all-purpose finite element formulation for arbitrarily shaped crossed-gratings embedded in a...

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878 J. Opt. Soc. Am. A/Vol. 27, No. 4 /April 2010 Demésy et al.

All-purpose finite element formulationfor arbitrarily shaped crossed-gratings embedded

in a multilayered stack

Guillaume Demésy,* Frédéric Zolla, André Nicolet, and Mireille Commandré

Institut Fresnel, Université Aix-Marseille, École Centrale Marseille, Campus de Saint-Jérôme,13013 Marseille, France

*Corresponding author: [email protected]

Received November 6, 2009; revised February 5, 2010; accepted February 10, 2010;posted February 16, 2010 (Doc. ID 119677); published March 29, 2010

We propose a novel formulation of the finite element method adapted to the calculation of the vector field dif-fracted by an arbitrarily shaped crossed-grating embedded in a multilayered stack and illuminated by an ar-bitrarily polarized plane wave under oblique incidence. A complete energy balance (transmitted and reflecteddiffraction efficiencies and losses) is deduced from field maps. The accuracy of the proposed formulation hasbeen tested using classical cases computed with independent methods. Moreover, to illustrate the indepen-dence of our method with respect to the shape of the diffractive object, we present the global energy balanceresulting from the diffraction of a plane wave by a lossy thin torus crossed-grating. Finally, computation timeand convergence as a function of the mesh refinement are discussed. As far as integrated energy values areconcerned, the presented method shows a remarkable convergence even for coarse meshes. © 2010 OpticalSociety of America

OCIS codes: 050.1950, 050.1755, 350.4238.

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. INTRODUCTIONhough the problem of diffraction of electromagneticaves by a monodimensional grating is abundantly

reated in the scalar and conical cases, one notes manyewer numerical methods allowing the calculation of theector field diffracted by a bidimensional grating (alsoalled bigrating or crossed-grating). We can refer to theork of Moharam et al. [1] for a description of the so-

alled rigorous coupled wave analysis method (RCWA), aethod also known as the Fourier modal method (FMM;

ee for instance Li [2] or Noponen and Turunen [3]). Theecent work of Schuster et al. [4] combines the approachesf Moharam et al. and Popov and Nevière [5] to improvehe convergence of the differential method (DM) intro-uced in 1978 [6,7]. These two similar methods (RCWAnd DM) are generally considered to be low-memory-onsuming and consequently mainly employed for thelectromagnetic modeling of crossed-gratings. We can alsoite the method of transformation of coordinates (or Cethod [8–10], whose most recent developments wereade by Harris et al. [11]); the Rayleigh method (RM)

12]; and the method of variation of boundaries (Brunond Reitich [13,14]). Finally, the finite-difference time-omain method (FDTD) [15,16] allows the computation oflectromagnetic vector fields. Its principle relies on theumerical propagation of a pulse along a temporal grid to-ether with a spatial one. Therefore, this method does isot well adapted to the harmonic domain that we are ad-ressing in the present paper.As far as the finite element method (FEM) is concerned,

1084-7529/10/040878-12/$15.00 © 2

his very general method dedicated to the solving of par-ial differential equations is massively used in mechanicsfluid mechanics, for instance) but not much in electro-agnetics at visible frequencies. Volakis et al. [17] wroteformulation adapted to tridimensional diffusion prob-

ems. Wei et al. recently described another formulation18] adapted to diffusion problems as well and suggestedt could be applied to periodical problems without givingny numerical illustrations.In this article, we propose a new formulation of the

EM dedicated to the modeling of vector diffraction byrossed-gratings and entirely based on use of second-rder edge elements. The main advantage of this methods its complete independence of the shape of the diffrac-ive element, whereas methods listed above require time-r memory-consuming adjustments depending onhether the geometry of the groove region presents ob-

ique edges (e.g., RCWA [19]), high permittivity contrasts,r inappropriate height-to-period ratios (e.g., RM [20]). Itsrinciple relies on a rigorous treatment of the plane waveources through an equivalence of the diffraction problemith a radiation problem whose sources are localized in-

ide the diffractive element itself, as proposed in the sca-ar case for monodimensional gratings [21,22].

This approach, combined with the use of second-orderdge elements, allows us to retrieve with good accuracyhe few numerical academic examples found in the litera-ure. Furthermore, we provide a new reference case com-ining major difficulties, such as a nontrivial toric geom-try together with strong losses and a high permittivity

010 Optical Society of America

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Demésy et al. Vol. 27, No. 4 /April 2010 /J. Opt. Soc. Am. A 879

ontrast. Finally, we discuss computation time and con-ergence as a function of the mesh refinement as well ashe choice of the direct solver.

