Abstract

We present analytic solutions obtained for the case of a stationary TE fundamental dark surface wave propagating at the interface between two dissimilar defocusing Kerr media. Expressions describing the wave are obtained, and certain characteristics of the solutions are discussed.

© 1991 Optical Society of America

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References

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  1. For a recent survey of this field see, e.g., Integrated Photonics Research, Vol. 5 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), and references therein.
  2. W. J. Tomlinson, “Surface waves at a nonlinear interface,” Opt. Lett. 5, 323–325 (1980).
    [Crossref] [PubMed]
  3. A. A. Maradudin, “s-polarized nonlinear surface polaritons,” Z. Phys. B 41, 341–344 (1981).
    [Crossref]
  4. A. D. Boardman and P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. QE-22, 319–324 (1986).
    [Crossref]
  5. 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); 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]
  6. A. D. Boardman and T. Twardowski, “Transverse-electric and transverse-magnetic waves in nonlinear isotropic waveguides,” Phys. Rev. A 39, 2481–2492 (1989).
    [Crossref] [PubMed]
  7. A. B. Aceves, P. Varatharajah, A. C. Newell, E. M. Wright, G. I. Stegeman, D. R. Heatley, J. V. Moloney, and H. Adachihara, “Particle aspects of collimated light channel propagation at nonlinear interfaces and in waveguides,” J. Opt. Soc. Am. B 7, 963–974 (1990).
    [Crossref]
  8. A. E. Kaplan, “Hysteresis reflection and refraction of light by a nonlinear boundary—a new class of effects in nonlinear optics,” JETP Lett. 24, 114–119 (1976); A. E. Kaplan, “Theory of hysteresis reflection and refraction of light by a boundary of a nonlinear medium,” Sov. Phys. JETP 45, 896–905 (1977).
  9. W. J. Tomlinson, J. P. Gordon, P. W. Smith, and A. E. Kaplan, “Reflection of a Gaussian beam at a nonlinear interface,” Appl. Opt. 21, 2041–2051 (1982).
    [Crossref] [PubMed]
  10. D. R. Andersen, “Surface wave excitation at the interface between Kerr-like nonlinear and linear media,” Phys. Rev. A 37, 189–193 (1988).
    [Crossref] [PubMed]
  11. D. R. Andersen and J. J. Regan, “Reflection and refraction of a three-dimensional Gaussian beam at a nonlinear interface,” J. Opt. Soc. Am. A 6, 1484–1492 (1989).
    [Crossref]
  12. A. Barthelemey, S. Maneuf, and C. Froehly, “Propagation soliton et auto-confinement de faisceaux laser par non linearite optique de Kerr,” Opt. Commun. 55, 201–206 (1985).
    [Crossref]
  13. J. S. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. M. Vogel, and P. W. E. Smith, “Observation of spatial optical solitons in a nonlinear glass waveguide,” Opt. Lett. 15, 471–473 (1990).
    [Crossref] [PubMed]
  14. P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1007 (1964).
    [Crossref]
  15. R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 15, 479–482 (1964).
    [Crossref]
  16. V. E. Zakharov and A. B. Shabat, “Interaction between solitons in a stable medium,” So. Phys. JETP 37, 823–828 (1973).
  17. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers, II. Normal dispersion,” Appl. Phys. Lett. 23, 171–172 (1973).
    [Crossref]
  18. G. A. Swartzlander, D. R. Andersen, J. J. Regan, and A. E. Kaplan, “Observation of spatial dark waves and solitons,” in OSA Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 272.
  19. D. R. Andersen, D. E. Hooton, G. A. Swartzlander, and A. E. Kaplan, “Direct measurement of the transverse velocity of dark spatial solitons,” Opt. Lett. 15, 783–785 (1990).
    [Crossref] [PubMed]
  20. G. R. Allan, S. R. Skinner, D. R. Andersen, and A. L. Smirl, “Observation of fundamental dark spatial solitons in semiconductors using picosecond pulses,” Opt. Lett. 16, 156–158 (1991).
    [PubMed]
  21. M. Sheik-bahae, D. J. Hagan, and E. W. VanStryland, “Dispersion and bandgap scaling of the electronic Kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65, 96–99 (1990).
    [Crossref] [PubMed]
  22. A. E. Kaplan, “Conditions of excitation of new waves (LITW) at nonlinear interface and diagram of wave states of the system,” IEEE J. Quantum Electron. QE-17, 336–340 (1981).
    [Crossref]

