E,7 Γ M K Γ JhT E,8 (h) 1.5 1.8 J 9= 0.508, Q 9= 0.258 h h (ωa/2πc)2 (ωa/2πc)2 M 1.4 0.5 Γ (i) Γ 0.8 Γ JhT E,5 J4 h = 0.969, Q h = 0.517 JhT E,4 (g) Γ 0.3 0.8 K 0.3 (d) 0.6 Γ 0.6 Γ J = 0.460, Q = 0.233 h h (e) K JhT E,3 0.9 M 0.9 (ωa/2πc)2 (ωa/2πc)2 0.8 (c) Γ JhT E,2 (b) 0.6 0.4 J = 0.126, Q = 0.063 h h 0.1 Γ JhT E,1 (a) 0.3 JhT E,9 M K 1.4 J10= 0.639, Q10= 0.328 h h 0.6 0.2 Γ Γ (j) M K Γ JhT E,10 Figure 4.11: Optimal structures with the first ten single TE band gap in hexagonal lattice. 93 0.5 0.2 (ωa/2πc)2 (ωa/2πc)2 0.3 = 0.475 J1 = 0.899, Q h h 0.1 Γ M K Γ (b) 0.4 0.2 Γ M K Γ JhT M,3 (d) (ωa/2πc)2 (ωa/2πc)2 0.2 Γ M K JhT M,5 Γ M K M K Γ J = 0.785, Q = 0.409 h h 0.5 Γ JhT M,7 Γ (h) M K Γ JhT M,8 1.6 1.6 J = 0.763, Q 9= 0.396 h h (ωa/2πc)2 (ωa/2πc)2 0.2 JhT M,6 (ωa/2πc)2 (ωa/2πc)2 J = 0.907, Q 7= 0.479 h h (i) 0.4 1.5 0.4 0.8 0.4 Γ 0.6 Γ (f) 0.8 1.2 K J = 0.684, Q = 0.353 h h 0.8 Γ 1.6 (g) M J = 0.568, Q = 0.290 h h 0.4 1.2 Γ JhT M,4 0.8 (e) Γ J = 0.884, Q4 = 0.466 h h 0.6 0.1 K 0.2 0.6 M JhT M,2 0.8 J = 0.787, Q 3= 0.410 h h (c) J = 0.584, Q = 0.299 h h 0.1 (ωa/2πc)2 (ωa/2πc)2 0.3 0.2 0.5 0.4 0.3 Γ JhT M,1 (a) 0.4 Γ JhT M,9 M K 1.2 0.8 0.4 Γ (j) J10= 0.695, Q10= 0.359 h h Γ M K Γ JhT M,10 Figure 4.12: Optimal structures with the first ten single TM band gap in hexagonal lattice. 94 4.3.6 Optimal structures with multiple band gaps In optimizing a weighted pair of band gaps, there is an intuitive trade-off between the size of one band gap versus the other band gap that can be computed by varying the weights associated with each band to yield a trade-off frontier between the two band gaps. Such a trade-off frontier is illustrated in Figure 4.3.6 for the problem of optimizing the weighted first and third band gaps in the hexagonal lattice. The points in the figure were produced by varying the weights (α1 , α2 ) associated with two band gaps (first and third bands, respectively) for a variety of values of (α1 , α2 ) ∈ [0, 1]2 . More specifically, we start by computing a solution for (α1 , α2 ) = (0.5, 0.5) on a uniform mesh 64 × 64. We then modify the weights (α1 , α2 ) ∈ [0, 1]2 slightly and use the previously obtained solution as the initiating distribution for computing the solution for the problem with modified weights. This step is repeated in order to track various local optimal branches. Figure 4.3.6 illustrates the trade-off frontier as well as the optimized structures A, B, C, and D corresponding to (α1 , α2 ) = (0.99, 0.01), (0.01, 0.99), (0.99, 0.01), and (0.01, 0.99), respectively. The figure shows two different frontiers A − B and C − D. The multiple branches of the frontiers represent multiple significant local optima corresponding to identical weights. Structure A favors the third band gap, whereas structure B favors the first band gap. Structure C exhibits a good compromise between the two band gaps since both are relatively large in this case. Structure C is particularly interesting in that it resembles D in overall design, but has larger disks with an air hole. The frontier A−B consist of structures having disks of two different radii but otherwise similar topology, while frontier C − D consists of structures having disks of different radii and different topology. We also studied the trade-off frontier for the first and third TE band gap in the hexagonal