. THEORETICAL DEVELOPMENTS. Setup of the Problem and Notatione denote by x, y, and z the unit vectors of the axes of an

rthogonal coordinate system Oxyz. We deal only withime-harmonic fields; consequently, electric and magneticelds are represented by the complex vector fields E and, with a time dependence in exp�−i�t�.In this paper, for the sake of simplicity, the materials

re assumed to be isotropic and therefore are opticallyharacterized by their relative permittivity � and relativeermeability � (note that the inverse of relative perme-bilities in this paper is denoted v). It is of importance toote that lossy materials can be studied, the relative per-ittivity and relative permeability being represented by

omplex-valued functions. The crossed-gratings that were addressing in this paper can be split into the followingegions as suggested in Fig. 1:

• The superstrate �z�z0� is assumed to be homoge-eous, isotropic, and lossless, and therefore characterizedy its relative permittivity �+ and its relative permeability+�=1/v+� and we denote k+

ªk0��+�+, where k0ª� /c.

ig. 1. (Color online) Scheme and notation of the studiedi-gratings.

• The multilayered stack �zN�z�z0� is made of N lay-rs that are assumed to be homogeneous and isotropic,nd therefore characterized by their relative permittivityn, their relative permeability �n�=1/vn�, and their thick-ess en. We denote knªk0��n�n for n integers between 1nd N.• The groove region �zg�z�zg−1�, which is embedded

n the layer indexed g��g ,�g� of the previously describedomain, is heterogeneous. Moreover the method used inhis paper works irrespective of whether the diffractivelements are homogeneous. The permittivity and perme-bility can vary continuously (gradient index gratings) oriscontinuously (step index gratings). This region is thusharacterized by the scalar fields �g��x ,y ,z� andg��x ,y ,z��=1/vg��x ,y ,z��. The groove periodicity along

he x-axis, respectively (resp.) y-axis, is denoted dx, resp.y, in the sequel.• The substrate �z�zN� is assumed to be homogeneous

nd isotropic and therefore characterized by its relativeermittivity �− and its relative permeability �−�=1/v−�,nd we denote k−

ªk0��−�−.

Let us emphasize that the method principles remainnchanged in the case of several diffractive patternsade of distinct geometry and/or material.The incident field on this structure is denoted

Einc = A0e exp�ik+ · r�, �1�

ith

k+ = ��0

�0

�0� = k+�

− sin 0 cos 0

− sin 0 sin 0

− cos 0� �2�

nd

A0e = �

Ex0

Ey0

Ez0� = Ae�

cos �0 cos 0 cos 0 − sin �0 sin 0

cos �0 cos 0 sin 0 + sin �0 cos 0

− cos �0 sin 0� , �3�

here 0� �0,2��, 0� �0,� /2�, and �0� �0,�� (polariza-ion angle).

The problem of diffraction that we address in this pa-er is therefore to find the solution of Maxwell’s equationsn the harmonic regime, i.e., the unique solution �E ,H� of

curl E = i��0�H, �4a�

curl H = − i��0�E, �4b�

uch that the diffracted field satisfies the so-called outgo-ng waves condition (OWC) [23], and where E and H areuasi-bi-periodic functions with respect to x and y coordi-ates.One can choose to calculate arbitrarily E, since H can

e deduced from Eq. (4a). The diffraction problemmounts to looking for the unique solution E of the so-alled vectorial Helmholtz propagation equation, deducedrom Eqs. (4a) and (4b):

sEc

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ble

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M�,v ª − curl�v curl E� + k02�E = 0, �5�

uch that the diffracted field satisfies an OWC, and whereis a quasi-bi-periodic function with respect to x and y

oordinates.

. From a Diffraction Problem to a Radiative One withocalized Sourcesccording to Fig. 1, the scalar relative permittivity � and

nverse permeability v fields associated with the studiediffractive structure can be written using complex-valuedunctions defined by part and taking into account the no-ation adopted in Subsection 2.A:

�x,y,z� ª� + for z � z0

n for zn−1 � z � zn with 1 � n � g

g��x,y,z� for zg−1 � z � zg

n for zn−1 � z � zn with g � n � N

− for z � zN

,

�6�

ith = � ,v�, z0=0, and zn=−�l=1n el for 1�n�N.

It is now convenient to introduce two functions definedy part �1 and v1 corresponding to the associated multi-ayered case (i.e., the same stack without any diffractivelement) constant over Ox and Oy:

1�x,y,z� ª � + for z � 0

n for zn−1 � z � zn with 1 � n � N

− for z � zN ,

�7�

ith = � ,v�. We denote by E0 the restriction of Einc tohe superstrate region:

E0 ª Einc for z � z0

0 for z � z0� . �8�

We are now in a position to define more explicitly theectorial diffraction problem that we are dealing with inhis paper. It amounts to looking for the unique vectoreld E solution of

M�,v�E� = 0, such that Edª E − E0 satisfies an OWC.

�9�

n order to reduce this diffraction problem to a radiationne, an intermediary vector field denoted E1 is necessarynd is defined as the unique solution of

M�1,v1�E1� = 0 such that E1

dª E1 − E0 satisfies an OWC.

