Downloaded 06/07/17 to 131.193.178.201. Redistribution...

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

SIAM J. NUMER. ANAL. c© 2017 Society for Industrial and Applied MathematicsVol. 55, No. 1, pp. 144–167

NUMERICAL SOLUTION OF DIFFRACTION PROBLEMS: AHIGH-ORDER PERTURBATION OF SURFACES ANDASYMPTOTIC WAVEFORM EVALUATION METHOD∗

DAVID P. NICHOLLS†

Abstract. The rapid and robust simulation of linear waves interacting with layered periodicmedia is a crucial capability in many areas of scientific and engineering interest. High-order pertur-bation of surfaces (HOPS) algorithms are interfacial methods which recursively estimate scatteringquantities via perturbation in the interface shape heights/slopes. For a single incidence wavelengthsuch methods are the most efficient available in the parameterized setting we consider here. In thecurrent contribution we generalize one of these HOPS schemes by incorporating a further expansionin the wavelength about a base configuration which constitutes an “asymptotic waveform evaluation”(AWE). We not only provide a detailed specification of the algorithm, but also verify the schemeand point out its benefits and shortcomings. With numerical experiments we show the remarkableefficiency, fidelity, and high-order accuracy one can achieve with an implementation of this algorithm.

Key words. high-order perturbation of surfaces methods, asymptotic waveform evaluation,high-order spectral methods, Helmholtz equation, diffraction gratings, layered media

AMS subject classifications. 65N35, 78A45, 78B22

DOI. 10.1137/16M1059679

1. Introduction. The rapid and robust simulation of linear waves interactingwith layered periodic media (a diffraction or scattering problem) is a crucial capabilityin many areas of scientific and engineering interest. Examples abound in areas suchas geophysics [VO09, BR09], oceanography [BL82], materials science [God92], imag-ing [NW01], and nanoplasmonics [Rae88, Mai07, EB12]. For this latter topic, onecan investigate topics as diverse as extraordinary optical transmission [ELG98], sur-face enhanced spectroscopy [Mos85], and surface plasmon resonance (SPR) biosensing[Hom08, ILW11] and [LJJ12, JJJ13, RJOM13, NRJO14]. Regardless of the physicalproblem, in each it is necessary to approximate the scattering returns of such modelsin a fast, highly accurate, and reliable fashion.

While all of the classical numerical algorithms have been utilized for simulationof this problem, we have recently argued [AN14, Nic16, Nic15, NOJR16] that suchvolumetric approaches (such as finite differences and finite/spectral element methods)are greatly disadvantaged with an unnecessarily large number of unknowns for thelayered media problems we consider here. Interfacial methods based upon integralequations (IEs) [CK98] are a natural candidate but, as we have pointed out [AN14,Nic16, Nic15, NOJR16], these also face difficulties. Most of these have been addressedin recent years through (i) the use of sophisticated quadrature rules to deliver high-order spectral accuracy, (ii) the design of preconditioned iterative solvers with suitableacceleration [GR87], and (iii) new strategies to avoid periodizing the Green function[BG11, CB15, LKB15]. Consequently, they are a compelling alternative (see, e.g.,the survey article of [RT04] for more details); however, two properties render them

∗Received by the editors February 2, 2016; accepted for publication (in revised form) September9, 2016; published electronically January 24, 2017.

http://www.siam.org/journals/sinum/55-1/M105967.htmlFunding: This work was supported by the National Science Foundation through grant DMS–

1522548.†Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago,

Chicago, IL 60607 ([email protected]).

144

Dow

nloa

ded

06/0

7/17

to 1

31.1

93.1

78.2

01. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p

http://www.siam.org/journals/sinum/55-1/M105967.html

mailto:[email protected]


DIFFRACTION PROBLEMS: A HOPS/AWE METHOD 145

noncompetitive for the parameterized problems we consider as compared with themethods we advocate here:

1. For configurations parameterized by the real value ε (for us the height/slopeof the irregular interface), an IE solver will return the scattering returns onlyfor a particular value of ε. If this value is changed then the solver must berun again.

2. The dense, nonsymmetric positive definite systems of linear equations whichmust be inverted with each simulation.

As we advocated in [Nic16, Nic15, NOJR16] a “high-order perturbation of sur-faces” (HOPS) approach can effectively address these concerns. More specifically, in[Nic15, NOJR16] we argued for the method of field expansions (FE) which trace theirroots to the low-order calculations of Rayleigh [Ray07] and Rice [Ric51]. Their high-order incarnation was first introduced by Bruno and Reitich [BR93a, BR93b, BR93c]and later enhanced and stabilized by the author and Reitich [NR04a, NR04b, NR08],and the author and Malcolm [MN11]. These formulations are particularly compellingas they maintain the advantageous properties of classical IE formulations (e.g., sur-face formulation and exact enforcement of far-field and quasi-periodicity conditions)while avoiding the shortcomings listed above:

1. Since the methods are built upon expansions in the boundary parameter,ε, once the Taylor coefficients are known for the scattering quantities, it issimply a matter of summing these (rather than beginning a new simulation)for any given choice of ε to recover the returns.

2. Due to the perturbative nature of the scheme, at every perturbation order oneneed only invert a single, sparse operator corresponding to the flat-interface,order-zero approximation of the problem.

For a single incidence wavelength such methods are the most efficient availablein the parameterized setting we consider here. We now discuss a generalization ofthe HOPS approach of Bruno and Reitich which incorporates a further expansion inthe wavelength about a base configuration that constitutes an “asymptotic waveformevaluation” (AWE) [PR90, KSN96, RDCB98, SLL01]. We will not only provide adetailed specification of the algorithm, but also verify the scheme and point out itsbenefits and shortcomings. With numerical experiments we shall show the remarkableefficiency, fidelity, and high-order accuracy one can achieve with an implementation ofthis algorithm. In this initial contribution we will defer several natural generalizationsto focus upon the idea of the algorithm rather than solving all possible problems. Insection 6 we discuss several of these future directions including three-dimensional ge-ometries and vectorial scattering (section 6.1), frequency-dependent materials (section6.2), and the occurrence of Rayleigh singularities [Ray07] (section 6.3). We note thatthese singularities are commonly referred to as “Wood’s anomalies,” but we encouragethe interested reader to find the fascinating article of Maystre in Chapter 1 of [EB12]which argues that this term is, in fact, a misnomer. Beyond this we note that nothingabout our algorithm limits the configuration to two layers and, up to notational com-plications, the developments of [MN11, Nic12, NRJO14, AN14, Nic15, Nic16] couldbe extended, which we intend to complete in a forthcoming publication.

The paper is organized as follows: In section 2 we briefly recall the equations whichgovern the propagation of linear waves in a two-dimensional periodic structure, andin section 2.1 we recall the Rayleigh expansions. In section 3 we recall the FE methodfor numerically approximating solutions to these governing equations, together withthe Taylor expansions necessary to specify an implementation (in section 3.1). Insection 4 we describe in some detail our new HOPS/AWE algorithm, including forms

Dow

nloa

ded

06/0

7/17

to 1

31.1

93.1

78.2

01. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p


146 DAVID P. NICHOLLS

x

z

z = g(x)

vinc = exp(−iγuz)

Su

Sw

Fig. 1. Plot of two–layer structure with periodic interface.

for the Taylor terms required of the algorithm in sections 4.1 and 4.2. In section 5 wepresent detailed numerical results (see sections 5.1–5.5) to validate our implementationversus exact solutions, the previously tested FE recursions, and a boundary IE (BIE)simulation [CB15]. These illustrate the accuracy and computational efficiency of ournew method, the latter of which we make precise in section 5.6. In section 6 we giveconcluding remarks and discuss future directions.

