Rigorous integral equation analysis of nonsymmetric coupled grating slab waveguides

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2888 J. Opt. Soc. Am. A/Vol. 23, No. 11 /November 2006 Tsitsas et al.

Rigorous integral equation analysis ofnonsymmetric coupled grating

slab waveguides

Nikolaos L. Tsitsas, Dimitra I. Kaklamani, and Nikolaos K. Uzunoglu

School of Electrical and Computer Engineering, National Technical University of Athens, Heroon Polytechniou 9,GR-15773 Zografou, Athens, Greece

Received March 8, 2006; accepted May 16, 2006; posted June 2, 2006 (Doc. ID 68762)

A rigorous integral equation formulation in conjunction with Green’s function theory is used to analyze thewaveguiding and coupling phenomena in nonsymmetric (composed of dissimilar slabs) optical couplers withgratings etched on both slabs. The resulting integral equation is solved by applying an entire-domain Galerkintechnique based on a Fourier series expansion of the unknown electric field on the grating regions. The pro-posed analysis actually constitutes a special type of the method of moments and provides high numerical sta-bility and controllable accuracy. The singular points of the system’s matrix accurately determine the complexpropagation constants of the guided waves. The results obtained improve on those derived by coupled-modemethods in the cases of large grating perturbations and highly dissimilar slabs. Numerical results referring tothe evolution of the propagation constants as a function of the grating’s characteristics are presented. Optimalgrating parameters with respect to minimum coupling length and maximum coupling efficiency are reported.The coupler’s efficient operation as an optical bandpass filter is thoroughly investigated. © 2006 Optical So-ciety of America

OCIS codes: 230.3120, 050.2770, 230.7370, 230.7400.

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. INTRODUCTIONirectional couplers were traditionally constructed bylacing two slab waveguides in close proximity to eachther. Efficient power exchange between two (nonidenti-al) asymmetric slabs occurs only when the propagationonstants of the two modes are almost equal,1–3 i.e., thewo slabs are in synchronism. However, even in the casen which the two modes have significantly different propa-ation constants (asynchronous slabs), efficient power ex-hange between the two slabs is possible when a periodiciffraction grating is etched on one or both slabs.4 Theole of the periodic grating is to assist the coupling pro-ess, matching the geometrical and physical characteris-ics of the different slabs. The structure thus obtained iseferred in what follows as an asymmetric grating-ssisted directional coupler (AGADC).AGADCs constitute the fundamental guided-wave com-

onents, used widely in several applications of integratedptoelectronics, such as wavelength selective filtering,5

unable semiconductor lasers,6 multiplexing andemultipexing,7 and mode conversion of ultrafast pulses.8

he analysis of the propagation and coupling phenomenan AGADCs has been developed by applying a number ofpproaches. The initial methods combine perturbation ar-uments with coupled-wave concepts9 and provide physi-ally intuitive but approximate results. The most widelysed method is that based on the coupled-mode theorysee Refs. 3, 4, and 10–12 and references cited therein).owever, this method incorporates several approxima-

ions and gives accurate results only for weak grating per-urbations (e.g., small grating thickness) and slabs withery similar index profiles and thicknesses. Also, a trans-

1084-7529/06/112888-18/$15.00 © 2

er matrix method, using a mode-matching technique, haseen proposed,13 but its results suffer from the same ap-roximations as those of the coupled-mode theory. Inractice, the proper computations of the propagation con-tants, the coupling length, the partition of energy be-ween the coupled slabs, and the radiation losses requireore accurate techniques. To this end, rigorous methods,

nitiated by Chang et al.14 and based on the Floquet–loch theory, have been employed.15–19 These methodstilize rigorous waveguide modes, derived as exact solu-ions of the corresponding boundary-value problem, andhey can accurately access the performance of gratingouplers. Their disadvantage consists in the fact that theccuracy of the results depends on the numerical evalua-ion of the resulting integrals or the numerical solution ofhe pertinent linear system of first-order differentialquations. Besides, it is worth noting that the propaga-ion in grating waveguides has also been investigated bypplying the transmission-line theory20 and time-domainechniques.21

Furthermore, integral equation techniques have alsoeen utilized for the investigation of coupling phenomenan AGADCs. In Refs. 22 and 23, subdomain integral equa-ion formulations have been employed and solved by ap-lying the boundary element method. However, an accu-ate computation of the propagation constants dependstrongly on the discretization in boundary elements. Onhe other hand, a rigorous integral equation method, us-ng the entire domain Galerkin’s technique, has been de-eloped in Ref. 24 for the analysis of symmetric (com-osed of identical slabs) grating-assisted directionalouplers. This method actually constitutes a special type

006 Optical Society of America

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Tsitsas et al. Vol. 23, No. 11 /November 2006 /J. Opt. Soc. Am. A 2889

f the method of moments and provides semianalytic so-utions with high numerical stability and controllableccuracy.25,26

In this paper, by modifying the concepts and extendinghe integral equation techniques of Ref. 24, we investigatehe coupling and waveguiding phenomena in an infiniteeriodic asymmetric grating-assisted optical coupler. Therating region is modeled as a periodic layer containingectangular discontinuities. The standard electric field in-egral equation is employed for the electric field on theiscontinuities. The integral equation is solved by apply-ng an entire domain Galerkin’s technique based on aourier series expansion of the electric field on the discon-inuities. The analysis concerns TE wave propagation,ut it can also be extended to TM wave propagation. Ithould be emphasized that the present paper has signifi-ant additional research compared with our previousork,24 since the asymmetry of an AGADC imposes high

omplexity in the analysis and computations concerninghe analytical expression of the Green’s function and thelocks of the linear system’s matrix as well as several nu-erical implementation aspects.The proposed method has high efficiency since accurate

esults can be obtained by using only a few expansionerms.25,26 Furthermore, the Green’s function is analyti-ally expressed and all the involved integrations are ana-ytically carried out. Thus, the computational cost is re-uced, the accuracy increases, and a solution is providedith no other approximation than the final truncation of

he expansion and test functions sets. It is also worth not-ng that the present method gives accurate results evenor large grating thicknesses and very dissimilar slabs,hile the methods mentioned above impose serious limi-

ations on the grating parameters. Besides, the presentormulation requires no discretization of the integralquation involved, as opposed to the case in the boundarylement method.22,23 In addition, the Green’s function ofhe coupled slab geometry used in this paper provides aore compact formulation, inherently satisfying the

ig. 1. (Color online) Geometrical configuration of the asymmere located on the top of slab 1 and on the bottom of slab 2. Th

6 7

oundary conditions. This property does not hold in theethod of Refs. 22 and 23, where the free-space Green’s

unction is used.The numerical results of this paper indicate the opti-al design of the grating’s characteristics, providing aoderate coupling length while preserving a small dis-

ance between the propagation constants. Also, it is ex-lained how the grating’s thickness, refractive index, pe-iod, and duty cycle may act as additional controlechanisms of the coupler’s optimal behavior with re-

pect to the desired specifications. The optimal AGADCharacteristics for its efficient operation as an opticalandpass filter are thoroughly investigated. On the otherand, the symmetric coupler cannot exhibit this filteringehavior, as explained in Ref. 24.

. MATHEMATICAL FORMULATION OFHE PROPAGATION PHENOMENAN ASYMMETRIC GRATING-ASSISTEDOUPLERShe geometrical configuration of the AGADC, depicted inig. 1, contains two dissimilar parallel dielectric slabsith refractive indices n2 and n4, thicknesses d2 and d4,nd separation distance 2d3. The plane regions above, be-ween, and below the slabs are assumed homogeneousith respective refractive indices n1, n3, and n5. On the

labs have been etched periodic rectangular gratingswith refractive indices n6 and n7, thicknesses w2 and w1,engths li and si, and distances from the x axis bi and ai,=1, . . . , t) comprising a �-periodic structure along theaveguiding axis z. The entire structure has constantagnetic permeability �0 and is assumed uniform along

he direction y. Thus, the refractive index distribution ofhe structure is determined by the periodic extension withespect to z of the step function n�x ,z�, defined on �−� ,�� 0,�� by

ating-assisted optical coupler. The rectangular periodic gratingsing parameters are period �, thicknesses w1 ,w2, and refractive

tric gre grat

ndices n ,n .

Iohntt

as

ATAmt

sa�

w

et

(gtt

wIf

t

t

wc

M

Ffi


n�x,z� =�n1, x � d3 + d2, z � �0,��

n2, d3 � x � d3 + d2, z � �bi,bi + li�

n6, d3 � x � d3 + w2, z � �bi,bi + li�

n3, − d3 � x � d3, z � �0,��

n7, − d3 − w1 � x � − d3, z � �ai,ai + si�

n4, − d3 − d4 � x � − d3, z � �ai,ai + si�

n5, x � − d3 − d4, z � �0,��

� .

