On the complexity of aperiodic Fourier modal methodsfor finite periodic structures

M. Pisarencoa, I.D. Setijaa

aDepartment of Research, ASML Netherlands B.V., De Run 6501, 5504DR Veldhoven, TheNetherlands

Abstract

The Fourier modal method (FMM) is based on Fourier expansions of the elec-tromagnetic field and is inherently built for infinitely periodic structures. Whenthe infinite periodicity assumption is not realistic, the finiteness of the structurehas to be incorporated into the model. In this paper we discuss the recent ex-tensions of the FMM for finite periodic structures and analyze their complexityboth with respect to the main discretization parameter N as well as with re-spect to the number of periods R. We show that among the three FMM-basedapproaches able to represent finiteness, the aperiodic Fourier modal methodwith alternative discretization has the lowest computational cost given by ei-ther O(N3 log2R) or O(N2R) depending on the values of N and R. This resultdemonstrates that the method is highly suited for rigorous modeling of scatter-ing from large periodic structures. For instance, for N = 100 and R < 1000 thecomplexity of the aperiodic Fourier modal method with alternative discretiza-tion is comparable to the complexity of the standard FMM.

Keywords: Fourier-modal method, FMM, rigorous coupled-wave analysis,RCWA, perfectly matched layers, PML, computational costs, logarithmiccomplexity, aperiodic Fourier-modal method, AFMM-CFF, alternativediscretization

1. Introduction

Periodicity encountered in a structure is often responsible for its special elec-tromagnetic properties. Examples are the creation of higher orders during wavepropagation in diffraction gratings or effective imitation of a negative refractiveindex [1] or even invisibility [2, 3] by metamaterials. In the process of modelingperiodic structures and solving the underlying Maxwell’s equations, an impor-tant assumption is routinely made: it is considered that the periodic structure,which is finite in reality, is approximated reasonably well by an idealized in-finitely periodic structure. In this case the field is quasi-periodically repeatingfrom one period to another and it suffices to compute the solution in a singleperiod of the structure as shown in Figure 1 (a). Obviously, this leads to lowcomputational costs. The disadvantage however lies in the ”infinite periodicity”

Preprint submitted to Journal of Computational Physics December 22, 2013

(a) FMM

(b) Supercell FMM

(c) Aperiodic FMM

(d) Aperiodic FMM with

alternative discretization

Figure 1: Overview of FMM-based methods for modeling finite periodic structures. (a) Clas-sical FMM employing the ”infinite periodicity” assumption. Only the field in the highlightedcell needs to be computed. (b) Supercell FMM. (c) AFMM-CFF with classical discretization.(d) AFMM-CFF with alternative discretization.

assumption. As the periodic structure gets smaller (e.g. due to being drivenby Moore’s law in the semiconductor industry) the approximation becomes lessaccurate and a different approach is required.

If infinite periodicity can be assumed then standard numerical techniquesfor Maxwell’s equations may be applied to solve for a single period of the struc-ture. These techniques include the finite-element method (FEM) [4, 5, 6], thefinite-difference time-domain method (FDTD) [7, 8, 9] and the integral equationmethods (IEM), which include the boundary element method (BEM) [10, 11, 12]and the volume integral method (VIM) [13, 14, 15, 16, 17].

The enumerated methods are derived from general-purpose numerical dis-cretization schemes. In the field of computational diffractive optics one spe-cialized method has gained a large popularity due to its simplicity and naturalinterpretation of the field expansions. It is known under several names andabbreviations: rigorous coupled-wave analysis (RCWA), modal method withFourier expansion (MMFE), and the Fourier-modal method (FMM). The lattername is used throughout this paper. The FMM [18] was proposed in 1978 by

2

Knop [19]. The method is analytical in one spatial direction and (for a broadrange of applications) outperforms general-purpose approaches relying on a fullnumerical discretization of all directions. In the past decades the FMM hasmatured due to fundamental studies and improvements to its stability [20, 21]and convergence [22, 23, 24]. Other important contributions to the evolutionof the method are the techniques of adaptive spatial resolution [25] and normalvector fields [26, 27, 28]. Ref. [29] gives a mathematical perspective of the chal-lenges that have been overcome in the FMM and of the open problems still tobe addressed. For possible improvements of the method see Chapter 7 in [30].

Recently, several extensions of the FMM have been proposed which do notrely on the infinite periodicity assumption and rigorously model all the periodsof the structure of interest. We illustrate the FMM-based approaches whichincorporate finiteness in Figures 1 (b), (c), and (d). The most straightforwardextension of the FMM to finite structures is the so-called supercell approach,depicted in Figure 1 (b). The computational domain is extended to includeall periods of the structure. Moreover, extra space is left on both sides of thestructure in order to decrease the coupling/reflections via the periodic bound-ary conditions which are built-in in the method due to the use of Fourier-modeexpansions. The more empty space is left on the sides, the better the modelrepresents finiteness. We note that, unlike the methods described below, thesupercell FMM is not rigorous in the sense that the radiation condition is im-posed indirectly by expanding the computational domain. The computationalcost significantly increases both as the number of periods in the structure be-comes larger and as the empty spaces are chosen wider. In practice, this meansthat supercell FMM can only be used for small structures (with a size of severaltens of wavelengths).

A more efficient approach is to replace the wide (and expensive) emptyspaces by narrow artificial absorbers as done in the aperiodic Fourier modalmethod in contrast-field formulation (AFMM-CFF) and depicted in Figure 1(c). The AFMM-CFF [31, 32, 33] is a recent extension of the FMM to aperi-odic structures. It builds upon the aperiodic Fourier modal method developedand refined by Lalanne and co-workers [34, 35, 36, 37, 38] with the key dif-ference that the AFMM-CFF solves Maxwell’s equations formulated in termsof a contrast (scattered) field instead of a total field. This reformulation al-lows prescription of arbitrary incoming fields onto the structure of interest [31].Aperiodicity is achieved by using perfectly matched layers (PMLs) [39] on thevertical boundaries in order to annihilate the periodic boundary condition.

Finally, it turns out that the computational cost for AFMM-CFF can befurther reduced by applying an alternative discretization [40], which can also beviewed as a classical FMM discretization applied to a rotated geometry as shownin Figure 1 (d). This alternative discretization is chosen based on complexityarguments and will be explained in Section 5.

Since in practical applications the number of periods can be quite large, animportant criterion for comparing these methods is the scaling of computationalcosts with the number of periods and with the number of basis functions perperiod. We will discuss the complexity in terms of time and memory for all

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methods presented.This paper is structured as follows. We describe the four FMM-based ap-

proaches in Sections 2 - 5. Computational results which illustrate the charac-teristics of the methods are presented in Section 6. In Section 7 the complexityof the AFMM-CFF with alternative discretization is analyzed and comparedto the complexity of other methods. Finally, our conclusions are presented inSection 8.

