Abstract

We derive the system of equations describing the nonlinear interaction, associated with the optical Kerr effect, among the four forward- and backward-propagating modes in a straight single-mode fiber. This allows us, in particular, to obtain the set of equations governing nonlinear evolution in a highly twisted fiber of the corresponding copropagating and counterpropagating right and left circularly polarized modes.

© 1988 Optical Society of America

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  1. A. Hasegawa, Y. Kodama, Proc. IEEE 69, 1145 (1981).
    [CrossRef]
  2. R. H. Stolen, J. Botineau, A. Ashkin, Opt. Lett. 7, 512 (1982).
    [CrossRef] [PubMed]
  3. K. Kitayama, Y. Kimura, S. Seikai, Appl. Phys. Lett. 46, 317, 623 (1985).
    [CrossRef]
  4. H. G. Winful, Appl. Phys. Lett. 47, 213 (1985); Opt. Lett. 11, 33 (1986).
    [CrossRef]
  5. B. Daino, G. Gregori, S. Wabnitz, Opt. Lett. 11, 42 (1986).
    [CrossRef]
  6. See, e.g., N. Tzoar, M. Jain, Phys. Rev. A 23, 1266 (1981).
    [CrossRef]
  7. B. Crosignani, A. Cutolo, P. Di Porto, J. Opt. Soc. Am. 72, 1136 (1982).
    [CrossRef]
  8. B. Crosignani, P. Di Porto, Opt. Acta 32, 1251 (1985).
  9. B. Crosignani, B. Daino, P. Di Porto, J. Opt. Soc. Am. B 3, 1120 (1986).
    [CrossRef]
  10. C. R. Menyuk, IEEE J. Quantum Electron. QE-23, 174 (1987).
    [CrossRef]
  11. A. J. Barlow, J. J. Ramskov-Hansen, D. N. Payne, Appl. Opt. 20, 2962 (1981).
    [CrossRef] [PubMed]
  12. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
  13. A. Yariv, Optical Electronics (Holt, Rinehart and Winston, New York, 1985).
  14. B. Crosignani, in New Directions in Guided Wave and Coherent Optics, D. B. Ostrowsky, E. Spitz, eds. (Nijhoff, The Hague, 1984), Vol. 1, p. 1.
  15. P. D. Maker, R. W. Terhune, Phys. Rev. 137, 801 (1965).
    [CrossRef]
  16. See, e.g., B. Crosignani, P. Di Porto, J. Opt. Soc. Am. 72, 1553 (1982).
    [CrossRef]
  17. F. Matera, S. Wabnitz, Opt. Lett. 11, 467 (1986).
    [CrossRef] [PubMed]
  18. A. E. Kaplan, C. T. Law, IEEE J. Quantum Electron. QE-21, 1529 (1985).
    [CrossRef]

1987 (1)

C. R. Menyuk, IEEE J. Quantum Electron. QE-23, 174 (1987).
[CrossRef]

1986 (3)

1985 (4)

B. Crosignani, P. Di Porto, Opt. Acta 32, 1251 (1985).

K. Kitayama, Y. Kimura, S. Seikai, Appl. Phys. Lett. 46, 317, 623 (1985).
[CrossRef]

H. G. Winful, Appl. Phys. Lett. 47, 213 (1985); Opt. Lett. 11, 33 (1986).
[CrossRef]

A. E. Kaplan, C. T. Law, IEEE J. Quantum Electron. QE-21, 1529 (1985).
[CrossRef]

1982 (3)

1981 (3)

See, e.g., N. Tzoar, M. Jain, Phys. Rev. A 23, 1266 (1981).
[CrossRef]

A. Hasegawa, Y. Kodama, Proc. IEEE 69, 1145 (1981).
[CrossRef]

A. J. Barlow, J. J. Ramskov-Hansen, D. N. Payne, Appl. Opt. 20, 2962 (1981).
[CrossRef] [PubMed]

1965 (1)

P. D. Maker, R. W. Terhune, Phys. Rev. 137, 801 (1965).
[CrossRef]

Ashkin, A.

Barlow, A. J.

Botineau, J.

