Abstract

The poles and zeros of the scattering operator of a corrugated waveguide and of a bare grating are studied mathematically and numerically. An initial tutorial section recalls how their use can explain grating anomalies and other curious phenomena in linear optics. This approach is then used in nonlinear optics to understand and predict curious efficiency-curve shapes observed in the study of second-harmonic generation and optical bistability enhanced by a corrugated surface.

© 1995 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 diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).
  2. J. W. S. Rayleigh, “Note on the remarkable case of diffraction spectra described by Prof. Wood,” Philos. Mag. 14, 60–65 (1907).
  3. U. Fano, “The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld’s waves),” J. Opt. Soc. Am. 31, 213–222 (1941).
    [CrossRef]
  4. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  5. D. Maystre, M. Nevière, “Sur une méthode d’étude quantitative des anomalies de Wood des réseaux de diffraction: application aux anomalies de plasmon,” J. Opt. (Paris) 88, 165–174 (1977).
    [CrossRef]
  6. M. Nevière, D. Maystre, P. Vincent, “Application du calcul des modes de propagation à l’étude théorique des anomalies des réseaux recouverts de diélectrique,” J. Opt. (Paris) 8, 231–242 (1977).
  7. M. Nevière, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 123–157.
    [CrossRef]
  8. P. Vincent, M. Nevière, “Corrugated dielectric waveguides: a numerical study of the second-order stop bands,” Appl. Opt. 20, 345–351 (1979).
  9. E. Popov, “Light diffraction by relief gratings: a macroscopic and microscopic view,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. XXXI, pp. 139–187.
    [CrossRef]
  10. E. Popov, L. Tsonev, “Electromagnetic field enhancement in deep metallic gratings,” Opt. Commun. 69, 193–198 (1989).
    [CrossRef]
  11. D. W. Pohl, D. Gourjon, eds., Near Field Optics, Vol. 242 of NATO Advanced Sciences Institutes Series E (Kluver, Dordrecht, The Netherlands, 1993).
    [CrossRef]
  12. R. Reinisch, G. Vitrant, M. Nevière, “Electromagnetic resonance induced nonlinear optical phenomena,” in Nonlinear Waves in Solid State Physics, A. D. Boardman, N. Bertolotti, T. Twardowski, eds. (Plenum, New York, 1990), pp. 435–461.
    [CrossRef]
  13. M. Nevière, M. Cadilhac, R. Petit, “Applications of conformal mappings to the diffraction of electromagnetic waves by a grating,” IEEE Trans. Antennas Propag. AP-21, 37–46 (1973).
    [CrossRef]
  14. E. Popov, L. Mashev, D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
    [CrossRef]
  15. E. G. Loewen, M. Nevière, “Dielectric coated gratings: a curious property,” Appl. Opt. 16, 3009–3011 (1977).
    [CrossRef] [PubMed]
  16. M. C. Hutley, D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
    [CrossRef]
  17. D. Maystre, M. Nevière, P. Vincent, “On a general theory of anomalies and energy absorption of diffraction gratings and their relation with surface waves,” Opt. Acta 25, 905–915 (1978).
    [CrossRef]
  18. R. Reinisch, M. Nevière, “Electromagnetic theory of diffraction in nonlinear optics and surface-enhanced nonlinear optical effects,” Phys. Rev. B 28, 1870–1885 (1983).
    [CrossRef]
  19. E. Popov, M. Nevière, “Surface-enhanced second-harmonic generation in nonlinear corrugated dielectrics: new theoretical approaches,” J. Opt. Soc. Am. B 11, 1555–1564 (1994).
    [CrossRef]
  20. R. Reinisch, G. Vitrant, J. L. Coutaz, P. Vincent, M. Nevière, H. Akhouayri, “Modal analysis of grating couplers for nonlinear waveguides,” Nonlin. Opt. 5, 111–118 (1993).
  21. R. Reinisch, M. Nevière, P. Vincent, G. Vitrant, “Radiated diffracted orders in Kerr-type grating couplers,” Opt. Commun. 91, 51–56 (1992).
    [CrossRef]
  22. P. Vincent, N. Paraire, M. Nevière, A. Koster, R. Reinisch, “Gratings in nonlinear optics and optical bistability,” J. Opt. Soc. Am. B 2, 1106–1116 (1985).
    [CrossRef]
  23. H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, Berlin, 1975), pp. 13–81.
    [CrossRef]

1994 (1)

1993 (1)

R. Reinisch, G. Vitrant, J. L. Coutaz, P. Vincent, M. Nevière, H. Akhouayri, “Modal analysis of grating couplers for nonlinear waveguides,” Nonlin. Opt. 5, 111–118 (1993).

