Abstract

The propagation and coupling phenomena in grating-assisted optical couplers are analyzed by using an integral equation formulation and applying an entire-domain Galerkin technique. The proposed method constitutes a special type of the method of moments and provides high numerical stability and controllable accuracy. The electric field in the grating region is the unknown quantity and the resulting integral equation is subsequently solved by using Galerkin’s method. The propagation constants of the guided waves are computed accurately by determining the singular points of the corresponding system’s matrix. Numerical results regarding the propagation constants are presented for various coupler parameters, and the effect of the grating’s physical and geometric characteristics on the coupling process is investigated.

© 2006 Optical Society of America

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References

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  1. D. Marcuse, Integrated Optics (IEEE Press, 1973).
  2. T. Tamir, Integrated Optics (Springer-Verlag, 1975).
  3. F. Liu, H. Hier, and T. Worchesky, "Dual-side processed demultiplexer using grating-assisted codirectional coupler," IEEE Photon. Technol. Lett. 17, 600-602 (2005).
    [CrossRef]
  4. R. C. Alferness, T. L. Kock, L. L. Buhl, F. Storz, F. Heismann, and M. J. R. Martyak, "Grating assisted InGaAsP/InP vertical co-directional coupler filter," Appl. Phys. Lett. 55, 2011-2013 (1989).
    [CrossRef]
  5. K. Ogawa, W. Chang, B. Sopori, and F. Rosenbaum, "A theoretical analysis of edged grating couplers for integrated optics," IEEE J. Quantum Electron. QE-9, 29-42 (1973).
    [CrossRef]
  6. S. Zhang and T. Tamir, "Analysis and design of broadband grating couplers," IEEE J. Quantum Electron. 29, 2813-2824 (1993).
    [CrossRef]
  7. D. Marcuse, "Directional couplers made of nonidentical asymmetric slabs. Part I: Synchronous couplers," J. Lightwave Technol. LT-5, 113-118 (1987).
    [CrossRef]
  8. D. Marcuse, "Directional couplers made of nonidentical assymetrical slabs. Part II: Grating-assisted couplers," J. Lightwave Technol. LT-5, 268-273 (1987).
    [CrossRef]
  9. G. Griffel, M. Itzkovich, and A. A. Hardy, "Coupled mode formulation for directional couplers with longitudinal perturbation," IEEE J. Quantum Electron. 27, 985-994 (1991).
    [CrossRef]
  10. W. Huang, B. E. Little, and S. K. Chaudhuri, "A new approach to grating-assisted couplers," J. Lightwave Technol. 9, 721-727 (1991).
    [CrossRef]
  11. W. P. Huang, "Coupled-mode theory for optical waveguides: an overview," J. Opt. Soc. Am. A 11, 963-983 (1994).
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  12. S. Zhang and T. Tamir, "Rigorous theory of grating-assisted couplers," J. Opt. Soc. Am. A 13, 2403-2413 (1996).
    [CrossRef]
  13. N. Sun, J. Butler, G. Evans, L. Pang, and P. Congdon, "Analysis of grating-assisted directional couplers using the Floquet-Bloch theory," J. Lightwave Technol. 15, 2301-2315 (1997).
    [CrossRef]
  14. J. Butler, N. Sun, G. Evans, L. Pang, and P. Congdon, "Grating-assisted coupling of light between semiconductor and glass waveguides," J. Lightwave Technol. 16, 1038-1048 (1998).
    [CrossRef]
  15. V. Passaro, "Optimal design of grating-assisted directional couplers," J. Lightwave Technol. 18, 973-984 (2000).
    [CrossRef]
  16. P. G. Dinesen, J. S. Hesthaven, J. P. Lynov, and L. Lading, "Pseudospectral method for the analysis of diffractive optical elements," J. Opt. Soc. Am. A 16, 1124-1130 (1999).
    [CrossRef]
  17. U. Levy, M. Nezhad, H.-C. Kim, C.-H. Tsai, L. Pang, and Y. Fainman, "Implementation of a graded-index medium by use of subwavelength structures with graded fill factor," J. Opt. Soc. Am. A 22, 724-733 (2005).
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    [CrossRef]
  21. J. Butler, W. Ferguson, G. Evans, P. Stabile, and A. Rosen, "A boundary element technique applied to the analysis of waveguides with periodic surface corrugations," IEEE J. Quantum Electron. 28, 1701-1709 (1992).
    [CrossRef]
  22. J. S. Bagby, D. P. Nyquist, and B. C. Drachman, "Integral formulation for analysis of integrated dielectric waveguides," IEEE Trans. Microwave Theory Tech. MTT-33, 906-915 (1985).
    [CrossRef]
  23. A. Sommerfeld, Partial Differential Equations in Physics (Academic, 1949).
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  25. I. D. Chremmos and N. K. Uzunoglu, "Analysis of coupling between two slab waveguides in the presence of ring resonators," J. Opt. Soc. Am. A 21, 267-279 (2004).
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  27. S. T. Peng, "Rigorous formulation of scattering and guidance by dielectric grating waveguides: general case of oblique incidence," J. Opt. Soc. Am. A 6, 1869-1883 (1989).
    [CrossRef]
  28. T. E. Rozzi and G. H. In't Veld, "Field and network analysis of interacting step discontinuities in planar dielectric waveguides," IEEE Trans. Microwave Theory Tech. MTT-27, 303-309 (1979).
    [CrossRef]
  29. M. Nishimoto and H. Ikuno, "Analysis of electromagnetic wave diffraction by a semi-infinite strip grating and evaluation of end effects," in Proceedings of Progress in Electromagnetic Research (PIERS) (1999), Vol. 23, pp. 39-58; http://emacademy.org/piers.
    [CrossRef]
  30. J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, and D. L. Brundrett, "Guided-mode resonant subwavelength gratings: effects of finite beams and finite gratings," J. Opt. Soc. Am. A 18, 1912-1928 (2001).
    [CrossRef]

2005 (3)

2004 (1)

2001 (1)

2000 (1)

1999 (1)

1998 (1)

1997 (1)

N. Sun, J. Butler, G. Evans, L. Pang, and P. Congdon, "Analysis of grating-assisted directional couplers using the Floquet-Bloch theory," J. Lightwave Technol. 15, 2301-2315 (1997).
[CrossRef]

1996 (1)

1994 (1)

1993 (1)

S. Zhang and T. Tamir, "Analysis and design of broadband grating couplers," IEEE J. Quantum Electron. 29, 2813-2824 (1993).
[CrossRef]

1992 (1)