�10�

he vector field E1 corresponds to an ancillary problemssociated with the general vectorial case of a multilay-red stack which can be calculated independently. Thiseneral calculation is seldom treated in the literature, soe present a development in Appendix A. Thus E1 is fromow on considered a known vector field. It is now appro-riate to introduce the unknown vector field E2

d, simplyefined as the difference between E and E1, which can fi-ally be calculated thanks to the FEM and

E2dª E − E1 = Ed − E1

d. �11�

t is of importance to note that the presence of the super-cript d is not fortuitous: As a difference between two dif-racted fields (Eq. (11)), E2

d satisfies an OWC that is ofrime importance in our formulation. By taking into ac-ount these new definitions, Eq. (9) can be written

M�,v�E2d� = − M�,v�E1�, �12�

here the right-hand member is a vector field that can benterpreted as a known vectorial source term −S1�x ,y ,z�hose support is localized inside the diffractive element it-

elf. To prove it, let us introduce the null term defined inq. (10) and make the use of the linearity of M, which

eads to

�13�

. Quasi-Periodicity and Weak Formulationhe weak form is obtained by multiplying scalarly Eq. (9)y weighted vectors E� chosen among the ensemble ofuasi-bi-periodic vector fields of L2�curl� (denoted2�curl , �dx ,dy� ,k�) in �:

R�,v�E,E�� =��

− curl�v curl E� · E� + k02�E · E�d�.

�14�

ntegrating by part Eq. (14) and making use of thereen–Ostrogradsky theorem lead to

R�,v�E,E�� =��

− v curl E · curl E� + k02�E · E�d�

−��

�n � �v curl E�� · E�dS, �15�

here n refers to the exterior unit vector normal to theurface �� enclosing �.

The first term of this sum concerns the volume behav-or of the unknown vector field, while the right-hand terman be used to set boundary conditions (Dirichlet,eumann, or so called quasi-periodic Bloch–Floquet con-itions).The solution E2

d of the weak form associated with theiffraction problem, expressed in its previously definedquivalent radiative form at Eq. (12), is the element of2�curl , �d ,d � ,k� such that
x y

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∀E� � L2�curl,dx,dy,k�,R�,v�E2d,E�� = − R�−�1,v−v1

�E1,E��.

�16�

n order to rigorously truncate the computation a set ofloch boundary conditions are imposed on the pairf planes defined by (y=−dy /2, y=dy /2) and (x=−dx /2,=dx /2). One can refer to [24] for a detailed implementa-ion of Bloch conditions adapted to the FEM. A set of per-ectly matched layers is used in order to truncate the sub-trate and the superstrate along the z-axis (see [25] forractical implementation of a PML adapted to the FEM).ince the proposed unknown E2

d is quasi-bi-periodic andatisfies an OWC, this set of boundary conditions is per-ectly reasonable: E2

d is radiated from the diffractive ele-ent towards the infinite regions of the problem and de-

ays exponentially inside the PMLs along the z-axis. Theotal field associated with the diffraction problem E is de-uced at once from Eq. (11).

. Edge or Whitney 1-form Second-Order Elementsn the vectorial case, edge elements (or Whitney forms)ake a much more relevant choice [26] than nodal ele-ents. Note that a lot of work (see for instance [27]) has

een done on higher-order edge elements since their in-roduction by Bossavit [28]. These elements are suitableo the representation of vector fields such as E2

d by lettingheir normal component be discontinuous and imposinghe continuity of their tangential components. Instead ofinking the degrees of freedom (DOF) of the final algebraicystem to the nodes of the mesh, the DOF associated withdge (resp. face) elements are the circulations (resp. flux)f the unknown vector field along (resp. across) its edgesresp. faces).

Let us consider the computation cell � together with itsxterior boundary ��. This volume is sampled in a finiteumber of tetrahedra (see Fig. 2) according to the follow-

ng rules: Two distinct tetrahedra have to share a node,n edge, or a face, or have no contact. Let us denote by T

he set of tetrahedra, F the set of faces, E the set of edgesnd N the set of nodes. In the sequel, one will refer to theode n= i�, the edge e= i , j�, the face f= i , j ,k�, and theetrahedron t= i , j ,k , l�.

ig. 2. (Color online) DOF of a second-order tetrahedrallement.

Twelve DOF (two for each of the six edges of a tetrahe-ron) are classically derived from the line integral of theeighted projection of the field E2

d on each oriented edge= i , j�:

��ij =�i

j

E2d · tij�idl

�ji =�j

i

E2d · tji�jdl , �17�

here tij is the unit vector and �i, the barycentric coordi-ate of node i, is the chosen weight function.According to Yioultsis and Tsiboukis [29], a judicious

hoice for the remaining DOF is to make use of a tangen-ial projection of the 1-form E2

d on the face f= i , j ,k�:

��ijk =� �f

�E2d � nijk

+ � · grad �jds

�ikj =� �f

�E2d � nijk

− � · grad �kds . �18�

he expressions for the shape functions, or basis vectors,f the second order 1-form Whitney element are given by

wij = �8�i2 − 4�i�grad �j + �− 8�i�j + 2�j�grad �i

wijk = 16�i�j grad �k − 8�j�k grad �i − 8�k�i grad �j� .