2. Classical FMM

We are concerned with solving the time-harmonic Maxwell equations fornon-magnetic materials on the domain (x, y, z) ∈ [0,Λ]× R× R.

∇× e(x) = −k0h(x), (1a)

∇× h(x) = −k0ε(x, z)e(x), (1b)

where x = (x, y, z) is the position vector, e = (ex, ey, ez) is the electric field

and h = (hx, hy, hz) is the magnetic field scaled by −i√ε0/µ0. The temporal

frequency ω is incorporated into the constant k0 = ω√ε0µ0. The electric per-

mittivity ε is assumed y-invariant. See Figure 2 for a sample geometry. Theincident field is given by

einc(x) = ae−ikinc·x, (1c)

where kinc = (kincx , kincy , kincz ) is the wavevector and a = (ax, ay, az) is the am-

plitude vector. Note that the Maxwell equations require that kinc · a = 0.For simplicity, we consider the special case of a planar TE incident wave

(only the y-component of the field is non-zero)

eincy (x, z) = aye−i(kinc

x x+kincz z), (2)

such that Maxwell’s equations are reduced to a single scalar equation for the ycomponent of the electric field,

∂2

∂x2ey +

∂2

∂z2ey + k20ε(x, z)ey = 0. (3)

We now describe the classical FMM for this equation. The first step is todivide the computational domain into layers such that the permittivity may beconsidered z-independent in each single layer. Thus, the profile of the scattereris approximated by a staircase as in Figure 2. Now in each layer j = 1, ...,Mthe permittivity εj(x) is independent of z.

∂2

∂z2ey,j(x, z) = −Ljey,j(x, z), with Lj =

∂2

∂x2+ k20εj(x). (4)

4

slice 2

slice 4

slice 3

slice 1

slice M − 2

slice M − 1

slice M

x

z

h1

Λ

h2

h3

h4

...

hM−2

hM−1

...

Figure 2: Slicing of the scatterer profile

with matching conditions

Bjey,j(x, hj) = Bj+1ey,j+1(x, hj), with Bj =

(1

k−10∂∂z

). (5)

where hj is the z-coordinate of the top interface of layer j. The next step is todiscretize the equations in the x-direction by using a Galerkin approach with”shifted” Fourier harmonics as basis functions and test functions,

φn(x) = e−ikxnx, where kxn = kincx + n2π

Λ, for n = −N, ...,+N. (6)

In each layer the electric fields are expanded as

ey,j(x, z) =

N∑n=−N

uj,n(z)φn(x). (7)

Here the functions uj,n(z) can be interpreted as unknown z-dependent Fouriercoefficients of the solution. The standard inner product on the interval [0,Λ] isused. After the application of the Galerkin method, Equations (4) become

u′′j (z) = −Ljuj(z), with Lj = −K2x + Ej . (8)

Here −K2x ∈ R(2N+1)×(2N+1) and Ej ∈ R(2N+1)×(2N+1) are discrete versions

of the respective continuous operators ∂2

∂x2 and k20εj(x). Equation (8) is a ho-mogeneous second-order ordinary differential equation whose general solution isgiven by

uj(z) = Wj(e−Qj(z−hj)c+j + eQj(z−hl+1)c−j ), (9)

5

where Wj ∈ R(2N+1)×(2N+1) is the matrix of eigenvectors of Lj and Qj ∈R(2N+1)×(2N+1) is a diagonal matrix with square roots of the correspondingeigenvalues on its diagonal. An eigenvalue decomposition based on the QR-algorithm costs O(N3) operations [41, Algorithm 7.5.2]. For all layers, the costof computing the general solutions is O(MN3).

The coefficients c+j , c−j ∈ R2N+1 are determined by applying the matching

conditions (5) to the analytical solution (9). The top and bottom layers are ofinfinite thickness and require a special treatment in order to impose the radiationcondition. Note that the c+j and c−j coefficients are associated with respectivelydownward and upward propagating waves. Therefore, the radiation conditionsimply c+1 = c−M = 0. Care has to be taken such that the application of thematching condition does not introduce instabilities in the algorithm [21]. Aftersome algebraic manipulations we arrive at expressions of this form(

c+j+1

c−j

)= Sj,j+1

(c+jc−j+1

), (10)

where the matrix Sj,j+1 depends on the layer matrices Wj ,Wj+1,Qj ,Qj+1 (seerelations (48) and (49) in [40]). This is a system of recursive linear equations.It can be solved by eliminating the unknown coefficients for the intermediarylayers or, equivalently, by computing the global S-matrix for all layers

S1,M = S1,2 ∗ S2,3 ∗ . . . ∗ SM−1,M . (11)

Here the ∗ symbol denotes the (homogeneous) Redheffer star product [42]. Fora < b < c, is is defined as

Sa,b ∗ Sb,c =

[S11b,cH

−11 S11

a,b S12b,c + S11

b,cS12a,bH

−12 S22

b,c

S21a,b + S22

a,bS21b,cH

−11 S11

a,b S22a,bH

−12 S22

b,c

], (12a)

with

H1 = (I− S12a,bS

21b,c), (12b)

H2 = (I− S21b,cS

12a,b). (12c)

Since a Redheffer star product is computed in O(N3) operations, computingthe global S-matrix will cost O(MN3). Given the global S-matrix we directlyevaluate the expression (

c+Mc−1

)= S1,M

(c+1c−M

), (13)

where the vectors c+1 and c−M on the right-hand side represent the incident fieldabove and below the stack respectively. At this point, the partial solution hasbeen computed.

Finally, if the near field is required, the intermediary coefficients can becomputed recursively using the expression (a < b < c)(

c+bc−b

)=

(H−11 S11

a,b S12a,bH

−12 S22

b,c

S21b,cH

−11 S11

a,b H−12 S22b,c

)(c+ac−c

). (14)

6

If the inverses of H1 and H2 (or their LU-decompositions) computed for theRedheffer star product (partial solution step) are properly reused, then onlymatrix-vector products are required in this step. Therefore, the computationalcost for obtaining the full solution scales as O(MN2). Depending on the appli-cation, the full solution might not be necessary.