J. Butler, W. Ferguson, G. Evans, P. Stabile, and A. Rosen, "A boundary element technique applied to the analysis of waveguides with periodic surface corrugations," IEEE J. Quantum Electron. 28, 1701-1709 (1992).
[CrossRef]

1991 (2)

G. Griffel, M. Itzkovich, and A. A. Hardy, "Coupled mode formulation for directional couplers with longitudinal perturbation," IEEE J. Quantum Electron. 27, 985-994 (1991).
[CrossRef]

W. Huang, B. E. Little, and S. K. Chaudhuri, "A new approach to grating-assisted couplers," J. Lightwave Technol. 9, 721-727 (1991).
[CrossRef]

1990 (1)

G. Hadjicostas, J. Butler, G. Evans, N. Carlson, and R. Amantea, "A numerical investigation of wave interactions in dielectric waveguides with periodic surface corrugations," IEEE J. Quantum Electron. 26, 893-902 (1990).
[CrossRef]

1989 (2)

R. C. Alferness, T. L. Kock, L. L. Buhl, F. Storz, F. Heismann, and M. J. R. Martyak, "Grating assisted InGaAsP/InP vertical co-directional coupler filter," Appl. Phys. Lett. 55, 2011-2013 (1989).
[CrossRef]

S. T. Peng, "Rigorous formulation of scattering and guidance by dielectric grating waveguides: general case of oblique incidence," J. Opt. Soc. Am. A 6, 1869-1883 (1989).
[CrossRef]

1987 (2)

D. Marcuse, "Directional couplers made of nonidentical asymmetric slabs. Part I: Synchronous couplers," J. Lightwave Technol. LT-5, 113-118 (1987).
[CrossRef]

D. Marcuse, "Directional couplers made of nonidentical assymetrical slabs. Part II: Grating-assisted couplers," J. Lightwave Technol. LT-5, 268-273 (1987).
[CrossRef]

1985 (1)

J. S. Bagby, D. P. Nyquist, and B. C. Drachman, "Integral formulation for analysis of integrated dielectric waveguides," IEEE Trans. Microwave Theory Tech. MTT-33, 906-915 (1985).
[CrossRef]

1982 (1)

1979 (1)

T. E. Rozzi and G. H. In't Veld, "Field and network analysis of interacting step discontinuities in planar dielectric waveguides," IEEE Trans. Microwave Theory Tech. MTT-27, 303-309 (1979).
[CrossRef]

1973 (1)

K. Ogawa, W. Chang, B. Sopori, and F. Rosenbaum, "A theoretical analysis of edged grating couplers for integrated optics," IEEE J. Quantum Electron. QE-9, 29-42 (1973).
[CrossRef]

Alferness, R. C.

R. C. Alferness, T. L. Kock, L. L. Buhl, F. Storz, F. Heismann, and M. J. R. Martyak, "Grating assisted InGaAsP/InP vertical co-directional coupler filter," Appl. Phys. Lett. 55, 2011-2013 (1989).
[CrossRef]

Amantea, R.

G. Hadjicostas, J. Butler, G. Evans, N. Carlson, and R. Amantea, "A numerical investigation of wave interactions in dielectric waveguides with periodic surface corrugations," IEEE J. Quantum Electron. 26, 893-902 (1990).
[CrossRef]

Bagby, J. S.

J. S. Bagby, D. P. Nyquist, and B. C. Drachman, "Integral formulation for analysis of integrated dielectric waveguides," IEEE Trans. Microwave Theory Tech. MTT-33, 906-915 (1985).
[CrossRef]

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, 1989).

Bendickson, J. M.

Brundrett, D. L.

Buhl, L. L.

R. C. Alferness, T. L. Kock, L. L. Buhl, F. Storz, F. Heismann, and M. J. R. Martyak, "Grating assisted InGaAsP/InP vertical co-directional coupler filter," Appl. Phys. Lett. 55, 2011-2013 (1989).
[CrossRef]

Butler, J.

J. Butler, N. Sun, G. Evans, L. Pang, and P. Congdon, "Grating-assisted coupling of light between semiconductor and glass waveguides," J. Lightwave Technol. 16, 1038-1048 (1998).
[CrossRef]

N. Sun, J. Butler, G. Evans, L. Pang, and P. Congdon, "Analysis of grating-assisted directional couplers using the Floquet-Bloch theory," J. Lightwave Technol. 15, 2301-2315 (1997).
[CrossRef]

J. Butler, W. Ferguson, G. Evans, P. Stabile, and A. Rosen, "A boundary element technique applied to the analysis of waveguides with periodic surface corrugations," IEEE J. Quantum Electron. 28, 1701-1709 (1992).
[CrossRef]

G. Hadjicostas, J. Butler, G. Evans, N. Carlson, and R. Amantea, "A numerical investigation of wave interactions in dielectric waveguides with periodic surface corrugations," IEEE J. Quantum Electron. 26, 893-902 (1990).
[CrossRef]

Carlson, N.

G. Hadjicostas, J. Butler, G. Evans, N. Carlson, and R. Amantea, "A numerical investigation of wave interactions in dielectric waveguides with periodic surface corrugations," IEEE J. Quantum Electron. 26, 893-902 (1990).
[CrossRef]

Chang, W.

K. Ogawa, W. Chang, B. Sopori, and F. Rosenbaum, "A theoretical analysis of edged grating couplers for integrated optics," IEEE J. Quantum Electron. QE-9, 29-42 (1973).
[CrossRef]

Chaudhuri, S. K.

W. Huang, B. E. Little, and S. K. Chaudhuri, "A new approach to grating-assisted couplers," J. Lightwave Technol. 9, 721-727 (1991).
[CrossRef]

Chremmos, I. D.

Congdon, P.

J. Butler, N. Sun, G. Evans, L. Pang, and P. Congdon, "Grating-assisted coupling of light between semiconductor and glass waveguides," J. Lightwave Technol. 16, 1038-1048 (1998).
[CrossRef]

N. Sun, J. Butler, G. Evans, L. Pang, and P. Congdon, "Analysis of grating-assisted directional couplers using the Floquet-Bloch theory," J. Lightwave Technol. 15, 2301-2315 (1997).
[CrossRef]

Dinesen, P. G.

Drachman, B. C.

J. S. Bagby, D. P. Nyquist, and B. C. Drachman, "Integral formulation for analysis of integrated dielectric waveguides," IEEE Trans. Microwave Theory Tech. MTT-33, 906-915 (1985).
[CrossRef]

Evans, G.