�19�

his choice of shape function ensures [30] the followingundamental property: every DOF associated with ahape function should be zero for any other shape func-ion. Finally, an approximation of the unknown E2

d pro-ected on the shape functions of the mesh m �E2

d,m� can beerived:

E2d,m = �

e�E

�ewe + �f�F

�fwf. �20�

eight functions E� [see Eq. (16)] are chosen in the samepace as the unknown E2

d, L2�curl , �dx ,dy� ,k�. Accordingo the Galerkin formulation, this choice is made so thatheir restriction to one bi-period belongs to the set ofhape functions mentioned above. Inserting the decompo-ition of E2

d of Eq. (20) in Eq. (16) leads to the final alge-raic system, which is solved in the following numericalxamples by means of direct solvers.

. ENERGETIC CONSIDERATIONS:IFFRACTION EFFICIENCIES AND LOSSESontrary to modal methods based on the determination ofayleigh coefficients, the rough results of the FEM are

hree complex components of the vector field Ed interpo-ated over the mesh of the computation cell. Diffraction ef-ciencies are deduced from this field maps as follows.As a difference between two quasi-periodic vector fields

see Eq. (9)], Ed is quasi-bi-periodic and its componentsan be expanded as a double Rayleigh sum:

w

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wT(sT

E(tsmsTt

rE

w

t

Tppviaps


Exd�x,y,z� = �

�n,m��Z2

un,md,x �z�ei��nx+�my�, �21�

ith �n=�0+2� /dxn, �m=�0+2� /dym and

un,md,x �z� =

1

dxdy�

−dx/2

dx/2 �−dy/2

dy/2

Exd�x,y,z�e−i��nx+�my�dxdy.

�22�

y inserting the decomposition of Eq. (21), which is satis-ed by Ex

d everywhere but in the groove region, into theelmholtz propagation equation, one can express Ray-

eigh coefficients in the substrate and the superstrate asollows:

un,md,x �z� = en,m

x,p e−i�n,m+ z + en,m

x,c ei�n,m+ z, �23�

ith �n,m±2

=k±2−�n

2 −�m2 , where �n,m (or −i�n,m) is positive.

he quantity un,md,x is the sum of a propagative plane wave

which propagates towards decreasing values of z, super-cript p) and a counterpropagative one (superscript c).he OWC verified by Ed imposes

∀�n,m� � Z2 en,mx,p = 0 for z � z0

en,mx,c = 0 for z � zN

� . �24�

quation (22) allows us to evaluate numerically en,mx,c

resp. en,mx,p ) by double trapezoidal integration of a slice of

he complex component Exd at an altitude zc fixed in the

uperstrate (resp. substrate). It is well known that theere trapezoidal integration method is very efficient for

mooth and periodic functions (integration on one period).he same holds for the Ey

d and Ezd components as well as

heir coefficients en,my,c,p� and en,m

z,c,p�.The dimensionless expression of the efficiency of each

eflected and transmitted �n ,m� order [3] is deduced fromqs. (23) and (24):

Fig. 4. (Color online) (a) Diffractive ele

�Rn,m =1

�Ae�2�n,m

+

�0en,m

c �zc� · en,mc �zc�, for zc � z0

Tn,m =1

�Ae�2�n,m

−

�0en,m

p �zc� · en,mp �zc�, for zc � zN ,

�25�

ith en,mc,p�=en,m

x,c,p�x+en,my,c,p�y+en,m

z,c,p�z.Furthermore, normalized losses Q can be obtained

hrough the computation of the following ratio:

Q =

�V

1

2��0Im��g��E · EdV

�S

1

2ReE0 � H0� · ndS

. �26�

he numerator in Eq. (26) clarifies losses in watts by bi-eriod of the considered crossed-grating; they are com-uted by integrating the Joule effect loss density over theolume V of the lossy element. The denominator normal-zes these losses to the incident power, i.e., the time-veraged incident Poynting vector flux across one bi-eriod (a rectangular surface S of area dxdy in theuperstrate parallel to Oxy, whose normal oriented along

Fig. 3. (Color online) Configuration of the studied cases.

with vertical edges. (b) ReE � in V/m.
ment x

dbtnlm

ctl

tdrnn�pmp

4ATicscsFbicobivt

1Im4t�s(ttgst

2Idsi�Rmtd

�


ecreasing values of z is denoted n). Since E0 is nothingut the plane wave defined at Eqs. (2) and (3), this lasterm is equal to �Ae

