Abstract

We study the s-polarized nonlinear guided and surface electromagnetic waves supported by a three-layer structure consisting of a nonlinear dielectric film, characterized by a Kerr-like dielectric constant, surrounded symmetrically by identical, linear dielectric media. We present numerical results for the effective wave number of these modes as a function of the intensity of the electric field at the lower surface of the nonlinear film for both self-focusing and self-defocusing nonlinear films. A double-valued relation between the effective wave number and the electric field intensity is obtained for certain ranges of the parameters characterizing the structure under investigation. Asymmetric modes in this symmetric structure are found to occur when the nonlinear film is made from a self-defocusing dielectric material.

© 1988 Optical Society of America

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  1. Yu. R. Alanakyan, “Surface waves in a plasma,” Zh. Tekh. Fiz. 37, 817–821 (1967) [Sov. Phys. Tech. Phys. 12, 587–589 (1967)].
  2. A. G. Litvak and V. A. Mironov, “Surface waves at the interface of nonlinear media,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 11, 1911–1912 (1968).
  3. A. G. Boev and A. V. Prokopov, “Contribution to the nonlinear theory of surface electromagnetic waves in an ionizing plasma,” Zh. Eksp. Teor. Fiz. 69, 1208–1217 (1975) [Sov. Phys. JETP 42, 617–621 (1975)].
  4. A. G. Boev, “On the theory of nonlinear surface waves in a plasma,” Zh. Eksp. Teor. Fiz. 77, 92–100 (1979) [Sov. Phys. JETP 50, 47–51 (1979)].
  5. W. J. Tomlinson, “Surface wave at a nonlinear interface,” Opt. Lett. 5, 323–325 (1980).
    [Crossref] [PubMed]
  6. A. G. Boev, “Nonlinear theory of penetration of p-polarized electromagnetic waves in a plasma,” Zh. Eksp. Teor. Fiz. 79, 134–142 (1980) [Sov. Phys. JETP 52, 67–71 (1980)].
  7. V. M. Agranovich, V. S. Babichenko, and V. Ya. Chernyak, “Nonlinear surface polaritons,” Pis’ma Zh. Eksp. Teor. Fiz. 32, 532–535 (1980) [Sov. Phys. JETP Lett. 32, 512–515 (1980)].
  8. A. A. Maradudin, “S-polarized nonlinear surface polaritons,” Z. Phys. B 41, 341–348 (1981).
    [Crossref]
  9. V. M. Agranovich and V. Ya. Chernyak, “Perturbation theory for weakly nonlinear p-polarized surface polaritons,” Solid State Commun. 44, 1309–1311 (1982).
    [Crossref]
  10. V. K. Fedyanin and D. Mihalache, “P-polarized nonlinear surface polaritons in layered structures,” Z. Phys. B 47, 167–173 (1982).
    [Crossref]
  11. N. N. Akhmediev, “Nonlinear theory of surface polaritons,” Zh. Eksp. Teor. Fiz. 84, 1907–1917 (1983) [Sov. Phys. JETP 57, 1111–1116 (1983)].
  12. D. Mihalache and V. K. Fedyanin, “P-polarized nonlinear surface and bounded (guided) waves in layered structures,” Teor. Mat. Fiz. 54, 443–445 (1983) [Theor. Math. Phys. 54,289–297 (1983)].
  13. M. Y. Yu, “Surface polaritons in nonlinear media,” Phys. Rev. A 28, 1855–1856 (1983).
    [Crossref]
  14. G. I. Stegeman, J. D. Valera, C. T. Seaton, J. E. Sipe, and A. A. Maradudin, “Nonlinear s-polarized surface plasmon polaritons,” Solid State Commun. 52, 293–297 (1984).
    [Crossref]
  15. K. M. Leung, “Propagation of nonlinear surface polaritons,” Phys. Rev. A 31, 1189–1192 (1985).
    [Crossref] [PubMed]
  16. K. M. Leung, “P-polarized nonlinear surface polaritons in materials with intensity-dependent dielectric functions,” Phys. Rev. B 32, 5093–5101 (1985).
    [Crossref]
  17. G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudip, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 (1985).
    [Crossref]
  18. A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 (1987).
    [Crossref] [PubMed]
  19. D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, “Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface,” Opt. Lett. 12, 187–189 (1987).
    [Crossref] [PubMed]
  20. A. J. Lomtev, “A new class of nonlinear surface waves,” Pis’ma Zh. Eksp. Teor. Fiz. 34, 64–67 (1981) [Sov. Phys. JETP Lett. 34,60–63 (1981)].
  21. A. A. Maradudin, “Nonlinear surface electromagnetic waves,” in Optical and Acoustic Waves in Solids—Modern Topics, M. Borissov, ed. (World Scientific, Singapore, 1983), pp. 72–142.
  22. A. I. Lomtev, “New class of s-polarized nonlinear surface waves,” Opt. Spektrosk. 55, 1079–1081 (1983) [Opt. Spectrosc. (USSR) 55,656–657 (1984)].
  23. N. N. Akhmediev, “Novel class of nonlinear surface waves: asymmetric modes in a symmetric layered structure,” Zh. Eksp. Teor. Fiz. 83, 545–553 (1982) [Sov. Phys. JETP 56, 299–303 (1982)].
  24. D. J. Robbins, “TE modes in a slab waveguide bounded by nonlinear media,” Opt. Commun. 47, 309–312 (1983).
    [Crossref]
  25. F. Lederer, U. Langbein, and H.-E. Ponath, “Nonlinear waves guided by a dielectric slab: I. TE-polarization,” Appl. Phys. B 31, 69–73 (1983).
    [Crossref]
  26. F. Lederer, U. Langbein, and H.-E. Ponath, “Nonlinear waves guided by a dielectric slab: II. TM-polarization,” Appl. Phys. B 31, 187–190 (1983).
    [Crossref]
  27. A. D. Boardman, P. Egan, and A. Shivarova, “TE modes of a layered nonlinear optical wave-guide,” Appl. Sci. Res. 41, 345–353 (1983).
    [Crossref]
  28. D. Mihalache, D. Mazilu, and M. Tortia, “Bistable states of s-polarized nonlinear waves guided by an asymmetric three-layer dielectric structure,” Phys. Scr. 30, 335–340 (1984).
    [Crossref]
  29. U. Langbein, F. Lederer, H.-E. Ponath, and U. Trutschel, “Dispersion relations for nonlinear guided waves,” J. Mol. Struct. 115, 493–496 (1984).
    [Crossref]
  30. D. Mihalache, R. G. Nazmitdinov, and V. K. Fedyanin, “P-polarized nonlinear surface waves in symmetric layered structures,” Phys. Scr. 29, 269–275 (1984).
    [Crossref]
  31. G. I. Stegeman and C. T. Seaton, “Nonlinear surface plasmons guided by thin metal films,” Opt. Lett. 9, 235–237 (1984).
    [Crossref] [PubMed]
  32. G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, “Nonlinear waves guided by thin films,” Appl. Phys. Lett. 44, 830–832 (1984).
    [Crossref]
  33. C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. Chilwell, and S. D. Smith, “Anomalous nonlinear guided wave cut-off phenomena,” Appl. Phys. Lett. 45, 1162–1163 (1984).
    [Crossref]
  34. J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by metal films,” J. Appl. Phys. 58, 2460–2466 (1985).
    [Crossref]
  35. C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of Nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. QE-21, 774–783 (1985).
    [Crossref]
  36. G. I. Stegeman, C. T. Seaton, and H. G. Winful, “Applications of guided waves to nonlinear optics,” Philos. Trans. R. Soc. London A 313, 321–326 (1984).
    [Crossref]
  37. A. D. Boardman and P. Egan, “Theory of optical hysteresis for TE guided modes,” Philos. Trans. R. Soc. London Ser. A 313, 363–369 (1984).
    [Crossref]
  38. D. Mihalache and H. Totia, “S-polarized nonlinear surface and guided waves in an asymmetric layered structure,” Rev. Roum. Phys. 29, 365–371 (1984).
  39. C. T. Seaton, J. D. Valera, B. Svenson, and G. I. Stegeman, “Comparison of solutions for TM-polarized guided waves,” Opt. Lett. 10, 149–150 (1985).
    [Crossref] [PubMed]
  40. U. Langbein, F. Lederer, H.-E. Ponath, and U. Trutschel, “Analysis of the dispersion relations of nonlinear slab-guided waves,” Appl. Phys. B 36, 187–193 (1985).
    [Crossref]
  41. U. Langbein, F. Lederer, H.-E. Ponath, and U. Trutschel, “Analysis of the dispersion relation of nonlinear slab-guided waves. Part II: Symmetrical configuration,” Appl. Phys. B 38, 263–268 (1985).
    [Crossref]
  42. A. D. Boardman and P. Egan, “S-polarized waves in a thin dielectric film asymmetrically bounded by optically nonlinear media,” IEEE J. Quantum Electron. QE-21, 1701–1713 (1985).
    [Crossref]
  43. L. Wendler, “S-polarized nonlinear surface polaritons. Effects of a transition layer,” Phys. Status Solidi B 135, 759–774 (1986).
    [Crossref]
  44. F. Lederer and D. Mihalache, “An additional kind of nonlinear s-polarized surface plasmon polaritons,” Solid State Commun. 59, 151–153 (1986).
    [Crossref]
  45. D. Mihalache and D. Mazilu, “Stability and instability of nonlinear guided waves in saturable media,” Solid State Commun. 63, 215–217 (1987).
    [Crossref]
  46. V. M. Eleonskii and V. P. Silin, “Propagation of electromagnetic waves in an inhomogeneous nonlinear medium,” Zh. Eksp. Teor. Fiz. 66, 146–153 (1974) [Sov. Phys. JETP 39, 67–70 (1974)].
  47. N. N. Achmediev, K. O. Bolter, and W. M. Eleonsky, “Optical dielectric waveguide with nonlinear permittivity,” Opt. Spektrosk. 53, 906–909 (1982) [Opt. Spectrosc. (USSR) 53, 540–542 (1982)].
  48. N. N. Achmediev, K. O. Bolter, and W. M. Eleonsky, “Dielectric optical waveguide with nonlinear susceptibility. Asymmetric refractive-index profiles,” Opt. Spektrosk. 53, 1097–1103 (1982) [Opt. Spectrosc. (USSR) 53,654–658 (1982)].
  49. U. Langbein, F. Lederer, and H.-E. Ponath, “A new type of nonlinear slab-guided waves,” Opt. Commun. 46, 167–169 (1983).
    [Crossref]
  50. A. D. Boardman and P. Egan, “Nonlinear surface and guided polaritons of a general layered dielectric structure,” J. Phys. (Paris) C5, Suppl. 45, C5-291–C5-303 (1984).
    [Crossref]
  51. U. Langbein, F. Lederer, and H.-E. Ponath, “Generalized dispersion relations for nonlinear slab-guided waves,” Opt. Commun. 53, 417–420 (1985).
    [Crossref]
  52. A. D. Boardman and P. Egan, “Nonlinear electromagnetic surface and guided waves: theory,” in Surface Waves in Plasmas and Solids, S. Vukovic, ed. (World Scientific, Singapore, 1986), pp. 3–77.
  53. A. D. Boardman and P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. QE-22, 319–324 (1986).
    [Crossref]
  54. W. R. Holland, “Nonlinear guided waves in low-index, self-focusing thin films: transverse electric case,” J. Opt. Soc. Am. B 3, 1529–1534 (1986).
    [Crossref]
  55. A. D. Boardman, T. Twardowski, A. Shivarova, and G. I. Stegeman, “Surface guided nonlinear TM waves in planar waveguides,” Proc. Inst. Electr. Eng. 134, 152–160 (1987).
  56. See, for example, A. A. Maradudin, “Surface waves,” in Modern Problems of Surface Physics,” I. J. Lalov, ed. (Bulgarian Academy of Sciences, Sofia, 1981), pp. 11–399.
  57. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Formulas (Dover, New York, 1968), p. 596, formula (17.4.52).
  58. Ref. 57, p. 596, formula (17.4.45).
  59. P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists (Springer-Verlag, Berlin, 1954), p. 145, formula (266.00).
  60. Ref. 57, p. 596, formula (17.4.49).
  61. Ref. 57, p. 596, formula (17.4.51).
  62. Ref. 57, p. 596, formula (17.4.43).
  63. Ref. 57, p. 596, formula (17.4.41).
  64. Ref. 59, p. 141, formula (263.00).
  65. Ref. 57, Secs. 16.4 and 17.6.
  66. Ref. 57, Sec. 17.2.
  67. Ref. 57, Sec. 17.5.
  68. A good reference dealing with the numerical calculation of Jacobian elliptic functions and inverse Jacobian elliptic functions is L. V. King, On the Direct Numerical Calculation of Elliptic Functions and Integrals (Cambridge U. Press, Cambridge, 1924).