J. Butler, N. Sun, G. Evans, L. Pang, and P. Congdon, "Grating-assisted coupling of light between semiconductor and glass waveguides," J. Lightwave Technol. 16, 1038-1048 (1998).
[CrossRef]

N. Sun, J. Butler, G. Evans, L. Pang, and P. Congdon, "Analysis of grating-assisted directional couplers using the Floquet-Bloch theory," J. Lightwave Technol. 15, 2301-2315 (1997).
[CrossRef]

J. Butler, W. Ferguson, G. Evans, P. Stabile, and A. Rosen, "A boundary element technique applied to the analysis of waveguides with periodic surface corrugations," IEEE J. Quantum Electron. 28, 1701-1709 (1992).
[CrossRef]

G. Hadjicostas, J. Butler, G. Evans, N. Carlson, and R. Amantea, "A numerical investigation of wave interactions in dielectric waveguides with periodic surface corrugations," IEEE J. Quantum Electron. 26, 893-902 (1990).
[CrossRef]

Fainman, Y.

Ferguson, W.

J. Butler, W. Ferguson, G. Evans, P. Stabile, and A. Rosen, "A boundary element technique applied to the analysis of waveguides with periodic surface corrugations," IEEE J. Quantum Electron. 28, 1701-1709 (1992).
[CrossRef]

Fikioris, J. G.

Gaylord, T. K.

Glytsis, E. N.

Griffel, G.

G. Griffel, M. Itzkovich, and A. A. Hardy, "Coupled mode formulation for directional couplers with longitudinal perturbation," IEEE J. Quantum Electron. 27, 985-994 (1991).
[CrossRef]

Hadjicostas, G.

G. Hadjicostas, J. Butler, G. Evans, N. Carlson, and R. Amantea, "A numerical investigation of wave interactions in dielectric waveguides with periodic surface corrugations," IEEE J. Quantum Electron. 26, 893-902 (1990).
[CrossRef]

Hardy, A. A.

G. Griffel, M. Itzkovich, and A. A. Hardy, "Coupled mode formulation for directional couplers with longitudinal perturbation," IEEE J. Quantum Electron. 27, 985-994 (1991).
[CrossRef]

Heismann, F.

R. C. Alferness, T. L. Kock, L. L. Buhl, F. Storz, F. Heismann, and M. J. R. Martyak, "Grating assisted InGaAsP/InP vertical co-directional coupler filter," Appl. Phys. Lett. 55, 2011-2013 (1989).
[CrossRef]

Hesthaven, J. S.

Hier, H.

F. Liu, H. Hier, and T. Worchesky, "Dual-side processed demultiplexer using grating-assisted codirectional coupler," IEEE Photon. Technol. Lett. 17, 600-602 (2005).
[CrossRef]

Huang, W.

W. Huang, B. E. Little, and S. K. Chaudhuri, "A new approach to grating-assisted couplers," J. Lightwave Technol. 9, 721-727 (1991).
[CrossRef]

Huang, W. P.

Ichikawa, H.

Ikuno, H.

M. Nishimoto and H. Ikuno, "Analysis of electromagnetic wave diffraction by a semi-infinite strip grating and evaluation of end effects," in Proceedings of Progress in Electromagnetic Research (PIERS) (1999), Vol. 23, pp. 39-58; http://emacademy.org/piers.
[CrossRef]

In't Veld, G. H.

T. E. Rozzi and G. H. In't Veld, "Field and network analysis of interacting step discontinuities in planar dielectric waveguides," IEEE Trans. Microwave Theory Tech. MTT-27, 303-309 (1979).
[CrossRef]

Itzkovich, M.

G. Griffel, M. Itzkovich, and A. A. Hardy, "Coupled mode formulation for directional couplers with longitudinal perturbation," IEEE J. Quantum Electron. 27, 985-994 (1991).
[CrossRef]

Jones, D. S.

D. S. Jones, Theory of Electromagnetism (Pergamon, 1964).

Kikuta, H.

Kim, H.-C.

Kock, T. L.

R. C. Alferness, T. L. Kock, L. L. Buhl, F. Storz, F. Heismann, and M. J. R. Martyak, "Grating assisted InGaAsP/InP vertical co-directional coupler filter," Appl. Phys. Lett. 55, 2011-2013 (1989).
[CrossRef]

Lading, L.

Levy, U.

Little, B. E.

W. Huang, B. E. Little, and S. K. Chaudhuri, "A new approach to grating-assisted couplers," J. Lightwave Technol. 9, 721-727 (1991).
[CrossRef]

Liu, F.

F. Liu, H. Hier, and T. Worchesky, "Dual-side processed demultiplexer using grating-assisted codirectional coupler," IEEE Photon. Technol. Lett. 17, 600-602 (2005).
[CrossRef]

Lynov, J. P.

Marcuse, D.

D. Marcuse, "Directional couplers made of nonidentical asymmetric slabs. Part I: Synchronous couplers," J. Lightwave Technol. LT-5, 113-118 (1987).
[CrossRef]

D. Marcuse, "Directional couplers made of nonidentical assymetrical slabs. Part II: Grating-assisted couplers," J. Lightwave Technol. LT-5, 268-273 (1987).
[CrossRef]

D. Marcuse, Integrated Optics (IEEE Press, 1973).

Martyak, M. J. R.

R. C. Alferness, T. L. Kock, L. L. Buhl, F. Storz, F. Heismann, and M. J. R. Martyak, "Grating assisted InGaAsP/InP vertical co-directional coupler filter," Appl. Phys. Lett. 55, 2011-2013 (1989).
[CrossRef]

Nezhad, M.

Nishimoto, M.

M. Nishimoto and H. Ikuno, "Analysis of electromagnetic wave diffraction by a semi-infinite strip grating and evaluation of end effects," in Proceedings of Progress in Electromagnetic Research (PIERS) (1999), Vol. 23, pp. 39-58; http://emacademy.org/piers.
[CrossRef]

Nyquist, D. P.

J. S. Bagby, D. P. Nyquist, and B. C. Drachman, "Integral formulation for analysis of integrated dielectric waveguides," IEEE Trans. Microwave Theory Tech. MTT-33, 906-915 (1985).
[CrossRef]

Ogawa, K.

K. Ogawa, W. Chang, B. Sopori, and F. Rosenbaum, "A theoretical analysis of edged grating couplers for integrated optics," IEEE J. Quantum Electron. QE-9, 29-42 (1973).
[CrossRef]

Pang, L.

Passaro, V.

Peng, S. T.

Rosen, A.

