Abstract

Propagation of light waves in a thin, self-defocusing, nonlinear film bounded by an infinite self-defocusing medium of a different nonlinearity is investigated. It is shown that both gray and dark solitary waves can be trapped in the film when the linear refractive index of the film, nf, is smaller than that of the cladding, no, and the wave’s effective index, ne, is nf < ne < no. On the other hand, a series of trapped dark oscillatory solitary waves results when ne < min{nf, no}.

© 1992 Optical Society of America

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References

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  1. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  2. R. Y. Chiao, E. Garmire, C. H. Townes, “Self trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
    [CrossRef]
  3. V. E. Zakharov, A. B. Shabat, “Interaction between solitons in a stable medium,” Sov. Phys. JETP 37, 823–828 (1973).
  4. G. I. Stegeman, C. T. Seaton, “Nonlinear integrated optics,” J. Appl. Phys. 58, R57–R78 (1985).
    [CrossRef]
  5. A. W. Snyder, Y. Chen, L. Poladian, D. J. Mitchell, “Fundamental mode of highly nonlinear fibers,” Electron. Lett. 15, 535–537 (1990).
  6. A. Barthelemy, S. Maneuf, C. Froehly, “Propagation solution et autoconfinement de faisceaux laser par non linearite optique de Kerr,” Opt. Commun. 55, 193–206 (1985);S. Maneuf, R. Desailly, C. Froehly, “Stable self-trapping of laser beams: observation in a nonlinear planar waveguide,” Opt. Commun. 65, 193–198 (1988);S. Maneuf, F. Reyneud, “Quasi-steady state self-trapping of first, second and third order subnanosecond soliton beams,” Opt. Commun. 66, 325–328 (1988).
    [CrossRef]
  7. J. S. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. M. Vogel, P. W. Smith, “Observation of spatial optical solitons in a nonlinear planar waveguide,” Opt. Lett. 15, 471–473 (1990).
    [CrossRef] [PubMed]
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  10. D. R. Andersen, S. R. Skinner, “Stationary fundamental dark surface waves,” J. Opt. Soc. Am. B 8, 759–764 (1991).
    [CrossRef]
  11. D. R. Andersen, S. R. Skinner, “Stability analysis of the fundamental dark surface wave,” J. Opt. Soc. Am. B 8, 2265–2268 (1991).
    [CrossRef]
  12. N. N. Akhmediev, “Novel class of nonlinear surface waves: asymmetric modes in a symmetric layered structure,” Sov. Phys. JETP 56, 299–303 (1982).
  13. C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, 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);G. I. Stegeman, C. T. Seaton, “Nonlinear waves guided by thin films,” Appl. Phys. Lett. 44, 830–832 (1984);H. Vach, C. T. Seaton, G. I. Stegeman, I. C. Khoo, “Observation of intensity-dependent guided waves,” Opt. Lett. 9, 238–240 (1984).
    [CrossRef] [PubMed]
  14. A. D. Boardman, 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]
  15. P. F. Byrd, M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer-Verlag, Berlin, 1971).
    [CrossRef]
  16. Y. Chen, “Vector dark spatial solitons,” Electron. Lett. 27, 1346–1348 (1991).
    [CrossRef]

1991 (4)

1990 (2)

1985 (4)

A. Barthelemy, S. Maneuf, C. Froehly, “Propagation solution et autoconfinement de faisceaux laser par non linearite optique de Kerr,” Opt. Commun. 55, 193–206 (1985);S. Maneuf, R. Desailly, C. Froehly, “Stable self-trapping of laser beams: observation in a nonlinear planar waveguide,” Opt. Commun. 65, 193–198 (1988);S. Maneuf, F. Reyneud, “Quasi-steady state self-trapping of first, second and third order subnanosecond soliton beams,” Opt. Commun. 66, 325–328 (1988).
[CrossRef]

G. I. Stegeman, C. T. Seaton, “Nonlinear integrated optics,” J. Appl. Phys. 58, R57–R78 (1985).
[CrossRef]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, 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);G. I. Stegeman, C. T. Seaton, “Nonlinear waves guided by thin films,” Appl. Phys. Lett. 44, 830–832 (1984);H. Vach, C. T. Seaton, G. I. Stegeman, I. C. Khoo, “Observation of intensity-dependent guided waves,” Opt. Lett. 9, 238–240 (1984).
[CrossRef] [PubMed]

A. D. Boardman, 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]

1982 (1)

N. N. Akhmediev, “Novel class of nonlinear surface waves: asymmetric modes in a symmetric layered structure,” Sov. Phys. JETP 56, 299–303 (1982).

