fast convergence with spectral volume integral equation for crossed block-shaped gratings with...

Fast convergence with spectral volume integral equationfor crossed block-shaped gratings with improved

material interface conditions

Martijn C. van Beurden

Faculty of Electrical Engineering, Eindhoven University of Technology,Den Dolech 2, 5612 AZ Eindhoven, The Netherlands ([email protected])

Received April 27, 2011; revised August 26, 2011; accepted September 12, 2011;posted September 13, 2011 (Doc. ID 146148); published October 12, 2011

For block-shaped dielectric gratings with two-dimensional periodicity, a spectral-domain volume integralequation is derived in which explicit Fourier factorization rules are employed. The Fourier factorization rulesare derived from a projection-operator framework and enhance the numerical accuracy of themethod, while main-taining a low computational complexity of OðN logNÞ or better and a low memory demand of OðNÞ. © 2011Optical Society of America

OCIS codes: 050.1755, 050.1950.

1. INTRODUCTIONElectromagnetic scattering by periodic structures has beenstudied since the early twentieth century. The ability of actu-ally constructing such structures from radio to opticalwavelengths has dramatically increased the interest in theirdesign and analysis during the past two decades. Applicationsrange from absorbing wedges in anechoic chambers at radiofrequencies [1], to electromagnetic bandgap structures to en-hance antenna performance [2–4] and phase gratings used asmetrology targets for monitoring IC manufacturing [5].

Many techniques have been developed to analyze periodicscattering problems, ranging from finite-element and finite-difference methods, to modal techniques and integral equa-tions. One of the most popular and successful techniques forgrating applications is the rigorous coupled wave analysis(RCWA) [6–8], also known as the Fourier modal method.RCWA is a very flexible technique because of its modularapproach. Its rapid numerical convergence for low dielectriccontrast has resulted in a very effective approach for periodicstructures with one-dimensional (1D) periodicity. However,for two-dimensional (2D) periodicity, the computationaladvantages are no longer obvious, due to the solution of aneigenvalue problem that scales with the sixth power of thenumber of Fourier modes per periodic dimension. Moreover,the memory advantage that goes hand in hand with the mod-ular approach is losing its momentum for 2D periodicity.

Here, we focus on a formulation based on a volume integralequation (VIE), to approach the scattering problem involvingdielectric objects with 2D periodicity. The three-dimensional(3D) nature of the formulation does not provide an immedi-ately obvious point of departure to overcome the obstaclesencountered by RCWA. Volume integral equations becomeexpedient when the discretized numerical system yields anefficient computational structure. An early development wasthe conjugate-gradient fast Fourier transform (CGFFT) meth-od [9,10] that led toOðNÞmemory requirements and high com-putational efficiency for the matrix–vector product of

OðN logNÞ via FFTs. In a spatial formulation, the uniformityof the grid avoids a time-consuming meshing procedure,thereby making the preprocessing stage typically very short.At the same time, the grid uniformity is also the largest draw-back and subsequent developments, like the adaptive integralmethod [11] and the multilevel fast multipole method (MLFMA[12]), were directed toward the capability of handling generalmeshes while retaining the computation efficiency. Thesemethods aim at efficient representations for the Green’s func-tion, e.g., the MLFMA and its variants employ rapidly conver-ging series of product kernels. In the case of periodic mediaand a spatial discretization as in [13], the infinite series repre-sentations in either the spatial or the spectral domain forthe Green’s function require further attention since the seriesconverge notoriously slowly and the Ewald transformationcan be applied to accelerate the convergence of the series;see, e.g., [14]. Nevertheless, the case of periodic scatterersin layered media is not fully handled by the Ewald transforma-tion and additional work is needed to arrive at efficient repre-sentations everywhere [15].

In the periodic spectral domain, the situation is ratherdifferent for VIEs, in the sense that the Green’s function repre-sentation is naturally truncated by the number of modes takeninto account and the Green’s function only acts per Fouriermode, which automatically translates into a computationaladvantage [3–5,16,17] and the computational load regardingthe Green’s function can be reduced to OðNÞ by exploitingthe semiseparability of the Green’s function per Fourier mode.On the other hand, the field–material interactions, which areof OðNÞ in the spatial domain, become an important computa-tional factor, especially for 3D problems. A straightforwardtransformation of the spatial field–material interaction to thespectral domain gives rise to discrete convolutions that can behandled with FFTs. Unfortunately, this approach suffers frompoor convergence in the computed results as a function ofthe number of Fourier modes when medium discontinuitiesappear in the transverse plane. The situation is then similar

Martijn C. van Beurden Vol. 28, No. 11 / November 2011 / J. Opt. Soc. Am. A 2269

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to that for RCWA before the introduction of the inverse rule,discovered by Lalanne, and subsequent systematic develop-ments by Li that led to the Fourier factorization rules[7,8,18,19]. The direct application of the inverse rule to theVIE has a detrimental effect on the computational efficiencyof the field–material interactions and, therefore, on the entirecomputational efficiency of the method since it does not leadto a representation in the form of a convolution acting on aFourier basis. It is the aim of the present paper to demonstratethe effectiveness of reformulated Fourier factorization rulesfor block-shaped gratings within a spectral formulation ofthe VIE. These factorization rules maintain the low computa-tional complexity of a matrix–vector product in the form ofdiscrete convolutions that can be efficiently handled by 1Dand 2D FFTs.