lattice, see Figure 4.3.6. This frontier has proven to be more complicated than its TM mode counterpart shown previously in Figure 4.3.6. In fact, no distinctive frontier can be observed, which is undoubtedly due to numerous local optima in this case. We employed a similar computational strategy to that used to produce Figure 4.3.6 as described in the previous subsection. In Figure 4.3.6 we display four (locally) optimized structures along the envelope 95 Figure 4.13: The trade-off frontier for the first (horizontal axis) and third (vertical axis) TM band gaps in the hexagonal lattice. The frontiers are traced by varying the weights corresponding to each band gap. Multiple frontiers can be attributed to multiple local optima. A − B − C − D of the apparent frontier, where A, B, C, and D correspond to solutions for (α1 , α2 ) = (0.99, 0.01), (0.5, 0.5), (0.5, 0.5), and (0.01, 0.99), respectively. Structure A favors the third band, whereas structure D favors the first band. Structures B and C offer a compromise between the two bands. Note further that the two structures B and C are very different despite the fact that they have similar objective values. (We also observed many local optimal solutions in our previous work on the optimal design of photonic crystals with single band gap [44].) Figure 4.15 shows solutions for optimizing the first and fourth TE band gaps for both the square and hexagonal lattices (with equal weights for the two bands). We observe that the optimized structure in the square lattice is not connected, while the optimized structure in the hexagonal lattice is connected. Moreover, we see for both band structures that the midgap frequency of the fourth band is roughly twice that of the first band. These designs can prohibit electromagnetic waves at both frequencies ω and 2ω. 96 1.0 Branch Branch Branch 0.8 J h3 0.6 0.4 0.2 0.2 0.4 J 1h 0.6 0.8 1.0 Figure 4.14: The trade-off frontier for the first (horizontal axis) and third (vertical axis) TE band gaps in the hexagonal lattice. The frontiers are traced by varying the weights corresponding to each band gap. Multiple frontiers can be attributed to multiple local optima. 97 (ωa/2πc)2 0.6 0.4 0.2 (a) J4 h = 0.390, Q h = 0.197 JhT E,1 J1 h = 0.412, Q h = 0.208 X Γ and M Γ JhT E,4 (ωa/2πc)2 0.6 0.4 J4 h = 0.560, Q h = 0.286 0.2 J1 h= 0.592, Q h = 0.303 (b) JhT E,1 M Γ and K Γ JhT E,4 Figure 4.15: Optimization results show the final optimal crystal structure (left), and frequency bands (right) for the first and fourth TE band gaps in (a) the square lattice and (b) the hexagonal lattice. Shown in Figures 4.16, 4.17, and 4.18 are optimized crystal structures with two or three optimal band gaps of TE polarization in square lattice as well as hexagonal lattice. In Figures 4.19 and 4.20 we show crystal structures with dual optimal band gaps of TM polarization in square and hexagonal lattices respectively. Figures 4.21 and 4.22 show crystal structures with three optimal band gaps of TM polarization in square and hexagonal lattices respectively. Finally in Figure 4.23, a structure with even four optimal band gaps of TM polarization in square lattice is shown. All these band gaps are assigned with equal weights (= 1). Unfortunately, the rule of thumb concluded on connectivity and isolation of dielectric materials in single band gap problems no longer holds for the multiple band gap structures. 