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2648 J. Opt. Soc. Am. A/Vol. 26, No. 12 /December 2009 Hudelist et al.

Analysis of crossed gratings with large periodsand small feature sizes by stitching of the

electromagnetic field

Florian Hudelist,* Andrew J. Waddie, and Mohammad R. Taghizadeh

Heriot-Watt University, School of Engineering and Physical Sciences, EH14 4AS Edinburgh, UK*Corresponding author: fh22@hw.ac.uk

Received September 30, 2009; accepted October 24, 2009;posted October 28, 2009 (Doc. ID 117983); published November 24, 2009

We present a new algorithm that enables the analysis of large two-dimensional optical gratings with verysmall feature sizes using the Fourier modal method (FMM). With the conventional algorithm such structurescannot be solved because of limitations in computer memory and calculation time. By dividing the grating intoseveral smaller subgratings and solving them sequentially, both memory requirement and calculation time canbe reduced dramatically. We have calculated a grating with 32�32 pixels for a different number of subgrat-ings. We show that the increased performance is directly related to the size of the subgratings. The field-stitched calculations prove to be very accurate and agree well with the predictions from the standard FMMapproach. © 2009 Optical Society of America

OCIS codes: 050.1755, 050.1960, 050.2770, 350.3950.

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. INTRODUCTIONlectromagnetic modeling of 3D volume gratings withubwavelength feature sizes is extremely demanding inoth computational time and memory. The Fourier modalethod (FMM) for crossed gratings [1–5], for example, re-

uires very high memory because of the large matriceshat can easily exceed 107 elements.

Many efforts have been undertaken to improve the con-ergence of the FMM in order to get reliable results withlower number of Fourier orders. Bai and Li exploit sym-etries of the investigated gratings [6–9]. Granet andlumey introduced a parametric formulation of the FMMhat increases the spatial sampling rates around discon-inuities by means of a coordinate transform [10].chuster et al. introduced a normal vector method inhich a vector field is defined for each application to in-

rease the convergence rate [11]. These methods however,re not feasible for large diffraction gratings with no sym-etric features.In this paper we introduce a field stitching method us-

ng the FMM for crossed gratings. This method is a gen-ralization of the 1D field stitching method for lamellarratings [12–14]. The large grating is split into severalverlapping subgratings that can be solved separately.rom the results of the subgratings, the global transmis-ion and reflection coefficients can be calculated. Theverlap of the subgratings ensures that no error is intro-uced at the edges from the new boundary conditions ofach small grating, as only the central part of the fieldill be taken into account.In Section 3 we derive the equations that connect the

ubgratings with the whole structure. The incident fieldor the subgratings as well as the transmitted and re-ected field for the main grating are calculated.In Section 4 we show how rotation symmetries and mir-

1084-7529/09/122648-6/$15.00 © 2

or symmetries can be exploited to speed up the simula-ions. Subgratings that can be transformed into eachther by a reflection or rotation need to be solved onlynce.

In Section 5 we show numerical evidence of thisethod by analyzing a 10�10 beam splitter consisting of

2 pixels in each direction.

. STATEMENT OF THE PROBLEMonsider a nonmagnetic crossed grating G with an arbi-

rary distribution of the permittivity ��x ,y�. The gratingan consist of several layers with ��z� being constant inach of the layers. It is periodic in x and y with large pe-iods dx and dy compared with the features within therating and is of thickness t, which should be small withespect to dx and dy. The grating divides two homoge-eous semi-infinite half spaces R1 �z�0� and R2 �z�h�ith refractive indices n1 and n2, respectively. The elec-

romagnetic field in R1 and R2 can be decomposed as a se-ies of plane waves. Inside the grating the field is ex-ressed by a superposition of Floquet modes which areound with the FMM. To connect the field of R1 with R2 acattering matrix S of the grating is calculated. A mono-hromatic linearly polarized light from R1 with vacuumavelength � and an incidence angles � to the z axis andto the x axis, respectively, illuminates the grating.If the grating does not scatter light strongly into the xy

lane, only areas in the immediate neighborhood influ-nce the field at each point in the grating. In the followingections we will use this property to divide the problem ofsingle large grating into several problems of small sub-

ratings.