2��0 /�0dxdy� / �2 cos�0��. Volumes andormals to surfaces being explicitly defined, normalized

osses losses Q are quickly computed once E is deter-ined and interpolated between mesh nodes.Finally, the accuracy and self-consistency of the whole

alculation can be evaluated by summing the real part ofransmitted and reflected efficiencies �n ,m� to normalizedosses:

Q + ��n,m��Z2

ReRn,m� + ��n,m��Z2

ReTn,m�,

he quantity to be compared to 1. The sole diffraction or-ers taken into account in this conservation criterion cor-espond to propagative orders whose efficiencies have aon-null real part. Indeed, diffraction efficiencies of eva-escent orders, corresponding to pure imaginary values of

n,m± for higher values of �n ,m� [see Eq. (23)], are alsoure imaginary values as appears clearly in Eq. (25). Nu-erical illustrations of such global energy balances are

resented in Section 4.

. ACCURACY AND CONVERGENCE. Classical Crossed-Gratingshere are only a few references in the literature contain-

ng numerical examples. For each of them, the problemonsists of only three regions (superstrate, grooves, andubstrate), as summed up on Fig. 3. For the four selectedases among the six found in the literature, published re-ults are compared to ones given by our formulation of theEM. Moreover, in each case, a satisfying global energyalance is detailed. Finally a new validation case combin-ng all the difficulties encountered when modelingrossed-gratings is proposed as follows: A nontrivial ge-metry for the diffractive pattern (a torus), made of an ar-itrary lossy material leading to a large step of index, andlluminated by a plane wave with oblique incidence. Con-ergence of the FEM calculation as well as computationime will be discussed in Subsection 4.B.1.

Fig. 5. (Color online) (a) Diffractive el

. Checkerboard Gratingn this example worked out by Li [2], the diffractive ele-ent is a rectangular parallelepiped as shown on Fig.

(a), and the grating parameters highlighted in Fig. 3 arehe following: 0=0=0�, �0=45�, dx=dy=5�0�2/4, h=�0,+=�g�=2.25 and �−=�g=1. Our formulation of the FEMhows good agreement with the FMM developed by Li[2], 1997) since the maximal relative difference betweenhe array of values presented in Table 1 remains lowerhan 10−3. Moreover, the sum of the efficiencies of propa-ative orders given by the FEM is very close to one inpite of the addition of all errors of determination uponhe efficiencies.

. Pyramidal Crossed-Gratingn this example first worked out by Derrick et al. [8], theiffractive element is a pyramid with rectangular basis ashown Fig. 5(a), and the grating parameters highlightedn Fig. 3 are the following: �0=1.533, 0=45�, 0=30�,

0=0�, dx=1.5, dy=1, h=0.25, �+=�g=1 and �−=�g�=2.25.esults given by the FEM in Table 2 show good agree-ent with those of the C method [8,10], the RM [12], and

he RCWA [31]. Note that, in this case, some edges of theiffractive element are oblique.

Table 1. Energy Balance [2]

FMM [2] FEM

T−1,−1 0.04308 0.04333T−1,0 0.12860 0.12845T−1,+1 0.06196 0.06176T0,−1 0.12860 0.12838T0,0 0.17486 0.17577T0,+1 0.12860 0.12839T+1,−1 0.06196 0.06177T+1,0 0.12860 0.12843T+1,+1 0.04308 0.04332

�n,m��ZReRn,m� — 0.10040

TOTAL — 1.00000

with oblique edges. (b) ReE � in V/m.
ement y

3Isd

Tl=ftcov

4Ifsi=+

TtRb

ete

5WmbfcrfR

wimeia

B

1WfemeMyd


. Bi-Sinusoidal Gratingn this example worked out by Bruno and Reitich [13], theurface of the grating is bi-sinusoidal [see Fig. 6(a)] andescribed by the function f defined by

f�x,y� =h

4�cos�2�x

d � + cos�2�y

d �� . �27�

he grating parameters highlighted in Fig. 3 are the fol-owing: �0=0.83, 0=0=�0=0�, dx=dy=1, h=0.2, �+=�g

1 and �−=�g�=4. Note that in order to define this sur-ace, the bi-sinusoid was first sampled (15�15 points),hen converted to a 3D file format. This sampling can ac-ount for the slight differences from the results in Table 3btained using the method of variation of boundaries de-eloped by Bruno and Reitich ([13], 1993).