1987 (4)

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 (1987).
[Crossref] [PubMed]

D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, “Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface,” Opt. Lett. 12, 187–189 (1987).
[Crossref] [PubMed]

D. Mihalache and D. Mazilu, “Stability and instability of nonlinear guided waves in saturable media,” Solid State Commun. 63, 215–217 (1987).
[Crossref]

A. D. Boardman, T. Twardowski, A. Shivarova, and G. I. Stegeman, “Surface guided nonlinear TM waves in planar waveguides,” Proc. Inst. Electr. Eng. 134, 152–160 (1987).

1986 (4)

A. D. Boardman and P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. QE-22, 319–324 (1986).
[Crossref]

W. R. Holland, “Nonlinear guided waves in low-index, self-focusing thin films: transverse electric case,” J. Opt. Soc. Am. B 3, 1529–1534 (1986).
[Crossref]

L. Wendler, “S-polarized nonlinear surface polaritons. Effects of a transition layer,” Phys. Status Solidi B 135, 759–774 (1986).
[Crossref]

F. Lederer and D. Mihalache, “An additional kind of nonlinear s-polarized surface plasmon polaritons,” Solid State Commun. 59, 151–153 (1986).
[Crossref]

1985 (10)

U. Langbein, F. Lederer, and H.-E. Ponath, “Generalized dispersion relations for nonlinear slab-guided waves,” Opt. Commun. 53, 417–420 (1985).
[Crossref]

K. M. Leung, “Propagation of nonlinear surface polaritons,” Phys. Rev. A 31, 1189–1192 (1985).
[Crossref] [PubMed]

K. M. Leung, “P-polarized nonlinear surface polaritons in materials with intensity-dependent dielectric functions,” Phys. Rev. B 32, 5093–5101 (1985).
[Crossref]

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudip, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 (1985).
[Crossref]

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by metal films,” J. Appl. Phys. 58, 2460–2466 (1985).
[Crossref]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of Nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. QE-21, 774–783 (1985).
[Crossref]

C. T. Seaton, J. D. Valera, B. Svenson, and G. I. Stegeman, “Comparison of solutions for TM-polarized guided waves,” Opt. Lett. 10, 149–150 (1985).
[Crossref] [PubMed]

U. Langbein, F. Lederer, H.-E. Ponath, and U. Trutschel, “Analysis of the dispersion relations of nonlinear slab-guided waves,” Appl. Phys. B 36, 187–193 (1985).
[Crossref]

U. Langbein, F. Lederer, H.-E. Ponath, and U. Trutschel, “Analysis of the dispersion relation of nonlinear slab-guided waves. Part II: Symmetrical configuration,” Appl. Phys. B 38, 263–268 (1985).
[Crossref]

A. D. Boardman and P. Egan, “S-polarized waves in a thin dielectric film asymmetrically bounded by optically nonlinear media,” IEEE J. Quantum Electron. QE-21, 1701–1713 (1985).
[Crossref]

1984 (11)

G. I. Stegeman, C. T. Seaton, and H. G. Winful, “Applications of guided waves to nonlinear optics,” Philos. Trans. R. Soc. London A 313, 321–326 (1984).
[Crossref]

A. D. Boardman and P. Egan, “Theory of optical hysteresis for TE guided modes,” Philos. Trans. R. Soc. London Ser. A 313, 363–369 (1984).
[Crossref]

D. Mihalache and H. Totia, “S-polarized nonlinear surface and guided waves in an asymmetric layered structure,” Rev. Roum. Phys. 29, 365–371 (1984).

D. Mihalache, D. Mazilu, and M. Tortia, “Bistable states of s-polarized nonlinear waves guided by an asymmetric three-layer dielectric structure,” Phys. Scr. 30, 335–340 (1984).
[Crossref]

U. Langbein, F. Lederer, H.-E. Ponath, and U. Trutschel, “Dispersion relations for nonlinear guided waves,” J. Mol. Struct. 115, 493–496 (1984).
[Crossref]

D. Mihalache, R. G. Nazmitdinov, and V. K. Fedyanin, “P-polarized nonlinear surface waves in symmetric layered structures,” Phys. Scr. 29, 269–275 (1984).
[Crossref]

G. I. Stegeman and C. T. Seaton, “Nonlinear surface plasmons guided by thin metal films,” Opt. Lett. 9, 235–237 (1984).
[Crossref] [PubMed]

G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, “Nonlinear waves guided by thin films,” Appl. Phys. Lett. 44, 830–832 (1984).
[Crossref]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. Chilwell, and S. D. Smith, “Anomalous nonlinear guided wave cut-off phenomena,” Appl. Phys. Lett. 45, 1162–1163 (1984).
[Crossref]

G. I. Stegeman, J. D. Valera, C. T. Seaton, J. E. Sipe, and A. A. Maradudin, “Nonlinear s-polarized surface plasmon polaritons,” Solid State Commun. 52, 293–297 (1984).
[Crossref]

A. D. Boardman and P. Egan, “Nonlinear surface and guided polaritons of a general layered dielectric structure,” J. Phys. (Paris) C5, Suppl. 45, C5-291–C5-303 (1984).
[Crossref]

1983 (9)

U. Langbein, F. Lederer, and H.-E. Ponath, “A new type of nonlinear slab-guided waves,” Opt. Commun. 46, 167–169 (1983).
[Crossref]

A. I. Lomtev, “New class of s-polarized nonlinear surface waves,” Opt. Spektrosk. 55, 1079–1081 (1983) [Opt. Spectrosc. (USSR) 55,656–657 (1984)].

N. N. Akhmediev, “Nonlinear theory of surface polaritons,” Zh. Eksp. Teor. Fiz. 84, 1907–1917 (1983) [Sov. Phys. JETP 57, 1111–1116 (1983)].

D. Mihalache and V. K. Fedyanin, “P-polarized nonlinear surface and bounded (guided) waves in layered structures,” Teor. Mat. Fiz. 54, 443–445 (1983) [Theor. Math. Phys. 54,289–297 (1983)].

M. Y. Yu, “Surface polaritons in nonlinear media,” Phys. Rev. A 28, 1855–1856 (1983).
[Crossref]

D. J. Robbins, “TE modes in a slab waveguide bounded by nonlinear media,” Opt. Commun. 47, 309–312 (1983).
[Crossref]

F. Lederer, U. Langbein, and H.-E. Ponath, “Nonlinear waves guided by a dielectric slab: I. TE-polarization,” Appl. Phys. B 31, 69–73 (1983).
[Crossref]

F. Lederer, U. Langbein, and H.-E. Ponath, “Nonlinear waves guided by a dielectric slab: II. TM-polarization,” Appl. Phys. B 31, 187–190 (1983).
[Crossref]

A. D. Boardman, P. Egan, and A. Shivarova, “TE modes of a layered nonlinear optical wave-guide,” Appl. Sci. Res. 41, 345–353 (1983).
[Crossref]

1982 (5)

V. M. Agranovich and V. Ya. Chernyak, “Perturbation theory for weakly nonlinear p-polarized surface polaritons,” Solid State Commun. 44, 1309–1311 (1982).
[Crossref]

V. K. Fedyanin and D. Mihalache, “P-polarized nonlinear surface polaritons in layered structures,” Z. Phys. B 47, 167–173 (1982).
[Crossref]

N. N. Akhmediev, “Novel class of nonlinear surface waves: asymmetric modes in a symmetric layered structure,” Zh. Eksp. Teor. Fiz. 83, 545–553 (1982) [Sov. Phys. JETP 56, 299–303 (1982)].