�1�

t is worth noting that thus far in the literature10–12,14–19

nly binary gratings (with one rectangle in each iteration)ave been considered. In this paper and in Ref. 24, theumber t and the geometrical characteristics li and si ofhe rectangles inside each grating iteration may be arbi-rarily chosen.

An exp�j�t� time dependence of the field quantities isssumed and suppressed throughout the following analy-is.

. Green’s Function of the Nongrating Couplerhe investigation of the waveguiding phenomena in theGADC of Fig. 1 by applying an entire domain method ofoments requires an appropriate analytic expression of

he Green’s function of the nongrating coupler.To this end, we consider the homogeneous coupled-slab

tructure of Fig. 2, excited by a two-dimensional infinitelong the y-axis line source, located at an arbitrary pointx� ,z�� inside slab 1, with normalized current density

J�x,z� = −1

j��0�x − x��z − z��y,

− d3 − d4 � x� � − d3, z� � R, �2�

here �·� is the Dirac function.

ig. 2. (Color online) Geometry of the homogeneous (nongratingnite line source with current density J�x ,z� is located inside sla

By applying the Sommerfeld method,27 hereafter wexpress Green’s function G1 as an appropriate Fourier in-egral. First, we consider the functions

gi�� = �2 − k02ni

2�1/2 �i = 1, . . . ,5�, �3�

k0 is the free space wavenumber). The primary field Gp,enerated by the line source [Eq. (2)] under the assump-ions that the two slabs are absent and that R2 is filled byhe material of slab 1, is expressed as28

Gp�x,z;x�,z�� =1

4��

−�

+�

de−j�z−z��e−g4��x−x��

g4��, �x,z� � R2,

�4�

here the sign of g4 corresponds to Reg4�0 andmg4�0, so that Gp is outgoing and decreasing to zeroor �x�→ +�.

On the other hand, the secondary field Gs induced inhe coupled-slab structure is also expressed as

Gs�x,z;x�,z�� =1

4��

−�

+�

de−j�z−z��,x,x��. �5�

The spectral function � under determination satisfieshe radiation condition as well as the wave equation

�2��,x,x��

�x2 − �2 − k02nh

2�x��,x,x�� = 0, �6�

here the refractive index distribution nh�x� of theoupled-slab structure is given by

nh�x� =�n1, x � d3 + d2,

n2, d3 � x � d3 + d2,

n3, − d3 � x � d3,

n4, − d3 − d4 � x � − d3,

n5, x � − d3 − d4.� �7�

oreover, � is expressed as a linear combination

lem, utilized for the computation of the Green’s function. An in-
) probb 1.

owsmf

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f

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ii

w

BCio


��,x,x�� =�A1��exp�− g1�x − d3 − d2��, x d3 + d2

A2��cosh�g2�x − d3 −d2

2 � + A3��sinh�g2�x − d3 −d2

2 � , d3 � x � d3 + d2

A4��exp�− g3x� + A5��exp�g3x�, − d3 � x � d3

A6��cosh�g4�x + d3 +d4

2 � + A7��sinh�g4�x + d3 +d4

2 � , − d3 − d4 � x � − d3

A8��exp�g5�x + d3 + d4��, x � − d3 − d4

� , �8�

at

svi

wtratFgt2

�p

wuoo

Nf

f the fundamental solutions of the differential Eq. (6),here the signs of g1 and g5 are selected as that of g4. Be-

ides, the unknown spectral coefficients A1–A8 are deter-ined by imposing the boundary conditions at the sur-

aces x=d3+d2, x= ±d3, and x=−d3−d4.Now, as an application of Sommerfeld’s method, we con-

lude that the total field inside slab 1 is the superpositionf the primary and the secondary fields, and moreover theotal field outside slab 1 is the secondary field. Hence, car-ying out certain suitable algebraic calculations, we ob-ain in every region the integral representation

G1�x,z;x�,z�� =1

4��

−�

+�

de−j�z−z��1�,x,x��, �9�

here the kernel 1 is defined in Appendix A.It is worth noting that the poles = ±�i of the integrant

unction in Eq. (9) are the roots of the algebraic equation

��g3,g5,g4,d4��g1,g3,g2,d2�

− K�g3,g5,g4,d4�K�g3,g1,g2,d2��2 = 0 �10�

K, �, and � are defined in Appendix A) and correspond tohe surface modes, propagating in the coupled-slab struc-ure. These poles constitute a finite set of real numbers,ocated in the intervals

min1�j�5

nj � ��i�/k0 � max1�j�5

nj. �11�

In the particular case where n1=n3=n5, n2=n4, and2=d4 the asymmetric coupled-slab structure is reducedo the respective symmetric structure studied in Ref. 24nd the Green’s function, given by Eq. (9) for the abovehoice of parameters, coincides with Eq. (11) of Ref. 24.lso, from Eq. (A1) it follows that the function G1 of Eq.

9) approaches as d3→��→0� the Green’s function of thene isolated slab.

Furthermore, in the case in which the source is locatednside slab 2 in a similar way we see that the total field G2s expressed by

G2�x,z;x�,z�� =1

4��

−�

+�

de−j�z−z��2�,x,x��, �12�

here 2 is defined in Appendix A.

. Integral Representation of the Electric Fieldonsider TEz polarized electromagnetic waves, propagat-

ng parallel to the z axis. The respective electric field hasnly a y component

E = ��x,z�y, �13�

nd hence the problem is reduced to the determination ofhe unknown scalar electric field factor �.

In the absence of external excitation, the waveguidetructure of an AGADC supports propagating waves pro-ided that the electric field factor � of the structure sat-sfies the integral representation

��x,z� = k02�n7

2 − n42� � �

Sd1

G1�x,z;x�,z��1�x�,z��dx�dz�

+ k02�n6

2 − n22� � �

Sd2

G2�x,z;x�,z��2�x�,z��dx�dz�,

�x,z� � R2, �14�

here Sd1 and Sd

2 are the total transverse cross sections ofhe rectangles, �1 and �2 the electric field factors on theectangles of slabs 1 and 2 (i.e., the restriction of � on Sd

1

nd Sd2, respectively), and G1 and G2 the Green’s func-

ions. Equation (14) actually represents a homogeneousredholm integral equation of the second kind. The inte-ral representation of Eq. (14) is justified by extendinghe techniques of the respective representation (2) of Ref.4.According to Floquet’s theorem, the electric field factorof a propagating mode along the direction z has the ex-

ression

��x,z� = u�x,z�exp�− �z�, �15�

here � is the complex propagation constant of the modender determination, and u�x ,z� is a �- periodic functionf z. Therefore, by considering the restrictions u1 and u2f u on Sd

1 and Sd2 and combining Eqs. (14) and (15) we get

u�x,z� = k02�n7

2 − n42� � �

Sd1

G1�x,z;x�,z��u1�x�,z��

�exp��z − z��dx�dz� + k02�n6

2 − n22�

�� Sd

2G2�x,z;x�,z��u2�x�,z��

�exp��z − z��dx�dz�, �x,z� � R2. �16�

ow, by combining Eqs. (9), (12), and (16), we obtain theollowing reformulation of Eq. (16)

wi

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oi

wf

c

tp

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(rg

pet


u�x,z� =k0

2�n72 − n4

2�

4� �r=−�

+� � �Sr

1��

−�

+�

d exp�− j�z

− z��1�,x,x��u1�x�,z��exp�− ��z� − z��dx�dz�

+k0

2�n62 − n2

2�

4� �r=−�

+� � �Sr

2��

−�

+�

d exp�− j�z

− z��2�,x,x��u2�x�,z��exp�− ��z� − z��dx�dz�,

�17�

here the transverse cross sections Sr1 and Sr

2 of the grat-ng’s rth iteration in slabs 1 and 2 are defined by

Sr1 = �

i=1

t

�− d3 − w1,− d3� � �ai + r�,ai + si + r��,

Sr2 = �

i=1

t

�d3,d3 + w2� � �bi + r�,bi + li + r��. �18�

Furthermore, the transformation ��=z�−r� reduceshe double integrals of Eq. (17) to integrals on the basicterations S0

