Electromagnetic scattering from multiple sub-wavelength ...oramahi/P-JOSA-Alavikia-March...FINITE-ELEMENT FORMULATION OF THE PROBLEM Figure 2 shows a 2-D hole having an arbitrary shape

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B. Alavikia and O. M. Ramahi Vol. 27, No. 4 /April 2010 /J. Opt. Soc. Am. A 815

Electromagnetic scattering from multiplesub-wavelength apertures in metallic screensusing the surface integral equation method

Babak Alavikia* and Omar M. Ramahi

Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, Ontario, N2 L3G1, Canada*Corresponding author: [email protected]

Received October 29, 2009; revised February 1, 2010; accepted February 2, 2010;posted February 2, 2010 (Doc. ID 119228); published March 24, 2010

This work presents a novel finite-element solution to the problem of scattering from multiple two-dimensionalholes with side grating in infinite metallic walls. The formulation is based on using the surface integral equa-tion with free-space Green’s function as the boundary constraint. The solution region is divided into interiorregions containing each hole or cavity as a side grating and exterior region. The finite-element formulation isapplied inside the interior regions to derive a linear system of equations associated with nodal field values. Thesurface integral equation is then applied at the opening of the holes as a boundary constraint to connect nodeson the boundaries to interior nodes. The technique presented here is highly efficient in terms of computingresources, versatile and accurate in comparison with previously published methods. The near and far fields aregenerated for different single and multiple hole examples. © 2010 Optical Society of America

OCIS codes: 050.1940, 290.0290, 050.1220, 050.1755.

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. INTRODUCTIONxtensive recent research in the problem of extraordinary

ransmission of light through sub-wavelength apertures1,2] and plasmonic resonance [3], has renewed interest in

odeling of wave scattering from grating surfaces. Thenefficiency and inaccuracy of the generic full-wave solv-rs, however, has renewed attention to the efficient solu-ion of the problem of scattering from cavities and holesngraved in conducting structures.

Grating couplers may consist of finite periodic sub-avelength cavities or holes engraved in infinite-sized

urfaces coated by conducting layers. An accurate calcu-ation for near and far fields scattered from grating cou-lers allows manipulation and localization of light inovel applications such as near-field microscopy, surfaceefect detection, sub-wavelength lithography, developingunable optical filters, and improving the efficiency of so-ar cell devices.

The problem of scattering from a two-dimensional (2-D)ingle hole in a perfect electric conductor (PEC) can beolved by decoupling the fields inside of the hole from out-ide by closing the apertures with a PEC surface and in-roducing equivalent magnetic currents over the openingsn [4]. Fields on the apertures were expressed by forcinghe continuity condition on the tangential components ofhe fields. An appropriate Green’s function was derived toxpress the fields due to equivalent magnetic currents in-ide and outside the hole. The Method of Moment (MoM)as used to calculate the equivalent magnetic current on

he apertures in [4]. Although using MoM is a powerfulethod to calculate the magnetic current on the aper-

ures, deriving the Green’s function inside the hole limits

1084-7529/10/040815-12/$15.00 © 2

his method to canonical shapes and only homogenousnd isotropic hole fillings.The modal-based method was used to solve the problem

f scattering from a single hole in [5,6]. The fields insidehe hole were expressed in terms of Fourier series of thearallel-plate waveguide modes. In the space exterior tohe hole, the scattered field was expressed in Fouriererms. Matching the modes inside and outside the hole athe apertures determined the unknown modes’ coeffi-ients. The same method was used to solve the problem ofcattering from multiple slits in parallel-plate waveguide7]. Although the modal method is very efficient and effec-ive in solving the problem of scattering from homoge-eously filled holes, it cannot be used when encounteringoles having inhomogeneous or anisotropic fillings.Methods based on numerical approaches to solve the

oundary value problems such as the finite elementethod (FEM) and the finite-difference time-domainethod (FDTD) are suitable for the problem of scattering

rom general-shape holes with complex fillings. In thease of infinite scatterers such as a hole within an infiniteround plane, it is impossible to fully enclose the scatter-r’s geometry by an absorbing boundary condition (ABC)r perfectly matched layers (PMLs) to truncate the solu-ion region (see Fig. 1). Therefore, the behavior of thecattered field due to the infinite PEC wall outside theomputational domain boundary cannot be modeled prop-rly. As shown in Fig. 1, a portion of the scattered fieldrom the PEC wall propagates into the solution region,hich causes errors. To minimize these errors, the do-ain truncating boundaries should be far enough from

he hole to enclose a larger segment of the PEC wall in

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816 J. Opt. Soc. Am. A/Vol. 27, No. 4 /April 2010 B. Alavikia and O. M. Ramahi

ddition to the hole. However, placing the boundary of theomputational domain far from the hole leads to prohibi-ive increase in the computational cost, in addition to in-ccuracy in the solution due to the exclusion of a largeart of the scatterer.To address the shortcomings of the generic type of FEM

olutions incorporating ABCs or PMLs to solve the prob-em of scattering from infinite structures, the boundaryntegral method (BIM) using the free-space Green’s func-ion was introduced in [8,9]. In these works, the mesh re-ion is truncated at the openings of the hole and the BIMs applied to derive an explicit form of a boundary condi-ion of the third kind (mixed boundary condition) at thepenings of the hole.