J. Butler, W. Ferguson, G. Evans, P. Stabile, and A. Rosen, "A boundary element technique applied to the analysis of waveguides with periodic surface corrugations," IEEE J. Quantum Electron. 28, 1701-1709 (1992).
[CrossRef]

Rosenbaum, F.

K. Ogawa, W. Chang, B. Sopori, and F. Rosenbaum, "A theoretical analysis of edged grating couplers for integrated optics," IEEE J. Quantum Electron. QE-9, 29-42 (1973).
[CrossRef]

Rozzi, T. E.

T. E. Rozzi and G. H. In't Veld, "Field and network analysis of interacting step discontinuities in planar dielectric waveguides," IEEE Trans. Microwave Theory Tech. MTT-27, 303-309 (1979).
[CrossRef]

Sommerfeld, A.

A. Sommerfeld, Partial Differential Equations in Physics (Academic, 1949).

Sopori, B.

K. Ogawa, W. Chang, B. Sopori, and F. Rosenbaum, "A theoretical analysis of edged grating couplers for integrated optics," IEEE J. Quantum Electron. QE-9, 29-42 (1973).
[CrossRef]

Stabile, P.

J. Butler, W. Ferguson, G. Evans, P. Stabile, and A. Rosen, "A boundary element technique applied to the analysis of waveguides with periodic surface corrugations," IEEE J. Quantum Electron. 28, 1701-1709 (1992).
[CrossRef]

Storz, F.

R. C. Alferness, T. L. Kock, L. L. Buhl, F. Storz, F. Heismann, and M. J. R. Martyak, "Grating assisted InGaAsP/InP vertical co-directional coupler filter," Appl. Phys. Lett. 55, 2011-2013 (1989).
[CrossRef]

Sun, N.

J. Butler, N. Sun, G. Evans, L. Pang, and P. Congdon, "Grating-assisted coupling of light between semiconductor and glass waveguides," J. Lightwave Technol. 16, 1038-1048 (1998).
[CrossRef]

N. Sun, J. Butler, G. Evans, L. Pang, and P. Congdon, "Analysis of grating-assisted directional couplers using the Floquet-Bloch theory," J. Lightwave Technol. 15, 2301-2315 (1997).
[CrossRef]

Tamir, T.

S. Zhang and T. Tamir, "Rigorous theory of grating-assisted couplers," J. Opt. Soc. Am. A 13, 2403-2413 (1996).
[CrossRef]

S. Zhang and T. Tamir, "Analysis and design of broadband grating couplers," IEEE J. Quantum Electron. 29, 2813-2824 (1993).
[CrossRef]

T. Tamir, Integrated Optics (Springer-Verlag, 1975).

Tsai, C.-H.

Uzunoglu, N. K.

Worchesky, T.

F. Liu, H. Hier, and T. Worchesky, "Dual-side processed demultiplexer using grating-assisted codirectional coupler," IEEE Photon. Technol. Lett. 17, 600-602 (2005).
[CrossRef]

Zhang, S.

S. Zhang and T. Tamir, "Rigorous theory of grating-assisted couplers," J. Opt. Soc. Am. A 13, 2403-2413 (1996).
[CrossRef]

S. Zhang and T. Tamir, "Analysis and design of broadband grating couplers," IEEE J. Quantum Electron. 29, 2813-2824 (1993).
[CrossRef]

Appl. Phys. Lett. (1)

R. C. Alferness, T. L. Kock, L. L. Buhl, F. Storz, F. Heismann, and M. J. R. Martyak, "Grating assisted InGaAsP/InP vertical co-directional coupler filter," Appl. Phys. Lett. 55, 2011-2013 (1989).
[CrossRef]

IEEE J. Quantum Electron. (5)

K. Ogawa, W. Chang, B. Sopori, and F. Rosenbaum, "A theoretical analysis of edged grating couplers for integrated optics," IEEE J. Quantum Electron. QE-9, 29-42 (1973).
[CrossRef]

S. Zhang and T. Tamir, "Analysis and design of broadband grating couplers," IEEE J. Quantum Electron. 29, 2813-2824 (1993).
[CrossRef]

G. Griffel, M. Itzkovich, and A. A. Hardy, "Coupled mode formulation for directional couplers with longitudinal perturbation," IEEE J. Quantum Electron. 27, 985-994 (1991).
[CrossRef]

G. Hadjicostas, J. Butler, G. Evans, N. Carlson, and R. Amantea, "A numerical investigation of wave interactions in dielectric waveguides with periodic surface corrugations," IEEE J. Quantum Electron. 26, 893-902 (1990).
[CrossRef]

J. Butler, W. Ferguson, G. Evans, P. Stabile, and A. Rosen, "A boundary element technique applied to the analysis of waveguides with periodic surface corrugations," IEEE J. Quantum Electron. 28, 1701-1709 (1992).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

F. Liu, H. Hier, and T. Worchesky, "Dual-side processed demultiplexer using grating-assisted codirectional coupler," IEEE Photon. Technol. Lett. 17, 600-602 (2005).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

J. S. Bagby, D. P. Nyquist, and B. C. Drachman, "Integral formulation for analysis of integrated dielectric waveguides," IEEE Trans. Microwave Theory Tech. MTT-33, 906-915 (1985).
[CrossRef]

T. E. Rozzi and G. H. In't Veld, "Field and network analysis of interacting step discontinuities in planar dielectric waveguides," IEEE Trans. Microwave Theory Tech. MTT-27, 303-309 (1979).
[CrossRef]

J. Lightwave Technol. (6)

W. Huang, B. E. Little, and S. K. Chaudhuri, "A new approach to grating-assisted couplers," J. Lightwave Technol. 9, 721-727 (1991).
[CrossRef]

N. Sun, J. Butler, G. Evans, L. Pang, and P. Congdon, "Analysis of grating-assisted directional couplers using the Floquet-Bloch theory," J. Lightwave Technol. 15, 2301-2315 (1997).
[CrossRef]

D. Marcuse, "Directional couplers made of nonidentical asymmetric slabs. Part I: Synchronous couplers," J. Lightwave Technol. LT-5, 113-118 (1987).
[CrossRef]

D. Marcuse, "Directional couplers made of nonidentical assymetrical slabs. Part II: Grating-assisted couplers," J. Lightwave Technol. LT-5, 268-273 (1987).
[CrossRef]

V. Passaro, "Optimal design of grating-assisted directional couplers," J. Lightwave Technol. 18, 973-984 (2000).
[CrossRef]

J. Butler, N. Sun, G. Evans, L. Pang, and P. Congdon, "Grating-assisted coupling of light between semiconductor and glass waveguides," J. Lightwave Technol. 16, 1038-1048 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

Other (6)

D. S. Jones, Theory of Electromagnetism (Pergamon, 1964).

M. Nishimoto and H. Ikuno, "Analysis of electromagnetic wave diffraction by a semi-infinite strip grating and evaluation of end effects," in Proceedings of Progress in Electromagnetic Research (PIERS) (1999), Vol. 23, pp. 39-58; http://emacademy.org/piers.
[CrossRef]

D. Marcuse, Integrated Optics (IEEE Press, 1973).

T. Tamir, Integrated Optics (Springer-Verlag, 1975).

A. Sommerfeld, Partial Differential Equations in Physics (Academic, 1949).

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, 1989).