1 and S02. Hence, by means of Poisson’s sum-

ation formula for the Dirac function, Eq. (17) is reducedo

u�x,z� =k0

2�n72 − n4

2�

2��

p=−�

+�

exp�− j2�p

�z

�� S0

1u1�x�,��exp�j

2�p

��

�1�− j� +2�p

�,x,x� dx�d��

+k0

2�n62 − n2

2�

2��

p=−�

+�

exp�− j2�p

�z

�� S0

2u2�x�,��exp�j

2�p

��

�2�− j� +2�p

�,x,x� dx�d�� . �19�

. SOLUTION OF THE INTEGRALQUATIONhe solution of Eq. (19) is obtained by applying an entireomain Galerkin technique as outlined in the followingteps:

1. The electric field factors u1=u1�x ,z� and u2=u2�x ,z�n the grating’s basic iterations S0

1 and S02 are expanded

n the Fourier series with respect to z

ui�x,z� = �n=−�

+�

�i,n�x�exp�− j2�n

�z , �x,z� � S0

i , �i = 1,2�,

�20�

here the Fourier coefficients are the space harmonicunctions

�1,n�x� = cn1+ exp�g7,n�x + d3 +

w1

2 �+ cn

1− exp�− g7,n�x + d3 +w1

2 � ,

�2,n�x� = cn2+ exp�g6,n�x − d3 −

w2

2 �+ cn

2− exp�− g6,n�x − d3 −w2

2 � , �21�

n1±, cn

2± are under determination coefficients and

gi,n = gi�− j� +2�n

� = ��− j� +

2�n

� 2

− k02ni

2�1/2

�i = 1, . . . ,7�. �22�

Now, by substituting Eqs. (20) into (19) and consideringhe constants Jp−n and the functions Qnp

± , defined in Ap-endix B, we find

u�x,z� =k0

2�n72 − n4

2�

2 �p=−�

+�

�n=−�

+� �Jp−n1 exp�− j

2�p

�z

��cn1+Qnp

1+�x� + cn1−Qnp

1−�x��+

k02�n6

2 − n22�

2 �p=−�

+�

�n=−�

+� �Jp−n2 exp�− j

2�p

�z

��cn2+Qnp

2+�x� + cn2−Qnp

2−�x��, �x,z� � R2. �23�

quation (23) indicates that the electric field factor u isxpressed as a double series of products of two functionsf one variable, the first depending on the grating’s geom-try along the z axis and the second on the grating’s ge-metry along the x axis.

2. To determine the unknown coefficients cn1±, cn

2± in Eq.23), we consider the observation vector �x ,z� in Eq. (23)estricted on the domain of the discontinuities, by distin-uishing the following two cases:

Case 1 �x ,z��Sd1. The function u=u1 in Eq. (23) is ex-

anded in the Fourier series [Eq. (20)]. Then, by consid-ring the inner products of both sides of Eq. (23) with theest functions

exp�±g7,m�x + d3 +w1

2 �exp�j2�m

�z ,

a

Tfi

tcw

w

Td

ht�cg

diwct

us

A

(a

trsbtg

n

t→

se

(gst(pn+

F(t


�x,z� � Sd1, �m � Z�,

nd carrying out the resulting integrations, we get

�n=−�

+�

Jm−n1 �Kmn

1±+cn1+ + Kmn

1±−cn1−�

= k02�n7

2 − n42� �

n=−�

+�

�p=−�

+�

Jp−n1 Jm−p

1

��cn1+Qmnp

11±+ + cn1−Qmnp

11±−�

+ k02�n6

2 − n22� �

n=−�

+�

�p=−�

+�

Jp−n2 Jm−p

1

��cn2+Qmnp

12±+ + cn2−Qmnp

12±−�. �24�

he infinite matrices �Kmn1±±� and �Qmnp

1i±±� �i=1,2� are de-ned in Appendix B.Case 2 �x ,z��Sd

2. As in case 1, by expanding the func-ion u=u2 in Eq. (23) in the Fourier series [Eq. (20)] andonsidering the inner products of both sides of Eq. (23)ith the test functions

exp�±g6,m�x − d3 −w2

2 �exp�j2�m

�z ,

�x,z� � Sd2 �m � Z�,

e obtain

�n=−�

+�

Jm−n2 �Kmn

2±+cn2+ + Kmn

2±−cn2−�

= k02�n7

2 − n42� �

n=−�

+�

�p=−�

+�

Jp−n1 Jm−p

2

��cn1+Qmnp

21±+ + cn1−Qmnp

21±−�

+ k02�n6

2 − n22� �

n=−�

+�

�p=−�

+�

Jp−n2 Jm−p

2

��cn2+Qmnp

22±+ + cn2−Qmnp

22±−�. �25�

he infinite matrices �Kmn2±±� and �Qmnp

2i±±� �i=1,2� are alsoefined in Appendix B.3. Equations (24) and (25) constitute an infinite square

omogeneous linear system of coupled algebraic equa-ions with respect to the unknown coefficients cn

1±, cn2± �n

Z�. For the numerical solution of this infinite system weonsider the respective truncated �8N+4�� 8N+4� homo-eneous linear system

�A11

++ A11+− A12

++ A12+−

A11−+ A11

−− A12−+ A12

−−

A21++ A21

+− A22++ A22

+−

A21−+ A21

−− A21−+ A22

−−��

c1+

c1−

c2+

c2−� = 0, �26�

erived by taking into account the terms of the expansionn Eq. (20) and the test functions in the inner productsith maximum absolute order N. However, a convergence

ontrol for increasing N should be applied to the solu-ions. A basic advantage of our method is that small val-

es of N �N�5� provide sufficient convergence (for detailsee Section 4).

The elements �Aij±±�mn of the �2N+1�� 2N+1� matrices

ij±± �i , j=1,2� are given by

�A11±±�mn = −�Jm−n

1 + �p=−�

+� k02�n7

2 − n42�

g7,n2 − g4,p

2 Jp−n1 Jm−p

1 �Kmn1±±

+ �p=−�

+� k02�n7

2 − n42�

g7,n2 − g4,p

2 Jp−n1 Jm−p

1 �Rmnp1±± + Hmnp

11±±�,

�A12±±�mn = − �

p=−�

+� k02�n6

2 − n22�

g6,n2 − g2,p

2 Jp−n2 Jm−p

1 Hmnp12±±,

�A21±±�mn = − �

p=−�

+� k02�n7

2 − n42�

g7,n2 − g4,p

2 Jp−n1 Jm−p

2 Hmnp21±±,

�A22±±�mn = −�Jm−n

2 + �p=−�

+� k02�n6

2 − n22�

g6,n2 − g2,p

2 Jp−n2 Jm−p

2 �Kmn2±±

+ �p=−�

+� k02�n6

2 − n22�

g6,n2 − g2,p

2 Jp−n2 Jm−p

2 �Rmnp2±± + Hmnp

22±±�.

�27�

Rmnpi±± and Hmnp

ij±± are defined in Appendix B) and c1±, c2±

re 2N+1 column vectors of the coefficients cn1±, cn

2±.The possible propagating modes correspond to the non-

rivial solutions of Eq. (26), i.e., to the values of the pa-ameter � for which the corresponding system’s matrix isingular. Finally, the nontrivial solutions of Eq. (26) maye used for the computation of the electric field’s distribu-ion of the propagating modes by means of the basic inte-ral representation of Eq. (14).

Referring to the truncated linear system of Eq. (26) weote the following:(i) The matrices �Hmnp

ij±± �, representing the coupling be-ween the two slab waveguides, approach matrix 0 as d3

��→0�, leading to a decoupling of linear system (26).(ii) In the particular case of a symmetric grating as-

isted coupler (where n1=n3=n5, n2=n4, and d2=d4) lin-ar system (26) is reduced to system (25) of Ref. 24.

(iii) The particular cases of (a) an isolated grating slabFig. 1 in the absence of slab 2) and (b) a coupler withrating on one slab (Fig. 1 in the absence of the grating oflab 2), which have been analyzed separately in the litera-ure [see Refs. 29 and 30 for (a) and Refs. 4, 11, and 17 forb)], are unified here as special cases of our method. Morerecisely, applications of an entire domain Galerkin tech-ique lead in both cases to the truncated �4N+2�� 4N2� linear system

�A++ A+−

A−+ A−−��c+

c−� = 0. �28�

or case (a) A±± are the blocks A11±± of the system in Eq.

26) for �=0, and for case (b) A±± are the blocks A11±± of sys-

em (26).

4Tioonptnmiotc1igowt

diw

Tspiccwl(octTw

ti

T�ttiio

s−g

aiscmc

c

TN+otsFertemmatilptt

qs

Tpss(t

ttwwttmpa

gtgrtb


. NUMERICAL RESULTShe analysis of the propagation and coupling phenomena

n the AGADC and the comparison of our approach to thatf the coupled-mode theory require first the introductionf the uncoupled modes propagating in the respectiveongrating coupler and then the consideration of theresence of the grating in order to study the evolution ofhe uncoupled modes into the coupled ones. The tech-iques based on the coupled-mode theory provide approxi-ate expressions of the propagation constants of the grat-

ng structure in terms of the constants of the nongratingne. These expressions are valid under certain restric-ions referring to the geometry and the materials of theoupler (see also the related discussion in Section 4 of Ref.2). On the other hand, the present semianalytic methods applied for the computation of the coupled modes in therating structure without involving the uncoupled modesf the nongrating structure. Thus, this method is appliedithout the above restrictions and is far more accurate

han the coupled-mode theory.To this end, first we consider the nonperiodic coupler,

erived from the AGADC of Fig. 1 by replacing the grat-ng layers with two homogeneous layers with thicknesses

1, w2 and average refractive indices

nh,1 =

n7�i=1

t

si + n4�i=1

t

ai

�, nh,2 =

n6�i=1

t

li + n2�i=1

t

bi

�.

�29�

he modes of this structure are computed accurately byolving the corresponding homogeneous boundary valueroblem, and the computations indicate that they appearn pairs referred as compound modes.3 In the case of aoupler with lossless layers the respective effective indi-es of the two compound modes become purely imaginaryith values jN1 /k0 and jN2 /k0. A significant exchange of

ight occurs only when the two slabs are in synchronismthat is, when N1 /k0 and N2 /k0 are almost equal). On thether hand, the modes of a coupler composed of nonsyn-hronous slabs resemble those of the isolated slabs, wherehe largest amount of power is concentrated in each slab.hus, no substantial exchange of light is taking placehen the two slabs lack phase synchronism.3,12

Furthermore, the presence of the gratings implies thathe compound modes gradually evolve into a pair of grat-ng modes with complex propagation constants

�i = �i + j�i �i = 1,2�. �30�

he attenuation factors �i are nonzero for large values of/, even when all layers are lossless. This fact is due to

he radiation losses caused by grating diffraction, so thathe modes are of the leaky variety.1,16 For AGADCs usedn practice the effective indices �1 /k0 and �2 /k0 are closen value to N1 /k0 and N2 /k0, while ��1 /k0� and ��2 /k0� aref the order of at most 10−3 (see also Ref. 16).

Now, according to Eqs. (15) and (20), each �i corre-ponds to an electric field factor ��x ,z� behaving as exp��i+ j�2�n /��z along the longitudinal direction z of therating. The concrete values

�i,n = �i + j��i + �2�n/�� i = 1,2� �n � Z� �31�

re the propagation constants of the different countablenfinite Floquet harmonics, imposed in an infinite periodictructure by Floquet’s theorem. In the following numeri-al results, we consider only the first-order Floquet har-onics (i.e., n= ±1) for comparison with the results of the

oupled-mode theory and the Floquet–Bloch theory.The coupled-mode theory predicts the phase-matching

ondition

�/ = k0/�N1 − N2�. �32�

he physical meaning of this condition is that the curves1/k0 and N2 /k0 with respect to � / intersect N2 /k0 /� and N1 /k0− /�, respectively.12,16 Thus, the first-rder Floquet harmonics are matched in the sense thathe +1 �−1� Floquet harmonic of N2 /k0 �N1 /k0� has theame phase as N1 /k0 �N2 /k0�. Note that the first-orderloquet harmonics of the nonperiodic coupler are consid-red in the limit situations where the periodic structureeduces to the nonperiodic one.16 Equation (32) may be in-erpreted in the following two ways. First, for a fixed op-rating wavelength the solution �=�w of Eq. (32) deter-ines the optimal grating period, corresponding to theatching of the first-order Floquet harmonics. Second, forfixed grating period � the solution =w of Eq. (32) de-

ermines the respective optimal wavelength. The follow-ng numerical results indicate that the optimal wave-ength and grating period are different from thoseredicted by the coupled-mode theory, and, moreover,hey are very close in value to those of the Floquet–Blochheory.

Furthermore, the coupling length Lc, i.e., the length re-uired for a total exchange of power between the twolabs, can be estimated by17,18

Lc =�

min��1 − �2�.

he coupling length constitutes an additional significantarameter, which should be taken into account in the de-ign of an AGADC because, although the attenuation con-tants � /k0 are reasonably small, the total leakage lossesexpressed by �Lc) could be so large as to become prohibi-ive for the coupler’s efficient performance.

Applying the techniques of Sections 2–4, we concludehat the effective indices of the AGADC are the roots ofhe linear system’s [Eq. (26)] determinant [for a couplerith the grating located on one slab, instead of Eq. (26)e consider the system in Eq. (28)]. The roots of the de-

erminant are computed accurately by applying the itera-ive complex Müller method. The essential part of thisethod concerns the determination of appropriate initial

oints, obtained by following the graphical procedure,nalyzed in Ref. 24.The following numerical results exhibit the effect of the

rating on the coupling process by plotting the propaga-ion constants with respect to the grating’s physical andeometrical characteristics. For each set of coupler pa-ameters the required number of space harmonic func-ions in the Fourier expansions in Eq. (20) is determinedy applying a convergence check on the computed propa-

gst++ttc1fdAaCl1

pn=t

ATwpia4tctFwtNfpswscolc

�Mt�

sFcsncomlvtm

d

==

Fctpmw


ation constants. A representative convergence pattern ishown in Table 1. A high accuracy (of the order of 10−7) ofhe present method is achieved by considering only 2N1=11 expansion coefficients in Eq. (20). The order 8N4=8�5+4=44 is by far smaller than that of the respec-

ive system of the integral equation–boundary elementechniques,23 requiring 500 boundary elements for theonvergence of the propagation constant up to the order of0−4. This efficiency of the present technique is due to theact that the unknown electric field factor and the entireomain expansion terms satisfy the same physical laws.lso, since the present method involves integrals, whichre analytically computed, it is very efficient in terms ofPU time. For the calculation of the determinant of the

inear system’s matrix, 0.5 s (Pentium IV, 2.80 GHz withGB of RAM) were sufficient.In the following Subsections 4.A–4.D we consider a cou-

ler with constant parameters n1=1, n2=3.2, n3=n5=3,4=3.5, d2=0.5 �m, d4=0.22 �m, �=10.748 �m, t=1, b1l1=� /2, and the grating located on slab 2 (see Fig. 1 in

he absence of the grating of slab 1).

. Resonant Wavelengthhe dispersion curves of the grating and compound modesith respect to the normalized wavelength � / are de-icted in Fig. 3(a) and the attenuation curves of the grat-ng modes in Fig. 3(b). The curves of Figs. 3(a) and 3(b)re in excellent agreement with those in Figs. 4(a) and(b) of Ref. 16. The resonant wavelength, correspondingo the minimization of the distance �� /k0= ��1−�2� /k0, oc-urs at opt=1.4983 �m. This value of opt is very close tohe 1.49775 �m, computed in Ref. 16 by application of theloquet–Bloch theory. The respective of the resonantavelength, as predicted by the coupled-mode theory (i.e.,

he intersection point of the curves N1 /k0+ /� and2/k0), is cm=1.5 �m. This small difference is due to the

act that the coupled-mode theory gives approximate ex-ressions of the propagation constants of the gratingtructure in terms of the constants of the nongrating one,hile the present method actually provides accurate re-

ults for the periodic structure. However, there are con-rete cases such as that of the grating located on the topf slab 2 when the difference between the accurate valueopt and the approximate value cm may become veryarge (see also the discussion in Ref. 16). Figure 3(a) indi-ates the asymptotic behavior of the dispersion curves

Table 1. Convergence Pattern of the NormalizedImaginary Parts of the Propagation Constants for

the Grating Coupler of Fig. 1a

N �1 /k0 ��1,N−�1,N−1� /�1,N−1b �2 /k0 ��2,N−�2,N−1� /�2,N−1

b

1 3.2245 3.23012 3.2175 −2.17�10−3 3.2235 −2.04�10−3

3 3.2158 −5.28�10−4 3.2238 9.30�10−5

4 3.2159 3.10�10−5 3.2241 9.80�10−5

5 3.2159 −5.10�10−7 3.2241 3.34�10−7

an1=1, n2=3.2, n3=n5=n6=n7=3, n4=3.5, d2=0.5 �m, 2d3=0.55 �m, d40.22 �m, w1=0.1 �m, w2=0.05 �m, �=10.748 �m, =1.5 �m, t=1, a1=s1=b1l1=� /2.

bRelative difference between the solutions for successive values of N.

1 /k0, �2 /k0 with respect to the lines N1 /k0+ /�, N2 /k0.ore precisely, the curve �1 /k0 ��2 /k0� has asymptotes,

he lines N1 /k0+ /� �N2 /k0� for large � / and N2 /k0N1 /k0+ /�� for small � /.

In addition, a comparative examination of the disper-ion curves of a symmetric grating-assisted coupler [seeigs. 5(a) and 6 in Ref. 24] with that of an asymmetricoupler [Fig. 3(a)] demonstrates that although the disper-ion curves �e /k0, �o /k0 of the symmetric coupler areearly parallel (with large distance ��e−�o� /k0), theurves �1 /k0, �2 /k0 of the asymmetric coupler approachne another near the resonant wavelength opt (withinimum distance ��1−�2� /k0 at opt) and diverge far from

opt (waist behavior). Thus, since an efficient exchange ofight between the slabs occurs only at those wavelengthsery close to opt, where the propagation constants of thewo grating modes become almost equal,3 the AGADCay be efficiently used as an optical bandpass filter.Furthermore, the attenuation constants �1 /k0, �2 /k0,

epicted in Fig. 3(b), are of sufficiently small order for all

ig. 3. (Color online) (a) Imaginary parts �1 /k0, �2 /k0 (solidurves) and N1 /k0+ /�, N2 /k0 (dashed curves) of the propaga-ion constants of the grating and compound modes and (b) realarts �1 /k0, �2 /k0 of the propagation constants of the gratingodes as functions [in both cases (a) and (b)] of the normalizedavelength � /, for 2d =0.55 �m, w =0.05 �m, n =3, n =3.1.
3 2 6 h,2

c(

BF�ci

==fanctdpi

w

mw

nrOst=

oc

sac�ww

Ffc

Ff(=


and intersect at =1.49 �m. However, the attenuationonstants of a symmetric grating coupler do not intersectsee Fig. 4(b) in Ref. 24).

. Influence of Grating Thicknessigures 4(a) and 4(b) show the imaginary parts �1 /k0,2 /k0 and the real parts �1 /k0, �2 /k0 of the propagationonstants of the grating modes as functions of the grat-ng’s thickness w2 for wavelengths =1.47 �m and=1.53 �m.The distance �� /k0 attains its minimal value at w2,opt

0.03 �m for =1.47 �m and at w2,opt=0.073 �m for 1.53 �m (according to Subsection 4.A, w2,opt=0.05 �m

or =1.4983 �m). Thus, the grating thickness may act asn additional control mechanism of the coupler’s reso-ance condition. In particular, the optimal value w2,opt in-reases with the operating wavelength . Also, the dis-ance �� /k0 increases and the coupling efficiencyecreases with w2�w2,opt. This last fact has also been re-orted in Refs. 18 and 19, without conducting a furthernvestigation on the dependence of w2,opt with respect to. Besides, for small enough w2�w2,opt and large enough

ig. 4. (Color online) (a) �1 /k0, �2 /k0 and (b) �1 /k0, �2 /k0 asunctions of the grating thickness w2, for =1.47 �m (dashed

3 6

2�w2,opt the respective parts of the curves for=1.47 �m and =1.53 �m are parallel lines. In the sym-etric case, it has been observed24 that wopt=0 �m for allavelengths (corresponding to the nongrating coupler).Furthermore, it is known18 that the grating thick-

esses that produce the best coupling efficiency lie in theanges of w2, where �� /k0= ��1−�2� /k0 has a minimum.n each of those ranges the attenuation attains the

mallest values. The curves of Fig. 5(b) show that the at-enuation constants �1 /k0 and �2 /k0 intersect at w2,opt0.031 �m for =1.47 �m and at w2,opt=0.0745 �m for=1.53 �m. These values of w2,opt are very close to thenes corresponding to the minimization of �� /k0 indi-ated in Fig. 4(a). Although the curves �1 /k0 and �2 /k0 for=1.47 �m intersect also at w2=0.1105 �m, the couplerhould not be designed with this grating thickness, sincet w2=0.1105 �m the distance �� /k0 is very large andonsequently the coupling efficiency very small. Moreover,1 /k0 attains a minimum and �2 /k0 a maximum near2,opt=0.031 �m, while the reverse situation occurs near

=0.0745 �m. This behavior is typical for a resonant

ig. 5. (Color online) (a) �1 /k0, �2 /k0 and (b) �1 /k0, �2 /k0 asunctions of the grating refractive index n6, for =1.5 �mdashed curves) and =1.53 �m (solid curves) with 2d30.55 �m, w2=0.05 �m.

2,opt
urves) and =1.53 �m (solid curves) with 2d =0.55 �m, n =3.

sti

CIptFm=pswart

a=ila

DTttttc6Nnrps

m�cd

e(−t(−tsd�3Ninect

EC==�ogtt−=bhctSim

tod

FNs


ystem and interprets the fact that near the resonanthickness w2,opt a stop band occurs for the one propagat-ng mode and a passband for the other.18

. Influence of Grating Refractive Indexn Figs. 5(a) and 5(b) the imaginary and real parts of theropagation constants as functions of the grating’s refrac-ive index n6 for =1.5 �m and =1.53 �m are depicted.or both wavelengths the distance �� /k0 attains its mini-um at n6=3�=n3�. The minimum distance 0.0008 for 1.5 �m is 1 order of magnitude smaller than 0.0039 for=1.53 �m. Thus, the grating provides more efficient cou-ling for =1.5 �m. Besides, �1 /k0 remains almost con-tant with respect to n6, while �2 /k0 strictly increasesith n6. Hence, the coupler becomes more asynchronouss n6 increases. In the case of the symmetric coupler24 theesonant grating refractive index coincides with that ofhe nongrating coupler.

Moreover, Fig. 5(b) indicates that the distance �� /k0ttains near n6=3 local minima 2�10−5 and 10−4 for 1.5 �m and =1.53 �m. Also, since for n6=n2 the grat-

ng coupler is reduced to the nongrating one composed ofossless media, in the neighborhood of n6=n2 both �1 /k0nd �2 /k0 approach zero.

. Natural Coupling between the Slabshe two distinct processes: (i) the natural coupling be-ween the two slabs (as in the case of the nonperiodic orhe nongrating coupler) and (ii) the periodic coupling dueo the grating perturbations (illustrated in Figs. 3–5), dic-ate the coupling of electromagnetic power in the gratingoupler.12 To study the natural coupling we plot in Fig.(a) the imaginary parts �1 /k0, �2 /k0 and N1

ng/k0,

2ng/k0 of the propagation constants of the grating and

ongrating (i.e., b1=0, l1=�, n6=n2) coupler modes withespect to the separation distance 2d3. Also, Fig. 6(b) de-icts the real parts �1 /k0, �2 /k0 of the propagation con-tants of the grating modes.

The distances ��1−�2� /k0 and ��1−�2� /k0 attain theirinimum value at the optimal separation distance

2d3�opt=0.537 �m. On the other hand, in the symmetricoupler, ��1−�2� /k0 is not minimized for any separationistance 2d3 [see Fig. 7(a) in Ref. 24].Now, referring to Fig. 6(b) we observe that for small

nough 2d3, ��1−�2� /k0 and �N1ng−N2

ng� /k0 are very closeas in the symmetric case). For large enough 2d3, �N1

ng

N2ng� /k0 is stabilized close to 0.013, the absolute value of

he difference of the propagation constants of the twonongrating) slabs, considered in isolation. However, ��1�2� /k0 is stabilized close to 0.005, the absolute value of

he difference of the propagation constants of the isolatedlab (with n4, d4) and the isolated grating slab (with n2,2; see Fig. 1 in the absence of slab 2). More precisely,1 /k0 and N2

ng/k0 approach the propagation constant.196 of the isolated slab (with n4, d4), while �2 /k0 and

1ng/k0 approach the propagation constants 3.201 of the

solated grating slab and 3.209 of the isolated slab (with2, d2). Furthermore, Fig. 6(b) indicates that, for largenough 2d3, �1 /k0 and �2 /k0 approach the attenuationonstants 0 of the isolated slab, composed of lossless ma-erial, and 3�10−4 of the isolated grating slab.