009 Optical Society of America

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Hudelist et al. Vol. 26, No. 12 /December 2009 /J. Opt. Soc. Am. A 2649

. 2D FIELD STITCHING METHOD. Representation of the External Field with the FMM

n the FMM the electromagnetic field is represented byloquet modes. In regions R1 and R2 (before and after thelement) the field can be written as a Rayleigh expansion:

E�1 = �

mn�I�,mn + R�,mn� � exp�i�mx + ny + �mn

−1 ��, �1�

E�2 = �

mnT�,mn exp�i�mx + ny + �mn

1 ��, �2�

ith m=kx+1+m2� /dx, n=ky

+1+n2� /dy, �mn�a� = �k�a�2−m

2

n2�1/2. � is the polarization of the electric field and can

e either x or y. I�,mn are the Fourier coefficients for theransmitted light, I�,mn for the incident light, and R�,mnor the reflected light.

To calculate the field propagation of an element large inize with many, very small features, a very high numberf Raleigh orders is required to get a accurate represen-ation of the field as well as for the refractive index in therating. The computational cost increases dramaticallyith the number of Raleigh orders, which pushes evenery modern computers to their limits.

. Derivation of the Transmission and Reflection Modeshe grating is divided into N adjacent rectangular re-ions (see Fig. 1). Each of the regions is extended in the xnd y directions to ensure that all areas of the gratinghat influence the field at the edges are included (see Fig.). For the calculation of the transmission and reflectionrders of the whole grating only the central part of theeld of each subgrating will be taken into account. To sim-lify the calculations each subgrating has a coordinateystem attached to it with the origin in the lower left cor-er. The transformation from the global coordinate sys-em to the coordinate system of the subgratings is de-oted by the translation vector rn.The origin of the coordinate system of subgrating n is

ig. 1. The grating is divided into adjacent rectangular subgrat-ngs numbered from 1 to N. They do not necessarily have to havehe same size and shape.

rn = �x1n − xs

n,y1n − ys

n,0�, �3�

here �xn ,yn� is the lower left corner of the nonoverlap-ing part of the grating, and xs

n and ysn are the overlaps in

he x and y directions. The size of subgrating n is

dxn = �x2

n + xsn� − �x1

n − xsn�,

dyn = �y2

n + ysn� − �y1

n − ysn�, �4�

here x2n and y2

n are the upper limits of the nonoverlap-ing area.The spatial frequencies in the computational cell are

kpq = �p,q� �5�

=�0 + 2�pdx−1,0 + 2�qdy

−1� �6�

or the large grating and

kpqn = �0

n + 2�p�dxn�−1,0

n + 2�q�dyn�−1� �7�

or the subgratings.The transform of the electric and magnetic field be-

ween the coordinate systems is

Un�r� = U�r + rn�, �8�

here r defines the position within the subregion, Un ishe electric or magnetic field representation with respecto the subarea, and U is the field representation with re-pect to the whole element.

The field in the nth subregion can be expressed as aourier expansion for both the subregion and the wholerating, for which both have to give the same result. Wese Eq. (8) to calculate the incident modes of each subre-ion. By putting the Fourier expansion of the incidenteld into Eq. (8) we get

ig. 2. Each subgrating is extended by a frame with dimensions and ys. This ensures that the fields at the edges are calculatedorrectly.

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2650 J. Opt. Soc. Am. A/Vol. 26, No. 12 /December 2009 Hudelist et al.

�pq

Ipqn exp�ikpq

n r� = �st

Ist exp�ikst · �r + rn��

=f�r�

, �9�

hich has the form of a discrete Fourier transform withhe Fourier coefficients Ipq

n and which can be calculated by

Ipqn = �dx

ndyn�−1�

0

dxn�

0

dyn

f�r�exp�− ikpqn r�dxdy

= �dxndy

n�−1�st

Ist exp�ikstrn�

��0

dxn�

0

dyn

exp�i�kst − kpqn �r�dxdy. �10�

The reflection and transmission orders can be calcu-ated as a combination of the subregions in a similar wayy using the reverse translation relation between the co-rdinate systems:

U�r� = Un�r − rn��x1n � x � x2

n

y1n � y � y2

n� . �11�

or this calculation it is important to choose the correctubregion that contains r.

Putting the Fourier expansion of the reflected andransmitted field into Eq. (11) we can identify the trans-ission and reflection modes as coefficients of a discreteourier transform that are calculated by

�pq

Tpq exp�ikpqr� = �st

Tstn exp�ikst

n �r − rn��

=f�r� �12�

nd

�pq

�Rpq + Ipq�exp�ikpqr� = �st

�Rstn + Ist

n �exp�ikstn �r − rn��

=f�r�

.