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Figures (11)

Fig. 1
Fig. 1

Geometry of the grating-assisted optical coupler. The rectangular periodic gratings are located on the top of slab 1 and on the bottom of slab 2. The gratings’ parameters are period Λ, thickness w, and diffraction index n 2 .

Fig. 2
Fig. 2

Geometry of the homogeneous (without discontinuities) problem. An infinite line source with current density J ( x , z ) is located inside slab 1.

Fig. 3
Fig. 3

Typical curve of the determinant’s absolute value with respect to the normalized imaginary part β k 0 .

Fig. 4
Fig. 4

(a) Imaginary parts β e k 0 and β o k 0 and (b) real parts a e k 0 and a o k 0 of the propagation constants of the grating modes as functions of the grating’s thickness w, for n 2 = 3.2 , H = 2 μ m , λ = 1.5 μ m , and d 1 = l 1 = Λ 2 .

Fig. 5
Fig. 5

(a) β e k 0 and β o k 0 (solid curves) and N e k 0 and N 0 k 0 (dashed curves) and (b) a e k o and a o k 0 as functions of the normalized wavelength Λ λ with n 2 = 3.2 , H = 2 μ m , w = 0.2 μ m , and d 1 = l 1 = Λ 2 .

Fig. 6
Fig. 6

Imaginary parts β e k 0 and β o k 0 (solid curves) of the propagation constants of the periodic coupler with d 1 = l 1 = Λ 2 and k e k 0 and k o k 0 (dashed curves) of the nonperiodic coupler with d 1 = 0 and l 1 = Λ as functions of Λ λ , for n 2 = 3.2 , H = 2 μ m , and w = 0.2 μ m .

Fig. 7
Fig. 7

(a) β e s k 0 and β o s k 0 (solid curves) and β e w k 0 and β o w k 0 (dashed lines) and (b) α e s k 0 and a o s k 0 (solid curves) and a e w k 0 and a o w k 0 (dashed curves) as functions of Λ λ , for n 2 = 3.2 (w, weak perturbation) and n 2 = 3.4 (s, strong perturbation), H = 2 μ m , w = 0.2 μ m , and d 1 = l 1 = Λ 2 .

Fig. 8
Fig. 8

(a) β e k 0 and β o k 0 and (b) α e k 0 and α e k 0 as functions of the grating’s normalized diffraction index n 2 n 1 with w = 0.2 μ m , H = 2 μ m , Λ = 1.5 μ m , and d 1 = l 1 = Λ 2 .

Fig. 9
Fig. 9

(a) β e k 0 and β o k 0 (solid curves) and N e k 0 and N o k 0 (dashed curves) and (b) α o k 0 and α o k 0 as functions of the separation distance H for w = 0.2 μ m , n 2 = 3.2 , λ = 1.5 μ m , and d 1 = l 1 = Λ 2 .

Fig. 10
Fig. 10

(a) β e k 0 and β o k 0 and α e k 0 and α o k 0 as functions of n 1 n 2 for the grating-assisted coupler with grating on both slabs with M = 1 , d 1 = l 1 = Λ 2 , and λ = 1.5 μ m .

Fig. 11
Fig. 11

Rectangular partition of the grating-assisted coupler.

Tables (4)

Tables Icon

Table 1 Normalized Propagation Constant Convergence Pattern of the Isolated Grating Slab in Subsection 5A

Tables Icon

Table 2 Distance of the Propagation Constants of the Grating Modes with Respect to n 2 n 1 for λ = 1.5 μ m

Tables Icon

Table 3 β e k 0 with Respect to λ for the Cases ( n 2 = 3.2 ) a

Tables Icon

Table 4 β o k 0 with Respect to λ for the Five Cases of Table 3

Equations (59)

