Optimal design of photonic crystals 51

Chapter Single-Polarization Single-Mode Photonic Crystal Fiber A waveguide that is single-polarization single-mode (SPSM) is one that guides only one polarization of the fundamental modes, while the orthogonal polarization is eliminated. The design objective of the SPSM waveguide is to optimize the frequency range in which only one mode is guided. Shown in Figure 5.1, the frequency range, also known as the band width, starts with the frequency G (β ) when the fundamental mode falls below the light line and becomes λW L effectively guided , while the second mode still lies inside the light cone, until G (β ) at which the second mode eventually becomes guided the frequency λW R when it falls below the light line. This problem is analogous to the band gap optimization problem of the two-dimensional photonic crystals, but instead of maximizing the difference between the maximum and minimum frequencies over the entire irreducible Brillouin zone (a set of k⊥ ), the band width optimization attempts to maximize the difference between two frequencies over a certain range of propagation constants (a set of kz ). Hence, it is natural to extend the algorithm developed for the band gap optimization problem, to the design of the SPSM photonic crystal fiber (PCF) with optimal band width. In this chapter, we will first state the optimization problem, and then pro- 105 (β) ( βR ) CL,1 = λh ( β R ) λ guided mode λhWG ,2 ( β ) λh CL,1 (β) WG ,2 h e fundamental ht lin guided mode λhWG ,1 ( β ) lig eigenvalue λ high order λhCL,2 light cone λhWG ,1 ( β L ) = λhCL,1 ( β L ) βmin = βL β max βR propagation constant β Figure 5.1: Example of dispersion relation of a three-dimensional photonic crystal fiber. The inset illustrates an example of the cross section of the waveguide. βL and βR are the propagation constants at which the first and second waveguide modes intersect the light line respectively. pose several formulations,starting with a very intuitive formulation. However, this first attempt presents some serious limitations, so a second formulation is proposed in the hope of overcoming the identified issues. A trust region approach is also incorporated into the optimization formulation to increase the validity of our model. 5.1 The Optimal Design Problem The resonance problem of PCF is governed by two eigenvalue problems derived from the Maxwell equations. The first one models the infinitely periodic photonic crystal that makes up the cladding of the PCF without defect: CL ACL , β)uCL,j = λCL,j MhCL uCL,j , h (ε h h h j = 1, . . . , N , β ∈ Ph , in Ω. (5.1.1) The operator ACL h can be assembled according to (3.2.19) over the computation domain of only one primitive cell Ω with periodic boundary condition, and the 106 dielectric function is determined by εCL (as in Figure 3.17 of subsection 3.2.2). The second eigenvalue problem models the case when a core is introduced as the defect to the otherwise periodic photonic crystal cladding: G,j G,j G,j G CL CO AW ; ε ], β)uW = λW MhW G uW , h ([ε h h h j = 1, . . . , Ns , β ∈ Ph , in Ωs . (5.1.2) G can be assembled according to (3.2.19) over the Similarly, the operator AW h computation domain of the super cell Ωs with periodic boundary condition, and the dielectric function is determined by εW G = [εCL ; εCO ] (as in Figure G and ACL are affine 3.17 of subsection 3.2.2). Recall that both operators AW h h −1 i ,i with respect to the reciprocal of the dielectric variables: = 1, . . . , nε ,(see (3.2.19)). We introduce a change of variables and define γ as the design variables of the optimization problem: −1 i ,i γi = = 1, . . . , nγ = nε , γL ≤ γi ≤ γH , γL = 1/ H, γH = 1/ L . (5.1.3) CL , β) We shall only work with γ for the rest of this chapter. The operators ACL h (γ CL ; γ CO ], β) are now linear in their respective design variables. G and AW h ([γ 5.1.1 Formulation I The most straightforward strategy to the optimization problem is to directly maximize the difference of the first two frequencies of the waveguide at the intersections with the light line (fundamental space filling mode of the periodic cladding), as illustrated in Figure 5.1. It can be stated mathematically as follows: max G [γ CL ;γ CO ]∈QW h s.t. G,2 G,1 λW ([γ CL ; γ CO ], βR ) − λW ([γ CL ; γ CO ], βL ) , h h G,j G,j G,j G AW ([γ CL ; γ CO ], β)uW = λW MhW G uW , h h h h β = βL , βR , G,1 CL ACL , β)uCL,1 = λCL,1 MhW G uW , h (γ h h h β = βL , βR , j = 1, 2, G,1 βL = arg λW [γ CL ; γ CO ], β = λCL,1 (γ CL , β) , h h G,1 βR = arg λW [γ CL ; γ CO ], β = λCL,1 (γ CL , β) . h h (5.1.4) We use arg(f (x) = g(x)) to mean “the argument x∗ ” such that f (x∗ ) = g(x∗ ). The superscript W G on the admissible range Qh indicates that the decision variables being considered are those defining the cross-section of the waveguide (su- 107 G per cell Ωs ), i.e., γ W G = [γ CL ; γ CO ] ∈ QW h . We will also encounter another admissible range denoted by QCL h , where the superscript CL indicates that the decision variables being considered are those defining the periodic cladding (only one primitive unit cell Ω), i.e., γ CL ∈ QCL := {γ : γ ∈ [γL , γH ]nγ CL }; analoh gously, γ CO ∈ QCO = {γ : γ ∈ [γL , γH ]nγ CO }. h Besides being a non-convex optimization problem in the design variables, formulation (5.1.4) is also nonlinear due to the design-variable-dependent βL and βR . The strategy is therefore to apply the subspace approximation and reduction algorithm developed in chapter to convert it to a convex semi-definite program, and to solve the resulting approximate subproblems at the approximately fixed {βL , βR } range at each linearization. We first rediscretize the propagation constant between {βL , βR } as Ph := β | β ∈ {β1 (= βL ), . . . , βnβ (= βR )} . (5.1.5) We also