Abstract

The behavior of nonlinear waves guided by a film configuration with a nonlinear cladding is investigated. The intensity-dependent dielectric function of the cladding is characterized by saturation effects. The field patterns that arise are discussed, and the dispersion relations are derived. Several dispersion curves are plotted. The results are compared with those familiar from nonlinear guided-wave propagation along configurations with Kerr-like cladding dielectric functions.

© 1985 Optical Society of America

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References

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  1. N. N. Akhmediev, Sov. Phys. JETP 56, 299 (1982).
  2. F. Lederer, U. Langbein, H.-E. Ponath, Appl. Phys. B 31, 69 (1983).
    [CrossRef]
  3. G. I. Stegeman, C. T. Seaton, J. Chilwell, S. D. Smith, Appl. Phys. Lett. 44, 830 (1984).
    [CrossRef]
  4. D. Mihalache, D. Mazila, H. Totia, Phys. Scr. 30, 335 (1984).
    [CrossRef]
  5. A. Boardman, P. Egan, J. Phys. Colloq. CS 45, 291 (1984).
  6. H. Vach, G. I. Stegeman, C. T. Seaton, I. C. Khoo, Opt. Lett. 9, 238 (1984).
    [CrossRef] [PubMed]
  7. U. Langbein, F. Lederer, H.-E. Ponath, Opt. Commun. 53, 417 (1985).
    [CrossRef]
  8. A. A. Maradudin, “Surface electromagnetic waves,” presented at Second International School on Condensed Matter Physics, Varna, Bulgaria, 1982.
  9. D. Marcuse, Appl. Opt. 19, 3130 (1980).
    [CrossRef] [PubMed]
  10. L. D. Landau, E. M. Lifshiz, Mechanics (Pergamon, Aberdeen, U.K., 1975).
  11. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
  12. U. Langbein, F. Lederer, H.-E. Ponath, U. Trutschel, Appl. Phys. B (to be published).

1985 (1)

U. Langbein, F. Lederer, H.-E. Ponath, Opt. Commun. 53, 417 (1985).
[CrossRef]

1984 (4)

G. I. Stegeman, C. T. Seaton, J. Chilwell, S. D. Smith, Appl. Phys. Lett. 44, 830 (1984).
[CrossRef]

D. Mihalache, D. Mazila, H. Totia, Phys. Scr. 30, 335 (1984).
[CrossRef]

A. Boardman, P. Egan, J. Phys. Colloq. CS 45, 291 (1984).

H. Vach, G. I. Stegeman, C. T. Seaton, I. C. Khoo, Opt. Lett. 9, 238 (1984).
[CrossRef] [PubMed]

1983 (1)

F. Lederer, U. Langbein, H.-E. Ponath, Appl. Phys. B 31, 69 (1983).
[CrossRef]

1982 (1)

N. N. Akhmediev, Sov. Phys. JETP 56, 299 (1982).

1980 (1)

Akhmediev, N. N.

N. N. Akhmediev, Sov. Phys. JETP 56, 299 (1982).

Boardman, A.

A. Boardman, P. Egan, J. Phys. Colloq. CS 45, 291 (1984).

Chilwell, J.

G. I. Stegeman, C. T. Seaton, J. Chilwell, S. D. Smith, Appl. Phys. Lett. 44, 830 (1984).
[CrossRef]

Egan, P.

A. Boardman, P. Egan, J. Phys. Colloq. CS 45, 291 (1984).

Khoo, I. C.

Landau, L. D.

L. D. Landau, E. M. Lifshiz, Mechanics (Pergamon, Aberdeen, U.K., 1975).

Langbein, U.

U. Langbein, F. Lederer, H.-E. Ponath, Opt. Commun. 53, 417 (1985).
[CrossRef]

F. Lederer, U. Langbein, H.-E. Ponath, Appl. Phys. B 31, 69 (1983).
[CrossRef]

U. Langbein, F. Lederer, H.-E. Ponath, U. Trutschel, Appl. Phys. B (to be published).

Lederer, F.

U. Langbein, F. Lederer, H.-E. Ponath, Opt. Commun. 53, 417 (1985).
[CrossRef]

F. Lederer, U. Langbein, H.-E. Ponath, Appl. Phys. B 31, 69 (1983).
[CrossRef]

U. Langbein, F. Lederer, H.-E. Ponath, U. Trutschel, Appl. Phys. B (to be published).

Lifshiz, E. M.

