Abstract

A new ray picture model based on the multiple interference of light waves in dielectric resonant grating–waveguide structures is presented. The model clearly elucidates the phase relationship between the incident plane wave and the waves diffracted from the resonant grating structure that is responsible for the interference of these waves. As a result of this interference process the incident wave can be totally reflected at a certain wavelength and orientation angle. The model is used to describe and analyze this resonance behavior of the grating–waveguide structures as a function of wavelength and incidence angle. The analysis is verified experimentally with semiconductor (InGaAsP/InP) structures at wavelengths of 1.55 μm and also with dielectric (silicon nitride/SiO2) structures at wavelengths of 0.6 μm. All of the structures were formed by electron beam lithography and chemical vapor deposition. The measured results reveal that subnanometer resonance bandwidths and finesses as large as 6000 can be achieved at contrast ratios of 50 with relatively compact structures.

© 1997 Optical Society of America

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References

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  1. R. W. Wood, “On a remarkable case of uneven distribution of light in a diffractiongrating spectrum,” Philos. Mag. 4, 396–402 (1902).
    [CrossRef]
  2. Rayleigh, “Note on the remarkable case of diffraction spectrum described by Prof. Wood,” Philos. Mag. 14, 399–416 (1907).
    [CrossRef]
  3. C. H. Palmer, “Parallel diffraction grating anomalies,” J. Opt. Soc. Am. 42, 269–276 (1952).
    [CrossRef]
  4. A. Hessel, A. A. Oliner, “A new theory of Wood's anomalies,” Appl. Opt. 4, 1275–1297 (1965).
    [CrossRef]
  5. M. Nevière, Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 5.
  6. V. A. Sychugov, A. V. Tishchenko, “Propagation and conversion of light waves in corrugated waveguide structures,” Sov. J. Quantum Electron. 12, 923–926 (1982).
    [CrossRef]
  7. L. Li, “Analysis of planar waveguide grating coupler with double surface corrugationsof identical period,” Opt. Commun. 114, 406–412 (1995).
    [CrossRef]
  8. A. Yariv, M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. QE-13, 233–253 (1977).
    [CrossRef]
  9. E. G. Loewen, M. Nevière, “Dielectric coated gratings: a curious property,” Appl. Opt. 16, 3009–3011 (1977).
    [CrossRef] [PubMed]
  10. E. Popov, L. Mashev, D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta 32, 607–629 (1986).
    [CrossRef]
  11. S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 7, 1470–1474 (1990).
    [CrossRef]
  12. R. Magnusson, S. S. Wang, “Optical grating-waveguide filters,” in International Conference on Holography, CorrelationOptics, and Recording Materials, O. V. Angelsky, ed., Proc. SPIE2108, 380–391 (1993).
    [CrossRef]
  13. M. T. Gale, K. Knop, R. Morf, “Zero order diffraction microstructures for security applications,”, in Optical Security and Anticounterfeiting Systems, W. F. Fugan, ed., Proc. SPIE1210, 83–89 (1990).
    [CrossRef]
  14. R. Magnusson, S. S. Wang, T. D. Black, A. Sohn, “Resonance properties of dielectric waveguide grating: theory and experimentsat 4-18 GHz,” IEEE Trans. Antennas Propag. 42, 567–569 (1994).
    [CrossRef]
  15. S. Peng, G. M. Morris, “Experimental demonstration of resonant anomalies in diffraction fromtwo dimensional gratings,” Opt. Lett. 21, 548 (1996).
    [CrossRef]
  16. A. Sharon, S. Glasberg, D. Rosenblatt, A. A. Friesem, “Metal-based resonant grating-waveguide structures,” J. Opt. Soc. Am. A 14, 588–595 (1997).
    [CrossRef]
  17. S. Glasberg, A. Sharon, D. Rosenblatt, A. A. Friesem, “Long-range surface plasmon resonances in grating-waveguide structures,” Appl. Phys. Lett. 70, 1210–1212 (1997).
    [CrossRef]
  18. A. Sharon, D. Rosenblatt, A. A. Friesem, H. G. Weber, H. Engel, R. Steingrueber, “Light modulation with resonant grating-waveguide structures,” Opt. Lett. 21, 1564–1566 (1996).
    [CrossRef] [PubMed]
  19. P. K. Tien, R. Ulrich, “Theory of prism-film coupler and thin-film light guides,” J. Opt. Soc. Am. 60, 1325–1337 (1970).
    [CrossRef]
  20. P. Sheng, R. S. Stepleman, P. N. Sanda, “Exact eigenfunctions for square wave gratings: applications to diffractionand surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
    [CrossRef]
  21. A. Sharon, D. Rosenblatt, A. A. Friesem, “Narrow spectral bandwidth with grating-waveguide structures,” Appl. Phys. Lett. 69, 4154–4156 (1996).
    [CrossRef]
  22. P. W. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