prescribe a left-hand limit βmin = 0, and right-hand limit βmax for the propagation constant, whose purpose is related to the computation of βL and βR , which will be discussed later. Let us also define two additional variables: U := max β∈[βL ,βR ] λhW G,2 ([γ CL , γ CO ], β), L := G,1 ([γ CL ; γ CO ], β). λW h β∈[β1 ,β2 ] (5.1.6) 108 The original problem (5.1.4) can be rewritten as P0 : max γ CL ,γ CO ,u, s.t. U −L , G,1 λW [γ CL ; γ CO ], β ≥ λCL,1 (γ CL , β), h h ∀β ∈ [βmin , βL ], G,1 (γ CL , β), λW [γ CL ; γ CO ], β ≤ λCL,1 h h ∀β ∈ [βL , βmax ], G,2 λW [γ CL ; γ CO ], β ≥ λCL,1 (γ CL , β), h h ∀β ∈ [βmin , βR ], G,2 (γ CL , β), λW [γ CL ; γ CO ], β ≤ λCL,1 h h ∀β ∈ [βR , βmax ], G,1 λW [γ CL ; γ CO ], βL ≤ L, h G,2 [γ CL ; γ CO ], βR ≥ U, λW h γL ≤ γiCL , γjCO ≤ γH , i = 1, . . . , nγ CL , j = 1, . . . , nγ CO , U, L ≥ 0, G,1 (γ CL , β) , βL = arg λW [γ CL ; γ CO ], β = λCL,1 h h G,2 βR = arg λW [γ CL ; γ CO ], β = λCL,1 (γ CL , β) . h h (5.1.7) G,j , j = 1, 2, are to satisfy both (5.1.1) and (5.1.2) and λW The eigenvalues λCL,1 h h at all times. For succinctness, these eigenvalue-related equality constraints are not explicitly included in P0 and all the subsequent formulations. The fifth and G,j , j = 1, 2. sixth constraints are derived from the monotonicity of λW h Depending on various design requirements, we have formulated three different scenarios: (a) cladding γ CL are known, and core γ CO are variables; (b) core γ CO are known, and cladding γ CL are variables; (c) both cladding γ CL and core γ CO are variables. Each scenario is described in detail below. For notational clarity, an overline • is used to denote known quantities, or quantities that are independent of the design variables; an overhat ˆ• is used to denote temporarily fixed quantities at each linearization. Scenario a: [γ CL ; γ CO ] At the beginning of each linearization, a given γ CO is assumed. Together with the known cladding γ CL , the intersections with the fundamental space-filling mode of the waveguide cladding λCL,1 can be calculated h 109 using various root-finding algorithms (e.g., bisection method1 ): CL,1 G,1 βL = arg λW (γ CL , γ CO β) = λh h (γ CL , β) , CL,1 G,2 βR = arg λW (γ CL , γ CO β) = λh h (γ CL , β) . (5.1.8) We relax (5.1.7) and obtain the following: max γ CO ,U,L s.t. U − L, CL,1 (γ CL , βj ), j = 1, . . . , nβ , CL,1 (γ CL , βj ), j = 1, . . . , nβ , λhW G,1 [γ CL ; γ CO ], β ≤ λh λhW G,2 [γ CL ; γ CO ], β ≥ λh λhW G,1 [γ CL ; γ CO ], β1 ≤ L, λhW G,2 [γ CL ; γ CO ], βnβ ≥ U, γL ≤ γiCO ≤ γH , i = 1, . . . , nγ CO , U, L ≥ 0. (5.1.9) It is important to note that we eliminated all the constraints associated with β < βL and β > βR , because these two impose very tight restriction on the eigenvalues at the current approximate βL and βR , and no progress could be made during the optimization process if they are to be satisfied. It also turns out the first two constraints are crucial in ensuring the validity of the physical problem: relaxing these two constraints might lead to various “false optimal” scenarios. For example, we could get into the trouble of lacking intersections βL G,2 or βR : λhW G,1 and λW are well separated but are both below the light line h λCL,1 for β ∈ [β1 , βnβ ]. This is an undesirable physical situation, as the first two h modes of the waveguide become both guided, rather than a single-guided mode. Next, we introduce the approximate and reduced matrices that eventually The bisection method requires two initial points at which the function evaluations have opposite signs. In our case, βmin = and βmax are used. Although function values are not guaranteed to be of opposite signs, one of the two limits will be taken as a reasonable “approximate” root in such case. 110 span the subspaces upon which our convex optimization models are built: G CL G G CL ΘW , γ CO , βj ) := [ΦW (γ CL , γ CO , βj ) | ΨW , γ CO , βj )] 1+aj (γ aj (γ W G,2+aj G,1 G,2 := [uW (γ CL , γ CO , βj ) | uW (γ CL , γ CO , βj ), . . . , uh h h (γ CL , γ CO , βj )], (5.1.10) for j = 1, . . . , nβ . The construction of these matrices has been discussed in detail in section 4.2.1. Notice that in this problem, the matrix that spans the lower subspace at any propagation constant is of rank 1, and consists of G,1 CL precisely only one eigenvector uW (γ , γ CO , βj ); the matrix that spans the h upper subspace at propagation constant βj is of dimension aj . We obtain the following equivalent convex formulation with semi-definite inclusions: CO γ PIa : max γ CO ,U,L s.t. U − L, nγ CO G∗ ΦW (γ CL , γ CO , βj ) i=1 CL,1 −λh G WG CL γiCO AW , βj ) h,i (βj ) + Ah,0 (γ G (γ CL , βj )MhW G ΦW (γ CL , γ CO , βj ) 0, j = 1, . . . , nβ , nγ CO G∗ ΨW (γ CL , γ CO , βj ) aj i=1 CL,1 −λh G WG CL , βj ) γiCO AW h,i (βj ) + Ah,0 (γ G CL (γ CL , βj )MhW G ΨW , γ CO , βj ) aj (γ 0, j = 1, . . . , nβ , G∗ ΦW (γ CL , γ CO , β1 ) nγ CO i=1 G WG CL , β1 ) γiCO AW h,i (β1 ) + Ah,0 (γ G −LMhW G ΦW (γ CL , γ CO , β1 ) CL G∗ , γ CO , βnβ ) ΨW anβ (γ nγ CO i=1 0, WG G CL γiCO AW , β nβ ) h,i (βnβ ) + Ah,0 (γ G CL −U MhW G ΨW , γ CO , βnβ ) anβ (γ γL ≤ γiCO ≤ γH , 0, i = 1, . . . , nγ CO , U, L ≥ 0. (5.1.11) G CL ; γ CO ], β) is decomposed into the γ CO −independent The discrete operator AW h ([γ G CL , β), plus a summation of the γ CL − and γ CO − independent mamatrix AW h,0 (γ G CO WG trices AW h,i (β) multiplying each decision variable γi . The mass matrix Mh is independent of both decision variable and propagation constant. Therefore CO γ PIa is a tractable convex program containing 2nβ + semi-definite inclusions, and 2nγ CO + linear constraints. 