optimization of scattering resonances
TRANSCRIPT
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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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446 P. Heider et al.
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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Optimization of scattering resonances 449
(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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Optimization of scattering resonances 451
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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452 P. Heider et al.
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)
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Optimization of scattering resonances 453
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
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454 P. Heider et al.
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
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Optimization of scattering resonances 455
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 =
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456 P. Heider et al.
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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