98 (ωa/2πc)2 0.35 0.25 J = 0.386, Q = 0.195 h h 0.15 0.05 Γ J1 h = 0.294, Q h= 0.148 X M Γ Figure 4.16: Optimization results show the final computational grid (left), optimal crystal structure (middle), and frequency bands (right) for the first and fourth TE band gaps in the square lattice. (ωa/2πc)2 0.9 J = 0.319, Q = 0.160 h h 0.6 J3 h = 0.319, Q h= 0.160 0.3 Γ M K Γ Figure 4.17: Optimization results show the final computational grid (left), optimal crystal structure (middle), and frequency bands (right) for the third, and fifth TE band gaps in the hexagonal lattice. (ωa/2πc)2 0.8 0.6 J = 0.194, Q h = 0.097 h 0.4 J 4= 0.198, Q h = 0.099 h 0.2 Γ J = 0.190, Q = 0.095 h h X M Γ Figure 4.18: Optimization results show the final computational grid (left), optimal crystal structure (middle), and frequency bands (right) for the second, fourth, and sixth TE band gaps in the square lattice. (ωa/2πc)2 0.6 0.4 J4 h = 0.443, Q h = 0.224 0.2 J = 0.497, Q = 0.252 h h Γ X M Γ Figure 4.19: Optimization results show the final computational grid (left), optimal crystal structure (middle), and frequency bands (right) for the second and fourth TM band gaps in the square lattice. 99 (ωa/2πc)2 0.12 0.08 0.04 Γ J = 0.598, Q = 0.306 h h J = 0.618, Q = 0.317 h h M K Γ Figure 4.20: Optimization results show the final computational grid (left), optimal crystal structure (middle), and frequency bands (right) for the third and ninth TM band gaps in the hexagonal lattice. 0.8 (ωa/2πc)2 0.6 0.4 0.2 Γ J3 h = 0.386, Q h = 0.195 J = 0.367, Q = 0.185 h h 1= 0.193 J1 = 0.383, Q h h X M Γ Figure 4.21: Optimization results show the final computational grid (left), optimal crystal structure (middle), and frequency bands (right) for the first, second, and fourth TM band gaps in the square lattice. (ωa/2πc)2 1.2 J = 0.417, Q = 0.211 h h 0.8 J = 0.449, Q 5= 0.228 h h 0.4 J1 h = 0.569, Q h= 0.291 M K Γ Γ Figure 4.22: Optimization results show the final computational grid (left), optimal crystal structure (middle), and frequency bands (right) for the first, fifth, and eighth TM band gaps in the hexagonal lattice. 1.2 (ωa/2πc)2 0.8 J = 0.186, Q = 0.093 h h J6 h = 0.239, Q h = 0.120 0.4 Γ J = 0.254, Q = 0.127 h h = 0.113 J1 = 0.226, Q h h X M Γ Figure 4.23: Optimization results show the final computational grid (left), optimal crystal structure (middle), and frequency bands (right) for the first, third, sixth and ninth TM band gaps in the square lattice. 100 (ωa/2πc)2 0.3 (ωa/2πc)2 0.2 J1 h = 0.331, Q h= 0.166 0.1 M Γ (a) min(JhT E,1 , K 0.1 (b) min(JhT E,1 J = 0.376, Q 1= 0.190 h h 0.3 (c) 0.8 (ωa/2πc)2 (ωa/2πc)2 0.9 0.6 Γ min(JhT E,4 , M JhT M,8 ) K M Γ Γ JhT M,2 ) J = 0.307, Q 1= 0.154 h h 0.2 0.6 (d) Γ J = 0.305, Q 1= 0.154 h h 0.4 0.2 Γ K JhT M,3 ) M Γ min(JhT E,4 , K Γ JhT M,11 ) Figure 4.24: Optimization results for single complete band gap in the hexagonal lattice. In the band structure subplots, solid lines represent TM bands, while broken lines represent TE bands. 