Abstract

We show that the electric field components Ex and Ez of a transverse magnetic (TM) wave propagating in a nonlinear medium can be formally decomposed through a conservation law, provided that ∂xx/∂Ez2 = ∂zz/∂Ex2. If this condition is satisfied, then the electromagnetic field distribution can be exactly obtained by quadrature. For the case of a guided nonlinear TM mode, our procedure can be employed to establish the eigenvalue equation directly.

© 1987 Optical Society of America

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  1. G. I. Stegeman, C. T. Seaton, J. Appl. Phys. 58, R.57 (1985).
    [Crossref]
  2. A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, E. M. Wright, Phys. Rev. A 35, 1159 (1987).
    [Crossref] [PubMed]
  3. A. E. Kaplan, JETP Lett. 24, 114 (1976);Sov. Phys. JETP 45, 896 (1977).
  4. V. M. Agranovich, V. S. Babichenko, V. Ya Chernyak, JETP Lett. 32, 512 (1980).
  5. N. N. Akhmediev, Sov. Phys. JETP 56, 299 (1982).
  6. K. M. Leung, Phys. Rev. B 32, 5093 (1985).
    [Crossref]
  7. M. Miyagi, S. Nishida, Sci. Rep. Res. Inst. Tohoku Univ. Ser. B 24, 53 (1972);Sci. Rep. Res. Inst. Tohoku Univ. Ser. B 25, 53 (1973).
  8. W. J. Tomlinson, Opt. Lett. 5, 323 (1980).
    [Crossref] [PubMed]
  9. U. Langbein, F. Lederer, T. Peschel, H.-E. Ponath, Opt. Lett. 10, 571 (1985).
    [Crossref] [PubMed]
  10. D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, T. Twardowski, Opt. Lett. 12, 187 (1987).
    [Crossref] [PubMed]
  11. A. D. Boardman, T. Twardowski, A. Shivarova, G. I. Stegeman, Proc. Inst. Electr. Eng. Part J 134, 152 (1987).
  12. A. L. Berkhoer, V. E. Zakharov, Sov. Phys. JETP 31, 486 (1970).
  13. V. N. Eleonskii, V. P. Silin, JETP Lett. 13, 117 (1971).
  14. M. Miyagi, S. Nishida, Radio Sci. 10, 833 (1975).
    [Crossref]
  15. V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).
  16. S. V. Vladimirov, Sov. J. Plasma Phys. 12, 552 (1986).
  17. C. G. Kuper, An Introduction to the Theory of Superconductivity (Clarendon, Oxford, 1968).
  18. F. S. Liu, R. I. Joseph, Radio Sci. 18, 532 (1983).
    [Crossref]
  19. P. D. Maker, R. Terhune, Phys. Rev. 137, A801 (1965).
    [Crossref]

1987 (3)

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, E. M. Wright, Phys. Rev. A 35, 1159 (1987).
[Crossref] [PubMed]

A. D. Boardman, T. Twardowski, A. Shivarova, G. I. Stegeman, Proc. Inst. Electr. Eng. Part J 134, 152 (1987).

D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, T. Twardowski, Opt. Lett. 12, 187 (1987).
[Crossref] [PubMed]

1986 (1)

S. V. Vladimirov, Sov. J. Plasma Phys. 12, 552 (1986).

1985 (3)

U. Langbein, F. Lederer, T. Peschel, H.-E. Ponath, Opt. Lett. 10, 571 (1985).
[Crossref] [PubMed]

K. M. Leung, Phys. Rev. B 32, 5093 (1985).
[Crossref]

G. I. Stegeman, C. T. Seaton, J. Appl. Phys. 58, R.57 (1985).
[Crossref]

1983 (1)

F. S. Liu, R. I. Joseph, Radio Sci. 18, 532 (1983).
[Crossref]

1982 (1)

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

1980 (2)

W. J. Tomlinson, Opt. Lett. 5, 323 (1980).
[Crossref] [PubMed]

V. M. Agranovich, V. S. Babichenko, V. Ya Chernyak, JETP Lett. 32, 512 (1980).

1976 (1)

A. E. Kaplan, JETP Lett. 24, 114 (1976);Sov. Phys. JETP 45, 896 (1977).

1975 (1)

M. Miyagi, S. Nishida, Radio Sci. 10, 833 (1975).
[Crossref]

1972 (2)

V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

M. Miyagi, S. Nishida, Sci. Rep. Res. Inst. Tohoku Univ. Ser. B 24, 53 (1972);Sci. Rep. Res. Inst. Tohoku Univ. Ser. B 25, 53 (1973).

1971 (1)

V. N. Eleonskii, V. P. Silin, JETP Lett. 13, 117 (1971).

1970 (1)

A. L. Berkhoer, V. E. Zakharov, Sov. Phys. JETP 31, 486 (1970).

1965 (1)

P. D. Maker, R. Terhune, Phys. Rev. 137, A801 (1965).
[Crossref]

Agranovich, V. M.

V. M. Agranovich, V. S. Babichenko, V. Ya Chernyak, JETP Lett. 32, 512 (1980).

Akhmediev, N. N.

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

Babichenko, V. S.

V. M. Agranovich, V. S. Babichenko, V. Ya Chernyak, JETP Lett. 32, 512 (1980).

Berkhoer, A. L.

A. L. Berkhoer, V. E. Zakharov, Sov. Phys. JETP 31, 486 (1970).

Boardman, A. D.

D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, T. Twardowski, Opt. Lett. 12, 187 (1987).
[Crossref] [PubMed]

A. D. Boardman, T. Twardowski, A. Shivarova, G. I. Stegeman, Proc. Inst. Electr. Eng. Part J 134, 152 (1987).

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, E. M. Wright, Phys. Rev. A 35, 1159 (1987).
[Crossref] [PubMed]

Eleonskii, V. N.

V. N. Eleonskii, V. P. Silin, JETP Lett. 13, 117 (1971).

Joseph, R. I.

F. S. Liu, R. I. Joseph, Radio Sci. 18, 532 (1983).
[Crossref]

Kaplan, A. E.

A. E. Kaplan, JETP Lett. 24, 114 (1976);Sov. Phys. JETP 45, 896 (1977).

Kuper, C. G.

C. G. Kuper, An Introduction to the Theory of Superconductivity (Clarendon, Oxford, 1968).

Langbein, U.

Lederer, F.

Leung, K. M.

K. M. Leung, Phys. Rev. B 32, 5093 (1985).
[Crossref]

Liu, F. S.

F. S. Liu, R. I. Joseph, Radio Sci. 18, 532 (1983).
[Crossref]

Maker, P. D.

P. D. Maker, R. Terhune, Phys. Rev. 137, A801 (1965).
[Crossref]

Maradudin, A. A.

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, E. M. Wright, Phys. Rev. A 35, 1159 (1987).
[Crossref] [PubMed]

Mihalache, D.

Miyagi, M.

M. Miyagi, S. Nishida, Radio Sci. 10, 833 (1975).
[Crossref]

M. Miyagi, S. Nishida, Sci. Rep. Res. Inst. Tohoku Univ. Ser. B 24, 53 (1972);Sci. Rep. Res. Inst. Tohoku Univ. Ser. B 25, 53 (1973).

Nishida, S.

M. Miyagi, S. Nishida, Radio Sci. 10, 833 (1975).
[Crossref]

M. Miyagi, S. Nishida, Sci. Rep. Res. Inst. Tohoku Univ. Ser. B 24, 53 (1972);Sci. Rep. Res. Inst. Tohoku Univ. Ser. B 25, 53 (1973).

Peschel, T.

Ponath, H.-E.

Seaton, C. T.

Shabat, A. B.

V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

Shivarova, A.