The main steps of the method together with their complexity are listed inTable 1. The total computational costs in terms of time and memory are given

Step Description Complexity1. Spatial discretization Slice the domain into layers,

so that ε is z-independent ineach layer.

negligible

2. Spectral discretization Apply Galerkin approachfor discretisation in the x-direction.

negligible

3. General solution Derive/compute the generalsolution in each layer.

O(MN3)

4. Partial solution Match the solutions at layerinterfaces using internalboundary conditions.

O(MN3)

5. Full solution Given the coefficients ofthe partial solution, com-pute the intermediary coef-ficients.

O(MN2)

Table 1: List of steps performed by the FMM together with their complexity. Here M denotesthe number of slices, N - number of harmonics.

respectively by

TFMM = (cTdiag + cTS )MN3 + cTc MN2, (15a)

MFMM = (cMdiag + cMS )MN2 + cMc MN . (15b)

Here, cTdiag, cTS , cTc , cMdiag, c

MS , cMc are machine-dependent constants. We have

used the variables N and M which give respectively the number of harmonicsand number of slices per period. Since the standard FMM considers a singleperiod of the structure, N = N and M = M . This differentiation is used inorder to facilitate the comparison of complexities of methods that discretize

more than one period. Since the constants cT /Mdiag , c

T /MS , c

T /Mc are expected to

be of the same order of magnitude, the second term in the above expressionscan be neglected. Then estimate (15) reduces to

TFMM = O(MN3), (16a)

MFMM = O(MN2). (16b)

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3. Supercell FMM

The supercell version of the FMM requires the user only to define a differentgeometry without having to modify the method itself. The new geometry willinclude all the periods of the scatterer as well as empty space on the sides (seeFigure 1 (b)) where the field can decay such that the effect of the periodicboundary condition is minimized. We introduce the domain expansion factorS as the ratio of the widths of the computational domain and of the structure.For an accurate representation of finiteness this factor should be large, S � 1.Also let R be the number of periods of the scatterer. In order to achieve thesame accuracy per period as in the standard FMM, we choose N = SRN andthe computational costs are given by

TSFMM = O(S3R3MN3), (17a)

MSFMM = O(S2R2MN2). (17b)

4. AFMM-CFF with classical discretization

The periodic basis functions (6) force the solution to be periodic. For anaperiodic scatterer we need to implement the radiation BC at the lateral bound-aries. One way of achieving this without changing the basis is to place perfectlymatched layers (PMLs) [39] of a certain thickness just before the boundary. Thisapproach has been previously used to apply the FMM to waveguide problems[34, 35, 37]. The PML changes the x-derivative in the differential equations (4)as follows

∂

∂x→ 1

f ′(x)

∂

∂x, with f(x) = x+ iβ(x). (18)

The function β(x) is continuous. Moreover, it has support only in the PMLsplaced in the intervals [0, xl] and [xr,Λ]. An example of such a function is shownin Figure 3. The modified version of the Lj operator where the x-derivative

has been changed according to (18) will be denoted by Lj . In order to avoidmodifications of the incident field by the PML, the computed solution shouldconsist of only outgoing waves [31]. Therefore the equations are reformulatedsuch that the incident field, i.e. the known part of the solution, is moved intothe right-hand side of the differential equation. In addition to the total fieldproblem

∂2

∂z2ey,j(x, z) = −Lj ey,j(x, z), with Lj =

1

f ′(x)

∂

∂x

1

f ′(x)

∂

∂x+ k20εj(x),

(19a)

eincy (x, z) = aye−i(kinc

x x+kincz z), (19b)

8

0 1 2 3 4 5−1

−0.5

0

0.5

1

x

β(x

)xl xr Λ

Figure 3: An example of β(x) to be used in (18).

we define a background problem with PMLs

∂2

∂z2eby,j(x, z) = −Lb

j eby,j(x, z), with Lb

j =1

f ′(x)

∂

∂x

1

f ′(x)

∂

∂x+ k20ε

bj (x),

(20a)

eb,incy (x, z) = aye−i(kinc

x x+kincz z), (20b)

and a background problem without PMLs

∂2

∂z2eby,j(x, z) = −Lb

j eby,j(x, z), with Lb

j =∂

∂x

∂

∂x+ k20ε

bj (x), (21a)

eb,incy (x, z) = aye−i(kinc

x x+kincz z). (21b)

Note that εbj is chosen to be x-independent, i.e. it represents a multilayer stack.This implies that the background problem without PMLs has an analyticalsolution. Since the PML effectively implements the radiation conditions, for anideal PML the two background problems have equal solutions in the physicaldomain (the region between the PMLs).

eby,j(x, z) = eby,j(x, z), for x ∈ [xl, xr]× R. (22)

Subtracting (20) from (19) and defining the contrast field ec = e− eb, we get

∂2

∂z2ecy,j(x, z) = −Lj e

cy,j(x, z)− (Lj − Lb

j )eby,j(x, z), ec,incy (x, z) = 0. (23)

The incident field has been removed, and a non-homogeneous term appears onthe right-hand side. The background permittivity is chosen such that ε(x, z)−εb(z) vanishes in the PML and has compact support. As a consequence, Lj −Lbj 6= 0 only for x ∈ [xl, xr], which according to (22) entitles the substitution of

eby,j(x) by eby,j(x) in (23).

9

Once the source-term is determined, Equation (23) may be solved. For thispurpose, the source term must also be expanded into Fourier modes. After trun-cation a non-homogeneous system of ordinary differential equations is obtainedfor each layer. The field is found by matching the general solutions at the layerinterfaces. This leads to non-homogeneous equations of the form(

c+j+1

c−j

)= Sj,j+1

(c+jc−j+1

)+

(f1j,j+1

f2j,j+1

). (24)

The system of recursive linear equations is solved by computing the so-calledglobal matrix-vector pair

(S1,M , f1,M ) = (S1,2, f1,2) ∗ (S2,3, f2,3) ∗ . . . ∗ (SM−1,M , fM−1,M ). (25)

Here the ∗ symbol denotes the non-homogeneous Redheffer star product [40].The computation of the matrix-vector pair is dominated by the matrix com-putation. For all layers, the cost again amounts to O(MN3). Note that thenumber of harmonics N required in the discretization will depend on the numberof periods R, i.e. N = RN . The time and memory complexity of the method isgiven by

TAFMM-CFF = O(R3MN3), (26a)

MAFMM-CFF = O(R2MN2). (26b)

5. AFMM-CFF with alternative discretization

It follows from (26) that the x-direction (for which harmonics are used) is”more expensive” both in terms of time and memory than the z-direction (forwhich slices are used). For rectangular scatterers/domains that are much longerin the x-direction it is reasonable to choose an alternative discretization: makethe longer direction ”cheaper” by using spatial discretization into layers andapply spectral discretization in the shorter direction.

