Abstract

The interaction between a periodic and conventional dielectric waveguides is investigated theoretically for a two-dimensional model system. A modified coupled-mode theory is formulated for the considered system and found to agree well with rigorous numerical calculations. It is shown that in a certain wavelength range the contra-directional coupling between the two waveguides can be achieved with high efficiency. But the spectrum of the coupling efficiency is blue-shifted and thus the strongest coupling does not occur in the case when two individual waveguides have the same propagation constant. For such a contra-directional coupling system, the coupling efficiency grows with the coupling length and it tends to 100% (excluding insertion loss) when the coupling length is larger than a certain value, and the coupling window can be largely broaden by reducing the distance between the coupled waveguides.

© 2010 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
  3. B.-I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, “Guided modes with imaginary transverse wave number ,” J. Appl. Phys. 93, 9386–9388 (2003).
    [Crossref]
  4. A. Alu and N. Engheta, “Anomalous mode coupling in guided-wave structures containing metamaterials with negative permittivity and permeability,” in Proc. IEEE Nanotechnology, Washington, DC, Aug. 26-28, 2002, pp. 233–234.
    [Crossref]
  5. R. Islam, F. Elek, and G. V. Eleftheriades, “Coupled-line metamaterial coupler having co-directional phase but contra-directional power flow,” Electron. Lett. 40, 315–317 (2004).
    [Crossref]
  6. C. Caloz, A. Sanada, and T. Itoh, “A novel composite right-/left-handed coupled-line directional coupler with arbitrary coupling level and broad bandwidth,” IEEE Trans. Microwave Theory Tech. 52, 980–992 (2004).
    [Crossref]
  7. Y. Yuan, L. Ran, H. Chen, J. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Backward coupling waveguide coupler using left-handed material,” Appl. Phys. Lett. 88, 211903 (2006).
    [Crossref]
  8. W. Yan, L. F. Shen, Y. Yuan, and T. J. Yang, “Interaction between negative and positive index medium waveguides,” J. Lightwave Technol. 26, 3560–3566 (2008).
    [Crossref]
  9. J.-L. Archambault, P. St. J. Russell, S. Barcelos, P. Hua, and L. Reekie, “Grating-frustrated coupler: a novel channel-dropping filter in single-mode optical fiber,” Opt. Lett. 19, 180–182 (1994).
    [Crossref] [PubMed]
  10. L. Dong, P. Hua, T. A. Birks, L. Reekie, and P. St. J. Russell, “Novel add/drop filters for wavelength-Division-multiplexing optical fiber systems using a Bragg grating assisted mismatched coupler,” IEEE Photon. Technol. Lett. 8, 1656–1658 (1996).
    [Crossref]
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    [Crossref]
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2008 (3)

2006 (1)

Y. Yuan, L. Ran, H. Chen, J. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Backward coupling waveguide coupler using left-handed material,” Appl. Phys. Lett. 88, 211903 (2006).
[Crossref]

2005 (1)

S. Foteinopoulou and C. M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects,” Phys. Rev. B 72, 165112 (2005).
[Crossref]

2004 (2)

R. Islam, F. Elek, and G. V. Eleftheriades, “Coupled-line metamaterial coupler having co-directional phase but contra-directional power flow,” Electron. Lett. 40, 315–317 (2004).
[Crossref]

C. Caloz, A. Sanada, and T. Itoh, “A novel composite right-/left-handed coupled-line directional coupler with arbitrary coupling level and broad bandwidth,” IEEE Trans. Microwave Theory Tech. 52, 980–992 (2004).
[Crossref]

2003 (2)

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003).
[Crossref]

B.-I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, “Guided modes with imaginary transverse wave number ,” J. Appl. Phys. 93, 9386–9388 (2003).
[Crossref]

2002 (3)

2000 (1)

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
[Crossref]

1997 (1)

1996 (3)

S. Z. Zhang and T. Tamir, “Rigorous theory of grating-assisted couplers,” J. Opt. Soc. Am. A 12, 2403–2413 (1996).
[Crossref]

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
[Crossref]

L. Dong, P. Hua, T. A. Birks, L. Reekie, and P. St. J. Russell, “Novel add/drop filters for wavelength-Division-multiplexing optical fiber systems using a Bragg grating assisted mismatched coupler,” IEEE Photon. Technol. Lett. 8, 1656–1658 (1996).
[Crossref]

1995 (1)

S. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B: Opt. Phys. 12, 1267–1272 (1995).
[Crossref]

1994 (1)

1990 (1)

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[Crossref] [PubMed]

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys.  10, 509–514 (1968).
[Crossref]

Alu, A.