This paper is organized as follows. In Section 2, boththe Green’s function and the field–material interactions arediscussed, followed by an outline of the employed verticaldiscretization scheme in Section 3. Section 4 explains howto compute scattering coefficients, such as a reflection coeffi-cient, from the proposed discretization. Section 5 providesnumerical examples to demonstrate the performance and ac-curacy obtained by employing a reformulation of the Fourierfactorization rules within the context of the VIE, as well asto show the numerical efficiency of the proposed method.Finally, conclusions are drawn in Section 6.

2. FORMULATIONThe VIE consists of a set of two equations. The first one isthe integral representation that describes the total electricfield in terms of the incident field and contrast current density,where the latter is convolved with the Green’s function toyield the scattered field. The second equation is a relation be-tween the total electric field and the contrast current density,which is essentially a constitutive relation defined by thematerials present in the configuration. The relation betweenthe total electric field and the contrast current density isderived from Maxwell’s equations and the notion of a back-ground configuration. The choice of the background is relatedto the ability of finding the Green’s function for this back-ground. Hence, typically, the background is a simplified con-figuration, such as a planarly stratified medium. Here, we willassume that the stratification takes place in the z directionand consists of materials with constant permeability through-out and, therefore, variation only in the permittivity.

In the absence of primary sources, the Ampere–Maxwellequation with expðjωtÞ time convention reads

∇ ×H ¼ jωD ¼ jωεE; ð1Þ

where H is the magnetic field strength, ω is the angularfrequency, D is the generalized electric flux density, whichincludes Ohmic current densities, ε is the generalized permit-tivity including Ohmic losses, and E is the electric fieldstrength. The permittivity of the background medium, whichis used in the definition of the Green’s function, is indicated byεb and the contrast current density J is then defined throughthe equation

∇ ×H ¼ jωðε − εbÞEþ jωεbE ¼ J þ jωεbE: ð2Þ

A. Integral RepresentationFor a periodic setup with 2D periodicity, we can define aunit cell spanned by the vectors a1 and a2 and a correspondingBravais lattice that indicates the periodicity. We assume thatthe 2D periodicity lies in the transverse x–y plane, orthogonalto the direction of the stratification (z) of the backgroundmedium. Further, we define reciprocal vectors b1 and b2, suchthat ai · bj ¼ δi;j , the Kronecker delta. To obtain the integralrepresentation of the total electric field, we first introducethe relations between the vector fields in the spatial andthe spectral domain, i.e.,

Eðx; y; zÞ ¼X∞

m1¼−∞

X∞m2¼−∞

eðm1;m2; zÞ expð−jkmT · rT Þ;

Hðx; y; zÞ ¼X∞

m1¼−∞

X∞m2¼−∞

hðm1;m2; zÞ expð−jkmT · rT Þ;

Jðx; y; zÞ ¼X∞

m1¼−∞

X∞m2¼−∞

jðm1;m2; zÞ expð−jkmT · rT Þ; ð3Þ

where rT ¼ xux þ yuy and kmT ¼ kiT þ 2πm1b1 þ 2πm2b2.Further, kiT is the transverse part of the wave vector of theincident plane wave. Since Maxwell’s equations for stratifiedmedia diagonalize with respect to the spectral decomposition,Maxwell’s equations become an infinite set of equations perspectral component, i.e.,

− jkmT × hðm1;m2; zÞ þ uz ×d

dzhðm1;m2; zÞ

¼ jðm1;m2; zÞ þ jωε0εrðzÞeðm1;m2; zÞ;

− jkmT × eðm1;m2; zÞ þ uz ×d

dzeðm1;m2; zÞ

¼ −jωμ0hðm1;m2; zÞ; ð4Þ

for each m1 and m2. Subsequently, following Felsen andMarcuvitz [20] (see also [21]), both these equations and thefields are separated into their transverse and longitudinalparts, and the transverse electromagnetic fields are furtherdecomposed as

eT ¼ jukve − jðuz × ukÞvh; hT ¼ jukih þ jðuz × ukÞie; ð5Þ

where we have suppressed the dependency of all quantities onðm1;m2; zÞ. Further, uk ¼ kmT =∥k

mT ∥. The final results are the

transmission-line equations, two decoupled sets of ordinarydifferential equations:

jωεðzÞ d

dzve ¼ −γ2ie þ kT jz; ð6Þ

d

dzie ¼ −jωεðzÞve þ juk · jT ; ð7Þ

d

dzvh ¼ −jωμ0ih; ð8Þ

jωμ0d

dzih ¼ −γ2vh þ ωμ0ðuz × ukÞ · jT ; ð9Þ

2270 J. Opt. Soc. Am. A / Vol. 28, No. 11 / November 2011 Martijn C. van Beurden

where kT ¼ ∥kmT ∥ and γ2 ¼ k2T − ω2εðzÞμ0. From the abovedecomposition, the longitudinal components of the electro-magnetic field are given by

ez ¼kT

jωεðzÞ ie −

1jωε jz; ð10Þ

hz ¼kT

jωμ0vh: ð11Þ

Hence, the spectral electric field can be expressed as

e ¼ jukve − jðuz × ukÞvh þ�

kT

jωεðzÞ ie −

1jωεðzÞ jz

�uz: ð12Þ

We are now in a position to define the dyadic Green’s func-tion. We define the up component of the spectral dyadicGreen’s function, i.e., ��Gðm1;m2; z; z

0Þ · up, as the spectral elec-tric field eðm1;m2; zÞ due to a up oriented spectral currentdensity with a unit-amplitude delta distribution in the z direc-tion, i.e., jðm1;m2; z; z

0Þ ¼ δðz − z0Þup. This definition is sup-plemented with the radiation conditions for jzj → ∞, whichmeans that the Green’s function contains only the outgoingwaves due to the source j. The incoming waves, includingthe waves directly scattered by the background, are indicatedby the superscript i, e.g., eiðm1;m2; zÞ. From these definitions,the total electric field eðm1;m2; zÞ per spectral component canbe written as

eðm1;m2; zÞ ¼ eiðm1;m2; zÞ

þZz0∈R

��Gðm1;m2; z; z0Þjðm1;m2; z

0Þdz0; ð13Þ

which holds for all m1, m2, and z. The construction of theGreen’s function requires the solution of the transmission-lineequations, given above. For a homogeneous medium with per-mittivity ε1, the dyadic Green’s function for the electric field isobtained as

��Gh ¼�−kTkT − jkTuz

d

dz− juzkT

d

dz

þ uzuzd2

dz2

�1

2jγωε1expð−γjz − z0jÞ; ð14Þ

where the concatenated vectors are understood as dyads.Further, the square root for γ is taken according to thestandard branch cut, to satisfy the radiation conditions. Forisotropic piecewise-homogeneous media, the solutions can befound via the standard scattering-matrix formalism, e.g., [22].For E-polarized waves in a 2D scattering setup with 1Dperiodicity, Eqs. (13) and (14) reduce to Eq. (6) of [16].

B. Field–Material Interactions: Implication of theFourier Factorization RulesLet us consider a configuration for which the backgroundmedium at a fixed z position is constant and the contrast func-tion is a nonzero continuous function, e.g., a constant, on arectangular domain within a rectangular unit cell of a periodicconfiguration in the x–y plane. Further, we assume that thematerials are isotropic and that the rectangular domain has

the same orientation as the unit cell. The unit cell is orientedalong the x and y directions and has dimensions ∥a1∥ and∥a2∥, respectively. The top and side views of the situationare depicted in Fig. 1.

To define the shape of the support of χ, we introduce

ΠΔðxÞ ¼�1 x ∈ ½−Δ=2;Δ=2�0 elsewhere

: ð15Þ

At a fixed position along the z direction, we can nowdefine the permittivity with respect to the background and itsinverse as

ε=εb ¼ 1þ χcΠΔxðx − x0ÞΠΔyðy − y0Þ; ð16Þ

ðε=εbÞ−1 ¼ 1þ χ̂cΠΔxðx − x0ÞΠΔyðy − y0Þ; ð17Þ

where χc is a continuous function or a constant, ðx0; y0Þ is thecenter position of the rectangular support, and

χ̂c ¼ −χc

1þ χc; ð18Þ

which follows from the condition ε · ε−1 ¼ 1.Further, the relation between the x components of the

electric flux and the electric field is

Dx ¼ jωεb½1þ χcΠΔxðx − x0ÞΠΔyðy − y0Þ�Ex: ð19Þ

Unfortunately, the spectral-domain counterpart of this for-mula, in the form of a convolution between the permittivityand the electric field, yields a poor approximation for thecontinuous component of the flux density in terms of a finiteFourier series. The remedy for this poor approximation wasfirst put forward by Lalanne and Morris [18] in terms of theinverse rule for the 2D TM case. The more general case wassystematically worked out by Li [7,8] in the form of Fourierfactorization rules. The Fourier factorization rules indicatethat the product of a continuous function and a discontinuousfunction in the spatial domain can be factorized in the finitespectral basis by a straightforward convolution, i.e., theLaurent rule. Unfortunately, the product of two discontinuousfunctions in the spatial domain cannot be factorized in afinite spectral basis in the form of a convolution. However,if the jumps in the two functions are complementary, i.e.,the product of the discontinuous functions is continuous, thenthe inverse rule can be applied to obtain a factorization in afinite spectral basis.