An ideal PML yields an effective radiation boundary condition. Thus, unlikein the (periodic) FMM, effectively the same boundary condition is imposed onall boundaries. This fact facilitates the exchange of discretization directions.If the problem is first rotated from the (x, y, z) coordinates to the (z,−y, x)coordinates, the same algorithm as described in the previous section can be used.For a discussion on the projection of the analytically computed background fieldonto the new basis see [40].

Due to the periodicity in the vertical direction, the S-matrices are also peri-odically repeated. As before, the non-homogeneous S-matrix algorithm [32, 40]uses the iteration

(S1,j+1, f1,j+1) = (S1,j , f1,j) ∗ (Sj,j+1, fj,j+1). (27)

The global matrix-vector pair for M − 1 interfaces (M layers) is given by

(S1,M , f1,M ) = [...[[(S1,2, f1,2) ∗ (S2,3, f2,3)] ∗ (S3,4, f3,4)] ∗ ... ∗ (SM−1,M , fM−1,M )].(28)

10

x

z

...

(S1,3,f 1

,3)

(S1,5,f 1

,5)

(S1,9,f 1

,9)

(S1,5,f 1

,5)

(S1,4,f 1

,4)

(S1,3,f 1

,3)

(S1,9,f 1

,9)

O(MR) O(M log2 R)

...

Figure 4: A schematic representation of the nonhomogeneous S-matrix algorithm applied toa grating with R = 4 periods and M = 2 interfaces per period: classical linear recursion (leftside) and fast exponential recursion (right side).

The left side of Figure 4 gives a visual representation of the merging process.The cost of computing (28) scales linearly with the number of periods,

TAD,S-lin = O(RMN3), (29a)

MAD,S-lin = O(RMN2). (29b)

Due to the associativity property of the non-homogeneous Redheffer product, wecan regroup the multiplication operations. We observe that once we have com-puted the matrix-vector corresponding to the stack {1, ..., M + 1}, the matrix-vector pair corresponding to the stack {M + 1, ..., 2M + 1} can be computedwith a minimal effort

(SM+1,2M+1, fM+1,2M+1) = (S1,M+1, νf1,M+1), (30)

where ν = exp(−k0q(hM+1 − h1)) is a phase factor which appears due to thescalar z-dependence of f . Thus, we can directly compute the matrix-vector paircorresponding to the stack {1, ..., 2M + 1}

(S1,2M+1, f1,2M+1) = (S1,M+1, f1,M+1) ∗ (S1,M+1, νf1,M+1). (31)

The iteration which computes the matrix-vector pair for an exponentially in-creasing number of layers is depicted on the right side of Figure 4 and defined

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by

(S1,2k+1M+1, f1,2k+1M+1) = (S1,2kM+1, f1,2kM+1) ∗ (S1,2kM+1, ν2k f1,2kM+1),

(32)

for k = 0, ..., R− 1. By exploiting the local periodicity the computational costsfor obtaining a partial solution have been reduced down to

TAD,S-log = O((M + log2(R))N3), (33a)

MAD,S-log = O((M + log2(R))N2). (33b)

The costs for diagonalization (general solution) and full solution (see the list ofsteps in Table 1) are respectively given by

TAD,diag = O(MN3), (34a)

MAD,diag = O(MN2), (34b)

and

TAD,c = O(RMN2), (35a)

MAD,c = O(RMN2). (35b)

By combining the estimates (34), (33) and (35) we obtain the total computa-tional costs,

TAD = ((cTdiag + cTS )M + cTS log2(R))N3 + cTc RMN2, (36a)

MAD = ((cMdiag + cMS )M + cMS log2(R))N2 + cMc RMN. (36b)

6. Illustrative computational results

The purpose of this section is to give a taste of the problems that can besolved using FMM-based methods while demonstrating the efficiency of PMLsand the distinct features exhibited by the solutions of infinite and finite periodicstructures. Consider the problem of a radiating infinitely long line in free space.In a two-dimensional setting the infinitely long line is modeled as a point source.For a source term with vanishing x- and z-components, the following equationholds,

∆ey(x, z) + εbey(x, z) = −δ(x, z). (37)

The solution of the above equation is referred to as the Green’s function of thisparticular PDE and is given by

ey(x, z) =i

4H

(1)0

(√εb(x2 + z2)

). (38)

Here H(1)0 is the Hankel function of the first kind. The solution of (37) can

also be computed numerically using either the supercell approach or PMLs (for

12

Figure 5: Plot of <(ey) for the radiating line problem solved using various approaches: (a)supercell FMM, Λ = 10, (b) supercell FMM, Λ = 20, (c) exact solution, (d) FMM with PMLs(indicated by hatched areas).

this particular problem it does not matter whether the classical or alternativediscretization is used in the AFMM-CFF). The plots (a) and (b) in Figure 5show the supercell solution for domain widths Λ = 10 and Λ = 20 respectively.Interference caused by the periodic boundary condition decreases for larger sizesof the computational domain. We note that larger computational domains re-quire more harmonics and imply higher computational costs. However, even forΛ = 20 the supercell solution is still far from the exact one given by (38) andplotted in Figure 5 (c). On the other hand, the PML solution shown in Figure5 (d) does not require large computational domains and closely resembles theexact solution in the area between the PMLs.

Figure 6 shows solutions computed with the four FMM-based methods fora 32 line grating. We observe that the fields in the center are similar for allmethods (although a weak global variation due to finiteness may be observedover the whole structure for the aperiodic methods). The effects of finitenessare most prominent on the edges as can be seen in Figure 6 (b), (c), (d). Theclassical FMM computes the solution on a single period. For comparison withthe other methods the one-period solution is repeated periodically over theextent of the finite grating. For the supercell FMM a larger domain (S = 5)is used and only the part of the domain containing the grating is shown. Thecomputational domains used in the AFMM-CFF with classical and alternativediscretization have a size comparable to the size of the grating and are shown

13

0 1 2 3 4 5 6

0

0.5

x

z

(c)

0

0.1

0.2

0 1 2 3 4 5 6

0

0.5

x

z

(a)

0

0.1

0.2

0 1 2 3 4 5 6

0

0.5

x

z(b)

0

0.1

0.2

0

0.50 1 2 3 4 5 6

z

x

(d)

0

0.1

0.2

Figure 6: An example of solutions obtained with the FMM-based methods described in thispaper for a 32 line grating (permittivities from top to bottom: ε1 = 1, ε2 = 2.25, ε3 = 4)illuminated by a plane wave incident at an angle θ = 45o, wavelength λ = 6.28 and unitamplitude. The absolute value of the solution is plotted. (a) FMM. (b) Supercell FMM. (c)AFMM-CFF. (d) AFMM-CFF with alternative discretization.

in Figure 6 (c), (d) without modifications.