2. The governing equations. The geometry we consider is displayed inFigure 1: a y-invariant, doubly layered structure. Dielectrics occupy both domains,one (with refractive index nu) fills the region above the graph z = g(x),

Su := {z > g(x)} ,

while the other (with index of refraction nw) fills

Sw := {z < g(x)} .

The superscripts are chosen to conform to the notation of previous work by the author[NOJR16, NT16, Nic12]. The grating is d-periodic so that g(x + d) = g(x). Thestructure is illuminated from above by monochromatic plane-wave incident radiationof frequency ω and wavenumber ku = nuω/c0 = ω/cu (c0 is the speed of light), alignedwith the grooves. We consider the reduced incident fields

Einc(x, z) = Aeiαx−iγuz, Hinc(x, z) = Beiαx−iγ

uz,

α = ku sin(θ), γu = ku cos(θ),

where time dependence of the form exp(−iωt) has been factored out. The reducedelectric and magnetic fields {E,H}, like the reduced scattered fields, are α-quasi-periodic due to the incident radiation [Pet80]. To close the problem, we specifythat the scattered radiation is “outgoing” (upward propagating in Su and downwardpropagating in Sw).

It is well-known (see, e.g., Petit [Pet80]) that in this two-dimensional setting, thetime-harmonic Maxwell equations decouple into two scalar Helmholtz problems whichgovern the transverse electric (TE) and transverse magnetic (TM) polarizations. Wedefine the invariant (y) directions of the scattered (electric or magnetic) fields by{u(x, z), w(x, z)} in Su and Sw, respectively, and the incident radiation in the upper

Dow

nloa

ded

06/0

7/17

to 1

31.1

93.1

78.2

01. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p



layer by uinc(x, z). For all three we factor out the phase factor exp(iαx) leavingfunctions d-periodic in the x direction.

In light of all of this, we are led to seek outgoing, d-periodic solutions of

∆u+ (2iα)∂xu+ (γu)2u = 0, z > g(x),(1a)

∆w + (2iα)∂xw + (γw)2w = 0, z < g(x),(1b)

u− w = ζ, z = g(x),(1c)

∂Nu− (iα)(∂xg)u− τ2 {∂Nw − (iα)(∂xg)w} = ψ, z = g(x),(1d)

where the Dirichlet and Neumann data are

ζ(x) := −e−iγug(x), ψ(x) := (iγu + iα(∂xg)) e−iγ

ug(x).

In these N = (−∂xg, 1)T and

τ2 =

{1, TE,

(ku/kw)2 = (nu/nw)2, TM,

where γw = kw cos(θ). For various reasons the case of TM polarization is of extraor-dinary importance (e.g., the classical study of SPRs [Rae88]) and thus we concentrateour attention on the TM case from here.

2.1. The Rayleigh expansions. The Rayleigh expansions, which can be de-rived from separation of variables [Pet80], are the periodic, outgoing solutions of (1a)and (1b). More specifically, they express the fields as

(2) u(x, z) =

∞∑p=−∞

apeipxeiγ

up z, w(x, z) =

∞∑p=−∞

dpeipxe−iγ

wp z,

where, for p ∈ Z and q ∈ {u,w},

p :=

(2π

d

)p, αp := α+ p, γqp :=

√

(kq)2 − α2

p, p ∈ Uq,

i√α2p − (kq)

2, p 6∈ Uq,

and

Uq ={p ∈ Z | α2

p < (kq)2},

which are the “propagating modes” in the upper and lower layers. Notice that apand dp are the upward and downward propagating Rayleigh amplitudes. Quantitiesof great interest are the efficiencies

eup = (γup /γu) |ap|2 , ewp = (γwp /γ

u)∣∣∣dp∣∣∣2 ,

which give the “reflectivity map”

(3) R :=∑p∈Uu

eup .

Dow

nloa

ded

06/0

7/17

to 1

31.1

93.1

78.2

01. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p



3. Field expansions. Before we discuss our new algorithm, we review the classi-cal HOPS methodology due to Bruno and Reitich, the “FE method” [BR93a, BR93b,BR93c]. Our viewpoint is that the FE algorithm is a perturbative approach to en-

forcing the boundary conditions (1c) and (1d) with the {ap, dp} from the Rayleighexpansions (2) as unknowns.

For this we define

(4) a(x) := u(x, 0) =

∞∑p=−∞

apeipx, d(x) := w(x, 0) =

∞∑p=−∞

dpeipx,

which are the “flat interface” field traces. We recall the definition of a Fourier multi-plier, m(D),

m(D)ξ(x) :=

∞∑p=−∞

m(p)ξpeipx,

where ξp is the pth Fourier coefficient of ξ(x). With this we can define the Fouriermultipliers

(5) U := iγuD, W := iγwD, A := iαD,

where the final operator is simply classical differentiation of a α-quasi-periodic func-tion. Now, follow [Nic15, NOJR16] and define the Dirichlet trace operators

Du : a→ u(x, g(x)), Dw : d→ w(x, g(x)),

and their Neumann counterparts

N u : a→ (∂zu− (∂xg)∂xu) (x, g(x)), Nw : d→ (∂zw − (∂xg)∂xw) (x, g(x)).

These operators map, respectively, the function pair (a, d) to the upper and lowerDirichlet and Neumann traces. It can be shown [Nic15, NOJR16] that these operatorshave the form

(6a) Du = exp(gU), Dw = exp(−gW ),

and

N u = exp(gU)U − (∂xg) exp(gU)A,

Nw = − exp(−gW )W − (∂xg) exp(−gW )A.(6b)

To clarify, we note that the meaning of Du is given by

Du[ξ] =

∞∑p=−∞

exp(g(x)iγup )ξpeipx.

In terms of these, the Dirichlet boundary condition, (1c), becomes

(7) Du [a]−Dw [d] = ζ,

while the Neumann condition, (1d), becomes

(8) N u [a]− (iα)(∂xg)Du [a]− τ2 {Nw [d]− (iα)(∂xg)Dw [d]} = ψ.

Dow

nloa

ded

06/0

7/17

to 1

31.1

93.1

78.2

01. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p



We state our governing equations, the boundary conditions (7) and (8), abstractly as

(9) Mv = b,

where

M =

(Du −Dw

N u − (iα)(∂xg)Du −τ2 {Nw − (iα)(∂xg)Dw}

), v =

(ad

), b =

(ζψ

).

3.1. Taylor expansions. The FE methodology considers interface deformationsof the form g(x) = εf(x) (f = O(1)) and notes that, for f sufficiently smooth (Lips-chitz) and ε sufficiently small, the linear operator M and inhomogeneity b are bothanalytic in ε [NR01, NR03]. Furthermore, an analytic solution v can be shown toexist so that the following Taylor expansions are convergent:

{M,v,b}(εf) =

∞∑n=0

{Mn(f),vn(f),bn(f)}εn.