111 Scenario b: [γ CL ; γ CO ] When γ CL are the design variables and are allowed to vary, the light line is now design-variable-dependent. We need to introduce more variables Lj , Uj , j = 1, . . . , nβ such that (γ CL , βj ), λhW G,1 ([γ CL , γ CO ], βj ) ≤ Lj ≤ λCL,1 h λhW G,2 ([γ CL , γ CO ], βj ) ≥ Uj ≥ j = 1, . . . , nβ λCL,1 (γ CL , βj ), h (5.1.12) j = 1, . . . , nβ . In addition, more reduced and approximate matrices for the periodic cladding modes are defined as CL CL CL ΘCL , βj ) := [ΦCL , βj ) | ΨCL , βj )] (γ 1+bj (γ bj (γ CL,2+bj := [uCL,1 (γ CL , βj ) | uCL,2 (γ CL , βj ), . . . , uh h h (γ CL , βj )], (5.1.13) for j = 1, . . . , nβ . Similar to the subspaces for the waveguide modes, the matrix that spans the lower subspace of the cladding modes is also of rank at any propagation constant, and consists of precisely only one eigenvector uCL,1 (γ CL , βj ); h the matrix that spans the upper subspace is of dimension bj at propagation constant βj . We obtain the following equivalent convex formulation: CL γ PIb : max Unβ − L1 γ CL , L1 , . . . , Lnβ , U1 , . . . , Unβ s.t. G∗ ΦW (γ CL , γ CO , βj ) nγ CL i=1 G WG CO , βj ) γiCL AW h,i (βj ) + Ah,0 (γ G −Lj MhW G ΦW (γ CL , γ CO , βj ) G∗ ΨW (γ CL , γ CO , βj ) aj nγ CL i=1 nγ CL i=1 nγ CL i=1 Uj , Lj ≥ 0, j = 1, . . . , nβ , 0, j = 1, . . . , nβ , CL γiCL ACL h,i (βj ) + Ah,0 (βj ) CL −Uj MhCL ΨCL , βj ) bj (γ γL ≤ γiCL ≤ γH , 0, CL γiCL ACL h,i (βj ) + Ah,0 (βj ) CL −Lj MhCL ΦCL , βj ) (γ CL ΨCL∗ , βj ) bj (γ j = 1, . . . , nβ , G WG CO γiCL AW , βj ) h,i (βj ) + Ah,0 (γ G CL −Uj MhW G ΨW , γ CO , βj ) aj (γ ΦCL∗ (γ CL , βj ) 0, 0, j = 1, . . . , nβ , i = 1, . . . , nγ CL , j = 1, . . . , nβ . (5.1.14) CL , β) can also be decomposed into the γ CL The discrete operator ACL h (γ 112 CL independent matriindependent matrix ACL h,0 (β), plus a summation of the γ CL CL is ces ACL h,i (β) multiplying each decision variable γi . The mass matrix Mh independent of both decision variable and propagation constant too as MhW G . CO CL γ γ Compared to PIa , program PIb contains 2nβ more semi-definite inclusions to account for the light line mode. Scenario c: [γ CL ; γ CO ] The extension from scenario (b) to (c), where both γ CL and γ CO are the design variables, is minor with the inclusion of γ CO being part of the decision variables: CL γ PIc ,γ CO : max γ CL , L1 , . . . , Lnβ , Unβ − L1 , γ CO , U1 , . . . , Unβ G∗ (γ CL , γ CO , β ) ΦW j s.t. nγ CL i=1 G γiCL AW h,i (βj ) + G W G ΦW G (γ CL , γ CO , β ) +AW j h,0 (βj ) − Lj Mh G∗ (γ CL , γ CO , β ) ΨW j aj nγ CL i=1 G γiCL AW h,i (βj ) + G W G ΨW G (γ CL , γ CO , β ) +AW j aj h,0 (βj ) − Uj Mh nγ CO ΦCL∗ (γ CL , βj ) i=1 ΨCL∗ (γ CL , βj ) b i=1 j i=1 0, nγ CO i=1 0, G (β ) γiCO AW h,nCO +i j γ j = 1, . . . , nβ , G γiCO AW (β ) h,nCO +i j γ j = 1, . . . , nβ , CL γiCO ACL h,i (βj ) + Ah,0 (βj ) CL , β ) −Lj MhCL ΦCL j (γ nγ CO nγ CO 0, j = 1, . . . , nβ , CL γiCO ACL h,i (βj ) + Ah,0 (βj ) CL , β ) −Uj MhCL ΨCL j b (γ j 0, j = 1, . . . , nβ , γL ≤ γiCL , γjCO ≤ γH , i = 1, . . . , nγCL , j = 1, . . . , nγCO , Uj , Lj ≥ 0, j = 1, . . . , nβ . (5.1.15) CL ,γ CO γ Program PIc has the most decision variables and constraints. Compared γ CL to PIb , the increase is nγCO more decision variables, and 2nγCO more linear inequality constraints. Despite being the most direct formulation of the optimization problem, formulation I has some fundamental drawbacks: • Unlike the previous band gap optimization problem, where a dimensionless quantity gap-midgap ratio is modeled as the objective function, one must have noticed that the absolute band width is chosen instead as the objective function in this formulation. This is caused by the possibility of L (or L1 in scenarios (b) and (c)) being zero, in which case the objective value would have turned out to be unity whatever value U or (Unβ ) takes. Thus it defeats the purpose of band width optimization. 113 • Probably the most serious caveat with this formulation lies in the difficulty of accurate computation of the intersections βL and βR (or the corresponding eigenfrequencies). As explained before, when a waveguide mode is guided, it decays exponentially away from the core into the cladding. This modal diameter increases rapidly with wavelength, i.e., when the frequency approaches the light line, the transverse decay rate slows down. In fact, it is explained in [38] that the modal diameter seems to increase exponentially with the wavelength. Given that any real structure has a finite cladding, this makes it difficult, both numerically and experimentally, to study the long-wavelength regime, especially the behavior of the G,1 fundamental guided mode (λW ) as it approaches the less than rigorh ously defined intersection βL with light line (λCL,1 ). The situation for the h G,2 second guided mode (λW ) becomes more subtle if it has a cut-off with h G,2 as the light line away from long-frequency. One can loosely view λW h a perturbed mode to the second light line, λCL,2 (a degenerate light line h if the cladding symmetry had not been broken). By the same token, it is G,2 hard to capture the regime when λW is approaching λCL,2 with finite h h cladding, even if more sophisticated boundary condition treatment had been prescribed, e.g., perfect matching layer. However, βR is defined as the intersection of the seconded guided mode with the first light line. So as long the degeneracy of two light lines is broken, βR can be numerically computed. This brings us to the next formulation, where we avoid computing βL by picking a fixed propagation constant β . 