7. Complexity analysis

In this section we focus on the AFMM-CFF with alternative discretizationand study the complexity exhibited by the underlying algorithm. Figure 7 showsa break-down of the computational costs as a function of the number of periodsR for a fixed number of harmonics per period N = 30. As discussed in theprevious sections, the three major contributions to the total costs are comingfrom the steps 3-5 in Table 1. The graph confirms the constant R-independentbehavior of the ”general solution” step, logarithmic scaling withR of the ”partialsolution” step, and linear scaling with R of the ”full solution” step. Dependingon the value of R either the ”partial solution” step or the ”full solution” stepdominates the computational cost. This corresponds to respectively the firstor second term in (36a) being larger. For the specific number of harmonicsper period N = 30, we see in Figure 7 that the switching point is aroundR = 100. Thus, if the structures to be simulated have less than 100 periods,the computation time will scale logarithmically with R. In fact in this case thecosts will be very close to the ones incurred when simulating a single period ofthe structure and assuming infinite periodicity. If R > 100 the total costs willscale linearly with R. Note that as the complexity with respect to R changesfrom logarithmic to linear, the complexity with respect to N must change from

14

100

101

102

103

10−3

10−2

10−1

R

T,[s]

TotalGeneral solutionPartial solutionFull solutionO(1)O(log2 R)O(R)

Figure 7: Number-of-periods time complexity obtained with the AFMM-CFF with alternativediscretization for a binary grating (N = 30, M = 2).

cubic to quadratic. The location of the switching point depends on R and N .These observations are formalized in the following proposition.

Proposition 1. The time complexity TAD(N , R) of the AFMM-CFF with al-ternative discretization exhibits one of the following behaviors:

TAD(N , R) =

{O(N2R), for N < fN (R) (⇔ R > fR(N)),

O(N3 log2R), for N > fN (R) (⇔ R < fR(N)).(39)

Here

fN (R) =bR

a+ log2R, (40a)

fR(N) =

{2(ln 2)−1−a, for N < b2−ae ln 2,

2−W−1(−bN−12−a ln 2)(ln 2)−1−a, for N ≥ b2−ae ln 2,(40b)

with

a =(cTdiag + cTS )M

cTS, (41a)

b =cTc M

cTS. (41b)

Proof. The complexity of the AFMM-CFF with alternative discretization isgiven by (36a). Introducing the notation

r = log2R, (42)

and using the definitions of a, b in (41), expression (36a) is rewritten as

TAD = (acTS + cTS r)N3 + bcTS 2rN2. (43)

15

For a specific combination of N and r, one of the two terms in the expressionabove dominates, and hence determines the overall complexity of the algorithm.By balancing the terms we can determine the frontier between the two regionswith distinct behavior,

a+ r = bN−12r. (44)

If solved for N this equation yields the definition for fN (R)

fN (R) =b2r

a+ r=

bR

a+ log2R. (45)

This expression is to be evaluated for any R ≥ 1, R ∈ Z. Derivation of an explicitexpression for R requires a bit more work. Equation (44) is manipulated suchthat all unknown terms are on one side and form a product logarithm:

−(a+ r) ln 2e−(a+r) ln 2 = −bN−12−a ln 2. (46)

Using the (implicit) definition of the product logarithm (also called Lambert Wfunction [43, 44])

z = W (z)eW (z), (47)

yields the relation

−(a+ r) ln 2 = W (−bN−12−a ln 2), (48)

such that the solution is given by

r = −W (−bN−12−a ln 2)(ln 2)−1 − a. (49)

We have the following bounds on the variables in the argument of W :

M ≥ 2, a > 2, b > 0, N ≥ 0. (50)

Therefore, the argument of W is negative. The Lambert W function has no val-ues on the interval (−∞,−e−1) and has two branches on the interval [−e−1, 0).With an additional constraint W ≥ −1 we define a single-valued functionW0(x), while the lower branch has W ≤ −1 and is denoted W−1(x). Thus, forN ≥ b2−ae ln 2 the two values W0(−bN−12−a ln 2) and W−1(−bN−12−a ln 2)give the coordinates of the two intersections of the functions in (44). Since weare interested in the intersection with largest r we choose the lower branch ofW . Returning to the original variable via (42) for N ≥ b2−ae ln 2 we obtain

fR(N) = 2−W−1(−bN−12−a ln 2)(ln 2)−1−a. (51)

Finally, for N < b2−ae ln 2 we continuously extend fR(N) by giving it a constantvalue that matches the value fR(b2−ae ln 2),

fR(N) = 2(ln 2)−1−a. (52)

From a > 2 follows that 2(ln 2)−1−a < 1. Since R ≥ 1, the exact location of thefrontier below unity becomes irrelevant. This concludes the proof.

16

log10 N

log 1

0R

log10 T

0 0.5 1 1.5 2 2.5 3 3.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

−5

−4

−3

−2

−1

0

1

2

3

4

5

O(log2 R)

O(R)

O(N2) O(N3)

Figure 8: Contour plot of the theoretical complexity TAD(N, R) given in (36a) with values forthe constants cTdiag, cTS , cTc given in (55). The striped line indicates the frontier between the

two complexity regimes defined in Proposition 1: O(N3 log2R) and O(N2R).

Proposition 2. The memory complexity MAD(N , R) of the AFMM-CFF withalternative discretization exhibits one of the following behaviors:

MAD(N , R) =

{O(NR), for N < fN (R) (⇔ R > fR(N)),

O(N2 log2R), for N > fN (R) (⇔ R < fR(N)).(53)

The functions fN (R), fR(N) are defined in (40) with

a =(cMdiag + cMS )M

cMS, (54a)

b =cMc M

cMS. (54b)

Proof. The proof follows the same steps as for Proposition 1.