In this paper, we develop a new FEM-based method toolve the problem of scattering from multiple holes withide grating in infinite metallic structures. The solutionegion is divided to interior and exterior sub-regions. TheEM formulation is used to obtain the solution of theelmholtz equation inside the interior region. The sur-

ace integral equation using the free-space Green’s func-ion, which was reported in [10,11], is applied at the open-ngs of the holes as a global boundary condition. Thisoundary condition determines the behavior of the nodesn the boundary of the interior region in terms of the in-erior nodes. The Neumann or Dirichlet boundary condi-ion is applied on the walls of the holes for the transverselectric (TE) case (where the magnetic field has a compo-ent only along the axis of the hole) or transverse mag-etic (TM) case (where the electric field has a componentnly along the axis of the hole), respectively.

We emphasize that the boundary condition presentedn this work is different from those reported in [8,9]. In8,9], the boundary condition was derived by forcing theontinuity of the fields at the aperture of the hole. Theelds outside the hole were calculated via the surface in-egral equation. However, in our method the surface inte-ral equation using the Green’s function is used to derivelinear system of equation as a constraint that connects

he field values on the boundary to the field values on thepertures of the hole. In our formulation, no enforcementf the continuity of the fields at the apertures takes place.he attractive feature of this boundary condition is thato singularity arises in the Green’s function while apply-

ng the surface integral equation as a boundary con-traint.

ig. 1. (Color online) Schematic of the scattering problem fromhole with arbitrary shape in an infinite PEC surface. An ABC

r PML is used to truncate the computational domain.

. FINITE-ELEMENT FORMULATION OFHE PROBLEMigure 2 shows a 2-D hole having an arbitrary shape in aerfect-conductor surface and illuminated by an obliquelyncident plane wave. The angle �inc represents the anglef the incident wave, and uinc, uref, us, and utrans denotehe incident, reflected, scattered, and transmitted fieldslong the hole axis, respectively. Next, we divide the prob-em space into three regions. Region I and II denote thepper and lower half-spaces of the PEC slab. Region IIIepresents the inside of the hole. Let �B

I and �BII represent

he contour at the interface of the openings of the holeith region I and II, respectively. Also let �O

I and �OII be

xterior to the hole and in close proximity to�BI and �B

II,espectively as shown in Fig. 2. Let �in denote the inte-ior region of the hole, region III, including the layer be-ween �B and �O in the region I and II. We use the finite-lement formulation inside �in to obtain the weak form ofelmholtz’s equation:

� · � 1

p�x,y�� ut� + k0

2q�x,y�ut = g, �1�

here ut is the total field and g is the impressed source.he time harmonic factor exp�j�t� is assumed and sup-ressed throughout. p�x ,y� and q�x ,y� are defined asr�x ,y� and �r�x ,y�, respectively, for TM polarization, orr�x ,y� and �r�x ,y�, respectively, for TE polarization. k0 ishe propagation constant of the wave in free space. Theeighted residual Ri is defined as

Ri =��in

wi�� · � 1

p�x,y�� ut� + k0

2q�x,y�ut − g�d� = 0,

�2�

here wi is the weighting function. Using Green’s firstdentity, Eq. (2) can be rewritten as

Ri = −��in

� 1

p�x,y�� wi · �ut − k0

2q�x,y�wiut + gwi�d�

+� wi

p�x,y�� ut · d� = 0. �3�

ollowing the method reported in [11], the finite element

ig. 2. (Color online) Schematic of the scattering problem fromhole in a PEC surface. The dashed line represents the bounded

egion that contains all inhomogeneities and anisotropies.

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ormulation is applied only inside the solution region. In-ide the solution domain, Ri is reduced to

Ri = −��in

� 1

p�x,y�� wi · �ut − k0

2q�x,y�wiut + gwi�d� = 0.

�4�

ext, the solution domain �in is discretized into triangu-ar elements. The unknown field over each element is de-cribed by a set of interpolating functions given by

ut = �i=1

m

uit�i�x,y�, i = 1,2,3, . . . ,m, �5�

here m is the number of nodes at which the unknowneld is defined, and �i�x ,y� are interpolation functions. ui

t

epresents unknown field values at each node. Usingalerkin’s method, we set wi=�i�x ,y�. Then, R can be ex-ressed in matrix form,

R = �M�U − �F = 0, �6�

here U represents the unknown field value at each node.he elements of the m�m matrix Mand the m�1 matrixare given by

Mij =�Element

� 1

p�x,y�� i�x,y� · ��j�x,y�

− k02q�x,y��i�x,y��j�x,y��d�,

Fi = −�Element

g�i�x,y�d�. �7�

quation (6) can be represented symbolically as

Mii MibI MibII 0 0

MbIi MbIbI 0 MbIoI 0

MbIIi 0 MbIIbII 0 MbIIoII

0 MoIbI 0 MoIoI 0

0 0 MoIIbII 0 MoIIoII

�ui

ubI

ubII

uoI

uoII

� = Fi

FbI

FbII

FoI

FoII

� ,

�8�

here ui, ubI, ubII, uoI, and uoII represent the nodal fieldalues inside the hole on �B

I , �BII and on �O

I , �OII, respec-

ively. The �F matrix represents any possible impressedources at each node; therefore, �F is zero in this prob-em.

The linear system of equations in Eq. (8) represents theelationship between the nodal field values without anyxternal constraint. The imposition of a specific excitationepresented by the incident plane wave has to be takennto consideration through a boundary constraint that es-ablishes a relationship between the incident field, theoundary nodes, and the interior nodes. In Sections 3 and, the surface integral equation will be developed for TMnd TE polarization, respectively, and used as a boundaryonstraint in lieu of the last term in Eq. (3), which cannote implemented directly since the field on �B

I and �BII is

ot known.

. SURFACE INTEGRAL EQUATION FORM POLARIZATIONhe surface integral equation using the free-spacereen’s function will be used to express the nodal fieldalues on �O in terms of the nodal field values on �B inegions I and II, respectively.