L. D. Landau, E. M. Lifshiz, Mechanics (Pergamon, Aberdeen, U.K., 1975).

Maradudin, A. A.

A. A. Maradudin, “Surface electromagnetic waves,” presented at Second International School on Condensed Matter Physics, Varna, Bulgaria, 1982.

Marcuse, D.

D. Marcuse, Appl. Opt. 19, 3130 (1980).
[CrossRef] [PubMed]

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

Mazila, D.

D. Mihalache, D. Mazila, H. Totia, Phys. Scr. 30, 335 (1984).
[CrossRef]

Mihalache, D.

D. Mihalache, D. Mazila, H. Totia, Phys. Scr. 30, 335 (1984).
[CrossRef]

Ponath, H.-E.

U. Langbein, F. Lederer, H.-E. Ponath, Opt. Commun. 53, 417 (1985).
[CrossRef]

F. Lederer, U. Langbein, H.-E. Ponath, Appl. Phys. B 31, 69 (1983).
[CrossRef]

U. Langbein, F. Lederer, H.-E. Ponath, U. Trutschel, Appl. Phys. B (to be published).

Seaton, C. T.

H. Vach, G. I. Stegeman, C. T. Seaton, I. C. Khoo, Opt. Lett. 9, 238 (1984).
[CrossRef] [PubMed]

G. I. Stegeman, C. T. Seaton, J. Chilwell, S. D. Smith, Appl. Phys. Lett. 44, 830 (1984).
[CrossRef]

Smith, S. D.

G. I. Stegeman, C. T. Seaton, J. Chilwell, S. D. Smith, Appl. Phys. Lett. 44, 830 (1984).
[CrossRef]

Stegeman, G. I.

G. I. Stegeman, C. T. Seaton, J. Chilwell, S. D. Smith, Appl. Phys. Lett. 44, 830 (1984).
[CrossRef]

H. Vach, G. I. Stegeman, C. T. Seaton, I. C. Khoo, Opt. Lett. 9, 238 (1984).
[CrossRef] [PubMed]

Totia, H.

D. Mihalache, D. Mazila, H. Totia, Phys. Scr. 30, 335 (1984).
[CrossRef]

Trutschel, U.

U. Langbein, F. Lederer, H.-E. Ponath, U. Trutschel, Appl. Phys. B (to be published).

Vach, H.

Appl. Opt. (1)

Appl. Phys. B (1)

F. Lederer, U. Langbein, H.-E. Ponath, Appl. Phys. B 31, 69 (1983).
[CrossRef]

Appl. Phys. Lett. (1)

G. I. Stegeman, C. T. Seaton, J. Chilwell, S. D. Smith, Appl. Phys. Lett. 44, 830 (1984).
[CrossRef]

J. Phys. Colloq. CS (1)

A. Boardman, P. Egan, J. Phys. Colloq. CS 45, 291 (1984).

Opt. Commun. (1)

U. Langbein, F. Lederer, H.-E. Ponath, Opt. Commun. 53, 417 (1985).
[CrossRef]

Opt. Lett. (1)

Phys. Scr. (1)

D. Mihalache, D. Mazila, H. Totia, Phys. Scr. 30, 335 (1984).
[CrossRef]

Sov. Phys. JETP (1)

N. N. Akhmediev, Sov. Phys. JETP 56, 299 (1982).

Other (4)

A. A. Maradudin, “Surface electromagnetic waves,” presented at Second International School on Condensed Matter Physics, Varna, Bulgaria, 1982.

L. D. Landau, E. M. Lifshiz, Mechanics (Pergamon, Aberdeen, U.K., 1975).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

U. Langbein, F. Lederer, H.-E. Ponath, U. Trutschel, Appl. Phys. B (to be published).

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Figures (4)

Fig. 1
Fig. 1

Phase paths (ϕi)1/2 (indicated by arrows) and the associated field-pattern schemes of a (ZO) NOW: Filled arrows, NGW with a real evanescent field maximum; open arrows, NGW with a virtual evanescent field maximum.