N. N. Achmediev, K. O. Bolter, and W. M. Eleonsky, “Optical dielectric waveguide with nonlinear permittivity,” Opt. Spektrosk. 53, 906–909 (1982) [Opt. Spectrosc. (USSR) 53, 540–542 (1982)].

N. N. Achmediev, K. O. Bolter, and W. M. Eleonsky, “Dielectric optical waveguide with nonlinear susceptibility. Asymmetric refractive-index profiles,” Opt. Spektrosk. 53, 1097–1103 (1982) [Opt. Spectrosc. (USSR) 53,654–658 (1982)].

1981 (2)

A. A. Maradudin, “S-polarized nonlinear surface polaritons,” Z. Phys. B 41, 341–348 (1981).
[Crossref]

A. J. Lomtev, “A new class of nonlinear surface waves,” Pis’ma Zh. Eksp. Teor. Fiz. 34, 64–67 (1981) [Sov. Phys. JETP Lett. 34,60–63 (1981)].

1980 (3)

W. J. Tomlinson, “Surface wave at a nonlinear interface,” Opt. Lett. 5, 323–325 (1980).
[Crossref] [PubMed]

A. G. Boev, “Nonlinear theory of penetration of p-polarized electromagnetic waves in a plasma,” Zh. Eksp. Teor. Fiz. 79, 134–142 (1980) [Sov. Phys. JETP 52, 67–71 (1980)].

V. M. Agranovich, V. S. Babichenko, and V. Ya. Chernyak, “Nonlinear surface polaritons,” Pis’ma Zh. Eksp. Teor. Fiz. 32, 532–535 (1980) [Sov. Phys. JETP Lett. 32, 512–515 (1980)].

1979 (1)

A. G. Boev, “On the theory of nonlinear surface waves in a plasma,” Zh. Eksp. Teor. Fiz. 77, 92–100 (1979) [Sov. Phys. JETP 50, 47–51 (1979)].

1975 (1)

A. G. Boev and A. V. Prokopov, “Contribution to the nonlinear theory of surface electromagnetic waves in an ionizing plasma,” Zh. Eksp. Teor. Fiz. 69, 1208–1217 (1975) [Sov. Phys. JETP 42, 617–621 (1975)].

1974 (1)

V. M. Eleonskii and V. P. Silin, “Propagation of electromagnetic waves in an inhomogeneous nonlinear medium,” Zh. Eksp. Teor. Fiz. 66, 146–153 (1974) [Sov. Phys. JETP 39, 67–70 (1974)].

1968 (1)

A. G. Litvak and V. A. Mironov, “Surface waves at the interface of nonlinear media,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 11, 1911–1912 (1968).

1967 (1)

Yu. R. Alanakyan, “Surface waves in a plasma,” Zh. Tekh. Fiz. 37, 817–821 (1967) [Sov. Phys. Tech. Phys. 12, 587–589 (1967)].

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Formulas (Dover, New York, 1968), p. 596, formula (17.4.52).

Achmediev, N. N.

N. N. Achmediev, K. O. Bolter, and W. M. Eleonsky, “Optical dielectric waveguide with nonlinear permittivity,” Opt. Spektrosk. 53, 906–909 (1982) [Opt. Spectrosc. (USSR) 53, 540–542 (1982)].

N. N. Achmediev, K. O. Bolter, and W. M. Eleonsky, “Dielectric optical waveguide with nonlinear susceptibility. Asymmetric refractive-index profiles,” Opt. Spektrosk. 53, 1097–1103 (1982) [Opt. Spectrosc. (USSR) 53,654–658 (1982)].

Agranovich, V. M.

V. M. Agranovich and V. Ya. Chernyak, “Perturbation theory for weakly nonlinear p-polarized surface polaritons,” Solid State Commun. 44, 1309–1311 (1982).
[Crossref]

V. M. Agranovich, V. S. Babichenko, and V. Ya. Chernyak, “Nonlinear surface polaritons,” Pis’ma Zh. Eksp. Teor. Fiz. 32, 532–535 (1980) [Sov. Phys. JETP Lett. 32, 512–515 (1980)].

Akhmediev, N. N.

N. N. Akhmediev, “Nonlinear theory of surface polaritons,” Zh. Eksp. Teor. Fiz. 84, 1907–1917 (1983) [Sov. Phys. JETP 57, 1111–1116 (1983)].

N. N. Akhmediev, “Novel class of nonlinear surface waves: asymmetric modes in a symmetric layered structure,” Zh. Eksp. Teor. Fiz. 83, 545–553 (1982) [Sov. Phys. JETP 56, 299–303 (1982)].

Alanakyan, Yu. R.

Yu. R. Alanakyan, “Surface waves in a plasma,” Zh. Tekh. Fiz. 37, 817–821 (1967) [Sov. Phys. Tech. Phys. 12, 587–589 (1967)].

Ariyasu, J.

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudip, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 (1985).
[Crossref]

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by metal films,” J. Appl. Phys. 58, 2460–2466 (1985).
[Crossref]

Babichenko, V. S.

V. M. Agranovich, V. S. Babichenko, and V. Ya. Chernyak, “Nonlinear surface polaritons,” Pis’ma Zh. Eksp. Teor. Fiz. 32, 532–535 (1980) [Sov. Phys. JETP Lett. 32, 512–515 (1980)].

Boardman, A. D.

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 (1987).
[Crossref] [PubMed]

D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, “Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface,” Opt. Lett. 12, 187–189 (1987).
[Crossref] [PubMed]

A. D. Boardman, T. Twardowski, A. Shivarova, and G. I. Stegeman, “Surface guided nonlinear TM waves in planar waveguides,” Proc. Inst. Electr. Eng. 134, 152–160 (1987).

A. D. Boardman and P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. QE-22, 319–324 (1986).
[Crossref]

A. D. Boardman and P. Egan, “S-polarized waves in a thin dielectric film asymmetrically bounded by optically nonlinear media,” IEEE J. Quantum Electron. QE-21, 1701–1713 (1985).
[Crossref]

A. D. Boardman and P. Egan, “Nonlinear surface and guided polaritons of a general layered dielectric structure,” J. Phys. (Paris) C5, Suppl. 45, C5-291–C5-303 (1984).
[Crossref]

A. D. Boardman and P. Egan, “Theory of optical hysteresis for TE guided modes,” Philos. Trans. R. Soc. London Ser. A 313, 363–369 (1984).
[Crossref]

A. D. Boardman, P. Egan, and A. Shivarova, “TE modes of a layered nonlinear optical wave-guide,” Appl. Sci. Res. 41, 345–353 (1983).
[Crossref]

A. D. Boardman and P. Egan, “Nonlinear electromagnetic surface and guided waves: theory,” in Surface Waves in Plasmas and Solids, S. Vukovic, ed. (World Scientific, Singapore, 1986), pp. 3–77.

Boev, A. G.

A. G. Boev, “Nonlinear theory of penetration of p-polarized electromagnetic waves in a plasma,” Zh. Eksp. Teor. Fiz. 79, 134–142 (1980) [Sov. Phys. JETP 52, 67–71 (1980)].

A. G. Boev, “On the theory of nonlinear surface waves in a plasma,” Zh. Eksp. Teor. Fiz. 77, 92–100 (1979) [Sov. Phys. JETP 50, 47–51 (1979)].

A. G. Boev and A. V. Prokopov, “Contribution to the nonlinear theory of surface electromagnetic waves in an ionizing plasma,” Zh. Eksp. Teor. Fiz. 69, 1208–1217 (1975) [Sov. Phys. JETP 42, 617–621 (1975)].

Bolter, K. O.

N. N. Achmediev, K. O. Bolter, and W. M. Eleonsky, “Dielectric optical waveguide with nonlinear susceptibility. Asymmetric refractive-index profiles,” Opt. Spektrosk. 53, 1097–1103 (1982) [Opt. Spectrosc. (USSR) 53,654–658 (1982)].

N. N. Achmediev, K. O. Bolter, and W. M. Eleonsky, “Optical dielectric waveguide with nonlinear permittivity,” Opt. Spektrosk. 53, 906–909 (1982) [Opt. Spectrosc. (USSR) 53, 540–542 (1982)].

Byrd, P. F.

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists (Springer-Verlag, Berlin, 1954), p. 145, formula (266.00).

Chernyak, V. Ya.

V. M. Agranovich and V. Ya. Chernyak, “Perturbation theory for weakly nonlinear p-polarized surface polaritons,” Solid State Commun. 44, 1309–1311 (1982).
[Crossref]

V. M. Agranovich, V. S. Babichenko, and V. Ya. Chernyak, “Nonlinear surface polaritons,” Pis’ma Zh. Eksp. Teor. Fiz. 32, 532–535 (1980) [Sov. Phys. JETP Lett. 32, 512–515 (1980)].

Chilwell, J.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. Chilwell, and S. D. Smith, “Anomalous nonlinear guided wave cut-off phenomena,” Appl. Phys. Lett. 45, 1162–1163 (1984).
[Crossref]

G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, “Nonlinear waves guided by thin films,” Appl. Phys. Lett. 44, 830–832 (1984).
[Crossref]

Chilwell, J. T.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of Nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. QE-21, 774–783 (1985).
[Crossref]

Egan, P.