The FE approach recovers vn using regular perturbation theory. To see this we write(9) as ( ∞∑

n=0

Mnεn

)( ∞∑`=0

v`ε`

)=

∞∑n=0

bnεn,

and, equating at each perturbation order, we find

(10) M0vn = bn −n−1∑`=0

Mn−`v`.

At order zero we recover the flat-interface solution, giving the Fresnel coefficients,while higher-order corrections, vn, can be computed by appealing to (10). Of greatimportance is the fact that one only need invert the same linear operator, M0 atevery perturbation order. All that remains is a specification of the terms {Mn,bn}.

Regarding the Dirichlet trace operators, upon defining

Fn(x) := f(x)n/n!,

it has been shown that [Nic15, NOJR16]

Dun = FnUn, Dwn = Fn(−W )n.

For their Neumann counterparts we have

N un = FnU

n+1 − (∂xf)Fn−1Un−1A, Nw

n = Fn(−W )n+1 − (∂xf)Fn−1(−W )n−1A.

Finally, for the surface data, bn, it is easy to show that

ζn = −Fn (−iγu)n

and

ψn = Fn (iγu) (−iγu)neiαx + (∂xf)Fn−1 (iα) (−iγu)

n−1,

where F−1(x) ≡ 0 and F0(x) ≡ 1. To clarify, we note that the meaning of Dun is givenby

Dun[ξ] =f(x)n

n!

∞∑p=−∞

(iγup )nξpeipx.

Dow

nloa

ded

06/0

7/17

to 1

31.1

93.1

78.2

01. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p



4. A HOPS/AWE approach. To describe our new HOPS/AWE method werecall the governing equations, (7)–(8),

Du [a]−Dw [d] = ζ,

N u [a]− (iα)(∂xg)Du [a]− τ2 {Nw [d]− (iα)(∂xg)Dw [d]} = ψ,

which we wrote compactly as Mv = b; c.f. (9). We now make two smallness assump-tions:

1. Boundary perturbation: g(x) = εf(x), ε ∈ R, ε� 1.2. Frequency perturbation: ω = (1 + δ)ω = ω + δω, δ ∈ R, δ � 1.

(We suspect that more careful analysis will reveal that neither ε nor δ need be in-finitesimal for our method to be applicable.) We note that the latter of these has anumber of consequences:

ku = ω/cu = (1 + δ)ω/cu =: (1 + δ)ku = ku + δku,

kw = ω/cw = (1 + δ)ω/cw =: (1 + δ)kw = kw + δkw,

α = ku sin(θ) = (1 + δ)ku sin(θ) =: α+ δα,

γu = ku cos(θ) = (1 + δ)ku cos(θ) =: γu + δγu,

γw = kw cos(θ) = (1 + δ)kw cos(θ) =: γw + δγw.

These in turn imply

αp = α+ (2π/d)p = α+ δα+ (2π/d)p =: αp + δα.

Akin to the FE method outlined above, we will assume for the moment the jointanalyticity of the operator M and inhomogeneity b with respect to both boundaryand frequency deviations. With these we postulate that the joint analyticity of v canbe established so that the following Taylor series can be shown to be convergent:

{M,v,b}(εf, ω + δω) =

∞∑n=0

∞∑m=0

{Mn,m(f, ω),vn,m(f, ω),bn,m(f, ω)}εnδm.

It is a goal of future research to make this mathematically precise. Our new HOPS/AWE algorithm finds the vn,m at each perturbation order using regular perturbationtheory. Now we write (9) as( ∞∑

n=0

∞∑m=0

Mn,mεnδm

)( ∞∑`=0

∞∑r=0

v`,rε`δr

)=

∞∑n=0

∞∑m=0

bn,mεnδm,

and, equating at each perturbation order, we find

M0,0vn,m = bn,m −n−1∑`=0

Mn−`,0v`,m −m−1∑r=0

M0,m−rvn,r(11)

−n−1∑`=0

m−1∑r=0

Mn−`,m−rv`,r.

As before, the key is to discover forms for the {Mn,m,bn,m} and we now begin thisprocess.

Dow

nloa

ded

06/0

7/17

to 1

31.1

93.1

78.2

01. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p



4.1. Expansions of γup , γw

p , U , and W in frequency. The first step in ourdevelopment is to derive the Taylor series expansion for γqp , q ∈ {u,w},

(12) γqp = γqp(δ) =

∞∑m=0

γqp,mδm.

We begin by using the fundamental relationship

α2p + (γqp)2 = (kq)2,

which implies( ∞∑m=0

γqp,mδm

)( ∞∑r=0

γqp,rδr

)= (1 + δ)

2(kq)2 −

(αp + δα

)2.

This delivers∞∑m=0

δmm∑r=0

γqp,m−rγqp,r =

{(ku)2 − (αp)

2}

+ 2δ{

(kq)2 − α αp}

+ δ2{

(kq)2 − (α)2}

= (γup)2 + 2δ

{(kq)2 − α αp

}+ δ2(γq)2,

so that at order O(δ0) we require

(13) γqp,0 = ±γqp,

while at order O(δ1) we need

(14) γqp,1 =2((kq)2 − α αp)

2γqp,0, γqp,0 6= 0.

Here we now see that it is crucial for the validity of expansion (12) that γqp6= 0 for

all p. We now make this assumption, and report upon the case γqp

= 0 for some p in

section 6.3. Continuing our development to O(δ2) we further set

(15) γqp,2 =(γq)2 − (γqp,1)2

2γqp,0, γqp,0 6= 0,

and for O(δm), m > 2, we demand

(16) γqp,m =−∑m−1r=1 γqp,m−rγ

qp,r

2γqp,0, γqp,0 6= 0.

To close, recall the definitions of U , W , and A:

U = iγuD, W = iγwD, A = iαD;

c.f. (5). In light of our expansions for γu(δ) and γw(δ), we expand

(17) {U,W,A} = {U,W,A} (δ) =

∞∑m=0

{Um,Wm, Am} δm,

and it is a simple matter to show that

Um = iγuD,m, Wm = iγwD,m, Am =

iαD, m = 0,

iα, m = 1,

0, m > 1.

Dow

nloa

ded

06/0

7/17

to 1

31.1

93.1

78.2

01. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p



4.2. Expansion of the Dirichlet and Neumann trace operators. We arenow in a position to find expressions for the Taylor series terms {Mn,m,bn,m} whichrequire the forms {Dun,m,Dwn,m,N u

n,m,Nwn,m} from

{Du,Dw,N u,Nw} = {Du,Dw,N u,Nw} (ε, δ)(18)

=

∞∑n=0

∞∑m=0

{Dun,m,Dwn,m,N u

n,m,Nwn,m

}εnδm.