A. Alu and N. Engheta, “Anomalous mode coupling in guided-wave structures containing metamaterials with negative permittivity and permeability,” in Proc. IEEE Nanotechnology, Washington, DC, Aug. 26-28, 2002, pp. 233–234.
[Crossref]

Archambault, J.-L.

Barcelos, S.

Birks, T. A.

L. Dong, P. Hua, T. A. Birks, L. Reekie, and P. St. J. Russell, “Novel add/drop filters for wavelength-Division-multiplexing optical fiber systems using a Bragg grating assisted mismatched coupler,” IEEE Photon. Technol. Lett. 8, 1656–1658 (1996).
[Crossref]

Caloz, C.

C. Caloz, A. Sanada, and T. Itoh, “A novel composite right-/left-handed coupled-line directional coupler with arbitrary coupling level and broad bandwidth,” IEEE Trans. Microwave Theory Tech. 52, 980–992 (2004).
[Crossref]

Chan, C. T.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[Crossref] [PubMed]

Chen, H.

Y. Yuan, L. Ran, H. Chen, J. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Backward coupling waveguide coupler using left-handed material,” Appl. Phys. Lett. 88, 211903 (2006).
[Crossref]

Chen, J. C.

S. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B: Opt. Phys. 12, 1267–1272 (1995).
[Crossref]

Devenyi, A.

S. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B: Opt. Phys. 12, 1267–1272 (1995).
[Crossref]

Dong, L.

L. Dong, P. Hua, T. A. Birks, L. Reekie, and P. St. J. Russell, “Novel add/drop filters for wavelength-Division-multiplexing optical fiber systems using a Bragg grating assisted mismatched coupler,” IEEE Photon. Technol. Lett. 8, 1656–1658 (1996).
[Crossref]

Economou, E. N.

C. M. Soukoulis, J. Zhou, T. Koschny, M. Kafesaki, and E. N. Economou, “The science of negative index materials,” J. Phys.: Condens. Matter 20, 304217 (2008).
[Crossref]

Eleftheriades, G. V.

R. Islam, F. Elek, and G. V. Eleftheriades, “Coupled-line metamaterial coupler having co-directional phase but contra-directional power flow,” Electron. Lett. 40, 315–317 (2004).
[Crossref]

Elek, F.

R. Islam, F. Elek, and G. V. Eleftheriades, “Coupled-line metamaterial coupler having co-directional phase but contra-directional power flow,” Electron. Lett. 40, 315–317 (2004).
[Crossref]

Engheta, N.

A. Alu and N. Engheta, “Anomalous mode coupling in guided-wave structures containing metamaterials with negative permittivity and permeability,” in Proc. IEEE Nanotechnology, Washington, DC, Aug. 26-28, 2002, pp. 233–234.
[Crossref]

Essen, S. V.

Fan, S.

S. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B: Opt. Phys. 12, 1267–1272 (1995).
[Crossref]

Foteinopoulou, S.

S. Foteinopoulou and C. M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects,” Phys. Rev. B 72, 165112 (2005).
[Crossref]

Grzegorczyk, T. M.

Y. Yuan, L. Ran, H. Chen, J. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Backward coupling waveguide coupler using left-handed material,” Appl. Phys. Lett. 88, 211903 (2006).
[Crossref]

B.-I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, “Guided modes with imaginary transverse wave number ,” J. Appl. Phys. 93, 9386–9388 (2003).
[Crossref]

He, S.

Ho, K. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[Crossref] [PubMed]

Hua, P.

