Abstract

Typically the grating problem is formulated for TE and TM polarizations by using, respectively, the electric and magnetic fields aligned with the grating wall and perpendicular to the plane of incidence, and this leads to a one-field-component problem. For some grating profiles such as metallic gratings with a triangular profile, the prediction of TM polarization by using a standard finite-element method experiences a slower convergence rate, and this reduces the accuracy of the computed results and also introduces a numerical polarization effect. This discrepancy cannot be seen as a simple numerical issue, since it has been observed for different types of numerical methods based on the classical formulation. Hence an alternative formulation is proposed, where the grating problem is modeled by taking the electric field as unknown for TM polarization. The application of this idea to both TE and TM polarizations leads to a two-field-component problem. The purpose of the paper is to propose an edge finite-element method to solve this wave problem. A comparison of the results of the proposed formulation and the classical formulation shows improvement and robustness in the new approach.

© 2005 Optical Society of America

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References

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  1. E. Popov, M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  4. A. Delâge, K. Dossou, “Polarisation dependent loss calculation in echelle gratings using finite element method and Rayleigh expansion,” Opt. Quantum Electron. 36, 223–238 (2004).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  8. M. Koshiba, S. Maruyama, K. Hirayama, “A vector finite element method with the high-order mixed-interpolation-type triangular elements for optical waveguiding problems,” J. Lightwave Technol. 12, 495–502 (1994).
    [CrossRef]
  9. L. Demkowicz, L. Vardapetyan, “Modeling of electromagnetic absorption/scattering problems using hp-adaptive finite elements,” Comput. Methods Appl. Mech. Eng. 152, 103–124 (1998).
    [CrossRef]
  10. K. Dossou, M. Fontaine, “A high order isoparametric finite element method for the computation of waveguide modes,” Comput. Methods Appl. Mech. Eng. 194, 837–858 (2005).
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    [CrossRef]
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    [CrossRef]
  13. D. Chowdhuri, “Design of low-loss and polarization-insensitive reflection grating-based planar demultiplexers,” IEEE J. Sel. Top. Quantum Electron. 6, 233–239 (2000).
    [CrossRef]
  14. P. Clemens, G. Heise, R. Marz, H. Michel, A. Reichelt, H. Schneider, “8-channel optical demultiplexer realized as SiO2/Si flat-field spectrograph,” IEEE Photonics Technol. Lett. 6, 1109–1111 (1994).
    [CrossRef]
  15. J.-J. He, B. Lamontagne, A. Delage, L. Erickson, M. Davies, E. Koteles, “Monolithic integrated wavelength demultiplexer based on a waveguide Rowland circle grating in InGaAsP/lnP,” J. Lightwave Technol. 16, 631–638 (1998).
    [CrossRef]
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    [CrossRef] [PubMed]

2005 (1)

K. Dossou, M. Fontaine, “A high order isoparametric finite element method for the computation of waveguide modes,” Comput. Methods Appl. Mech. Eng. 194, 837–858 (2005).

2004 (2)

A. Delâge, K. Dossou, “Polarisation dependent loss calculation in echelle gratings using finite element method and Rayleigh expansion,” Opt. Quantum Electron. 36, 223–238 (2004).
[CrossRef]

S. Janz, A. Balakrishnan, S. Charbonneau, P. Cheben, M. Cloutier, A. Delage, K. Dossou, L. Erickson, M. Gao, P. Krug, B. Lamontagne, M. Packirisamy, M. Pearson, D.-X. Xu, “Planar waveguide echelle gratings in silica-on-silicon,” IEEE Photonics Technol. Lett. 16, 503–505 (2004).
[CrossRef]

2000 (2)

D. Chowdhuri, “Design of low-loss and polarization-insensitive reflection grating-based planar demultiplexers,” IEEE J. Sel. Top. Quantum Electron. 6, 233–239 (2000).
[CrossRef]

E. Popov, M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
[CrossRef]

1999 (1)