It is clear from the formulas (6a)–(6b), if appropriate forms can be found for

Eu := exp(gU) = exp(εfU(ω + δω)) =

∞∑n=0

∞∑m=0

Eun,mεnδm,(19a)

Ew := exp(−gW ) = exp(−εfW (ω + δω)) =

∞∑n=0

∞∑m=0

Ewn,mεnδm(19b)

then

Dun,m = Eun,m, Dwn,m = Ewn,m,

while a little more work delivers

N un,m =

m∑r=0

Eun,m−rUr − (∂xf)Eun−1,mA0 − (∂xf)Eun−1,m−1A1,

Nwn,m = −

m∑r=0

Ewn,m−rWr − (∂xf)Ewn−1,mA0 − (∂xf)Ewn−1,m−1A1.

In order to find the Taylor series terms for Eu we insert the expansion

U = U(δ) =

∞∑m=0

Umδm

into (19a), and since

Eu(0, δ) = exp(0 U(δ)) = I,

we have

(20) Eu0,m =

{I, m = 0,

0, m > 0.

Next,

Eu(ε, 0) = exp(εfU(0)) =

∞∑n=0

FnU(0)n,

so that

Eun,0 = FnU(0)n.Dow

nloa

ded

06/0

7/17

to 1

31.1

93.1

78.2

01. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p



Finally, we follow the technique of Pourahmadi [Pou84] (see also the works of Roberts[Rob83] and Marchant and Roberts [MR87]) who uses the fact that

∂εEu = fEuU,

to equate

∞∑n=0

∞∑m=0

Eun+1,m(n+ 1)εnδm = f

( ∞∑n=0

∞∑m=0

Eun,mεnδm)( ∞∑

m=0

Umδm

).

Upon equating at like orders we have

Eun+1,m =f

n+ 1

m∑r=0

Eun,m−rUr.

So, to discover the coefficient at order (n + 1,m), one only needs (n, 0), . . . , (n,m).For instance, we have Eu0,m from (20) which gives Eu1,m and one can proceed to recoverall of the Eun,m. Clearly, the same procedure can be used to deduce that

Ew0,m =

{I, m = 0,

0, m > 0,

Ewn,0 = Fn(−W (0))n,

Ewn+1,m = − f

n+ 1

m∑r=0

Ewn,m−rWr.

5. Numerical results. We are now in a position to test a numerical implemen-tation of this algorithm and demonstrate its advantageous computational complexity.For this we compare our novel HOPS/AWE method to the carefully studied and val-idated classical FE scheme of Bruno and Reitich [BR93a, BR93b, BR93c] outlined insection 3. Both the FE and our new HOPS/AWE schemes are high–order spectralapproaches, where nonlinearities are approximated with convolutions implementedvia the fast Fourier transform (FFT) algorithm [GO77, CHQZ88].

To demonstrate the convergence of our algorithm we take several steps as we feelit helps illuminate not only the accuracy of our new scheme, but also its range ofvalidity. To begin we consider convergence of partial sums of the Taylor series for γupgiven in (12), and then move to approximation of the operator U by the partial sumsof its Taylor series (17). We then proceed to study the convergence of partial Taylorsums of the Dirichlet and Neumann trace operators from (18). To close, of course, weconsider computations of the full reflectivity map (3) from (10) or (11).

5.1. Approximation of γup . To begin our demonstrations we recall that the

frequency perturbation assumption

ω = ω(δ) = ω(1 + δ)

led to the Taylor expansion of the quantity

γup = γup (δ) =

∞∑m=0

γum,pδm;

Dow

nloa

ded

06/0

7/17

to 1

31.1

93.1

78.2

01. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p



c.f. (12). Our numerical approximation will be to truncate this Taylor series after afinite number of terms,

(21) γup (δ) ≈ γu,Mp (δ) :=

M∑m=0

γum,pδm

with forms for the γum,p given in (13)–(16).As we noted before, the expansion (12) ceases to be valid when γu

p= 0 for any

p which we term a “Rayleigh singularity” (commonly called a Wood’s anomaly), andwe now quantify what this means for our computations. Since

α2p + (γu

p)2 = (ku)2,

a singularity occurs when α2p = (ku)2 for any integer p (notice that this cannot occur

for p = 0). To simplify matters we assume that

α = 0, d = 2π,

and recall that ku = ω/cu = nuω/c0 (where c0 is the speed of light which we chooseto be unity). In this case αp = p so that the Rayleigh singularity condition becomes,if nu = 1 (vacuum),

p2 = (ku)2 = ω2/(cu)2 = (nu)2ω2 = ω2.

Thus “resonance” occurs at integer values of the frequency ω. To maximize the radiusof convergence of our algorithm in our tests we define

ωq := q +1

2, q = 0, 1, 2, 3, . . .

and choose (q +

1

2

)− σ

2< ω(δ) <

(q +

1

2

)+σ

2

to sample at a fraction 0 < σ < 1 of the “allowable” frequencies implying, after somesimplification, that

− σ

2q + 1< δ <

σ

2q + 1.

In Figure 2 we display results of the comparison between a numerical implemen-tation of the approximation γu,Mp (c.f. (21)), versus an exact computation of γup . Morespecifically, we compute

(22) Error :=max{−Nx/2≤p≤Nx/2−1}

∣∣γu,Mp − γup∣∣

max{−Nx/2≤p≤Nx/2−1}∣∣γup ∣∣

for parameter choices

q = 1, σ = 0.99, nu = 1, Nx = 32, M = 4, 8, 16, 32, 64Dow

nloa

ded

06/0

7/17

to 1

31.1

93.1

78.2

01. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p



1.2 1.4 1.6 1.8

ω

10-15

10-10

10-5

Error

Error versus ω

M = 4

M = 8

M = 16

M = 32

M = 64

Fig. 2. Relative maximum norm error, (22), versus frequency ω in approximation of γup by

γu,Mp (−Nx/2 ≤ p ≤ Nx/2 − 1) for various perturbation orders M . Parameter choices are q = 1,σ = 0.99, nu = 1, Nx = 32, and M = 4, 8, 16, 32, 64.

5.2. Approximation of U . We now repeat the computations from the previoussection for the operator U = U(δ), the Fourier multiplier which has iγup (δ) as itssymbol. As before, we will use the analyticity of U as a function of δ to approximateits action, beginning with

U = U(δ) =

∞∑m=0

Umδ

(c.f. (17)); we will simulate U by truncating this series

(23) U(δ) ≈ UM (δ) :=

M∑m=0

Umδ.

In section 4.1 we saw that the symbol of the Fourier multiplier Um is given by iγup,mso we may simply use the formulas (13)–(16) together with the FFT to approximatethe action of Um and thus UM .

Figure 3 shows the outcomes of the comparison between a numerical implemen-tation of the approximation UM (c.f. (23)), versus an exact computation of U . Moreprecisely, we compute (at Nx collocation points on [0, 2π])

(24) Error :=

∣∣UM [ξ]− U [ξ]∣∣L∞

|U [ξ]|L∞

for

ξ(x) = ecos(x) + iesin(x),

Dow

nloa

ded

06/0

7/17

to 1

31.1

93.1

78.2

01. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p



1.2 1.4 1.6 1.8

ω

10-15

10-10

10-5

Error

Error versus ω

M = 4

M = 8

M = 16

M = 32

M = 64

Fig. 3. Relative supremum norm error, (24), versus frequency ω in approximation of U by UM

for various perturbation orders M . Parameter choices are q = 1, σ = 0.99, nu = 1, Nx = 32, andM = 4, 8, 16, 32, 64.

and parameter choices

q = 1, σ = 0.99, nu = 1, Nx = 32, M = 4, 8, 16, 32, 64.