Fig. 1. Side and top views of the unit cell with a block-shaped gratingon top of a layered medium.


In the case of a VIE, the fundamental unknown is typicallythe electric field, and the contrast current density can be de-rived from the product between the contrast function ðε − εbÞand the electric field. However, both the contrast function andthe electric field have discontinuities and these are not com-plementary. To express the contrast current density in termsof the electric field, we rewrite the contrast current density as

J ¼ jωðε − εbÞE ¼ jωðD − εbEÞ; ð20Þ

where the background permittivity εb is constant along theinterval in z that covers the height of the grating. This formulaallows us to exploit the continuity of the flux field D in thenormal direction, to arrive at Fourier factorization rules forthe contrast current density.

We now follow Li’s line of reasoning to reformulate theabove equation such that it will be suitable for approximationin the spectral domain. The starting point is to select the fieldcomponents that are continuous along one of the Cartesiandirections. For a fixed y position, the x component of the elec-tric flux density, observed along the x direction, is continuous,as is the y component of the electric field. Further, the jumpdiscontinuities of the x component of the electric field and they component of the flux density are the complement of thejumps in the permittivity and inverse permittivity functionfor fixed y and z.

We choose to start from the relation

jωεbEx ¼ ½1þ χ̂cΠΔxðx − x0ÞΠΔyðy − y0Þ�Dx; ð21Þ

where Dx is continuous in the x direction and, therefore, themultiplication by ΠΔx can be directly replaced by its spectral-domain counterpart, which yields a 1D convolution along thex direction. However, in the y direction, we observe comple-mentary jumps in the above formula. Instead of directly apply-ing the Laurent and inverse rules in the spectral domain, wechoose to remain in the spatial domain and derive the factor-ization rules there. This allows us to further explain the natureof the Fourier factorization rules for piecewise constant per-mittivity profiles in terms of spatial projection operators.

Since the multiplication by a pulse functionΠΔy is a spatialprojection operator, we can construct its inverse using theidempotency property and the fact thatΠΔx acts only as a con-stant multiplier in the y direction and commutes withΠΔy. Forsuch a case, the inverse of I þ AΠΔy is of the form I þ BΠΔy,where B follows from the algebraic property

ðI þ BΠΔyÞðI þ AΠΔyÞ ¼ I þ ðBþ Aþ BAÞΠΔy; ð22Þ

from which it follows that ðBþ Aþ BAÞ ¼ 0, i.e.,B ¼ −AðI þ AÞ−1, and I denotes the identity operator.

With these notations and following the above line of reason-ing, we arrive at

jωεb½I − χ̂cΠΔxΠΔyðI þ χ̂cΠΔxÞ−1�Ex ¼ Dx; ð23Þ

owing to the fact that ΠΔx and ΠΔy act on orthogonal direc-tions and, therefore, the operator ðI þ χ̂cΠΔxÞ−1 commuteswith ΠΔy. We note that Eqs. (19) and (23) are identical inthe spatial domain. In the spectral domain, these formulascan only be identical if the projection property of idempo-tency is maintained, which occurs only in the infinite Fourier

basis. This explains why these formulations give differentresults in a finite spectral basis. Further, we observe that theoperator ðI þ χ̂cΠΔxÞ−1 can be interpreted as an operator thatlifts the x component of the electric field on the inside ofthe block-shaped grating to the level outside, thereby makingthe product ðI þ χ̂cΠΔxÞ−1Ex continuous at the material inter-face. The subsequent multiplication by ΠΔy prevents an arti-ficially introduced discontinuity in the product away from thematerial interface from appearing outside the support of theblock-shaped grating.

At this point, the multiplication operators ΠΔx and ΠΔy

are replaced by their spectral-domain counterparts in termsof the finite Laurent rule (discrete convolution), which we de-note by Px and Py. Further, the inverse operator ðI þ χ̂cΠΔxÞ−1is replaced by its spectral representative in the form of theinverse rule. This brings us to the following relation betweenthe electric flux density and the electric field, both expressedin terms of a finite Fourier series:

dx ¼ jωεb½I − χ̂cPxPyðI þ χ̂cPxÞ−1�ex; ð24Þ

dy ¼ jωεb½I − χ̂cPxPyðI þ χ̂cPyÞ−1�ey; ð25Þ

dz ¼ jωεbχcPxPyez; ð26Þ

where the z component follows from the observation that thez component of the electric field is continuous in the x–y

plane owing to the shape of the grating.For the VIE, we are interested only in the total electric field

e and the normalized contrast current density j. For the latterwe have

jx ¼ dx − jωεbex ¼ jωεbχcPxPy

�1

1þ χcðI þ χ̂cPxÞ−1

�ex;

jy ¼ dy − jωεbey ¼ jωεbχcPxPy

�1

1þ χcðI þ χ̂cPyÞ−1

�ey;

jz ¼ dz − jωεbez ¼ jωεbχcPxPyez;

