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 and Y. Kodama, Proc. IEEE 69, 1145 (1981).
    [Crossref]
  2. R. H. Stolen, J. Botineau, and A. Ashkin, Opt. Lett. 7, 512 (1982).
    [Crossref] [PubMed]
  3. K. Kitayama, Y. Kimura, and 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, and S. Wabnitz, Opt. Lett. 11, 42 (1986).
    [Crossref]
  6. See, e.g., N. Tzoar and M. Jain, Phys. Rev. A 23, 1266 (1981).
    [Crossref]
  7. B. Crosignani, A. Cutolo, and P. Di Porto, J. Opt. Soc. Am. 72, 1136 (1982).
    [Crossref]
  8. B. Crosignani and P. Di Porto, Opt. Acta 32, 1251 (1985).
  9. B. Crosignani, B. Daino, and 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, and 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 and E. Spitz, eds. (Nijhoff, The Hague, 1984), Vol. 1, p. 1.
  15. P. D. Maker and R. W. Terhune, Phys. Rev. 137, 801 (1965).
    [Crossref]
  16. See, e.g., B. Crosignani and P. Di Porto, J. Opt. Soc. Am. 72, 1553 (1982).
    [Crossref]
  17. F. Matera and S. Wabnitz, Opt. Lett. 11, 467 (1986).
    [Crossref] [PubMed]
  18. A. E. Kaplan and 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 and P. Di Porto, Opt. Acta 32, 1251 (1985).

K. Kitayama, Y. Kimura, and 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 and C. T. Law, IEEE J. Quantum Electron. QE-21, 1529 (1985).
[Crossref]

1982 (3)

1981 (3)

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

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

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

1965 (1)

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

Ashkin, A.

Barlow, A. J.

Botineau, J.

Crosignani, B.

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

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

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

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

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

Cutolo, A.

Daino, B.

Di Porto, P.

Gregori, G.

Hasegawa, A.

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

Jain, M.

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

Kaplan, A. E.

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

Kimura, Y.

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

Kitayama, K.

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

Kodama, Y.

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

Law, C. T.

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

Maker, P. D.

P. D. Maker and 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, and S. Seikai, Appl. Phys. Lett. 46, 317, 623 (1985).
[Crossref]

Stolen, R. H.

Terhune, R. W.

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

Tzoar, N.

See, e.g., N. Tzoar and 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, and 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 and 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 and P. Di Porto, Opt. Acta 32, 1251 (1985).

Opt. Lett. (3)

Phys. Rev. (1)

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

Phys. Rev. A (1)

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

Proc. IEEE (1)

A. Hasegawa and 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 and 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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