1998 (2)

J.-J. He, B. Lamontagne, A. Delage, L. Erickson, M. Davies, E. Koteles, “Monolithic integrated wavelength demultiplexer based on a waveguide Rowland circle grating in InGaAsP/lnP,” J. Lightwave Technol. 16, 631–638 (1998).
[CrossRef]

L. Demkowicz, L. Vardapetyan, “Modeling of electromagnetic absorption/scattering problems using hp-adaptive finite elements,” Comput. Methods Appl. Mech. Eng. 152, 103–124 (1998).
[CrossRef]

1995 (2)

1994 (2)

M. Koshiba, S. Maruyama, K. Hirayama, “A vector finite element method with the high-order mixed-interpolation-type triangular elements for optical waveguiding problems,” J. Lightwave Technol. 12, 495–502 (1994).
[CrossRef]

P. Clemens, G. Heise, R. Marz, H. Michel, A. Reichelt, H. Schneider, “8-channel optical demultiplexer realized as SiO2/Si flat-field spectrograph,” IEEE Photonics Technol. Lett. 6, 1109–1111 (1994).
[CrossRef]

1993 (1)

1986 (1)

J.-C. Nédélec, “A new family of mixed finite elements in R3,” Numer. Math. 50, 57–81 (1986).
[CrossRef]

1980 (1)

J.-C. Nédélec, “Mixed finite elements in R3,” Numer. Math. 35, 315–341 (1980).
[CrossRef]

Balakrishnan, A.

S. Janz, A. Balakrishnan, S. Charbonneau, P. Cheben, M. Cloutier, A. Delage, K. Dossou, L. Erickson, M. Gao, P. Krug, B. Lamontagne, M. Packirisamy, M. Pearson, D.-X. Xu, “Planar waveguide echelle gratings in silica-on-silicon,” IEEE Photonics Technol. Lett. 16, 503–505 (2004).
[CrossRef]

Bao, G.

Chandezon, J.

Charbonneau, S.

S. Janz, A. Balakrishnan, S. Charbonneau, P. Cheben, M. Cloutier, A. Delage, K. Dossou, L. Erickson, M. Gao, P. Krug, B. Lamontagne, M. Packirisamy, M. Pearson, D.-X. Xu, “Planar waveguide echelle gratings in silica-on-silicon,” IEEE Photonics Technol. Lett. 16, 503–505 (2004).
[CrossRef]

Cheben, P.

S. Janz, A. Balakrishnan, S. Charbonneau, P. Cheben, M. Cloutier, A. Delage, K. Dossou, L. Erickson, M. Gao, P. Krug, B. Lamontagne, M. Packirisamy, M. Pearson, D.-X. Xu, “Planar waveguide echelle gratings in silica-on-silicon,” IEEE Photonics Technol. Lett. 16, 503–505 (2004).
[CrossRef]

Chowdhuri, D.

D. Chowdhuri, “Design of low-loss and polarization-insensitive reflection grating-based planar demultiplexers,” IEEE J. Sel. Top. Quantum Electron. 6, 233–239 (2000).
[CrossRef]

Clemens, P.

P. Clemens, G. Heise, R. Marz, H. Michel, A. Reichelt, H. Schneider, “8-channel optical demultiplexer realized as SiO2/Si flat-field spectrograph,” IEEE Photonics Technol. Lett. 6, 1109–1111 (1994).
[CrossRef]

Cloutier, M.

S. Janz, A. Balakrishnan, S. Charbonneau, P. Cheben, M. Cloutier, A. Delage, K. Dossou, L. Erickson, M. Gao, P. Krug, B. Lamontagne, M. Packirisamy, M. Pearson, D.-X. Xu, “Planar waveguide echelle gratings in silica-on-silicon,” IEEE Photonics Technol. Lett. 16, 503–505 (2004).
[CrossRef]

Cox, J. A.

Davies, M.

Delage, A.