. Upper Half-Space (Region I)et us consider the domain �

I representing the half-pace above the PEC (see Fig. 3). In �

I and for the TMolarization case, the electric field vector has only a-component satisfying Helmholtz’s equation,

�2Ez�� + k02Ez�� = j��Jz��, � �

I , �9�

here Jz�� is an electric current source inside �I . Let us

ntroduce the Green’s function GeI� ,��, which is the solu-

ion due to an electric current filament located at � andoverned by Helmholtz’s equation,

�2GeI�,�� + k0

2GeI�,�� = − �� − ��, ,� � �

I ,

�10�

nd satisfying the boundary condition �GeI� ,��y=0=0

i.e., GeI =0 on �I) and the Sommerfeld radiation condition

t infinity. GeI� ,�� is easily found to be the zeroth-order

ankel function of the second kind given as

GeI�,�� = −

j

4H0

2�k0� − �source�� +j

4H0

2�k0� − �image source��.

�11�

ultiplying both sides of Eq. (9) by GeI� ,�� and integrat-

ng over �I yields

��

IGe

I�,��2Ez�� + k02Ez��d�I

= j��

IJz��Ge

I�,��d�I �12�

nvoking Green’s second identity

ig. 3. (Color online) Schematic of the surface integral contourn the upper half-space and lower half-space of the hole.

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��

I�Ez�

2GeI − Ge

I�2Ez�d�I =��I+�

I�Ez

�GeI

�n− Ge

I�Ez

�n �d�I,

�13�

here �I+�I is the contour enclosing �

I , Eq. (12) can beritten as

��

IEz��2Ge

I + k02Ge

I�d�I = j��

IJz��Ge

Id�I

+��I+�

I�Ez

�GeI

�n− Ge

I�Ez

�n �d�I. �14�

ubstituting Eq. (10) in Eq. (14), we have

Ez�� = − j��

IJz��Ge

I�,��d�I

−��I+�

I�Ez��

�GeI�,��

�n− Ge

I�,��Ez��

�n �d�I.

�15�

ince both Ez and GeI satisfy the Sommerfeld radiation

ondition at infinity, integration over �I (see Fig. 3) on the

ight-hand side of Eq. (15) vanishes. Notice that GeI is zero

n �I [see Eq. (11)]. Additionally, Ez�� is zero over theEC ground plane except on aperture of the hole. On in-erchanging primed and unprimed coordinates, Eq. (15)an be simplified to

Ez�� = − j��

IJz��Ge

I�,��d�I

−�ApertureI

Ez��Ge

I�,��

�n�d�I. �16�

n Eq. (16), the first term of the right-hand side repre-ents the electric field generated by the current filamentnd its image in the vicinity of the PEC ground plane. Theecond term in Eq. (16) represents the field perturbationue to the aperture of the hole. In other words, the totallectric field at each point in the upper half-space is theum of the incident field, reflected field due to the PECurface and the scattered field due to the aperture of theole as

Ez�� = Ezinc�� + Ez

ref�� −�ApertureI

Ez��Ge

I�,��

�n�d�I.

�17�

eferring to Fig. 2, let and � be designated the positionf nodes on �O

I and �BI , respectively. Therefore the incident

nd the reflected waves can be written as

Ezinc = exp�jk0�x sin � − y cos ��,

Ezref = − exp�jk0�x sin � + y cos ��, �18�

here x and y are Cartesian components of . To calculatehe last term in Eq. (17), the aperture, �B

I , is discretizednto n segments with length of �x . We then expand E � �
� z �
ver �BI in terms of piecewise linear interpolating func-

ions as

Ez�� = �j=1

n

Ezj�k=1

2

jk�xj��, �19�

here x� and y� are Cartesian components of � and

jk�x�� = xj�

�x�, k = 1;

1 −xj�

�x�, k = 2.� �20�

quation (17) can be represented in matrix notation as

�uoI = �TI + �SI�ubI, �21�

here the elements of �uoI, �ubI, and �TI matrices repre-ent Ez��, Ezj, and �Ez

inc+Ezref�, respectively, at each node.

oting that n�ˆ =−y�̂, the elements of �SI are defined as

SijI =�

xj�−�x�

xj� j1�xj�� Ge

I�xi,y,xj�,y��

�y��

y�=0

dx�

+�xj�

xj�+�x� j2�xj�� Ge

I�xi,y,xj�,y��

�y��

y�=0

dx�, �22�

here

� �GeI�xi,y,xj�,y��

�y��

y�=0

=− jk0y

2��xi − xj��2 + y2

H12�k0��xi − xj��

2 + y2�. �23�

quation (21) represents the boundary condition on thepening of the hole.

. Lower Half-Space (Region II)et us consider the domain �

II representing the half-pace below the PEC slab (see Fig. 3). In the source-freeegion �

II and for the TM polarization case, the transmit-ed electric field vector has only a z-component satisfyinghe homogenous Helmholtz’s equation:

�2Ez�� + k02Ez�� = 0, � �

II. �24�

et us introduce the Green’s function GeII� ,�� governed

y Helmholtz’s equation:

�2GeII�,�� + k0

2GeII�,�� = − �� − �� ,� � �

II.