1991 (1)

1990 (4)

1989 (2)

D. R. Andersen and J. J. Regan, “Reflection and refraction of a three-dimensional Gaussian beam at a nonlinear interface,” J. Opt. Soc. Am. A 6, 1484–1492 (1989).
[Crossref]

A. D. Boardman and T. Twardowski, “Transverse-electric and transverse-magnetic waves in nonlinear isotropic waveguides,” Phys. Rev. A 39, 2481–2492 (1989).
[Crossref] [PubMed]

1988 (1)

D. R. Andersen, “Surface wave excitation at the interface between Kerr-like nonlinear and linear media,” Phys. Rev. A 37, 189–193 (1988).
[Crossref] [PubMed]

1987 (1)

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); 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]

1986 (1)

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

1985 (1)

A. Barthelemey, S. Maneuf, and C. Froehly, “Propagation soliton et auto-confinement de faisceaux laser par non linearite optique de Kerr,” Opt. Commun. 55, 201–206 (1985).
[Crossref]

1982 (1)

1981 (2)

A. A. Maradudin, “s-polarized nonlinear surface polaritons,” Z. Phys. B 41, 341–344 (1981).
[Crossref]

A. E. Kaplan, “Conditions of excitation of new waves (LITW) at nonlinear interface and diagram of wave states of the system,” IEEE J. Quantum Electron. QE-17, 336–340 (1981).
[Crossref]

1980 (1)

1976 (1)

A. E. Kaplan, “Hysteresis reflection and refraction of light by a nonlinear boundary—a new class of effects in nonlinear optics,” JETP Lett. 24, 114–119 (1976); A. E. Kaplan, “Theory of hysteresis reflection and refraction of light by a boundary of a nonlinear medium,” Sov. Phys. JETP 45, 896–905 (1977).

1973 (2)

V. E. Zakharov and A. B. Shabat, “Interaction between solitons in a stable medium,” So. Phys. JETP 37, 823–828 (1973).

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers, II. Normal dispersion,” Appl. Phys. Lett. 23, 171–172 (1973).
[Crossref]

1964 (2)

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1007 (1964).
[Crossref]

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 15, 479–482 (1964).
[Crossref]

Aceves, A. B.

Adachihara, H.

Aitchison, J. S.

Allan, G. R.

Andersen, D. R.

Barthelemey, A.

A. Barthelemey, S. Maneuf, and C. Froehly, “Propagation soliton et auto-confinement de faisceaux laser par non linearite optique de Kerr,” Opt. Commun. 55, 201–206 (1985).
[Crossref]

Boardman, A. D.

A. D. Boardman and T. Twardowski, “Transverse-electric and transverse-magnetic waves in nonlinear isotropic waveguides,” Phys. Rev. A 39, 2481–2492 (1989).
[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); 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 and P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. QE-22, 319–324 (1986).
[Crossref]

Chiao, R. Y.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 15, 479–482 (1964).
[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]

Froehly, C.

A. Barthelemey, S. Maneuf, and C. Froehly, “Propagation soliton et auto-confinement de faisceaux laser par non linearite optique de Kerr,” Opt. Commun. 55, 201–206 (1985).
[Crossref]

Garmire, E.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 15, 479–482 (1964).
[Crossref]

Gordon, J. P.

Hagan, D. J.

M. Sheik-bahae, D. J. Hagan, and E. W. VanStryland, “Dispersion and bandgap scaling of the electronic Kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65, 96–99 (1990).
[Crossref] [PubMed]

Hasegawa, A.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers, II. Normal dispersion,” Appl. Phys. Lett. 23, 171–172 (1973).
[Crossref]

Heatley, D. R.

Hooton, D. E.

Jackel, J. L.