S. Janz, A. Balakrishnan, S. Charbonneau, P. Cheben, M. Cloutier, A. Delage, K. Dossou, L. Erickson, M. Gao, P. Krug, B. Lamontagne, M. Packirisamy, M. Pearson, D.-X. Xu, “Planar waveguide echelle gratings in silica-on-silicon,” IEEE Photonics Technol. Lett. 16, 503–505 (2004).
[CrossRef]

J.-J. He, B. Lamontagne, A. Delage, L. Erickson, M. Davies, E. Koteles, “Monolithic integrated wavelength demultiplexer based on a waveguide Rowland circle grating in InGaAsP/lnP,” J. Lightwave Technol. 16, 631–638 (1998).
[CrossRef]

Delâge, A.

A. Delâge, K. Dossou, “Polarisation dependent loss calculation in echelle gratings using finite element method and Rayleigh expansion,” Opt. Quantum Electron. 36, 223–238 (2004).
[CrossRef]

Delort, T.

Demkowicz, L.

L. Demkowicz, L. Vardapetyan, “Modeling of electromagnetic absorption/scattering problems using hp-adaptive finite elements,” Comput. Methods Appl. Mech. Eng. 152, 103–124 (1998).
[CrossRef]

Dobson, D. C.

Dossou, K.

K. Dossou, M. Fontaine, “A high order isoparametric finite element method for the computation of waveguide modes,” Comput. Methods Appl. Mech. Eng. 194, 837–858 (2005).

A. Delâge, K. Dossou, “Polarisation dependent loss calculation in echelle gratings using finite element method and Rayleigh expansion,” Opt. Quantum Electron. 36, 223–238 (2004).
[CrossRef]

S. Janz, A. Balakrishnan, S. Charbonneau, P. Cheben, M. Cloutier, A. Delage, K. Dossou, L. Erickson, M. Gao, P. Krug, B. Lamontagne, M. Packirisamy, M. Pearson, D.-X. Xu, “Planar waveguide echelle gratings in silica-on-silicon,” IEEE Photonics Technol. Lett. 16, 503–505 (2004).
[CrossRef]

Erickson, L.

S. Janz, A. Balakrishnan, S. Charbonneau, P. Cheben, M. Cloutier, A. Delage, K. Dossou, L. Erickson, M. Gao, P. Krug, B. Lamontagne, M. Packirisamy, M. Pearson, D.-X. Xu, “Planar waveguide echelle gratings in silica-on-silicon,” IEEE Photonics Technol. Lett. 16, 503–505 (2004).
[CrossRef]

J.-J. He, B. Lamontagne, A. Delage, L. Erickson, M. Davies, E. Koteles, “Monolithic integrated wavelength demultiplexer based on a waveguide Rowland circle grating in InGaAsP/lnP,” J. Lightwave Technol. 16, 631–638 (1998).
[CrossRef]

Fontaine, M.

K. Dossou, M. Fontaine, “A high order isoparametric finite element method for the computation of waveguide modes,” Comput. Methods Appl. Mech. Eng. 194, 837–858 (2005).

Gao, M.

S. Janz, A. Balakrishnan, S. Charbonneau, P. Cheben, M. Cloutier, A. Delage, K. Dossou, L. Erickson, M. Gao, P. Krug, B. Lamontagne, M. Packirisamy, M. Pearson, D.-X. Xu, “Planar waveguide echelle gratings in silica-on-silicon,” IEEE Photonics Technol. Lett. 16, 503–505 (2004).
[CrossRef]

Granet, G.

He, J.-J.

Heise, G.

P. Clemens, G. Heise, R. Marz, H. Michel, A. Reichelt, H. Schneider, “8-channel optical demultiplexer realized as SiO2/Si flat-field spectrograph,” IEEE Photonics Technol. Lett. 6, 1109–1111 (1994).
[CrossRef]

Hirayama, K.

M. Koshiba, S. Maruyama, K. Hirayama, “A vector finite element method with the high-order mixed-interpolation-type triangular elements for optical waveguiding problems,” J. Lightwave Technol. 12, 495–502 (1994).
[CrossRef]

Janz, S.

S. Janz, A. Balakrishnan, S. Charbonneau, P. Cheben, M. Cloutier, A. Delage, K. Dossou, L. Erickson, M. Gao, P. Krug, B. Lamontagne, M. Packirisamy, M. Pearson, D.-X. Xu, “Planar waveguide echelle gratings in silica-on-silicon,” IEEE Photonics Technol. Lett. 16, 503–505 (2004).
[CrossRef]

Koshiba, M.

M. Koshiba, S. Maruyama, K. Hirayama, “A vector finite element method with the high-order mixed-interpolation-type triangular elements for optical waveguiding problems,” J. Lightwave Technol. 12, 495–502 (1994).
[CrossRef]

Koteles, E.

Krug, P.

S. Janz, A. Balakrishnan, S. Charbonneau, P. Cheben, M. Cloutier, A. Delage, K. Dossou, L. Erickson, M. Gao, P. Krug, B. Lamontagne, M. Packirisamy, M. Pearson, D.-X. Xu, “Planar waveguide echelle gratings in silica-on-silicon,” IEEE Photonics Technol. Lett. 16, 503–505 (2004).
[CrossRef]

Lamontagne, B.

S. Janz, A. Balakrishnan, S. Charbonneau, P. Cheben, M. Cloutier, A. Delage, K. Dossou, L. Erickson, M. Gao, P. Krug, B. Lamontagne, M. Packirisamy, M. Pearson, D.-X. Xu, “Planar waveguide echelle gratings in silica-on-silicon,” IEEE Photonics Technol. Lett. 16, 503–505 (2004).
[CrossRef]

J.-J. He, B. Lamontagne, A. Delage, L. Erickson, M. Davies, E. Koteles, “Monolithic integrated wavelength demultiplexer based on a waveguide Rowland circle grating in InGaAsP/lnP,” J. Lightwave Technol. 16, 631–638 (1998).
[CrossRef]

Li, L.

Loewen, E.

Maruyama, S.

M. Koshiba, S. Maruyama, K. Hirayama, “A vector finite element method with the high-order mixed-interpolation-type triangular elements for optical waveguiding problems,” J. Lightwave Technol. 12, 495–502 (1994).
[CrossRef]

Marz, R.

P. Clemens, G. Heise, R. Marz, H. Michel, A. Reichelt, H. Schneider, “8-channel optical demultiplexer realized as SiO2/Si flat-field spectrograph,” IEEE Photonics Technol. Lett. 6, 1109–1111 (1994).
[CrossRef]

Maystre, D.

Michel, H.

P. Clemens, G. Heise, R. Marz, H. Michel, A. Reichelt, H. Schneider, “8-channel optical demultiplexer realized as SiO2/Si flat-field spectrograph,” IEEE Photonics Technol. Lett. 6, 1109–1111 (1994).
[CrossRef]

Nédélec, J.-C.

J.-C. Nédélec, “A new family of mixed finite elements in R3,” Numer. Math. 50, 57–81 (1986).
[CrossRef]

J.-C. Nédélec, “Mixed finite elements in R3,” Numer. Math. 35, 315–341 (1980).
[CrossRef]

Nevière, M.

Packirisamy, M.

S. Janz, A. Balakrishnan, S. Charbonneau, P. Cheben, M. Cloutier, A. Delage, K. Dossou, L. Erickson, M. Gao, P. Krug, B. Lamontagne, M. Packirisamy, M. Pearson, D.-X. Xu, “Planar waveguide echelle gratings in silica-on-silicon,” IEEE Photonics Technol. Lett. 16, 503–505 (2004).
[CrossRef]

Pearson, M.