1992 (1)

R. Reinisch, M. Nevière, P. Vincent, G. Vitrant, “Radiated diffracted orders in Kerr-type grating couplers,” Opt. Commun. 91, 51–56 (1992).
[CrossRef]

1989 (1)

E. Popov, L. Tsonev, “Electromagnetic field enhancement in deep metallic gratings,” Opt. Commun. 69, 193–198 (1989).
[CrossRef]

1986 (1)

E. Popov, L. Mashev, D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

1985 (1)

1983 (1)

R. Reinisch, M. Nevière, “Electromagnetic theory of diffraction in nonlinear optics and surface-enhanced nonlinear optical effects,” Phys. Rev. B 28, 1870–1885 (1983).
[CrossRef]

1979 (1)

P. Vincent, M. Nevière, “Corrugated dielectric waveguides: a numerical study of the second-order stop bands,” Appl. Opt. 20, 345–351 (1979).

1978 (1)

D. Maystre, M. Nevière, P. Vincent, “On a general theory of anomalies and energy absorption of diffraction gratings and their relation with surface waves,” Opt. Acta 25, 905–915 (1978).
[CrossRef]

1977 (3)

D. Maystre, M. Nevière, “Sur une méthode d’étude quantitative des anomalies de Wood des réseaux de diffraction: application aux anomalies de plasmon,” J. Opt. (Paris) 88, 165–174 (1977).
[CrossRef]

M. Nevière, D. Maystre, P. Vincent, “Application du calcul des modes de propagation à l’étude théorique des anomalies des réseaux recouverts de diélectrique,” J. Opt. (Paris) 8, 231–242 (1977).

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

1976 (1)

M. C. Hutley, D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
[CrossRef]

1973 (1)

M. Nevière, M. Cadilhac, R. Petit, “Applications of conformal mappings to the diffraction of electromagnetic waves by a grating,” IEEE Trans. Antennas Propag. AP-21, 37–46 (1973).
[CrossRef]

1941 (1)

1907 (1)

J. W. S. Rayleigh, “Note on the remarkable case of diffraction spectra described by Prof. Wood,” Philos. Mag. 14, 60–65 (1907).

1902 (1)

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

Akhouayri, H.

R. Reinisch, G. Vitrant, J. L. Coutaz, P. Vincent, M. Nevière, H. Akhouayri, “Modal analysis of grating couplers for nonlinear waveguides,” Nonlin. Opt. 5, 111–118 (1993).

Cadilhac, M.

M. Nevière, M. Cadilhac, R. Petit, “Applications of conformal mappings to the diffraction of electromagnetic waves by a grating,” IEEE Trans. Antennas Propag. AP-21, 37–46 (1973).
[CrossRef]

Coutaz, J. L.

R. Reinisch, G. Vitrant, J. L. Coutaz, P. Vincent, M. Nevière, H. Akhouayri, “Modal analysis of grating couplers for nonlinear waveguides,” Nonlin. Opt. 5, 111–118 (1993).

Fano, U.

Hutley, M. C.

M. C. Hutley, D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, Berlin, 1975), pp. 13–81.
[CrossRef]

Koster, A.

Loewen, E. G.

Mashev, L.

E. Popov, L. Mashev, D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

Maystre, D.