Fig. 2
Fig. 2

Dispersion curves β(V, 1NL) (solid lines) for ZO and FO NGW’s (dashed lines, forbidden branches). Dashed–dotted lines illustrate the condition α1NL = 0 (transition points) (dotted lines, forbidden branches). 1, 1NL = 2.255; 2, 1NL = 2.33.

Fig. 3
Fig. 3

Dispersion curves β(P/P0, V) for ZO, FO, and (SO) NGW’s, where the dielectric function of the cladding is given by Eq. (1a). Small arrows indicate transition points. Solid lines, a/b → ∞ (Kerr-like nonlinearity); dashed lines, a/b = 0.33, dotted lines, a/b = 0.2; fixed parameters: ¯ 1 = 2.25, 2 = 2.5, 3 = 2.1, V = 8.

Fig. 4
Fig. 4

Same dispersion curves and notation as in Fig. 3 but for the dielectric function of the cladding given by Eq. (1b).

Equations (17)

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1 ( E 1 2 ) = ¯ 1 + a E 1 2 1 + b E 1 2 ,
1 ( E 1 2 ) = ¯ 1 + a b [ 1 - exp ( - b E 1 2 ) ] ,             ¯ 1 > 3 .
E ¯ ( x , z , t ) = ½ { E ( x ) exp [ i ( β z - ω t ) ] + c . c . } .
ϕ 1 = α 1 ( E 1 2 ) E 1 2 ,             ϕ 2 = C 2 - γ 2 2 E 2 2 ,             ϕ 3 = α 3 2 E 3 2 ,
α 1 ( E 1 2 ) = [ β 2 - 1 E 1 2 0 E 1 2 1 ( E ˜ 2 ) d ( E ˜ 2 ) ] 1 / 2 , α 3 = ( β 2 - ω 2 c 2 3 ) 1 / 2 , γ 2 = ( ω 2 2 c 2 - β 2 ) 1 / 2             for β 2 ω 2 c 2 2 , = i Γ 2 = i ( β 2 - ω 2 c 2 2 ) 1 / 2             for β 2 > ω 2 c 2 2 .
C 2 = ( ω 2 / c 2 ) [ 2 E ( 1 ) 2 - 0 E ( 1 ) 2 1 ( E ˜ 2 ) d ( E ˜ 2 ) ]
ϕ 1 ( E 1 2 ) = α 1 2 E 1 2 - ω 2 a c 2 b { E 1 2 - 1 / b [ ln ( 1 + b E 1 2 ) ] } ,
ϕ 1 ( E 1 2 ) = α 1 2 E 1 2 - ω 2 a c 2 b { E 1 2 - 1 / b [ 1 - exp ( - b E 1 2 ) ] }
tan ( γ 2 d ) = γ 2 ( α 1 NL + α 3 ) γ 2 2 - α 1 NL α 3 ,
α 1 NL = ± ( β 2 - ω 2 c 2 1 NL ) 1 / 2 ,
1 NL = 1 E ( 1 ) 2 0 E ( 1 ) 2 1 ( E ˜ 2 ) d ( E ˜ 2 ) .
1 NL = ¯ 1 + a b { 1 - ln [ 1 + b E ( 1 ) 2 ] b E ( 1 ) 2 } ,
1 NL = ¯ 1 + a b { 1 - 1 - exp [ - b E ( 1 ) 2 ] b E ( 1 ) 2 } ,
P 1 = β 2 μ 0 ω d E 1 2 ( x ) d x .
P 1 = β 2 μ 0 ω [ 0 E ^ 1 E ˜ 2 d E ˜ ( ϕ 1 ) 1 / 2 + ( - 1 ) M E ^ 1 E ( 1 ) E ˜ 2 d E ˜ ( ϕ 1 ) 1 / 2 ] .
P 2 = β 2 μ 0 ω 0 d E 2 2 ( x ) d x ,             P 3 = β 2 μ 0 ω - 0 E 3 2 ( x ) d x ,
V V μ = ( 2 - ¯ 1 2 - ¯ 1 - a / b ) 1 / 2 × [ arctan ( ¯ 1 + a / b - 3 2 - ¯ 1 - a / b ) 1 / 2 + μ π ] ,             μ = 0 , 1 , 2 ,

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