Kaplan, A. E.

D. R. Andersen, D. E. Hooton, G. A. Swartzlander, and A. E. Kaplan, “Direct measurement of the transverse velocity of dark spatial solitons,” Opt. Lett. 15, 783–785 (1990).
[Crossref] [PubMed]

W. J. Tomlinson, J. P. Gordon, P. W. Smith, and A. E. Kaplan, “Reflection of a Gaussian beam at a nonlinear interface,” Appl. Opt. 21, 2041–2051 (1982).
[Crossref] [PubMed]

A. E. Kaplan, “Conditions of excitation of new waves (LITW) at nonlinear interface and diagram of wave states of the system,” IEEE J. Quantum Electron. QE-17, 336–340 (1981).
[Crossref]

A. E. Kaplan, “Hysteresis reflection and refraction of light by a nonlinear boundary—a new class of effects in nonlinear optics,” JETP Lett. 24, 114–119 (1976); A. E. Kaplan, “Theory of hysteresis reflection and refraction of light by a boundary of a nonlinear medium,” Sov. Phys. JETP 45, 896–905 (1977).

G. A. Swartzlander, D. R. Andersen, J. J. Regan, and A. E. Kaplan, “Observation of spatial dark waves and solitons,” in OSA Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 272.

Kelley, P. L.

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1007 (1964).
[Crossref]

Leaird, D. E.

Maneuf, S.

A. Barthelemey, S. Maneuf, and C. Froehly, “Propagation soliton et auto-confinement de faisceaux laser par non linearite optique de Kerr,” Opt. Commun. 55, 201–206 (1985).
[Crossref]

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); 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. A. Maradudin, “s-polarized nonlinear surface polaritons,” Z. Phys. B 41, 341–344 (1981).
[Crossref]

Moloney, J. V.

Newell, A. C.

Oliver, M. K.

Regan, J. J.

D. R. Andersen and J. J. Regan, “Reflection and refraction of a three-dimensional Gaussian beam at a nonlinear interface,” J. Opt. Soc. Am. A 6, 1484–1492 (1989).
[Crossref]

G. A. Swartzlander, D. R. Andersen, J. J. Regan, and A. E. Kaplan, “Observation of spatial dark waves and solitons,” in OSA Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 272.

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Interaction between solitons in a stable medium,” So. Phys. JETP 37, 823–828 (1973).

Sheik-bahae, M.

M. Sheik-bahae, D. J. Hagan, and E. W. VanStryland, “Dispersion and bandgap scaling of the electronic Kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65, 96–99 (1990).
[Crossref] [PubMed]

Silberberg, Y.

Skinner, S. R.

Smirl, A. L.

Smith, P. W.

Smith, P. W. E.

Stegeman, G. I.

A. B. Aceves, P. Varatharajah, A. C. Newell, E. M. Wright, G. I. Stegeman, D. R. Heatley, J. V. Moloney, and H. Adachihara, “Particle aspects of collimated light channel propagation at nonlinear interfaces and in waveguides,” J. Opt. Soc. Am. B 7, 963–974 (1990).
[Crossref]

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); 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]

Swartzlander, G. A.

D. R. Andersen, D. E. Hooton, G. A. Swartzlander, and A. E. Kaplan, “Direct measurement of the transverse velocity of dark spatial solitons,” Opt. Lett. 15, 783–785 (1990).
[Crossref] [PubMed]

G. A. Swartzlander, D. R. Andersen, J. J. Regan, and A. E. Kaplan, “Observation of spatial dark waves and solitons,” in OSA Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 272.

Tappert, F.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers, II. Normal dispersion,” Appl. Phys. Lett. 23, 171–172 (1973).
[Crossref]

Tomlinson, W. J.

Townes, C. H.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 15, 479–482 (1964).
[Crossref]

Twardowski, T.

A. D. Boardman and T. Twardowski, “Transverse-electric and transverse-magnetic waves in nonlinear isotropic waveguides,” Phys. Rev. A 39, 2481–2492 (1989).
[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); 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]

VanStryland, E. W.