E. Popov, L. Mashev, D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

D. Maystre, M. Nevière, P. Vincent, “On a general theory of anomalies and energy absorption of diffraction gratings and their relation with surface waves,” Opt. Acta 25, 905–915 (1978).
[CrossRef]

D. Maystre, M. Nevière, “Sur une méthode d’étude quantitative des anomalies de Wood des réseaux de diffraction: application aux anomalies de plasmon,” J. Opt. (Paris) 88, 165–174 (1977).
[CrossRef]

M. Nevière, D. Maystre, P. Vincent, “Application du calcul des modes de propagation à l’étude théorique des anomalies des réseaux recouverts de diélectrique,” J. Opt. (Paris) 8, 231–242 (1977).

M. C. Hutley, D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
[CrossRef]

Nevière, M.

E. Popov, M. Nevière, “Surface-enhanced second-harmonic generation in nonlinear corrugated dielectrics: new theoretical approaches,” J. Opt. Soc. Am. B 11, 1555–1564 (1994).
[CrossRef]

R. Reinisch, G. Vitrant, J. L. Coutaz, P. Vincent, M. Nevière, H. Akhouayri, “Modal analysis of grating couplers for nonlinear waveguides,” Nonlin. Opt. 5, 111–118 (1993).

R. Reinisch, M. Nevière, P. Vincent, G. Vitrant, “Radiated diffracted orders in Kerr-type grating couplers,” Opt. Commun. 91, 51–56 (1992).
[CrossRef]

P. Vincent, N. Paraire, M. Nevière, A. Koster, R. Reinisch, “Gratings in nonlinear optics and optical bistability,” J. Opt. Soc. Am. B 2, 1106–1116 (1985).
[CrossRef]

R. Reinisch, M. Nevière, “Electromagnetic theory of diffraction in nonlinear optics and surface-enhanced nonlinear optical effects,” Phys. Rev. B 28, 1870–1885 (1983).
[CrossRef]

P. Vincent, M. Nevière, “Corrugated dielectric waveguides: a numerical study of the second-order stop bands,” Appl. Opt. 20, 345–351 (1979).

D. Maystre, M. Nevière, P. Vincent, “On a general theory of anomalies and energy absorption of diffraction gratings and their relation with surface waves,” Opt. Acta 25, 905–915 (1978).
[CrossRef]

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

D. Maystre, M. Nevière, “Sur une méthode d’étude quantitative des anomalies de Wood des réseaux de diffraction: application aux anomalies de plasmon,” J. Opt. (Paris) 88, 165–174 (1977).
[CrossRef]

M. Nevière, D. Maystre, P. Vincent, “Application du calcul des modes de propagation à l’étude théorique des anomalies des réseaux recouverts de diélectrique,” J. Opt. (Paris) 8, 231–242 (1977).

M. Nevière, M. Cadilhac, R. Petit, “Applications of conformal mappings to the diffraction of electromagnetic waves by a grating,” IEEE Trans. Antennas Propag. AP-21, 37–46 (1973).
[CrossRef]

R. Reinisch, G. Vitrant, M. Nevière, “Electromagnetic resonance induced nonlinear optical phenomena,” in Nonlinear Waves in Solid State Physics, A. D. Boardman, N. Bertolotti, T. Twardowski, eds. (Plenum, New York, 1990), pp. 435–461.
[CrossRef]

M. Nevière, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 123–157.
[CrossRef]

Paraire, N.

Petit, R.

M. Nevière, M. Cadilhac, R. Petit, “Applications of conformal mappings to the diffraction of electromagnetic waves by a grating,” IEEE Trans. Antennas Propag. AP-21, 37–46 (1973).
[CrossRef]

Popov, E.

E. Popov, M. Nevière, “Surface-enhanced second-harmonic generation in nonlinear corrugated dielectrics: new theoretical approaches,” J. Opt. Soc. Am. B 11, 1555–1564 (1994).
[CrossRef]

E. Popov, L. Tsonev, “Electromagnetic field enhancement in deep metallic gratings,” Opt. Commun. 69, 193–198 (1989).
[CrossRef]

E. Popov, L. Mashev, D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

E. Popov, “Light diffraction by relief gratings: a macroscopic and microscopic view,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. XXXI, pp. 139–187.
[CrossRef]

Rayleigh, J. W. S.