. Optimal Grating Period and Duty Cycleonsider the coupler with n1=1, n2=3.3, n3=n6=3.2, n43.5, n5=3, d2=1.05 �m, 2d3=0.55 �m, d4=0.3 �m, w20.1 �m, and =1.5 �m. The imaginary parts �1 /k0,2 /k0 and N1 /k0− /�, N2 /k0 of the propagation constantsf the grating and compound modes with respect to therating’s period � are depicted in Fig. 7. It is worth notinghe excellent agreement between the curves of Fig. 7 andhose of Fig. 2 in Ref. 15. The distance �� /k0= ��1�2� /k0 attains its minimum value 0.0011 at �opt31.287 �m (very close to 31.268 �m computed in Ref. 15y applying the Floquet–Bloch theory). On the otherand, the optimal grating period, predicted by theoupled-mode theory, occurs at �cm=31.463 �m, wherehe line N2 /k0 intersects N1 /k0− /�. As explained inubsection 4.A, the small difference between �opt and �cm

s due to the approximations imposed by the coupled-ode theory.Furthermore, to study the dependence of the propaga-

ion constants on the geometrical characteristics �bi , li , t�f the grating along the z axis, we introduce the gratinguty cycle

ig. 6. (Color online) (a) �1 /k0, �2 /k0 (solid curves) and N1ng/k0,

2ng/k0 (dashed curves), and (b) �1 /k0, �2 /k0 as functions of the

eparation distance 2d3, for w2=0.05 �m, =1.5 �m, n6=3.

Ftnmmm=ttid=pa

odctl

FIi==Mgtp=t

ptaatpms

F−f

Fps

Foo


dc =�i=1

t

li

�100 % .

igure 8 depicts the distance �� /k0= ��1−�2� /k0 as func-ion of the grating period � for varying duty cycles dc andumbers t of rectangles. Also, Table 2 presents the opti-al periods �opt and coupling lengths Lc, determining theinimum value of �� /k0, as indicated by Fig. 8. Theinimum value 2�10−4 of min�� /k0� corresponds to dc25%, t=2 and the minimum coupling length 0.682 mm

o dc=50%, t=1 (binary grating). Since efficient powerransfer is ensured only when the two modes have nearlydentical propagation constants, it is reasonable to choosec=25% and t=2. However, the large coupling length Lc3.75 mm, corresponding to dc=25% and t=2, might berohibitive (due to the limitations imposed by the modalttenuation) for the coupling of a substantial amount of

ig. 7. (Color online) �1 /k0, �2 /k0 (solid curves) and N1 /k0 /�, N2 /k0 (dashed curves) as functions of the grating period �,

or =1.5 �m, t=1, b1= l1=� /2, and nh,2=3.25.

ig. 8. (Color online) Distance �� /k0= ��1−�2� /k0 as a functionf the grating period �, for varying duty cycles dc and numbers tf rectangles.

ptical power between the two slabs. Thus, the choicesc=75% and t=1 or 2 seem in certain cases to be moreonvenient, because they combine a relatively small dis-ance �� /k0=8�10−4 with a not extremely large couplingength Lc=0.938 mm.

. Gratings on Both Slabsn this final Section, we investigate the coupler with grat-ngs on both slabs and n1=1, n2=3.3, n3=n5=3.2, n43.5, n6=n7=3.2, d2=1 �m, 2d3=1 �m, d4=0.3 �m, w1w2=0.1 �m, t=1, b1= l1=a1=s1=� /2, and =1.5 �m.arcuse3 has already analyzed the corresponding non-

rating coupler and noted that the fundamental modes ofhe two nongrating slabs have significantly differentropagation constants (N1

ng/k0=3.2559, N2ng/k0

3.3243), so that very little power can be exchanged be-ween the two slabs.

In Fig. 9 the imaginary parts �1 /k0 and �2 /k0 of theropagation constants of the grating modes with respecto the grating’s period � are depicted. The distance �� /k0ttains its minimum value 0.0011 at �opt=31.842 �m,nd the respective coupling length is Lc=0.683 mm. Onhe other hand, it is worth noting that the AGADC pro-osed by Marcuse4 exhibits a coupling length of approxi-ately Lc=5 mm [see Fig. 3(a) in Ref. 4]. However, it

eems that this large coupling length would not result in

ig. 9. (Color online) �1 /k0 and �2 /k0 as functions of the gratingeriod � for the grating-assisted coupler with gratings on bothlabs, considered in Subsection 4.F.

Table 2. Optimal Periods and Coupling Lengthsfor Varying Grating Duty Cycles and Number ofRectangles per Iteration as Indicated by Fig. 8

dc % t min�� /k0��opt

��m�Lc

(mm)

25 1 9�10−4 32.316 0.833 ��26 �opt�2 2�10−4 32.105 3.75 ��117 �opt�

50 1 1.1�10−3 31.263 0.682 ��22 �opt�2 3�10−4 31.684 2.5 ��79 �opt�

75 1 8�10−4 30.842 0.938 ��31 �opt�2 8�10−4 30.842 0.938 ��31 �opt�

astpnpsm

5Tgamhpc

rasgScaposttT

ATG


n efficient coupling of optical power between the twolabs, even in cases of couplers with moderate attenua-ion constants. The attenuation losses of the grating cou-ler proposed here with Lc=0.683 mm do not affect sig-ificantly the amount of transferred energy. Thus, theresence of the two gratings provides flexibility in the de-ign of grating couplers, leading to a significant improve-ent of the efficiency of the energy transfer mechanism.

. CONCLUSIONShe waveguiding and coupling phenomena in asymmetricrating-assisted optical couplers have been investigatednalytically by applying a rigorous integral equationethod. The basic advantages of this method include itsigh accuracy, numerical efficiency, and the analytic com-utation of the integrals involved. In contrast to theoupled-mode methods, the developed approach can accu-

ately address couplers with large grating thicknessesnd highly dissimilar slabs. The complex propagation con-tants of the two guided modes are determined versus therating period, duty cycle, thickness, and refractive index.everal numerical results reveal the optimal grating’sharacteristics, leading to maximum coupling efficiencynd moderate coupling length. Finally, we note that theroposed method may be extended to couplers consistingf finite gratings31 by considering a Fourier series expan-ion different from that in Eq. (20) in each grating’s itera-ion and a special type of basis functions in the grating’serminations to satisfy the boundary edge conditions.his problem will be investigated in a future work.

PPENDIX Ahe kernels 1 and 2 of the Fourier integrals of G1 and

are given by
2
1�,x,x�� = ��g3,g5,g4,d4��g1,g3,g2,d2� − K�g3,g5,g4,d4�K�g3,g1,g2,d2��2�−1��g3g2Po�g5,g4,d4�Pe�g5,g4,d4�

�� cosh�g4�x� + d3 +d4

2 �Pe�g5,g4,d4�

+

sinh�g4�x� + d3 +d4

2 �Po�g5,g4,d4�

�exp�− g1�x − d3 − d2��, x d3 + d2,

Po�g1,g2,d2�Pe�g1,g2,d2�Po�g5,g4,d4�Pe�g5,g4,d4�g3�� cosh�g4�x� + d3 +d4

2 �Pe�g5,g4,d4�

+


2 �Po�g5,g4,d4�

�� cosh�g2�x − d3 −

d2

2 �Pe�g1,g2,d2�

−

sinh�g2�x − d3 −d2

2 �Po�g1,g2,d2�

�, d3 � x � d3 + d2,

��g3,g1,g2,d2�exp�− g3�x + d3�� + �K�g3,g1,g2,d2�exp�g3�x − d3��Po�g5,g4,d4�Pe�g5,g4,d4�� cosh�g4�x� + d3 +d4

2 �Pe�g5,g4,d4�

+


2 �Po�g5,g4,d4�

�, − d3 � x � d3,

�g4�−1K�g3,g1,g2,d2�Qe�g3,g4,d4�Pe�g5,g4,d4�Qo�g3,g4,d4�Po�g5,g4,d4��2� cosh�g4�x� + d3 +d4

2 �Pe�g5,g4,d4�

+


2 �Po�g5,g4,d4�

�� cosh�g4�x + d3 +

d4

2 �Qe�g3,g4,d4�

+

sinh�g4�x + d3 +d4

2 �Qo�g3,g4,d4�

�+ �g4�−1��g1,g3,g2,d2�Pe�g3,g4,d4�Pe�g5,g4,d4�Po�g5,g4,d4�Po�g3,g4,d4�� cosh�g4�x� + d3 +

d4

2 �Pe�g5,g4,d4�


+


2 �Po�g5,g4,d4�

�� cosh�g4�x + d3 +d4

2 �Pe�g3,g4,g4�

−


2 �Po�g3,g4,d4�

�, − d3 − d4 � x� � x � − d3,

�g4�−1K�g3,g1,g2,d2�Qe�g3,g4,d4�Pe�g5,g4,d4�Qo�g3,g4,d4�Po�g5,g4,d4��2� cosh�g4�x� + d3 +d4