A. D. Boardman and P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. QE-22, 319–324 (1986).
[Crossref]

A. D. Boardman and P. Egan, “S-polarized waves in a thin dielectric film asymmetrically bounded by optically nonlinear media,” IEEE J. Quantum Electron. QE-21, 1701–1713 (1985).
[Crossref]

A. D. Boardman and P. Egan, “Nonlinear surface and guided polaritons of a general layered dielectric structure,” J. Phys. (Paris) C5, Suppl. 45, C5-291–C5-303 (1984).
[Crossref]

A. D. Boardman and P. Egan, “Theory of optical hysteresis for TE guided modes,” Philos. Trans. R. Soc. London Ser. A 313, 363–369 (1984).
[Crossref]

A. D. Boardman, P. Egan, and A. Shivarova, “TE modes of a layered nonlinear optical wave-guide,” Appl. Sci. Res. 41, 345–353 (1983).
[Crossref]

A. D. Boardman and P. Egan, “Nonlinear electromagnetic surface and guided waves: theory,” in Surface Waves in Plasmas and Solids, S. Vukovic, ed. (World Scientific, Singapore, 1986), pp. 3–77.

Eleonskii, V. M.

V. M. Eleonskii and V. P. Silin, “Propagation of electromagnetic waves in an inhomogeneous nonlinear medium,” Zh. Eksp. Teor. Fiz. 66, 146–153 (1974) [Sov. Phys. JETP 39, 67–70 (1974)].

Eleonsky, W. M.

N. N. Achmediev, K. O. Bolter, and W. M. Eleonsky, “Optical dielectric waveguide with nonlinear permittivity,” Opt. Spektrosk. 53, 906–909 (1982) [Opt. Spectrosc. (USSR) 53, 540–542 (1982)].

N. N. Achmediev, K. O. Bolter, and W. M. Eleonsky, “Dielectric optical waveguide with nonlinear susceptibility. Asymmetric refractive-index profiles,” Opt. Spektrosk. 53, 1097–1103 (1982) [Opt. Spectrosc. (USSR) 53,654–658 (1982)].

Fedyanin, V. K.

D. Mihalache, R. G. Nazmitdinov, and V. K. Fedyanin, “P-polarized nonlinear surface waves in symmetric layered structures,” Phys. Scr. 29, 269–275 (1984).
[Crossref]

D. Mihalache and V. K. Fedyanin, “P-polarized nonlinear surface and bounded (guided) waves in layered structures,” Teor. Mat. Fiz. 54, 443–445 (1983) [Theor. Math. Phys. 54,289–297 (1983)].

V. K. Fedyanin and D. Mihalache, “P-polarized nonlinear surface polaritons in layered structures,” Z. Phys. B 47, 167–173 (1982).
[Crossref]

Friedman, M. D.

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists (Springer-Verlag, Berlin, 1954), p. 145, formula (266.00).

Holland, W. R.

King, L. V.

A good reference dealing with the numerical calculation of Jacobian elliptic functions and inverse Jacobian elliptic functions is L. V. King, On the Direct Numerical Calculation of Elliptic Functions and Integrals (Cambridge U. Press, Cambridge, 1924).

Langbein, U.

U. Langbein, F. Lederer, and H.-E. Ponath, “Generalized dispersion relations for nonlinear slab-guided waves,” Opt. Commun. 53, 417–420 (1985).
[Crossref]

U. Langbein, F. Lederer, H.-E. Ponath, and U. Trutschel, “Analysis of the dispersion relation of nonlinear slab-guided waves. Part II: Symmetrical configuration,” Appl. Phys. B 38, 263–268 (1985).
[Crossref]

U. Langbein, F. Lederer, H.-E. Ponath, and U. Trutschel, “Analysis of the dispersion relations of nonlinear slab-guided waves,” Appl. Phys. B 36, 187–193 (1985).
[Crossref]

U. Langbein, F. Lederer, H.-E. Ponath, and U. Trutschel, “Dispersion relations for nonlinear guided waves,” J. Mol. Struct. 115, 493–496 (1984).
[Crossref]

F. Lederer, U. Langbein, and H.-E. Ponath, “Nonlinear waves guided by a dielectric slab: II. TM-polarization,” Appl. Phys. B 31, 187–190 (1983).
[Crossref]

F. Lederer, U. Langbein, and H.-E. Ponath, “Nonlinear waves guided by a dielectric slab: I. TE-polarization,” Appl. Phys. B 31, 69–73 (1983).
[Crossref]

U. Langbein, F. Lederer, and H.-E. Ponath, “A new type of nonlinear slab-guided waves,” Opt. Commun. 46, 167–169 (1983).
[Crossref]

Lederer, F.

F. Lederer and D. Mihalache, “An additional kind of nonlinear s-polarized surface plasmon polaritons,” Solid State Commun. 59, 151–153 (1986).
[Crossref]

U. Langbein, F. Lederer, H.-E. Ponath, and U. Trutschel, “Analysis of the dispersion relation of nonlinear slab-guided waves. Part II: Symmetrical configuration,” Appl. Phys. B 38, 263–268 (1985).
[Crossref]

U. Langbein, F. Lederer, and H.-E. Ponath, “Generalized dispersion relations for nonlinear slab-guided waves,” Opt. Commun. 53, 417–420 (1985).
[Crossref]

U. Langbein, F. Lederer, H.-E. Ponath, and U. Trutschel, “Analysis of the dispersion relations of nonlinear slab-guided waves,” Appl. Phys. B 36, 187–193 (1985).
[Crossref]

U. Langbein, F. Lederer, H.-E. Ponath, and U. Trutschel, “Dispersion relations for nonlinear guided waves,” J. Mol. Struct. 115, 493–496 (1984).
[Crossref]

F. Lederer, U. Langbein, and H.-E. Ponath, “Nonlinear waves guided by a dielectric slab: II. TM-polarization,” Appl. Phys. B 31, 187–190 (1983).
[Crossref]

F. Lederer, U. Langbein, and H.-E. Ponath, “Nonlinear waves guided by a dielectric slab: I. TE-polarization,” Appl. Phys. B 31, 69–73 (1983).
[Crossref]

U. Langbein, F. Lederer, and H.-E. Ponath, “A new type of nonlinear slab-guided waves,” Opt. Commun. 46, 167–169 (1983).
[Crossref]

Leung, K. M.

K. M. Leung, “Propagation of nonlinear surface polaritons,” Phys. Rev. A 31, 1189–1192 (1985).
[Crossref] [PubMed]

K. M. Leung, “P-polarized nonlinear surface polaritons in materials with intensity-dependent dielectric functions,” Phys. Rev. B 32, 5093–5101 (1985).
[Crossref]

Litvak, A. G.

A. G. Litvak and V. A. Mironov, “Surface waves at the interface of nonlinear media,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 11, 1911–1912 (1968).

Lomtev, A. I.

A. I. Lomtev, “New class of s-polarized nonlinear surface waves,” Opt. Spektrosk. 55, 1079–1081 (1983) [Opt. Spectrosc. (USSR) 55,656–657 (1984)].

Lomtev, A. J.

A. J. Lomtev, “A new class of nonlinear surface waves,” Pis’ma Zh. Eksp. Teor. Fiz. 34, 64–67 (1981) [Sov. Phys. JETP Lett. 34,60–63 (1981)].

Maradudin, A. A.

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 (1987).
[Crossref] [PubMed]

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by metal films,” J. Appl. Phys. 58, 2460–2466 (1985).
[Crossref]

G. I. Stegeman, J. D. Valera, C. T. Seaton, J. E. Sipe, and A. A. Maradudin, “Nonlinear s-polarized surface plasmon polaritons,” Solid State Commun. 52, 293–297 (1984).
[Crossref]

A. A. Maradudin, “S-polarized nonlinear surface polaritons,” Z. Phys. B 41, 341–348 (1981).
[Crossref]

A. A. Maradudin, “Nonlinear surface electromagnetic waves,” in Optical and Acoustic Waves in Solids—Modern Topics, M. Borissov, ed. (World Scientific, Singapore, 1983), pp. 72–142.

See, for example, A. A. Maradudin, “Surface waves,” in Modern Problems of Surface Physics,” I. J. Lalov, ed. (Bulgarian Academy of Sciences, Sofia, 1981), pp. 11–399.

Maradudip, A. A.

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudip, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 (1985).
[Crossref]

Mazilu, D.

D. Mihalache and D. Mazilu, “Stability and instability of nonlinear guided waves in saturable media,” Solid State Commun. 63, 215–217 (1987).
[Crossref]

D. Mihalache, D. Mazilu, and M. Tortia, “Bistable states of s-polarized nonlinear waves guided by an asymmetric three-layer dielectric structure,” Phys. Scr. 30, 335–340 (1984).
[Crossref]

Mihalache, D.

D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, “Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface,” Opt. Lett. 12, 187–189 (1987).
[Crossref] [PubMed]

D. Mihalache and D. Mazilu, “Stability and instability of nonlinear guided waves in saturable media,” Solid State Commun. 63, 215–217 (1987).
[Crossref]

F. Lederer and D. Mihalache, “An additional kind of nonlinear s-polarized surface plasmon polaritons,” Solid State Commun. 59, 151–153 (1986).
[Crossref]

D. Mihalache and H. Totia, “S-polarized nonlinear surface and guided waves in an asymmetric layered structure,” Rev. Roum. Phys. 29, 365–371 (1984).