L. Dong, P. Hua, T. A. Birks, L. Reekie, and P. St. J. Russell, “Novel add/drop filters for wavelength-Division-multiplexing optical fiber systems using a Bragg grating assisted mismatched coupler,” IEEE Photon. Technol. Lett. 8, 1656–1658 (1996).
[Crossref]

J.-L. Archambault, P. St. J. Russell, S. Barcelos, P. Hua, and L. Reekie, “Grating-frustrated coupler: a novel channel-dropping filter in single-mode optical fiber,” Opt. Lett. 19, 180–182 (1994).
[Crossref] [PubMed]

Huangfu, J.

Y. Yuan, L. Ran, H. Chen, J. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Backward coupling waveguide coupler using left-handed material,” Appl. Phys. Lett. 88, 211903 (2006).
[Crossref]

Islam, R.

R. Islam, F. Elek, and G. V. Eleftheriades, “Coupled-line metamaterial coupler having co-directional phase but contra-directional power flow,” Electron. Lett. 40, 315–317 (2004).
[Crossref]

Itoh, T.

C. Caloz, A. Sanada, and T. Itoh, “A novel composite right-/left-handed coupled-line directional coupler with arbitrary coupling level and broad bandwidth,” IEEE Trans. Microwave Theory Tech. 52, 980–992 (2004).
[Crossref]

Joannopoulos, J. D.

S. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B: Opt. Phys. 12, 1267–1272 (1995).
[Crossref]

Kafesaki, M.

C. M. Soukoulis, J. Zhou, T. Koschny, M. Kafesaki, and E. N. Economou, “The science of negative index materials,” J. Phys.: Condens. Matter 20, 304217 (2008).
[Crossref]

Kim, C.

Kivshar, Y. S.

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003).
[Crossref]

Kong, J. A.

Y. Yuan, L. Ran, H. Chen, J. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Backward coupling waveguide coupler using left-handed material,” Appl. Phys. Lett. 88, 211903 (2006).
[Crossref]

B.-I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, “Guided modes with imaginary transverse wave number ,” J. Appl. Phys. 93, 9386–9388 (2003).
[Crossref]

Koschny, T.

C. M. Soukoulis, J. Zhou, T. Koschny, M. Kafesaki, and E. N. Economou, “The science of negative index materials,” J. Phys.: Condens. Matter 20, 304217 (2008).
[Crossref]

Kuang, W.

Li, L.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Meade, R. D.

S. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B: Opt. Phys. 12, 1267–1272 (1995).
[Crossref]

Notomi, M.

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
[Crossref]

O’Brien, J. D.

Orlov, S. S.

Ran, L.

Y. Yuan, L. Ran, H. Chen, J. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Backward coupling waveguide coupler using left-handed material,” Appl. Phys. Lett. 88, 211903 (2006).
[Crossref]

Reekie, L.

L. Dong, P. Hua, T. A. Birks, L. Reekie, and P. St. J. Russell, “Novel add/drop filters for wavelength-Division-multiplexing optical fiber systems using a Bragg grating assisted mismatched coupler,” IEEE Photon. Technol. Lett. 8, 1656–1658 (1996).
[Crossref]

J.-L. Archambault, P. St. J. Russell, S. Barcelos, P. Hua, and L. Reekie, “Grating-frustrated coupler: a novel channel-dropping filter in single-mode optical fiber,” Opt. Lett. 19, 180–182 (1994).
[Crossref] [PubMed]

Russell, P. St. J.

L. Dong, P. Hua, T. A. Birks, L. Reekie, and P. St. J. Russell, “Novel add/drop filters for wavelength-Division-multiplexing optical fiber systems using a Bragg grating assisted mismatched coupler,” IEEE Photon. Technol. Lett. 8, 1656–1658 (1996).
[Crossref]

J.-L. Archambault, P. St. J. Russell, S. Barcelos, P. Hua, and L. Reekie, “Grating-frustrated coupler: a novel channel-dropping filter in single-mode optical fiber,” Opt. Lett. 19, 180–182 (1994).
[Crossref] [PubMed]

Sanada, A.

C. Caloz, A. Sanada, and T. Itoh, “A novel composite right-/left-handed coupled-line directional coupler with arbitrary coupling level and broad bandwidth,” IEEE Trans. Microwave Theory Tech. 52, 980–992 (2004).
[Crossref]

Shadrivov, I. V.

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003).
[Crossref]

Shen, L. F.

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Soukoulis, C. M.