1997 (2)

S. Glasberg, A. Sharon, D. Rosenblatt, A. A. Friesem, “Long-range surface plasmon resonances in grating-waveguide structures,” Appl. Phys. Lett. 70, 1210–1212 (1997).
[CrossRef]

A. Sharon, S. Glasberg, D. Rosenblatt, A. A. Friesem, “Metal-based resonant grating-waveguide structures,” J. Opt. Soc. Am. A 14, 588–595 (1997).
[CrossRef]

1996 (3)

S. Peng, G. M. Morris, “Experimental demonstration of resonant anomalies in diffraction fromtwo dimensional gratings,” Opt. Lett. 21, 548 (1996).
[CrossRef]

A. Sharon, D. Rosenblatt, A. A. Friesem, “Narrow spectral bandwidth with grating-waveguide structures,” Appl. Phys. Lett. 69, 4154–4156 (1996).
[CrossRef]

A. Sharon, D. Rosenblatt, A. A. Friesem, H. G. Weber, H. Engel, R. Steingrueber, “Light modulation with resonant grating-waveguide structures,” Opt. Lett. 21, 1564–1566 (1996).
[CrossRef] [PubMed]

1995 (1)

L. Li, “Analysis of planar waveguide grating coupler with double surface corrugationsof identical period,” Opt. Commun. 114, 406–412 (1995).
[CrossRef]

1994 (1)

R. Magnusson, S. S. Wang, T. D. Black, A. Sohn, “Resonance properties of dielectric waveguide grating: theory and experimentsat 4-18 GHz,” IEEE Trans. Antennas Propag. 42, 567–569 (1994).
[CrossRef]

1990 (1)

1986 (1)

E. Popov, L. Mashev, D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta 32, 607–629 (1986).
[CrossRef]

1982 (2)

P. Sheng, R. S. Stepleman, P. N. Sanda, “Exact eigenfunctions for square wave gratings: applications to diffractionand surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
[CrossRef]

V. A. Sychugov, A. V. Tishchenko, “Propagation and conversion of light waves in corrugated waveguide structures,” Sov. J. Quantum Electron. 12, 923–926 (1982).
[CrossRef]

1977 (2)

A. Yariv, M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. QE-13, 233–253 (1977).
[CrossRef]

E. G. Loewen, M. Nevière, “Dielectric coated gratings: a curious property,” Appl. Opt. 16, 3009–3011 (1977).
[CrossRef] [PubMed]

1970 (1)

1965 (1)

1952 (1)

1907 (1)

Rayleigh, “Note on the remarkable case of diffraction spectrum described by Prof. Wood,” Philos. Mag. 14, 399–416 (1907).
[CrossRef]

1902 (1)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffractiongrating spectrum,” Philos. Mag. 4, 396–402 (1902).
[CrossRef]

Bagby, J. S.

Black, T. D.

R. Magnusson, S. S. Wang, T. D. Black, A. Sohn, “Resonance properties of dielectric waveguide grating: theory and experimentsat 4-18 GHz,” IEEE Trans. Antennas Propag. 42, 567–569 (1994).
[CrossRef]

Engel, H.

Feshbach, H.