D. Mihalache, D. Mazilu, and M. Tortia, “Bistable states of s-polarized nonlinear waves guided by an asymmetric three-layer dielectric structure,” Phys. Scr. 30, 335–340 (1984).
[Crossref]

D. Mihalache, R. G. Nazmitdinov, and V. K. Fedyanin, “P-polarized nonlinear surface waves in symmetric layered structures,” Phys. Scr. 29, 269–275 (1984).
[Crossref]

D. Mihalache and V. K. Fedyanin, “P-polarized nonlinear surface and bounded (guided) waves in layered structures,” Teor. Mat. Fiz. 54, 443–445 (1983) [Theor. Math. Phys. 54,289–297 (1983)].

V. K. Fedyanin and D. Mihalache, “P-polarized nonlinear surface polaritons in layered structures,” Z. Phys. B 47, 167–173 (1982).
[Crossref]

Mironov, V. A.

A. G. Litvak and V. A. Mironov, “Surface waves at the interface of nonlinear media,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 11, 1911–1912 (1968).

Nazmitdinov, R. G.

D. Mihalache, R. G. Nazmitdinov, and V. K. Fedyanin, “P-polarized nonlinear surface waves in symmetric layered structures,” Phys. Scr. 29, 269–275 (1984).
[Crossref]

Ponath, H.-E.

U. Langbein, F. Lederer, H.-E. Ponath, and U. Trutschel, “Analysis of the dispersion relations of nonlinear slab-guided waves,” Appl. Phys. B 36, 187–193 (1985).
[Crossref]

U. Langbein, F. Lederer, H.-E. Ponath, and U. Trutschel, “Analysis of the dispersion relation of nonlinear slab-guided waves. Part II: Symmetrical configuration,” Appl. Phys. B 38, 263–268 (1985).
[Crossref]

U. Langbein, F. Lederer, and H.-E. Ponath, “Generalized dispersion relations for nonlinear slab-guided waves,” Opt. Commun. 53, 417–420 (1985).
[Crossref]

U. Langbein, F. Lederer, H.-E. Ponath, and U. Trutschel, “Dispersion relations for nonlinear guided waves,” J. Mol. Struct. 115, 493–496 (1984).
[Crossref]

F. Lederer, U. Langbein, and H.-E. Ponath, “Nonlinear waves guided by a dielectric slab: II. TM-polarization,” Appl. Phys. B 31, 187–190 (1983).
[Crossref]

F. Lederer, U. Langbein, and H.-E. Ponath, “Nonlinear waves guided by a dielectric slab: I. TE-polarization,” Appl. Phys. B 31, 69–73 (1983).
[Crossref]

U. Langbein, F. Lederer, and H.-E. Ponath, “A new type of nonlinear slab-guided waves,” Opt. Commun. 46, 167–169 (1983).
[Crossref]

Prokopov, A. V.

A. G. Boev and A. V. Prokopov, “Contribution to the nonlinear theory of surface electromagnetic waves in an ionizing plasma,” Zh. Eksp. Teor. Fiz. 69, 1208–1217 (1975) [Sov. Phys. JETP 42, 617–621 (1975)].

Robbins, D. J.

D. J. Robbins, “TE modes in a slab waveguide bounded by nonlinear media,” Opt. Commun. 47, 309–312 (1983).
[Crossref]

Seaton, C. T.

D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, “Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface,” Opt. Lett. 12, 187–189 (1987).
[Crossref] [PubMed]

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudip, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 (1985).
[Crossref]

C. T. Seaton, J. D. Valera, B. Svenson, and G. I. Stegeman, “Comparison of solutions for TM-polarized guided waves,” Opt. Lett. 10, 149–150 (1985).
[Crossref] [PubMed]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of Nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. QE-21, 774–783 (1985).
[Crossref]

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by metal films,” J. Appl. Phys. 58, 2460–2466 (1985).
[Crossref]

G. I. Stegeman and C. T. Seaton, “Nonlinear surface plasmons guided by thin metal films,” Opt. Lett. 9, 235–237 (1984).
[Crossref] [PubMed]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. Chilwell, and S. D. Smith, “Anomalous nonlinear guided wave cut-off phenomena,” Appl. Phys. Lett. 45, 1162–1163 (1984).
[Crossref]

G. I. Stegeman, C. T. Seaton, and H. G. Winful, “Applications of guided waves to nonlinear optics,” Philos. Trans. R. Soc. London A 313, 321–326 (1984).
[Crossref]

G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, “Nonlinear waves guided by thin films,” Appl. Phys. Lett. 44, 830–832 (1984).
[Crossref]

G. I. Stegeman, J. D. Valera, C. T. Seaton, J. E. Sipe, and A. A. Maradudin, “Nonlinear s-polarized surface plasmon polaritons,” Solid State Commun. 52, 293–297 (1984).
[Crossref]

Shivarova, A.

A. D. Boardman, T. Twardowski, A. Shivarova, and G. I. Stegeman, “Surface guided nonlinear TM waves in planar waveguides,” Proc. Inst. Electr. Eng. 134, 152–160 (1987).

A. D. Boardman, P. Egan, and A. Shivarova, “TE modes of a layered nonlinear optical wave-guide,” Appl. Sci. Res. 41, 345–353 (1983).
[Crossref]

Shoemaker, R. L.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of Nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. QE-21, 774–783 (1985).
[Crossref]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. Chilwell, and S. D. Smith, “Anomalous nonlinear guided wave cut-off phenomena,” Appl. Phys. Lett. 45, 1162–1163 (1984).
[Crossref]

Silin, V. P.

V. M. Eleonskii and V. P. Silin, “Propagation of electromagnetic waves in an inhomogeneous nonlinear medium,” Zh. Eksp. Teor. Fiz. 66, 146–153 (1974) [Sov. Phys. JETP 39, 67–70 (1974)].

Sipe, J. E.

G. I. Stegeman, J. D. Valera, C. T. Seaton, J. E. Sipe, and A. A. Maradudin, “Nonlinear s-polarized surface plasmon polaritons,” Solid State Commun. 52, 293–297 (1984).
[Crossref]

Smith, S. D.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of Nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. QE-21, 774–783 (1985).
[Crossref]

G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, “Nonlinear waves guided by thin films,” Appl. Phys. Lett. 44, 830–832 (1984).
[Crossref]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. Chilwell, and S. D. Smith, “Anomalous nonlinear guided wave cut-off phenomena,” Appl. Phys. Lett. 45, 1162–1163 (1984).
[Crossref]

Stegeman, G. I.

A. D. Boardman, T. Twardowski, A. Shivarova, and G. I. Stegeman, “Surface guided nonlinear TM waves in planar waveguides,” Proc. Inst. Electr. Eng. 134, 152–160 (1987).

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 (1987).
[Crossref] [PubMed]

D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, “Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface,” Opt. Lett. 12, 187–189 (1987).
[Crossref] [PubMed]

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudip, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 (1985).
[Crossref]

C. T. Seaton, J. D. Valera, B. Svenson, and G. I. Stegeman, “Comparison of solutions for TM-polarized guided waves,” Opt. Lett. 10, 149–150 (1985).
[Crossref] [PubMed]

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by metal films,” J. Appl. Phys. 58, 2460–2466 (1985).
[Crossref]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of Nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. QE-21, 774–783 (1985).
[Crossref]

G. I. Stegeman, C. T. Seaton, and H. G. Winful, “Applications of guided waves to nonlinear optics,” Philos. Trans. R. Soc. London A 313, 321–326 (1984).
[Crossref]

G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, “Nonlinear waves guided by thin films,” Appl. Phys. Lett. 44, 830–832 (1984).
[Crossref]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. Chilwell, and S. D. Smith, “Anomalous nonlinear guided wave cut-off phenomena,” Appl. Phys. Lett. 45, 1162–1163 (1984).
[Crossref]

G. I. Stegeman, J. D. Valera, C. T. Seaton, J. E. Sipe, and A. A. Maradudin, “Nonlinear s-polarized surface plasmon polaritons,” Solid State Commun. 52, 293–297 (1984).
[Crossref]

G. I. Stegeman and C. T. Seaton, “Nonlinear surface plasmons guided by thin metal films,” Opt. Lett. 9, 235–237 (1984).
[Crossref] [PubMed]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Formulas (Dover, New York, 1968), p. 596, formula (17.4.52).

Svenson, B.

Tomlinson, W. J.

Tortia, M.

D. Mihalache, D. Mazilu, and M. Tortia, “Bistable states of s-polarized nonlinear waves guided by an asymmetric three-layer dielectric structure,” Phys. Scr. 30, 335–340 (1984).
[Crossref]

Totia, H.

D. Mihalache and H. Totia, “S-polarized nonlinear surface and guided waves in an asymmetric layered structure,” Rev. Roum. Phys. 29, 365–371 (1984).

Trutschel, U.

U. Langbein, F. Lederer, H.-E. Ponath, and U. Trutschel, “Analysis of the dispersion relations of nonlinear slab-guided waves,” Appl. Phys. B 36, 187–193 (1985).
[Crossref]

U. Langbein, F. Lederer, H.-E. Ponath, and U. Trutschel, “Analysis of the dispersion relation of nonlinear slab-guided waves. Part II: Symmetrical configuration,” Appl. Phys. B 38, 263–268 (1985).
[Crossref]

U. Langbein, F. Lederer, H.-E. Ponath, and U. Trutschel, “Dispersion relations for nonlinear guided waves,” J. Mol. Struct. 115, 493–496 (1984).
[Crossref]

Twardowski, T.

D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, and T. Twardowski, “Exact dispersion relations for transverse magnetic polarized guided waves at a nonlinear interface,” Opt. Lett. 12, 187–189 (1987).
[Crossref] [PubMed]

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 (1987).
[Crossref] [PubMed]

A. D. Boardman, T. Twardowski, A. Shivarova, and G. I. Stegeman, “Surface guided nonlinear TM waves in planar waveguides,” Proc. Inst. Electr. Eng. 134, 152–160 (1987).

Valera, J. D.