5.3. Approximation of the trace operators. Next, we consider the Dirichlettrace operator Du = Du(ε, δ) and its Neumann counterpartN u = N u(ε, δ). In section3 we saw that (c.f. (6)),

Du = exp(gU), N u = exp(gU)U − (∂xg) exp(gU)A,

and, in section 4.2, we posited the expansions

{Du,N u} = {Du,N u} (ε, δ) =

∞∑n=0

∞∑m=0

{Dun,m,N u

n,m

}εnδm

(c.f. (18)), and recovered expressions for the HOPS/AWE terms {Dun,m,N un,m}. As

we have done in the previous two sections we simulate {Du,N u} by the truncatedTaylor series

(25) {Du,N u} ≈{Du,N,M ,N u,N,M

}(ε, δ) :=

N∑n=0

M∑m=0

{Dun,m,N u

n,m

}εnδm.

Figure 4 shows the outcomes of the comparison between a numerical implemen-tation of the approximation Du,N,M (c.f. (25)), versus an exact computation of Du.More precisely we compute (at Nx collocation points on [0, 2π])

(26) Error :=

∣∣Du,N,M (εmaxf)[ξ]−Du(εmaxf)[ξ]∣∣L∞

|Du(εmaxf)[ξ]|L∞

Dow

nloa

ded

06/0

7/17

to 1

31.1

93.1

78.2

01. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p



ω

1.2 1.3 1.4 1.5 1.6 1.7 1.8

Error

10-16

10-14

10-12

10-10

10-8

10-6

10-4

Error at εmax versus ω

M = 4

M = 8

M = 16

M = 32

M = 64

Fig. 4. Relative supremum norm error, (26), versus frequency ω in approximation of Du byDu,N,M for various perturbation orders N = M . Parameter choices are q = 1, σ = 0.99, nu = 1,Nx = 32, and M = 4, 8, 16, 32, 64.

for

ξ(x) = ecos(x) + iesin(x), f(x) = cos(x), εmax = 0.2,


q = 1, σ = 0.99, nu = 1, Nx = 32, M = 4, 8, 16, 32, 64.

Figure 5 shows the outcomes of the comparison between a numerical implemen-tation of the approximation N u,N,M (c.f. (25)), versus an exact computation of N u.More precisely we compute (at Nx collocation points on [0, 2π])

(27) Error :=

∣∣N u,N,M (εmaxf)[ξ]−N u(εmaxf)[ξ]∣∣L∞

|N u(εmaxf)[ξ]|L∞

for

ξ(x) = ecos(x) + iesin(x), f(x) = cos(x), εmax = 0.2,


q = 1, σ = 0.99, nu = 1, Nx = 32, M = N = 4, 8, 16, 32, 64.

Dow

nloa

ded

06/0

7/17

to 1

31.1

93.1

78.2

01. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p



ω

1.2 1.3 1.4 1.5 1.6 1.7 1.8

Error

10-16

10-14

10-12

10-10

10-8

10-6

10-4

Error at εmax versus ω

M = 4

M = 8

M = 16

M = 32

M = 64

Fig. 5. Relative supremum norm error, (27), versus frequency ω in approximation of Nu byNu,N,M for various perturbation orders N = M . Parameter choices are q = 1, σ = 0.99, nu = 1,Nx = 32, and M = 4, 8, 16, 32, 64.

5.4. Approximation of the reflectivity map. To close, we consider our origi-nal object of study, the reflectivity mapR = R(ε, λ), (3). Using both the FE recursionsand their HOPS/AWE counterparts, we compute

RN,Nx

FE ≈ R, RN,M,Nx

HOPS/AWE ≈ R,

and display, in Figure 6, the error

(28) Error :=∣∣∣RN,Nx

FE −RN,M,Nx

HOPS/AWE

∣∣∣L∞

for

f(x) = cos(x), εmax = 0.2,


q = 1, σ = 0.75, nu = 1, nw = 1.1, Nx = 32, M = N = 4, 8, 16, 32, 64.

We point out that σ = 0.75 was chosen in order to avoid the Rayleigh singularitiescoming from both the top and bottom layers.

To close this section we show the kind of simulations which our new HOPS/AWEmethod can produce with very high fidelity and quite modest computational effort.We revisit the calculations above in the cases q = 1, 2, 3, 4, 5, 6 with the followingfrequency and wavelength ranges (σ = 0.99):

Dow

nloa

ded

06/0

7/17

to 1

31.1

93.1

78.2

01. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p



3.5 4 4.5 5 5.5

λ

10-15

10-10

10-5

Error

Error at εmax versus λ

M = 4

M = 8

M = 16

M = 32

M = 64

Fig. 6. Relative supremum norm difference, (28), between FE and HOPS/AWE algorithmsversus wavelength λ in computation of the reflectivity map, R(εmax, λ) for various perturbationorders N = M . Parameter choices are q = 1, σ = 0.75, nu = 1, nw = 1.1, Nx = 32, andM = 4, 8, 16, 32, 64.

q = 1 : ω ∈ [1.005, 1.995] =⇒ λ ∈ [3.14947, 6.25193],

q = 2 : ω ∈ [2.005, 2.995] =⇒ λ ∈ [2.09789, 3.13376],

q = 3 : ω ∈ [3.005, 3.995] =⇒ λ ∈ [1.57276, 2.09091],

q = 4 : ω ∈ [4.005, 4.995] =⇒ λ ∈ [1.25789, 1.56884],

q = 5 : ω ∈ [5.005, 5.995] =⇒ λ ∈ [1.04807, 1.25538],

q = 6 : ω ∈ [6.005, 6.995] =⇒ λ ∈ [0.89824, 1.04633].

Once again we select

f(x) = cos(x), εmax = 0.2,


nu = 1, nw = 1.1, Nx = 32, M = N = 4.

In Figure 7(a) we plot the six “subsets” of the reflectivity map, R, all together on oneset of axes. In Figure 7(b) we insert blue lines at the edges of the subsets showinghow the entire approximation was built one piece at a time.

5.5. A rough interface. Finally, we consider the reflectivity map R = R(ε, λ),(3), generated by a grating with a rough interface. In this setting we further validateour algorithm by making a comparison with the BIE solver for quasi-periodic gratingsof Cho and Barnett [CB15] (see also the related work in [BG11, LKB15]). We mentionthat we are greatly indebted to both Barnett and Cho for not only providing us with animplementation [Bar16] of the algorithm in MATLAB [MAT10], but also for extensive

Dow

nloa

ded

06/0

7/17

to 1

31.1

93.1

78.2

01. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p



R

1 2 3 4 5 6

λ

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

ε

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

R

1 2 3 4 5 6

λ

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

ε

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

Fig. 7. (a) The reflectivity map, R(ε, λ), computed with six invocations of our new HOPS/AWEalgorithm. Computed with N = M = 4 and a granularity of Nε = Nδ = 101 per invocation.Parameter choices are σ = 0.99, nu = 1, nw = 1.1, and Nx = 32. (b) Blue lines included toindicate boundaries between the six runs.

assistance with its use. Now, using both the HOPS/AWE recursions and this BIEmethodology we compute

RNx

BIE ≈ R, RN,M,Nx

HOPS/AWE ≈ R,

and display, in Figure 8, the error

(29) Error :=∣∣∣RNx

BIE −RN,M,Nx

HOPS/AWE

∣∣∣L∞

for

f(x) = fL,P (x) :=

P∑p=1

8

π2(2p− 1)2cos(2πpx), P = 10, εmax = 0.01, d = 1,


q = 1, σ = 0.75, nu = 1, nw = 1.1, Nx = 32, M = N = 4, 8, 16, 32, 64.