M. Sheik-bahae, D. J. Hagan, and E. W. VanStryland, “Dispersion and bandgap scaling of the electronic Kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65, 96–99 (1990).
[Crossref] [PubMed]

Varatharajah, P.

Vogel, E. M.

Weiner, A. M.

Wright, E. M.

A. B. Aceves, P. Varatharajah, A. C. Newell, E. M. Wright, G. I. Stegeman, D. R. Heatley, J. V. Moloney, and H. Adachihara, “Particle aspects of collimated light channel propagation at nonlinear interfaces and in waveguides,” J. Opt. Soc. Am. B 7, 963–974 (1990).
[Crossref]

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); 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]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, “Interaction between solitons in a stable medium,” So. Phys. JETP 37, 823–828 (1973).

Appl. Opt. (1)

Appl. Phys. Lett. (1)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers, II. Normal dispersion,” Appl. Phys. Lett. 23, 171–172 (1973).
[Crossref]

IEEE J. Quantum Electron. (2)

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

A. E. Kaplan, “Conditions of excitation of new waves (LITW) at nonlinear interface and diagram of wave states of the system,” IEEE J. Quantum Electron. QE-17, 336–340 (1981).
[Crossref]

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

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

JETP Lett. (1)

A. E. Kaplan, “Hysteresis reflection and refraction of light by a nonlinear boundary—a new class of effects in nonlinear optics,” JETP Lett. 24, 114–119 (1976); A. E. Kaplan, “Theory of hysteresis reflection and refraction of light by a boundary of a nonlinear medium,” Sov. Phys. JETP 45, 896–905 (1977).

Opt. Commun. (1)

A. Barthelemey, S. Maneuf, and C. Froehly, “Propagation soliton et auto-confinement de faisceaux laser par non linearite optique de Kerr,” Opt. Commun. 55, 201–206 (1985).
[Crossref]

Opt. Lett. (4)

Phys. Rev. A (3)

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); 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 and T. Twardowski, “Transverse-electric and transverse-magnetic waves in nonlinear isotropic waveguides,” Phys. Rev. A 39, 2481–2492 (1989).
[Crossref] [PubMed]

D. R. Andersen, “Surface wave excitation at the interface between Kerr-like nonlinear and linear media,” Phys. Rev. A 37, 189–193 (1988).
[Crossref] [PubMed]

Phys. Rev. Lett. (3)

M. Sheik-bahae, D. J. Hagan, and E. W. VanStryland, “Dispersion and bandgap scaling of the electronic Kerr effect in solids associated with two-photon absorption,” Phys. Rev. Lett. 65, 96–99 (1990).
[Crossref] [PubMed]

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1007 (1964).
[Crossref]

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 15, 479–482 (1964).
[Crossref]

So. Phys. JETP (1)

V. E. Zakharov and A. B. Shabat, “Interaction between solitons in a stable medium,” So. Phys. JETP 37, 823–828 (1973).

Z. Phys. B (1)

A. A. Maradudin, “s-polarized nonlinear surface polaritons,” Z. Phys. B 41, 341–344 (1981).
[Crossref]

Other (2)

G. A. Swartzlander, D. R. Andersen, J. J. Regan, and A. E. Kaplan, “Observation of spatial dark waves and solitons,” in OSA Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 272.

For a recent survey of this field see, e.g., Integrated Photonics Research, Vol. 5 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), and references therein.

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Figures (2)

Fig. 1
Fig. 1

Nature of the intensity profile for the FDSW solution as E2 is varied from near the minimum value to near the maximum value for (top) a nonlinear–nonlinear interface and (bottom) a linear–nonlinear interface.

Fig. 2
Fig. 2

Nature of the intensity profile for the FDSW solution as A is varied from near the minimum value to near the maximum value for (top) a nonlinear–nonlinear interface and (bottom) a linear–nonlinear interface.