C. T. Seaton, J. D. Valera, B. Svenson, and G. I. Stegeman, “Comparison of solutions for TM-polarized guided waves,” Opt. Lett. 10, 149–150 (1985).
[Crossref] [PubMed]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of Nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. QE-21, 774–783 (1985).
[Crossref]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. Chilwell, and S. D. Smith, “Anomalous nonlinear guided wave cut-off phenomena,” Appl. Phys. Lett. 45, 1162–1163 (1984).
[Crossref]

G. I. Stegeman, J. D. Valera, C. T. Seaton, J. E. Sipe, and A. A. Maradudin, “Nonlinear s-polarized surface plasmon polaritons,” Solid State Commun. 52, 293–297 (1984).
[Crossref]

Wallis, R. F.

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudip, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 (1985).
[Crossref]

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by metal films,” J. Appl. Phys. 58, 2460–2466 (1985).
[Crossref]

Wendler, L.

L. Wendler, “S-polarized nonlinear surface polaritons. Effects of a transition layer,” Phys. Status Solidi B 135, 759–774 (1986).
[Crossref]

Winful, H. G.

G. I. Stegeman, C. T. Seaton, and H. G. Winful, “Applications of guided waves to nonlinear optics,” Philos. Trans. R. Soc. London A 313, 321–326 (1984).
[Crossref]

Wright, E. M.

Yu, M. Y.

M. Y. Yu, “Surface polaritons in nonlinear media,” Phys. Rev. A 28, 1855–1856 (1983).
[Crossref]

Zanoni, R.

Appl. Phys. B (4)

F. Lederer, U. Langbein, and H.-E. Ponath, “Nonlinear waves guided by a dielectric slab: I. TE-polarization,” Appl. Phys. B 31, 69–73 (1983).
[Crossref]

F. Lederer, U. Langbein, and H.-E. Ponath, “Nonlinear waves guided by a dielectric slab: II. TM-polarization,” Appl. Phys. B 31, 187–190 (1983).
[Crossref]

U. Langbein, F. Lederer, H.-E. Ponath, and U. Trutschel, “Analysis of the dispersion relations of nonlinear slab-guided waves,” Appl. Phys. B 36, 187–193 (1985).
[Crossref]

U. Langbein, F. Lederer, H.-E. Ponath, and U. Trutschel, “Analysis of the dispersion relation of nonlinear slab-guided waves. Part II: Symmetrical configuration,” Appl. Phys. B 38, 263–268 (1985).
[Crossref]

Appl. Phys. Lett. (2)

G. I. Stegeman, C. T. Seaton, J. Chilwell, and S. D. Smith, “Nonlinear waves guided by thin films,” Appl. Phys. Lett. 44, 830–832 (1984).
[Crossref]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. Chilwell, and S. D. Smith, “Anomalous nonlinear guided wave cut-off phenomena,” Appl. Phys. Lett. 45, 1162–1163 (1984).
[Crossref]

Appl. Sci. Res. (1)

A. D. Boardman, P. Egan, and A. Shivarova, “TE modes of a layered nonlinear optical wave-guide,” Appl. Sci. Res. 41, 345–353 (1983).
[Crossref]

IEEE J. Quantum Electron. (3)

A. D. Boardman and P. Egan, “S-polarized waves in a thin dielectric film asymmetrically bounded by optically nonlinear media,” IEEE J. Quantum Electron. QE-21, 1701–1713 (1985).
[Crossref]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and S. D. Smith, “Calculations of Nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. QE-21, 774–783 (1985).
[Crossref]

A. D. Boardman and P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. QE-22, 319–324 (1986).
[Crossref]

Izv. Vyssh. Uchebn. Zaved. Radiofiz. (1)

A. G. Litvak and V. A. Mironov, “Surface waves at the interface of nonlinear media,” Izv. Vyssh. Uchebn. Zaved. Radiofiz. 11, 1911–1912 (1968).

J. Appl. Phys. (2)

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by metal films,” J. Appl. Phys. 58, 2460–2466 (1985).
[Crossref]

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudip, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 (1985).
[Crossref]

J. Mol. Struct. (1)

U. Langbein, F. Lederer, H.-E. Ponath, and U. Trutschel, “Dispersion relations for nonlinear guided waves,” J. Mol. Struct. 115, 493–496 (1984).
[Crossref]

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

J. Phys. (Paris) (1)

A. D. Boardman and P. Egan, “Nonlinear surface and guided polaritons of a general layered dielectric structure,” J. Phys. (Paris) C5, Suppl. 45, C5-291–C5-303 (1984).
[Crossref]

Opt. Commun. (3)

U. Langbein, F. Lederer, and H.-E. Ponath, “Generalized dispersion relations for nonlinear slab-guided waves,” Opt. Commun. 53, 417–420 (1985).
[Crossref]

D. J. Robbins, “TE modes in a slab waveguide bounded by nonlinear media,” Opt. Commun. 47, 309–312 (1983).
[Crossref]

U. Langbein, F. Lederer, and H.-E. Ponath, “A new type of nonlinear slab-guided waves,” Opt. Commun. 46, 167–169 (1983).
[Crossref]

Opt. Lett. (4)

Opt. Spektrosk. (3)

N. N. Achmediev, K. O. Bolter, and W. M. Eleonsky, “Optical dielectric waveguide with nonlinear permittivity,” Opt. Spektrosk. 53, 906–909 (1982) [Opt. Spectrosc. (USSR) 53, 540–542 (1982)].

N. N. Achmediev, K. O. Bolter, and W. M. Eleonsky, “Dielectric optical waveguide with nonlinear susceptibility. Asymmetric refractive-index profiles,” Opt. Spektrosk. 53, 1097–1103 (1982) [Opt. Spectrosc. (USSR) 53,654–658 (1982)].

A. I. Lomtev, “New class of s-polarized nonlinear surface waves,” Opt. Spektrosk. 55, 1079–1081 (1983) [Opt. Spectrosc. (USSR) 55,656–657 (1984)].

Philos. Trans. R. Soc. London A (1)

G. I. Stegeman, C. T. Seaton, and H. G. Winful, “Applications of guided waves to nonlinear optics,” Philos. Trans. R. Soc. London A 313, 321–326 (1984).
[Crossref]

Philos. Trans. R. Soc. London Ser. A (1)

A. D. Boardman and P. Egan, “Theory of optical hysteresis for TE guided modes,” Philos. Trans. R. Soc. London Ser. A 313, 363–369 (1984).
[Crossref]

Phys. Rev. A (3)

K. M. Leung, “Propagation of nonlinear surface polaritons,” Phys. Rev. A 31, 1189–1192 (1985).
[Crossref] [PubMed]

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright, “Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 (1987).
[Crossref] [PubMed]

M. Y. Yu, “Surface polaritons in nonlinear media,” Phys. Rev. A 28, 1855–1856 (1983).
[Crossref]

Phys. Rev. B (1)

K. M. Leung, “P-polarized nonlinear surface polaritons in materials with intensity-dependent dielectric functions,” Phys. Rev. B 32, 5093–5101 (1985).
[Crossref]

Phys. Scr. (2)

D. Mihalache, R. G. Nazmitdinov, and V. K. Fedyanin, “P-polarized nonlinear surface waves in symmetric layered structures,” Phys. Scr. 29, 269–275 (1984).
[Crossref]

D. Mihalache, D. Mazilu, and M. Tortia, “Bistable states of s-polarized nonlinear waves guided by an asymmetric three-layer dielectric structure,” Phys. Scr. 30, 335–340 (1984).
[Crossref]

Phys. Status Solidi B (1)

L. Wendler, “S-polarized nonlinear surface polaritons. Effects of a transition layer,” Phys. Status Solidi B 135, 759–774 (1986).
[Crossref]

Pis’ma Zh. Eksp. Teor. Fiz. (2)

A. J. Lomtev, “A new class of nonlinear surface waves,” Pis’ma Zh. Eksp. Teor. Fiz. 34, 64–67 (1981) [Sov. Phys. JETP Lett. 34,60–63 (1981)].

V. M. Agranovich, V. S. Babichenko, and V. Ya. Chernyak, “Nonlinear surface polaritons,” Pis’ma Zh. Eksp. Teor. Fiz. 32, 532–535 (1980) [Sov. Phys. JETP Lett. 32, 512–515 (1980)].

Proc. Inst. Electr. Eng. (1)

A. D. Boardman, T. Twardowski, A. Shivarova, and G. I. Stegeman, “Surface guided nonlinear TM waves in planar waveguides,” Proc. Inst. Electr. Eng. 134, 152–160 (1987).

Rev. Roum. Phys. (1)

D. Mihalache and H. Totia, “S-polarized nonlinear surface and guided waves in an asymmetric layered structure,” Rev. Roum. Phys. 29, 365–371 (1984).

Solid State Commun. (4)

F. Lederer and D. Mihalache, “An additional kind of nonlinear s-polarized surface plasmon polaritons,” Solid State Commun. 59, 151–153 (1986).
[Crossref]

D. Mihalache and D. Mazilu, “Stability and instability of nonlinear guided waves in saturable media,” Solid State Commun. 63, 215–217 (1987).
[Crossref]

V. M. Agranovich and V. Ya. Chernyak, “Perturbation theory for weakly nonlinear p-polarized surface polaritons,” Solid State Commun. 44, 1309–1311 (1982).
[Crossref]

G. I. Stegeman, J. D. Valera, C. T. Seaton, J. E. Sipe, and A. A. Maradudin, “Nonlinear s-polarized surface plasmon polaritons,” Solid State Commun. 52, 293–297 (1984).
[Crossref]

Teor. Mat. Fiz. (1)

D. Mihalache and V. K. Fedyanin, “P-polarized nonlinear surface and bounded (guided) waves in layered structures,” Teor. Mat. Fiz. 54, 443–445 (1983) [Theor. Math. Phys. 54,289–297 (1983)].