S. Janz, A. Balakrishnan, S. Charbonneau, P. Cheben, M. Cloutier, A. Delage, K. Dossou, L. Erickson, M. Gao, P. Krug, B. Lamontagne, M. Packirisamy, M. Pearson, D.-X. Xu, “Planar waveguide echelle gratings in silica-on-silicon,” IEEE Photonics Technol. Lett. 16, 503–505 (2004).
[CrossRef]

Plumey, J.-P.

Popov, E.

Reichelt, A.

P. Clemens, G. Heise, R. Marz, H. Michel, A. Reichelt, H. Schneider, “8-channel optical demultiplexer realized as SiO2/Si flat-field spectrograph,” IEEE Photonics Technol. Lett. 6, 1109–1111 (1994).
[CrossRef]

Schneider, H.

P. Clemens, G. Heise, R. Marz, H. Michel, A. Reichelt, H. Schneider, “8-channel optical demultiplexer realized as SiO2/Si flat-field spectrograph,” IEEE Photonics Technol. Lett. 6, 1109–1111 (1994).
[CrossRef]

Tsonev, L.

Vardapetyan, L.

L. Demkowicz, L. Vardapetyan, “Modeling of electromagnetic absorption/scattering problems using hp-adaptive finite elements,” Comput. Methods Appl. Mech. Eng. 152, 103–124 (1998).
[CrossRef]

Xu, D.-X.

S. Janz, A. Balakrishnan, S. Charbonneau, P. Cheben, M. Cloutier, A. Delage, K. Dossou, L. Erickson, M. Gao, P. Krug, B. Lamontagne, M. Packirisamy, M. Pearson, D.-X. Xu, “Planar waveguide echelle gratings in silica-on-silicon,” IEEE Photonics Technol. Lett. 16, 503–505 (2004).
[CrossRef]

Appl. Opt. (2)

Comput. Methods Appl. Mech. Eng. (2)

L. Demkowicz, L. Vardapetyan, “Modeling of electromagnetic absorption/scattering problems using hp-adaptive finite elements,” Comput. Methods Appl. Mech. Eng. 152, 103–124 (1998).
[CrossRef]

K. Dossou, M. Fontaine, “A high order isoparametric finite element method for the computation of waveguide modes,” Comput. Methods Appl. Mech. Eng. 194, 837–858 (2005).

IEEE J. Sel. Top. Quantum Electron. (1)

D. Chowdhuri, “Design of low-loss and polarization-insensitive reflection grating-based planar demultiplexers,” IEEE J. Sel. Top. Quantum Electron. 6, 233–239 (2000).
[CrossRef]

IEEE Photonics Technol. Lett. (2)

P. Clemens, G. Heise, R. Marz, H. Michel, A. Reichelt, H. Schneider, “8-channel optical demultiplexer realized as SiO2/Si flat-field spectrograph,” IEEE Photonics Technol. Lett. 6, 1109–1111 (1994).
[CrossRef]

S. Janz, A. Balakrishnan, S. Charbonneau, P. Cheben, M. Cloutier, A. Delage, K. Dossou, L. Erickson, M. Gao, P. Krug, B. Lamontagne, M. Packirisamy, M. Pearson, D.-X. Xu, “Planar waveguide echelle gratings in silica-on-silicon,” IEEE Photonics Technol. Lett. 16, 503–505 (2004).
[CrossRef]

J. Lightwave Technol. (2)

J.-J. He, B. Lamontagne, A. Delage, L. Erickson, M. Davies, E. Koteles, “Monolithic integrated wavelength demultiplexer based on a waveguide Rowland circle grating in InGaAsP/lnP,” J. Lightwave Technol. 16, 631–638 (1998).
[CrossRef]

M. Koshiba, S. Maruyama, K. Hirayama, “A vector finite element method with the high-order mixed-interpolation-type triangular elements for optical waveguiding problems,” J. Lightwave Technol. 12, 495–502 (1994).
[CrossRef]

J. Opt. Soc. Am. A (3)

Numer. Math. (2)