1980 (1)

1973 (1)

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

1964 (1)

R. Y. Chiao, E. Garmire, C. H. Townes, “Self trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

Aitchison, J. S.

Akhmediev, N. N.

N. N. Akhmediev, “Novel class of nonlinear surface waves: asymmetric modes in a symmetric layered structure,” Sov. Phys. JETP 56, 299–303 (1982).

Allan, G. R.

Andersen, D. R.

Barthelemy, A.

A. Barthelemy, S. Maneuf, C. Froehly, “Propagation solution et autoconfinement de faisceaux laser par non linearite optique de Kerr,” Opt. Commun. 55, 193–206 (1985);S. Maneuf, R. Desailly, C. Froehly, “Stable self-trapping of laser beams: observation in a nonlinear planar waveguide,” Opt. Commun. 65, 193–198 (1988);S. Maneuf, F. Reyneud, “Quasi-steady state self-trapping of first, second and third order subnanosecond soliton beams,” Opt. Commun. 66, 325–328 (1988).
[CrossRef]

Boardman, A. D.

A. D. Boardman, 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]

Byrd, P. F.

P. F. Byrd, M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer-Verlag, Berlin, 1971).
[CrossRef]

Chen, Y.

Y. Chen, “Vector dark spatial solitons,” Electron. Lett. 27, 1346–1348 (1991).
[CrossRef]

A. W. Snyder, Y. Chen, L. Poladian, D. J. Mitchell, “Fundamental mode of highly nonlinear fibers,” Electron. Lett. 15, 535–537 (1990).

Chiao, R. Y.

R. Y. Chiao, E. Garmire, C. H. Townes, “Self trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

Chilwell, J. T.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, 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);G. I. Stegeman, C. T. Seaton, “Nonlinear waves guided by thin films,” Appl. Phys. Lett. 44, 830–832 (1984);H. Vach, C. T. Seaton, G. I. Stegeman, I. C. Khoo, “Observation of intensity-dependent guided waves,” Opt. Lett. 9, 238–240 (1984).
[CrossRef] [PubMed]

Egan, P.

A. D. Boardman, 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]

Friedman, M. D.

P. F. Byrd, M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer-Verlag, Berlin, 1971).
[CrossRef]

Froehly, C.

A. Barthelemy, S. Maneuf, C. Froehly, “Propagation solution et autoconfinement de faisceaux laser par non linearite optique de Kerr,” Opt. Commun. 55, 193–206 (1985);S. Maneuf, R. Desailly, C. Froehly, “Stable self-trapping of laser beams: observation in a nonlinear planar waveguide,” Opt. Commun. 65, 193–198 (1988);S. Maneuf, F. Reyneud, “Quasi-steady state self-trapping of first, second and third order subnanosecond soliton beams,” Opt. Commun. 66, 325–328 (1988).
[CrossRef]

Garmire, E.

R. Y. Chiao, E. Garmire, C. H. Townes, “Self trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

Jackel, J. L.

Leaird, D. E.

Love, J. D.

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

Maneuf, S.

A. Barthelemy, S. Maneuf, C. Froehly, “Propagation solution et autoconfinement de faisceaux laser par non linearite optique de Kerr,” Opt. Commun. 55, 193–206 (1985);S. Maneuf, R. Desailly, C. Froehly, “Stable self-trapping of laser beams: observation in a nonlinear planar waveguide,” Opt. Commun. 65, 193–198 (1988);S. Maneuf, F. Reyneud, “Quasi-steady state self-trapping of first, second and third order subnanosecond soliton beams,” Opt. Commun. 66, 325–328 (1988).
[CrossRef]

Mitchell, D. J.

A. W. Snyder, Y. Chen, L. Poladian, D. J. Mitchell, “Fundamental mode of highly nonlinear fibers,” Electron. Lett. 15, 535–537 (1990).

Oliver, M. K.