Crosignani, B.

B. Crosignani, B. Daino, P. Di Porto, J. Opt. Soc. Am. B 3, 1120 (1986).
[CrossRef]

B. Crosignani, P. Di Porto, Opt. Acta 32, 1251 (1985).

See, e.g., B. Crosignani, P. Di Porto, J. Opt. Soc. Am. 72, 1553 (1982).
[CrossRef]

B. Crosignani, A. Cutolo, P. Di Porto, J. Opt. Soc. Am. 72, 1136 (1982).
[CrossRef]

B. Crosignani, in New Directions in Guided Wave and Coherent Optics, D. B. Ostrowsky, E. Spitz, eds. (Nijhoff, The Hague, 1984), Vol. 1, p. 1.

Cutolo, A.

Daino, B.

Di Porto, P.

Gregori, G.

Hasegawa, A.

A. Hasegawa, Y. Kodama, Proc. IEEE 69, 1145 (1981).
[CrossRef]

Jain, M.

See, e.g., N. Tzoar, M. Jain, Phys. Rev. A 23, 1266 (1981).
[CrossRef]

Kaplan, A. E.

A. E. Kaplan, C. T. Law, IEEE J. Quantum Electron. QE-21, 1529 (1985).
[CrossRef]

Kimura, Y.

K. Kitayama, Y. Kimura, S. Seikai, Appl. Phys. Lett. 46, 317, 623 (1985).
[CrossRef]

Kitayama, K.

K. Kitayama, Y. Kimura, S. Seikai, Appl. Phys. Lett. 46, 317, 623 (1985).
[CrossRef]

Kodama, Y.

A. Hasegawa, Y. Kodama, Proc. IEEE 69, 1145 (1981).
[CrossRef]

Law, C. T.

A. E. Kaplan, C. T. Law, IEEE J. Quantum Electron. QE-21, 1529 (1985).
[CrossRef]

Maker, P. D.

P. D. Maker, R. W. Terhune, Phys. Rev. 137, 801 (1965).
[CrossRef]

Marcuse, D.

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

Matera, F.

Menyuk, C. R.

C. R. Menyuk, IEEE J. Quantum Electron. QE-23, 174 (1987).
[CrossRef]

Payne, D. N.

Ramskov-Hansen, J. J.

Seikai, S.

K. Kitayama, Y. Kimura, S. Seikai, Appl. Phys. Lett. 46, 317, 623 (1985).
[CrossRef]

Stolen, R. H.

Terhune, R. W.

P. D. Maker, R. W. Terhune, Phys. Rev. 137, 801 (1965).
[CrossRef]

Tzoar, N.

See, e.g., N. Tzoar, M. Jain, Phys. Rev. A 23, 1266 (1981).
[CrossRef]

Wabnitz, S.

Winful, H. G.

H. G. Winful, Appl. Phys. Lett. 47, 213 (1985); Opt. Lett. 11, 33 (1986).
[CrossRef]

Yariv, A.

A. Yariv, Optical Electronics (Holt, Rinehart and Winston, New York, 1985).

Appl. Opt. (1)

Appl. Phys. Lett. (2)

K. Kitayama, Y. Kimura, S. Seikai, Appl. Phys. Lett. 46, 317, 623 (1985).
[CrossRef]

H. G. Winful, Appl. Phys. Lett. 47, 213 (1985); Opt. Lett. 11, 33 (1986).
[CrossRef]

IEEE J. Quantum Electron. (2)

C. R. Menyuk, IEEE J. Quantum Electron. QE-23, 174 (1987).
[CrossRef]

A. E. Kaplan, C. T. Law, IEEE J. Quantum Electron. QE-21, 1529 (1985).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Acta (1)

B. Crosignani, P. Di Porto, Opt. Acta 32, 1251 (1985).

Opt. Lett. (3)

Phys. Rev. (1)

P. D. Maker, R. W. Terhune, Phys. Rev. 137, 801 (1965).
[CrossRef]

Phys. Rev. A (1)

See, e.g., N. Tzoar, M. Jain, Phys. Rev. A 23, 1266 (1981).
[CrossRef]

Proc. IEEE (1)

A. Hasegawa, Y. Kodama, Proc. IEEE 69, 1145 (1981).
[CrossRef]

Other (3)

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

A. Yariv, Optical Electronics (Holt, Rinehart and Winston, New York, 1985).

B. Crosignani, in New Directions in Guided Wave and Coherent Optics, D. B. Ostrowsky, E. Spitz, eds. (Nijhoff, The Hague, 1984), Vol. 1, p. 1.