. Circular Apertures in a Lossy Layern this example worked out by Schuster et al. [4], the dif-ractive element is a circular aperture in a lossy layer ashown Fig. 7(a), and the grating parameters highlightedn Fig. 3 are the following: �0=500 nm, 0=0=0�, �0

45�, dx=dy=1 �m, h=500 nm, �+=�g=1, �g�=0.81255.2500i and �−=2.25.In this lossy case, results obtained with the FEM in

able 4 show good agreement with those obtained withhe FMM [2], the differential method [4,32] and theCWA [1]. Joule losses inside the diffractive element cane easily calculated, which allows us to provide a global

Table 2. Comparison with the Results Given in[8,10,12,31]

[8] [12] [31] [10] FEM

R−1,0 0.00254 0.00207 0.00246 0.00249 0.00251R0,0 0.01984 0.01928 0.01951 0.01963 0.01938

T−1,−1 0.00092 0.00081 0.00086 0.00086 0.00087T0,−1 0.00704 0.00767 0.00679 0.00677 0.00692T−1,0 0.00303 0.00370 0.00294 0.00294 0.00299T0,0 0.96219 0.96316 0.96472 0.96448 0.96447T1,0 0.00299 0.00332 0.00280 0.00282 0.00290

TOTAL 0.99855 1.00001 1.00008 0.99999 1.00004

Fig. 6. (Color online) (a) Diffractive el

nergy balance for this configuration. The convergence ofhe value R0,0 as a function of the mesh refinement will bexamined in Subsection 4.B.1.

. Lossy Torus Gratinge finally propose a new test case for crossed-grating nu-erical methods. The major difficulty of this case lies

oth in the nontrivial geometry [see Fig. 8(a)] of the dif-ractive object and in the fact that it is made of a materialhosen so that losses are optimal inside it. The grating pa-ameters highlighted in Fig. 3 and Fig. 8(a) are theollowing: �0=1, 0=�0=0�, dx=dy=0.3, a=0.1, b=0.05,=0.15, h=500 nm, �+=�g=1, �g�=−21+20i and �−=2.25.Table 5 illustrates the independence of our method

ith respect to the geometry of the diffractive element. �g�

s chosen so that the skin depth has the same order ofagnitude as b, which maximizes losses. Note that en-

rgy balances remain very accurate at normal and obliquencidence in spite of both the nontriviality of the geometrynd the strong losses.

. Convergence and Computation Time

. Convergence as a Function of Mesh Refinementhen using modal methods such as the RCWA or the dif-

erential method, based on the calculation of Rayleigh co-fficients, a number proportional to NR has to be be deter-ined a priori. Then, the unknown diffracted field is

xpanded as a Fourier series, injected under this form inaxwell’s equations, which annihilates x- and

-dependencies. This leads to a system of coupled partialifferential equations whose coefficients can be struc-

with oblique edges. (b) ReE � in V/m.

Table 3. Energy Balance [13]

[13] FEM

R−1,0 0.01044 0.01164R0,−1 0.01183 0.01165T−1,−1 0.06175 0.06299

��n,m��ZReRn,m� — 0.10685��n,m��ZReTn,m� — 0.89121

TOTAL — 0.99806

ement
z

tswzfchslfi

gnoegcn�

cdpaspFcmm

g

N[(rtua

2AwRgdPt

fmolmtatsct

5WFfiwlpcitaa

eleme


ured in a matrix formalism. The resulting matrix isometimes directly invertible (RCWA), depending onhether the geometry allows to suppress the

-dependence, which makes this method adapted to dif-ractive elements with vertically (or decomposed in stair-ase functions) shaped edges. In some other cases, oneas to make use of integral methods in order to solve theystem—as in the pyramidal case, for instance—whicheads to the so-called differential method. The diffractedeld map can be deduced from these coefficients.If the grating configuration calls for only a few propa-

ative orders, and if the field inside the groove region isot the main information sought, these two similar meth-ds allow us to determine the repartition of the incidentnergy quickly. However, if the field inside the groove re-ion is the main piece of information, it is advisable to cal-ulate many Rayleigh coefficients corresponding to eva-escent waves, which increases the computation time asNR�3 or even �NR�4.

The FEM relies on the direct calculation of the vectorialomponents of the complex field. Rayleigh coefficients areetermined a posteriori. The parameter limiting the com-utation time is the number of tetrahedral elementslong which the computational domain is split up. We as-ume that it is necessary to calculate at least two or threeoints (or mesh nodes) per period of the field ��0 /�Re��.igure 9 shows the convergence of the efficiency R0,0 [cir-ular apertures case; see Fig. 7(a)] as a function of theesh refinement characterized by the parameter NM: Theaximum size of each element is set to �0 / �NM�Re��.It is of interest to note that even if NM�3 the FEM still

ives pertinent diffraction efficiencies: R0,0=0.2334 for

Fig. 7. (Color online) (a) Lossy diffractive

Table 4. Comparison with [1,2,4] and EnergyBalance

R0,0

[1] [2] [4] FEM

0.24657 0.24339 0.24420 0.24415

��n,m��ZReTn,m� — — — 0.29110��n,m��ZReRn,m� — — — 0.26761

Q — — — 0.44148

TOTAL — — — 1.00019

M=1, and R0,0=0.2331 for NM=2. The Galerkin methodsee Eq. (15)] corresponds to a minimization of the errorbetween the exact solution and the approximation) withespect to a norm that can be physically interpreted inerms of energy-related quantities. Therefore, the FEMssually provide energy-related quantities that are moreccurate than the local values of the fields themselves.