4.3.7 Optimal structures with complete band gaps A photonic crystal exhibiting complete band gaps is able to prevent the propagation of light despite its polarization, which renders it the most useful structure in practice. In Figures 4.24 and 4.25, we present some optimal solutions with sizable single complete band gap for both the hexagonal and square lattices. Contrary to the observations from single polarization structures, both connected high−ε (Figures 4.24(a),(b),(c), and Figures 4.25(b),(d)) and isolated high−ε (Figures 4.24(d), and Figures 4.25(a),(c)) structures are shown to possess complete band gap. We are also able to compute multiple complete band gaps for both the hexagonal and square lattices; the band gaps in the hexagonal lattice lie in the first and third bands for the TE polarization, yet in the third and sixth bands for the TM polarization. Figure 4.26 illustrates the geometry and band structures for the hexagonal lattice. The corresponding frequency gap-midgap ratios of 5.76% and 6.94% are quite sizable – and are the first multiple and complete band gaps ever reported for photonic crystals in the hexagonal lattice. This photonic crystal structure has both connected and non-connected features. Moreover, it has a more complicated geometry than the previous structures shown herein. For the square lattice, complete band gaps lie in the third band for the TE 101 0.3 0.3 0.1 X Γ min(JhT E,1 , (a) (ωa/2πc)2 (ωa/2πc)2 1 0.2 J = 0.147, Q = 0.074 h h M (b) (ωa/2πc)2 (ωa/2πc)2 (c) J = 0.209, Q 1= 0.105 h h 0.1 X Γ min(JhT E,2 , M Γ JhT M,3 ) 0.9 0.2 X Γ min(JhT E,2 , 0.4 0.3 J = 0.232, Q 1= 0.116 h h 0.1 Γ JhT M,2 ) 0.2 JhT M,5 ) M 0.6 J = 0.335, Q 1= 0.169 h h 0.3 Γ Γ (d) min(JhT E,5 , X M Γ JhT M,9 ) Figure 4.25: Optimization results for single complete band gap in the square lattice. In the band structure subplots, solid lines represent the TM bands, while broken lines represent the TE bands. polarization, yet in the sixth and ninth bands for the TM polarization. Figure 4.27 illustrates our results. The corresponding frequency gap-midgap ratios of 7.59% and 13.5% are also large. We emphasize again that no other multiple and complete gaps have previously been found for photonic crystals in the square lattice. In general, the complete band gaps are smaller than either of the corresponding TE and TM band gaps because it is rather difficult to simultaneously achieve both the TE and TM band gaps. Of course, one can also widen the gap size for one band at the expense of narrowing the gap size of the other band by choosing appropriate weights in the optimization problem (4.2.12). It is interesting to note in this case that although the photonic crystal structure has a complicated geometry, it is nevertheless connected. 4.4 Conclusions In this chapter, we have introduced a novel approach, based on reduced eigenspaces and semidefinite programming, for the optimization of band gaps of two-dimensional photonic crystals on both square lattice and hexagonal lattice. Our numerical results convincingly show that the proposed method is very effective in producing a variety of structures with large band gaps at various frequency levels in the spectrum. 102 (ωa/2πc)2 0.6 J = 0.139, Q 2= 0.069 h h 0.4 J = 0.115, Q 1= 0.058 h h 0.2 Γ M K Γ Figure 4.26: Optimization results for two complete band gaps in the hexagonal lattice, max(min(JhT E,1 , JhT M,3 ), min(JhT E,3 , JhT M,6 )). In the band structure subplot, solid lines represent the TM bands, while broken lines represent the TE bands. (ωa/2πc)2 0.8 0.6 0.4 J = 0.268, Q 2= 0.135 h h J = 0.151, Q 1= 0.076 h h 0.2 Γ X M Γ Figure 4.27: Optimization results for two complete band gaps in the square lattice, max(min(JhT E,3 , JhT M,6 ), min(JhT E,3 , JhT M,9 )). In the band structure subplot, solid lines represent the TM bands, while broken lines represent the TE bands. 