C. M. Soukoulis, J. Zhou, T. Koschny, M. Kafesaki, and E. N. Economou, “The science of negative index materials,” J. Phys.: Condens. Matter 20, 304217 (2008).
[Crossref]

S. Foteinopoulou and C. M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects,” Phys. Rev. B 72, 165112 (2005).
[Crossref]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[Crossref] [PubMed]

Stapleton, A.

Sukhorukov, A. A.

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003).
[Crossref]

Tamir, T.

S. Z. Zhang and T. Tamir, “Rigorous theory of grating-assisted couplers,” J. Opt. Soc. Am. A 12, 2403–2413 (1996).
[Crossref]

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys.  10, 509–514 (1968).
[Crossref]

Winn, J. N.

S. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B: Opt. Phys. 12, 1267–1272 (1995).
[Crossref]

Wu, B.-I.

B.-I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, “Guided modes with imaginary transverse wave number ,” J. Appl. Phys. 93, 9386–9388 (2003).
[Crossref]

Yan, W.

Yang, T. J.

Yariv, A.

Yuan, Y.

W. Yan, L. F. Shen, Y. Yuan, and T. J. Yang, “Interaction between negative and positive index medium waveguides,” J. Lightwave Technol. 26, 3560–3566 (2008).
[Crossref]

Y. Yuan, L. Ran, H. Chen, J. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Backward coupling waveguide coupler using left-handed material,” Appl. Phys. Lett. 88, 211903 (2006).
[Crossref]

Zhang, S. Z.

S. Z. Zhang and T. Tamir, “Rigorous theory of grating-assisted couplers,” J. Opt. Soc. Am. A 12, 2403–2413 (1996).
[Crossref]

Zhang, Y.

B.-I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, “Guided modes with imaginary transverse wave number ,” J. Appl. Phys. 93, 9386–9388 (2003).
[Crossref]

Zhou, J.

C. M. Soukoulis, J. Zhou, T. Koschny, M. Kafesaki, and E. N. Economou, “The science of negative index materials,” J. Phys.: Condens. Matter 20, 304217 (2008).
[Crossref]

Appl. Phys. Lett. (1)

Y. Yuan, L. Ran, H. Chen, J. Huangfu, T. M. Grzegorczyk, and J. A. Kong, “Backward coupling waveguide coupler using left-handed material,” Appl. Phys. Lett. 88, 211903 (2006).
[Crossref]

Electron. Lett. (1)

R. Islam, F. Elek, and G. V. Eleftheriades, “Coupled-line metamaterial coupler having co-directional phase but contra-directional power flow,” Electron. Lett. 40, 315–317 (2004).
[Crossref]

IEEE Photon. Technol. Lett. (1)

L. Dong, P. Hua, T. A. Birks, L. Reekie, and P. St. J. Russell, “Novel add/drop filters for wavelength-Division-multiplexing optical fiber systems using a Bragg grating assisted mismatched coupler,” IEEE Photon. Technol. Lett. 8, 1656–1658 (1996).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

C. Caloz, A. Sanada, and T. Itoh, “A novel composite right-/left-handed coupled-line directional coupler with arbitrary coupling level and broad bandwidth,” IEEE Trans. Microwave Theory Tech. 52, 980–992 (2004).
[Crossref]

J. Appl. Phys. (1)

B.-I. Wu, T. M. Grzegorczyk, Y. Zhang, and J. A. Kong, “Guided modes with imaginary transverse wave number ,” J. Appl. Phys. 93, 9386–9388 (2003).
[Crossref]

J. Lightwave Technol. (1)

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

J. Opt. Soc. Am. B: Opt. Phys. (1)

S. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B: Opt. Phys. 12, 1267–1272 (1995).
[Crossref]

J. Phys.: Condens. Matter (1)

C. M. Soukoulis, J. Zhou, T. Koschny, M. Kafesaki, and E. N. Economou, “The science of negative index materials,” J. Phys.: Condens. Matter 20, 304217 (2008).
[Crossref]

Opt. Lett. (4)

Phys. Rev. B (2)

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696–10705 (2000).
[Crossref]

S. Foteinopoulou and C. M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects,” Phys. Rev. B 72, 165112 (2005).
[Crossref]

Phys. Rev. E (1)

I. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602 (2003).
[Crossref]

Phys. Rev. Lett. (1)

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[Crossref] [PubMed]

Proc. IEEE Nanotechnology (1)

A. Alu and N. Engheta, “Anomalous mode coupling in guided-wave structures containing metamaterials with negative permittivity and permeability,” in Proc. IEEE Nanotechnology, Washington, DC, Aug. 26-28, 2002, pp. 233–234.
[Crossref]

Sov. Phys. (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys.  10, 509–514 (1968).
[Crossref]

Other (1)

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

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Figures (6)

Fig. 1.
Fig. 1.