�25�

quation (25) is similar to Eq. (10), but the main differ-nce is in the location of the delta source. In Eq. (10) wessumed that the delta source is located on �o

I, whereas inq. (25) the delta source is assumed to be located on �o

II.

eII� ,�� satisfies the boundary condition �Ge

II� ,��y=−t0, where t is the thickness of the PEC slab (i.e., Ge

II=0 onII) and the Sommerfeld radiation condition at infinity.

eII� ,�� is found to be the zeroth-order Hankel function

f the second kind as Eq. (11). Multiplying both sides ofq. (24) byGII� , � and following a procedure similar to
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hat of region I, we obtain

Ez�� = −��II+�

II�Ez��

�GeII�,��

�n− Ge

II�,��Ez��

�n �d�II,

�26�

here �II+�II is the contour enclosing �

II in the counter-lockwise direction. Since both Ez and Ge

II satisfy the Som-erfeld radiation condition at infinity, integration over

II (see Fig. 3) on the right-hand side of Eq. (26) vanishes.otice that Ge

II is zero on �II. Additionally, Ez�� is zerover the PEC ground plane except on the aperture of theole. On interchanging primed and unprimed coordi-ates, Eq. (26) can be simplified to

Ez�� = −�ApertureII

Ez��Ge

II�,��

�n�d�II. �27�

quation (27) represents the transmitted field from theperture of the hole into the region II. Upon using x� and� as Cartesian components of �, Eq. (27) can be writtens

Ez�� = −�ApertureII

Ez��Ge

II�,��

�y�dx�. �28�

t is noticeable that the normal vector n�ˆ is in the y�̂ di-ection. To calculate Eq. (28), the aperture, �B

II, is dis-retized into n segments with length of �x� in the sameanner as for the upper half-space. Also we expand Ez��

ver �BII in terms of piecewise linear interpolating func-

ions as Eq. (19), where the interpolating function is de-ned as Eq. (20). Equation (28) can be represented in ma-rix notation as

�uoII = − �SII�ubII, �29�

here the elements of �uoII matrix represent Ez�� atach node. The elements of �SII are defined as

SijII =�

xj�−�x�

xj� j1�xj�� Ge

II�xi,y,xj�,y��

�y��

y�=−t

dx�

+�xj�

xj�+�x� j2�xj�� Ge

II�xi,y,xj�,y��

�y��

y�=−t

dx�, �30�

here

� �GeII�xi,y,xj�,y��

�y��

y�=−t

=− jk0�y + t�

2��xi − xj��2 + �y + t�2

H12�k0��xi − xj��

2 + �y + t�2�. �31�

quation (29) represents the boundary condition on theperture of the hole in region II.

. Reduced Matrix Form for TM Polarizationquations (21) and (29) represent the boundary conditionn the opening of the hole. Combining these equationsnd Eq. (8) in matrix form results in the reduced systematrix

Mii MibI MibII

MbIi MbIbI + MbIoISI 0

MbIIi 0 MbIIbII − MbIIoIISII�ui

ubI

ubII�

= Fi

FbI − MbIoITI

FbII� . �32�

quation (32) represents the modified system matrix thatan be solved using commonly used methods for solvinginear systems such as conjugate gradient method (CGM).

. SURFACE INTEGRAL EQUATION FOR TEOLARIZATIONy modifying the Green’s function, the surface integralquation is derived for the TE polarization in region I andegion II.

. Upper Half-Space (Region I)he surface integral equation for the TE polarization case

s defined by replacing the electric current filament with aagnetic current filament Mz. In �

I and for the TE po-arization case, the magnetic field vector has only a-component satisfying the Helmholtz equation,

�2Hz�� + k02Hz�� = j��Mz��, � �

I , �33�

here and �I have the same definition as in Section 3

see Fig. 3). Let us introduce the Green’s function

hI � ,��, which is the solution due to the magnetic cur-

ent filament located at � and governed by the Helmholtzquation,

�2GhI �,�� + k0

2GhI �,�� = − �� − ��, ,� � �

I .

�34�

ince the image of the magnetic current in the vicinity ofhe PEC surface is in the same direction as the originalurrent, therefore �Gh

I � ,��y=0�0 on the ground plane.n this case, the boundary condition on the PEC surface isepresented as

� �GhI �,��

�y�

y=0

= 0. �35�

n addition, GhI � ,�� satisfies the Sommerfeld radiation

ondition at infinity. Therefore GhI � ,�� can be repre-

ented in terms of the zeroth-order Hankel function of theecond kind as

GhI �,�� = −

j

4H0

2�k0� − �source�� −j

4H0

2�k0� − �image source��.

�36�

ultiplying both sides of Eq. (33) by GhI � ,�� and follow-

ng a procedure similar to that of the TM polarizationase, we obtain

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Hz�� = − j��

IMz��Gh

I �,��d�I

−��I+�

I�Hz��

�GhI �,��

�n− Gh

I �,��Hz��

�n �d�I.

�37�

oth Hz�� and GhI � ,�� satisfy the Sommerfeld radiation

ondition at infinity; therefore integration over �I on the

ight-hand side of Eq. (37) vanishes. Notice thatHz�� /�n is zero over the PEC ground plane except on theperture of the hole and �Gh

I � ,�� /�n is zero on the �I

see Eq. (35)]. On interchanging primed and unprimed co-rdinates, Eq. (37) reduces to

Hz�� = − j��

IMz��Gh

I �,��d�I

+�ApertureI

GhI �,��

�Hz��

�n�d�I. �38�

imilar to the TM case, the first term on the right-handide of Eq. (38) represents the magnetic field generated byhe current filament and its image in the vicinity of theEC ground plane. The second term in Eq. (38) repre-ents field perturbation due to the aperture of the hole.hen Eq. (38) can be rewritten as

Hz�� = Hzinc�� + Hz

ref�� +�ApertureI

GhI �,��

�Hz��

�n�d�I.