Poladian, L.

A. W. Snyder, Y. Chen, L. Poladian, D. J. Mitchell, “Fundamental mode of highly nonlinear fibers,” Electron. Lett. 15, 535–537 (1990).

Seaton, C. T.

G. I. Stegeman, C. T. Seaton, “Nonlinear integrated optics,” J. Appl. Phys. 58, R57–R78 (1985).
[CrossRef]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, 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);G. I. Stegeman, C. T. Seaton, “Nonlinear waves guided by thin films,” Appl. Phys. Lett. 44, 830–832 (1984);H. Vach, C. T. Seaton, G. I. Stegeman, I. C. Khoo, “Observation of intensity-dependent guided waves,” Opt. Lett. 9, 238–240 (1984).
[CrossRef] [PubMed]

Shabat, A. B.

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

Shoemaker, R. L.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, 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);G. I. Stegeman, C. T. Seaton, “Nonlinear waves guided by thin films,” Appl. Phys. Lett. 44, 830–832 (1984);H. Vach, C. T. Seaton, G. I. Stegeman, I. C. Khoo, “Observation of intensity-dependent guided waves,” Opt. Lett. 9, 238–240 (1984).
[CrossRef] [PubMed]

Silberberg, Y.

Skinner, S. R.

Smirl, A. L.

Smith, P. W.

Smith, S. D.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, 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);G. I. Stegeman, C. T. Seaton, “Nonlinear waves guided by thin films,” Appl. Phys. Lett. 44, 830–832 (1984);H. Vach, C. T. Seaton, G. I. Stegeman, I. C. Khoo, “Observation of intensity-dependent guided waves,” Opt. Lett. 9, 238–240 (1984).
[CrossRef] [PubMed]

Snyder, A. W.

A. W. Snyder, Y. Chen, L. Poladian, D. J. Mitchell, “Fundamental mode of highly nonlinear fibers,” Electron. Lett. 15, 535–537 (1990).

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

Stegeman, G. I.

G. I. Stegeman, C. T. Seaton, “Nonlinear integrated optics,” J. Appl. Phys. 58, R57–R78 (1985).
[CrossRef]

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, 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);G. I. Stegeman, C. T. Seaton, “Nonlinear waves guided by thin films,” Appl. Phys. Lett. 44, 830–832 (1984);H. Vach, C. T. Seaton, G. I. Stegeman, I. C. Khoo, “Observation of intensity-dependent guided waves,” Opt. Lett. 9, 238–240 (1984).
[CrossRef] [PubMed]

Tomlinson, W. J.

Townes, C. H.

R. Y. Chiao, E. Garmire, C. H. Townes, “Self trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

Valera, J. D.

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, 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);G. I. Stegeman, C. T. Seaton, “Nonlinear waves guided by thin films,” Appl. Phys. Lett. 44, 830–832 (1984);H. Vach, C. T. Seaton, G. I. Stegeman, I. C. Khoo, “Observation of intensity-dependent guided waves,” Opt. Lett. 9, 238–240 (1984).
[CrossRef] [PubMed]

Vogel, E. M.

Weiner, A. M.

Zakharov, V. E.

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

Electron. Lett. (2)

A. W. Snyder, Y. Chen, L. Poladian, D. J. Mitchell, “Fundamental mode of highly nonlinear fibers,” Electron. Lett. 15, 535–537 (1990).

Y. Chen, “Vector dark spatial solitons,” Electron. Lett. 27, 1346–1348 (1991).
[CrossRef]

IEEE J. Quantum Electron. (2)

C. T. Seaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, 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);G. I. Stegeman, C. T. Seaton, “Nonlinear waves guided by thin films,” Appl. Phys. Lett. 44, 830–832 (1984);H. Vach, C. T. Seaton, G. I. Stegeman, I. C. Khoo, “Observation of intensity-dependent guided waves,” Opt. Lett. 9, 238–240 (1984).
[CrossRef] [PubMed]

A. D. Boardman, 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]

J. Appl. Phys. (1)

G. I. Stegeman, C. T. Seaton, “Nonlinear integrated optics,” J. Appl. Phys. 58, R57–R78 (1985).
[CrossRef]

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

Opt. Commun. (1)