103 We have also demonstrated the usefulness of such a computational scheme based on conic (semidefinite) convex optimization to design photonic crystals with multiple and complete band gaps in two types of lattices. These photonic crystal patterns are different from existing photonic crystal designs [33, 50, 44] with a single band gap. In particular, we observe from our results that unlike optimal photonic crystals discovered in [50], photonic crystals with multiple band gaps not follow some simple geometric properties. Therefore, numerical optimization is crucial to the design of photonic crystals that support several band gaps. These photonic crystals may prove useful for the suppression of resonance at harmonic frequencies, as they prohibit the propagation of electromagnetic waves at several different frequencies. In addition, we have computed photonic structures with large complete band gaps. These results hopefully will open up a new arena for the design of photonic crystals. More importantly, we have also incorporated an efficient adaptive mesh refinement procedure to the optimization algorithm. The numerical results convincingly show that the proposed method is also effective in producing structures with multiple complete as well as combined band gaps. Moreover, using the adaptive mesh refinement predictably reduced the computational cost in both the single and multiple band gap optimization problems. 104 [...]... Outer Iterations 16 × 16 22 .1% 56 .4% 21. 5% 0 .15 6.5 32 × 32 21. 3% 57 .1% 21. 6% 0.32 8 .1 TE 64 × 64 19 .5% 61. 5% 19 .0% 1. 4 9.0 12 8 × 12 8 4. 9% 78 .4% 16 .7% 23.7 9.0 16 × 16 22.3% 57.3% 20 .4% 0.08 4. 0 32 × 32 22.0% 60.9% 17 .1% 0 .15 4. 5 TM 64 × 64 18 .5% 65 .1% 16 .4% 0.72 4 .1 128 × 12 8 5.9% 78.3% 15 .8% 10 .0 4. 5 Table 4. 2: Average computational cost (and the breakdown) of 10 runs as the mesh is refined uniformly... runs) Jh 1. 3 9.0 7 Average Execution time (min) Average Outer Iterations Successes (out of 10 runs) Jh 0 .42 3 .4 10 T E,2 Jh 2 .4 14 .1 4 T E,3 Jh 1. 7 7.7 3 T M,2 Jh 0. 81 5.0 6 T E ,4 Jh 2.9 14 .1 5 T M,3 Jh 1. 6 8 .4 2 T E,5 Jh 3.2 15 .5 1 T M ,4 Jh 1. 7 8.9 9 T E,6 Jh 3.0 13 .0 7 T M,5 Jh 2.2 11 .7 7 T E,7 Jh 3 .4 14 .2 2 T M,6 Jh 2.3 11 .0 3 T E,8 Jh 5 .1 23.5 1 T M,7 Jh 2.2 10 .9 3 T M,8 Jh 4. 6 22.5 2 Table 4 .1: Average... = a /12 8 (bottom panel) 87 1. 2 10 (ωa/2πc)2 J h = 0 .15 5, Q10 = 0.078 h 0.8 0 .4 0 Γ (ωa/2πc)2 1. 2 M K Γ J10 = 0.378, Q10 = 0 .19 1 h h 0.8 0 .4 0 Γ M K Γ (ωa/2πc)2 1. 6 J10 = 0.572, Q10= 0.292 h h 1. 2 0.8 0 .4 0 Γ M K Γ (ωa/2πc)2 1. 6 J10= 0.625, Q10= 0.320 h h 1. 2 0.8 0 .4 0 Γ M K Γ (ωa/2πc)2 1. 6 J10 = 0.639, Q10= 0.328 h h 1. 2 0.8 0 .4 0 Γ M K Γ Figure 4. 7: Mesh adaptivity results show the grids (left), optimal. .. E,3 2 .4 0.98 59 .1 T Jh E ,4 1. 7 0 .43 74. 5 T Jh E,5 2.9 1. 4 51. 7 T Jh E,6 3.2 2.0 37.5 T Jh E,7 3.0 2.2 26.7 T Jh E,8 3 .4 1. 4 58.8 Execution time (min) Uniform mesh Adaptive mesh Computational saving (%) T Jh M ,1 0 .42 0 .12 71. 4 T Jh M,2 0.72 0 .19 73.6 T Jh M,3 0. 81 0.22 72.8 T Jh M ,4 1. 