�39�

y the same definition of and � and assuming the co-rdinate system as in Fig. 2, the incident and reflectedaves can be written as

Hzinc = exp�jk0�x sin � − y cos ��,

Hzref = exp�jk0�x sin � + y cos ��. �40�

o calculate the last term in Eq. (39), the partial deriva-ive �Hz�� /�n� can be conveniently expressed as a first-rder finite difference as

�Hz��

�n�= −

Hz�x = x�,y� − Hz�x�,y��

y − y��41�

notice that the minus sign on the right-hand side of Eq.41) is due to n�ˆ =−y�̂); then the aperture �B

I , and �OI are

iscretized into n segments with length of �x�. By ex-anding both Hz�y� and Hz�y�� over the aperture of theole in terms of pulse functions as

Hz = �j=1

n

Hzj j�xj��, �42�

here

j�xj�� = 1, xj� −�xj�

2� xj� � xj� +

�xj�

2,

0, elsewhere.� �43�

he behavior of the fields at the edges of the hole can beaptured accurately. By replacing the field expansions inq. (39) and defining the matrix �SI as

SijI =�

xj�−�xj�/2

xj�+�xj�/2 GhI �xi,y,xj�,y� = 0� j�xj��

y − y�dx�, �44�

here

GhI �xi,y,xj�,y� = 0� =

− j

2H0

2�k0��xi − xj��2 + y2�, �45�

q. (39) can be represented in the matrix form as

�uoI = �TI − �SI��uoI − �ubI�, �46�

here the elements of �uoI, �ubI, and �TI matrices repre-ent Hz�x ,y�, Hz�x� ,y��, and Hz

inc�x ,y�+Hzref�x ,y�, respec-

ively, at each node. Equation (46) can be rearranged as

�uoI = ��1 + �SI�−1�TI + ��1 + �SI�−1�SI�ubI. �47�

quation (47) represents the boundary condition on thepening of the hole into region I.

. Lower Half-Space (Region II)et us consider the domain �

II representing the source-ree half-space below the PEC slab (see Fig. 3). In theource-free region �

II and for the TE polarization case,he transmitted magnetic field vector has only a-component satisfying the homogenous the Helmholtz’squation:

�2Hz�� + k02Hz�� = 0, � �

II �48�

et us introduce the Green’s function GhII� ,�� governed

y Helmholtz’s equation:

�2GhII�,�� + k0

2GhII�,�� = − �� − �� ,� � �

II.

�49�

otice that Eq. (49) differs from Eq. (34) in the location ofhe delta source. In Eq. (34) we assumed that delta sources located on �o

I, whereas in Eq. (49) the delta source isssumed to be located on �o

II. GhII� ,�� satisfies the Som-

erfeld radiation condition at infinity and the boundaryondition as

� �GhII�,��

�y�

y=−t

= 0, �50�

here t is the thickness of the PEC slab. GhII� ,�� is found

o be the zeroth-order Hankel function of the second kinds Eq. (36). Multiplying both sides of Eq. (48) by Gh

II� ,��nd following a procedure similar to that of region I, webtain

wcSoiaou

Eaato

T�Ba(fi

w

E

wHt

Eo

CEoam

5Itssrtw�

wbb

Thtitfif�tet(tFc

Fs


Hz�� = −��II+�

II�Hz��

�GhII�,��

�n− Gh

II�,��Hz��

�n �d�II,

�51�

here �II+�II is the contour enclosing �

II in the counter-lockwise direction. Since both Hz and Gh

II satisfy theommerfeld radiation condition at infinity, integrationver �

II (see Fig. 3) on the right-hand side of Eq. (51) van-shes. Notice that �Gh

II� ,�� /�n is zero on �II. Addition-lly, �Hz�� /�n is zero over the PEC ground plane exceptn aperture of the hole. On interchanging primed andnprimed coordinates, the Eq. (51) can be simplified to

Hz�� =�ApertureII

GhII�,��

�Hz��

�n�d�II. �52�

quation (52), represents the transmitted field from theperture of the hole into the region II. Upon using �x� ,y��nd �x ,y� as a Cartesian components of � and , respec-ively, and expressing the partial derivative as a first-rder finite difference, Eq. (52) can be written as

Hz�� =�ApertureII

GhII�,��Hz�x = x�,y� − Hz�x�,y��

y − y��dx�

�53�

o calculate the integral in Eq. (53), the aperture �BII, and

OII are discretized into n segments with length of �x�.oth Hz�y� and Hz�y�� are expanded as Eq. (42) over theperture of the hole in terms of pulse functions as Eq.43). By replacing the field expansions in Eq. (53) and de-ning of the matrix �SII as

SijII =�

xj�−�xj�/2

xj�+�xj�/2 GhII�xi,y,xj�,y� = − t� j�xj��

y − y�dx�, �54�

here

GhII�xi,y,xj�,y� = − t� =

− j

2H0

2�k0��xi − xj��2 + �y + t�2�,

�55�

q. (53) can be represented in the matrix form as

�uoII = �SII��uoII − �ubII�, �56�

here the elements of �uoII and �ubII matrices representz�x ,y� and Hz�x� ,y��, respectively, at each node. Equa-

ion (56) can be rearranged as

�uoII = − ��1 − �SII�−1�SII�ubII. �57�

quation (57) represents the boundary condition on thepening of the hole into region II.