J. W. S. Rayleigh, “Note on the remarkable case of diffraction spectra described by Prof. Wood,” Philos. Mag. 14, 60–65 (1907).

Reinisch, R.

R. Reinisch, G. Vitrant, J. L. Coutaz, P. Vincent, M. Nevière, H. Akhouayri, “Modal analysis of grating couplers for nonlinear waveguides,” Nonlin. Opt. 5, 111–118 (1993).

R. Reinisch, M. Nevière, P. Vincent, G. Vitrant, “Radiated diffracted orders in Kerr-type grating couplers,” Opt. Commun. 91, 51–56 (1992).
[CrossRef]

P. Vincent, N. Paraire, M. Nevière, A. Koster, R. Reinisch, “Gratings in nonlinear optics and optical bistability,” J. Opt. Soc. Am. B 2, 1106–1116 (1985).
[CrossRef]

R. Reinisch, M. Nevière, “Electromagnetic theory of diffraction in nonlinear optics and surface-enhanced nonlinear optical effects,” Phys. Rev. B 28, 1870–1885 (1983).
[CrossRef]

R. Reinisch, G. Vitrant, M. Nevière, “Electromagnetic resonance induced nonlinear optical phenomena,” in Nonlinear Waves in Solid State Physics, A. D. Boardman, N. Bertolotti, T. Twardowski, eds. (Plenum, New York, 1990), pp. 435–461.
[CrossRef]

Tsonev, L.

E. Popov, L. Tsonev, “Electromagnetic field enhancement in deep metallic gratings,” Opt. Commun. 69, 193–198 (1989).
[CrossRef]

Vincent, P.

R. Reinisch, G. Vitrant, J. L. Coutaz, P. Vincent, M. Nevière, H. Akhouayri, “Modal analysis of grating couplers for nonlinear waveguides,” Nonlin. Opt. 5, 111–118 (1993).

R. Reinisch, M. Nevière, P. Vincent, G. Vitrant, “Radiated diffracted orders in Kerr-type grating couplers,” Opt. Commun. 91, 51–56 (1992).
[CrossRef]

P. Vincent, N. Paraire, M. Nevière, A. Koster, R. Reinisch, “Gratings in nonlinear optics and optical bistability,” J. Opt. Soc. Am. B 2, 1106–1116 (1985).
[CrossRef]

P. Vincent, M. Nevière, “Corrugated dielectric waveguides: a numerical study of the second-order stop bands,” Appl. Opt. 20, 345–351 (1979).

D. Maystre, M. Nevière, P. Vincent, “On a general theory of anomalies and energy absorption of diffraction gratings and their relation with surface waves,” Opt. Acta 25, 905–915 (1978).
[CrossRef]

M. Nevière, D. Maystre, P. Vincent, “Application du calcul des modes de propagation à l’étude théorique des anomalies des réseaux recouverts de diélectrique,” J. Opt. (Paris) 8, 231–242 (1977).

Vitrant, G.

R. Reinisch, G. Vitrant, J. L. Coutaz, P. Vincent, M. Nevière, H. Akhouayri, “Modal analysis of grating couplers for nonlinear waveguides,” Nonlin. Opt. 5, 111–118 (1993).

R. Reinisch, M. Nevière, P. Vincent, G. Vitrant, “Radiated diffracted orders in Kerr-type grating couplers,” Opt. Commun. 91, 51–56 (1992).
[CrossRef]

R. Reinisch, G. Vitrant, M. Nevière, “Electromagnetic resonance induced nonlinear optical phenomena,” in Nonlinear Waves in Solid State Physics, A. D. Boardman, N. Bertolotti, T. Twardowski, eds. (Plenum, New York, 1990), pp. 435–461.
[CrossRef]

Wood, R. W.

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

Appl. Opt. (2)

P. Vincent, M. Nevière, “Corrugated dielectric waveguides: a numerical study of the second-order stop bands,” Appl. Opt. 20, 345–351 (1979).