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E = Ψ ( x , z ) y ̂
Ψ ( x , z ) = k 0 2 ( n 2 2 n 1 2 ) [ S d 1 G 1 ( x , z ; x , z ) Ψ 1 ( x , z ) d x d z + S d 2 G 2 ( x , z ; x , z ) Ψ 2 ( x , z ) d x d z ] , ( x , z ) R 2 ,
Ψ ( x , z ) = u ( x , z ) exp ( γ z ) ,
u ( x , z ) = k 0 2 ( n 2 2 n 1 2 ) [ S d 1 G 1 ( x , z ; x , z ) u 1 ( x , z ) exp ( γ ( z z ) ) d x d z + S d 2 G 2 ( x , z ; x , z ) u 2 ( x , z ) exp ( γ ( z z ) ) d x d z ] , ( x , z ) R 2 .
J ( x , z ) = 1 j ω μ 0 δ ( x x ) δ ( z z ) y ̂ ,
d x d , z R ,
G p ( x , z ; x , z ) = 1 4 π + d λ e j λ ( z z ) e g 1 x x g 1 ,
G s ( x , z ; x , z ) = 1 4 π + d λ e j λ ( z z ) γ ( λ , x , x ) .
2 γ ( λ , x , x ) x 2 ( λ 2 k 0 2 n 2 ( x ) ) γ ( λ , x , x ) = 0 ,
n ( x ) = { n 0 , x > H + d n 1 , H d < x < H + d n 0 , d < x < H d n 1 , d < x < d n 0 , x < d .
γ ( λ , x , x ) = { A 8 ( λ ) exp [ g 0 ( λ ) ( x H d ) ] , x > H + d A 6 ( λ ) cosh [ g 1 ( λ ) ( x H ) ] A 7 ( λ ) sinh [ g 1 ( λ ) ( x H ) ] , H d < x < H + d A 4 ( λ ) cosh [ g 0 ( λ ) ( x H 2 ) ] + A 5 ( λ ) sinh [ g 0 ( λ ) ( x H 2 ) ] , d < x < H d A 2 ( λ ) cosh [ g 1 ( λ ) x ] A 3 ( λ ) sinh [ g 1 ( λ ) x ] , d < x < d A 1 ( λ ) exp [ g 0 ( λ ) ( x + d ) ] , x < d
G 1 ( x , z ; x , z ) = 1 4 π + d λ e j λ ( z z ) δ 1 ( λ , x , x ) ,
( P o P e K ε ) ( P o P e + K ε ) = 0 ,
G 2 ( x , z ; x , z ) = 1 4 π + d λ e j λ ( z z ) δ 2 ( λ , x , x ) ,
u ( x , z ) = k 0 2 ( n 2 2 n 1 2 ) 4 π r = + { S r 1 [ + d λ exp [ j λ ( z z ) ] δ 1 ( λ , x , x ) ] u 1 ( x , z ) exp [ γ ( z z ) ] d x d z + S r 2 [ + d λ exp [ j λ ( z z ) ] δ 2 ( λ , x , x ) ] u 2 ( x , z ) exp [ γ ( z z ) ] d x d z } ,
S r 1 = i = 1 M [ d w , d ] × [ d i + r Λ , d i + l i + r Λ ] ,
S r 2 = i = 1 M [ H d , H d + w ] × [ d i + r Λ , d i + l i + r Λ ]
u ( x , z ) = k 0 2 ( n 2 2 n 1 2 ) 2 Λ p = + exp ( j 2 π p Λ z ) [ S 0 1 u 1 ( x , ζ ) exp ( j 2 π p Λ ζ ) δ 1 ( j γ + 2 π p Λ , x , x ) d x d ζ + S 0 2 u 2 ( x , ζ ) exp ( j 2 π p Λ ζ ) δ 2 ( j γ + 2 π p Λ , x , x ) d x d ζ ] .
u i ( x , ζ ) = n = + φ i , n ( x ) exp ( j 2 π n Λ ζ ) , ( x , ζ ) S 0 i , i = 1 , 2 ;
φ 1 , n ( x ) = c n 1 + exp [ g 2 , n ( x d + w 2 ) ] + c n 1 exp [ g 2 , n ( x d + w 2 ) ] ,
φ 2 , n ( x ) = c n 2 + exp [ g 2 , n ( x H + d w 2 ) ] + c n 2 exp [ g 2 , n ( x H + d w 2 ) ] ,
g i , n = g i ( j γ + 2 π n Λ ) = [ ( j γ + 2 π n Λ ) 2 k 0 2 n i 2 ] 1 2 , i = 0 , 1 , 2 .
u ( x , z ) = k 0 2 ( n 2 2 n 1 2 ) 2 p = + n = + { J p n exp ( j 2 π p Λ z ) [ c n 1 + Q n p 1 + ( x ) + c n 1 Q n p 1 ( x ) + c n 2 + Q n p 2 + ( x ) + c n 2 Q n p 2 ( x ) ] } , ( x , z ) R 2 ,
exp [ ± g 2 , m ( x d + w 2 ) ] exp ( + j 2 π m Λ z ) , ( x , z ) S d 1 , m Z
n = + J m n ( K m n ± + c n 1 + + K m n ± c n 1 ) = k 0 2 ( n 2 2 n 1 2 ) p = + n = + J m p J p n ( c n 1 + Q m n p 11 ± + + c n 1 Q m n p 11 ± + c n 2 + Q m n p 12 ± + + c n 2 Q m n p 12 ± ) , m Z .
n = + J m n ( K m n ± + c n 2 + + K m n ± c n 2 ) = k 0 2 ( n 2 2 n 1 2 ) × p = + n = + J m p J p n ( c n 1 + Q m n p 21 ± + + c n 1 Q m n p 21 ± + c n 2 + Q m n p 22 ± + + c n 2 Q m n p 22 ± ) , m Z .
[ A 11 + + A 11 + A 12 + + A 12 + A 11 + A 11 A 12 + A 12 A 21 + + A 21 + A 22 + + A 22 + A 21 + A 21 A 22 + A 22 ] [ c 1 + c 1 c 2 + c 2 ] = 0 ,
( A 11 ± ± ) m n = { J m n + p = + [ k 0 2 ( n 2 2 n 1 2 ) g 2 , n 2 g 1 , p 2 J m p J p n ] } K m n ± ± + p = + [ k 0 2 ( n 2 2 n 1 2 ) g 2 , n 2 g 1 , p 2 J m p J p n ( R m n p 1 ± ± + H m n p 11 ± ± ) ] ,
( A 12 ± ± ) m n = p = + [ k 0 2 ( n 2 2 n 1 2 ) g 2 , n 2 g 1 , p 2 J m p J p n H m n p 12 ± ± ] ,
( A 21 ± ± ) m n = p = + [ k 0 2 ( n 2 2 n 1 2 ) g 2 , n 2 g 1 , p 2 J m p J p n H m n p 21 ± ± ] ,
( A 22 ± ± ) m n = { J m n + p = + [ k 0 2 ( n 2 2 n 1 2 ) g 2 , n 2 g 1 , p 2 J m p J p n ] } K m n ± ± + p = + [ k 0 2 ( n 2 2 n 1 2 ) g 2 , n 2 g 1 , p 2 J m p J p n ( R m n p 2 ± ± + H m n p 22 ± ± ) ] ,