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Equations (34)

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E ( r , z , t ) = m σ = 1 2 E m σ ( r ) e ^ σ { exp [ i ω 0 t - i β m , σ ( ω 0 ) z ] Φ m , σ + ( z , t ) + exp [ i ω 0 t + i β m , σ ( ω 0 ) z ] Φ m , σ - ( z , t ) } ,
P ( 3 ) = 0 χ 3 ( E · E ) E ,
L m , σ + Φ m , σ + = n σ K m , n σ , σ { exp [ i ( β m , σ - β n , σ ) z ] Φ n , σ + + exp [ i ( β m , σ + β n , σ ) z ] Φ n , σ - } , L m , σ - Φ m , σ - = n σ K m , n σ , σ { exp [ - i ( β m , σ + β n , σ ) z ] Φ n , σ + + exp [ - i ( β m , σ - β n , σ ) z ] Φ n , σ - } ,
K m , n σ , σ = - ( 2 / 3 ) k n 2 - + - + E m σ ( r ) E n σ ( r ) e ^ σ · T · e ^ σ d x d y ,
T = [ E 2 + ( 1 / 2 ) E x 2 ( 1 / 2 ) E x * E y ( 1 / 2 ) E x E y * E 2 + ( 1 / 2 ) E y 2 ] ,
L m , σ ± = / z ± ( 1 / V m , σ ) / t ( i / 2 ! A m , σ ) 2 / t 2 ,
V m , σ = ( d β m , σ / d ω ) ω = ω 0 - 1 ,             A m , σ = ( d 2 β m , σ / d ω 2 ) ω = ω 0 - 1 ,
L 1 ± Φ 1 ± = i R 11 ( Φ 1 ± 2 + 2 Φ 1 2 ) Φ 1 ± ( 2 / 3 ) i R 12 ( Φ 2 + 2 + Φ 2 - 2 ) × Φ 1 ± ( 2 / 3 ) i R 12 Φ 1 * Φ 2 + Φ 2 - ( 2 / 3 ) i R 12 × exp ( ± 2 i δ β z ) Φ 1 Φ 2 * Φ 2 ± ( 1 / 3 ) i R 12 × exp ( ± 2 i δ β z ) Φ 1 ± * Φ 2 ± 2 ,
L 2 ± Φ 2 ± = i R 22 ( Φ 2 ± 2 + 2 Φ 2 2 ) Φ 2 ± ( 2 / 3 ) i R 12 ( Φ 1 + 2 + Φ 1 - 2 ) × Φ 2 ± ( 2 / 3 ) i R 12 Φ 2 * Φ 1 + Φ 1 - ( 2 / 3 ) i R 12 × exp ( 2 i δ β z ) Φ 2 Φ 1 * Φ 1 ± ( 1 / 3 ) i R 12 × exp ( 2 i δ β z ) Φ 2 ± * Φ 1 ± 2 ,
R σ , σ = k n 2 - + - + E 1 σ ( r ) 2 E 1 σ ( r ) 2 d x d y .
- + - + E n σ ( r ) 2 d x d y = 1.
L 1 + Φ 1 + = - i R 11 Φ 1 + 2 Φ 1 + - ( 2 / 3 ) i R 12 Φ 2 + 2 Φ 1 + - ( 1 / 3 ) i R 12 × exp ( 2 i δ β z ) Φ 1 + * Φ 2 + 2 , L 2 + Φ 2 + = - i R 22 Φ 2 + 2 Φ 2 + - ( 2 / 3 ) i R 12 Φ 1 + 2 Φ 2 + - ( 1 / 3 ) i R 12 × exp ( - 2 i δ β z ) Φ 2 + * Φ 1 + 2 .
L 2 ± Φ 2 ± = i R 22 ( Φ 2 ± 2 + 2 Φ 2 2 ) Φ 2 ± , L 1 ± Φ 1 ± = ( 2 / 3 ) i R 12 ( Φ 2 + 2 + Φ 2 - 2 ) Φ 1 ± ( 2 / 3 ) i R 12 Φ 1 * Φ 2 - Φ 2 + .