P. W. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Friesem, A. A.

S. Glasberg, A. Sharon, D. Rosenblatt, A. A. Friesem, “Long-range surface plasmon resonances in grating-waveguide structures,” Appl. Phys. Lett. 70, 1210–1212 (1997).
[CrossRef]

A. Sharon, S. Glasberg, D. Rosenblatt, A. A. Friesem, “Metal-based resonant grating-waveguide structures,” J. Opt. Soc. Am. A 14, 588–595 (1997).
[CrossRef]

A. Sharon, D. Rosenblatt, A. A. Friesem, H. G. Weber, H. Engel, R. Steingrueber, “Light modulation with resonant grating-waveguide structures,” Opt. Lett. 21, 1564–1566 (1996).
[CrossRef] [PubMed]

A. Sharon, D. Rosenblatt, A. A. Friesem, “Narrow spectral bandwidth with grating-waveguide structures,” Appl. Phys. Lett. 69, 4154–4156 (1996).
[CrossRef]

Gale, M. T.

M. T. Gale, K. Knop, R. Morf, “Zero order diffraction microstructures for security applications,”, in Optical Security and Anticounterfeiting Systems, W. F. Fugan, ed., Proc. SPIE1210, 83–89 (1990).
[CrossRef]

Glasberg, S.

S. Glasberg, A. Sharon, D. Rosenblatt, A. A. Friesem, “Long-range surface plasmon resonances in grating-waveguide structures,” Appl. Phys. Lett. 70, 1210–1212 (1997).
[CrossRef]

A. Sharon, S. Glasberg, D. Rosenblatt, A. A. Friesem, “Metal-based resonant grating-waveguide structures,” J. Opt. Soc. Am. A 14, 588–595 (1997).
[CrossRef]

Hessel, A.

Knop, K.

M. T. Gale, K. Knop, R. Morf, “Zero order diffraction microstructures for security applications,”, in Optical Security and Anticounterfeiting Systems, W. F. Fugan, ed., Proc. SPIE1210, 83–89 (1990).
[CrossRef]

Li, L.

L. Li, “Analysis of planar waveguide grating coupler with double surface corrugationsof identical period,” Opt. Commun. 114, 406–412 (1995).
[CrossRef]

Loewen, E. G.

Magnusson, R.

R. Magnusson, S. S. Wang, T. D. Black, A. Sohn, “Resonance properties of dielectric waveguide grating: theory and experimentsat 4-18 GHz,” IEEE Trans. Antennas Propag. 42, 567–569 (1994).
[CrossRef]

S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 7, 1470–1474 (1990).
[CrossRef]

R. Magnusson, S. S. Wang, “Optical grating-waveguide filters,” in International Conference on Holography, CorrelationOptics, and Recording Materials, O. V. Angelsky, ed., Proc. SPIE2108, 380–391 (1993).
[CrossRef]

Mashev, L.

E. Popov, L. Mashev, D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta 32, 607–629 (1986).
[CrossRef]

Maystre, D.

E. Popov, L. Mashev, D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta 32, 607–629 (1986).
[CrossRef]

Moharam, M. G.

Morf, R.

M. T. Gale, K. Knop, R. Morf, “Zero order diffraction microstructures for security applications,”, in Optical Security and Anticounterfeiting Systems, W. F. Fugan, ed., Proc. SPIE1210, 83–89 (1990).
[CrossRef]

Morris, G. M.

S. Peng, G. M. Morris, “Experimental demonstration of resonant anomalies in diffraction fromtwo dimensional gratings,” Opt. Lett. 21, 548 (1996).
[CrossRef]

Morse, P. W.

P. W. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Nakamura, M.

A. Yariv, M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. QE-13, 233–253 (1977).
[CrossRef]

Nevière, M.

E. G. Loewen, M. Nevière, “Dielectric coated gratings: a curious property,” Appl. Opt. 16, 3009–3011 (1977).
[CrossRef] [PubMed]

M. Nevière, Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 5.

Oliner, A. A.

Palmer, C. H.

Peng, S.

S. Peng, G. M. Morris, “Experimental demonstration of resonant anomalies in diffraction fromtwo dimensional gratings,” Opt. Lett. 21, 548 (1996).
[CrossRef]

Popov, E.

E. Popov, L. Mashev, D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta 32, 607–629 (1986).
[CrossRef]

Rayleigh,

Rayleigh, “Note on the remarkable case of diffraction spectrum described by Prof. Wood,” Philos. Mag. 14, 399–416 (1907).
[CrossRef]

Rosenblatt, D.