A 11 ± ± = A 22 , A 12 ± ± = A 21 .
[ A 11 + + A 11 + A 12 + + A 12 + A 11 + A 11 A 12 + A 12 A 12 A 12 + A 11 A 11 + A 12 + A 12 + + A 11 + A 11 + + ] [ c 1 + c 1 c 2 + c 2 ] = 0 .
[ A + + A + A + A ] [ c + c ] = 0 .
γ e = α e + j β e , γ o = α o + j β o .
M i [ Ψ ( r ) r 2 G ( r r ) G ( r r ) r 2 Ψ ( r ) ] ds
= M i [ Ψ ( r ) r G ( r r ) G ( r r ) r Ψ ( r ) ] n ̂ i dl ,
d i s i [ Ψ ( r ) r 2 G ( r r ) G ( r r ) r 2 Ψ ( r ) ] ds = L i [ Ψ ( r ) r G ( r r ) G ( r r ) r Ψ ( r ) ] n ̂ i dl I L i ,
k 0 2 ( n 2 2 n 1 2 ) s i Ψ ( r ) G ( r r ) ds d i s i Ψ ( r ) δ ( r r ) ds = I L i .
k 0 2 ( n 2 2 n 1 2 ) i = + s i Ψ ( r ) G ( r r ) ds i = + L i Ψ ( r ) δ ( r r ) ds = i = + I L i .
Ψ ( r ) = k 0 2 ( n 2 2 n 1 2 ) i = + s i Ψ ( r ) G ( r r ) ds , r R 2 ,
δ 1 ( λ , x , x ) = 1 P o 2 P e 2 K 2 ε 2 { P o P e g 0 g 1 ε [ cosh ( g 1 x ) P e + sinh ( g 1 x ) P o ] exp [ g 0 ( x H d ) ] , x H + d P o 2 P e 2 g 0 ε [ cosh ( g 1 x ) P e + sinh ( g 1 x ) P o ] { cosh [ g 1 ( x H ) ] P e sinh [ g 1 ( x H ) ] P o } , H d x H + d P o P e [ cosh ( g 1 x ) P e + sinh ( g 1 x ) P o ] { P o P e exp [ g 0 ( x d ) ] + K ε exp [ g 0 ( x H + d ) ] } , d x H d { P o 3 P e 3 g 1 [ cosh ( g 1 x ) P e + sinh ( g 1 x ) P o ] [ cosh ( g 1 x ) P e sinh ( g 1 x ) P o ] } + { K Q o Q e P o P e g 1 ε 2 [ cosh ( g 1 x ) P e + sinh ( g 1 x ) P o ] [ cosh ( g 1 x ) Q e + sinh ( g 1 x ) Q o ] } , d x x d { P o 3 P e 3 g 1 [ cosh ( g 1 x ) P e sinh ( g 1 x ) P o ] [ cosh ( g 1 x ) P e + sinh ( g 1 x ) P o ] } { + K Q o Q e P o P e g 1 ε 2 [ cosh ( g 1 x ) Q e + sinh ( g 1 x ) Q o ] [ cosh ( g 1 x ) P e + sinh ( g 1 x ) P o ] } , d x x d { P o 2 P e 2 [ cosh ( g 1 x ) P e sinh ( g 1 x ) P o ] + K Q o Q e ε 2 [ cosh ( g 1 x ) Q e + sinh ( g 1 x ) Q o ] } exp [ g 0 ( x + d ) ] , x d ; }
δ 2 ( λ , x , x ) = 1 P o 2 P e 2 K 2 ε 2 { P o 2 P e 2 { cosh [ g 1 ( H x ) ] P e sinh [ g 1 ( H x ) ] P o } exp [ g 0 ( x H d ) ] + K Q o Q e ε 2 { cosh [ g 1 ( H x ) ] Q e + sinh [ g 1 ( H x ) ] Q o } exp [ g 0 ( x H d ) ] , x H + d P o 3 P e 3 g 1 { cosh [ g 1 ( H x ) ] P e sinh [ g 1 ( H x ) ] P o } { cosh [ g 1 ( x H ) ] P e sinh [ g 1 ( x H ) ] P o } + K P o P e Q o Q e g 1 ε 2 { cosh [ g 1 ( H x ) ] Q e + sinh [ g 1 ( H x ) ] Q o } { cosh [ g 1 ( x H ) ] P e sinh [ g 1 ( x H ) ] P o } , H d x x H + d P o 3 P e 3 g 1 { cosh [ g 1 ( H x ) ] P e + sinh [ g 1 ( H x ) ] P o } { cosh [ g 1 ( x H ) ] P e + sinh [ g 1 ( x H ) ] P o } + K P o P e Q o Q e g 1 ε 2 { cosh [ g 1 ( H x ) ] P e + sinh [ g 1 ( H x ) ] P o } { cosh [ g 1 ( x H ) ] Q e sinh [ g 1 ( x H ) ] Q o } , H d x x H + d P o P e { cosh [ g 1 ( H x ) ] P e + sinh [ g 1 ( H x ) ] P o } { P o P e exp [ g o ( x H + d ) ] + K ε exp [ g 0 ( x d ) ] } , d x H d P o 2 P e 2 g 0 ε { cosh [ g 1 ( H x ) ] P e + sinh [ g 1 ( H x ) ] P o } [ cosh ( g 1 x ) P e + sinh ( g 1 x ) P o ] , d x d P o P e g 0 g 1 ε { cosh [ g 1 ( H x ) ] P e + sinh [ g 1 ( H x ) ] P o } exp [ g 0 ( x + d ) ] , x d .
P e = P e ( g 0 , g 1 , d ) = g 0 cosh ( g 1 d ) + g 1 sinh ( g 1 d ) ,
P o = P o ( g 0 , g 1 , d ) = g 0 sinh ( g 1 d ) + g 1 cosh ( g 1 d ) ,
Q e = Q e ( g 0 , g 1 , d ) = g 0 cosh ( g 1 d ) g 1 sinh ( g 1 d ) ,
Q o = Q o ( g , g 1 , d ) = g 1 cosh ( g 1 d ) g 0 sinh ( g 1 d ) ,
K = Q e P o Q o P e 2 , ε = exp [ g 0 ( H 2 d ) ] .
J q = 1 Λ i = 1 M d i d i + l i exp ( j 2 π q Λ ζ ) d ζ ,
Q n p 1 ± ( x ) = d w d exp [ ± g 2 , n ( x d + w 2 ) ] δ 1 ( j γ + 2 π p Λ , x , x ) d x ,
Q n p 2 ± ( x ) = H d H d + w exp [ ± g 2 , n ( x H + d w 2 ) ] δ 2 ( j γ + 2 π p Λ , x , x ) d x ,
K m n ± ± = d w d exp [ ± g 2 , m ( x d + w 2 ) ] exp [ ± g 2 , n ( x d + w 2 ) ] d x ,
Q m n p 1 i ± ± = 1 2 d w d exp [ ± g 2 , m ( x d + w 2 ) ] Q n p i ± ( x ) d x , i = 1 , 2 ,
Q m n p 2 i ± ± = 1 2 H d H d + w exp [ ± g 2 , m ( x H + d w 2 ) ] Q n p i ± ( x ) d x , i = 1 , 2 ,
