Matlab Implementation of the Extended Marcatili ...

Matlab Implementation of the Extended Marcatili approaches for rectangular silicon optical waveguides Wouter J. Westerveld1, 2 and H. Paul Urbach1 1

Optics Research Group, Faculty of Applied Sciences, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands 2 Nano-instrumenation Expertise Group, Technical Sciences, TNO, Stieltjesweg 1, 2628 CK Delft, The Netherlands (Dated: November 1, 2012) We recently extended Marcatili’s approximate analytical approach for the description of light propagation in rectangular waveguides to the regime of (silicon) high-index-contrast waveguides [1]. In this document, we explain the Matlab implementation of the equations in that paper. I.

We recently extended Marcatili’s approximate analytical approach for the description of light propagation in rectangular waveguides to the regime of high-indexcontrast waveguides [1]. The method is implemented in Matlab [2] and provides a very fast approximate modesolver for rectangular waveguides. The Matlab files are documented themselves on how to use them, so that this document only describes the equations that are used in the Matlab .m-files. The best starting point for this implementation is the file rectwg example.m, as it provides an example on how to use the functions (.m-files) of this implementation. A list of the most relevant Matlab functions (.m-files) with a brief explanation is given in Table I. The most relevant differences between the variable names in the .m-files and the theory are shown in Table II. Throughout the document, references to figures and equations in Ref. [1] are indicate with “JLT”. Understanding of that work is required prior to reading this document. In Sec. III, the functions related to the computation of the free parameters in the Ansatz are explained. This corresponds to the methods presented in Sec. JLT.II-B to D. In Sec. IV, the full electromagnetic field is described in terms of variable Ψ which plays an important role in the implementation. Then, the energy, flux, and energy of the difference field are computed from Ψ.

II.

III.

INTRODUCTION

WAVEGUIDE GEOMETRY

In the implementation, we follow the assumption that the dominant electric field component is Ex . The waveguide should be chosen such that this is the case for the mode under study. An example for a TM-like and a TElike mode is given in Figures (JLT.1c) and (JLT.1d), respectively. For a accurate comparison between the numerical and the analytical fields, the size of regions 2-5 is reduced not to extend into infinity. Regions 2 and 3 are chosen to extend wx outside the waveguide, while regions 4 and 5 extend wy outside the core of the waveguide.

COMPUTATION OF ANSATZ PARAMETERS

In Section JLT.II-B, C, and D, different methods to compute the free parameters of the Ansatz, are presented. These methods are implemented in the Matlab functions rectwg AnsatzMethods1 and rectwg bcfitfun. In this Section, we detail the theory that is used in these scripts: rectwg AnsatzMethods1: computes the free parameters in the Ansatz via various methods. rectwg bcfitfun: computes the residual mismatch Umm at the boundaries.

A.

Computation of mismatch Umm

The computation of the mismatch Umm via Eq. (JLT.53) is detailed in this Section. First of all, the contributions of the four interfaces and of the four tangential field components are separated, thus 16 contributions in total. The mismatch is the sum of the individual contributions. As example, we compute the contribution of Ex on the 1-4 interface. In region 1, Ex is given by Eq. (JLT.10) as Ex− =

βA1 kx + ωµ0 A2 ky cos[kx (x + ξ)] cos[ky (y + η)]. ıK12

Similarly, Ex+ in region 4 is computed from Eqs. (JTL.18), (JLT.19) and (JLT.2) as Ex+ =

βA7 kx + ωµ0 A8 γ4 cos[kx (x + ξ)] exp[γ4 (y + b/2)]. ıK42

At the 1-4 interface, y = −b/2, this gives Ex− =

βA1 kx + ωµ0 A2 ky cos[ky (η − b/2)] · cos[kx (x + ξ)], ıK12

and Ex+ =

βA7 kx + ωµ0 A8 γ4 · cos[kx (x + ξ)], ıK42

2 respectively. The contribution of this component on this interface to Umm is labeled number 9. Employing the (9) latter two Equations in Eq. (JLT.53) gives Umm