. Reduced Matrix Form for TE Polarizationquations (47) and (57) represent the boundary conditionn the openings of the hole. Combining these equationsnd Eq. (8) in matrix form results in the reduced systematrix as

Mii MibI MibII

MbIi MbIbI + MbIoI�1 + SI�−1SI 0

MbIIi 0 − MbIIoII�1 − SII�−1SII��

ui

ubI

ubII� =

Fi

FbI − MbIoI�1 + SI�−1TI

FbII�. �58�

. EXTENSION TO MULTIPLE HOLESn this section, we extend the method developed above tohe problem of scattering from multiple holes in a PECurface. As an example, a schematic representing thecattering problem for two holes is shown in Fig. 4. Theegions inside the holes are labeled �1 and �2. Extendinghe finite-element development in Section 2 to two holes,e generalize the system matrix for the domains �1 and2 as

�M�1��u�1� = �F�1�,

�M�2��u�2� = �F�2�, �59�

here each system of equations can be represented sym-olically as Eq. (8). Assembling the two systems using glo-al numbering of nodes gives

��M�1� 0

0 �M�2��u�1�

�u�2�� = ��F�1�

�F�2�� . �60�

he two system matrices arising from each of the twooles will be coupled through the surface integral equa-ion in the following manner. In region I, each node on �O

I

s connected via the Green’s function to all the nodes onhe aperture of the two holes, �B

I , via Eq. (17) and Eq. (39)or TM and TE polarization, respectively (see Fig. 4). Also,n region II, each node on �O

II is connected via the Green’sunction to all the nodes on the aperture of the two holes,

BII, via Eq. (27) and Eq. (52) for the TM and TE polariza-ion, respectively. In other words, the holes are coupled toach other only through the surface integral equation andhe Green’s function in each region. In (17) and (39), or in27) and (52), the integration is performed over the aper-ures of both holes in region I or region II, respectively.or instance, Eq. (21) and Eq. (29) for the TM polarizationan be represented symbolically in matrix form as

ig. 4. (Color online) Schematic showing the extension of theurface integral method to multiple holes with side cavities.

a

rnjiEt

wc

a

a

Np

w


��uoI�1�

�uoI�2�� = ��TI�1�

�TI�2�� + ��SI�11� �SI�12�

�SI�21� �SI�22��ubI�1�

�ubI�2��61�

nd

��uoII�1�

�uoII�2�� = − ��SII�11� �SII�12�

�SII�21� �SII�22��ubII�1�

�ubII�2�� , �62�

espectively. �S�ij� represents the connectivity betweenodes on �O of the ith hole �uo�i� and nodes on �B of the

th hole �ub�j� via the surface integral equation �i , j=1,2�n both region I and II. Combining Eq. (60), Eq. (61) andq. (62) in matrix form results in the reduced system ma-

rix as

��M��1� �C�12�

�C�21� �M��2��u��1�

�u��2�� = ��F��1�

�F��2�� , �63�

here �C�12� and �C�21� are matrices representing theoupling between the two holes and are given by

�C�12� = 0 0 0

0 �MbIoI�1��SI�12� 0

0 0 − �MbIIoII�1��SII�12�� ,

�C�21� = 0 0 0

0 �MbIoI�2��SI�21� 0

0 0 − �MbIIoII�2��SII�21�� ,

�64�

nd �M and �F are given by
� �
�M��1� = �Mii�1� �MibI�1� �MibII�1�

�MbIi�1� �MbIbI�1� + �MbIoI�1��SI�11� 0

�MbIIi�1� 0 �MbIIbII�1� − �MbIIoII�1��SII�11�� ,

�M��2� = �Mii�2� �MibI�2� �MibII�2�

�MbIi�2� �MbIbI�2� + �MbIoI�2��SI�22� 0

�MbIIi�2� 0 �MbIIbII�2� − �MbIIoII�2��SII�22�� ,

�F��1� = �Fi�1�

�FbI�1� − �MbIoI�1��TI�1�

�FbII�1� � ,

�F��2� = �Fi�2�

�FbI�2� − �MbIoI�2��TI�2�

�FbII�2� � , �65�

nd �u��k� is given by

�u��k� = �ui�k�

�ubI�k�

�ubII�k�� k = 1 & 2�. �66�

otice that the coupling matrices �C�12� and �C�21� are not necessarily identical. A similar procedure is applicable to TEolarization. Generalizing the formulation to N holes results in the following system matrix:

�M��1� �C�12� . . . �C�1N�

�C�21� �M��2� . . . �C�2N�

] � ]

�C�N1� �C�N2� . . . �M��N��

�u��1�

�u��2�

]

�u��N�� =

�F��1�

�F��2�

]

�F��N�� , �67�

here

�C�pq� = 0 0 0

0 �MbIoI�p��SI�pq� 0

0 0 − �M II II�p��SII�pq�� ,

b o

hsmthett

6OpioctdtHuFtce�tmlsfi

AI�faoaawqaTh

twg


�M��p� = �Mii�p� �MibI�p� �MibII�p�

�MbIi�p� �MbIbI�p� + �MbIoI�p��SI�pp� 0

�MbIIi�p� 0 �MbIIbII�p� − �MbIIoII�p��SII�pp�� ,

�F��p� = �Fi�p�

�FbI�p� − �MbIoI�p��TI�p�

�FbII�p� � �p,q = 1 ¯ N�. �68�

shasetl

Fo�(

Fo�(

The formulation presented in this work is applicable tooles with side cavities present in perfectly conductingurfaces. For multiple holes and cavities, Eq. (67) gives aathematical quantification of the coupling factors be-

ween the holes and cavities. Physically, we expect theoles and cavities to be coupled through surface currentsxisting on the segments connecting the holes and cavi-ies, as that is the only mechanism for energy transfer be-ween them.