R m n p 1 ± ± = P o 3 ( g 0 p , g 1 p , d ) P e 3 ( g 0 p , g 1 p , d ) 2 g 1 p [ P o 2 ( g 0 p , g 1 p , d ) P e 2 ( g 0 p , g 1 p , d ) K 2 ( g 0 p , g 1 p , d ) ε 2 ] ( g 2 m 2 g 1 p 2 ) × { e ( ± g 2 m ± g 2 n ) w 2 [ P e ( g 2 m , g 1 p , d ) P e ( g 0 p , g 1 p , d ) + P o ( g 2 m , g 1 p , d ) P o ( g 0 p , g 1 p , d ) ] [ P e ( g 2 n , g 1 p , d ) P e ( g 0 p , g 1 p , d ) P o ( g 2 n , g 1 p , d ) P o ( g 0 p , g 1 p , d ) ] e ( ± g 2 m g 2 n ) w 2 [ P e ( g 2 m , g 1 p , d ) P e ( g 0 p , g 1 p , d ) P o ( g 2 m , g 1 p , d ) P o ( g 0 p , g 1 p , d ) ] [ P e ( g 2 n , g 1 p , d w ) P e ( g 0 p , g 1 p , d ) + P o ( g 2 n , g 1 p , d w ) P o ( g 0 p , g 1 p , d ) ] e ( g 2 m ± g 2 n ) w 2 [ P e ( g 2 m , g 1 p , d w ) P e ( g 0 p , g 1 p , d ) + P o ( g 2 m , g 1 p , d w ) P o ( g 0 p , g 1 p , d ) ] [ P e ( g 2 n , g 1 p , d ) P e ( g 0 p , g 1 p , d ) P o ( g 2 n , g 1 p , d ) P o ( g 0 p , g 1 p , d ) ] + e ( g 2 m g 2 n ) w 2 [ P e ( g 2 m , g 1 p , d w ) P e ( g 0 p , g 1 p , d ) P o ( g 2 m , g 1 p , d w ) P o ( g 0 p , g 1 p , d ) ] [ P e ( g 2 n , g 1 p , d w ) P e ( g 0 p , g 1 p , d ) + P o ( g 2 n , g 1 p , d w ) P o ( g 0 p , g 1 p , d ) ] } ,
H m n p 11 ± ± = K ( g 0 p , g 1 p , d ) P o ( g 0 p , g 1 p , d ) P e ( g 0 p , g 1 p , d ) Q o ( g 0 p , g 1 p , d ) Q e ( g 0 p , g 1 p , d ) 2 g 1 p [ P o 2 ( g 0 p , g 1 p , d ) P e 2 ( g 0 p , g 1 p , d ) K 2 ( g 0 p , g 1 p , d ) ε 2 ] ( g 2 m 2 g 1 p 2 ) ε 2 × { e ( ± g 2 m ± g 2 n ) w 2 [ P e ( g 2 m , g 1 p , d ) P e ( g 0 p , g 1 p , d ) + P o ( g 2 m , g 1 p , d ) P o ( g 0 p , g 1 p , d ) ] [ P e ( g 2 n , g 1 p , d ) Q e ( g 0 p , g 1 p , d ) + P o ( g 2 n , g 1 p , d ) Q o ( g 0 p , g 1 p , d ) ] e ( ± g 2 m g 2 n ) w 2 [ P e ( g 2 m , g 1 p , d ) Q e ( g 0 p , g 1 p , d ) + P o ( g 2 m , g 1 p , d ) Q o ( g 0 p , g 1 p , d ) ] [ P e ( g 2 n , g 1 p , d w ) P e ( g 0 p , g 1 p , d ) + P o ( g 2 n , g 1 p , d w ) P o ( g 0 p , g 1 p , d ) ] e ( g 2 m ± g 2 n ) w 2 [ P e ( g 2 m , g 1 p , d w ) P e ( g 0 p , g 1 p , d ) + P o ( g 2 m , g 1 p , d w ) P o ( g 0 p , g 1 p , d ) ] [ P e ( g 2 n , g 1 p , d ) Q e ( g 0 p , g 1 p , d ) + P o ( g 2 n , g 1 p , d ) Q o ( g 0 p , g 1 p , d ) ] + e ( g 2 m g 2 n ) w 2 [ P e ( g 2 m , g 1 p , d w ) Q e ( g 0 p , g 1 p , d ) + P o ( g 2 m , g 1 p , d w ) Q o ( g 0 p , g 1 p , d ) ] [ P e ( g 2 n , g 1 p , d w ) P e ( g 0 p , g 1 p , d ) + P o ( g 2 n , g 1 p , d w ) P o ( g 0 p , g 1 p , d ) ] } ,
H m n p 12 ± ± = g 0 p P o 2 ( g 0 p , g 1 p , d ) P e 2 ( g 0 p , g 1 p , d ) 2 [ P o 2 ( g 0 p , g 1 p , d ) P e 2 ( g 0 p , g 1 p , d ) K 2 ( g 0 p , g 1 p , d ) ε 2 ] ( g 2 m 2 g 1 p 2 ) ε × { e ( ± g 2 m ± g 2 n ) w 2 [ P e ( g 2 m , g 1 p , d ) P e ( g 0 p , g 1 p , d ) + P o ( g 2 m , g 1 p , d ) P o ( g 0 p , g 1 p , d ) ] [ P e ( ± g 2 n , g 1 p , d w ) P e ( g 0 p , g 1 p , d ) + P o ( ± g 2 n , g 1 p , d w ) P o ( g 0 p , g 1 p , d ) ] e ( ± g 2 m g 2 n ) w 2 [ P e ( g 2 m , g 1 p , d ) P e ( g 0 p , g 1 p , d ) + P o ( g 2 m , g 1 p , d ) P o ( g 0 p , g 1 p , d ) ] [ P e ( ± g 2 n , g 1 p , d ) P e ( g 0 p , g 1 p , d ) + P o ( ± g 2 n , g 1 p , d ) P o ( g 0 p , g 1 p , d ) ] e ( g 2 m ± g 2 n ) w 2 [ P e ( g 2 m , g 1 p , d w ) P e ( g 0 p , g 1 p , d ) + P o ( g 2 m , g 1 p , d w ) P o ( g 0 p , g 1 p , d ) ] [ P e ( g 2 n , g 1 p , d w ) P e ( g 0 p , g 1 p , d ) + P o ( g 2 n , g 1 p , d w ) P o ( g 0 p , g 1 p , d ) ] + e ( g 2 m g 2 n ) w 2 [ P e ( g 2 m , g 1 p , d w ) P e ( g 0 p , g 1 p , d ) + P o ( g 2 m , g 1 p , d w ) P o ( g 0 p , g 1 p , d ) ] [ P e ( g 2 n , g 1 p , d ) P e ( g 0 p , g 1 p , d ) + P o ( g 2 n , g 1 p , d ) P o ( g 0 p , g 1 p , d ) ] } .
K m n ± ± = K m n , R m n p 2 = R m n p 1 ± ± ,
H m n p 22 = H m n p 11 ± ± , H m n p 21 = H m n p 12 ± ± ,

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