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

IEEE Trans. Antennas Propag. (1)

M. Nevière, M. Cadilhac, R. Petit, “Applications of conformal mappings to the diffraction of electromagnetic waves by a grating,” IEEE Trans. Antennas Propag. AP-21, 37–46 (1973).
[CrossRef]

J. Opt. (Paris) (2)

D. Maystre, M. Nevière, “Sur une méthode d’étude quantitative des anomalies de Wood des réseaux de diffraction: application aux anomalies de plasmon,” J. Opt. (Paris) 88, 165–174 (1977).
[CrossRef]

M. Nevière, D. Maystre, P. Vincent, “Application du calcul des modes de propagation à l’étude théorique des anomalies des réseaux recouverts de diélectrique,” J. Opt. (Paris) 8, 231–242 (1977).

J. Opt. Soc. Am. (1)

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

Nonlin. Opt. (1)

R. Reinisch, G. Vitrant, J. L. Coutaz, P. Vincent, M. Nevière, H. Akhouayri, “Modal analysis of grating couplers for nonlinear waveguides,” Nonlin. Opt. 5, 111–118 (1993).

Opt. Acta (2)

D. Maystre, M. Nevière, P. Vincent, “On a general theory of anomalies and energy absorption of diffraction gratings and their relation with surface waves,” Opt. Acta 25, 905–915 (1978).
[CrossRef]

E. Popov, L. Mashev, D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

Opt. Commun. (3)

M. C. Hutley, D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
[CrossRef]

E. Popov, L. Tsonev, “Electromagnetic field enhancement in deep metallic gratings,” Opt. Commun. 69, 193–198 (1989).
[CrossRef]

R. Reinisch, M. Nevière, P. Vincent, G. Vitrant, “Radiated diffracted orders in Kerr-type grating couplers,” Opt. Commun. 91, 51–56 (1992).
[CrossRef]

Philos. Mag. (2)

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

J. W. S. Rayleigh, “Note on the remarkable case of diffraction spectra described by Prof. Wood,” Philos. Mag. 14, 60–65 (1907).

Phys. Rev. B (1)

R. Reinisch, M. Nevière, “Electromagnetic theory of diffraction in nonlinear optics and surface-enhanced nonlinear optical effects,” Phys. Rev. B 28, 1870–1885 (1983).
[CrossRef]

Other (6)

H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, Berlin, 1975), pp. 13–81.
[CrossRef]

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
[CrossRef]

D. W. Pohl, D. Gourjon, eds., Near Field Optics, Vol. 242 of NATO Advanced Sciences Institutes Series E (Kluver, Dordrecht, The Netherlands, 1993).
[CrossRef]

R. Reinisch, G. Vitrant, M. Nevière, “Electromagnetic resonance induced nonlinear optical phenomena,” in Nonlinear Waves in Solid State Physics, A. D. Boardman, N. Bertolotti, T. Twardowski, eds. (Plenum, New York, 1990), pp. 435–461.
[CrossRef]

M. Nevière, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 123–157.
[CrossRef]

E. Popov, “Light diffraction by relief gratings: a macroscopic and microscopic view,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. XXXI, pp. 139–187.
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Schematic representation of a grating, with notation.

Fig. 2
Fig. 2

Loci of the pole δp and the zero δz of B0(δ, h) in the complex plane when h (in micrometers) is varied, for a 2400-groove/mm sinusoidal silver grating used in TM polarization. λ = 0.5 μm. Long-dashed curve, δp; short-dashed curve, δz.

Fig. 3
Fig. 3

Loci of the pole δp (upper curve) and the zero δz (lower curve) in the complex plane for a 2400-groove/mm, 10°22′ blaze angle aluminum grating used in TE polarization, at λ = 0.492 μm. The grating is coated with a layer of MgF2 with thickness e chosen as the parameter (in μm).

Fig. 4
Fig. 4

Zeroth-order efficiency curves for several groove depths of a 2400-groove/mm sinusoidal silver grating as function of δ. λ = 0.5 μm, TM polarization.

Fig. 5
Fig. 5

Zeroth-order efficiency of a sinusoidal corrugated waveguide with thickness 0.19 μm (ν1 = 1, ν2 = 2.3, ν3 = 1) and d = 0.37 μm, illuminated at λ = 0.6328 μm, in TE polarization, as a function of incidence. The values of groove depths are given in micrometers.