A. Barthelemy, S. Maneuf, C. Froehly, “Propagation solution et autoconfinement de faisceaux laser par non linearite optique de Kerr,” Opt. Commun. 55, 193–206 (1985);S. Maneuf, R. Desailly, C. Froehly, “Stable self-trapping of laser beams: observation in a nonlinear planar waveguide,” Opt. Commun. 65, 193–198 (1988);S. Maneuf, F. Reyneud, “Quasi-steady state self-trapping of first, second and third order subnanosecond soliton beams,” Opt. Commun. 66, 325–328 (1988).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. Lett. (1)

R. Y. Chiao, E. Garmire, C. H. Townes, “Self trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964).
[CrossRef]

Sov. Phys. JETP (2)

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

N. N. Akhmediev, “Novel class of nonlinear surface waves: asymmetric modes in a symmetric layered structure,” Sov. Phys. JETP 56, 299–303 (1982).

Other (2)

P. F. Byrd, M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer-Verlag, Berlin, 1971).
[CrossRef]

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

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Figures (8)

Fig. 1
Fig. 1

Dependence of the effective wave index ne and x1 on m for the dark solitary wave trapped in the structure with no/nf = 1.5 and kdν/(2 − m2)1/2 = 0.25π.

Fig. 2
Fig. 2

Field and intensity profiles of the dark solitary wave for the parameters of Fig. 1 and n2o/n2f = 1.5. The solid curves are for m = 0.1 and the dashed curves for m = 0.95, where Ē = E exp ( i β z ) n 2 o / γ.

Fig. 3
Fig. 3

(a) Dependence of ne and x1 on m for the gray solitary wave trapped in the structure with no/nf = 1.5, where kdν/(2m2 − 1)1/2 = 0.25π. (b) The corresponding field ( Ē = E exp ( i β z ) n 2 o / γ ) and intensity profiles with n2o/n2f = 1.5; the solid curves are for m = 0.75 and the dashed curves for m = 0.95.

Fig. 4
Fig. 4

Measurement of grayness = min(|Ē|2 = |E|2n2o2) of the gray solitary wave (a) versus ne for fixed film widths kdnf and (b) versus the film width kdnf for fixed ne, where no/nf = 1.5 and n2o/n2f = 1.5.

Fig. 5
Fig. 5

Characteristics of dark oscillatory waves (a) for the j = 1 mode and (b) for the j = 2, 3, 4 modes, where Ē = E exp ( i β z ) n 2 o / γ, no/nf = 0.8, n2o/n2f = 0.8, m = 0.45, and d/(1 + m2)1/2 = 1 for the j = 1, 2 modes and kκd/(1 + m2)1/2 = 4.24 for the j = 3, 4 modes.

Fig. 6
Fig. 6

Dependence of ne on m (a) for the j = 1 mode and (b) for the j = 2 mode, where kκd/(1 + m2)1/2 = 0.25π. n2o/n2f = 0.5 is identified by the solid curves and n2o/n2f = 1.5 by the dashed curves.

Fig. 7
Fig. 7

Demonstration of the field profiles of degenerate mode patterns with n2o/n2f = 0.5 and no/nf = 0.8. The insets are the corresponding relations between ne and m. (a) j = 2 mode (kdnf = 0.8) for ne/nf = 0.51, and m = 0.55 identified by the dashed curve and m = 0.95 by the solid curve; (b) j = 3 mode (kdnf = 2.4) for ne/nf = 0.56, with m = 0.41 identified by the dashed curve and m = 0.81 by the solid curve.

Fig. 8
Fig. 8

Regions for the existence of (a) j = 1, 3, 5 modes and (b) j = 2, 4, 6 modes in terms of m versus kκd/(1 + m2)1/2, where n2o/n2f = 0.5 and no/nf = 0.8.