Based on the timings reported in Figure 7, the constants in (36a) are deter-mined to be

cTdiag = 5.5× 10−7, (55a)

cTS = 2.2× 10−7, (55b)

cTc = 2.8× 10−7. (55c)

Figure 8 shows a contour plot of the theoretical complexity TAD(N , R) given in(36a) with values for the constants cTdiag, cTS , cTc given in (55). The striped line

17

100

101

102

103

10−2

100

R

T,[s]

(a)

Classical FMMSupercell FMMAFMM-CFFAFMM-CFF alt. dis.O(R3)/O(log2 R)/O(R)

100

101

102

103

10−2

100

102

R

T,[s]

(b)

Classical FMMSupercell FMMAFMM-CFFAFMM-CFF alt. dis.O(R3)/O(log2 R)/O(R)

100

101

102

103

10−2

100

102

R

T,[s]

(c)

Classical FMMSupercell FMMAFMM-CFFAFMM-CFF alt. dis.O(R3)/O(log2 R)/O(R)

Figure 9: A direct comparison of the number-of-periods time complexity for the FMM-basedmethods described in this paper for a binary grating (M = 2 slices per period) for various

numbers of harmonics per period. (a) N = 10. (b) N = 30. (c) N = 100.

indicates the frontier between the two complexity regimes defined in Proposition1. The values in (55) are specific to the machine on which the computationshave been performed. However the ratios cTc /c

TS and (cTdiag + cTS )/cTS , which de-

termine the complexity regime as defined by (40), are expected to stay constant.Therefore, on a different machine only the scale for log10 TAD will change whilethe qualitative shape and the position of the frontier will remain unchanged.The complexity in the region above the frontier is dominated by the computa-tion of the full solution which is O(N2R), while below the frontier the partialsolution dominates giving a complexity of O(N3 log2R). In the applicationswhere only the partial solution (above and below the scatterer) is required thecomplexity is O(N3 log2R) everywhere. Moreover, if instead of a full solutionthe solution in an arbitrary period/layer is computed then the complexity for afull but ’local’ solution can be reduced from O(N2R) to O(N2 log2R) such thatthe partial solution costs are always dominant, resulting again in a complexityof O(N3 log2R).

The consequences of Proposition 1 are confirmed by performing R sweepsfor three different values of N (10, 30 and 100). The computation times for the

18

four FMM-based methods described in this paper are reported in Figure 9. Thesupercell FMM and the AFMM-CFF scale cubically with R and are not suitedfor problems with more than 100 periods. On the other hand, the AFMM-CFF with alternative discretization has an ”optimal” scaling with respect toR and can be used for large-scale problems where R reaches 105. In fact, thecomputational cost for the latter method with a moderate R is quite closeto the cost of the classical (periodic) FMM. As expected, the transition fromlogarithmic to linear complexity in R occurs earlier for smaller N . It is easy tocheck that the transition point in Figure 9 is accurately predicted by the stripedfrontier line in Figure 8.

We summarize the properties of the FMM-based methods for finite periodicstructures in Table 2. It lists the time and memory complexity of the methods(based on (16), (17), (26), (36b), and (39)), as well as their ability to model theeffects of finiteness. The classical FMM appears to be the most efficient method

Method Time Memory Finiteness

FMM O(MN3) O(MN2) No

Supercell FMM O(S3R3MN3) O(S2R2MN2) if S � 1

AFMM-CFF O(R3MN3) O(R2MN2) Yes

AFMM-CFFwith alt. dis.

O((M + log2R)N3)or O(RMN2)

O((M + log2R)N2)or O(RMN)

Yes

Table 2: Overview of the FMM-based methods for finite structures. Here M denotes thenumber of slices per period, N - number of harmonics per period, R - number of periods, S -domain expansion factor for the supercell FMM.

in terms of complexity since its cost does not scale with R. This method ishowever unable to incorporate finiteness in the model. The supercell FMM isthe most simple approach among the ones able to model finiteness. The price forthe simplicity is paid in terms of computation time. For instance, simulatinga grating with R = 100 lines and a domain expansion factor S = 5 wouldtake S3R3 = 125× 106 more time than approximating the finite grating by aninfinite one and using the FMM. Here we assumed that the resolution per periodis kept constant. The AFMM-CFF with classical and alternative discretizationdrastically reduce these costs. The AFMM-CFF with classical discretizationremoves the need for domain expansion by employing PMLs, such that for ourexample it will be 125 times faster than supercell FMM but still about 106 timesslower than the FMM. Finally, the AFMM-CFF with alternative discretizationalso removes the cubic scaling with the number of periods by exploiting therepeating patterns in the geometry. For the specific case when R = 100 lines,the new complexity is O(R) for N < 20 and O(log2R) for N > 20. In the lattercase the complexity of the algorithm is brought very close to the complexity ofthe periodic FMM while still being able to represent finiteness.

19

8. Conclusions

The FMM-based methods offer a range of approaches for the numerical so-lution of scattering problems with finite periodic geometries. When the objectunder study is large and the edge effects due to finiteness are not of interestthe standard FMM should be applied for a single period of the structure. Ifthe edge effects are important, then one of the three other methods should beused. The choice depends on the desired computational cost, on one hand,and the implementation effort, on the other hand. The supercell FMM is triv-ially obtained from the FMM by including all the periods of the structure andexpanding the computational domain to include empty space around the struc-ture. The AFMM-CFF requires a certain implementation effort, with the rewardthat computational costs are reduced due to the fact that the computationaldomain only contains the scatterer itself. No empty space around the scattereris needed. Due to the cubic complexity exhibited by the supercell FMM and theAFMM-CFF the time and memory requirements increase rapidly with R andthe methods are capable of handling periodic structures with up to 100 periods.

The AFMM-CFF with alternative discretization stands apart from the firstthree methods. It discretizes the finite periodic geometry such that the S-matrix algorithm is subsequently applied in the periodic direction. This enablesa tremendous decrease of computational costs by merging the S-matrices of theindividual periods in a geometric progression. We have shown that the com-plexity exhibited by this method is given by O(N2R)/O(N3 log2R), in termsof computation time, and by O(N2R)/O(N3 log2R), in terms of memory.

The (sub-)linear scaling with R makes the AFMM-CFF with alternativediscretization particularly suited for large periodic but finite structures suchas metamaterials [45, 46]. Another potential application is the generation ofreference solutions for homogenization/upscaling approaches [47, 48]. Finally, alarge field of application for the AFMM-CFF with alternative discretization isthe solution of nonlinear inverse problems involving finite periodic geometrieswhere a fast solution of the forward problem is critical.