5.1.2 Formulation II In formulation II, we start with a prescribed propagation constant β , and try to optimize the corresponding frequency difference between the light line and the fundamental guided mode while requiring the second mode to be above the light line. The band width where only one mode is guided is defined below as a 114 dimensionless ratio for the band gap optimization problem: β Jh (γ CL , γ CO ) = G,2 G,1 min{λCL,1 (γ CL , β ), λW [γ CL ; γ CO ], β } − λW [γ CL ; γ CO ], β h h h . G,2 G,1 min{λCL,1 (γ CL , β ), λW [γ CL ; γ CO ], β } + λW [γ CL ; γ CO ], β h h h (5.1.16) G,1 Note that λW [γ CL ; γ CO ], β = 0, for β = 0, thus it is a well defined h (β) objective function. This concept is illustrated in Figure 5.2. The goal is to λhCL,2 light cone U1 e λh CL,1 (β) guided mode λhWG ,2 ( β ) lin fundamental guided mode λhWG ,1 ( β ) ht L lig eigenvalue λ U2 ( or U ) high order β1 β2 propagation constant β Figure 5.2: Illustration of formulation II for the optimal design of single-mode single-polarization photonic crystal fiber. The propagation constant β is prescribed, while β1 is a computed quantity. max γ CL ,γ CO β Jh (γ CL , γ CO ). 115 This can be described as P1 : max γ CL ,γ CO ,u, s.t. U −L U +L , G,1 λW [γ CL ; γ CO ], β ≤ L, h G,2 [γ CL ; γ CO ], β ≥ L, λW h λCL,1 (γ CL , β ) ≥ U, h G,2 λW [γ CL ; γ CO ], β ≥ λCL,1 (γ CL , β ), h h G,1 λW ([γ CL ; γ CO ], β1 ) ≤ L, h G,2 λW ([γ CL ; γ CO ], β1 ) ≥ L, h γL ≤ γiCL , γjCO ≤ γH , i = 1, . . . , nγ CL , j = 1, . . . , nγ CO , U, L ≥ 0, G,1 β1 = arg λCL,1 (γ CL , β) = λW ([γ CL ; γ CO ], β2 ) . h h (5.1.17) Here U and L are two auxiliary variables indicating the lower bound of the light line and upper bound of the fundamental guided mode at β . The second waveguide mode is also required to be unguided at β , and at β1 as a tighter constraint. β1 is defined as the propagation constant at which the light line G,1 takes the upper bound value of λW at β , and it should be that β1 < β . This h translates to the fifth inequality in P1 . We again formulate the programs into three different scenarios depending on the various design requirements. Same notations on prescribed and given quantities are used as formulation I, scenarios (a), (b), and (c). Scenario a: [γ CL ; γ CO ] Assuming a prescribed cladding γ CL throughout, and a given core γ CO at the beginning of each linearization, β1 defined as G,1 CL β1 = arg λCL,1 (γ CL , β) = λW (γ , γ CO , β ) , h h 116 can be computed using root-finding algorithms (e.g., the bisection method is chosen in our implementation). P1 is relaxed to max γ CO ,L s.t. U −L , U +L G,1 CL λW (γ , γ CO , β1 ) ≤ L , h λhW G,2 (γ CL , γ CO , β1 ) ≥ L , λhW G,1 (γ CL , γ CO , β ) (5.1.18) ≤L, G,2 CL λW (γ , γ CO , β ) ≥ U , h γL ≤ γjCO ≤ γH , j = 1, . . . , nγCO , L ≥ 0. Here U is a constant, and can be computed with given γ CL and β as (γ CL , β ). U = λLC,1 h The approximate and reduced matrices for the waveguide modes are defined as CL G G CL G , γ CO , βj ) := [ΦW (γ CL , γ CO , βj ) | ΨW , γ CO , βj )] ΘW aj (γ 1+aj (γ W G,2+aj G,2 G,1 (γ CL , γ CO , βj ), . . . , uh := [uW (γ CL , γ CO , βj ) | uW h h (γ CL , γ CO , βj )], j = 1, 2. (5.1.19) 117 The equivalent convex formulation with semi-definite inclusions can be written as CO γ PIIa γ CO ,L U −L U +L s.t. G∗ ΦW (γ CL , γ CO , β1 ) max : nγ CO i=1 G CL WG γiCO AW , β1 ) h,i (β1 ) + Ah,0 (γ G −LMhW G ΦW (γ CL , γ CO , β1 ) G∗ (γ CL , γ CO , β1 ) ΨW a1 nγ CO i=1 G CL WG γiCO AW , β1 ) h,i (β1 ) + Ah,0 (γ G CL −LMhW G ΨW , γ CO , β1 ) aj (γ G∗ ΦW (γ CL , γ CO , β ) nγ CO i=1 G∗ (γ CL , γ CO , β ) ΨW a2 i=1 0, G WG CL γiCO AW , β2) h,i (β ) + Ah,0 (γ CL G −U MhW G ΨW , γ CO , β ) a2 (γ γL ≤ γiCO ≤ γH , 0, G WG CL γiCO AW , β2) h,i (β ) + Ah,0 (γ G −LMhW G ΦW (γ CL , γ CO , β ) nγ CO 0, 0, i = 1, . . . , nγ CO , L ≥ 0. (5.1.20) The stiffness matrices can again be expressed as a linear combination of the variable independent CO γ parts, while the decision variables are the coefficients. PIIa is a convex program containing four SDP inclusions, and 2nγ CO + linear inequalities. This is a drastic reduction in terms of the number of constraints as compared to those in formulation I. 118 Scenario b: [γ CL ; γ CO ] When core γ CO is fixed, and cladding γ CL takes on a given value γ CL at the beginning of each linearization, we can compute G,1 β1 = arg λCL,1 (γ CL , β) = λW (γ CL , γ CO , β ) . h h The light line is now variable dependent, and we need to introduce two auxiliary variables as its upper bound U1 and lower bound U2 at β . To make sure the second waveguide mode is not guided, we require it to be above the upper bound of the light line at β , i.e., G,1 G,2 λW ≤ L ≤ U2 ≤ λCL,1 ≤ U1 ≤ λW . h h h These translate to the following formulation: max γ CL ,L,U1 ,U2 s.t. U2 −L , U2 +L G,1 λW (γ CL , γ CO , β1 ) ≤ L , h G,2 λW (γ CL , γ CO , β1 ) ≥ L , h G,1 λW (γ CL , γ CO , β ) ≤ L , h G,2 λW (γ CL , γ CO , β ) h (5.1.21) ≥ U1 , λhCL,1 (γ CL , β ) ≤ U1 , λhCL,1 (γ CL , β ) ≥ U2 , γL ≤ γiCL ≤ γU , i = 1, . . . , nγCL , L, U1 , U2 ≥ 0. G,2 The second inequality ensures the unguidedness of λW at β1 by remaining above the light h line. In addition to the matrices defined in II(a), more approximate and reduced matrices are necessary for the cladding modes: CL CL CL ΘCL , β2 ) := [ΦCL , β2 ) | ΨCL , β2 )] 1+b2 (γ (γ b2 (γ (5.1.22) G,1+b2 := [uCL,1 (γ CL , β2 | uCL,1 (γ CL , β2 ), . . . , uW (γ CL , β2 )]. h h h With these matrices, we can rewrite (5.1.21) into a convex program of the following form: 119 CL γ PIIb : max γ CL ,L,U1 ,U2 s.t. U2 −L U2 +L nγ CL G∗ ΦW (γ CL , γ CO , β1 ) i=1 G WG CO γiCL AW , β1 ) h,i (β1 ) + Ah,0 (γ G −LMhW G ΦW (γ CL , γ CO , β1 ) nγ CL G∗ ΨW (γ CL , γ CO , β1 ) a1 i=1 0, G WG CO γiCL AW , β1) h,i (β1 ) + Ah,0 (γ G CL −LMhW G ΨW , γ CO , β1 ) aj (γ nγ CL G∗ ΦW (γ CL , γ CO , β ) i=1 0, WG CO G γiCL AW , β2) h,i (β ) + Ah,0 (γ G −LMhW G ΦW (γ CL , γ CO , β ) nγ CL G∗ ΨW (γ CL , γ CO , β ) a2 i=1 0, WG CO G γiCL AW , β2) h,i (β ) + Ah,0 (γ G CL −U1 MhW G ΨW , γ CO , β ) a2 (γ ΦCL∗ (γ CL , β ) nγ CL i=1 CL CL γiCL ACL , β2) h,i (β ) + Ah,0 (γ CL −U1 MhCL ΦCL , β2) (γ ΨbCL∗ (γ CL , β ) nγ CL i=1 CL CL , β2) γiCL ACL h,i (β ) + Ah,0 (γ CL −U2 MhCL ΨCL , β2) a2 (γ γL ≤ γiCL ≤ γH , 0, 0, i = 1, . . . , nγ CL , L, U1 , U2 ≥ 0. (5.1.23) γ CL PIIb includes two additional decision variables U1 and U2 , as well as two more SDP inclusions CO γ accounting for the light line modes, compared to PIIa in which only the cladding region is to be designed and optimized. Scenario c: [γ CL ; γ CO ] Designing both the core and the cladding does not require significantly more effort than the design when only the cladding is considered. First, β1 can be computed almost the same way as before: G,1 β1 = arg λCL,1 (γ CL , β) = λW (γ CL , γ CO , β ) . h h The formulation is almost the same as (5.1.21), except that both γ CL , and γ CO are decision 120 0, variables: max γ CL ,γ CO ,L,U1 ,U2 U2 − L , U2 + L G,1 λW (γ CL , γ CO , β1 ) ≤ L , h s.t. G,2 λW (γ CL , γ CO , β1 ) ≥ L , h G,1 λW (γ CL , γ CO , β ) ≤ L , h G,2 λW (γ CL , γ CO , β ) ≥ U1 , h λCL,1 (γ CL , β ) ≤ U1 , h λCL,1 (γ CL , β ) ≥ U2 , h γL ≤ γiCL , γjCO ≤ γH , i = 1, . . . , nεCL , j = 1, . . . , nεCO , L, U1 , U2 ≥ 0. With all the approximate and reduced matrices defined in both scenarios (a) and (b), we derive the following convex program for scenario (c): CL ˆ γ PIIc ,ˆ γ CO : max γ CL , γ CO , U2 −L , U2 +L L, U1 , U2 s.t. nεCL i=1 G∗ CL ΦW (ε , εCO , β1 ) G γiCL AW h,i (β1 ) + G +AW h,0 (β1 ) nεCL i=1 G∗ CL ΦW (ε , εCO , β2 ) − LMhW G nεCO i=1 G γiCO AW h,nCO +i (β1 ) ε G CL ΦW (ε , εCO , β1 ) G γiCL AW h,i (β2 ) + nεCO i=1 G γiCO AW h,nCO +i (β2 ) ε G WG G CL +AW ΦW (ε , εCO , β2 ) h,0 (β2 ) − LMh nεCL i=1 G∗ CL ΨW (ε , εCO , β1 ) a1 G γiCL AW h,i (β1 ) + G +AW h,0 (β1 ) nεCL i=1 G∗ CL ΨW (ε , εCO , β2 ) a2 nεCO i=1 − U1 MhW G nεCO i=1 ε ε CL γiCO ACL h,i (β2 ) + Ah,0 (β2 ) 0, CL γiCO ACL h,i (β2 ) + Ah,0 (β2 ) CL −U2 MhCL ΨCL , β2 ) b2 (ε 0, γL ≤ γiCL , γjCO ≤ γH , i = 1, . . . , nγCL , j = 1, . . . , nγCO , L, U1 , U2 ≥ 0. CL ˆ γ As one can see, the only addition in PIIc ,ˆ γ CO 0, G γiCO AW h,nCO +i (β2 ) G CL ΨW , εCO , β2 ) a2 (ε CL −U1 MhCL ΦCL , β2 ) (ε CL ΨCL∗ , β2 ) b2 (ε nεCO i=1 0, G γiCO AW h,nCO +i (β1 ) G CL ΨW , εCO , β1 ) a1 (ε G γiCL AW h,i (β2 ) + G +AW h,0 (β2 ) ΦCL∗ (εCL , β2 ) − LMhW G nεCO i=1 0, is the inclusion of the decision variables γ CL representing the cladding design, as well as the upper and lower bounds on these variables. While scenarios (c) in both formulations I and II comprise the most decision variables and con- 121 0, straints, their extension from the simple scenarios (a) and (b) are natural and simply minimal; moreover, they provide more flexible design needs. 5.1.3 Trust region Computing a solution to a nonlinear and non-convex problem is nontrivial. Although the SDP and linear relaxation introduced via our algorithm have proven to be efficient for the band gap optimization of the two-dimensional photonic crystal, the linearization and approximation did not seem to be sufficiently reliable in this band width optimization problem with an underlying quasi-three-dimensional vectorial PDE. The idea behind a trust region method is very simple. A bound is levied on the step size of the solution to an approximate subproblem Bk sk ≤ ∆k , where Bk is a scaling matrix and ∆k is a positive scalar representing the trust region size. The step size between two sequential optimal solutions is denoted by sk . In our problem, the step size can be represented by the difference between the optimal solution and the linearizer, i.e.,γ − γ. Given the lower bound γL and the upper bound γH on the γ-related decision variables, we propose the following two trust region methods. Method I: nγ nγ j=1 γj − γj γj − γL nγ γj − γj γH − γj + j=1 ≤ ∆, (5.1.24) which gives a diagonal scaling matrix satisfying Bj,j = Dj,j = nγ γj − γL + γH − γj . (5.1.25) If we examine this method I, it is not hard to realize that it would limit the solution strictly inside the feasible region, and away from the boundary faces or vertices. This is due to the implicit non-zero requirement on the denominators. In practical photonic crystal design however, one wishes to achieve a solution comprising of either low or high dielectric materials. In other words, the solution should always be on the faces or vertices of the polyhedron defining the feasible region. This brings us to the following modified method II. Method II: nγ nγ j=1 γj − γj γH − γL ≤ ∆, (5.1.26) which gives another diagonal scaling matrix satisfying Bj,j = Dj,j = nγ γH − γL . (5.1.27) 122 We rewrite it in matrix-vector multiplication form (γ − γ)∗ D(γ − γ) = (γ − γ)∗ B ∗ B(γ − γ) ≤ ∆, (5.1.28) and simplify the notation γ ∗ Dγ + χγ + β ≤ ∆, (5.1.29) where χ = −2γ ∗ D, and σ = γ ∗ Dγ − ∆. Finally, we define a second-order cone KQ := {x = ¯ ] ∈ Rnγ +2 : x0 ≥ x ¯ }, and introduce a new variable x = [x0 ; x ¯ ] ∈ KQ such that [x0 ; x      x0    −     ¯ x     Q 0,  Bγ  ,  ¯= x    (χ∗ γ + σ + 1)       x = (1 − χ∗ γ − σ). (5.1.30) These new decision variables as well as the new constraints are to be incorporated to the previous formulations. Effectively, there will be nγ + additional decision variables, one more SCOP inclusion, and nγ + extra equalities (or 2(nγ + 2) equivalent inequalities). 