Equations (13)

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n 2 = { n f 2 n 2 f | E | 2 | x | < d n o 2 n 2 o | E | 2 | x | > d ,
2 E + k 2 n 2 E = 0 ,
E ( x , z ) = { ± exp ( i β z ) ( γ n 2 o ) tanh [ k γ ( x + x 1 ) 2 ] x < d exp ( i β z ) [ 2 ( 1 m ) 2 / n 2 f ] 1 / 2 η f ± ( x ) cn ( k η x | m ) | x | < d , exp ( i β z ) ( γ n 2 o ) tanh [ k γ ( x x 1 ) 2 ] x > d
γ 2 ν 2 2 ( n 2 o n 2 f ) 1 / 2 ( 1 m 2 ) 1 / 2 2 m 2 × { [ n 2 o n 2 f ( 1 m 2 ) ] 1 / 2 sn 2 [ k d ν ( 2 m 2 ) 1 / 2 | m ] + d n [ k d ν ( 2 m 2 ) 1 / 2 | m ] } / cn 2 [ k d ν ( 2 m 2 ) 1 / 2 | m ] = 0
γ 2 ν 2 2 ( n 2 o n 2 f ) 1 / 2 ( 1 m 2 ) 1 / 2 2 m 2 1 × { [ n 2 o n 2 f ( 1 m 2 ) ] 1 / 2 + sn [ k d ν ( 2 m 2 1 ) 1 / 2 | m ] × d n [ k d ν ( 2 m 2 1 ) 1 / 2 | m ] } / cn 2 ( k d ν ( 2 m 2 1 ) 1 / 2 | m ) = 0
x 1 = d + 1 2 k γ ln 1 [ 2 ( 1 m 2 ) n 2 o / n 2 f ] 1 / 2 ( η / γ ) f ± ( d ) / cn ( k η d | m ) 1 + [ 2 ( 1 m 2 ) n 2 o / n 2 f ] 1 / 2 ( η / γ ) f ± ( d ) / cn ( k η d | m ) .
min { | Ē | 2 = | E | 2 n 2 o / γ 2 } = 2 ( 1 m 2 ) ν 2 n 2 o ( 2 m 2 1 ) γ 2 n 2 f ,
E ( x , z ) = ( 1 ) ( j 1 ) / 2 ( 2 / n 2 f ) 1 / 2 [ κ m / ( 1 + m 2 ) 1 / 2 ] × sn [ k κ x / ( 1 + m 2 ) 1 / 2 | m ] exp ( i β z )
E ( x , z ) = ( 1 ) j / 2 ( 2 / n 2 f ) 1 / 2 [ κ m / ( 1 + m 2 ) 1 / 2 ] cn [ k κ x / ( 1 + m 2 ) 1 / 2 | m ] d n [ k κ x / ( 1 + m 2 ) 1 / 2 | m ] exp ( i β z )
γ 2 κ 2 2 ( n 2 o n 2 f ) 1 / 2 m 1 + m 2 { m ( n 2 o n 2 f ) 1 / 2 sn 2 [ k κ d ( 1 + m 2 ) 1 / 2 | m ] + ( 1 ) ( j 1 ) / 2 c n [ k κ d ( 1 + m 2 ) 1 / 2 | m ] dn [ k κ d ( 1 + m 2 ) 1 / 2 | m ] } = 0
γ 2 κ 2 2 ( n 2 o n 2 f ) 1 / 2 m 1 + m 2 { m ( n 2 o n 2 f ) 1 / 2 cn 2 [ k κ d ( 1 + m 2 ) 1 / 2 | m ] ( 1 ) j / 2 ( 1 m 2 ) sn [ k κ d ( 1 + m 2 ) 1 / 2 | m ] } / d n 2 [ k κ d ( 1 + m 2 ) 1 / 2 | m ] = 0
x 1 = d + 1 2 k γ ln 1 ( 1 ) ( j 1 ) / 2 ( 2 n 2 o / n 2 f ) 1 / 2 [ κ m / γ ( 1 + m 2 ) 1 / 2 ] sn [ k κ d / ( 1 + m 2 ) 1 / 2 | m ] 1 + ( 1 ) ( j 1 ) / 2 ( 2 n 2 o / n 2 f ) 1 / 2 [ κ m / γ ( 1 + m 2 ) 1 / 2 ] sn [ k κ d / ( 1 + m 2 ) 1 / 2 | m ]
E ( x , z ) = { exp ( i β z ) ( 2 / n 2 o ) 1 / 2 | γ | / sinh [ k | γ | ( x + x 1 ) ] x < d ± exp ( i β z ) ( 2 / n 2 o ) 1 / 2 | γ | / sinh [ k | γ | ( x x 1 ) ] x > d .

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