[1] R. A. Shelby, D. R. Smith, S. Schultz, Experimental Verificationof a Negative Index of Refraction, Science 292 (5514) (2001) 77–79.doi:10.1126/science.1058847.URL http://dx.doi.org/10.1126/science.1058847

[2] A. Alu, N. Engheta, Achieving transparency with plasmonic andmetamaterial coatings, Physical Review E 72 (1) (2005) 016623+.doi:10.1103/physreve.72.016623.URL http://dx.doi.org/10.1103/physreve.72.016623

[3] D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F.Starr, D. R. Smith, Metamaterial Electromagnetic Cloak at Microwave Fre-quencies, Science 314 (5801) (2006) 977–980. doi:10.1126/science.1133628.URL http://dx.doi.org/10.1126/science.1133628

20

[4] P. Monk, A finite element method for approximating the time-harmonicMaxwell equations, Numerische Mathematik 63 (1) (1992) 243–261.doi:10.1007/bf01385860.URL http://dx.doi.org/10.1007/bf01385860

[5] L. Zschiedrich, S. Burger, B. Kettner, F. Schmidt, Advanced Finite ElementMethod for Nano-Resonators, in: Proceedings of SPIE, Vol. 6115, 2006.arXiv:physics/0601025, doi:10.1117/12.646252.URL http://dx.doi.org/10.1117/12.646252

[6] P. Monk, Finite Element Methods for Maxwell’s Equations, OxfordUniversity Press, Oxford, 2003.URL http://www.oup.com/us/catalog/general/subject/Mathematics/AppliedMathematics/?view=usa&#38;ci=9780198508885

[7] K. Yee, Numerical solution of initial boundary value problems involvingMaxwell’s equations in isotropic media, Antennas and Propagation, IEEETransactions on 14 (3) (1966) 302–307. doi:10.1109/tap.1966.1138693.URL http://dx.doi.org/10.1109/tap.1966.1138693

[8] D. W. Prather, S. Shi, Formulation and application of the finite-difference time-domain method for the analysis of axially symmetric diffrac-tive optical elements, J. Opt. Soc. Am. A 16 (5) (1999) 1131–1142.doi:10.1364/josaa.16.001131.URL http://dx.doi.org/10.1364/josaa.16.001131

[9] A. Taflove, S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Third Edition, 3rd Edition, ArtechHouse, 2005.URL http://www.worldcat.org/isbn/1580538320

[10] B. H. Kleemann, A. Mitreiter, F. Wyrowski, Integral equation methodwith parametrization of grating profile theory and experiments, Journal ofModern Optics 43 (7) (1996) 1323–1349. doi:10.1080/09500349608232807.URL http://dx.doi.org/10.1080/09500349608232807

[11] D. W. Prather, M. S. Mirotznik, J. N. Mait, Boundary integral methodsapplied to the analysis of diffractive optical elements, J. Opt. Soc. Am. A14 (1) (1997) 34–43. doi:10.1364/josaa.14.000034.URL http://dx.doi.org/10.1364/josaa.14.000034

[12] P. Li, Coupling of finite element and boundary integral methods for elec-tromagnetic scattering in a two-layered medium, Journal of ComputationalPhysics 229 (2) (2010) 481–497. doi:10.1016/j.jcp.2009.09.040.URL http://dx.doi.org/10.1016/j.jcp.2009.09.040

[13] A. P. M. Zwamborn, P. M. van den Berg, A weak form of the conjugategradient FFT method for plate problems, Antennas and Propagation, IEEETransactions on 39 (2) (1991) 224–228. doi:10.1109/8.68186.URL http://dx.doi.org/10.1109/8.68186

21

[14] M. Botha, Solving the volume integral equations of electromagneticscattering, Journal of Computational Physics 218 (1) (2006) 141–158.doi:10.1016/j.jcp.2006.02.004.URL http://dx.doi.org/10.1016/j.jcp.2006.02.004

[15] Y.-C. Chang, G. Li, H. Chu, J. Opsal, Efficient finite-element, Green’sfunction approach for critical-dimension metrology of three-dimensionalgratings on multilayer films, J. Opt. Soc. Am. A 23 (3) (2006) 638–645.doi:10.1364/josaa.23.000638.URL http://dx.doi.org/10.1364/josaa.23.000638

[16] M. C. van Beurden, Fast convergence with spectral volume integralequation for crossed block-shaped gratings with improved material in-terface conditions, J. Opt. Soc. Am. A 28 (11) (2011) 2269–2278.doi:10.1364/josaa.28.002269.URL http://dx.doi.org/10.1364/josaa.28.002269

[17] A. A. Shcherbakov, A. V. Tishchenko, New fast and memory-sparingmethod for rigorous electromagnetic analysis of 2D periodic dielectric struc-tures, Journal of Quantitative Spectroscopy and Radiative Transfer 113 (2)(2012) 158–171. doi:10.1016/j.jqsrt.2011.09.019.URL http://dx.doi.org/10.1016/j.jqsrt.2011.09.019

[18] M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, Formula-tion for stable and efficient implementation of the rigorous coupled-waveanalysis of binary gratings, J. Opt. Soc. Am. A 12 (5) (1995) 1068–1076.doi:10.1364/josaa.12.001068.URL http://dx.doi.org/10.1364/josaa.12.001068

[19] K. Knop, Rigorous diffraction theory for transmission phase gratings withdeep rectangular grooves, J. Opt. Soc. Am. 68 (9) (1978) 1206–1210.doi:10.1364/josa.68.001206.URL http://dx.doi.org/10.1364/josa.68.001206

[20] M. G. Moharam, D. A. Pommet, E. B. Grann, T. K. Gaylord, Stable imple-mentation of the rigorous coupled-wave analysis for surface-relief gratings:enhanced transmittance matrix approach, J. Opt. Soc. Am. A 12 (5) (1995)1077–1086. doi:10.1364/josaa.12.001077.URL http://dx.doi.org/10.1364/josaa.12.001077

[21] L. Li, Formulation and comparison of two recursive matrix algorithms formodeling layered diffraction gratings, J. Opt. Soc. Am. A 13 (5) (1996)1024–1035. doi:10.1364/josaa.13.001024.URL http://dx.doi.org/10.1364/josaa.13.001024

[22] G. Granet, B. Guizal, Efficient implementation of the coupled-wave methodfor metallic lamellar gratings in TM polarization, J. Opt. Soc. Am. A 13 (5)(1996) 1019–1023. doi:10.1364/josaa.13.001019.URL http://dx.doi.org/10.1364/josaa.13.001019

22

[23] P. Lalanne, G. M. Morris, Highly improved convergence of the coupled-wavemethod for TM polarization, J. Opt. Soc. Am. A 13 (4) (1996) 779–784.doi:10.1364/josaa.13.000779.URL http://dx.doi.org/10.1364/josaa.13.000779

[24] L. Li, Use of Fourier series in the analysis of discontinuous pe-riodic structures, J. Opt. Soc. Am. A 13 (9) (1996) 1870–1876.doi:10.1364/josaa.13.001870.URL http://dx.doi.org/10.1364/josaa.13.001870

[25] G. Granet, Reformulation of the lamellar grating problem through the con-cept of adaptive spatial resolution, J. Opt. Soc. Am. A 16 (10) (1999) 2510–2516. doi:10.1364/josaa.16.002510.URL http://dx.doi.org/10.1364/josaa.16.002510