(9) Umm

ǫ0 (n1 + n4 )2 = 4 = =

ǫ0 (n1 + n4 )2 4 ǫ0 (n1 + n4 )2 4

Z

d/2 −d/2

− Ex − Ex+ 2 dx

(1)

(βA1 kx + ωµ0 A2 ky ) cos[ky (η − b/2)] βA7 kx + ωµ0 A8 γ4 − K12 K42 (βA1 kx + ωµ0 A2 ky ) cos[ky (η − b/2)] βA7 kx + ωµ0 A8 γ4 − K12 K42

Likewise, the other contributions to Umm are computed, and shown in Table III. The summation reads Umm

cos2 [kx (x + ξ)]dx

(4)

Many of the functions in the implementation use the variable Ψ which describes the modal fields. Although it might seem a bit cumbersome, the computations of, for example, the energy and the flux become very small after this introduction. Also the computation of the field components, for example for plotting, becomes easy. In this section, we detail the Equations used in these .m-files: rectwg getPsi: compute Ψ from the parameters of the Ansatz rectwg getFieldsPsi: compute the full electromagnetic fields from Ψ, e.g. used for plotting. rectwg getEnergyPsi: compute energy carried by the modal field. rectwg getFluxPsi: compute power flux through the mode. rectwg getEnergyDifferencePsiFields: compute energy in the differnce field between an approximate analytically computed mode and a rigorous numerically computed mode. rectwg getIntegralsI2: compute integrals in Table IV rectwg getIntegralsI1: compute integrals in Table V

Description of the fields with Ψ

For a convenient numerical implementation of the Equations in this work, we define the rather abstract

(2)

−d/2

2

d cos[2kx ξ] sin[kx d] + 2 2kx

tc ,m

Ψc,m (x, y) = C c,m Txx

IV. COMPUTATION OF THE FLUX, THE ENERGY, THE ENERGY DIFFERENCE, ETC.

A.

d/2

. (3)

variable Ψ which describes the field in all regions

16

1 X (j) = U . l j=1 mm

2 Z

tc ,m

(x)Ty y

(y),

(5)

in which c

index of the field component. Ex = 1, Ey = 2, Ez = 3, Hx = 4, Hy = 5, Hz = 6. m index of the region (medium) 1 - 5. tx label (sin or cos), indicating the the shape of the field component c in the x-direction in the core of the waveguide. ty likewise, but describing the shape in the ydirection. C amplitude coefficient of a field coefficient c in region m. Tx (x) trigonometric function describing the shape of the field in the x-direction. Ty (y) trigonometric function describing the shape of the field in the y-direction. The function Tx is defined as Txsin,1 = Txsin,4 = Txsin,5 = sin[kx (x + η)],

(6)

Txsin,2 Txsin,3 Txcos,5 Txcos,2 Txcos,3

= exp[γ2 (x + d/2)],

(7)

= exp[−γ3 (x − d/2)],

(8)

= cos[kx (x + ξ)],

(9)

Txcos,1

=

Txcos,4

=

= exp[γ2 (x + d/2)],

(10)

= exp[−γ3 (x − d/2)],

(11)

while Ty is defined as Tysin,1 = Tysin,2 = Tysin,3 = sin[ky (y + ξ)],

(12)

Tysin,4 Tysin,5 Tycos,5 Tycos,4 Tycos,5

= exp[γ4 (y + b/2)],

(13)

= exp[−γ5 (y − d/2)],

(14)

= cos[kx (x + ξ)],

(15)

= exp[γ4 (y + b/2)],

(16)

= exp[−γ5 (y − b/2)].