�13�

he coefficients Tpq and Rpq can be calculated in the sameanner as the Ipq

n in Eq. (10):

Tst = �dxdy�−1�pq

Tpqn exp�ikpq

n rn�

��0

dx�0

dy

exp�i�kpqn − kst�r�dxdy. �14�

n this equation the parameters of the correct subgratingave to be chosen according to the x and y in the integral.hus the integral has to be broken up into a sum of N

ntegrals—one for each subgrating—to get the right coef-cients for each point of the structure:

Tst = �dxdy�−1�n

�pq

Tpqn exp�ikpq

n rn�

��x1

n

x2n�

y1n

y2n

exp�i�kpqn − kst�r�dxdy �15�

nd

�Rst + Ist� = �dxdy�−1�n

�pq

�Rpqn + Ipq

n �exp�ikpqn rn�

��x1

n

x2n�

y1n

y2n

exp�i�kpqn − kst�r�dxdy. �16�

. Special case: Small Modulated Area in Blackurroundingn order to calculate the far field of of an isolated element,he periodic effects of the transmitted field have to be re-uced. This can be done by embedding the structure in aarge nonemitting padding so that light of diffracted or-ers reaches the boundary farther away from the ele-ent. By choosing an appropriate size of the padding, it is

ossible to remove the interference effects for the regionf interest. We have introduced this method for lamellarratings [15]. Here we show a generalization to 2D grat-ngs. The new Fourier orders for the larger area can bealculated using the above introduced field stitchingethod:

Tst = �dxdy�−1�pq

Tpq� exp�ikpq� r��

��x1

x2�y1

y2

exp�i�kpq� − kst�r�dxdy, �17�

x and dy are the dimensions of the larger area with spa-ial vectors kpq. The primed variables refer to the smalloordinate system of the element. The coordinates of theower left and upper right corners of the element arex1 ,y1� and �x2 ,y2�, respectively.

. EXPLOITING SYMMETRIES OF THETRUCTURE FOR FIELD STITCHING

f the structure to be investigated shows mirror symmetryC2 or C4) or rotational symmetry (SO2 or SO4) eachroup of symmetric subgratings need to be solved onlynce. The transmission coefficients of the symmetric part-ers can be calculated with coordinate transformations.he field is written generally as U�n��x ,y� with the corre-ponding Fourier coefficients Xpq

�n�. The field U can be re-laced with the reflected, transmitted, and incident fields.or ease of reading, kpq and dx and dy are defined as thearameters of the subgratings in quadrant 1. For the fol-owing calculations we assume the grating to be illumi-ated from R2 with an incidence angle of �=0°.The calculation of the transmitted and reflected fields

nvolves many coordinate transforms of the vectors con-aining the Fourier coefficients of the electric field. Theeld vectors containing the Fourier coefficients are trans-ormed in such a way that the calculated field is mirroredr rotated by multiples of 90°. When using the FMM withsquare truncation scheme and truncation order M, the

eld vectors have 2N2 elements, N2 for x polarization and2 for y polarization with N=2M+1. The elements of the

ector are defined by three indices [see Eqs. (1) and (2)]:

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s

Ttm

R

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BIIpaetsttl

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mt

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Hudelist et al. Vol. 26, No. 12 /December 2009 /J. Opt. Soc. Am. A 2651

X�mn = X�N2+mN+n, �18�

here � is 0 for x polarization and 1 for y polarization. mnd n go from 0 to N denoting the coefficients from −M toM.If the grating has C4 mirror symmetry [Fig. 3(a)], the

ransformations of the field in region 1 to regions 2, 3, andare

U�2��x,y� = U�1��x,− y�, �19�

U�3��x,y� = U�1��− x,− y�, �20�

U�4��x,y� = U�1��− x,y�. �21�

For SO4 rotational symmetry [Fig. 3(b)] we get

U�2��x,y� = U�1��y,− x�, �22�

U�3��x,y� = U�1��− x,− y�, �23�

U�4��x,y� = U�1��− y,x�. �24�

To get an equation for the Fourier coefficients we sub-titute the Fourier series of the field in the original sub-rating and the symmetric subgrating into Eqs. (19)–(24).