where we have employed the relation between χ̂c andχc. These equations should be compared to the classicalapproach, in which the factorization rules are ignored asin Eq. (19). This leads to the classical spectral-domaincounterpart:

jx ¼ jωεbχcPxPyex; ð27Þ

jy ¼ jωεbχcPxPyey; ð28Þ

jz ¼ jωεbχcPxPyez: ð29Þ

C. Explicit Factorization RulesWhen an iterative solver is used to solve the set of VIEs, thepresence of the inverse operators ðI þ χ̂cPxÞ−1 and ðI þχ̂cPyÞ−1 in Eq. (27), in which j is obtained from e, leads toa serious increase in the computational complexity of thematrix–vector product of the overall VIEs. Therefore, it is


highly desirable to avoid these inverses, without sacrificingthe increased accuracy that they bring. Therefore, we proposeto introduce an artificial vector field in the spectral domain,denoted f , that has the following relation to e:

ex ¼ ð1þ χ̂cÞðI þ χ̂cPxÞf x; ð30Þ

ey ¼ ð1þ χ̂cÞðI þ χ̂cPyÞf y; ð31Þ

ez ¼ f z: ð32Þ

These additional equations are 1D convolutions that canbe implemented via 1D FFTs along the x direction for thefirst equation and along the y direction for the second equa-tion, once f is accepted as the fundamental unknown. Theoperations in the above set of equations are summarized inthe operator L, i.e.,

eðm1;m2; zÞ ¼ ½Lf �ðm1;m2; zÞ: ð33Þ

Then, the relation between j and f becomes

j ¼ jωεbχcPxPyf ¼ Mf ; ð34Þ

which can be implemented via 2D FFTs in the x–y planeand avoids the execution of the inverse rule. To distinguishthis formulation from the one in Eq. (27), we will refer to thesystem in Eqs. (33) and (34) as the explicit factorization rules.Now both the relation between e and f and the relation be-tween j and f contain only convolution operators accordingto the finite Laurent rule, which makes their matrix–vectorproducts efficient, since they can be implemented by (a com-bination of) 1D and 2D FFTs.

In summary, we solve the VIE for the vector field f insteadof for e and, via an additional postprocessing step, we obtainthe electric field e from Eq. (30). The change in this procedurecompared to the original one is that we now require two field–material interaction operations (i.e., one for e and one for j)instead of one (only for j), and a postprocessing step to obtaine from f . However, these two operations in the new procedurehave a much more efficient implementation than the singleone in the procedure with the inverse rule as in Eq. (27).Hence, we solve a set of VIEs of the form

eiðm1;m2; zÞ ¼ ½Lf �ðm1;m2; zÞ

−

Zz0∈R

��Gðm1;m2; z; z0Þ½Mf �ðm1;m2; z

0Þdz0:ð35Þ

Also note that, when the operator L in Eq. (13) is replaced bythe identity operator, the classical VIE is obtained, in whichthe factorization rules are ignored.

3. DISCRETIZATION IN z FOR A BINARYGRATINGLet us now assume that the contrast source is restricted toa single layer of the stratified medium and that this layeroccupies the interval z ∈ ½0; h�. The functional behavior of the

Green’s function in the z direction, restricted to the layer inwhich the contrast source resides, can be summarized as

��Gðz; z0Þ ¼ ��Ghðjz − z0jÞ þ ��Ru exp½−γðzþ z0Þ�þ ��Rl exp½γðzþ z0 − 2hÞ�; ð36Þ

which contains a convolution term due to the presence of thesource in the layer. This term is the Green’s function for ahomogeneous medium with the same properties as the layer,as discussed in Subsection 2.A. The other two terms representthe reflection of the mode at the superstrate and substrate,respectively, and have the form of a product kernel.

To arrive at a numerical scheme for the integral equationsper mode in the z direction, we employ a discretization that isidentical for all modes. We employ a uniform mesh over thesupport of the contrast source in the z direction and thebehavior of the contrast source is approximated by piecewiselinear functions on its support. We assume that there are nochanges in the medium properties along the support of thecontrast source. Consequently, the assumption of continuityalong the z direction of the contrast source is valid, as all elec-tric field and flux components are continuous along this inter-val. Subsequently, a collocation scheme at the same meshpoints is used to arrive at a square system of linear equations.The uzuz component of the dyadic Green’s function contains adelta distribution, for which the collocation scheme is ambig-uous at the end points of the interval ½0; h�. For these points,we take the collocation points as the limit from inside the in-terval toward the end points, which results in a unique limit.The merit of the collocation scheme is that it avoids the gen-eration of a tridiagonal matrix for the identity operator for theelectric field, in the case of a Galerkin procedure with bilinearfunctions. This tridiagonal matrix often deteriorates the con-vergence of an iterative solver and can be overcome by anextra preconditioning step as in [5] or a lumping procedureas in [3,4].