(17)

Tycos,1

=

Tycos,4

=

3 The shape of the fields tx and ty is defined by the Ansatz, and the amplitudes C c,m are computed from the free parameters. In region 1, these shapes and amplitudes can be red from Eqs. (JLT.8) to (JLT.13), while the amplitudes in the other regions can be computed from Eqs. (JLT.14) to (JLT.21) by employing Eqs. (JLT.2) to (JLT.5). The shapes tx (c) and ty (c) are t(1) x = cos

t(1) y = cos

(Ex ),

(18)

t(2) x = sin

t(2) y = sin

(Ey ),

(19)

t(3) x = sin

t(3) y = cos

(Ez ),

(20)

t(4) x = sin

t(4) y = sin

(Hx ),

t(5) x = cos

t(5) y = cos

(Hy ),

(22)

t(6) x = cos

t(6) y = sin

(Hz ).

(23)

The amplitudes C

c,m

C

1,2

C

1,3

= −(βA3 γ2 +

ωµ0 A4 ky )/K22 ,

(24)

(27)

C 2,1 = −(−βA1 ky + ωµ0 A2 kx )/K12 ,

(29)

C 1,4 = C 1,5 =

(26) (28)

for Ey

C

2,2

=

C

2,3

=

C

2,4

=

C

2,5

=

−(−βA3 ky − ωµ0 A4 γ2 )/K22 , −(−βA5 ky + ωµ0 A6 γ3 )/K32 , −(βA7 γ4 + ωµ0 A8 kx )/K42 , −(−βA9 γ5 + ωµ0 A10 kx )/K52 ,

(30) (31) (32) (33)

C

5,2

=

C

5,3

=

C

5,4

=

C

5,5

=

−(βA4 ky + ωǫ0 n22 A3 γ2 )/K22 , −(βA6 ky − ωǫ0 n23 A5 γ3 )/K32 , −(βA8 γ4 + ωǫ0 n24 A7 kx )/K42 , −(−βA10 γ5 + ωǫ0 n25 A9 kx )/K52 ,

(44) (45) (46) (47) (48)

for Hz C 3,1 = A2 ,

(49)

C

3,2

= A4 ,

(50)

C

3,3

= A6 ,

(51)

C

3,4

= A8 ,

(52)

C

3,5

= A10 .

(53)

Note that the amplitudes C are chosen real. This needs attention for e.g. the computation of the energy flux. As example, we show how Ex is described in Ψ, i.e.

(25)

−(−βA5 γ3 + ωµ0 A6 ky )/K32 , −(βA7 kx + ωµ0 A8 γ4 )/K42 , −(βA9 kx − ωµ0 A10 γ5 )/K52 ,

=

C 5,1 = −(βA2 ky + ωǫ0 n21 A1 kx )/K12 ,

(21)

are for Ex

C 1,1 = −(βA1 kx + ωµ0 A2 ky )/K12 ,

for Hy

Ψ1,m (x, y) = C 1,m Txcos,m (x)Tycos,m (y).

(54)

The Ex field component has a cosinusoidal shape in the x- and the y-direction as shown in Fig. JLT.1 and in Eq. (JLT.10). In the Ψ notation, this is indicated in (1) (1) Eqs. (18) by the labels tx = cos and ty = cos. The shape of Ex in all regions 1-5 is given by Eqs. (9)-(11) and Eqs. (15)-(17), which can be compared with Fig. JLT.1. Although the full electromagnetic field can be described by the parameters in the Ansatz, we explicitly computed the amplitudes of all field components from the parameters in Eq. (24)-(28). Additional to Ψ, we also introduce ǫ0 n2m E components (c=1-3), medium m Ωc,m ≡ . µ0 H components (c=4-6) (55)

for Ez B.

C 3,1 = A1 ,

(34)

C 3,2 = A3 ,

(35)

C 3,3 = A5 ,

(36)

C

3,4

= A7 ,

(37)

C

3,5

= A9 ,

(38)

Description of the numerical field

The numerically computed electrical field E N is discretized as X E N (x, y) = Φi,j lxi (x)lyj (y), (56) i,j

with x − xi , ∆x y − yi lyj (y) = Π , ∆y

lxi (x) = Π

for Hx C 4,1 = −(−βA2 kx + ωǫ0 n21 A1 ky )/K12 , C

4,2

=

C

4,3

=

C 4,4 = C

4,5

=

−(βA4 γ2 + ωǫ0 n22 A1 ky )/K22 , −(−βA6 γ3 + ωǫ0 n23 A5 ky )/K32 , −(−βA8 kx − ωǫ0 n24 A7 γ4 )/K42 , −(−βA10 kx + ωǫ0 n25 A9 γ5 )/K12 ,

(39) (40) (41) (42) (43)

(57) (58)

where xi = x0 + i∆x, yj = y0 + j∆y,

(59) (60)

4 and Π is the rect function   0 if |t| > 12 , 1 Π(t) = if |t| = 21 ,  2 1 if |t| < 21 .