With the incidence angle of the light being normal tohe grating surface, i.e., �=0, we can simplify to Eq. (25).n this case we get 0=0=0, m=−−m, and m=−−m.he field in region 2 of Fig. 3(b) is calculated from theeld in region 1 with the relation

U�2��x,y� =1

dxdy�pq

Xpq�2� exp�i�px + qy��

=1

dxdy�pq

Xpq�1� expi�px + q�− y��

=1

dxdy�pq

Xp,−q�1� exp�i�px + qy��, �25�

o

�pq

�Xpq�2� − Xp,−q

�1� �exp�ikr� = 0. �26�

he exponential functions in Eq. (26) form a set of or-hogonal functions so the coefficients of each exp�ikr�ust add to 0.

ig. 3. (Color online) Subgratings that can be transformed intoach each other by reflection (left) or rotation (right) need to beolved only once. The transmission coefficients and reflection co-fficients for all partners can easily be calculated.

We now have an equation for Xp,q�2� :

Xp,q�2� = X−p,q

�1� . �27�

egions 3 and 4 are solved the same way:

Xpq�3� = X−p,−q

�1� , �28�

Xpq�4� = X−p,q

�1� . �29�

For a S04 symmetry the shape of the subgratings muste quadratic, which means p=p. The relations are then

Xpq�2� = Xq,−p

�1� , �30�

Xpq�3� = X−p,−q

�1� , �31�

Xpq�4� = X−q,p

�1� . �32�

We define a transform C that tells how to swap the el-ments in the field vectors according to Eqs. (27)–(32):

Cb,a�X�,m,n� = X�,n,m, �33�

Ca,−b�X�,m,n� = X�,m,−n. �34�

−a,b, C−a,−b, and C−b,−a are defined likewise. Ca,b is thedentity.

In the following subsections we describe how to calcu-ate the field vector of the transmitted field for subgrat-ngs with a symmetric partner. The calculations for theeflected field are identical; to get the equations the T iseplaced by R.

. Plane-Wave Incident Fieldf the incident light is just a plane wave, all subgratingsave the same incident field that is invariant to rotationsnd reflections. The easiest way to get the field vector ofhe transmitted field in regions 2 to 4 is by applying theppropriate transform C to the T vectors of region 1:

T�n� = C�T�1��. �35�

. Exploiting Symmetry for a Spatially Structuredncident Beamf the incident light is not a plane wave but has a certainhase and amplitude distribution the method describedbove cannot be applied because subgratings from differ-nt parts of the grating will have a different field. Theransmission coefficients can then not be calculated by aimple permutation rule. However it is still possible to ob-ain the transmission orders of symmetric subgratings, ashe scattering matrices of symmetric gratings are equiva-ent. In this case the calculation consists of three steps:

First, the incident field of the subgrating to be solvedeeds to be transformed to the coordinate system of thelready solved subgrating. The incident field is trans-ormed by

I�n� = C�I�1��. �36�

Second, the transmission coefficients are calculated byultiplying the transformed incident field vector I� with

he scattering matrix S of the subgrating in region 1:

tg

C

5Tttbftiic

TuspWrpp

Nrm1

gd

FtrpEl

ttttcsepo�s

go

Fe

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Ftsg(

2652 J. Opt. Soc. Am. A/Vol. 26, No. 12 /December 2009 Hudelist et al.

T� = S · I�. �37�

In the last step the transmission field vector T� isransformed back to the orientation of the symmetricrating with the same transform as in step 1:

T�n� = C�T��. �38�

ombining all steps into one concise equation gives

T�n� = C�S�1� · C�I�n���. �39�

. NUMERICAL PERFORMANCEhe underlying principle of the field stitching method iso increase the spatial resolution of the discrete Fourierransform not by increasing the number of Fourier ordersut by decreasing the period of the computational cell. If,or example, a grating of size dx�dy is solved using aruncation order N, the maximum spatial frequency in xs max=2�m /dx. If the grating is split into a�a subgrat-ngs and each grating is solved individually with a trun-ation order N, the maximum spatial frequency in x is

max� = 2�m/�dx/a� = amax. �40�

his means for the same spatial resolution the matricessed in the calculation of the scattering matrix are muchmaller. The increase of the spatial resolution is directlyroportional to the decrease of the size of the subgratings.e can define an effective truncation order Neff that cor-

esponds to the truncation order of the standard FMM ap-roach. For a grating that is divided into a�a regulararts, the effective truncation order is