S. Glasberg, A. Sharon, D. Rosenblatt, A. A. Friesem, “Long-range surface plasmon resonances in grating-waveguide structures,” Appl. Phys. Lett. 70, 1210–1212 (1997).
[CrossRef]

A. Sharon, S. Glasberg, D. Rosenblatt, A. A. Friesem, “Metal-based resonant grating-waveguide structures,” J. Opt. Soc. Am. A 14, 588–595 (1997).
[CrossRef]

A. Sharon, D. Rosenblatt, A. A. Friesem, H. G. Weber, H. Engel, R. Steingrueber, “Light modulation with resonant grating-waveguide structures,” Opt. Lett. 21, 1564–1566 (1996).
[CrossRef] [PubMed]

A. Sharon, D. Rosenblatt, A. A. Friesem, “Narrow spectral bandwidth with grating-waveguide structures,” Appl. Phys. Lett. 69, 4154–4156 (1996).
[CrossRef]

Sanda, P. N.

P. Sheng, R. S. Stepleman, P. N. Sanda, “Exact eigenfunctions for square wave gratings: applications to diffractionand surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
[CrossRef]

Sharon, A.

A. Sharon, S. Glasberg, D. Rosenblatt, A. A. Friesem, “Metal-based resonant grating-waveguide structures,” J. Opt. Soc. Am. A 14, 588–595 (1997).
[CrossRef]

S. Glasberg, A. Sharon, D. Rosenblatt, A. A. Friesem, “Long-range surface plasmon resonances in grating-waveguide structures,” Appl. Phys. Lett. 70, 1210–1212 (1997).
[CrossRef]

A. Sharon, D. Rosenblatt, A. A. Friesem, “Narrow spectral bandwidth with grating-waveguide structures,” Appl. Phys. Lett. 69, 4154–4156 (1996).
[CrossRef]

A. Sharon, D. Rosenblatt, A. A. Friesem, H. G. Weber, H. Engel, R. Steingrueber, “Light modulation with resonant grating-waveguide structures,” Opt. Lett. 21, 1564–1566 (1996).
[CrossRef] [PubMed]

Sheng, P.

P. Sheng, R. S. Stepleman, P. N. Sanda, “Exact eigenfunctions for square wave gratings: applications to diffractionand surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
[CrossRef]

Sohn, A.

R. Magnusson, S. S. Wang, T. D. Black, A. Sohn, “Resonance properties of dielectric waveguide grating: theory and experimentsat 4-18 GHz,” IEEE Trans. Antennas Propag. 42, 567–569 (1994).
[CrossRef]

Steingrueber, R.

Stepleman, R. S.

P. Sheng, R. S. Stepleman, P. N. Sanda, “Exact eigenfunctions for square wave gratings: applications to diffractionand surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
[CrossRef]

Sychugov, V. A.

V. A. Sychugov, A. V. Tishchenko, “Propagation and conversion of light waves in corrugated waveguide structures,” Sov. J. Quantum Electron. 12, 923–926 (1982).
[CrossRef]

Tien, P. K.

Tishchenko, A. V.

V. A. Sychugov, A. V. Tishchenko, “Propagation and conversion of light waves in corrugated waveguide structures,” Sov. J. Quantum Electron. 12, 923–926 (1982).
[CrossRef]

Ulrich, R.

Wang, S. S.

R. Magnusson, S. S. Wang, T. D. Black, A. Sohn, “Resonance properties of dielectric waveguide grating: theory and experimentsat 4-18 GHz,” IEEE Trans. Antennas Propag. 42, 567–569 (1994).
[CrossRef]

S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 7, 1470–1474 (1990).
[CrossRef]

R. Magnusson, S. S. Wang, “Optical grating-waveguide filters,” in International Conference on Holography, CorrelationOptics, and Recording Materials, O. V. Angelsky, ed., Proc. SPIE2108, 380–391 (1993).
[CrossRef]

Weber, H. G.

Wood, R. W.

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffractiongrating spectrum,” Philos. Mag. 4, 396–402 (1902).
[CrossRef]

Yariv, A.