(Here we have changed the period from 2π to 1 and the polarization from TM to TEto facilitate the BIE simulation.) The profile fL,P consists of the first P -many termsin a Fourier expansion of the Lipschitz profile

FL(x) =

{−4x+ 1, 0 ≤ x ≤ 1/2,

−3 + 4x, 1/2 ≤ x ≤ 1.

Again, σ = 0.75 was chosen in order to avoid the Rayleigh singularities coming fromboth the top and bottom layers. In regards to Figure 8 we chose a discretization in theBIE algorithm (Nx = 32 quadrature points on the interface and along the fictitiousboundaries; see [CB15] for more details) which had accuracy of approximately 10−12.

5.6. Computational complexity. Of course the true motivation for this en-tire algorithm is the very advantageous computational complexity the HOPS/AWEapproach has for computing quantities such as the reflectivity map, R = R(ε, λ),

Dow

nloa

ded

06/0

7/17

to 1

31.1

93.1

78.2

01. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p



0.55 0.6 0.65 0.7 0.75 0.8 0.85

λ

10-10

10-8

10-6

10-4

Error

Error at εmax versus λ (HOPS/AWE versus BIE)

M = 4M = 8M = 16M = 32M = 64

Fig. 8. Relative supremum norm difference, (29), between HOPS/AWE and BIE algorithmsversus wavelength λ in computation of the reflectivity map, R(εmax, λ) for various perturbationorders N = M . Parameter choices are q = 1, σ = 0.75, nu = 1, nw = 1.1, Nx = 32, andM = 4, 8, 16, 32, 64.

versus all other methods, even the highly efficient FE method. To summarize ourconclusions on this front we begin by fixing the problem of computing R for Nε manyvalues of ε and Nδ many values of λ. Using any surface numerical method requiresthe use of a number of discretization points which we denote Nx. Finally, for theFE approach we will retain N perturbation orders in ε, while our new HOPS/AWEalgorithm mandates the additional consideration of M Taylor orders in δ.

A careful study of the FE recursions (10) reveals that, for a single value of λ,forming the right-hand side at order n has cost

O(nNx log(Nx)).

Inverting the operator M0 has complexity O(Nx log(Nx)) so that the full cost ofcomputing the {an,p, dn,p} is therefore

O(N2Nx log(Nx)).

Once these are recovered, the cost of summing the series in ε is minimal, providedthat it is done in an efficient manner (e.g., by Horner’s rule [BF97, AH01]) so thatthe full cost of computing the reflectivity map by the FE method is

O(NδN2Nx log(Nx)).

Consideration of the HOPS/AWE recursions (11) shows that the computationalcomplexity in forming the right-hand side at order (n,m) has cost

O(nmNx log(Nx)).

Dow

nloa

ded

06/0

7/17

to 1

31.1

93.1

78.2

01. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p



As before, inverting the operator M0,0 has complexity O(Nx log(Nx)) so that the fullcost of computing the {an,m,p, dn,m,p} is therefore

O(N2M2Nx log(Nx)).

Again, once these coefficients are recovered, the cost of summing the series in (ε, δ) isminimal, provided that it is done in an efficient manner (e.g., by Horner’s rule [BF97,AH01]) so that the full cost of computing the reflectivity map by the HOPS/AWEmethod is

O(N2M2Nx log(Nx)).

Thus, once M2 � Nδ our new algorithm becomes prohibitively more efficient. Un-surprisingly this was a crucial consideration in the computations of section 5.4 as weoften found M = N = 4 to be sufficient to produce the entire reflectivity map, whiledesiring a sampling of 100, 1000, or even 10,000 values in both the ε and λ variables.

6. Conclusion and future directions. In this paper we have described in somedetail a novel, high-order spectral [GO77, CHQZ88] boundary/wavenumber pertur-bation method which, for problems akin to that of computing the reflectivity map,possesses optimal computational complexity and execution time. This HOPS/AWEalgorithm has been shown to be both highly accurate and robust. However, it isclear that it can be extended and enhanced in a number of directions. In this con-cluding section we comment on some of these avenues which we intend to explore inforthcoming publications.

6.1. Three dimensions and vectorial scattering. To begin, it is trivial tosee how our scheme could be extended to the three–dimensional problem of scatteringof scalar waves by a two–dimensional periodic grating shaped by, e.g.,

z = g(x, y), g(x+ d1, y + d2) = g(x, y).

In short, every relevant formula from section 3 and section 4 would simply be modifiedby replacing occurrences of (αx) by (αx+ βy) [MN11].

By contrast, two generalizations of interest to the author which are genuinelynontrivial are to the cases of vectorial scattering arising in electromagnetics [Jac75]and linear elastodynamics [Ach73], giving rise to Maxwell’s and Navier’s equations,respectively. In these situations (vector) Helmholtz equations (1a) and (1b) againgovern the frequency-domain scattering while straightforward generalizations of quasi-periodicity and the outgoing wave condition are again relevant. The new complicationscome from the interfacial boundary conditions which are no longer as simple as (1c)–(1d). However, we have recently shown in the setting of Maxwell’s equations [Nic15]how these conditions can be phrased in terms of trace operators akin to Du, Dw,N u, and Nw resulting in equations much like (7)–(8). By expanding these operators(and those relevant to linear elastodynamics) in power series in ε and δ, it is easy toimagine how our HOPS/AWE approach could be readily extended to these importantmodels.

6.2. Simulation of frequency-dependent materials. In this contribution wehave focused exclusively upon materials whose index of refraction n (speed c) is bothreal and independent of ω. While the generalization to the setting where these con-stants take on imaginary values (e.g., for modeling the propagation of electromagneticwaves in a metal [Rae88, Mai07, EB12]) is straightforward, it will be interesting to

Dow

nloa

ded

06/0

7/17

to 1

31.1

93.1

78.2

01. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p



investigate the case where n = n(ω). Here we envision an index of refraction whichcan be expressed as a convergent Taylor series

n = n(ω) = n(ω + δω) =

∞∑m=0

nm(ω)δm

in some disk |δ| < R. In this case the dependence of {Du,Dw,N u,Nw} upon δwill be more complicated and subtle; however, one can imagine that if the terms{Dun,m,Dwn,m,N u

n,m,Nwn,m} can be discovered, then a novel HOPS/AWE algorithm

could be built.

6.3. Rayleigh singularity. To close, we reconsider the fundamental obstructionto the convergence of our algorithm: The Rayleigh singularities (Wood’s anomalies)

γup

= 0 or γwp

= 0 for p 6= 0.

