Optimal design of photonic crystals 51

λW G1 βL when the fundamental mode falls below the light line and becomes effectively guided , while the second mode still lies inside the light cone, until the frequency λW G2 βR at whi

λW G1 (βL) when the fundamental mode falls below the light line and becomes effectively guided , while the second mode still lies inside the light cone, until the frequency λW G2 (βR) at which the second mode eventually becomes guided 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 optimiza- tion 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

is also incorporated into the optimization formulation to increase the validity of our model.

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:

ACLh (εCL, β)uCL,jh = λCL,jh MhCLuCL,jh , j = 1, , N , β ∈ Ph, in Ω.

(5.1.1) The operator ACLh can be assembled according to (3.2.19) over the computation domain of only one primitive cell Ω with periodic boundary condition, and the

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:

AW Gh ([εCL; εCO], β)uW G,jh = λW G,jh MhW GuW G,jh , j = 1, , Ns, β ∈ Ph, in Ωs.

(5.1.2) Similarly, the operator AW Gh can be assembled according to (3.2.19) over the

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

3.17 of subsection 3.2.2) Recall that both operators AW Gh and ACLh are affine

with respect to the reciprocal of the dielectric variables: −1i , i = 1, , nε,(see

(3.2.19)) We introduce a change of variables and define γ as the design variables

of the optimization problem:

γi= −1i , i = 1, , nγ = nε, γL≤ γi ≤ γH, γL= 1/H, γH = 1/L (5.1.3)

We shall only work with γ for the rest of this chapter The operators ACLh (γCL, β)

and AW Gh ([γCL; γCO], β) are now linear in their respective design variables.

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:

We use arg(f (x) = g(x)) to mean “the argument x∗” such that f (x∗) = g(x∗).

The superscriptW Gon the admissible range Qh indicates that the decision

vari-ables being considered are those defining the cross-section of the waveguide

(su-per cell Ωs), i.e., γW G = [γCL; γCO] ∈ QW Gh We will also encounter another admissible range denoted by QCLh , 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., γCL ∈ QCL

h := {γ : γ ∈ [γL, γH]nγCL}; gously, γCO ∈ QCO

The original problem (5.1.4) can be rewritten as

γ CL ,γ CO ,u,`

U − L , s.t λW G,1h [γCL; γCO], β
≥ λCL,1h (γCL, β), ∀β ∈ [βmin, βL],

at all times For succinctness, these eigenvalue-related equality constraints are not explicitly included in P0 and all the subsequent formulations The fifth and sixth constraints are derived from the monotonicity of λW G,jh , j = 1, 2.

Depending on various design requirements, we have formulated three different scenarios: (a) cladding γCLare known, and core γCOare variables; (b) core γCOare known, and cladding γCL are variables; (c) both cladding γCL and core

γCO 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: [γCL; γCO] At the beginning of each linearization, a given b γCO

is assumed Together with the known cladding γCL, the intersections with the fundamental space-filling mode of the waveguide cladding λCL,1h can be calculated

using various root-finding algorithms (e.g., bisection method1):

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

or βR: λW G,1h and λW G,2h are well separated but are both below the light line

λCL,1h for β ∈ [β1, βn β] This is an undesirable physical situation, as the first two

modes of the waveguide become both guided, rather than a single-guided mode Next, we introduce the approximate and reduced matrices that eventually

1

The bisection method requires two initial points at which the function evaluations have

guaranteed to be of opposite signs, one of the two limits will be taken as a reasonable imate” root in such case

“approx-span the subspaces upon which our convex optimization models are built:

ΘW Gj(γCL,γbCO, βj) := [ΦW G1 (γCL,γbCO, βj) | ΨW Gaj (γCL,bγCO, βj)]

:= [uW G,1h (γCL,γbCO, βj) | uW G,2h (γCL,bγCO, βj), , uW G,2+aj

h (γCL,bγCO, 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 precisely only one eigenvector uW G,1h (γCL, γ bCO, βj); the matrix that spans the upper subspace at propagation constant βj is of dimension aj We obtain the following equivalent convex formulation with semi-definite inclusions:

is independent of both decision variable and propagation constant Therefore

Pb γ CO

Ia is a tractable convex program containing 2nβ+ 2 semi-definite inclusions, and 2nγCO+ 2 linear constraints.

Scenario b: [γCL; γCO] When γCL 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

independent matrix ACLh,0(β), plus a summation of the γCL independent

matri-ces ACLh,i(β) multiplying each decision variable γiCL The mass matrix MhCL is

independent of both decision variable and propagation constant too as MhW G.

Compared to Pb γ CO

Ia , program Pγ b CL

Ib contains 2nβ more semi-definite inclusions

to account for the light line mode.

Scenario c: [γCL; γCO] The extension from scenario (b) to (c), where both

γCLand γCO are the design variables, is minor with the inclusion of γCObeing

part of the decision variables:

i=1 γ CO

i A W G h,n CO

γ +i (β j ) +A W G

i=1 γ CO

i A W G h,n CO

γ +i (β j ) +AW Gh,0 (β j ) − U j MhW G

i A CL h,i (β j ) + A CL

h,0 (β j )

−LjM CL h

i A CL h,i (β j ) + A CL

h,0 (β j )

−UjM CL h

Despite being the most direct formulation of the optimization problem,

for-mulation 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 L1in

scenarios (b) and (c)) being zero, in which case the objective value would

have turned out to be unity whatever value U or (Un β) takes Thus it

defeats the purpose of band width optimization.

• Probably the most serious caveat with this formulation lies in the culty of accurate computation of the intersections βLand βR(or the corre- sponding eigenfrequencies) As explained before, when a waveguide mode

diffi-is guided, it decays exponentially away from the core into the cladding This modal diameter increases rapidly with wavelength, i.e., when the fre- quency 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 experimen- tally, to study the long-wavelength regime, especially the behavior of the fundamental guided mode (λW G,1h ) as it approaches the less than rigor- ously defined intersection βL with light line (λCL,1h ) The situation for the second guided mode (λW G,2h ) becomes more subtle if it has a cut-off with the light line away from long-frequency One can loosely view λW G,2h as

a perturbed mode to the second light line, λCL,2h (a degenerate light line

if the cladding symmetry had not been broken) By the same token, it is hard to capture the regime when λW G,2h is approaching λCL,2h with finite 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 β2.

In formulation II, we start with a prescribed propagation constant β2, 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

dimensionless ratio for the band gap optimization problem:

( )

fundamental guided mode WG

( )

λ

β,2

max

γ CL ,γ CO Jβ2

h (γCL, γCO).

This can be described as

γ CL ,γ CO ,u,`

U −L

U +L, s.t λW G,1h [γCL; γCO], β2
≤ L,

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: [γCL; γCO] Assuming a prescribed cladding γCLthroughout, and

a given core γ bCO at the beginning of each linearization, b β1 defined as

b

β1 = arg  λCL,1h (γCL, β) = λW G,1h (γCL, b γCO, β2)  ,

can be computed using root-finding algorithms (e.g., the bisection method is chosen in our implementation) P1 is relaxed to

The equivalent convex formulation with semi-definite inclusions can be written as

the number of constraints as compared to those in formulation I

Scenario b: [γCL; γCO] When core γCO is fixed, and cladding γCL takes on a given

b

β1= argλCL,1h (γbCL, β) = λW G,1h (γbCL, γCO, β2)

The light line is now variable dependent, and we need to introduce two auxiliary variables as

line In addition to the matrices defined in II(a), more approximate and reduced matrices arenecessary for the cladding modes:

be designed and optimized.

Scenario c: [γCL; γCO] Designing both the core and the cladding does not require

computed almost the same way as before:

b

β1= argλCL,1h (γbCL, β) = λW G,1h (γbCL,γbCO, β2)

i=1 γCO

i AW G h,n CO

ε +i(β1)+AW Gh,0 (β1) − LMhW GiΦW G1 (εbCL,εbCO, β1)  0,

1 (εbCL,εbCO, β2)hPnεCL

i=1 γCL

i AW G h,i (β2) +PnεCO

i=1 γCO

i AW G h,n CO

ε +i(β2)+AW Gh,0 (β2) − LMhW GiΦW G1 (εbCL,εbCO, β2)  0,

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-

straints, their extension from the simple scenarios (a) and (b) are natural and simply minimal;moreover, they provide more flexible design needs.

Computing a solution to a nonlinear and non-convex problem is nontrivial Although the SDPand linear relaxation introduced via our algorithm have proven to be efficient for the band gapoptimization of the two-dimensional photonic crystal, the linearization and approximation didnot seem to be sufficiently reliable in this band width optimization problem with an underlyingquasi-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

step size can be represented by the difference between the optimal solution and the linearizer,

variables, we propose the following two trust region methods

+

1

We rewrite it in matrix-vector multiplication form

and simplify the notation

These new decision variables as well as the new constraints are to be incorporated to the

We consider a three-dimensional photonic crystal with z-invariant cross-section set up

de-composed into a uniform grid of the same mesh size, but of more elements The claddingconsists of two rings of Ω surrounding the core, hence, the total number of elements in Ω is

We use the same dielectric materials as in section 3.2.3 to construct the waveguide, i.e.,

at room temperature, connectivity is no longer a concern in our formulations The periodic

hole in the rhombic lattice

The trust region, if chosen appropriately, provides us with the confidence of the

approx-imations made in each run, and the effectiveness of the optimal solution If we go back andexamine, for example, the third inequality constraint in (5.1.17), we really wish it is neveractive In other words, the progress in each run should not be too radical to violate this in-equality, and to avoid the second mode being also guided As a rule of thumb, the size of thetrust region is chosen to be ∆ = 1%, which is shown to be sufficient in our implementation.

is shown at the top left corner It has been shown ([38]) that in this case, the first guidedmode is cut-off free, and is asymptotically close to the light line at the long wavelength limit,

optimization progresses, the optimal structure approaches a homogeneous configuration of thelow 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

that caused the overshoot of the objective value in the last panel of Figure 5.3 Recall that in

Formulation II

In this section, we will demonstrate some optimal structures obtained via formulation II, andvalidate the solutions by examining the confinement of the field variables Recall that the designobjective is to find an optimal frequency range, or band width (in terms of a dimensionlessratio), such that the fundamental mode of the waveguide is guided, or it lies below the lightline, while the second order mode is un-guided, or, above the light line In other words, we