. Computation Timell the calculations were performed on a server equippedith eight dual core Itanium1 processors and 256 Go ofAM. Tetrahedral quadratic edge elements were used to-ether with the direct solver PARDISO. Among differentirect solvers adapted to sparse matrix algebra (UMF-ACK, SPOOLES and PARDISO), PARDISO turned outo be the least time-consuming one, as shown in Table 6.

Figure 10 shows the computation time required to per-orm the whole FEM computational process for a systemade up of a number of DOF indicated on the right-hand

rdinate. It is of importance to note that for values of NMower than 3, the problem can be solved in less than a

inute on a standard laptop (4 Go RAM, 2�2 GHz) withhree significant digits on the diffraction efficiencies. Thisccuracy is more than sufficient in numerous experimen-al cases. Furthermore, as far as integrated values are attake, relatively coarse meshes �NM�1� can be used withonfidence, allowing fast geometric, spectral, or polariza-ion studies.

. CONCLUSIONe have established a new vectorial formulation of theEM allowing one to calculate the diffraction efficiencies

rom the electric field maps of an arbitrarily shaped grat-ng embedded in a multilayered stack lighted by a planeave of arbitrary incidence and polarization angle. It re-

ies on a rigorous treatment of the plane wave sourceroblem through an equivalent radiation problem with lo-alized sources. Bloch conditions and PMLs have beenmplemented in order to rigorously truncate the computa-ional domain. Nowadays, the efficiency of the numericallgorithms for sparse matrix algebra, together with thevailable power of computers and the fact that the prob-

nt with vertical edges. (b) ReEy� in V/m.

lop

gmgottApawstt

mttcws

ut

ATvpTcfpaedt

s. (b) C

Fc


em reduces to a basic cell with the size of a small numberf wavelengths, make the 3D problem very tractable, asroved here.The main advantage of this formulation is its complete

enerality with respect to the studied geometries and theaterial properties, as illustrated with the lossy torus

rating nontrivial case. Its principle remains independentf both the number of diffractive elements by period andhe number of stack layers. Finally, choosing fully aniso-ropic materials for the groove region or stack layers (seeppendix A for the principle of calculation of the ancillaryroblem in this case) is possible. The weak form associ-ted with the problem would involve more terms, but itould not add any degree of freedom to the final algebraic

ystem. Its flexibility allowed us to retrieve with accuracyhe few numerical academic examples found in the litera-ure and established with independent methods.

Its remarkable accuracy observed in the case of coarseeshes makes it a fast tool for the design and optimiza-

ion of diffractive optical components (e.g., reflection andransmission filters, polarizers, beam shapers, and pulseompression gratings). Finally, its complete independenceith respect to both the geometry and the isotropic con-

tituent materials of the diffractive elements makes it a

Table 5. Energy Balances at Normal and ObliqueIncidence

FEM 3D =0° =40°

R0,0 0.36376 0.27331T0,0 0.32992 0.38191Q 0.30639 0.34476

TOTAL 1.00007 0.99998

Fig. 8. (Color online) (a) Torus parameter

seful tool for the study of metamaterials, finite-size pho-onic crystals, periodic plasmonic structures, etc.

PPENDIX Ahis appendix is dedicated to the determination of theectorial electric field in a dielectric stack lighted by alane wave of arbitrary polarization and incidence angle.his calculation, abundantly treated in the 2D scalarase, is generally not presented in the literature, since, asar as isotropic cases are concerned, it is possible toroject the general vectorial case on the two reference TEnd TM cases. However, the presented formulation can bextended to a fully anisotropic case for which this TE/TMecoupling is no longer valid, and the three components ofhe field have to be calculated as follows.

Let us consider the ancillary problem mentioned in

oarse mesh of the computational domain.

ig. 9. Convergence of R0,0 as function of Nm (circular aperturesrossed-grating).

SntsldpaTr�r

1Bpst

w

Wgi

Tetb

a

Cp

Ba

ThF

N


ubsection 2.B, i.e., a dielectric stack made of N homoge-eous, isotropic, lossy layers characterized by their rela-ive permittivity denoted �j and their thickness ej. Thistack is deposited on a homogeneous, isotropic, possiblyossy substrate characterized by its relative permittivityenoted �N+1=�−. The superstrate is air and its relativeermittivity is denoted �+=1. Finally, we denote by zj theltitude of the interface between the jth and j+1th layers.he restriction of the incident field Einc to the superstrateegion is denoted E0. The problem amounts to looking forE1 ,H1� satisfying Maxwell’s equations in the harmonicegime [see Eqs. (4a) and (4b)].