Z. Phys. B (2)

V. K. Fedyanin and D. Mihalache, “P-polarized nonlinear surface polaritons in layered structures,” Z. Phys. B 47, 167–173 (1982).
[Crossref]

A. A. Maradudin, “S-polarized nonlinear surface polaritons,” Z. Phys. B 41, 341–348 (1981).
[Crossref]

Zh. Eksp. Teor. Fiz. (6)

N. N. Akhmediev, “Nonlinear theory of surface polaritons,” Zh. Eksp. Teor. Fiz. 84, 1907–1917 (1983) [Sov. Phys. JETP 57, 1111–1116 (1983)].

A. G. Boev, “Nonlinear theory of penetration of p-polarized electromagnetic waves in a plasma,” Zh. Eksp. Teor. Fiz. 79, 134–142 (1980) [Sov. Phys. JETP 52, 67–71 (1980)].

N. N. Akhmediev, “Novel class of nonlinear surface waves: asymmetric modes in a symmetric layered structure,” Zh. Eksp. Teor. Fiz. 83, 545–553 (1982) [Sov. Phys. JETP 56, 299–303 (1982)].

V. M. Eleonskii and V. P. Silin, “Propagation of electromagnetic waves in an inhomogeneous nonlinear medium,” Zh. Eksp. Teor. Fiz. 66, 146–153 (1974) [Sov. Phys. JETP 39, 67–70 (1974)].

A. G. Boev and A. V. Prokopov, “Contribution to the nonlinear theory of surface electromagnetic waves in an ionizing plasma,” Zh. Eksp. Teor. Fiz. 69, 1208–1217 (1975) [Sov. Phys. JETP 42, 617–621 (1975)].

A. G. Boev, “On the theory of nonlinear surface waves in a plasma,” Zh. Eksp. Teor. Fiz. 77, 92–100 (1979) [Sov. Phys. JETP 50, 47–51 (1979)].

Zh. Tekh. Fiz. (1)

Yu. R. Alanakyan, “Surface waves in a plasma,” Zh. Tekh. Fiz. 37, 817–821 (1967) [Sov. Phys. Tech. Phys. 12, 587–589 (1967)].

Other (15)

A. D. Boardman and P. Egan, “Nonlinear electromagnetic surface and guided waves: theory,” in Surface Waves in Plasmas and Solids, S. Vukovic, ed. (World Scientific, Singapore, 1986), pp. 3–77.

See, for example, A. A. Maradudin, “Surface waves,” in Modern Problems of Surface Physics,” I. J. Lalov, ed. (Bulgarian Academy of Sciences, Sofia, 1981), pp. 11–399.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Formulas (Dover, New York, 1968), p. 596, formula (17.4.52).

Ref. 57, p. 596, formula (17.4.45).

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists (Springer-Verlag, Berlin, 1954), p. 145, formula (266.00).

Ref. 57, p. 596, formula (17.4.49).

Ref. 57, p. 596, formula (17.4.51).

Ref. 57, p. 596, formula (17.4.43).

Ref. 57, p. 596, formula (17.4.41).

Ref. 59, p. 141, formula (263.00).

Ref. 57, Secs. 16.4 and 17.6.

Ref. 57, Sec. 17.2.

Ref. 57, Sec. 17.5.

A good reference dealing with the numerical calculation of Jacobian elliptic functions and inverse Jacobian elliptic functions is L. V. King, On the Direct Numerical Calculation of Elliptic Functions and Integrals (Cambridge U. Press, Cambridge, 1924).

A. A. Maradudin, “Nonlinear surface electromagnetic waves,” in Optical and Acoustic Waves in Solids—Modern Topics, M. Borissov, ed. (World Scientific, Singapore, 1983), pp. 72–142.

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Figures (9)

Fig. 1
Fig. 1

neff as a function of the intensity of the electric field at the lower boundary of the nonlinear film for positive values of the nonlinear coefficient.

Fig. 2
Fig. 2

The electric field component within the nonlinear slab, E(x3), in units of E0 [=E(0)], at Point 1 in Fig. 1.

Fig. 3
Fig. 3

The same as Fig. 2 but at Point 2 in Fig. 1.

Fig. 4
Fig. 4

The same as Fig. 2 but at Point 3 in Fig. 1.

Fig. 5
Fig. 5

neff as a function of the intensity of the electric field at the lower boundary of the nonlinear film for negative values of the nonlinear coefficient.

Fig. 6
Fig. 6

The electric field component within the nonlinear slab, E(x3), in units of E0 [=E(0)], at Point 1 in Fig. 5.

Fig. 7
Fig. 7

The same as Fig. 6 but at Point 2 in Fig. 5.

Fig. 8
Fig. 8

The same as Fig. 6 but at Point 3 in Fig. 5.

Fig. 9
Fig. 9

The same as Fig. 6 but at Point 4 in Fig. 5.

Equations (93)