Fig. 6
Fig. 6

Cross-sectional view of the two corrugated systems with a nonlinear middle layer: (a) modulation of the upper interface, (b) modulation of the lower interface. NL, nonlinear polarization.

Fig. 7
Fig. 7

(a) Trajectories of the poles (thick curves) and the zeros (thin curves) of the zeroth reflected order at 2ω for the system of Fig. 6(a), when the groove depth is varied. The values are given in micrometers. (b) The second-harmonic reflectivity as a function of angle of incidence in the region of waveguide-mode excitation. Solid curves, rigorous results; diamonds, phenomenological results from Eq. (23), with C ˜ = 1, h = 0.12 μm.

Fig. 8
Fig. 8

(a) Same as in Fig. 7(a) but for the system of Fig. 6(b); (b), (c) nonlinear reflectivity as a function of angle of incidence for (b) h = 0.004 μm and (c) h = 0.019 μm. Solid curves, rigorous results; diamonds, phenomenological results from Eq. (23); crosses, phenomenological results with a single pole δ 1 p and a single zero δ 1 z.

Fig. 9
Fig. 9

Schematic representation of a bistable corrugated waveguide.

Fig. 10
Fig. 10

Trajectories of the real parts of a pole (solid curve) and a zero (dashed curve) of the S matrix when A i 2 is varied.

Fig. 11
Fig. 11

Zeroth-order efficiency curve, E 0 = ( B 0 NL / A i ) 2, as a function of Ai: solid curve, results of rigorous computation; crosses, results derived from Eqs. (25′) and (26′).

Equations (58)