[26] E. Popov, M. Neviere, Grating theory: new equations in Fourier spaceleading to fast converging results for TM polarization, J. Opt. Soc. Am. A17 (10) (2000) 1773–1784. doi:10.1364/josaa.17.001773.URL http://dx.doi.org/10.1364/josaa.17.001773

[27] E. Popov, M. Neviere, Maxwell equations in Fourier space: fast-convergingformulation for diffraction by arbitrary shaped, periodic, anisotropic media,J. Opt. Soc. Am. A 18 (11) (2001) 2886–2894. doi:10.1364/josaa.18.002886.URL http://dx.doi.org/10.1364/josaa.18.002886

[28] T. Schuster, J. Ruoff, N. Kerwien, S. Rafler, W. Osten, Normal vectormethod for convergence improvement using the RCWA for crossed gratings,J. Opt. Soc. Am. A 24 (9) (2007) 2880–2890. doi:10.1364/josaa.24.002880.URL http://dx.doi.org/10.1364/josaa.24.002880

[29] J. J. Hench, Z. Strakos, The RCWA method - A case study with openquestions and perspectives of algebraic computations, Electronic Transac-tions on Numerical Analysis 31 (2008) 331–357.URL http://etna.mcs.kent.edu/vol.31.2008/pp331-357.dir/pp331-357.html

[30] M. Pisarenco, Scattering from finite structures: An extended Fouriermodal method, Ph.D. thesis, Eindhoven University of Technology (Sep.2011).URL http://www.narcis.nl/publication/RecordID/oai%3Alibrary.tue.nl%3A716275/coll/person/id/2

[31] M. Pisarenco, J. Maubach, I. Setija, R. Mattheij, Aperiodic Fouriermodal method in contrast-field formulation for simulation of scatteringfrom finite structures, J. Opt. Soc. Am. A 27 (11) (2010) 2423–2431.doi:10.1364/josaa.27.002423.URL http://dx.doi.org/10.1364/josaa.27.002423

[32] M. Pisarenco, J. Maubach, I. Setija, R. Mattheij, Modified S-matrix algo-rithm for the aperiodic Fourier modal method in contrast-field formulation,J. Opt. Soc. Am. A 28 (7) (2011) 1364–1371. doi:10.1364/josaa.28.001364.URL http://dx.doi.org/10.1364/josaa.28.001364

23

[33] M. Pisarenco, J. Maubach, I. Setija, R. Mattheij, A Numerical Method forthe Solution of Time-Harmonic Maxwell Equations for Two-DimensionalScatterers, AIP Conference Proceedings 1281 (1) (2010) 2049–2052.doi:10.1063/1.3498350.URL http://dx.doi.org/10.1063/1.3498350

[34] P. Lalanne, E. Silberstein, Fourier-modal methods applied to waveg-uide computational problems, Opt. Lett. 25 (15) (2000) 1092–1094.doi:10.1364/ol.25.001092.URL http://www.opticsinfobase.org/ol/abstract.cfm?&#38;uri=ol-25-15-1092

[35] E. Silberstein, P. Lalanne, J.-P. Hugonin, Q. Cao, Use of grating theo-ries in integrated optics, J. Opt. Soc. Am. A 18 (11) (2001) 2865–2875.doi:10.1364/josaa.18.002865.URL http://dx.doi.org/10.1364/josaa.18.002865

[36] Q. Cao, P. Lalanne, J.-P. Hugonin, Stable and efficient Bloch-mode compu-tational method for one-dimensional grating waveguides, J. Opt. Soc. Am.A 19 (2) (2002) 335–338. doi:10.1364/josaa.19.000335.URL http://dx.doi.org/10.1364/josaa.19.000335

[37] J. P. Hugonin, P. Lalanne, Perfectly matched layers as nonlinear coordinatetransforms: a generalized formalization, J. Opt. Soc. Am. A 22 (9) (2005)1844–1849.URL http://www.opticsinfobase.org/abstract.cfm?id=85042

[38] G. Lecamp, J. P. Hugonin, P. Lalanne, Theoretical and computational con-cepts for periodic optical waveguides, Opt. Express 15 (18) (2007) 11042–11060. doi:10.1364/oe.15.011042.URL http://dx.doi.org/10.1364/oe.15.011042

[39] J.-P. Berenger, A perfectly matched layer for the absorption of electromag-netic waves, Journal of Computational Physics 114 (2) (1994) 185–200.doi:10.1006/jcph.1994.1159.URL http://dx.doi.org/10.1006/jcph.1994.1159

[40] M. Pisarenco, J. M. L. Maubach, I. D. Setija, R. M. M. Mattheij, Effi-cient solution of Maxwell’s equations for geometries with repeating pat-terns by an exchange of discretization directions in the aperiodic Fouriermodal method, Journal of Computational Physics 231 (24) (2012) 8209–8228. doi:10.1016/j.jcp.2012.07.049.URL http://dx.doi.org/10.1016/j.jcp.2012.07.049

[41] G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins Stud-ies in Mathematical Sciences)(3rd Edition), 3rd Edition, The Johns Hop-kins University Press, 1996.URL http://www.worldcat.org/isbn/0801854148

24

[42] R. Redheffer, Difference equations and functional equations intransmission-line theory, McGraw-Hill, New York, 1961, Ch. 12, pp. 282–337.

[43] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, D. E. Knuth, Onthe Lambert W function, Advances in Computational Mathematics 5 (1)(1996) 329–359. doi:10.1007/bf02124750.URL http://dx.doi.org/10.1007/bf02124750

[44] B. Hayes, Why W?, American Scientist 93 (2) (2005) 104+.doi:10.1511/2005.2.104.URL http://dx.doi.org/10.1511/2005.2.104

[45] N. Engheta, R. W. Ziolkowski (Eds.), Electromagnetic Metamaterials:Physics and Engineering Explorations, 1st Edition, Wiley-IEEE Press,2006.URL http://www.worldcat.org/isbn/0471761028

[46] J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal,X. Zhang, Three-dimensional optical metamaterial with a negative refrac-tive index, Nature 455 (7211) (2008) 376–379. doi:10.1038/nature07247.URL http://dx.doi.org/10.1038/nature07247

[47] S. Guenneau, R. Craster, T. Antonakakis, K. Cherednichenko,S. Cooper, Homogenization Techniques for Periodic Structures (Apr. 2013).arXiv:1304.7519.URL http://arxiv.org/abs/1304.7519

[48] X. Zhi-Jie, Homogenization and Upscaling for Diffusion, Heat Conduction,and Wave Propagation in Heterogeneous Materials, Communications inTheoretical Physics 57 (3) (2012) 348+. doi:10.1088/0253-6102/57/3/04.URL http://dx.doi.org/10.1088/0253-6102/57/3/04

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