Owing to the uniform mesh for the discretization, the con-volution structure of the Green’s function is retained in thediscrete system. This allows for an FFT implementation tocompute the interaction between the Green’s function andthe contrast source, except for the contribution of the con-trast currents at the end points of the interval. Alternatively,the semiseparability [23] of the 1D kernel, i.e.,

��Ghðjz − z0jÞ ¼(��g1ðzÞ��g2ðz0Þ z < z0

��g3ðzÞ��g4ðz0Þ z > z0; ð37Þ

can be used to build two update equations: one integratingfrom bottom to top and one from top to bottom, as, e.g., dis-cussed in [16]. The integrals that result from both reflectionterms can be computed by reusing the result of the discreteconvolution at the top and bottom of the integration interval,owing to the separability of product kernels.

A. Computational ComplexityTo express the computational complexity of the entire algo-rithm, we denote the number of Fourier modes in the x andy direction by Mx and My, respectively, and the number ofsample points in the z direction by Nz. For the Green’s func-tion representation, the above discussion yields a complexity


of OðNz logNzÞ per Fourier mode for the discrete convolutionalgorithm and a complexity of OðNzÞ per Fourier mode whenthe semiseparability of the kernel is used. Hence the complex-ity regarding the Green’s function is OðMxMyNz logNzÞ orOðMxMyNzÞ.

For the field–material interactions, the operator M has acomplexity dominated by 2D FFTs per sample point, whichyields OðMxMyNz logðMxMyÞÞ. Finally, the operator L has acomputational complexity of repeated 1D FFTs in one ofthe modal directions, per sample point and per Fourier modein the other direction. This corresponds to a complexity ofOðMxMyNz logðMxÞ and OðMxMyNz logðMyÞ in the x and y

directions, respectively, and, hence, a total complexity ofOðMxMyNz logðMxMyÞÞ.

The complexity of the total VIE, the contributions of theGreen’s function, and the field–material interactions lead toan asymptotic complexity of OðMxMyNz logðMxMyNzÞÞ whenthe discrete convolution for the Green’s function is used orOðMxMyNz logðMxMyÞÞ when the semiseparability of theGreen’s function is used. By denoting the total number ofunknowns by N ¼ MxMyNz, we arrive at a complexity esti-mate of OðN logNÞ or better.

4. COMPUTING THE REFLECTIONCOEFFICIENTThe algorithm outlined above is aimed at computing a solutionfor the spectral variable f ðm1;m2; zÞ, instead of eðm1;m2; zÞor jðm1;m2; zÞ. However, this variable does not directly yieldthe most interesting parameters of the scattering problem, i.e.,the reflection or transmission coefficients of the structure.The reflection coefficients, for example, can be obtainedin two ways: via the contrast current density relationjðm1;m2; zÞ ¼ Mf ðm1;m2; zÞ in combination with the integralrepresentation in Eq. (13), as proposed in [5], or via theelectric field at the top of the grating and the relationeðm1;m2; zÞ ¼ Lf ðm1;m2; zÞ, similar to the approach inRCWA. In both cases, an additional postprocessing step isneeded to obtain the contrast current or the electric field from

the artificial field f . This postprocessing step has the compu-tational complexity ofMf or Lf , which are composed of multi-plications and 1D or 2D FFTs. However, the route for theelectric field is more efficient, since it requires no additionalintegration in the z direction, as is the case for the integralrepresentation. Moreover, the second route requires only thetransverse components of the electric field at the top or bot-tom of the grating, whereas the contrast current density alongthe entire grating is needed in the former case.

Once we have obtained the transverse electric fields at thetop of the grating for each Fourier mode from Lf , we use thesame line of reasoning as in RCWA to obtain the electric fieldjust above the grating, i.e., we employ the continuity of thetangential electric field. The tangential electric field can thenbe propagated per Fourier mode to the top material interfaceand there we obtain the longitudinal component of the electricfield per Fourier mode via the divergence of the electric fluxdensity.

5. NUMERICAL RESULTSAll the computations discussed below have been per-formed using BiCGstab(2) [24,25] to compute the solutionof the linear system with a tolerance of 10−7, without anypreconditioner. The computer was a Pentium M single-corelaptop operating at a clock frequency of 1:86GHz and with1Gbyte RAM. The mentioned computation times concernthe total time required for setting up and solving the linearsystem, as well as computing the reflection or transmissioncoefficients.

As a first example, we consider an array of square air holes,as shown in Fig. 2(a), with edge length Δx ¼ Δy ¼ 500nmand period of 1000 nm in the x and y directions. The holesare embedded in a metal layer with relative permittivity0:8125 − j5:25 and thickness h ¼ 50nm on a half-space withrelative permittivity εr ¼ 2:25. The employed wavelength λis 500 nm and the plane wave is normally incident and polar-ized along the diagonal of the array, Ei

x ¼ Eiy. This setup has

been characterized using RCWA in [26]. The diffraction

10 12 14 16 18 20 22 24 260.215

0.22

0.225

0.23

0.235

# Fourier modes per direction

DE

0,0

VIE Fourier factorizationSchuster et al.VIE classical

Fig. 2. (a) Top and side views of the array of holes and (b) the corresponding convergence of the zeroth-order diffraction efficiency for an array ofrectangular holes compared to the RCWA reference solution [26] versus the number of Fourier modes per direction.