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u ( x , y ) = A 0 exp [ i ( γ 0 x - β 0 y ) ] + n = - + B n exp [ i ( γ n x + β n y ) ] .
u ( x , y ) = n = - + T n exp [ i ( γ n x - β n y ) ] ,
γ n = k 1 sin θ + n K ,
K = 2 π d , β n = k 0 2 ν 1 2 - γ n 2             or             i γ n 2 - k 0 2 ν 1 2 ,
β n = k 0 2 ν 2 2 - γ n 2 ,             Re ( β n ) + Im ( β n ) > 0 , k 0 = ω / c ,
B n ( θ ) = m = - + S n m ( θ ) A m
B n ( θ ) = m U ( θ ) s n m ( θ ) A m ,
B n ( θ ) β n 1 / 2 = m U ( θ ) s n m ( θ ) A m β m 1 / 2 .
[ B ( δ ) ] = [ S ( δ ) ] [ A ] ,
[ A ] = [ M ( δ ) ] [ B ( δ ) ] ,
[ M ] [ B ] = 0.
det [ M ( δ ) ] = 0.
B n ( δ ) = m = - + C n m ( δ ) δ - δ p A m ,
B n ( δ ) C δ - δ p ;
B n ( δ ) 2 C 2 ( δ - δ p ) 2 + ( δ p ) 2 .
β n = k 0 2 ν 1 2 - γ n 2 ,             Re ( β n ) + Im ( β n ) > 0.
B n ( δ , h ) = u ( δ , h ) δ - δ p ( h ) ,             u [ δ p ( h ) , h ] 0.
B 0 ( δ , 0 ) = r ( δ ) = u ( δ , 0 ) δ - δ p ( 0 ) ,
u ( δ , 0 ) = [ δ - δ p ( 0 ) ] r ( δ ) .
B 0 ( δ , h ) = C ( h ) + D ( h ) δ - δ p ( h ) ,
lim h 0 C ( h ) = r ( δ ) ,             lim h 0 D ( h ) = 0.
B 0 ( δ , h ) = C ( h ) δ - δ p ( h ) + D ( h ) / C ( h ) δ - δ p ( h )
B 0 ( δ , h ) = C ( h ) δ - δ z ( h ) δ - δ p ( h ) ,
δ z ( h ) = δ p ( h ) - D ( h ) / C ( h ) .
det [ s ] = C d ( δ - δ p ¯ ) ,
s 0 , 0 = C 0 , 0 ( δ - δ 0 , 0 z ) ,
s - 1 , - 1 = C - 1 , - 1 ( δ - δ - 1 , - 1 z ) .
C - 1 , - 1 ( δ - δ - 1 , - 1 z ) C 0 , 0 ( δ - δ 0 , 0 z ) - C d ( δ - δ p ¯ ) = s 0 , - 1 s - 1 , 0 .
B 0 ( δ , h ) = w ( δ , h ) δ - δ z ( h ) δ - δ p ( h ) ,
B 0 ( δ , h ) r ( δ ) δ - δ z ( h ) δ - δ p ( h ) .
E 0 ( δ , h ) = B 0 2 = R ( δ ) δ - δ z ( h ) 2 δ - δ p ( h ) 2 ,
P NL ( 2 ω ) = χ ( 2 ω ) E ( ω ) E ( ω ) ,
B ˜ = S ˜ ( 2 ω ) A ˜ .
B ˜ 0 ( δ , h ) = C ˜ ( h ) + D ˜ ( h ) [ δ - δ 1 p ( h ) ] 2 [ δ - δ 2 p ( h ) ] ,
lim h 0 C ˜ ( h ) = r NL ( δ ) ,             lim h 0 D ˜ ( h ) = 0.
B ˜ 0 ( δ , h ) = C ˜ ( h ) δ 3 + a δ 2 + b δ + c [ δ - δ 1 p ( h ) ] 2 [ δ - δ 2 p ( h ) ] ,
B ˜ 0 ( δ , h ) = C ˜ ( h ) ( δ - δ 1 z ) ( δ - δ 3 z ) ( δ - δ 2 z ) ( δ - δ 1 p ) 2 ( δ - δ 2 p ) ,
lim h 0 δ 1 z ( h ) = δ 1 p = lim h 0 δ 3 z ( h ) ,             lim h 0 δ 2 z ( h ) = δ 2 p .
B ˜ 0 ( δ , h ) = a + b ( δ - δ 1 p ) 2 + c δ - δ 2 p + d ( δ - δ 1 p ) 2 ( δ - δ 2 p ) ,
χ x x x = ɛ 0 for the 2 ω TM case , χ z z z = ɛ 0 for the 2 ω TE case .
Re ( δ 1 z ) - 1.093 = ν 1 ( ω ) sin ( - 45.44 ° ) , Re ( δ 3 z ) - 1.801 = ν 1 ( ω ) sin ( - 44.8 ° ) , Re ( δ 1 p ) Re ( δ 2 p ) - 1.077 = ν 1 ( ω ) sin ( - 44.6 ° ) ,
ν 2 2 = ν 2 , 0 2 [ 1 + α E 2 ( x , y ) ] ,
B n = C n δ - δ n z δ - δ p A i ,
A g w = t p δ - δ p A i ,
B n = C n A i + C n δ p - δ n z δ - δ p A i = C n A i + g n A g w ,
g n = C n t p ( δ p - δ n z ) .
B n NL = C n A i + g n A g w NL .
C n ( δ p - δ n z ) t p = C n ( δ p , NL - δ n z , NL ) t p .
δ n z , NL - δ n z = δ p , NL - δ p .
δ p , NL = δ p + ξ p A g w NL 2 ,             δ n z , NL = δ n z + ξ p A g w NL 2 ,
A g w NL = t p δ - δ p , NL A i ,
B n NL = C n A i + g n A g w NL .
B 0 ( δ , h ) = C ( δ , h ) + D ( δ , h ) P ( δ , h ) ,
P ( δ , h ) = 0 for δ = δ n p , n = 1 , 2 , 3 , N , lim h 0 D ( δ , h ) = 0 ,
C ( δ , 0 ) 0.
B 0 ( δ , h ) = C ( δ , h ) R ( δ , h ) P ( δ , h ) ,
R ( δ , h ) = P ( δ , h ) + D ( δ , h ) C ( δ , h ) .
lim h 0 R ( δ , h ) = lim h 0 P ( δ , h ) = P ( δ , 0 ) .

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