. NUMERICAL RESULTSnce the system of linear equations, Eq. (32) for the TMolarization or Eq. (58) for the TE polarization, is derived,ts solution can be obtained using commonly used meth-ds for solving linear systems. In this work, we use theonjugate gradient method (CGM) to solve the linear sys-em of equations. In this section, we provide examples ofifferent holes and provide comparison with results ob-ained using the commercial finite-element simulatorFSS [12]. Throughout this work, the solution obtainedsing the method presented in this paper is referred to asEM-TFSIE. Without loss of generality, the magnitude ofhe incident electric field is assumed to be unity. To dis-retize the solution domain, we use first-order triangle el-ments with mesh density of approximately 20 nodes perfor the TM case. To accurately capture the behavior of

he field at the edges of the holes in the TE case, we useesh density of 100 nodes per �. Notice that in such po-

arization, the H field on the PEC surface corresponds tourface current flowing in directions orthogonal to the Held.

. Single-Hole Casen the first example, we considered a 0.8��0.5� �widthdepth� rectangular hole, where � is the wavelength in

ree space. Figure 5 shows the total electric field at bothpertures of the hole for TM incident plane wave for anblique incident angle. The results in Fig. 5 show stronggreement between the calculations using FEM-TFSIEnd those obtained using HFSS. In the case of HFSS,here an absorbing boundary condition is used, the re-uired computational domain needed to achieve similarccuracy was approximately 18�2. Notice that the FEM-FSIE solution space was confined only to the area of theole of 0.4�2.Figure 6 shows the total magnetic field at both aper-

ures of the same hole, for the case of TE incident planeave. We observe that results from our method are inood agreement with those obtained by HFSS despite a

mall deviation that is pronounced at the edges of theole. The reason for this deviation is that we used tri-ngle elements of a constant size since our meshingcheme is performed without any automated mesh gen-rator. If an adaptive mesh scheme is employed as washe case in HFSS, the deviation in the field at the cornerocation can be eliminated.

ig. 5. (Color online) Amplitude of total E field at the openingsf the hole for a 0.8��0.5� air-filled rectangular hole, TM case,=30, calculated using the method introduced in this workFEM-TFSIE), and HFSS (FEM-HFSS).

ig. 6. (Color online) Amplitude of total H field at the openingsf the hole for a 0.8��0.5� air-filled rectangular hole, TE case,=30, calculated using the method introduced in this workFEM-TFSIE), and HFSS (FEM-HFSS).

0fi=atet

BWsaptttc

1rtsFd

CHasrcat

owF

Foht�h

Filia

Filia

FclT


As the next example of a single-hole case, we consider a.7��0.35� rectangular hole shown in Fig. 7. The hole islled with silicon having a relative permittivity of �r11.9 and including a 0.42��0.07� PEC strip positionedt the geometric center of the hole. The electric field athe aperture for the TM case is calculated. This numericalxample shows the versatility of FEM-TFSIE for solvinghe problem of holes with complex structures and fillings.

. Multiple Holese consider five identical 0.4��0.2� rectangular holes

eparated by 0.4�. Figure 8 shows the total electric fieldt the apertures of the five holes for oblique incidentlane waves with TM polarization. Close agreement be-ween FEM-TFSIE and HFSS is observed. Figure 9 showshe total magnetic field at the aperture of the same struc-ure for the TE case, for oblique incidence. We observelose agreement between FEM-TFSIE and HFSS. In Figs.

ig. 7. (Color online) Amplitude of total E-Field at the openingsf the hole for a 0.7��0.35� Silicon��r=11.9� filled rectangularole, TM case, �=30, calculated using the method introduced inhis work (FEM-TFSIE), and HFSS (FEM-HFSS). The 0.42�0.07� PEC strip is positioned at the geometric center of the

ole.

ig. 8. (Color online) Amplitude of the total E field at the open-ngs of the holes for five identical 0.4��0.2� air-filled rectangu-ar holes, TM case, �=30, calculated using the method introducedn this work (FEM-TFSIE), and HFSS (FEM-HFSS). The holesre separated by 0.4�.

0 and 11 we present the far-field in region II for the ar-ay of five holes. The far field is calculated from the aper-ure fields using the surface integral equation. Figure 10hows the far field for TM-polarized normal incidence andig. 11 shows the far field for TE-polarized oblique inci-ence.

. Single Hole with Side Gratingere we consider a single 0.5��0.8� rectangular holend three 0.5��0.3� cavities as a side grating in the PEClab (see inset of Fig. 12). The hole and cavities are sepa-ated by 0.5�. The two sides of the PEC slab have identi-al gratings. The analysis of this structure is importantnd highly relevant in the study of the phenomenon of ex-raordinary transmission of light.

Figure 12 shows the total electric field at the aperturesf the hole and side cavities for oblique incident planeaves with TM polarization. Close agreement betweenEM-TFSIE and HFSS is observed. However in the case

ig. 9. (Color online) Amplitude of the total H field at the open-ngs of the holes for five identical 0.4��0.2� air-filled rectangu-ar holes, TE case, �=30, calculated using the method introducedn this work (FEM-TFSIE), and HFSS (FEM-HFSS). The holesre separated by 0.4�.

ig. 10. (Color online) Amplitude of the far field for five identi-al 0.4��0.2� air-filled rectangular holes, TM case, �=0, calcu-ated using the method introduced in this work (FEM-TFSIE).he holes are separated by 0.4�.