efficiency is plotted in Fig. 2(b) for an increasing number ofFourier modes in the x and y directions and 32 samples in z,by using both the classic [L in Eq. (35) is replaced by the iden-tity operator] and the proposed VIE formulation, where L isdefined by Eq. (30). In both cases, the number of matrix–vector products required to solve the linear system variesbetween 36 and 46. As can be seen, the results obtained withthe Fourier factorization method combined with the volumeintegral method accurately follow the published results,whereas the results without the Fourier factorization are notas accurate. This effect corresponds to similar observationsfor RCWA with and without taking into account the Fourierfactorization rules.

To further examine the convergence of the scheme pre-sented above, we consider a biperiodic grating consistingof pillars, similar to the one given in Fig. 1. The pillars havea relative permittivity εr ¼ 2:32 and are embedded in avacuum upper half-space. The lower half-space has relativepermittivity εr ¼ 18:4 − j0:403, e.g., silicon at a wavelengthof 500 nm, and a thin dielectric layer (εr ¼ 2:47 and thickness0:18λ) is placed on top of the lower half-space. The dimen-sions of the unit cell are λ × λ, where λ denotes the wavelengthof the incident plane wave in vacuum. The grating structure

has a square footprint with an edge length of 0:15λ and theheight of the pillar is 0:436λ. The incident plane wave origi-nates from the upper half-space at incident angles θ ¼ 8:13°and ϕ ¼ 45°, with parallel polarization.

As a reference solution, reflection coefficients have beencomputed via an independent RCWA calculation, includingthe Fourier factorization rules, by employing 25 Fouriermodes per direction (from −12 to þ12). The obtained zeroth-order parallel reflection coefficient has the numerical value−9:8376839 · 10−3 þ j3:0279232 · 10−1 and both first-orderparallel reflection coefficients have the numerical value7:2092448 · 10−3 þ j1:02461116 · 10−2. The advantage of com-paring to an RCWA solution is that a direct comparison withrespect to the number of Fourier modes in the transverseplane is possible and the dependence on the discretizationin the z direction can be studied independently, since RCWAemploys exact solutions in the z direction.

First, we consider the convergence of the classical spectral-domain volume integral formulation, in which the Fourier fac-torization rules have not been taken into account and which isthe formulation where L is the identity operator. The relativeerror in the first-order parallel reflection coefficient is shownin Fig. 3(a) for an increasing number of Fourier modes per

0 5 10 15 20 2510

−2

10−1

100

rela

tive

err

or

number of Fourier modes per direction

48163264

0 5 10 15 20 2510

−4

10−3

10−2

10−1

100

rela

tive

err

or


48163264

Fig. 3. Convergence of the first-order parallel reflection coefficient obtained via (a) the classical spectral volume integral formulation and (b) thespectral volume integral formulation with explicit Fourier factorization rules for an increasing number of Fourier modes, compared to the RCWAreference solution. The number of samples in the z direction is indicated in the legend.

0 20 40 60 80 10010

−7

10−6

10−5

10−4

10−3

10−2


rela

tive

err

or

explicit FFclassic

0 20 40 60 80 10010

−7

10−6

10−5

10−4

10−3

10−2


rela

tive

err

or

explicit FFclassic

Fig. 4. Self-consistent relative error in the (a) zeroth-order and (b) first-order parallel reflection coefficients obtained via the spectral volumeintegral formulation with explicit Fourier factorization rules and via the classical volume integral formulation, when the number of samples in the zdirection is given by Eq. (38).


periodic direction and for an increasing number of samplingpoints in the z direction. As can be observed from this figure,the convergence of the reflection coefficients is very slow as afunction of the number of Fourier modes. Moreover, the con-vergence is only mildly influenced by increasing the numberof sampling points in the z direction. This observation is inagreement with the statement made in [5] that the numberof expansion functions in the z direction has hardly anyimpact on the accuracy beyond 15 points per wavelength.We continue by considering the formulation of Eq. (35), whereL now incorporates the effect of the explicit Fourier factori-zation rules and we solve for the artificial vector field f .Subsequently, we postprocess this vector field with L to ob-tain the electric field and we extract again the first-orderreflection coefficient. The convergence of this reflection coef-ficient is shown in Fig. 3(b) for the same set of discretizations.We clearly observe a much better convergence and we ob-serve that a relation between the number of samples in thez direction and the number of Fourier modes per periodic di-rection exists for optimal convergence. This is, of course, notsurprising, since the resolution in each direction needs to be

refined to obtain more accurate results. At this point, it is clearthat Fourier factorizations rules play an important role in pro-ducing numerically accurate results for field–material interac-tions with jumps, even in the case where these jumps are notcomplementary, such as in this case of e and j. For bothformulations, we note that we need 18 to 20 matrix–vectorproducts to reach convergence, independent of the size ofthe system.