E ( r , z , t ) = E ( r ) x [ exp ( - i β 1 z + i ω 0 t ) ϕ 1 + ( z , t ) + exp ( + i β 1 z + i ω 0 t ) ϕ 1 - ( z , t ) ] + E ( r ) y [ exp ( - i β 2 z + i ω 0 t ) ϕ 2 + ( z , t ) + exp ( + i β 2 z + i ω 0 t ) ϕ 2 - ( z , t ) ] ,
[ / z ± ( 1 / V 1 ) / t ] ϕ 1 ± = ± ( i K + τ ) exp ( ± i δ β z ) ϕ 2 ± , [ / z ± ( 1 / V 2 ) / t ] ϕ 2 ± = ± ( i K - τ ) exp ( ± i δ β z ) ϕ 1 ± ,
V i = d β i / d ω ω 0 - 1             ( i = 1 , 2 )
[ / z ± 1 / ( V 1 ) / t ] ϕ 1 ± = ± ( i K + τ ) exp ( ± i δ β z ) ϕ 2 ± + NLT             [ as in Eq . ( 8 a ) ] ,
[ / z ± 1 / ( V 2 ) / t ] ϕ 2 ± = ± ( i K - τ ) exp ( i δ β z ) ϕ 1 ± + NLT             [ as in Eq . ( 8 b ) ] .
e r = ( 1 / 2 ) ( x + i y ) ,             e l = 1 ( 2 ) ( x - i y ) ,
E ( r , z , t ) = E ( r ) { [ exp ( - i β r z + i ω 0 t ) ϕ r + ( z , t ) + exp ( + i β l z + i ω 0 t ) ϕ r - ( z , t ) ] e r + [ exp ( - i β l z + i ω 0 t ) × ϕ l + ( z , t ) + exp ( + i β r z + i ω 0 t ) ϕ l - ( z , t ) ] e l } ,
ϕ 1 + = ( 1 / 2 ) exp ( i β 1 z ) [ exp ( - i β r z ) ϕ r + + exp ( - i β l z ) ϕ l + ] , ϕ 2 + = ( i / 2 ) exp ( i β 2 z ) [ exp ( - i β r z ) ϕ r + - exp ( - i β l z ) ϕ l + ] , ϕ 1 - = ( 1 / 2 ) exp ( - i β 1 z ) [ exp ( i β l z ) ϕ r - + exp ( i β r z ) ϕ l - ] , ϕ 2 - = ( i / 2 ) exp ( - i β 2 z ) [ exp ( i β l z ) ϕ r - - exp ( i β r z ) ϕ l - ] .
{ - ( β 1 - β r ) + i / z + ( i / V 1 ) / t } ϕ r + + exp ( i Δ β z ) { - ( β 1 - β l ) + i / z + ( i / V 1 ) / t } ϕ l + = - ( i K + τ ) [ ϕ r + - exp ( i Δ β z ) ϕ l + ] + ( 2 / 3 ) R ( ϕ r + 2 + 2 ϕ l + 2 + 2 ϕ r - 2 + 2 ϕ l - 2 ) ϕ r + + ( 4 / 3 ) R ϕ r - ϕ l + ϕ l - * + ( 2 / 3 ) R exp ( i Δ β z ) ( ϕ l + 2 + 2 ϕ r + 2 + 2 ϕ r - 2 + 2 ϕ l - 2 ) ϕ l + + ( 4 / 3 ) R exp ( i Δ β z ) ϕ r - * ϕ r + ϕ l - ,