At this point it is unclear how to proceed in this setting to build a full HOPS/AWEalgorithm, however, there is something we can say which may be the foundation forfuture developments. We now revisit the Taylor series expansion for γup of section 4.1in the case γu

p= 0. As expansion (12) is no longer valid (γup is not analytic in δ in

this case) we seek a new form of “Puiseux type”

(30) γup = γup (δ) =

∞∑m=0

γup,mδm+1/2 = δ1/2

∞∑m=0

γup,mδm

inspired by the fact that γup is the solution of a quadratic equation. Inserting thisform into the relationship

α2p + (γup )2 = (ku)2,

we find

δ

( ∞∑m=0

γup,mδm

)( ∞∑r=0

γup,rδr

)= (1 + δ)

2(ku)2 −

(αp + δα

)2.

This delivers

δ

∞∑m=0

δmm∑r=0

γup,m−rγup,r =

{(ku)2 − (αp)

2}

+ 2δ{

(ku)2 − α αp}

+ δ2{

(ku)2 − (α)2}

= 2δ{

(ku)2 − α αp}

+ δ2(γu)2,

where we have used that

(ku)2 − (αp)2 = (γu

p)2 = 0.

Canceling a factor of δ, we find at order O(δ0)

γup,0 = ±√

2((ku)2 − α αp),Dow

nloa

ded

06/0

7/17

to 1

31.1

93.1

78.2

01. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p



which we point out can never be zero. To see that this is true recall that γup

= 0

implies (αp)2 = (ku)2 so

γup,0 = ±√

2((ku)2 − α αp) = ±√

2((αp)2 − α αp) = ±

√2αp

√αp − α 6= 0

since p 6= 0.At order O(δ1) we require

γup,1 =(γu)2

2γup,0, γup,0 6= 0.

For O(δm), m > 1, we demand

γup,m =−∑m−1r=1 γup,m−rγ

up,r

2γup,0, γup,0 6= 0.

We now revisit the calculations of section 5.1 and produce the analogue of Figure 2for the parameter choices

q = 3/2, σ = 0.99, nu = 1, Nx = 32, M = 4, 8, 16, 32, 64

in the case δ > 0. This delivers Figure 9 which displays the remarkable convergenceand stability our Puiseux series can deliver. Given the Puiseux series (30) it is clearhow one would approximate the operator U ; however, how the trace operators, e.g.,Du, would depend upon δ (the composition of an analytic function with a singularone) is unclear to the author.

ω

2 2.2 2.4 2.6 2.8

Error

10-15

10-10

10-5

Error versus ω

M = 4

M = 8

M = 16

M = 32

M = 64

Fig. 9. Relative supremum norm error, (22), versus frequency ω in approximation of γup by

γu,Mp (Nx/2 ≤ p ≤ Nx/2 − 1) for various perturbation orders M . Parameter choices are q = 3/2,σ = 0.99, nu = 1, Nx = 32, and M = 4, 8, 16, 32, 64

Dow

nloa

ded

06/0

7/17

to 1

31.1

93.1

78.2

01. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p



Acknowledgment. The author would like to thank F. Reitich for initial con-versations and encouragement on this project. In addition to this, he pointed out theconnection to the engineering literature, in particular, to that of the AWE.

REFERENCES

[Ach73] J. D. Achenbach, Wave Propagation in Elastic Solids, North–Holland, Amsterdam,1973.

[AH01] K. Atkinson and W. Han, Theoretical Numerical Analysis, Texts Appl. Math. 39,Springer, New York, 2001.

[AN14] D. Ambrose and D. P. Nicholls, Fokas integral equations for three dimensionallayered–media scattering, J. Comput. Phys., 276 (2014), pp. 1–25.

[Bar16] A. Barnett, mpspack, https://github.com/ahbarnett/mpspack (2016).[BF97] R. Burden and J. D. Faires, Numerical Analysis, 6th ed., Brooks/Cole, Pacific Grove,

CA, 1997.[BG11] A. Barnett and L. Greengard, A new integral representation for quasi-periodic

scattering problems in two dimensions, BIT, 51 (2011), pp. 67–90.[BL82] L. M. Brekhovskikh and Y. P. Lysanov, Fundamentals of Ocean Acoustics, Springer,

Berlin, 1982.[BR93a] O. Bruno and F. Reitich, Numerical solution of diffraction problems: A method of

variation of boundaries, J. Opt. Soc. Amer. A, 10 (1993), pp. 1168–1175.[BR93b] O. Bruno and F. Reitich, Numerical solution of diffraction problems: A method of

variation of boundaries. II. Finitely conducting gratings, Pade approximants, andsingularities, J. Opt. Soc. Amer. A, 10 (1993), pp. 2307–2316.

[BR93c] O. Bruno and F. Reitich, Numerical solution of diffraction problems: A method ofvariation of boundaries. III. Doubly periodic gratings, J. Opt. Soc. Amer. A, 10(1993), pp. 2551–2562.

[BR09] F. Bleibinhaus and S. Rondenay, Effects of surface scattering in full-waveform in-version, Geophysics, 74 (2009), pp. WCC69–WCC77.

[CB15] M. H. Cho and A. Barnett, Robust fast direct integral equation solver for quasi–periodic scattering problems with a large number of layers, Opt. Express, 23 (2015),pp. 1775–1799.

[CHQZ88] C. Canuto, M. Y. H, A. Quarteroni, and T. A. Zang, Spectral Methods in FluidDynamics, Springer, New York, 1988.

[CK98] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,2nd ed., Springer, Berlin, 1998.

[EB12] S. Enoch and N. Bonod, Plasmonics: From Basics to Advanced Topics, Springer Ser.Opt. Sci., Springer, New York, 2012.

[ELG98] T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, Extraordi-nary optical transmission through sub-wavelength hole arrays, Nature, 391 (1998),pp. 667–669.

[GO77] D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theoryand Applications, CBMS-NSF Regional Conf. Ser. in Appl. Math. 26, SIAM,Philadelphia, 1977.

[God92] C. Godreche, ed., Solids Far from Equilibrium, Cambridge University Press,Cambridge, 1992.

[GR87] L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comput.Phys., 73 (1987), pp. 325–348.

[Hom08] J. Homola, Surface plasmon resonance sensors for detection of chemical and biologicalspecies, Chem. Rev., 108 (2008), pp. 462–493.

[ILW11] H. Im, S. H. Lee, N. J. Wittenberg, T. W. Johnson, N. C. Lindquist, P. Nagpal,D. J. Norris, and S.-H. Oh, Template-stripped smooth Ag nanohole arrays withsilica shells for surface plasmon resonance biosensing, ACS Nano, 5 (2011), pp.6244–6253.

[Jac75] J. D. Jackson, Classical Electrodynamics, 2nd ed., Wiley, New York, 1975.[JJJ13] J. Jose, L. R. Jordan, T. W. Johnson, S. H. Lee, N. J. Wittenberg, and S.-

H. Oh, Topographically flat substrates with embedded nanoplasmonic devices forbiosensing, Adv. Funct. Mater, 23 (2013), pp. 2812–2820.

[KSN96] M. Kolbehdar, M. Srinivasan, M. Nakhla, Q.-J. Zhang , and R. Achar, Si-multaneous time and frequency domain solutions of EM problems using finiteelement and CFH techniques, IEEE Trans. Microwave Theory Tech., 44 (1996),pp. 1526–1534.

Dow

nloa

ded

06/0

7/17

to 1

31.1

93.1

78.2

01. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p

https://github.com/ahbarnett/mpspack



[LJJ12] N. C. Lindquist, T. W. Johnson, J. Jose, L. M. Otto, and S.-H. Oh, Ultrasmoothmetallic films with buried nanostructures for backside reflection-mode plasmonicbiosensing, Ann. Phys., 524 (2012), pp. 687–696.

[LKB15] J. Lai, M. Kobayashi, and A. Barnett, A fast and robust solver for the scatteringfrom a layered periodic structure containing multi-particle inclusions, J. Comput.Phys., 298 (2015), pp. 194–208.

[Mai07] S. A. Maier, Plasmonics: Fundamentals and Applications, Springer, New York, 2007.[MAT10] The MathWorks Inc., MATLAB, version 7.10.0 (R2010a), Natick, MA, 2010.[MN11] A. Malcolm and D. P. Nicholls, A field expansions method for scattering by periodic

multilayered media, J. Acoust. Soc. Amer., 129 (2011), pp. 1783–1793.[Mos85] M. Moskovits, Surface–enhanced spectroscopy, Rev. Modern Phys., 57 (1985), pp.

783–826.[MR87] T. R. Marchant and A. J. Roberts, Properties of short-crested waves in water of

finite depth, J. Austr. Math. Soc. Ser. B, 29 (1987), pp. 103–125.[Nic12] D. P. Nicholls, Three–dimensional acoustic scattering by layered media: A novel

surface formulation with operator expansions implementation, R. Soc. Lond. Proc.Ser. A Math. Phys. Eng. Sci., 468 (2012), pp. 731–758.

[Nic15] D. P. Nicholls, A method of field expansions for vector electromagnetic scatter-ing by layered periodic crossed gratings, J. Opt. Soc. Amer. A, 32 (2015),pp. 701–709.

[Nic16] D. P. Nicholls, A high–order perturbation of surfaces (HOPS) approach to Fokas in-tegral equations: Three–dimensional layered media scattering, Quart. Appl. Math.,74 (2016), pp. 61–87.

[NOJR16] D. P. Nicholls, S.-H. Oh, T. W. Johnson, and F. Reitich, Launching surface plas-mon waves via vanishingly small periodic gratings, J. Opt. Soc. Amer. A, 33 (2016),pp. 276–285.

[NR01] D. P. Nicholls and F. Reitich, A new approach to analyticity of Dirichlet-Neumannoperators, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), pp. 1411–1433.

[NR03] D. P. Nicholls and F. Reitich, Analytic continuation of Dirichlet-Neumann opera-tors, Numer. Math., 94 (2003), pp. 107–146.

[NR04a] D. P. Nicholls and F. Reitich, Shape deformations in rough surface scattering:Cancellations, conditioning, and convergence, J. Opt. Soc. Amer. A, 21 (2004),pp. 590–605.

[NR04b] D. P. Nicholls and F. Reitich, Shape deformations in rough surface scattering:Improved algorithms, J. Opt. Soc. Amer. A, 21 (2004), pp. 606–621.

[NR08] D. P. Nicholls and F. Reitich, Boundary perturbation methods for high–frequencyacoustic scattering: Shallow periodic gratings, J. Acoust. Soc. Amer., 123 (2008),pp. 2531–2541.

[NRJO14] D. P. Nicholls, F. Reitich, T. W. Johnson, and S.-H. Oh, Fast high–order perturba-tion of surfaces (HOPS) methods for simulation of multi–layer plasmonic devicesand metamaterials, J. Opt. Soc. Amer. A, 31 (2014), pp. 1820–1831.

[NT16] D. P. Nicholls and V. Tammali, A high–order perturbation of surfaces (HOPS) ap-proach to Fokas integral equations: Vector electromagnetic scattering by periodiccrossed gratings, Appl. Numer. Methods, 101 (2016), pp. 1–17.

[NW01] F. Natterer and F. Wubbeling, Mathematical Methods in Image Reconstruction,Math. Model. Comput., SIAM, Philadelphia, 2001.

[Pet80] R. Petit, ed., Electromagnetic Theory of Gratings, Springer, Berlin, 1980.[Pou84] M. Pourahmadi, Taylor expansion of exp(

∑∞k=0 akz

k) and some applications, Amer.Math. Monthly, 91 (1984), pp. 303–307.

[PR90] L. Pillage and R. Rohrer, Asymptotic waveform evaluation for timing analysis,IEEE Trans. Comput.-Aided Des., 9 (1990), pp. 352–366.

[Rae88] H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings,Springer, Berlin, 1988.

[Ray07] L. Rayleigh, On the dynamical theory of gratings, R. Soc. Lond. Proc. Ser. A Math.Phys. Eng. Sci., A, 79 (1907), pp. 399–416.

[RDCB98] C. J. Reddy, M. D. Deshpande, C. R. Cockrell, and F. B. Beck, Fast RCS compu-tation over a frequency band using method of moments in conjunction with asymp-totic waveform evaluation technique, IEEE Trans. Antennas and Propagation, 46(1998), pp. 1229–1233.

[Ric51] S. O. Rice, Reflection of electromagnetic waves from slightly rough surfaces, Comm.Pure Appl. Math., 4 (1951), pp. 351–378.D

ownl

oade

d 06

/07/

17 to

131

.193

.178

.201

. Red

istr

ibut

ion

subj

ect t

o SI

AM

lice

nse

or c

opyr

ight

; see

http

://w

ww

.sia

m.o

rg/jo

urna

ls/o

jsa.

php



[RJOM13] F. Reitich, T. W. Johnson, S.-H. Oh, and G. Meyer, A fast and high-order accurateboundary perturbation method for characterization and design in nanoplasmonics,J. Opt. Soc. Amer. A, 30 (2013), pp. 2175–2187.

[Rob83] A. J. Roberts, Highly nonlinear short-crested water waves, J. Fluid Mech., 135 (1983),pp. 301–321.

[RT04] F. Reitich and K. Tamma, State–of–the–art, trends, and directions in computationalelectromagnetics, CMES Comput. Model. Eng. Sci., 5 (2004), pp. 287–294.

[SLL01] R. Sloane, R. Lee, and J.-F. Lee, Multipoint Galerkin asymptotic waveform evalua-tion for model order reduction of frequency domain FEM electromagnetic radiationproblems, IEEE Trans. Antennas and Propagation, 49 (2001), pp. 1504–1513.

[VO09] J. Virieux and S. Operto, An overview of full-waveform inversion in explorationgeophysics, Geophysics, 74 (2009), pp. WCC1–WCC26.

Dow

nloa

ded

06/0

7/17

to 1

31.1

93.1

78.2

01. R

edis

trib

utio

n su

bjec

t to

SIA

M li

cens

e or

cop

yrig

ht; s

ee h

ttp://

ww

w.s

iam

.org

/jour

nals

/ojs

a.ph

p

Downloaded 06/07/17 to 131.193.178.201. Redistribution...

Documents

Transcript of Downloaded 06/07/17 to 131.193.178.201. Redistribution...