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E ( x ; t ) = [ 0 , E 2 ( k ω x 3 ) , 0 ] exp i ( k x 1 - ω t ) ,
H ( x ; t ) = [ H 1 ( k ω x 3 ) , 0 , H 3 ( k ω x 3 ) ] exp i ( k x 1 - ω t ) ,
H 1 ( k ω x 3 ) = - c i ω x 3 E 2 ( k ω x 3 ) ,
H 3 ( k ω x 3 ) = c k ω E 2 ( k ω x 3 ) .
22 N L = d + λ E 2 ( k ω x 3 ) 2 ,
[ d 2 d x 3 2 - α 2 ( k ω ) ] E 2 ( k ω x 3 ) = 0         x 3 > d ,
[ d 2 d x 3 2 + β 2 ( k ω ) + λ ω 2 c 2 E 2 ( k ω x 3 ) 2 ] E 2 ( k ω x 3 ) = 0             0 < x 3 < d ,
[ d 2 d x 3 2 - α 2 ( k ω ) ] E 2 ( k ω x 3 ) = 0             x 3 < 0.
α 2 ( k ω ) = k 2 - ω 2 c 2 ,             β 2 ( k ω ) = d ω 2 c 2 - k 2 .
E 2 ( k ω d + ) = E 2 ( k ω d - ) ,
E 2 ( k ω d + ) = E 2 ( k ω d - ) ,
E 2 ( k ω 0 + ) = E 2 ( k ω 0 - ) ,
E 2 ( k ω 0 + ) = E 2 ( k ω 0 - ) ,
E 2 ( k ω x 3 ) = E d exp [ - α ( k ω ) ( x 3 - d ) ]             x 3 > d ,
= E 0 exp [ α ( k ω ) x 3 ]             x 3 < 0.
E 2 ( k ω x 3 ) = E ( k ω x 3 ) exp [ i ϕ ( k ω x 3 ) ] ,
E - E ϕ 2 + β 2 E + λ ω 2 c 2 E 3 = 0 ,
2 E ϕ + E ϕ = 0 ,
ϕ = W / E 2 ,
E - W 2 E 3 + β 2 E + λ ω 2 c 2 E 3 = 0.
S = c 8 π Re ( E * × H ) = c 2 8 π ω Re ( x ^ 1 1 i E 2 * E 2 x 1 + x 3 1 i E 2 * E 2 x 3 ) ,
S 1 = c 2 k 8 π ω E 2 ( k ω x 3 ) 2 = c 2 k 8 π ω E 2 ( k ω x 3 ) ,
S 3 = c 2 8 π ω E 2 ( k ω x 3 ) d ϕ ( k ω x 3 ) d x 3 = c 2 8 π ω W .
E + β 2 E + λ ω 2 c 2 E 3 = 0.
( d E d x 3 ) 2 + β 2 E 2 + 1 2 λ ω 2 c 2 E 4 = K ,
E ( x 03 ) E ( x 3 ) d t ( K - β 2 t 2 - ½ λ ω 2 c 2 t 4 ) 1 / 2 = ± ( x 3 - x 03 ) .
α 2 E d 2 + β 2 E d 2 + 1 2 λ ω 2 c 2 E d 4 = K ,
α 2 E 0 2 + β 2 E 0 2 + 1 2 λ ω 2 c 2 E 0 4 = K .
ω 2 c 2 ( d - ) E d 2 + 1 2 λ ω 2 c 2 E d 4 = K ,
ω 2 c 2 ( d - ) E 0 2 + 1 2 λ ω 2 c 2 E 0 4 = K .
( E d 2 - E 0 2 ) [ d - + ½ λ ( E d 2 + E 0 2 ) ] = 0 ,
E d = ± E 0
d - + ½ λ ( E d 2 + E 0 2 ) = 0
E ( 0 ) E ( x 3 ) d t ( K - β 2 t 2 ) 1 / 2 = ± x 3 .
sin - 1 [ β E ( x 3 ) K ] - sin - 1 [ β E ( 0 ) K ] = β x 3
E ( x 3 ) = E ( 0 ) ( cos β x 3 + α β sin β x 3 ) ,
tan ½ β d = α β
E ( x 3 ) = E ( 0 ) cos ½ β d cos β ( x 3 - ½ d ) ,
tan ½ β d = - β α
E ( x 3 ) = - E ( 0 ) sin ½ β d sin β ( x 3 - ½ d ) ,
E ( 0 ) E ( x 3 ) d t ( a 2 + t 2 ) 1 / 2 ( b 2 - t 2 ) 1 / 2 = ± ( λ ω 2 2 c 2 ) 1 / 2 x 3 ,
a 2 = [ β 4 + ( 2 λ ω 2 / c 2 ) K ] 1 / 2 + β 2 ( λ ω 2 / c 2 ) ,
b 2 = [ β 4 + ( 2 λ ω 2 / c 2 ) K ] 1 / 2 - β 2 ( λ ω 2 / c 2 ) ,
E ( 0 ) E ( x 3 ) = E ( 0 ) b - E ( x 3 ) b ,
cn - 1 [ E ( 0 ) b | b 2 a 2 + b 2 ] - cn - 1 [ E ( x 3 ) b | b 2 a 2 + b 2 ] = [ ( a 2 + b 2 ) ( λ ω 2 2 c 2 ) ] 1 / 2 x 3 ,
E ( x 3 ) = b cn { cn - 1 [ E ( 0 ) b | b 2 a 2 + b 2 ] - [ ( a 2 + b 2 ) ( λ ω 2 2 c 2 ) ] 1 / 2 x 3 | b 2 a 2 + b 2 } .
E ( 0 ) E ( x 3 ) d t ( a 2 - t 2 ) 1 / 2 ( b 2 - t 2 ) 1 / 2 = ± ( λ ω 2 2 c 2 ) 1 / 2 x 3
a 2 = β 2 + [ β 4 - ( 2 λ ω 2 K / c 2 ) ] 1 / 2 ( λ ω 2 / c 2 ) ,
b 2 = β 2 - [ β 4 - ( 2 λ ω 2 K / c 2 ) ] 1 / 2 ( λ ω 2 / c 2 ) ,
sn - 1 [ E ( x 3 ) b | b 2 a 2 ] - sn - 1 [ E ( 0 ) b | b 2 a 2 ] = a ( λ ω 2 2 c 2 ) 1 / 2 x 3
E ( x 3 ) = b sn { a ( λ ω 2 2 c 2 ) 1 / 2 x 3 + sn - 1 [ E ( 0 ) b | b 2 a 2 ] | b 2 a 2 } .
E ( 0 ) E ( x 3 ) d t ( t 4 - b t 2 + a 4 ) 1 / 2 = ( λ ω 2 2 c 2 ) 1 / 2 x 3 ,
a 4 = 2 c 2 K λ ω 2 ,             b = 2 c 2 β 2 λ ω 2 .
E ( 0 ) E ( x 3 ) = E ( 0 ) - E ( x 3 ) ,
cn - 1 [ E 2 ( 0 ) - a 2 E 2 ( 0 ) + a 2 | ( 2 a 2 + b 4 a 2 ) ] - cn - 1 [ E 2 ( x 3 ) - a 2 E 2 ( x 3 ) + a 2 | ( 2 a 2 + b 4 a 2 ) ] = 2 a ( λ ω 2 2 c 2 ) 1 / 2 x 3 ,
E 2 ( x 3 ) = a 2 1 + cn { } 1 - cn { } ,
cn { } = cn { cn - 1 [ E 2 ( 0 ) - a 2 E 2 ( 0 ) + a 2 | 2 a 2 + b 4 a 2 ] - 2 ( λ ω 2 2 c 2 ) 1 / 2 x 3 | 2 a 2 + b 4 a 2 } .
E ( 0 ) E ( x 3 ) d t ( t 2 + a 2 ) 1 / 2 ( t 2 - b 2 ) 1 / 2 = ± ( λ ω 2 2 c 2 ) 1 / 2 x 3 ,
a 2 = [ β 4 + ( 2 λ ω 2 K / c 2 ) 1 / 2 - β 2 ( λ ω 2 / c 2 ) ,
b 2 = [ β 4 + ( 2 λ ω 2 K / c 2 ) ] 1 / 2 + β 2 ( λ ω 2 / c 2 ) ,
E ( 0 ) E ( x 3 ) = b E ( x 3 ) - b E ( 0 ) ,
nc - 1 [ E ( x 3 ) b | a 2 a 2 + b 2 ] - nc - 1 [ E ( 0 ) b | a 2 a 2 + b 2 ] = [ ( a 2 + b 2 ) ( λ ω 2 2 c 2 ) ] 1 / 2 x 3 .
E ( x 3 ) = b nc { [ ( a 2 + b 2 ) ( λ ω 2 2 c 2 ) ] 1 / 2 x 3 + nc - 1 [ E ( 0 ) b | a 2 a 2 + b 2 ] | a 2 a 2 + b 2 } .
β 2 ( k ω ) - k 2 = d ω 2 c 2
E ( 0 ) E ( x 3 ) d t ( K + β 2 t 2 - ½ λ ω 2 c 2 t 4 ) 1 / 2 = ± x 3 .
α 2 E d 2 - β 2 E d 2 + ½ λ ω 2 c 2 E d 4 = K ,
α 2 E 0 2 - β 2 E 0 2 + ½ λ ω 2 c 2 E 0 4 = K ,
E ( 0 ) E ( x 3 ) d t ( a 2 + t 2 ) 1 / 2 ( b 2 - t 2 ) 1 / 2 = ± ( λ ω 2 2 c 2 ) 1 / 2 x 3 ,
a 2 = [ β 4 + ( 2 λ ω 2 K / c 2 ) ] 1 / 2 - β 2 ( λ ω 2 / c 2 ) ,
b 2 = [ β 4 + ( 2 λ ω 2 K / c 2 ) ] 1 / 2 + β 2 ( λ ω 2 / c 2 ) ,
E ( 0 ) E ( x 3 ) = 0 E ( x 3 ) - 0 E ( 0 ) ,
sd - 1 [ E ( x 3 ) ( a 2 + b 2 ) 1 / 2 a b | b 2 a 2 + b 2 ] - sd - 1 [ E ( 0 ) ( a 2 + b 2 ) 1 / 2 a b | b 2 a 2 + b 2 ] = [ ( a 2 + b 2 ) λ ω 2 c 2 ] 1 / 2 x 3 .
E ( x 3 ) = a b ( a 2 + b 2 ) 1 / 2 sd { [ ( a 2 + b 2 ) λ ω 2 c 2 ] 1 / 2 x 3 + sd - 1 [ E ( 0 ) ( a 2 + b 2 ) 1 / 2 a b | b 2 a 2 + b 2 ] | b 2 a 2 + b 2 } .
E ( 0 ) E ( x 3 ) d t ( a 2 - t 2 ) 1 / 2 ( t 2 - b 2 ) 1 / 2 = ± ( λ ω 2 2 c 2 ) 1 / 2 x 3 ,
a 2 = β 2 + [ β 4 - ( 2 λ ω 2 K / c 2 ) ] 1 / 2 ( λ ω 2 / c 2 ) ,
b 2 = β 2 - [ β 4 - ( 2 λ ω 2 K / c 2 ) ] 1 / 2 ( λ ω 2 / c 2 )
nd - 1 [ E ( x 3 ) b | a 2 - b 2 a 2 ] - nd - 1 [ E ( 0 ) b | a 2 - b 2 a 2 ] = a ( λ ω 2 2 c 2 ) 1 / 2 x 3 .
E ( x 3 ) = b nd { a ( λ ω 2 2 c 2 ) 1 / 2 x 3 + nd - 1 [ E ( 0 ) b | a 2 - b 2 a 2 ] | a 2 - b 2 a 2 } .
E ( 0 ) E ( x 3 ) d t ( a 2 + t 2 ) 1 / 2 ( b 2 + t 2 ) 1 / 2 = ± ( λ ω 2 2 c 2 ) 1 / 2 x 3
a 2 = β 2 + [ β 4 - ( 2 λ ω 2 K / c 2 ) ] 1 / 2 ( λ ω 2 / c 2 ) ,
b 2 = β 2 - [ β 4 - ( 2 λ ω 2 K / c 2 ) ] 1 / 2 ( λ ω 2 / c 2 ) ,
sc - 1 [ E ( x 3 ) b | a 2 - b 2 a 2 ] - sc - 1 [ E ( 0 ) b | a 2 - b 2 a 2 ] = a ( λ ω 2 2 c 2 ) 1 / 2 x 3 .
E ( x 3 ) = b sc { a ( λ ω 2 2 c 2 ) 1 / 2 x 3 + sc - 1 [ E ( 0 ) b | a 2 - b 2 a 2 ] | a 2 - b 2 a 2 } .
E ( 0 ) E ( x 3 ) d t ( t 4 + 2 b 2 t 2 + a 4 ) 1 / 2 = ( λ ω 2 2 c 2 ) 1 / 2 x 3 ,
a 4 = 2 c 2 K λ ω 2 ,             b 2 = c 2 β 2 λ ω 2 .
cn - 1 [ E 2 ( 0 ) - a 2 E 2 ( 0 ) + a 2 | a 2 - b 2 2 a 2 ] - cn - 1 [ E 2 ( x 3 ) - a 2 E 2 ( x 3 ) + a 2 × | a 2 - b 2 2 a 2 ] = 2 a ( λ ω 2 2 c 2 ) 1 / 2 x 3 ,
E 2 ( x 3 ) = a 2 1 + cn { } 1 - cn { } ,
cn { } = cn { cn - 1 [ E 2 ( 0 ) - a 2 E 2 ( 0 ) + a 2 | a 2 - b 2 2 a 2 ] - 2 a ( λ ω 2 2 c 2 ) 1 / 2 x 3 | a 2 - b 2 2 a 2 } .
E ( 0 ) E ( x 3 ) d t ( a 2 + t 2 ) 1 / 2 ( t 2 - b 2 ) 1 / 2 = ± ( λ ω 2 2 c 2 ) 1 / 2 x 3 ,
a 2 = [ β 4 + ( 2 λ ω 2 K / c 2 ) ] 1 / 2 + β 2 ( λ ω 2 / c 2 ) ,
b 2 = [ β 4 + ( 2 λ ω 2 K / c 2 ) ] 1 / 2 - β 2 ( λ ω 2 / c 2 ) ,
nc - 1 [ E ( x 3 ) b | a 2 a 2 + b 2 ] - nc - 1 [ E ( 0 ) b | a 2 a 2 + b 2 ] = [ ( a 2 + b 2 ) ( λ ω 2 2 c 2 ) ] 1 / 2 x 3 ,
E ( x 3 ) = b nc { [ ( a 2 + b 2 ) ( λ ω 2 2 c 2 ) ] 1 / 2 x 3 + nc - 1 [ E ( 0 ) b | a 2 a 2 + b 2 ] | a 2 a 2 + b 2 } .

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