A. Yariv, M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. QE-13, 233–253 (1977).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

S. Glasberg, A. Sharon, D. Rosenblatt, A. A. Friesem, “Long-range surface plasmon resonances in grating-waveguide structures,” Appl. Phys. Lett. 70, 1210–1212 (1997).
[CrossRef]

A. Sharon, D. Rosenblatt, A. A. Friesem, “Narrow spectral bandwidth with grating-waveguide structures,” Appl. Phys. Lett. 69, 4154–4156 (1996).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. Yariv, M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. QE-13, 233–253 (1977).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

R. Magnusson, S. S. Wang, T. D. Black, A. Sohn, “Resonance properties of dielectric waveguide grating: theory and experimentsat 4-18 GHz,” IEEE Trans. Antennas Propag. 42, 567–569 (1994).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Acta (1)

E. Popov, L. Mashev, D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta 32, 607–629 (1986).
[CrossRef]

Opt. Commun. (1)

L. Li, “Analysis of planar waveguide grating coupler with double surface corrugationsof identical period,” Opt. Commun. 114, 406–412 (1995).
[CrossRef]

Opt. Lett. (2)

S. Peng, G. M. Morris, “Experimental demonstration of resonant anomalies in diffraction fromtwo dimensional gratings,” Opt. Lett. 21, 548 (1996).
[CrossRef]

A. Sharon, D. Rosenblatt, A. A. Friesem, H. G. Weber, H. Engel, R. Steingrueber, “Light modulation with resonant grating-waveguide structures,” Opt. Lett. 21, 1564–1566 (1996).
[CrossRef] [PubMed]

Philos. Mag. (2)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffractiongrating spectrum,” Philos. Mag. 4, 396–402 (1902).
[CrossRef]

Rayleigh, “Note on the remarkable case of diffraction spectrum described by Prof. Wood,” Philos. Mag. 14, 399–416 (1907).
[CrossRef]

Phys. Rev. B (1)

P. Sheng, R. S. Stepleman, P. N. Sanda, “Exact eigenfunctions for square wave gratings: applications to diffractionand surface-plasmon calculations,” Phys. Rev. B 26, 2907–2916 (1982).
[CrossRef]

Sov. J. Quantum Electron. (1)

V. A. Sychugov, A. V. Tishchenko, “Propagation and conversion of light waves in corrugated waveguide structures,” Sov. J. Quantum Electron. 12, 923–926 (1982).
[CrossRef]

Other (4)

M. Nevière, Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 5.

R. Magnusson, S. S. Wang, “Optical grating-waveguide filters,” in International Conference on Holography, CorrelationOptics, and Recording Materials, O. V. Angelsky, ed., Proc. SPIE2108, 380–391 (1993).
[CrossRef]

M. T. Gale, K. Knop, R. Morf, “Zero order diffraction microstructures for security applications,”, in Optical Security and Anticounterfeiting Systems, W. F. Fugan, ed., Proc. SPIE1210, 83–89 (1990).
[CrossRef]

P. W. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

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Figures (8)

Fig. 1
Fig. 1

Basic ray picture propagation in a GWS.

Fig. 2
Fig. 2

Calculated transmitted intensities ratio as a function of modified dephasing Δ / S , for three different modified absorbencies α / S .

Fig. 3
Fig. 3

GWS with optical and geometric parameters: (a) With dielectric materials, (b) with semiconductor materials.

Fig. 4
Fig. 4

Numerical calculations for the normalized transmitted and reflected intensities as a function of wavelength for a dielectric GWS of Fig. 3(a).

Fig. 5
Fig. 5

Numerical and analytical calculations of the spectral bandwidth at FWHM as a function of grating height. All other semiconductor GWS parameters are as in Fig. 3(b).

Fig. 6
Fig. 6

Experimental setup for evaluating the spectral response of GWS.

Fig. 7
Fig. 7

Experimental results for the transmitted and reflected intensities as a function of wavelength for the dielectric GWS of Fig. 3(a).

Fig. 8
Fig. 8

Experimental results for the reflected intensity as a function of wavelength for the semiconductor GWS of Fig. 3(b).

