Abstract

We examine the nonlinear interaction between the linearly polarized modes on a few-mode elliptical-core optical fiber. Three modes are involved in this process: the fundamental (LP01) mode and the two second-order symmetry modes, LP11 (even) and LP11 (odd), which have slightly different propagation constants. In practical situations the large δβ = β01β11(β is the propagation constant) precludes any significant coupling between the fundamental and either of the LP11 modes; however, coupling between the two LP11 symmetry modes may occur for sufficient pump intensities. Exact and approximate analytic expressions are given.

© 1992 Optical Society of America

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  1. H. G. Winful, “Self-induced polarization changes in birefringent optical fibers,” Appl. Phys. Lett. 47, 213–215 (1985).
    [CrossRef]
  2. S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, G. I. Stegeman, “Experimental observation of polarization instability in a birefringent optical fiber,” Appl. Phys. Lett. 49, 1224–1226 (1987).
    [CrossRef]
  3. R. J. Black, A. Henault, L. Gagnon, F. Gonthier, S. Lacroix, “Intensity-dependent few-mode light guide interferometry,” in Nonlinear Guided Wave Phenomena: Physics and Applications, Vol. 2 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989) pp. 66–69.
  4. H. G. Park, C. C. Pohalski, B. Y. Kim, “Optical Kerr switch using elliptical core two-mode fiber,” Opt. Lett. 9, 776–778 (1988).
    [CrossRef]
  5. R. V. Plenty, I. H. White, A. R. L. Travis, “Non-linear, two-moded, single-fiber, interferometric switch,” Electron. Lett. 24, 1338–1339 (1988).
    [CrossRef]
  6. S. J. Garth, C. Pask, “Nonlinear polarization behaviour on circular core bimodal optical fibres,” Electron. