optimization of scattering resonances

8/4/2019 Optimization of Scattering Resonances

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Struct Multidisc Optim (2008) 36:443456

DOI 10.1007/s00158-007-0201-8

RESEARCH PAPER

Optimization of scattering resonances

P. Heider D. Berebichez R. V. Kohn M. I. Weinstein

Received: 15 August 2007 / Accepted: 9 October 2007 / Published online: 20 February 2008 Springer-Verlag 2008

Abstract The increasing use of micro- and nano-scale

components in optical, electrical, and mechanical sys-tems makes the understanding of loss mechanisms andtheir quantification issues of fundamental importance.

In many situations, performance-limiting loss is due to

scattering and radiation of waves into the surroundingstructure. In this paper, we study the problem of sys-

tematically improving a structure by altering its design

so as to decrease the loss. We use sensitivity analysis

and local gradient optimization, applied to the scat-tering resonance problem, to reduce the loss within

the class of piecewise constant structures. For a class

of optimization problems where the material parame-

ters are constrained by upper and lower bounds, it is

This work was supported in part by the Humboldt Stiftung(PH), NSF-GOALI grant DMS-0313890 (DB, RVK, MIW),NSF grant DMS-0313744 (RVK), and NSF grantsDMS-0412305 and DMS-0707850 (MIW).

P. HeiderMathematisches Institut der Universitt zu Kln,50931 Kln, Germanye-mail: [email protected]

P. Heider D. Berebichez M. I. Weinstein

Department of Applied Physics and Applied Mathematics,Columbia University, New York, NY, USA

D. Berebicheze-mail: [email protected]

M. I. Weinsteine-mail: [email protected]

D. Berebichez R. V. Kohn (B)Courant Institute of Mathematical Sciences,New York University, New York, NY, USAe-mail: [email protected]

observed that an optimal structure is piecewise constant

with values achieving the bounds.

Keywords Scattering resonance Quality factor

Sensitivity analysis

1 Introduction and outline

There is great current interest in the design of micro-

and nano-structures in dielectric materials for storage,

channeling, amplification, compression, filtering, or, ingeneral, manipulation of light pulses. Such structures

have a broad range of applications from optical com-

munication technologies to quantum information sci-

ence. Energy loss is a performance-limiting concernin the design of micro- and nano-scale components.

Thus, the question of how to design such components

with very low radiative loss is a fundamental question.Periodic structures are important classes of structures

(Joannopoulos et al. 1995). In practice, these photoniccrystals (PCs) are structures with piecewise constantmaterial properties. The ability of these structures to

influence light propagation is achieved through varia-

tion of the period, the choice of material contrasts, and

through the introduction of defects.We study the problem of scattering loss from a

PC with defects, for a class of one-dimensional wave

equations:

n2(x) 2t (x, t) = x(x)x (x, t) (1)

The functions n(x) and (x) are strictly positive, as-

sumed to be variable within some compact set, con-tained in a bounded open interval, a < x < b , and

constant outside of it. Without loss of generality, we


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444 P. Heider et al.

assume (x) = n(x) = 1 for x < a and x > b . This cor-

responds to a normalization of the wave speed c = 1 in

the uniform medium. The region where these functions

vary with x is also referred to as the cavity. As the waveequation (1) has real-valued and time-independent co-

efficients, it models a system which conserves energy.

Therefore, by cavity loss, we mean scattering loss; that

is, loss due to leakage of energy from the cavity. Thisis in contrast to loss due to processes such as material

absorption.

Energy leakage or scattering loss from the cavity isgoverned by the scattering resonances associated with

the cavity. Scattering resonances are solutions to the

eigenvalue equation satisfied by time-harmonic solu-tions of (1) subject to outgoing radiation conditions,

imposed outside the cavity:

The scattering resonance problem (SRP) Seek non-trivial u(x; k), such that

x(x)xu(x) + k2 n2(x)u(x) = 0, (2)

(x + ik) u = 0, x = a

(x ik) u = 0, x = b (3)At a jump discontinuity, , of (x) or n(x), (2) isinterpreted via the flux continuity relation, obtained by

integration across the discontinuity:

( +)xu(+) = ()xu(

), (4)

where F( ) = lim0 F( ). Corresponding to a so-lution, u(x, k) of SRP is an outgoing time-dependent

solution (x, t) = eitu(x; k), = ck = k.

Remark 1 The above one-dimensional SRP governsscattering resonances ofslab type structures. It is a con-

sequence of Maxwells equations, under the assumption

of time-harmonic solutions. Variation in n(x) with

constant corresponds to the case of TM polarization;variation in (x) with n constant corresponds to TEpolarization.

SRP is a non-self-adjoint boundary value problem

having a sequence of complex eigenvalues {kj} satis-

fying Im kj < 0 and corresponding resonance modesu(x; kj). The modes u(x; kj) are locally square inte-

grable but not square integrable over all space. Assum-

ing there are no bound states (non-decaying in time, L2

states), the evolution of an arbitrary initial conditionfor (1) admits a resonance expansion in terms of time-

exponentially decaying states of the form eickjtu(x; kj),where u(; kj) L2loc. In particular, for any A > 0 andcompact set K, there exist (A, K) >0 and (A, K) >0,

such that for any compactly supported smooth ini-

tial conditions u(x, 0) and tu(x, 0), the solution u(x, t)

satisfies:

u(x, t)

{km :Im kmA}cm e

ickmt u(x; km)

L2(K)

= O(eA(1+)t), t ; (5)

see, for example, Tang and Zworski (2000). Therefore,

the rate at which energy escapes from the cavity, mea-sured for example by the rate of decay of field energy

within the cavity

cavity |u(x, t)|2dx

, is controlled by

the resonance, k, with largest imaginary part. Thetime it takes for the energy, associated with a gen-eral initial condition, localized in the defect, to radiate

away is

= (c|Im k

|)1. In practice, for example, inexperiments, initial conditions can be quite spectrallyconcentrated, and therefore, the observed time-decayrate is determined by the imaginary parts of resonances

whose real parts lie near the spectral support of the

initial condition.Results such as the resonance expansion (5) imply

that, to understand the dynamics of scattering loss,

it suffices to consider the time-independent spectralproblem SRP.

Our goal is to apply sensitivity analysis and local gra-

dient optimization methods to the scattering resonance

problem in the class of piecewise constant structures,to systematically decrease a cavitys loss in a particular

frequency range. Our measure of cavity loss is the mag-

nitude of the imaginary part of a scattering resonance.More specifically, we proceed as follows. Starting

with a particular scattering resonance, k(0, n0), of a

piecewise constant structure 0(x), n0(x), we deform

the cavity structure, (0, n0) (1 = 0 + , n1 =n0 + n), within the class of piecewise constant struc-

tures, so as to increase the imaginary part (decrease| Im k |), i.e., Im k(1, n1) > Im k(0, n0). That is, weincrease the lifetime of the mode. In this study,

and n are chosen in the direction of the gradient

of Im k(0, n0), with respect to the design parameters,which is computed using sensitivity analysis. For each

structure, along the constructed sequence of improv-

ing structures, the associated scattering resonance is

computed via Newton iteration. In some eigenvalue op-timization problems, complications arise when eigen-

values coalesce or have multi-dimensional eigenspaces.

Such complications cannot arise in our setting because,in one space dimension, our eigenvalue problem (SRP)

is easily seen to have simple eigenvalues.


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frequencies, associated with defect eigenstates of the

infinite structure, approach eigenvalues on the real axis

and become infinitely long-lived (non-decaying) states.

1.2 Outline

The article is structured as follows. In Section 2, we

define N, the class of admissible piecewise constantstructures, with at most N jump discontinuities in (x)

and n(x). Section 3 concerns sensitivity analysis, the

computation of variations in a scattering resonance

with respect to changes in the design parameters.Section 4 describes two approaches to the computation

of scattering resonances: one based on use of the exact

solution in each interval where the material proper-ties are constant, the other based on finite differences.

Section 5 contains a discussion of numerical optimiza-

tion results for the three classes of problems discussedabove.

2 Admissible piecewise constant structures:

the class N

We shall investigate the scattering resonance problemfor a class of piecewise constant structures, which we

denote by N. A schematic diagram of a typical struc-

ture is presented in Fig. 1. The set N consists ofpiecewise constant structures, ((x), n(x)), which for

j= 2, . . . , N, take on the values (j, nj) on the intervals(xj

1,xj). The endpoints x1 = a and xN = b are fixed

as are the values (x) = 1 = N+1 = 1 and n(x) = n1 =nN+1 = 1, for x < a and x > b .

n(x) =

n1 = 1 : x < x1 = a

nj : xj1


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Optimization of scattering resonances 447

in the setting of general structures and, then, specialize

to the discrete setting associated with the class, N, of

piecewise constant structures.

3.1 Sensitivity analysisgeneral setting

To avoid cumbersome notation, where it causes no con-fusion, we shall denote a resonance, kres, simply by k.Furthermore, when studying kres/, we suppress the

dependence ofkres on n(x), and similarly, we suppress

the dependence on (x), when we study kres/n.

Computation ofkres/ Denote by 0 an initial struc-ture and k(0), one of its scattering resonance frequen-

cies. Suppose we make a small perturbation in , withn(x) = n0(x) fixed:

0(x)

0(x) + (x). (9)

To compute kres/, we expand the scattering reso-

nance for the structure 0 + about the scatteringresonance for the structure, 0:

= 0 +

u(x; ) = u0(x) + u

k() = k(0) + k (10)

Substituting (10) into the Helmholtz equation (2) and

the outgoing boundary conditions (3), and keeping only

linear terms in the increments , u, and k, we obtainthe system:

x0xu+n2

0k2(0) u =x x u02k(0) k n20 u0,

(11)

(x + ik(0)) u = i k u0, x = a (12)(x ik(0)) u = +i k u0, x = b . (13)

An expression for k can be found by deriving the

compatibility equation for the system (1113). To de-

rive the compatibility condition, multiply (11) by u0,integrate by parts and use (a) the equation for u0,(2), (b) the boundary conditions for u0 and u, (c)

that (a) = (b ) = 1 and (d) that (a) = (b ) = 0 .

This yields

k =

ba

(xu0(x))

2

2k(0)b

an20

u20

+ i

u20

(b ) + u20

(a)

(x) dx,

(14)

whence

k

(0) =

(xu0(x))2

2k(0)b

an20

u20

+ i

u20

(b ) + u20

(a) . (15)

We are interested in increasing the value of Im k()

(decreasing |Im k()|). Note that, to first order in ,

k() = Im k(0) + k

(0)

,

. (16)

Thus a small perturbation, of 0, will increase the

value of Im k most rapidly, provided we choose tobe in the direction ofsteepest ascent:

=

k

(0)

= (xu0(x))22k(0)

ba

n20

u20

+ i

u20

(b ) + u20

(a) . (17)

Computation of k/n We now consider variations

in k due to changes in n(x), with = 0(x) fixed. To

compute k/n, we expand

n(x) = n0(x) + n(x)

u(x; n) = u0(x) + u(x)

k(n) = k(n0) + k. (18)

Proceeding in a manner analogous to the above compu-

tation, we obtain

k =

ba

2k2(n0) n0(x) u20(x)

2k(n0)b

an20

u20

+ i

u20

(b ) + u20

(a)

n(x) dx.

(19)

Therefore

kn

(n0) = 2

k2

(n0) n0(x) u2

0(x)2k(n0)

ba

n20

u20

+ i

u20

(b ) + u20

(a) ,

and the direction of steepest ascent for n is given by

n =

k

n(n0)

=

2k2(n0) n0(x) u20(x)2k(n0)

ba

n20

u20

+ i

u20

(b ) + u20

(a)

.


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3.2 Sensitivity analysis and gradient ascent in N

In this subsection, we restrict the results of the previoussection on k/ and k/n to structures (, n) in the

class N.

As introduced in Section 2, the structure of agiven PC structure is specified by 3N

4 design

parameters:

x = (x2, . . . ,xN1) possible points of discontinuity of

(x), n(x)

s = (2, . . . , N) values of(x) on interval (xj1,xj),

n = (n2, . . . , nN) values ofn(x) on interval (xj1,xj).

Given a resonance, kold = k(xold,sold, nold), associated

with a structure defined by (x,s, n)old = (xold,sold, nold),we obtain an improved structure by deforming the

initial structure in the direction of the gradient of the

objective functional evaluated at (x,s, n)old. Thus, weobtain a new or updated structure as follows:

(x,s, n)update = (x,s, n)old + ( Im k)old= (x,s, n)old

+ x Im k,s Im k,n Im kold .

In this paper, we use the notation k to denotethe gradient with respect to all parameters: x,s, n, andx k, s k, n k to denote gradients with respect to thespecified parameters. Equivalently and explicitly,

xnew = xold +

x2, . . . ,

xN1

Im k

(xold,sold,nold)

snew = sold +

2, . . . ,

N

Im k

(xold,sold,nold)

nnew = nold +

n2, . . . ,

nN

Im k

(xold,sold,nold)

(20)

We discuss the choice of later in this section.

The following proposition gives expressions for the

partial derivatives of a scattering resonance k(x,s, n),with respect to the design parameters:

Proposition 1 In the above notation, we have the follow-

ing equations for the partial derivatives of k(x,s, n) with

respect to the design parameters:

(1) Variations in k with respect to the jump locations,

x= (x2, . . . ,xN1), for j=2, . . . , N1, are given by

k(x,s, n)

xj=

(j j+1) xu(xj )xu(x+j ) + (n2j+1 n2j) k2(x,s, n) u2(xj, x,s, n)

i u2(a, x,s, n) + u2(b , x,s, n) + 2k(x,s, n)b

a

n2

(x, x,s, n)u2

(x, x,s, n)dx

, (21)

(2) Variations in k with respect to s = (2, . . . , N)

satisfy, for j= 2, . . . N,

k(x,s, n)

j=

xjxj1

(xu(x; x,s, n))2 dx

i

u2(a, x,s, n) + u2(b , x,s, n)

+ 2k(x,s, n)

ba

n2(x, x,s, n)u2(x, x,s, n)dx

. (22)


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(3) Variations in k with respect to n = (n2, . . . , nN), for

j= 2, . . . , N are given by:

k(x,s, n)

nj

=

2k2(x,s, n)xj

xj1

nju2(x, x,s, n)dx

i

u2(a, x,s, n) + u2(b , x,s, n)

+ 2k(x,s, n)

ba

n2(x, x,s, n)u2(x, x,s, n)dx

. (23)

Proof To prove (22) and (23), we can apply the formu-las from Section 3.1 with specially chosen perturbations

of(0, n0). The proof of (21) requires some more care

because (x) and xu(x) are possibly discontinuous atx =xj.

Choose a < a and b > b . Multiply (2) by u, inte-

grate over [a, b ], and then integrate by parts. Thisgives

uxu|b a

b a

(xu)2 +

b a

k2n2u2 = 0. (24)

The dependence of the terms in this equation on the

design parameter xj has been suppressed. Denote byz the derivative of a function z with respect to xj. We

write the integrals in (24) as the sum of integrals over

[a,xj) and (xj, b ]. Note that = 0, n = 0 for x = xj.Differentiation of (24) with respect to xj yields:

xj

uxu|

b a

2

xja

xuxu + 2

b xj

xuxu

+ j(xu(xj ))

2 j+1(xu(x+j ))2

+ 2

b a

(kkn2u2 + k2n2uu)

+ (k2n2ju(xj)2 k2n2j+1u(xj)2) = 0.

After a second integration by parts and using (2), we

find

xj

uxu|

b a

2u xu|b

a + 2

b a

kkn2u2

= j+1(xu(x+j ))

2 j(xu(xj ))2 + k2n2j+1u(xj)2

k2

n2

ju(xj)2

,

where we used the flux condition (4) at xj. Remem-bering (a) = (b ) = 1, the outgoing boundary condi-tions (3), and the derivative of the boundary conditions,

the above equation simplifies to

ik(u2(a) + u2(b )) + 2

b a

kkn2u2

= (j j+1) xu(xj )xu(x+j ) + k2n2j+1u(xj)2

k2n2ju(xj)

2,

where we use the identity

j+1(xu(x+j ))

2 j(xu(xj ))2

=

j+1xu(x

+j ) jxu(xj )

= 0by (4)

xu(x

+j ) + xu(x

j )

+ (j j+1) xu(xj )xu(x+j ). (25)Finally, we note using the explicit exponential form ofu on [a, a] and [b , b ], that the expression in (25) isindependent of a < a and b > b . Letting a

a and

b b , we get the expression in (21).For expressions (22) and (23), we begin by comput-

ing, for structures in N, variations of a resonance with

respect to the design parameters x,s, n.


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Let 0(x), n0(x) denote a structure in N. Admissible

perturbations of (0, n0) are achieved through varia-

tions of the values of, s, and n, n, on each subinterval.(, n), the total variations of the coefficient functions, and n, due to these two kinds of variations, can

therefore be expressed as:

=

Nj=2

j

j (26)

n =

Nk=j

n

njnj (27)

The terms on the right hand side of (26), (27) are

computed as follows. Note that general variation of(, n), within N, is of the form:

, = +

N

j=2(j + j) [xj1,xj],

n, = n +

Nj=2

(nj + nj) [xj1,xj],

where [a,b ](x) denotes the characteristic function of

the interval [a, b ]. Thus,

j j =

d

d

=0

(j + j) [xj1,xj] = [xj1,xj] j

(28)

n

nj nj =

d

d =0(nj + nj) [xj1,xj] = [xj1,xj] nj.

(29)

To prove (22) a n d (23), for each fixed j {2,. . . , N 1}, consider the perturbation j j + , re-spectively, nj nj + nj, with other parameters heldunchanged. Then, by (2628) and (29), we have

= [xj1,xj], n = [xj1,xj]

Substitution into (14) and (19) yields the expression(21) for k/j and k/nj.

3.3 Gradient ascent algorithm in N

Using Proposition 1, we can now compute an updated

structure and approximate corresponding resonance

from the old structure:

(x,s, n)update = (x,s, n)old + Imkold (30)kapprox,update = kold + (kold, Imkold) ; (31)see (20).

The parameter, , is carefully chosen so to ensure

convergence of the Newton method resonance finder

with kapprox,update as initial guess, see Section 4.2. To

achieve this, we control the relative distance:

p =

kapprox,update kk .

If p is fixed as control parameter for the step length,then one finds

= p

k(k, Imk) .

Algorithm 1 Steepest Ascent

Input: k0 start guess, (x,s, n)0 start structure, p control

parameter, eps accuracy for Newtons method, Mnum-

ber of steepest descent steps.Output: (x,s, n)opt optimized structure, kopt optimized

resonance.

1 kopt result of Newtons method with accuracyeps and initial guess k = k0 for the initial structure(x,s, n)0.

2 For count = 1, . . . , M

3 Let p kopt

(kopt ),Imkopt)

, where kopt is com-puted by Proposition (1) with kopt

4 kapprox,update kopt + kopt, Imkopt

5 (x,s, n)opt (x,s, n)opt + Imkopt6 kopt result of Newtons method with accuracy

eps and initial guess kapprox,update for the structure(x,s, n)opt.

7 end for

4 Numerical computation of scattering resonances

In this section, we focus on the numerical determina-tion of scattering resonances of structures in the classN. Then, we outline a gradient ascent method, which

finds locally optimal structures of class N. For ournumerical ascent scheme, we need three different ingre-

dients. First, we need a discretization of the scattering

resonance problem (SRP) (2, 3). Second, we need a

numerical routine to solve the discretized problem.The starting point for this latter step is typically an

approximate solution of the discretized problem and

a routine that corrects this guess. Finally, we performa gradient ascent step, which provides an approximate

improved structure.


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We will present two different ways for the dis-

cretization of (2). The first approach (see Section

4.1) exploits the property that, for piecewise con-

stant one-dimensional structures, the equation is ex-actly solvable on each subinterval. The solution is

determined by matching conditions for the functions

and fluxes at the subinterval endpoints. The second

approach (see Section 4.3) uses a finite differencediscretization for (2). The corrector routine for both

methods is a Newton root finder (Section 4.2),

which is adapted for the particular discretization. The

gradient ascent step is the same for both methods and

is based on Proposition 1.

4.1 Matching method

We introduce the following simplifying notation:

rj :=nj

j, j= 1, . . . , N+ 1. (32)

The solution of (2) is given by

u(x) =

c1eikx : x < a

c2jeirj+1kx + c2j+1e

irj+1kx : xj < x < xj+1, j= 1, . . . , N 1c2Ne

ikx : b < x

, (33)

where the complex coefficients c1, . . . , c2N

are deter-mined from the conditions that u(x) be continuous on

R and such that the flux continuity relations (4) are

satisfied at the jump locations, xj.

Writing down these matching conditions gives thefollowing set of equations at x1:

c1eikx1 = c2e

ir2kx1 + c3eir2kx1

c1eikx1 = r2

c2eir2kx1 c3eir2kx1

,

at xj for j= 2, . . . , N 1:

c2j2eirjkxj + c2j1eirjkxj = c2jeirj+1kxj + c2j+1eirj+1kxj

rj(c2j2eirjkxj c2j1eirjkxj) = rj+1

c2je

irj+1kxj

c2j+1eirj+1kxj

,

at xN:

c2N2eirNkxN + c2N1e

irNkxN = c2NeikxN

rN(c2N2eirNkxN c2N1eirNkxN) = c2NeikxN.

This is a linear system of2N equations of the form

A(k)c = 0.

The matrix A(k) C2N2N depends nonlinearly on thewavenumber k and has a narrow band structure, whichwill be exploited in the numerical scheme. The modeu(x) is recovered from k and the amplitude vector c.

To impose that the solution we find is nontrivial,we impose a normalization condition on the vector

c; for example, one can set the amplitude c1 of the

scattered part at the left end to 1. Thus, one arrives at

the following nonlinear discretized problem:

F(k, c) :=

A(k) c

eT1

c 1

= 0. (34)

This is a system of 2N+ 1 nonlinear equations in the2N+ 1 unknowns k, c1, . . . , c2N. Of course, one could

reduce this system by one dimension by eliminating the

fixed c1, but for notational ease, (34) will be used in thefollowing.

Equation (34) can be solved by any kind of nonlinear

solver. We use Newtons method, which is outlined in

the next subsection.

4.2 Newtons method

For a given structure in N, with resonance energy

k and mode amplitude vector c, we typically have an

approximate or guessed resonance k0 and amplitudevector c0 . If (k0, c0) (k, c) is sufficiently small,then Newtons method applied to (34) guarantees con-

vergence. With a good initial guess, Newtons method

converges quadratically, and few iteration steps arenecessary. Moreover, the error in the computation ofk

and c (and thus ofu(x)) can be estimated by Newtons

method.Newton iteration, applied to (34), reads

DF(kj, cj)

cj+1 cjkj+1 kj

= F(kj, cj),

where

DF(k, c) =

A(k) A(k)c

eT1

0


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is the Jacobian of F(k, c). Written out, this equation

yields

A(kj)cj+1 + (kj+1 kj) A(kj)cj = 0eT1

cj+1 = 1,

which is formulated in the Newton step

A(kj)yj+1 := A(kj)cj,

(NA) cj+1 :=yj+1

eT1

yj+1

kj+1 := kj 1eT1

yj+1.

Remarks:

The matrices A(k) and A(k) are explicitly known,and their computation can easily be implemented.

As A(k) is sparse, one can find yj+1

efficiently. The matrices A(k) and A(k) can be stored withO(8N) memory.

4.3 Finite difference discretization of resonance

problem

In this subsection, we present a second approach tocomputation of scattering resonances, based on a finite

difference solution of the scattering resonance prob-

lem. For simplicity, we take n(x) 1; the requiredmodifications for general n(x) are straightforward.

Let U(x; k) and V(x; k) denote solutions of (2) whichare outgoing at x = a, respectively, x = b :

(x + ik) U(x; k) |x=a = 0, U(a; k) = 1 (35)

(x ik) V(x; k) |x=b = 0, V(b ; k) = 1. (36)

For typical values ofk, U(x; k) and V(x; k) are linearly

independent. Let W(k) denote the Wronskian ofU andV:

W(k) = V(x; k)(x)xU(x; k) U(x; k)(x)x V(x; k).

(37)

W(k) is independent of x. U and V are linearly de-

pendent if and only if W(k) = 0. In this case, U andV are proportional. Thus, ifW(k) = 0, then U(x; k) isoutgoing at x = a and x = b , and the pair (U(x; k), k)is a scattering resonance. As remarked in Section 1,

scattering resonance energies (zeros ofW(k)) are in the

lower half plane, Im k < 0.We now turn to a finite difference implementa-

tion. Partition [a, b ] into N intervals of length x =

(b a)/N. We introduce a scheme for approximatingu(x) at discrete grid-points

uj = u(yj), yj = (j 1)x, j= 1, . . . , N. (38)

We use the symmetric discretization of (2) is

j+ 12 (uj+1 uj)(x)2

j 12 (uj uj1)(x)2

+ k2uj = 0 (39)

where yj = a + (j 1)x. This discretization respectsthe continuity of flux condition (4). The points of dis-

continuity of (x) are chosen to be grid points. The

scheme uses values of at staggered points yj+ 12

, where

yj +

1

2

= j+ 1

2

.

We shall use the scheme (39) to construct out fun-damental set {U(x; k), V(x; k)}. To do so, we require

appropriate discretizations of the outgoing conditions

(35) and (36). These discrete outgoing conditions atx = a and x = b are derived as follows.

At the ends of the domain, the equation becomes

uj+1 2uj + uj1(x)2

+ k2uj = 0. (40)

This constant coefficient difference equation has, forkx sufficiently small, oscillatory exponential solutions

uj = A j + B j, (41)

where is the root of the quadratic equation z2

2 (kx)2z + 1 = 0 corresponding to the choice ofpositive square root:

=

2 (kx)2

+ i

4 2 (kx)222

. (42)

The discrete solution uj is outgoing to the right pro-

vided B = 0 . Therefore, the discrete boundary condi-tion for a solution which is outgoing to the right x =

xN = b is

Vj+

1 = Vj

, j= N

1. (43)

We also have from (36) the normalization VN = 1,

which together with (43) gives

VN1 = 1. (44)

Analogously, the discrete outgoing to the left conditionat x = a is

Uj = 1 Uj1, j= 2, (45)


11/14


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

3

x

|u|2

(x)

Fig. 2 The initial structure for all three scenarios. It consists of22 barriers and a missing barrier in the middle. The outgoingwave conditions are imposed on the left resp. right end of thestructure. Below is the resonance mode (modulus squared) of thechosen initial resonance

which together with the normalization from (35), U1 =

1, gives

U2 = 1. (46)

To construct the approximation to U(x; k), we ob-tain U2 from (46) and use the expression for Uj+1obtained from (39) to propagate this value forward

to obtain Uj, j= 3, . . . , N. Similarly, to construct theapproximation to V(x; k) we obtain VN1 from (44)and then use the expression for Vj

1, obtained from

(39) to back-propagate the solution obtaining Vj forj= N 2, . . . , 1.

Finally, we require a discretized version of theWronskian, W(k). The discrete Wronskian is given by:

W(k) = j+ 12

Uj(k)(Vj+1(k) Vj(k)) Vj(k)(Uj+1(k) Uj(k))x

,

(47)

which is independent of j.Scattering resonance frequencies of our discrete ap-

proximation (39, 43, 45) are values of k for which the

discrete Wronskian is zero,

W(k) = 0. (48)

In this implementation, the numerical scheme for

finding a scattering resonance of a fixed structure,((x), n(x)), starts with an approximate scattering reso-

nance kres,0 (an approximate root of (48)) and improves

it by Newton iteration:

kres,r+1 = kres,r W(kres,r)

kW(kres,r), r = 0, 1, . . . (49)

Remark 2 There are serious issues of instability in

searching for approximate zeros of W(k) by seeking

zeros of the discrete Wronskian, W(k), constructedfrom the numerical approximations Uj and Vj. Though

in theory a constant in j, the numerically computed

Wronskian, Wcomputed (k; yj), oscillates with j due to

roundoff error. This can be overcome by applying

Newtons method to an averaged Wronskian: W(k) =N1

Nj=1 W

computed (k; yj).

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6

7

8

9

x

(x)

58 60 62 64 66 68 70-16

-14

-12

-10

-8

-6

-4

-2

0

x 10-3

Re k

Im

k

Fig. 3 Left optimal structure for Problem Opt together with optimized resonance mode. The optimized mode is more concentratedwithin the defect than the initial mode. Rightthe path of the resonance during optimization


12/14


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

x

|u|2

(x)

57 57.5 58 58.5 59 59.5 60 60.5 61-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

Re k

Imk

Fig. 4 Left optimal structure for Problem Optarea together with optimized resonance mode. Right the path of the resonance duringoptimization for scenario Optarea

45 50 55 60 65 700

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

k

|T|

60.5 60.6 60.7 60.8 60.9 61 61.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

k

|T|

50 55 60 65 70 75 80 850

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

k

|T|

66.55 66.5505 66.551 66.5515 66.552 66.5525 66.553 66.5535 66.554 66.5545

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

k

|T|

Fig. 5 Transmission diagram for the initial structure (top) andthe optimized structure (bottom) of problem Opt

max. The band-

gap shifted during the optimization, and the resonance is sitting

nearly in the middle of the band-gap of the optimized structure.To the right zoom of the resonance peak in each case. Aftersuccessful optimization, the resonance peak gets very sharp