Neff = aN. �41�

For a square truncation scheme with truncation orderthe S matrix has a size of �2N2�2. Figure 4 shows the

equired memory of the S matrix. Each element of theatrix is complex with double precision, which takes

6 bytes of memory.Based on the theory presented above, a computer pro-

ram was built to test the numerical performance. Theiffractive phase grating in Fig. 5 was solved with the

0 1 0 2 0 3 0 4 0 5 0

E f f e c t i v e T r u n c a t i o n O r d e r s

1 0- 7

1 0- 6

1 0- 5

1 0- 4

1 0- 3

1 0- 2

1 0- 1

1 00

1 01

1 02

1 03

SizeS-M

atrix

[MB]

1 x 1

2 x 2

4 x 4

ig. 4. (Color online) Size of the S matrix as a function of theffective truncation order.

MM. As a reference the whole structure was solved. Forhe field stitching method the grating was divided into aegular grid of 2�2 and 4�4 subgratings. The gratingarameters were n1=n2=1, �=�=0°, h=2�, d1=d2=32�.ach pixel in the grating was quadratic with side

ength �.Figure 6 shows the convergence of the three different

ransmission orders as a function of the effective trunca-ion order. For the FMM calculations a quadratic trunca-ion scheme was used. Using the field stitching methodhe convergence of the result can be improved signifi-antly, which means a much larger grating could beolved without increasing the memory requirement, asach subgrating is solved sequentially. This diagram alsorovides evidence as to the use of an effective truncationrder. The curves using 2�2 subgratings as well as 44 subgratings match nearly exactly the curve for the

tandard FMM approach.Figure 7 shows the calculation time for solving the

rating in Fig. 5 as a function of the effective truncationrder. The calculations were performed using a computer

0 5 1 0 1 5 2 0 2 5 3 00

5

0

5

0

5

0

1 . 0 0

1 . 0 5

1 . 1 0

1 . 1 5

1 . 2 0

1 . 2 5

1 . 3 0

1 . 3 5

1 . 4 0

ig. 5. Diffractive phase grating designed as 10�10 beam split-er. The grating consists of 32�32 pixels and seven refractive in-ex levels.

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0

E f f e c t i v e T r u n c a t i o n O r d e r

0 . 0 0

0 . 0 5

0 . 1 0

0 . 1 5

0 . 2 0

0 . 2 5

0 . 3 0

0 . 3 5

0 . 4 0

abs(T

)

T ( 0 , 0 )

T ( 3 , - 3 )

T ( 3 , 1 )

ig. 6. (Color online) Convergence of the absolute value of threeransmission orders. All three cases show the convergence for thetandard FMM approach (triangles) and for the field stitching al-orithm with 2�2 subgratings (circles) and 4�4 subgratingsstars).

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1

1

1

Ftuf

Hudelist et al. Vol. 26, No. 12 /December 2009 /J. Opt. Soc. Am. A 2653

ith two Intel Xeon 3000 double core processors with alock speed of 3 GHz. The computation time difference be-ween the standard approach and the 4�4 field stitchingpproach is more than two orders of magnitude.This example shows that for large thin gratings the

eld stitching method for crossed gratings reduces theemory requirements as well as the calculation time. If a

rating shows symmetries, the calculation time is re-uced by a further factor of 2 for C2 and SO2 symmetrynd a factor of 4 for C4 and S04 symmetry, as each groupf symmetric subgratings needs to be solved only once.

. CONCLUSIONe introduced a generalization of the field stitchingethod to crossed gratings. The relations between the

ransmission and reflection coefficients of the subgratingsnd the main grating were derived. For a further im-rovement of the field stitching method we exploitedroperties of gratings that show mirror symmetric or dis-rete rotational symmetric properties. On an example of aiffractive element that acts as a 10�10 beam splitter wehowed the numerical performance of the field stitchingethod. Both memory requirements and calculation time

an be reduced by orders of magnitude. This method can

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0

E f f e c t i v e T r u n c a t i o n O r d e r

1 0- 2

1 0- 1

1 00

1 01

1 02

1 03

1 04

1 05

1 06

CalculationTim

e[s]

1 x 1

2 x 2

4 x 4

ig. 7. (Color online) Calculation time as a function of the effec-ive truncation order. Curves with circles show the trend for thendivided structure, diamonds for a 2�2 division, and squaresor a 4�4 division.

e used not only for the FMM but for all modal methods,uch as the method of moments or the rigorous coupledave analysis.

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