(a) Dispersion relations for the individual PDWG (solid lines) and CDWG (dotted d = 0.75p.

Fig. 2.
Fig. 2.

(a)-(c) Real part of the propagation constant of the supermodes. (d)-(f) Imaginary

Fig. 3.
Fig. 3.

Electric field amplitudes of the supermode with propagation constants β + [(a), (b)] and the backward mode of the individual PDWG (c) at λ = 1.5 μm. (a) corresponds to results from the modified CMT and (b) to those from the PWEM.

Fig. 4.
Fig. 4.

(a) Coupling efficiency versus coupling length for different wavelengths. The value of d is fixed at 0.75p. (b) Coupling efficiency versus coupling length for different d values. The wavelength is fixed at λ = 1.55 μm. Lines in (a) and (b) are calculated with Eq. (11) and circles in (b) are obtained from the rigorous numerical calculations.

Fig. 5.
Fig. 5.

Simulated E field amplitude for the coupling between the PDWG and CDWG at different wavelengths. The initial power is injected into the left end of the CDWG.

Fig. 6.
Fig. 6.

Spectrum of coupling efficiency. Solid line corresponds to the results obtained from 11) and dotted line with circles to the accurate results from the numerical calculations. (a) d = p, (b) d = 0.75p, and (c) d = 0.5p.

Equations (11)

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+ F c ( x , z n + 1 ) · z ̂ d x + F c ( x , z n ) · z ̂ d x = i ω ε 0 z n z n + 1 d z + d x ( ε s ε m ) E · E m * ,
A 1 ( n + 1 ) A 1 ( n ) = i K 12 e i 2 Δ β z n + 1 / 2 A 2 ( n ) + A 2 ( n + 1 ) 2 ,
A 2 ( n + 1 ) A 2 ( n ) = i K 21 e i 2 Δ β z n + 1 / 2 A 1 ( n ) + A 1 ( n + 1 ) 2 ,
K 12 = ω ε 0 4 N 1 ( ε r 2 ε r 3 ) x c 2 w / 2 x c 2 + w / 2 d x p / 2 p / 2 d z ( e y u y * ) e i 2 Δ β z ,
K 21 = ω ε 0 4 N 2 ( ε r 1 ε r 3 ) x c 1 a / 2 x c 1 + a / 2 d x b / 2 b / 2 d z ( u y e y * ) e i 2 Δ β z ,
A ̅ 1 ( n + 1 ) e i 2 Δ β p A ̅ 1 ( n ) e i 2 Δ β p = i 2 K 12 [ A ̅ 2 ( n ) e i 2 Δ β p + A ̅ 2 ( n + 1 ) e i 2 Δ β p ] ,
A ̅ 1 ( n + 1 ) e i 2 Δ β p A ̅ 2 ( n ) e i 2 Δ β p = i 2 K 21 [ A ̅ 1 ( n ) e i 2 Δ β p + A ̅ 1 ( n + 1 ) e i 2 Δ β p ] .
q ± = ( 1 + K 2 ) cos ( Δ β p ) ± 4 K 2 ( 1 + K 2 ) 2 sin 2 ( Δ β p ) 1 K 2 ,
A 1 ( n ) = τ τ + q N q + N τ + q + N τ q N [ q ( n N ) q + ( n N ) ] e i Δ β z n A 0 ,
A 2 ( n ) = q N q + N τ + q + N τ q N [ τ + q ( n N ) τ q + ( n N ) ] e i Δ β z n A 0 ,
η = N 1 N 2 τ τ + ( q + N q N ) τ + q + N τ q N 2 .

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