oawhmrscvrscsgotcif

chtF

7Iishprfb

FclT

Fiai(

Fo�la0

Fo�la0c


f HFSS, the required computational domain needed tochieve the shown accuracy was approximately 32.2�2,hile the FEM-TFSIE solution space was confined to theole and cavities area of 2.2�2. This significant enhance-ent in efficiency facilitates further studies of plasmonic

esonance due to grating surfaces. In Figs. 13 and 14, wehow the total magnetic field at the apertures for the TEase, for normal and oblique incidence. We observe a de-iation between FEM-TFSIE and HFSS. Each cavity inegion II is coupled to the hole and the other cavities viaurface current excited on the PEC segments. The surfaceurrent is excited by the incident field in region I. Theurface current on the surface in region II supports propa-ating waves that graze the upper and lower boundariesf the computational domain when using ABC or PML toruncate the computational domain. To accurately ac-ount for such waves, the computational domain when us-ng the HFSS solver has to be enlarged appreciably. Inact, we show that by increasing the distance between the

ig. 11. (Color online) Amplitude of the far-field for five identi-al 0.4��0.2� air-filled rectangular holes, TE case, �=30, calcu-ated using the method introduced in this work (FEM-TFSIE).he holes are separated by 0.4�.

ig. 12. (Color online) Amplitude of the total E field at the open-ngs of a single 0.5��0.8� hole with three identical 0.5��0.3�ir-filled rectangular side cavities, TM case, �=30, calculated us-ng the method introduced in this work (FEM-TFSIE), and HFSSFEM-HFSS). The hole and cavities are separated by 0.5�.

omputational domain boundaries and the aperture of theole and cavities from h=1� in Fig. 14 to h=3� in Fig. 15,he deviation between the results obtained by HFSS andEM-TFSIE decreases significantly.

. CONCLUSIONn this study, the finite-element method and the surfacentegral equation were combined to solve the problem ofcattering from single and multiple holes and a singleole with side grating. The surface integral equation em-loying Green’s function was used to truncate the meshegion. In this formulation no singularities in Green’sunction arises while applying the surface integral as aoundary constraint. The formulation is based on the to-

ig. 13. (Color online) Amplitude of the total H field at thepenings of a single 0.5��0.8� hole with three identical 0.5�0.3� air-filled rectangular side cavities, TE case, �=0, calcu-

ated using the method introduced in this work (FEM-TFSIE),nd HFSS (FEM-HFSS). The hole and cavities are separated by.5�.


ated using the method introduced in this work (FEM-TFSIE),nd HFSS (FEM-HFSS). The hole and cavities are separated by.5�. The distance between ABC and aperture of the hole andavities is h=1�.

tbc

aifiiwtFmi

ATN

CC

R

1

1

1

Fo�la0c


al field and overcomes the limitations of earlier methodsased on the surface integral equation, and it is appli-able to both TM and TE polarizations.

Several numerical examples were presented for singlend multiple holes. The solutions using FEM-TFSIE weren close agreement with results obtained by a commercialnite-element simulator. Furthermore, the FEM-TFSIE

s very versatile in handling complex structures of holesith inhomogeneous fillings without any modification to

he algorithm. The run time and solution efficiency ofEM-TFSIE are two major attractive features of thisethod, making it well suited for optimization problems

nvolving scattering from gratings in metallic screens.

CKNOWLEDGMENTShis work was supported by Research in Motion and the


ated using the method introduced in this work (FEM-TFSIE),nd HFSS (FEM-HFSS). The hole and cavities are separated by.5�. The distance between ABC and aperture of the hole andavities, is h=3�.

ational Science and Engineering Research Council of

anada under the RIM/NSERC Industrial Researchhair Program.

EFERENCES1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A.

Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).

2. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaminglight from a subwavelength aperture,” Science 297, 820–823 (2002).

3. H. Raether, Surface Plasmons on Smooth and Rough Sur-faces and on Gratings (Springer, 1988).

4. D. T. Auckland and R. F. Harrington, “Electromagnetictransmission through a filled slit in a conducting plane offinite thickness, TE case,” IEEE Trans. Microwave TheoryTech. 26, 499–505 (1978).

5. S. H. Kang, H. J. Eom, and T. J. Park, “TM-scattering froma slit in a thick conducting screen: Revisited,” IEEE Trans.Microwave Theory Tech. 41, 895–899 (1993).

6. T. J. Park, S. H. Kang, and H. J. Eom, “TE scattering froma slit in a thick conducting screen: Revisited,” IEEE Trans.Antennas Propag. 42, 112–114 (1994).

7. H. J. Eom, “Slit array antenna,” in Electromagnetic WaveTheory for Boundary-Value Problems (Springer, 2004), pp.251–257.

8. J. M. Jin and J. L. Volakis, “TM scattering by an inhomo-geneously filled aperture in a thick conducting plane,” inIEE Proc., Part H: Microwaves, Antennas Propag. 137,153–159 (1990).

9. J. M. Jin and J. L. Volakis, “TE scattering by an inhomoge-neously filled thick conducting plane,” IEEE Trans. Anten-nas Propag. 38, 1280–1286 (1990).

0. O. M. Ramahi and R. Mittra, “Finite element solution for aclass of unbounded geometries,” IEEE Trans. AntennasPropag. 39, 244–250 (1991).

1. B. Alavikia and O. M. Ramahi, “Finite-element solution ofthe problem of scattering from cavities in metallic screensusing the surface integral equation as a boundary con-straint,” J. Opt. Soc. Am. A 26, 1915–1925 (2009).

2. Ansoft HFSS Version 10.1., Ansoft Corporation, http://
www.ansoft.com.
http://www.ansoft.com

http://www.ansoft.com

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Electromagnetic scattering from multiple sub-wavelength ...oramahi/P-JOSA-Alavikia-March...FINITE-ELEMENT FORMULATION OF THE PROBLEM Figure 2 shows a 2-D hole having an arbitrary shape