5.2 5.2.1 Results and Discussion Model setup We consider a three-dimensional photonic crystal with z-invariant cross-section set up schematically according to Figure 3.17, on either a rectangular lattice with pitch distance Λx a = 1.5a = and Λy a = a = 2, or a rhombic lattice with lattice constant a = 2. The primitive cell domain Ω is decomposed into a uniform grid of mesh size a/40, and the super cell domain Ωs is decomposed into a uniform grid of the same mesh size, but of more elements. The cladding consists of two rings of Ω surrounding the core, hence, the total number of elements in Ω is 40 × 60 = 2, 400, and in Ωs is 2, 400 × 25 = 60, 000 in a rectangular lattice; the total number of elements in Ω is 40 × 40 = 1, 600, and in Ωs is 1, 600 × 25 = 40, 000 in a rhombic lattice setup. We use the same dielectric materials as in section 3.2.3 to construct the waveguide, i.e., epoxy of L = 2.25, and silicon carbide of H = 7.02. Note that since both materials are solid at room temperature, connectivity is no longer a concern in our formulations. The periodic cladding in the rectangular lattice has cylinders of material background of material holes of material material H. L H; L of radius 0.485a L in the while the periodic cladding in the rhombic lattice has elliptical of major axis length 0.485a and minor axis 0.194a in the background of The initial configurations are chosen simply as a periodic photonic crystal with one defect, i.e., an H −filled circular hole in the rectangular lattice , and a H− filled elliptical hole in the rhombic lattice. The trust region, if chosen appropriately, provides us with the confidence of the approx- 123 imations made in each run, and the effectiveness of the optimal solution. If we go back and examine, for example, the third inequality constraint in (5.1.17), we really wish it is never active. In other words, the progress in each run should not be too radical to violate this inequality, and to avoid the second mode being also guided. As a rule of thumb, the size of the trust region is chosen to be ∆ = 1%, which is shown to be sufficient in our implementation. 5.2.2 Optimal structures Formulation I Despite it being unsuccessful, we would like to briefly report the results obtained with formulation I for completeness, and attempt an intuitive explanation for its misfortune. In Figure 5.3 we can examine the evolution of the optimization process of a sample problem, constructed with epoxy cylinders of radius 0.3a (in amber color) in silicon carbide background (in black color) on a rectangular lattice (Λx a = 1.5a, Λy a = a). The starting configuration is shown at the top left corner. It has been shown ([38]) that in this case, the first guided mode is cut-off free, and is asymptotically close to the light line at the long wavelength limit, or, the left intersection βL = 0. The second guided mode has a cut-off with the light line at βR > 0. To achieve wider band width, intuitively, βR should be pushed further to the right thanks to the monotonicity of the dispersion relations. Shown in Figure 5.3, as the optimization progresses, the optimal structure approaches a homogeneous configuration of the low dielectric material, which leads to the degeneracy of the light lines. At the same time, the second guided mode is approaching the (degenerated) light line closer from above, and becoming a cut-off free mode (βR → 0), while the fundamental guided mode is approaching from below. Numerically however, the first guided mode remains cut-off free: βL = while it is asymptotically approaching the light line at βL ; the second mode remains a false cut-off βR , and it is asymptotically approaching the light line at an increasing βR . Hence it is the increase of βR that is pushing the band width wider. It is also the failure of computing a physical βR that caused the overshoot of the objective value in the last panel of Figure 5.3. Recall that in the root finding procedure, an artificial maximum limit β max is imposed on the propagation constant. β max is set to 6π in our implementation for practical reasons. The last panel of Figure 5.3 corresponds to the situation when βR = β max , i.e., the bisection search procedure failed to compute a root within the given search range [β , β max ] = [0, 6π]. Formulation II In this section, we will demonstrate some optimal structures obtained via formulation II, and validate the solutions by examining the confinement of the field variables. Recall that the design objective is to find an optimal frequency range, or band width (in terms of a dimensionless ratio), such that the fundamental mode of the waveguide is guided, or it lies below the light line, while the second order mode is un-guided, or, above the light line. In other words, we 124 0.8 Width = U−L 0.7 0.6 0.5 0.4 0.3 0.2 0.1 1.4 Width = U−L Width = U−L 0.8 0.6 0.4 run 10 30 35 40 1.5 0.5 0.2 14 run 10 15 20 10 15 20 run 400 25 350 12 300 10 Width = U−L Width = U−L 250 200 150 100 2.5 1.2 0 50 10 20 run 30 40 50 10 20 30 run 40 50 60 Figure 5.3: Evolution of the optimization process based on PIc (5.1.15) with the initial configuration shown at the top left corner. The objective value: band width U − L (vertical axis) v.s. the number of runs (horizontal axis) are plotted. The insets show the current optimal structure at each run: run = 10, 20, 40, 50, 52. 125 would like to achieve a situation where the field variable (closely related to an eigenfunction) corresponding to the fundamental eigenvalue is highly confined within the core, and decays exponentially away into the cladding, while the second eigenfunction is attenuated inside Ωs , at least within the propagation constant range [βˆ1 , β ]. Keep in mind that in this formulation, β is prescribed and βˆ1 is approximated at each linearization. For the purpose of illustration, the intensity of the field variable in the following plots is calculated as u(x, y) = ux (x, y)∗ ux (x, y) + uy (x, y)∗ uy (x, y) + uz (x, y)∗ uz (x, y). In most of the simulations we performed, it seems that the optimization of the core is the least interesting case, especially when the core consists of only one primitive cell Ω. It is almost certain that the optimal scenario is when the core is filled up with the high dielectric material H. That is if one were to start with an initial configuration of a random distribution of the dielectric materials H and L, it would eventually converge to a uniform H