A. D. Boardman, T. Twardowski, A. Shivarova, G. I. Stegeman, Proc. Inst. Electr. Eng. Part J 134, 152 (1987).

Silin, V. P.

V. N. Eleonskii, V. P. Silin, JETP Lett. 13, 117 (1971).

Stegeman, G. I.

A. D. Boardman, T. Twardowski, A. Shivarova, G. I. Stegeman, Proc. Inst. Electr. Eng. Part J 134, 152 (1987).

D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, T. Twardowski, Opt. Lett. 12, 187 (1987).
[Crossref] [PubMed]

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, E. M. Wright, Phys. Rev. A 35, 1159 (1987).
[Crossref] [PubMed]

G. I. Stegeman, C. T. Seaton, J. Appl. Phys. 58, R.57 (1985).
[Crossref]

Terhune, R.

P. D. Maker, R. Terhune, Phys. Rev. 137, A801 (1965).
[Crossref]

Tomlinson, W. J.

Twardowski, T.

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, E. M. Wright, Phys. Rev. A 35, 1159 (1987).
[Crossref] [PubMed]

A. D. Boardman, T. Twardowski, A. Shivarova, G. I. Stegeman, Proc. Inst. Electr. Eng. Part J 134, 152 (1987).

D. Mihalache, G. I. Stegeman, C. T. Seaton, E. M. Wright, R. Zanoni, A. D. Boardman, T. Twardowski, Opt. Lett. 12, 187 (1987).
[Crossref] [PubMed]

Vladimirov, S. V.

S. V. Vladimirov, Sov. J. Plasma Phys. 12, 552 (1986).

Wright, E. M.

Ya Chernyak, V.

V. M. Agranovich, V. S. Babichenko, V. Ya Chernyak, JETP Lett. 32, 512 (1980).

Zakharov, V. E.

V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

A. L. Berkhoer, V. E. Zakharov, Sov. Phys. JETP 31, 486 (1970).

Zanoni, R.

J. Appl. Phys. (1)

G. I. Stegeman, C. T. Seaton, J. Appl. Phys. 58, R.57 (1985).
[Crossref]

JETP Lett. (3)

A. E. Kaplan, JETP Lett. 24, 114 (1976);Sov. Phys. JETP 45, 896 (1977).

V. M. Agranovich, V. S. Babichenko, V. Ya Chernyak, JETP Lett. 32, 512 (1980).

V. N. Eleonskii, V. P. Silin, JETP Lett. 13, 117 (1971).

Opt. Lett. (3)

Phys. Rev. (1)

P. D. Maker, R. Terhune, Phys. Rev. 137, A801 (1965).
[Crossref]

Phys. Rev. A (1)

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, E. M. Wright, Phys. Rev. A 35, 1159 (1987).
[Crossref] [PubMed]

Phys. Rev. B (1)

K. M. Leung, Phys. Rev. B 32, 5093 (1985).
[Crossref]

Proc. Inst. Electr. Eng. Part J (1)

A. D. Boardman, T. Twardowski, A. Shivarova, G. I. Stegeman, Proc. Inst. Electr. Eng. Part J 134, 152 (1987).

Radio Sci. (2)

M. Miyagi, S. Nishida, Radio Sci. 10, 833 (1975).
[Crossref]

F. S. Liu, R. I. Joseph, Radio Sci. 18, 532 (1983).
[Crossref]

Sci. Rep. Res. Inst. Tohoku Univ. Ser. B (1)

M. Miyagi, S. Nishida, Sci. Rep. Res. Inst. Tohoku Univ. Ser. B 24, 53 (1972);Sci. Rep. Res. Inst. Tohoku Univ. Ser. B 25, 53 (1973).