=

C.

− C 2,m C 4,m Ix2,sin,m Iy2,sin,m .

(61)

In general, the full numerical electromagnetic fields E N and H N are described by Φc,i,j with c the electromagnetic field component (Ex = 1, Ey = 2, etc) and i, j the discretized spatial coordinate. Computation of the energy carried by the fields, and the power flux of the fields is now an easy summation of the energy or the flux over all the spatial positions [i,j] times ∆x∆y.

5 1 X 1,m 5,m 2,cos,m 2,cos,m C C Ix Iy 2 m=1

(68)

The Ex and Hy field components have a cosinusoidal shape, see Eqs. (18) and (22), while the Ey and Hx components have a sinusoidal shape, see Eqs. (19) and (21), which allows employing the integrals in Eqs. (92) to (99). The amplitudes C in the Ψ description where chosen real, which needs to be taken into account in this computation. However, the amplitudes of the transversal electromagnetic fields are purely imaginary, such that this cancels out in Eq. (67) as ı · ı∗ = 1.

Computation of the energy of the fields

The energy U of the electromagnetic field ZZ ZZ µ0 |H|2 dxdy, ǫ0 n2 |E|2 dxdy + U= regions 1-5

regions 1-5

(62) =

6 5 X X

Ωc,m

m=1 c=1

=

5 X

6 X

ZZ

|Ψc,m |2 dxdy,

Ωc,m |C c,m |2 · Ix

2,tc ,m Iy y ,

Computation of the energy of the difference field

(63)

region m 2,tcx ,m

E.

The unnormalized energy of the difference field U∆ is given from Eq. (JLT.61) as (64)

m=1 c=1

in which 2,tcx ,m

Ix

2,tc ,m Iy y

= =

Z

Z

U∆ = tc ,m 2

(Txx

) dx,

(65)

region m tc ,m (Ty y )2 dy.

(66)

region m

For example, the integral for a cosine-shaped field component in region 1 reads Ix2,cos,1

=

Z

d/2

cos2 [kx (x+ξ)]dx = −d/2

regions 1-5

n2 ǫ0 |E A − E N |2 + µ0 |H A − H N |2 dxdy

(69) In the description that we have, and also in the numerical fields that are used, the fields are real. We will now write the last Equation is written in terms of Ψ and Φ. First, we introduce

d cos[2kx ξ] sin[kx d] + . 2 2kx

The full list of integrals, which depend on the shape (sin/cos) and the region m, is given in Table IV. D.

ZZ

Computation of the power flux

2,tc ,m Ix x (xi , yi )

=

xi +∆x/2 Z

Tx (x)dx,

(70)

yi +∆y/2 Z

Ty (y)dy..

(71)

xi −∆x/2 2,tc ,m Iy y (xi , yi )

=

yi −∆y/2

The power flux P carried by the fields through the waveguide (JLT.44) is given by   ZZ   1 (67) P = Re Ex Hy∗ − Ey Hx∗ dxdy  2  regions 1-5

These integrals are explicitly given in Table V. Equation (69) is formulated in terms of Ψ and Φ as

5

U∆ =

6 5 X X

Ω

m=1 c=1 5

=

m=1 c=1

=

ZZ

C

Tx (x)Ty (y) −

Ω

c,m

ZZ

X

(C

c,m 2

)

Tx2 (x)Ty2 (y)

+Ω

c,m

Ωc,m (C c,m )2

Z

Tx2 (x)dx

m

| V.