2 �Qe�g3,g4,d4�

+


2 �Qo�g3,g4,d4�

�� cosh�g4�x + d3 +

d4

2 �Pe�g5,g4,d4�

+


2 �Po�g5,g4,d4�

�+ �g4�−1��g1,g3,g2,d2�Pe�g3,g4,d4�Pe�g5,g4,d4�Po�g5,g4,d4�Po�g3,g4,d4�� cosh�g4�x� + d3 +

d4

2 �Pe�g3,g4,d4�

−


2 �Po�g3,g4,d4�

�� cosh�g4�x + d3 +d4

2 �Pe�g5,g4,d4�

+


2 �Po�g5,g4,d4�

�, − d3 − d4 � x � x� � − d3,

K�g3,g1,g2,d2�Qo�g3,g4,d4�Qe�g3,g4,d4��2� cosh�g4�x� + d3 +d4

2 �Qe�g3,g4,d4�

+


2 �Qo�g3,g4,d4�

�exp�g5�x + d3 + d4��

+ ��g1,g3,g2,d2�Po�g3,g4,d4�Pe�g3,g4,d4�� cosh�g4�x� + d3 +d4

2 �Pe�g3,g4,d4�

−


2 �Po�g3,g4,d4�

��exp�g5�x + d3 + d4��,� x � − d3 − d4, �A1�

2�,x,x�� = ��g3,g1,g2,d2��g5,g3,g4,d4� − K�g3,g1,g2,d2�K�g3,g5,g4,d4��2�−1�K�g3,g5,g4,d4��2

��Qo�g3,g2,d2�cosh�g2�x� − d3 −d2

2 � − Qe�g3,g2,d2�sinh�g2�x� − d3 −d2

2 ��exp�− g1�x − d3 − d2�� + ��g5,g3,g4,d4��Po�g3,g2,d2�cosh�g2�x� − d3 −

d2

2 �+ Pe�g3,g2,d2�sinh�g2�x� − d3 −

d2

2 ��exp�− g1�x − d3 − d2��, x d3 + d2,

�g2�−1K�g3,g5,g4,d4�Pe�g1,g2,d2�Po�g1,g2,d2�Qe�g3,g2,d2�Qo�g3,g2,d2��2� cosh�g2�x� − d3 −d2

2 �Qe�g3,g2,d2�

−

sinh�g2�x� − d3 −d2

2 �Qo�g3,g2,d2�

�� cosh�g2�x − d3 −d2

2 �Pe�g1,g2,d2�

−


2 �Po�g1,g2,d2�

�

T


+ �g2�−1��g5,g3,g4,d4�Pe�g1,g2,d2�Po�g1,g2,d2�Pe�g3,g2,d2�Po�g3,g2,d2�� cosh�g2�x� − d3 −d2

2 �Pe�g3,g2,d2�

+


2 �Po�g3,g2,d2�

�� cosh�g2�x − d3 −d2

2 �Pe�g1,g2,d2�

−


2 �Po�g1,g2,d2�

�, d3 � x� � x � d3 + d2,

�g2�−1K�g3,g5,g4,d4�Pe�g1,g2,d2�Po�g1,g2,d2�Qe�g3,g2,d2�Qo�g3,g2,d2��2� cosh�g2�x� − d3 −d2

2 �Pe�g1,g2,d2�

−


2 �Po�g1,g2,d2�

�� cosh�g2�x − d3 −d2

2 �Qe�g3,g2,d2�

−


2 �Qo�g3,g2,d2�

�+ �g2�−1��g5,g3,g4,d4�Pe�g1,g2,d2�Po�g1,g2,d2�Pe�g3,g2,d2�Po�g3,g2,d2�� cosh�g2�x� − d3 −

d2

2 �Pe�g1,g2,d2�

−


2 �Po�g1,g2,d2�

�� cosh�g2�x − d3 −d2

2 �Pe�g3,g2,d2�

+


2 �Po�g3,g2,d2�

�, d3 � x � x� � d3 + d2,

��g3,g5,g4,d4�exp�g3�x − d3�� + �K�g3,g5,g4,d4�exp�− g3�x + d3��Po�g1,g2,d2�Pe�g1,g2,d2�� cosh�g2�x� − d3 −d2

2 �Pe�g1,g2,d2�

−


2 �Po�g1,g2,d2�

�, − d3 � x � d3,

�g3Pe�g1,g2,d2�Po�g1,g2,d2�Pe�g5,g4,d4�Po�g5,g4,d4�� cosh�g2�x� − d3 −d2

2 �Pe�g1,g2,d2�

−


2 �Po�g1,g2,d2�

�� cosh�g4�x + d3 +

d4

2 �Pe�g5,g4,d4�

+


2 �Po�g5,g4,d4�

�, − d3 − d4 � x � − d3,

�g3g4Pe�g1,g2,d2�Po�g1,g2,d2�� cosh�g2�x� − d3 −d2

2 �Pe�g1,g2,d2�

−


2 �Po�g1,g2,d2�

�exp�g5�x + d3 + d4��, x � − d3 − d4.

�A2�

he functions Po, Pe, Qo, Qe, K, � and �, appearing in Eqs. (A1) and (A2), are defined by

Pe�gi,gj,d� = gi cosh�gj�d/2�� + gj sinh�gj�d/2��,

Po�gi,gj,d� = gj cosh�gj�d/2�� + gi sinh�gj�d/2��,

Qe�gi,gj,d� = gi cosh�gj�d/2�� − gj sinh�gj�d/2��,

Q �g ,g ,d� = g cosh�g �d/2�� − g sinh�g �d/2��,
o i j j j i j

w

AIu


K�gi,gj,gk,d� =Qe�gi,gk,d�Po�gj,gk,d� − Qo�gi,gk,d�Pe�gj,gk,d�

2,

��gi,gj,gk,d� =Pe�gi,gk,d�Po�gj,gk,d� + Po�gi,gk,d�Pe�gj,gk,d�

2,

� = exp�− 2g3d3�, �A3�

here 1� i, j, k�7.

PPENDIX Bn this appendix we define certain auxiliary functionssed for the formulation of the method’s linear system.

Jq1 =

1

��i=1

t �ai

ai+si

exp�j2�q

�� d��,

Jq2 =

1

��i=1

t �bi

bi+li

exp�j2�q

�� d��. �B1�

Qnp1±�x� =�

−d3−w1

−d3

exp�±g7,n�x� + d3 +w1

2 ��1�− j� +

2�p

�,x,x� dx�. �B2�

Qnp2±�x� =�

d3

d3+w2

exp�±g6,n�x� − d3 −w2

2 ��2�− j� +

2�p

�,x,x� dx�. �B3�

Kmn1±± =�

−d3−w1

−d3


2 ��exp�±g7,n�x + d3 +

w1

2 �dx. �B4�

Qmnp1i±± =

1

2�−d3−w1

−d3


2 ��Qnp

i± �x�dx �i = 1,2�. �B5�

Kmn2±± =�

d3

d3+w2


2 ��exp�±g6,n�x − d3 −

w2

2 �dx. �B6�

Qmnp2i±± =

1

2�d3

d3+w2


2 ��Qnp

i± �x�dx �i = 1,2�. �B7�

Rmnp1±± =

1

2�g7,m2 − g4,p

2 �

��g1,p,g3,p,g2,p,d2�Pe�g3,p,g4,p,d4�Pe�g5,p,g4,p,d4�Po�g5,p,g4,p,d4�Po�g3,p,g4,p,d4�

g4,p��g3,p,g5,p,g4,p,d4��g1,p,g3,p,g2,p,d2� − K�g3,p,g5,p,g4,p,d4�K�g3,p,g1,p,g2,p,d2��2�

��exp��±g7,m ± g7,n�w1

2 ��Pe��g7,m,g4,p,d4�

Pe�g5,p,g4,p,d4�+

Po��g7,m,g4,p,d4�

Po�g5,p,g4,p,d4� ��Pe��g7,n,g4,p,d4�

Pe�g3,p,g4,p,d4�

−Po��g7,n,g4,p,d4�

Po�g3,p,g4,p,d4� � − exp��±g7,m � g7,n�w1

2 ��Pe��g7,m,g4,p,d4�

Pe�g3,p,g4,p,d4�−


Po�g3,p,g4,p,d4� ��Pe��g7,n,g4,p,d4 − 2w1�


Po��g7,n,g4,p,d4 − 2w1�

Po�g5,p,g4,p,d4� � − exp��g7,m ± g7,n�w1

2 ��Pe��g7,m,g4,p,d4 − 2w1�


Po��g7,m,g4,p,d4 − 2w1�

Po�g5,p,g4,p,d4� ��Pe��g7,n,g4,p,d4�


Po��g7,n,g4,p,d4�

Po�g3,p,g4,p,d4� �+ exp��g7,m � g7,n�

w1

2 ��Pe��g7,m,g4,p,d4 − 2w1�


Po��g7,m,g4,p,d4 − 2w1�



Po��g7,n,g4,p,d4 − 2w1�

Po�g5,p,g4,p,d4� �� . �B8�


Rmnp2±± =

1

2�g6,m2 − g2,p

2 �

��g5,p,g3,p,g4,p,d4�Pe�g1,p,g2,p,d2�Po�g1,p,g2,p,d2�Pe�g3,p,g2,p,d2�Po�g3,p,g2,p,d2�


��exp��±g6,m ± g6,n�w2

2 ��Pe�±g6,m,g2,p,d2 − 2w2�


Po�±g6,m,g2,p,d2 − 2w2�

Po�g3,p,g2,p,d2� ��Pe�±g6,n,g2,p,d2 − 2w2�


+Po�±g6,n,g2,p,d2 − 2w2�


2 ��Pe�±g6,m,g2,p,d2 − 2w2�


+Po�±g6,m,g2,p,d2 − 2w2�

Po�g1,p,g2,p,d2� ��Pe�±g6,n,g2,p,d2�


Po�±g6,n,g2,p,d2�

Po�g3,p,g2,p,d2� � − exp��g6,m ± g6,n�w2

2 ��Pe�±g6,m,g2,p,d2�


Po�±g6,m,g2,p,d2�



Po�±g6,n,g2,p,d2 − 2w2�

Po�g1,p,g2,p,d2� �+ exp��g6,m � g6,n�

w2

2 ��Pe�±g6,m,g2,p,d2�



Po�g1,p,g2,p,d2� ��Pe�±g6,n,g2,p,d2�


−Po�±g6,n,g2,p,d2�

Po�g3,p,g2,p,d2� �� . �B9�

Hmnp11±± =

1

2�g7,m2 − g4,p

2 �

K�g3,p,g1,p,g2,p,d2�Qe�g3,p,g4,p,d4�Pe�g5,p,g4,p,d4�Qo�g3,p,g4,p,d4�Po�g5,p,g4,p,d4��2


��exp��±g7,m ± g7,n�w1

2 ��Pe��g7,m,g4,p,d4�



Po�g5,p,g4,p,d4� ��Po��g7,n,g4,p,d4�

Qo�g3,p,g4,p,d4�

+Pe��g7,n,g4,p,d4�

Qe�g3,p,g4,p,d4� � − exp��g7,m ± g7,n�w1

2 ��Pe��g7,m,g4,p,d4 − 2w1�


Po��g7,m,g4,p,d4 − 2w1�

Po�g5,p,g4,p,d4� ��Po��g7,n,g4,p,d4�

Qo�g3,p,g4,p,d4�+

Pe��g7,n,g4,p,d4�

Qe�g3,p,g4,p,d4� � − exp��±g7,m � g7,n�w1

2 ��Pe��g7,m,g4,p,d4�

Qe�g3,p,g4,p,d4�

+Po��g7,m,g4,p,d4�

Qo�g3,p,g4,p,d4� ��Pe��g7,n,g4,p,d4 − 2w1�


Po��g7,n,g4,p,d4 − 2w1�

Po�g5,p,g4,p,d4� � + exp��g7,m � g7,n�w1

2 ��Pe��g7,m,g4,p,d4 − 2w1�

Qe�g3,p,g4,p,d4�+

Po��g7,m,g4,p,d4 − 2w1�

Qo�g3,p,g4,p,d4� ��Pe��g7,n,g4,p,d4 − 2w1�


+Po��g7,n,g4,p,d4 − 2w1�

Po�g5,p,g4,p,d4� �� . �B10�

Hmnp12±± =

1

2�g7,m2 − g4,p

2 �

�g3,pPe�g1,p,g2,p,d2�Po�g1,p,g2,p,d2�Pe�g5,p,g4,p,d4�Po�g5,p,g4,p,d4�

��g3,p,g5,p,g4,p,d4��g1,p,g3,p,g2,p,d2� − K�g3,p,g5,p,g4,p,d4�K�g3,p,g1,p,g2,p,d2��2�

��exp�±g7,m�w1

2 ± g6,n�w2

2 ��Pe��g7,m,g4,p,d4�





+Po�±g6,n,g2,p,d2 − 2w2�

Po�g1,p,g2,p,d2� � − exp�±g7,m�w1

2 � g6,n�w2

2 ��Pe��g7,m,g4,p,d4�



Po�g5,p,g4,p,d4� ��Pe�±g6,n,g2,p,d2�



Po�g1,p,g2,p,d2� � − exp��g7,m�w1

2 ± g6,n�w2

2 ��Pe��g7,m,g4,p,d4 − 2w1�


+Po��g7,m,g4,p,d4 − 2w1�



Po�±g6,n,g2,p,d2 − 2w2�

Po�g1,p,g2,p,d2� �

af

R


+ exp��g7,m�w1

2 � g6,n�w2

2 ��Pe��g7,m,g4,p,d4 − 2w1�


Po��g7,m,g4,p,d4 − 2w1�

Po�g5,p,g4,p,d4� ��Pe�±g6,n,g2,p,d2�



Po�g1,p,g2,p,d2� �� . �B11�

Hmnp21±± =

1

2�g6,m2 − g2,p

2 �

Po�g1,p,g2,p,d2�Pe�g1,p,g2,p,d2�Po�g5,p,g4,p,d4�Pe�g5,p,g4,p,d4�g3,p�

��g3,p,g5,p,g4,p,d4��g1,p,g3,p,g2,p,d2� − K�g3,p,g5,p,g4,p,d4�K�g3,p,g1,p,g2,p,d2��2�

��exp�±g6,m�w2

2 ± g7,n�w1

2 ��Pe�±g6,m,g2,p,d2 − 2w2�


Po�±g6,m,g2,p,d2 − 2w2�

Po�g1,p,g2,p,d2� ��Pe��g7,n,g4,p,d4�


Po��g7,n,g4,p,d4�

Po�g5,p,g4,p,d4� � − exp�±g6,m�w2

2 � g7,n�w1

2 ��Pe�±g6,m,g2,p,d2 − 2w2�


+Po�±g6,m,g2,p,d2 − 2w2�



Po��g7,n,g4,p,d4 − 2w1�

Po�g5,p,g4,p,d4� �− exp��g6,m�w2

2 ± g7,n�w1

2 ��Pe�±g6,m,g2,p,d2�



Po�g1,p,g2,p,d2� ��Pe��g7,n,g4,p,d4�


+Po��g7,n,g4,p,d4�

Po�g5,p,g4,p,d4� � + exp��g6,m�w2

2 � g7,n�w1

2 ��Pe�±g6,m,g2,p,d2�





Po��g7,n,g4,p,d4 − 2w1�

Po�g5,p,g4,p,d4� �� . �B12�

Hmnp22±± =

1

2�g6,m2 − g2,p

2 �

K�g3,p,g5,p,d4�Pe�g1,p,g2,p,d2�Po�g1,p,g2,p,d2�Qe�g3,p,g2,p,d2�Qo�g3,p,g2,p,d2��2


��exp��±g6,m ± g6,n�w2

2 ��Pe�±g6,m,g2,p,d2 − 2w2�


Po�±g6,m,g2,p,d2 − 2w2�

Qo�g3,p,g2,p,d2� ��Pe�±g6,n,g2,p,d2 − 2w2�


+Po�±g6,n,g2,p,d2 − 2w2�


2 ��Pe�±g6,m,g2,p,d2 − 2w2�


+Po�±g6,m,g2,p,d2 − 2w2�

Po�g1,p,g2,p,d2� ��Pe�±g6,n,g2,p,d2�



Qo�g3,p,g2,p,d2� � − exp��g6,m ± g6,n�w2

2 ��Pe�±g6,m,g2,p,d2�



Qo�g3,p,g2,p,d2� ��Pe�±g6,n,g2,p,d2 − 2w2�


Po�±g6,n,g2,p,d2 − 2w2�

Po�g1,p,g2,p,d2� �+ exp��g6,m � g6,n�

w2

2 ��Pe�±g6,m,g2,p,d2�



Po�g1,p,g2,p,d2� ��Pe�±g6,n,g2,p,d2�

Qe�g3,p,g2,p,d2�

+Po�±g6,n,g2,p,d2�

Qo�g1,p,g2,p,d2� �� . �B13�

Address correspondence to Nikolaos L. Tsitsas at theddress on the title page or by phone, 30-210-772-2467;ax, 30-210-772-3557; or e-mail, [email protected].

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Rigorous integral equation analysis of nonsymmetric coupled grating slab waveguides