Equations (32)

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2 k 2 h + 2 ϕ 12 + 2 ϕ 23 = 2 m π ,
β 2 = 2 k 0 2 - k 2 2 ,
S t E y 0 | t , 0 E y 0 | i , S d 1 E y 1 | w , 1 E y 0 | i ,
S d 2 E y 0 | t , 1 E y 1 | w , 1 , S r E y 1 | w , 2 E y 1 | w , 1 .
S d 1 = exp ( i ϕ 12 ) exp ( - i π / 2 ) | S d 1 | ,
| S d 1 | = Δ k 0 2 ξ 1 d / 2 γ 1 ,
S d 2 = exp ( i ϕ 12 ) exp ( - i π / 2 ) | S d 2 | ,
| S d 2 | = Δ k 0 2 ξ - 1 d / 2 γ 2 ,
| S r | = 1 - S , S | S d 1 | | S d 2 | ,
E = S t E 0 + { | S d 1 | exp ( - i π / 2 ) × exp [ i ( 2 k 2 h + 2 ϕ 21 + 2 ϕ 23 + Δ ) ] × exp ( - i π / 2 ) | S d 2 | E 0 } + { | S d 1 | exp ( - i π / 2 ) × exp [ i ( 2 k 2 h + 2 ϕ 21 + 2 ϕ 23 + Δ ) ] | S r | × exp [ i ( 2 k 2 h + 2 ϕ 21 + 2 ϕ 23 + Δ ) ] × exp ( - i π / 2 ) | S d 2 | E 0 } + + ,
E = S t E 0 + exp ( - i π )   S   exp ( i Δ ) 1 - | S r | exp ( i Δ )   E 0 .
E E 0 1 - S   exp ( i Δ ) 1 - ( 1 - S ) exp ( i Δ ) .
T T 0 1 ( 1 - S )   sin 2 Δ 2 S 2 4 ( 1 - S ) + sin 2 Δ 2 ,
T T 0 = Δ 2 S 2 + Δ 2 .
R R 0 = S 2 S 2 + Δ 2 .
T T 0 = Δ 2 + α 2 Δ 2 + ( S + α ) 2 .
d 2 E y d x 2 + d 2 E y d z 2 + k 0 2 E y = 0 .
g ( x ,   z ) = 0 ( x ) - Δ m 0 ξ m ( x ) exp ( - imKz )
E y ( x ,   z ) = E y 0 ( x ) exp ( ik z 0 z ) + E y 1 ( x ) exp [ i ( k z 0 + K ) z ] ,
d 2 E y 1 d x 2 + k x 1 2 E y 1 = Δ k 0 2 ξ 1 E y 0 ,
d 2 E y 0 d x 2 + k x 0 2 E y 0 = Δ k 0 2 ξ - 1 E y 1 ,
γ 1 = ( | k x 1 | w 2 + | k x 1 | i 2 ) 1 / 2 / 2 , γ 2 = ( | k x 0 | w 2 + | k x 0 | i 2 ) 1 / 2 / 2 .
E y 1 ( x ) = Δ k 0 2 ξ 1 0 d E y 0 h ( x )   × exp [ i ϕ 12 - ik x 1 ( x - x ) ] 2 i γ 1   d x ,
S d 1 = exp ( i ϕ 12 ) exp - i   π 2 Δ k 0 2 ξ 1 d / 2 γ 1 .
E y 0 = E y 0 h - ( Δ 2 k 0 4 ξ 1 ξ - 1 d 2 / 4 γ 1 γ 2 ) exp ( i ϕ 12 ) E y 0 h .
S t = 1 - ( Δ 2 k 0 4 ξ 1 ξ - 1 d 2 / 4 γ 1 γ 2 ) exp ( i ϕ 12 ) .
d 2 E y 0 d x 2 + k x 0 2 E y 0 = Δ k 0 2 ξ - 1 E y 1 ,
d 2 E y 1 d x 2 + k x 1 2 E y 1 = Δ k 0 2 ξ 1 E y 0 ,
E y 0 ( x ) = Δ k 0 2 ξ 1 0 d E y 1 h ( x )   exp [ - ik x 0 ( x - x ) ] 2 i γ 2   d x .
S d 2 = exp ( i ϕ 12 ) exp - i   π 2 Δ k 0 2 ξ - 1 d / 2 γ 2 .
S r = [ 1 - ( Δ 2 k 0 4 ξ 1 ξ - 1 d 2 / 4 γ 1 γ 2 ) ] exp ( i 2 ϕ 12 ) .
| S r | = 1 - | S d 1 | S d 2 | .

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