X

2,tc ,m Ix x

}|

dxdy

!2

−Ω

c,m

2C

c,m

Ty2 (y)dy +Ωc,m

X i,j

{z

2,tc y ,m Iy

}

COPYRIGHT

Copyright (c) 2012, Wouter Westerveld (Delft University of Technology) Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the ”Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: Authors of scientific papers with results that benefitted from this Sofware shall cite the original paper: W.J. Westerveld, S.M. Leinders, K.W.A. van Dongen, H.P. Urbach, M. Yousefi, ”Extension of Marcatili’s Analytical Approach for Rectangular Silicon Optical Waveguides”, IEEE/OSA Journal of Lightwave Technology, vol. 30, pp. 2388-2401, July 2012. The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED ”AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR

[1] W. Westerveld, S. Leinders, K. van Dongen, H. Urbach, and M. Yousefi, IEEE/OSA Journal of Lightwave Technology 30, 2388 (2012).

Tx (x)Ty (y) ·

X

Φc,i,j lxi (x)lyj (y)

i,j

xi +∆x/2

reg. m

reg. m

Z

m

{z

!2


i,j

region m

m=1 c=1


i,j

6

XX

c,m

region m

6

XX 5

c,m

(Φc,i,j )2 ∆x∆y − Ωc,m C c,m

X i,j

Z

Φc,i,j

|

c

{z

1,t ,m Ix x (xi ,yi )

dxdy

yi +∆y/2

Tx (x)dx

xi −∆x/2

!

Z

Ty (y)dy

yi −∆y/2

}|

{z

1,tc y ,m

Iy

(xi ,yi )

IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

ACKNOWLEDGMENTS

We gratefully acknowledge TNO and the IOP Photonic Devices program of NL Agency for the financial support of the project which has led to Ref. [1]. The authors would like to thank all authors and contributors of this paper: ir. Suzanne M. Leinders, dr. Koen W.A. van Dongen, dr. Mirvais Yousefi, dr. Jose Pozo, dr. Omar El Gawhary, mr. Vincent Brulis, dr. Jos M. Thijssen, dr. Ad H. Verbruggen, and ir. Kevin A. van Hoogdalem.

[2] “Matlab - the language of technical computing,” The MathWorks, Inc., Natick, Massachusetts, USA (2010).

}

6 TABLE I. Matlab functions (.m-files) Description Example on how to use these Matlab scripts. computes the effective index a TE mode in a slap waveguide. Eq. (JLT.43). slab neff tm computes the effective index a TM mode in a slap waveguide. Eq. (JLT.35). rectwg neff computes the approximate effective index of a mode in a rectangular waveguide. Eq. (JLT.35), (JLT.43) and (JLT.23). rectwg ng compute approximate effective group index ng of a mode in the waveguide. Eqs. (JLT.55),(JLT.73)-(JLT.78). rectwg external effect compute influence of an external effect X on the effective index. Eqs. (JLT.58), (JLT.73)-(JLT.76), (JLT.79)-(JLT.82). rectwg getSoiWg built wg structure for typical SOI waveguide. rectwg getSoiWgEpsmu add epsmu (Ω) variable to wg structure. Eq. (55). computes the free parameters in the Ansatz via various methods. rectwg AnsatzMethods1 rectwg bcfitfun computes the residual mismatch Umm at the boundaries. Eq. (JLT.53) compute Ψ from the parameters of the Ansatz. Sec. IV A rectwg getPsi rectwg getFieldsPsi compute the full electromagnetic fields from Ψ, e.g. used for plotting. Sec. IV A. rectwg getEnergyPsi compute energy carried by the modal field. Sec. IV C. compute power flux through the mode. Sec. IV D. rectwg getFluxPsi rectwg getEnergyDifferencePsiFields compute energy in the differnce field between an approximate analytically computed mode and a rigorous numerically computed mode. Eq. (JLT.61), Sec. IV E compute integrals in Table IV rectwg getIntegralsI2 rectwg getIntegralsI1 compute integrals in Table V Normalize amplitude A1 such that the power flux through the waverectwg FluxNormalizeAnsatzParams guide equals unity, i.e. P = 1. rectwg getFluxFields compute energy in numerical field F (Φ). rectwg getFluxFields compute flux through waveguide for numerical field F (Φ). .m-file rectwg example slab neff te