J.-C. Nédélec, “Mixed finite elements in R3,” Numer. Math. 35, 315–341 (1980).
[CrossRef]

J.-C. Nédélec, “A new family of mixed finite elements in R3,” Numer. Math. 50, 57–81 (1986).
[CrossRef]

Opt. Quantum Electron. (1)

A. Delâge, K. Dossou, “Polarisation dependent loss calculation in echelle gratings using finite element method and Rayleigh expansion,” Opt. Quantum Electron. 36, 223–238 (2004).
[CrossRef]

Other (1)

R. Petit, ed., Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics (Springer-Verlag, New York, 1980).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Grating profile, where d is the grating period.

Fig. 2
Fig. 2

Nodes and degrees of freedom of an FEM mesh triangle.

Fig. 3
Fig. 3

Grating profile.

Fig. 4
Fig. 4

Perfect metallic conductor gratings (order -22): (a) efficiency curves, (b) PDL curves.

Fig. 5
Fig. 5

Grating with only perfectly conductive reflecting facets: (a) efficiency curves, (b) PDL curves.

Fig. 6
Fig. 6

Grating with finite-conductivity metallic facets: (a) efficiency curves, (b) PDL curves.

Fig. 7
Fig. 7

Grating with finite-conductivity metallic facets: convergence of the efficiency curves for the TE polarizations [(a) and (b)] and TM polarizations [(c) and (d)]. Circles, crosses, diamonds, and triangles correspond to the efficiency obtained by using the FEM meshes numbered (1), (2), (3), and (4), respectively, in Table 4.

Fig. 8
Fig. 8

Grating with finite-conductivity metallic facets: convergence of the PDL curves using (a) the vy formulation and (b) the vt formulation. The symbol notation is the same as that in Fig. 7.

Fig. 9
Fig. 9

Finite-conductivity metal only on reflecting facets: (a) efficiency curves, (b) PDL curves.

Tables (4)

Tables Icon

Table 1 Quadratic Element Basis Functionsa

Tables Icon

Table 2 Comparison of the Diffraction Efficiencies of the Grating Described in Subsection 5.A as Computed by the C Method, the FMM, and the vt and vz FEMs

Tables Icon

Table 3 Convergence of the vt and vz FEMs for the Grating Presented in Subsection 5.A

Tables Icon

Table 4 Grating with Finite-Conductivity Metallic Facets: Number of FEM Mesh Elements (NE), Number of Nodes (NN), and Truncation Parameters P1 and P2

Equations (46)

Equations on this page are rendered with MathJax. Learn more.