core at optimum. Nevertheless, the situation gets interesting when we allow a bigger design region by enlarging the core to include × primitive cells Ω. Shown in the top left corner of Figure 5.4, the initial configuration is one with a uniform H core of × Ω, surrounded by rings of Ω as the cladding. The final optimal core (top right corner) has an X−shaped H region. The initial (left) and optimal (right) dispersion relations in the second row of Figure 5.4 clearly illustrate the movement of the bands as a result of the optimization. In particular, G,2 before optimization, the second waveguide mode λW was a guided mode below the light h line λCL,1 between [βˆ1 , β ], which can be clearly observed from the last two intensity plots in h the left column. After optimization, the second waveguide mode λh W G, is now above the light line λCL,1 between [βˆ1 , β ] with a clear intersection at β (which indicates that the forth h constraint in (5.1.18) is active). The last two intensity plots in the right column of Figure 5.4 also demonstrate the attenuation of the corresponding field variables. At the same time, the confinement of the first guided mode has been enhanced thanks to the optimization, as shown in the intensity plots in the third and forth rows of Figure 5.4. The situation in the rectangular lattice however, is not quite as positive. The observation of the core optimization in a rectangular lattice when the core consists of one Ω, is the same as that of the rhombic lattice: the filling with H is the optimized structure. However, if the core region is enlarged to include 3×3 Ω, no positive band width is obtained. The results of cladding and cladding+core optimizations are shown in Figure 5.5 in which β = 1, and in Figure 5.6 in which β = 2. In each case, the initial configuration is shown in the left column, including the initial crystal structure (top), the dispersion relation (second row), the first eigenfunction intensity at βˆ1 and β respectively (third and forth rows), as well as the second eigenfunction intensity at βˆ1 and β respectively (fifth and last rows). The middle column illustrates the optimal configuration when only the cladding is the design region. Note that at the optimum, the value of βˆ1 is different and often smaller than the initial value. The last column on the 126 λCL,1 h λWG,1 h λWG,2 h 0.125 0.125 0.115 0.115 0.105 0.105 0.095 0.085 J h = 0.0063 ^ β1=0.74 _ β2=0.8 0.095 0.085 J h = 0.060 ^ β =0.75 _ β2=0.8 ^ ||H1(β1)||2 − ||H2(β1)||2 ^ ||H2(β1)||2 − ||H2(β2)||2 Figure 5.4: β = 0.8. The figures in the columns from left to right each represent the case: (left)initial configuration, with band width Jh = 0.0063 (βˆ1 = 0.74); (right) optimal configuration when only core is optimized, with band width Jh = 0.060 (βˆ1 = 0.75). The first row illustrates the PCF cross-sections. The second row shows the corresponding dispersion relations between [βˆ1 , β ]. From the third row to the last, the magnetic field intensities at both βˆ1 and β are shown in 3D surf plots. 127 right hand side contains the optimal configuration if both the cladding and the core, i.e., the whole cross-section of the waveguide is optimized. Again, the final value of βˆ1 is even smaller than the middle column, which allows one a wider range of operation. In both figures, we can observe a clear improvement of the band width, and the desired movement of the waveguide G,1 G,2 modes: first mode (λW ) being below the light line, while the second mode (λW ) being h h shifted above the light line. However, it is not hard to notice the contradictions in the intensity plots below. In Figure 5.5, besides exhibiting weak confinement strength of the fundamental guided mode after optimization, the second waveguide mode failed to get attenuated. At a larger propagation constant β = in Figure 5.6 we can observe a slight improvement. In the last column, the optimized structure provided better confinement for the fundamental mode, attenuated the second mode at βˆ1 = 1.7807, and weakened its localization at β = 2. Finally we return to the rhombic lattice and the optimization of the cladding, and the cladding+core. Shown in Figures 5.7 and 5.8 are the cases of β = 0.8, and β = 1.5 in a rhombic lattice respectively, with the same arrangement of the subplots as in the previous figures. In both Figures 5.7 and 5.8, the band widths have clearly been improved after the cladding optimization, and drastically improved after the cladding+core optimization. The confinement of the first mode and the attenuation of the second mode shown in the intensity subplots can further validate the desired results. Based on all these results, several crucial observations can be made: • Formulation I can generate infeasible crystal structures due to the numerical difficulty of computing accurate intersections βL and βR . This has been predicted theoretically and also shown experimentally. Formulation II is proposed as a remedy to address the flaws of formulation I, and it is shown to be a reasonable model for the purpose of optimal design of the single-mode single-polarization photonic crystal fiber. • Rectangular lattices setup as in our simulations did not provide satisfactory physical confinement and attenuation of the field variables, despite the numerically optimized dispersion relations. Photonic crystal fibers set on the rhombic lattice on the other hand, have demonstrated both optimal dispersion relations, as well as desirable fields confinement and attenuation. They are shown to be favorable candidates for the SPSM photonic crystal fibers. Interestingly enough, this observation is consistent with the band gap optimization problems in chapter 4, in which the crystal structures with the largest band gaps are also obtained in the hexagonal lattice (a special case of the rhombic lattice). 