TABLE II. Variable names and notation in the paper. Theory Script Theory Script Theory Script ic c k0 k0 A A1 ia m l0 λ0 B A2 epsmu(ic,ia) Ωc,m g1 .. g4 γ1 ..γ4 C A3 c(ic,ia) C c,m kx /k0 kx k0 M A4 K1 .. K5 K12 .. K52 ky k0 ky /k0 E A5 A1 .. A5 region 1..5 neff neff F A6 mX, mY x, y ng ng G A7 I2x(ia,1) Ix2,sin,m dn1dk0 ∂n1 /∂k0 H A8 L A9 I2x(ia,2) Ix2,cos,m J A10 I1x(1,i,j) Ix1,sin (xi , yi ) I1x(2,i,j) Ix1,cos (xi , yi ) F Φ

7 TABLE III. Contributions to Umm

(1) Umm

ǫ0 (n1 + n2 )2 = 4

(2) Umm =

(−βA1 ky + ωµ0 A2 kx ) sin[kx (ξ − d/2)] −βA3 ky − ωµ0 A4 γ2 − K12 K22

2 ǫ0 (n1 + n2 )2 (A1 sin[kx (ξ − d/2)] − A3 )2 · I cos y 4

(3) Umm = µ0

(βA2 ky + ωǫ0 n21 A1 kx ) cos[kx (ξ − d/2)] βA4 ky + ωǫ0 n22 A3 γ2 − 2 K1 K22

(4) Umm = µ0 (A2 cos[kx (ξ − d/2)] − A4 )2 · I sin

2

y

(5) Umm =

ǫ0 (n1 + n3 )2 4

(6) Umm =

2 ǫ0 (n1 + n3 )2 (A1 sin[kx (ξ + d/2)] − A5 )2 · I cos y 4

(7) Umm = µ0

(8) Umm = µ0 (A2 cos[kx (ξ + d/2)] − A6 )2 · I sin

2

y

(9) Umm =

ǫ0 (n1 + n4 )2 4

(10) Umm =

2 ǫ0 (n1 + n4 )2 (A1 cos[ky (η − b/2)] − A7 )2 · I sin x 4

(11) Umm = µ0

(−βA2 kx + ωǫ0 n21 A1 ky ) sin[ky (η − b/2)] −βA8 kx − ωǫ0 n24 A7 γ4 − 2 K1 K42 2

ǫ0 (n1 + n5 )2 = 4

(14) Umm =

2

2

· I cos

(βA1 kx + ωµ0 A2 ky ) cos[ky (η − b/2)] βA7 kx + ωµ0 A8 γ4 − K12 K42

(12) Umm = µ0 (A2 sin[ky (η − b/2)] − A8 )2 · I cos

(13) Umm

2

· I cos

x

2

2

2 ǫ0 (n1 + n5 )2 (A1 cos[ky (η + b/2)] − A9 )2 · I sin x 4

(15) Umm = µ0

(−βA2 kx + ωǫ0 n21 A1 ky ) sin[ky (η + b/2)] −βA10 kx + ωǫ0 n25 A9 γ5 − 2 K1 K52 2

(16) Umm = µ0 (A2 sin[ky (η + b/2)] − A10 )2 · I cos

x

2

2

2

· I sin

2

y

2

· I cos

2

· I sin

(βA1 kx + ωµ0 A2 ky ) cos[ky (η + b/2)] βA9 kx − ωµ0 A10 γ5 − K12 K52

· I sin

y

y

(−βA1 ky + ωµ0 A2 kx ) sin[kx (ξ + d/2)] −βA5 ky + ωµ0 A6 γ3 − K12 K32

(βA2 ky + ωǫ0 n21 A1 kx ) cos[kx (ξ + d/2)] βA6 ky − ωǫ0 n23 A5 γ3 − 2 K1 K32

2

2

2

x

x

2

· I cos

2

· I sin

x

x

y

Ey on 1-2

(72)