×(p×v)-k02qv=0,
t×(pt×vt)-k02qvt=0,
t×vt=vzx-vxz,t×vy=vyz-vyx.
Ω={(x, z)|0<x<d, f(x)<z<a},Ω+={(x, z)|0<x<d, a<z},
Γ={(x, f(x))|0<x<d},Γ+={(x, 0)|0<x<d}.
vtt=0ifvt=Et,t×vt=0ifvt=Ht.
vti=Rti exp(jα0x-jβ0z)
α0=n+k0 sin θ0,β0=n+k0 cos θ0,Rti=cos θ0sin θ0.
vtd=m=-+ Am+Bm+exp(jαmx+jβm+z),
αm=α0+m 2πd,βm+=[(n+)2k02-αm2]1/2if(n+)2k02-αm20j[αm2-(n+)2k02]1/2if(n+)2k02-αm2<0.
αmAm++βm+Bm+=0,
vtd=m=-+ Am+Rtm+ exp(jαmx+jβm+z),
Rtm+=1n+k0 -βm+αm.
em+=Am+(Am+)*βm+β0,
vtt=(Rtit)exp(jα0x)+m=-+ Am+ exp(jαmx)(Rtm+t),
(pt×vt)=jn+k0p+ exp(jα0x)+jn+k0p+m=-+ Am+ exp(jαmx).
t×[Rtm+ exp(jαmx+jβm+z)]=n+k0 exp(jαmx+jβm+z).
t×(pt×vt)-k02qvt=0inΩ,
vtt=0onΓifvt=Et, (ort×vt=0onΓifvt=Ht),
-vx=1n+k0 -β0 exp(jα0x)+m=-+ βm+Am+×exp(jαmx)                 onΓ+,
(pt×vt)=(jn+k0p+)exp(jα0x)+m=-+ Am+ exp(jαmx)                 onΓ+.
Ω[p(t×vt)(t×wt*)-k02qvtwt*]dxdz-ΓΓ+p(t×vt)(wt*t)ds=0.
Γ+p(t×vt)(wt*t)ds=jn+k0p+0d exp(jα0x)(-wx*)dx+jn+k0p+m=-+Am+0d exp(jαmx)(-wx*)dx,
Ω[p(t×vt)(t×wt*)-k02qvtwt*]dxdz+jn+k0p+m=-+Am+0d exp(jαmx)wx*dx=-(jn+k0p+)0d exp(jα0x)wx*dx.
1n+k0 βm+Am+d+0dvx(x, 0)exp(-jαmx)dx=1n+k0 β0dδm0,
vt=i=112 uivi.
MUUMUAMAUMAA UA=FUFA,
(MUU)ms=Ω[p(t×vs)(t×vm*)-k02qvsvm*]dxdz, m,s=1,,D,(MUA)ms=jn+k0p+0dvmx*(x, 0)exp(jαs+P1-1x)dx,m=1,,D,s=1,,(P2-P1+1),(MAU)ms=0dvsx(x, 0)exp(-jαm+P1-1x)dx,m=1,,(P2-P1+1),s=1,,D,(MAA)ms=βm+P1-1+dn+k0 δms,m,s=1,,(P2-P1+1).
(FU)m=-jn+k0p+0dvmx* exp(jα0x)dx,m=1,, D,(FA)m=β0dn+k0 δ(m+P1-1)0,m=1,,(P2-P1+1).
{U}=[MUU]-1{FU}-[MUU]-1[MUA]{A};
[MˆAA]{A}={FˆA},
[MˆAA]=[MAA]-[MAU][MUU]-1[MUA],
{FˆA}={FA}-[MAU][MUU]-1{FU}.
[MUU]{Uˆi}={FU},[MUU]{Uˆm}={Fˆm},
(MˆAA)ms=(MAA)ms-0dUˆsx(x, 0)exp(-jαm+P1-1x)dx,m, s=1,,(P2-P1+1),(FˆA)m=(FA)m-0dUˆxi(x, 0)exp(-jαm+P1-1x)dx,m=1,, (P2-P1+1).
{U}={Uˆi}-m=1P2-P1+1Am+{Uˆm}.
vt=uti+m=-+Am+utm,
E(PR)z=exp(jαd)[E(PL)z],H(PR)z=exp(jαd)[H(PL)z],
vt(PR)z=exp(jαd)[vt(PL)z],
(p×vt)(PR)=exp(jαd)(p×vt)(PL).
vtd=m=-+Am-Rtm- exp[jαmx-jβm-(z+h)],
αm=α+m 2πd,βm-=[(n-)2k02-αm2]1/2if(n-)2k02-αm20j[αm2-(n-)2k02]1/2if(n-)2k02-αm2<0,
Rtm-=1n-k0 βm-αm.
em-=Am-(Am-)*p-βm-p+β.
Ω[p(t×vt)(t×wt*)-k02qvtwt*]dxdz+(jn+k0p+)m=-+Am+0d exp(jαmx)wx*(x, 0)dx-(jn-k0p-)m=-+Am-0d exp(jαmx)wx*(x, -h)dx=-(jn+k0p+)0d exp(jα0x)wx*(x, 0)dx,
βm+dn+k0Am++0dvx(x, 0)exp(-jαmx)dx=β0dn+k0δm0,βm-dn-k0Am--0dvx(x, -h)exp(-jαmx)dx=0.

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