- exp ( i Δ β z ) { - ( β 2 - β l ) + i / z + ( i / V 2 ) / t } ϕ l + + { - ( β 2 - β r ) + i / z + ( i / V 2 ) / t } ϕ r + = ( i K - τ ) [ exp ( i Δ β z ) ϕ l + + ϕ r + ] - ( 2 / 3 ) R exp ( i Δ β z ) ( ϕ l + 2 + 2 ϕ r + 2 + 2 ϕ r - 2 + 2 ϕ l - 2 ) ϕ l + - ( 4 / 3 ) R exp ( i Δ β z ) ϕ l - ϕ r + ϕ r - * + ( 2 / 3 ) R ( ϕ r + 2 + 2 ϕ l + 2 + 2 ϕ r - 2 + 2 ϕ l - 2 ) ϕ r + + ( 4 / 3 ) R ϕ l - * ϕ l + ϕ r - ,
[ - ( β 1 - β r ) + i / z + ( i / V 1 ) / t ] ϕ r + = - ( i K + τ ) ϕ r + + ( 2 / 3 ) R ( ϕ r + 2 + 2 ϕ l + 2 + 2 ϕ r - 2 + 2 ϕ l - 2 ) ϕ r + + ( 4 / 3 ) R ϕ r - ϕ l + ϕ l - * ,
[ - ( β 2 - β r ) + i / z + ( i / V 2 ) / t ] ϕ r + = ( i K - τ ) ϕ r + + ( 2 / 3 ) R ( ϕ r + 2 + 2 ϕ l + 2 + 2 ϕ r - 2 + 2 ϕ l - 2 ) ϕ r + + ( 4 / 3 ) R ϕ r - ϕ l + ϕ l - * ,
- ( 1 / 2 ) ( β 1 + β 2 - 2 β r ) ϕ r + + i [ / z + ( 1 / 2 ) ( 1 / V 1 + 1 / V 2 ) / t ] ϕ r + = - τ ϕ r + + ( 2 / 3 ) R ( ϕ r + 2 + 2 ϕ l + 2 + 2 ϕ r - 2 + 2 ϕ l - 2 ) ϕ r + + ( 4 / 3 ) R ϕ r - ϕ l + ϕ l - * ,
β r = ( β 1 + β 2 ) / 2 - τ γ - τ
[ / z + ( 1 / V ) / t ] ϕ r + = - ( 2 / 3 ) i R ( ϕ r + 2 + 2 ϕ l + 2 + 2 ϕ r - 2 + 2 ϕ l - 2 ) ϕ r + - ( 4 / 3 ) i R ϕ r - ϕ l + ϕ l - * ,
β l = ( β 1 + β 2 ) / 2 + τ γ + τ
[ / z + ( 1 / V ) / t ] ϕ l + = - ( 2 / 3 ) i R ( ϕ l + 2 + 2 ϕ r + 2 + 2 ϕ r - 2 + 2 ϕ l - 2 ) ϕ l + - ( 4 / 3 ) i R ϕ l - ϕ r + ϕ r - * .
[ / z - ( 1 / V ) / t ] ϕ r - = ( 2 / 3 ) i R ( ϕ r - 2 + 2 ϕ l - 2 + 2 ϕ r + 2 + 2 ϕ l + 2 ) ϕ r - + ( 4 / 3 ) i R ϕ r + ϕ l - ϕ l + * ,
[ / z - ( 1 / V ) / t ] ϕ l - = ( 2 / 3 ) i R ( ϕ l - 2 + 2 ϕ r - 2 + 2 ϕ r + 2 + 2 ϕ l + 2 ) ϕ l - + ( 4 / 3 ) i R ϕ l + ϕ r - ϕ r + * .
x = cos ( τ z ) x - sin ( τ z ) y , y = sin ( τ z ) x + cos ( τ z ) y ,
E ( r , z , t ) = E ( r ) { [ exp ( - i γ z + i ω 0 t ) ϕ r + ( z , t ) + exp ( + i γ z + i ω 0 t ) ϕ r - ( z , t ) ] e r + [ exp ( - i γ z + i ω 0 t ) ϕ l + ( z , t ) + exp ( + i γ z + i ω 0 t ) ϕ l - ( z , t ) ] e l } ,

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