Ez on 1-2

(73)

Hy on 1-2

(74)

Hy on 1-2

(75)

Ey on 1-3

(76)

Ez on 1-3

(77)

Hy on 1-3

(78)

Hz on 1-3

(79)

Ex on 1-4

(80)

Ez on 1-4

(81)

Hx on 1-4

(82)

Hz on 1-4

(83)

Ex on 1-5

(84)

Ez on 1-5

(85)

Hx on 1-5

(86)

Hz on 1-5

(87)

where

I

cos2 x

I sin I

2

x

cos2 y

2

I sin

y

≡

≡

≡

≡

Z

Z

Z

Z

d/2

cos2 [kx (x + ξ)]dx =

cos[2kx ξ] sin[kx d] d + 2 2kx

(88)

sin2 [kx (x + ξ)]dx =

cos[2kx ξ] sin[kx d] d − 2 2kx

(89)

cos2 [ky (y + η)]dy =

cos[2ky η] sin[ky b] b + 2 2ky

(90)

sin2 [ky (y + η)]dx =

cos[2ky η] sin[ky b] b − 2 2ky

(91)

−d/2 d/2

−d/2 b/2

−b/2 b/2

−b/2

8 TABLE IV. Explicit form of integrals Ix2,sin,1 etc

Ix2,sin,1 = Ix2,sin,4 = Ix2,sin,5 = Ix2,cos,1 = Ix2,cos,4 = Ix2,cos,5 = Ix2,sin,2 = Ix2,cos,2 = Ix2,sin,3 = Ix2,cos,3 = Iy2,sin,1 = Iy2,sin,2 = Iy2,sin,3 = Iy2,cos,1 = Iy2,cos,2 = Iy2,cos,3 = Iy2,sin,4 = Iy2,cos,4 = Iy2,sin,3 = Iy2,cos,5 =

cos[2kx ξ] sin[kx d] d , − 2 2kx cos[2kx ξ] sin[kx d] d , + 2 2kx exp[−2γ2 wx ] 1 − 2γ2 2γ2 exp[−2γ3 wx ] 1 − 2γ3 2γ3 cos[2ky η] sin[ky b] b , − 2 2ky cos[2ky η] sin[ky b] b , + 2 2ky exp[−2γ4 wy ] 1 − 2γ4 2γ4 exp[−2γ5 wy ] 1 − 2γ5 2γ5

(92) (93) (94) (95) (96) (97) (98) (99)

TABLE V. Explicit form of integrals Ix1,sin,1 etc Ix1,sin,1 = Ix1,sin,4 = Ix1,sin,5 = Ix1,cos,1 = Ix1,cos,4 = Ix1,cos,5 = x1,sin,2 = x1,cos,2 = x1,sin,3 = x1,cos,3 = Iy1,sin,1 = Iy1,sin,2 = Iy1,sin,3 = Iy1,cos,1 = Iy1,cos,2 = Iy1,cos,3 = x1,sin,4 = x1,cos,4 = x1,sin,5 = x1,cos,5 =

2 sin[kx (xi + ξ)] sin[kx ∆x/2], kx 2 cos[kx (xi + ξ)] sin[kx ∆x/2], kx 2 exp[γ2 (xi + d/2)] sinh[γ2 ∆x/2] γ2 2 exp[−γ3 (xi − d/2)] sinh[γ3 ∆x/2] γ3 2 sin[ky (yi + η)] sin[ky ∆y/2], ky 2 cos[ky (yi + η)] sin[ky ∆y/2], ky 2 exp[γ4 (yi + b/2)] sinh[γ4 ∆y/2] γ4 2 exp[−γ5 (yi − b/2)] sinh[γ5 ∆y/2] γ5

(100) (101) (102) (103) (104) (105) (106) (107)