6 0. 31 80.6 T Jh M,5 1. 7 0 .47 72.3 T Jh M,6 2.2 0. 51 76.8 T Jh M,7 2.3 0. 81 64. 8 T Jh M,8 2.2 0.58 73.6 Table 4. 3: Comparison... 0.353 h h 0.8 0 Γ 1. 6 (g) M 1 J 5 = 0.568, Q 5 = 0.290 h h 0 .4 1. 2 Γ T Jh M ,4 0.8 (e) Γ J 4 = 0.8 84, Q4 = 0 .46 6 h h 0.6 0 .1 0 K 1 0.2 0.6 M T Jh M,2 0.8 J 3 = 0.787, Q 3= 0. 41 0 h h 0 (c) J 2 = 0.5 84, Q 2 = 0.299 h h 0 .1 (ωa/2πc)2 (ωa/2πc)2 0.3 0.2 0 0.5 0 .4 0.3 Γ T Jh M ,1 (a) 0 .4 Γ T Jh M,9 M K 1. 2 0.8 0 .4 0 Γ (j) 10 J10= 0.695, Q = 0.359 h h Γ M K Γ T Jh M ,10 Figure 4 .12 : Optimal structures with the... gap in square lattice 91 0.5 (ωa/2πc)2 (ωa/2πc)2 0.3 0.2 0 .1 0 (a) J1= 0. 748 , Q 1 = 0.388 h h Γ X M T Jh M ,1 0 .1 Γ X M Γ J 4 = 0.733, Q 4= 0.380 h h Γ (d) X M Γ T Jh M ,4 0.9 0.6 (ωa/2πc)2 0.8 J 5 = 0.806, Q 5= 0 .4 21 h h 0.6 0 .4 Γ X M 0 Γ Γ (f) M Γ 1. 5 J 7 = 0.880, Q 7 = 0 .46 3 h h J 8 = 0.789, Q 8 = 0. 41 1 h h 1 0.5 0 .4 Γ X M 0 Γ T Jh M,7 (h) Γ X M Γ T Jh M,8 1. 6 2 1. 6 (ωa/2πc)2 1. 2 X T Jh M,6 (ωa/2πc)2... 1. 4 0 0.5 0 Γ (i) Γ 0.8 Γ T Jh E,5 1 J 4 = 0.969, Q 4 = 0. 517 h h T Jh E ,4 1 (g) Γ 1 0.3 0.8 K 0.3 0 (d) 0.6 Γ 0.6 Γ J 5 = 0 .46 0, Q 5 = 0.233 h h 0 (e) K T Jh E,3 0.9 M 0.9 (ωa/2πc)2 (ωa/2πc)2 0.8 (c) Γ T Jh E,2 (b) 0.6 0 .4 J 2 = 0 .12 6, Q 2 = 0.063 h h 0 .1 Γ T Jh E ,1 (a) 0.3 T Jh E,9 M K 1. 4 1 J10= 0.639, Q10= 0.328 h h 0.6 0.2 0 Γ Γ (j) M K Γ T Jh E ,10 Figure 4 .11 : Optimal structures with the first ten... 0 Γ (h) X M Γ T Jh E,8 1. 5 1. 2 J 9 = 0. 642 , Q 9 = 0.330 h h (ωa/2πc)2 0.8 J 8 = 0.685, Q 8 = 0.353 h h 1 Γ T Jh E,7 (ωa/2πc)2 Γ T (d) Jh E ,4 0.9 Γ T Jh E,9 10 J10 = 0.5 64, Q = 0.288 h h 1 0.5 0 .4 0 (i) J 4 = 0. 643 , Q4 = 0.330 h h 0 .1 Γ (g) Γ 0.3 0 (e) M 0.7 0.3 0 X T Jh E,2 0 .4 (c) J 2 = 0.655, Q 2 = 0.337 h h 0 .1 Γ T Jh E ,1 (a) 0.3 X M 0 Γ (j) Γ X M Γ T Jh E ,10 Figure 4. 9: Optimal structures with... observe that the computational 82 T E,9 T M,9 8 8 Occurences 10 Occurences 10 6 4 2 0 −0 .4 6 4 2 −0.2 0 0.2 J TE,2 h 0 .4 0.6 0 −0 .4 0.8 25 0.2 J TE,9 h 0 .4 0.6 0.8 0.6 0.7 0.8 25 20 0 30 20 Occurences Occurences 30 −0.2 15 15 10 10 5 0 −0 .1 5 0 0 .1 0.2 0.3 0 .4 0.5 J TM,2 h 0.6 0.7 0 0.8 −0 .1 0 0 .1 0.2 0.3 0 .4 0.5 JTM,9 h Figure 4. 3: Histograms of gap-midgap ratios for 30 runs with random initial configurations,... frontiers can be attributed to multiple local optima 97 (ωa/2πc)2 0.6 0 .4 0.2 0 (a) J 4 = 0.390, Q 4 = 0 .19 7 h h T Jh E ,1 J 1 = 0. 41 2 , Q 1 = 0.208 h h X Γ and M Γ T Jh E ,4 (ωa/2πc)2 0.6 0 .4 J 4 = 0.560, Q 4 = 0.286 h h 0.2 J 1= 0.592, Q 1 = 0.303 h h 0 (b) T Jh E ,1 M Γ and K Γ T Jh E ,4 Figure 4 .15 : Optimization results show the final optimal crystal structure (left), and frequency bands (right) for the . solve 56 .4% 57 .1% 61. 5% 78 .4% 57.3% 60.9% 65 .1% 78.3% Other cost 21. 5% 21. 6% 19 .0% 16 .7% 20 .4% 17 .1% 16 .4% 15 .8% Total time (min) 0 .15 0.32 1. 4 23.7 0.08 0 .15 0.72 10 .0 Outer Iterations 6.5 8 .1 9.0. Execution time (min) 0 .42 0.72 0. 81 1.6 1. 7 2.2 2.3 2.2 4. 6 Average Outer Iterations 3 .4 4 .1 5.0 8 .4 8.9 11 .7 11 .0 10 .9 22.5 Successes (out of 10 runs) 10 8 6 2 9 7 3 3 2 Table 4 .1: Average computational. representation of the dielectric function, as well as the 83 J 2 h TE TM Mesh size 16 × 16 32 × 32 64 × 64 12 8 × 12 8 16 × 16 32 × 32 64 × 64 12 8 × 12 8 Eigenvalue solve 22 .1% 21. 3% 19 .5% 4. 9% 22.3% 22.0% 18 .5%