Subsequently, we consider the convergence of the reflec-tion coefficients for both the classical formulation and theformulation with explicit Fourier factorization, but now withrespect to their own reference for a large number of Fouriermodes and samples in the z direction. In both cases, the num-ber of samples in the z direction (Nz) is related to the totalnumber of Fourier modes per direction (M) as

Nz ¼�

2Mh

maxf∥a1∥; ∥a2∥g�: ð38Þ

This relation indicates a similar resolution in the basis ofFourier modes and the samples in the z direction. For thesame physical setup, the convergence for the zeroth- and

103

104

105

106

107

100

101

102

103

# unknowns

pea

k m

emo

ry u

sag

e [M

B]

103

104

105

106

107

10−2

10−1

100

101

102

103

# unknowns

cpu

tim

e [s

]

Fig. 5. (a) Peak-memory usage and (b) total CPU time versus the total number of unknowns, when the number of samples in the z direction isgiven by Eq. (38). The dashed curves indicate the lines (a) 3 · 10−6N logN and (b) 3 · 10−4N .

10 11 12 13 14 150.88

0.9

0.92

0.94

0.96

0.98

1

frequency [GHz]

|T|

4 layers8 layers12 layers16 layers20 layersShi et al.

Fig. 6. (a) Two side views of a four-layer woodpile structure of aluminum bars and (b) the transmission coefficients versus frequency for thecorresponding woodpile structure consisting of multiple layers together with the solution in [14] indicated by stars for the case of 20 layers.


first-order parallel reflection coefficients is shown in Fig. 4,where the reference is computed with 99 Fourier modesper direction. It is clear that even the self-consistent conver-gence of the classical formulation is much poorer than theconvergence of the formulation derived in this paper. Further,the improvement is not just a temporary improvement, butrather extends over the full discretization range. This obser-vation shows that it is advantageous to formulate the field–material interactions in such a way that the continuity ofthe tangential electric field and the continuity of the electricflux density at material interfaces are taken into account. Thecorresponding peak-memory usage and corresponding com-putation times for the above case are shown in Fig. 5, wherethe OðNÞmemory complexity is clearly demonstrated. For thecomputations, the FFT algorithm for the Green’s function hasbeen used, as explained in Section 3, and therefore the fullOðN logNÞ complexity in the computation time is expectedand observed in Fig. 5(b).

As a final example, we consider a woodpile structure thathas been analyzed in [14] at microwave frequencies. The arrayconsists of a stack of aluminum (εr ¼ 9:61) bars, with squarecross section, with alternating orientation along the x and y

directions; the bars are shifted over half a period for everytwo horizontal layers of the stack. The two side views ofthe setup are shown in Fig. 6(a) for the case of four horizontallayers, where the parameters are given by h ¼ 0:318mm andp ¼ 1:123mm. For a normally incident plane wave polarizedalong the x direction, the transmission coefficient along the x

direction is shown in Fig. 6(b) for an increasing number oflayers, using 15 Fourier modes (from −7 to þ7) per directionand four sample points per layer. Further, we have used alinear-complexity algorithm for the Green’s function, analo-gous to the update equations derived in [16]. As a reference,the results published in [14] for the case of 20 layers areindicated by stars and the dashed–dotted curve following thestars is the computed solution with the method presentedhere. Again we observe good agreement between the com-puted results and the literature. The data regarding the totalcomputation time are given in Table 1, which shows an almostlinear behavior in the computation time as expected, owing tothe fixed number of Fourier modes. For all cases, the numberof matrix–vector products varied between 36 and 40. Byincreasing the number of unknowns per direction by a factorof 2 per direction, i.e., the total number of unknowns isincreased by a factor of 8, we have observed maximum varia-tions in the transmission coefficient of less than 0.35%. Hence,relatively few unknowns are needed here to reach an accep-table accuracy level.

6. CONCLUSIONSFor block-shaped dielectric periodic gratings with 2D periodi-city, a spectral-domain VIE that takes into account explicitFourier factorization rules has been derived. The Fourierfactorization rules derived by Li that give rise to the inverserule have been rewritten in such a way that the correspondingmatrix inverses are avoided and that the computationalcomplexity of the matrix–vector product remains of orderOðN logNÞ or less and a memory complexity of OðNÞ, byexploiting the discrete convolution structure of the materialoperators involved. The Fourier factorization rules have beenderived within a projection-operator framework, which ex-plains the nature of the factorization rules and yields insightin how to deal with spectral variables with spatial jumps, evenif they are not complementary. Finally, the computationalperformance of this approach has been demonstrated byseveral examples.

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Table 1. Computation Times for the Woodpile

Structure Consisting of Several Layers

# Layers # Unknowns Max. CPU Time (s)

4 10800 1.18 21600 2.312 32400 3.516 43200 4.720 54000 5.8


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fast convergence with spectral volume integral equation for crossed block-shaped gratings with...

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