5.3 Conclusions In this chapter, we studied the band width optimization problem arising in the photonic crystal fibers due to the index guiding mechanism. Applying the eigenvalue design concept developed previously for the band gap optimization problem, we proposed several formal convex 128 λCL,1 h λWG,1 h λWG,2 h 0.3 0.3 0.3 0.25 0.25 0.25 0.2 0.2 0.2 ^ ||H1(β1)||2 − ||H2(β1)||2 ^ ||H2(β1)||2 − ||H2(β2)||2 Figure 5.5: β = 1. The figures in the columns from left to right each represent the case: (left)initial configuration, with band width Jh = 0.0111 (βˆ1 = 0.988); (middle) optimal configuration when only cladding is optimized, with band width Jh = 0.0387 (βˆ1 = 0.962); (right) optimal configuration when both cladding and core are optimized, with band width Jh = 0.0478 (βˆ1 = 0.953). The first row illustrates the PCF cross-sections. The second row shows the corresponding dispersion relations between [βˆ1 , β ]. From the third row to the last, the magnetic field intensities at both βˆ1 and β are shown in 3D surf plots. 129 [...]... remedy to address the flaws of formulation I, and it is shown to be a reasonable model for the purpose of optimal design of the single-mode single-polarization photonic crystal fiber • Rectangular lattices setup as in our simulations did not provide satisfactory physical confinement and attenuation of the field variables, despite the numerically optimized dispersion relations Photonic crystal fibers set... epoxy of L = 2.25, and silicon carbide of H = 7.02 Note that since both materials are solid at room temperature, connectivity is no longer a concern in our formulations The periodic cladding in the rectangular lattice has cylinders of material background of material holes of material material H L H; L of radius 0.485a L in the while the periodic cladding in the rhombic lattice has elliptical of major... the purpose of illustration, the intensity of the field variable in the following plots is calculated as u(x, y) = ux (x, y)∗ ux (x, y) + uy (x, y)∗ uy (x, y) + uz (x, y)∗ uz (x, y) In most of the simulations we performed, it seems that the optimization of the core is the least interesting case, especially when the core consists of only one primitive cell Ω It is almost certain that the optimal scenario... configuration of a random distribution of the dielectric materials H and L, it would eventually converge to a uniform H core at optimum Nevertheless, the situation gets interesting when we allow a bigger design region by enlarging the core to include 3 × 3 primitive cells Ω Shown in the top left corner of Figure 5.4, the initial configuration is one with a uniform H core of 3 × 3 Ω, surrounded by 2 rings of Ω... 6π] Formulation II In this section, we will demonstrate some optimal structures obtained via formulation II, and validate the solutions by examining the confinement of the field variables Recall that the design objective is to find an optimal frequency range, or band width (in terms of a dimensionless ratio), such that the fundamental mode of the waveguide is guided, or it lies below the light line, while... a/40, and the super cell domain Ωs is decomposed into a uniform grid of the same mesh size, but of more elements The cladding consists of two rings of Ω surrounding the core, hence, the total number of elements in Ω is 40 × 60 = 2, 400, and in Ωs is 2, 400 × 25 = 60, 000 in a rectangular lattice; the total number of elements in Ω is 40 × 40 = 1, 600, and in Ωs is 1, 600 × 25 = 40, 000 in a rhombic lattice... guided mode λh ,2 ( β ) lin fundamental WG guided mode λh ,1 ( β ) ht L lig eigenvalue λ U2 ( or U ) high order β1 β2 propagation constant β Figure 5.2: Illustration of formulation II for the optimal design of single-mode single-polarization photonic crystal fiber The propagation constant β 2 is prescribed, while β1 is a computed quantity max γ CL ,γ CO β Jh 2 (γ CL , γ CO ) 115 This can be described as... 0.194a in the background of The initial configurations are chosen simply as a periodic photonic crystal with one defect, i.e., an H −filled circular hole in the rectangular lattice , and a H− filled elliptical hole in the rhombic lattice The trust region, if chosen appropriately, provides us with the confidence of the approx- 123 imations made in each run, and the effectiveness of the optimal solution If we... configuration is one with a uniform H core of 3 × 3 Ω, surrounded by 2 rings of Ω as the cladding The final optimal core (top right corner) has an X−shaped H region The initial (left) and optimal (right) dispersion relations in the second row of Figure 5.4 clearly illustrate the movement of the bands as a result of the optimization In particular, before optimization, the second waveguide mode λW G,2 was a guided... plots in the right column of Figure 5.4 also demonstrate the attenuation of the corresponding field variables At the same time, the confinement of the first guided mode has been enhanced thanks to the optimization, as shown in the intensity plots in the third and forth rows of Figure 5.4 The situation in the rectangular lattice however, is not quite as positive The observation of the core optimization . problem of the two-dimensional photonic crystals, but instead of maximizing the difference between the maximum and minimum frequencies over the entire irreducible Brillouin zone (a set of k ⊥ ),. certain range of propagation constants (a set of k z ). Hence, it is natural to extend the algorithm developed for the band gap optimization problem, to the design of the SPSM photonic crystal. ) λ β ,2CL h Figure 5.1: Example of dispersion relation of a three-dimensional photonic crys- tal fiber. The inset illustrates an example of the cross section of the waveguide. β L and β R are