13/14


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

3

x

|u|2

(x)

58 59 60 61 62 63 64 65 66 67

-0.015

-0.01

-0.005

0

Re k

Imk

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.5

1

1.5

2

2.5

3

x

|u|2

(x)

58 59 60 61 62 63 64 65 66 67-0.018

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

Re k

Imk

Fig. 6 Top left optimal structure for Problem Optmax

togetherwith optimized resonance mode. Top right the path of the res-onance during optimization for problem Opt

max. Bottom left

optimized structure with fixed jump positions. Bottom right pathof the resonances during optimization, solid line with varying

jumps, scattered line with fixed jumps

5 Numerical results

We performed numerical gradient ascent for the three

optimization problems Opt, Optarea, and Optmax. For

all simulations, we use the same initial truncated pe-

riodic structure with a defect. Both methods, match-ing and finite difference, were used to solve for and

optimize scattering resonances and gave consistentresults. We performed our most extensive investiga-tions using the matching approach, due to its greater

efficiency.

The initial structure we used is shown in Fig. 2: (x)

is piecewise constant. There is a central defect region

of width 3 on which (x) = 1. Moving outward from

the defect, takes on the values = 2 (barriers) and = 1 on alternating intervals of width = 0.0324. Thisparticular example has 22 barriers.

We have chosen k0 = 60.81836306650.0163109133ias the initial resonance for all three optimization prob-

lems. The associated resonance mode is also plottedin Fig. 2. To find this k0, an initial approximation is

guessed from the value of k at which the transmission

diagram has a peak, see Fig. 5top. Starting with this

real value ofk as an initial guess, Newton iteration was

then used to find an accurate k0.For the problems Opt and Optarea, we kept the jump

positions fixed during the optimization, whereas forproblem Opt

max, we also allowed the jump positions to

vary.

5.1 Problem Opt

Gradient ascent was applied to the above struc-

ture and yielded a (local) optimal resonance kopt =


14/14


69.2633131254 0.0000004471i. The Q-factor is fourorders of magnitude larger than that of the initial

resonance. We stopped the ascent iteration withIm kopt = 4.17053 107.

In Fig. 3, the optimal structure and optimal reso-

nance mode are plotted. The path of the resonance dur-

ing the optimization in the complex plane is also plotted

in that figure. Observe the smooth turning point, indi-cating Re k(tur) = 0 for a particular structure turvisited during optimization.

5.2 Problem Optarea

Fixing the initial area of imposes an additional con-

straint. The optimal resonance was found to be karea =57.1639554364 0.0045894230i, whose Q-factor is onlya single order of magnitude larger than that of the

initial structure. The structure and resonance mode are

plotted in Fig. 4. Comparing the path of the resonancein Fig. 4 with the path in Problem Opt, one sees that

the additional constraint strongly influences the search

directions of the ascent method.

5.3 Problem Optmax

In this study, we constrain the values of (x) to lie

between upper and lower bounds:

1 = min

max = 3.

In contrast to the previous simulations, the jump po-

sitions can now also vary. The optimal resonance was

found to be kopt = 66.55233131 0.000071246i, whoseQ-factor is three orders of magnitude larger than that

of the initial structure. We stopped the ascent iteration

after the bounds for were reached and xIm kopt =1.69 107.

We also ran the simulation with fixed jumps to com-

pare the optimal structures and resonances. For this

case, the optimal resonance was found to be kopt =

62.0211038345

0.0002390987i.

In Fig. 5, the transmission diagram of the initial

structure (top panel) together with the optimized struc-

ture of Problem Optmax

(bottom panel) is shown. Asexpected, the peak corresponding to the optimized res-

onance kopt is much sharper than the peak belonging tok0. The band-gap shifted during the optimization, and

the resonance peak of the optimized structure is nearly

in the middle of the band-gap.

We note that, when allowing, in addition to the

values of , the locations of possible jumps, {xj}N1j=2 ,

to vary, that the optimal structure is found to be one

whose parameter values achieve the bounds min andmax, see Fig. 6. We have also observed, in our opti-

mization, the emergence of structures, similar to Braggresonators, whose intervals of constant are multiples

of one quarter of the wavelength in the local material.

These properties are currently under investigation.

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