Equations (30)

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2 E + ɛ ω 2 c 2 E = 0.
ɛ ( x , y , z ) = ɛ 0 - ɛ 1 E ( x , y , z ) 2 ( x < 0 ) , ɛ ( x , y , z ) = ɛ 0 + Δ ɛ - ɛ 2 E ( x , y , z ) 2 ( x > 0 ) .
E ( x , y , z ) = E 1 tanh [ k 1 ( x - x 1 ) ] exp ( i k z z ) ( x < 0 ) , E ( x , y , z ) = E 2 tanh [ k 2 ( x - x 2 ) ] exp ( i k z z ) ( x > 0 ) .
- k z 2 - 2 k 1 2 + k 0 2 + ( 2 k 1 2 E 1 2 - k 0 2 ɛ 1 ɛ 0 ) E 2 = 0 ,
k z 2 = k 0 2 - 2 k 1 2
k 1 2 = k 0 2 E 1 2 ɛ 1 2 Δ ɛ Ψ c 2 ,
- k 2 2 - 2 k 2 2 + k 0 2 + k 0 2 Ψ c 2 + ( 2 k 2 2 E 2 2 - k 0 2 ɛ 2 ɛ 0 ) E 2 = 0.
k z 2 = k 0 2 ( 1 + Ψ c 2 ) - 2 k 2 2
k 2 2 = k 0 2 Ψ c 2 E 2 2 ɛ 2 2 Δ ɛ .
k 1 2 = k 2 2 - k 0 2 Ψ c 2 2 .
ɛ 1 E 1 2 = ɛ 2 E 2 2 - Δ ɛ .
E 1 tanh ( k 1 x 1 ) = E 2 tanh ( k 2 x 2 )
k 1 E 1 sech 2 ( k 1 x 1 ) = k 2 E 2 sech 2 ( k 2 x 2 ) .
E 2 2 = 2 Δ ɛ ɛ 2 ( D + 1 2 ) .
k 2 = k 0 Ψ c ( D + 1 2 ) 1 / 2 ,
k 1 = k 0 Ψ c D 1 / 2 ,
k z = k 0 Ψ c ( Ψ c - 2 - 2 D ) 1 / 2 ,
E 1 2 = 2 Δ ɛ D ɛ 1 .
tanh ( k 1 x 1 ) = ± ( k 1 E 1 - k 2 E 2 k 1 E 1 - k 2 E 1 2 E 2 ) 1 / 2 = ± [ D + 0.5 D ( ɛ 1 ɛ 2 ) 1 / 2 - 1 ( ɛ 2 ɛ 1 ) 1 / 2 - 1 ] 1 / 2 ± β 1
tanh ( k 2 x 2 ) = ± ( k 2 E 2 - k 1 E 1 k 2 E 2 - k 1 E 2 2 E 1 ) 1 / 2 = ± [ D D + 0.5 ( ɛ 2 ɛ 1 ) 1 / 2 - 1 ( ɛ 1 ɛ 2 ) 1 / 2 - 1 ] 1 / 2 ± β 2 .
ɛ 2 ɛ 1 > 1 ,             D < 1 2 [ ( ɛ 2 ɛ 1 ) 1 / 2 - 1 ] - 1 ,             D > 1 2 ( ɛ 2 ɛ 1 - 1 ) - 1 .
x 1 = ± 1 2 k 1 ln ( 1 + β 1 1 - β 1 )
x 2 = ± 1 2 k 2 ln ( 1 + β 2 1 - β 2 ) .
E ( x , y , z ) = E 3 exp ( k 3 x ) exp ( i k z z )             ( x < 0 ) .
k 3 = k 0 Ψ c ( - 2 D ) 1 / 2 .
E 3 2 = Δ ɛ ɛ 2 { 1 - 2 [ - D ( D + 1 ) ] 1 / 2 }
x 2 = - 1 2 k 2 ln ( E 2 + E 3 E 2 - E 3 ) .
ɛ 2 ɛ 1 > Δ ɛ ɛ 0 + 1.
E 2 min 2 = E 1 min 2 = Δ ɛ ɛ 2 - ɛ 1             ( i . e . , D = D min )
E 2 max 2 = ( ɛ 1 ɛ 2 ) 1 / 2 E 1 max 2 = Δ ɛ ɛ 2 - ( ɛ 2 ɛ 1 ) 1 / 2             ( i . e . , D = D max ) .

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