Sov. J. Plasma Phys. (1)

S. V. Vladimirov, Sov. J. Plasma Phys. 12, 552 (1986).

Sov. Phys. JETP (3)

V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

A. L. Berkhoer, V. E. Zakharov, Sov. Phys. JETP 31, 486 (1970).

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

Other (1)

C. G. Kuper, An Introduction to the Theory of Superconductivity (Clarendon, Oxford, 1968).

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Equations (31)

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e ( r , t ) = { e x x ̂ + e z z ̂ } exp [ i ( k x ω t ) ]
h ( r , t ) = h y ŷ exp [ i ( k x ω t ) ] ,
P NL = A e ( e e * ) + B e * ( e e ) ,
= 0 [ x x ( | e x | 2 , | e z | 2 ) 0 0 z z ( | e x | 2 , | e z | 2 ) ] ,
e x z ( e z x + i k e z ) = i ω μ 0 h y ,
h y z = i ω 0 x x e x ,
i k h y + h y x = i ω 0 z z e z ,
d E x d η = ( 1 γ z z ) E z ,
d d η = ( z z E z ) = x x E x .
d E z d η [ z z + 2 E z 2 ( z z E z 2 ) ] = E x [ x x + 2 E z 2 ( γ z z 1 ) ( z z E x 2 ) ] .
Z E z 2 , X E x 2 ,
M d Z + N d X = 0 ,
M = z z + Z [ Z z z ( γ z z 2 ) ] ,
N = x x + X [ Z z z ( γ z z 2 ) ] .
x x E z 2 = z z E x 2 .
x x = W / X ,
z z = W / Z ,
F ( X , Z ) = Z z z ( γ z z 2 ) + 1 2 0 X d X [ x x ( X , Z ) + x x ( X , 0 ) ] + 1 2 0 Z d Z [ z z ( X , Z ) + z z ( 0 , Z ) ] = Z W Z ( γ W Z 2 ) + W ( X , Z ) W ( 0 , 0 ) = C 0 ,
X = R ( z ) or Z = R 1 ( X ) ,
η = ± E z 0 E z d E z × [ z z + 2 E z 2 ( z z E z 2 ) ] [ R ( E z 2 ) ] 1 / 2 [ x x + 2 E z 2 ( γ z z 1 ) ( z z E x 2 ) ] ,
E x = E x 0 exp [ η ( 1 γ L ) 1 / 2 ] ,
E z = E x 0 ( 1 γ L ) 1 / 2 exp [ η ( 1 γ L ) 1 / 2 ] ,
E z 0 z z ( X 0 , Z 0 ) = L E x 0 ( 1 γ L ) 1 / 2 ,
F ( X 0 , Z 0 ) = 0 ,
L X 0 + Z 0 z z ( X 0 , Z 0 ) [ 2 L 1 z z ( X 0 , Z 0 ) ] = W ( X 0 , Z 0 ) W ( 0 , 0 ) .
γ = L 1 [ 1 L 2 X 0 Z 0 z z 2 ( X 0 , Z 0 ) ]
S NL 0 Z 0 d Z z z ( Z / X ) 1 / 2 [ z z + 2 Z ( z z Z ) x x + 2 Z ( γ z z 1 ) ( x x Z ) ] .
x x = x + α 1 X + α 2 Z , z z = z + α 1 Z + α 2 X ,
Z ( z + α 1 Z + α 2 X ) [ 2 γ ( z + α 1 Z + α 2 X ) ] = x X + z Z + ( α 1 / 2 ) ( X 2 + Z 2 ) + α 2 X Z ,
[ ( α 1 / 2 ) + L 1 α 2 2 Z 0 ] X 0 2 + { x L + α 2 Z 0 [ 2 L 1 ( z + α 1 Z 0 ) 1 ] } X 0 + Z 0 [ L 1 ( z + α 1 Z 0 ) 2 z ( 3 / 2 ) α 1 Z 0 ] = 0 ,
γ = L 1 [ 1 L 2 X 0 Z 0 ( z + α 1 Z 0 + α 2 X 0 ) 2 ] ,

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