. Across the Interface z=zjy projection on the main axis of the vectorial Helmholtzropagation equation [Eq. (5)], the total electric field in-ide the jth layer can be written as the sum of a propaga-ive and a counterpropagative plane wave:

E1�x,y,z� = �E1

x,j,+

E1y,j,+

E1z,j,+�exp�j��0x + �0y + �jz��

+ �E1

x,j,−

E1y,j,−

E1z,j,−�exp�j��0x + �0y − �jz��, �A1�

here

�j2 = kj

2 − �02 − �0

2. �A2�

hat follows consists in writing the continuity of the tan-ential components of �E1 ,H1� across the interface z=zj,.e., the continuity of the vector field � defined by

Table 6. Computation Time Variations from Solverto Solver

SolverComputation time

41720 DOFComputation time

205198 DOF

SPOOLES 15 min 32 s 14 h 44 minUMFPACK 2 min 7 s 1 h 12 minPARDISO 57 s 16 min

ig. 10. Computation time and number of DOF as functions of
M
� = �E1

x

E1y

iH1x

iH1y� . �A3�

he continuity of � along Oz together with its analyticalxpression inside the jth and j+1th layers allows us to es-ablish a recurrence relation for the interface z=zj. Then,y projection of Eqs. (4a) and (4b) on Ox, Oy, and Oz,

� i�0H1z −

�H1y

dz�H1

x

dz − i�0H1z

i�0H1y − i�0H1

x� = − i��

E1x

E1y

E1z� �A4�

nd

� i�0E1z −

�E1y

�z�E1

x

�z − i�0E1z

i�0E1y − i�0E1

x� = i��

H1x

H1y

H1z� . �A5�

onsequently, tangential components of H1 can be ex-ressed as functions of tangential components of E1:

�� 0 �0

0 �� − �0

− �0 �0 − ��

B

�iH1

x

iH1y

iH1z� = �

�E1y

dz

−�E1

x

dz

0� . �A6�

y noting the invariance and linearity of the problemlong Ox and Oy, the following notations are adopted:

Uxj,± = E1

x,j,± exp� ± i�jz�

Uyj,± = E1

y,j,± exp� ± i�jz�� , �A7�

�j = �Ux

+,j

Ux−,j

Uy+,j

Uy−,j� . �A8�

hanks to Eq. (A1) and Eq. (A5) and letting M=B−1, weave for the jth layer
.

F

N(

2Uf

Bti

3Ttdcvtbskcb

Ta�le

R


��x,y,z� = exp�i��0x + �0y��

��1 1 0 0

0 0 1 1

�jM12j − �jM12

j − �jM11j �jM11

j

�jM22j − �jM22

j − �jM21j �jM21

j�

�j

�Ux

+,j

Ux−,j

Uy+,j

Uy−,j� .

�A9�

inally, the continuity of � at the interface z=zj leads to

1

1

1

1

1

1

�j+1�zj� = �j+1−1 �j�j�zj�. �A10�

ormal components can be deduced using Eqs. (A4) andA5).

. Traveling Inside the j+1th Layersing Eq. (A1), a simple phase shift allows one to travel

rom z=z to z=z =z −e :
j j+1 j j+1
�j+1�zj+1� = �exp�− i�j+1ej+1� 0 0 0

0 exp�+ i�j+1ej+1� 0 0

0 0 exp�− i�j+1ej+1� 0

0 0 0 exp�+ i�j+1ej+1��

Tj+1

�j+1�zj�. �A11�

y means of Eq. (A11) and Eq. (A10), a recurrence rela-ion can be formulated for the analytical expression of E1n each layer:

�j+1�zj+1� = Tj+1�j+1−1 �j�j�zj�. �A12�

. Reflection and Transmission Coefficientshe last step consists in the determination of the firsterm �0, which is not entirely known, since the problemefinition specifies only Ux

0,+ and Uy0,+, imposed by the in-

ident field E0. Let us make use of the OWC hypothesiserified by E1

d [see Eq. (10)]. This hypothesis directlyranslates the fact that none of the components of E1

d cane traveling either down in the superstrate or up in theubstrate: Uy

N+1,−=UxN+1,−=0. Therefore, the four un-

nowns Ux0,−, Uy

0,−, UyN+1,+, and Ux

N+1,+, i.e., the transverseomponents of the vector fields reflected and transmittedy the stack, verify the following equation system:

�N+1�zN� = ��N+1�−1�N�j=0

N−1

TN−j��N−j�−1�N−j−1�0�z0�.

�A13�

his allows one to extend the definition of transmissionnd reflection widely used in the scalar case. Finally,N+1 is entirely defined. Making use of the recurrence re-

ation of Eq. (A12) and of Eq. (A1) leads to an analyticalxpression for E1

d in each layer.

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all-purpose finite element formulation for arbitrarily shaped crossed-gratings embedded in a...

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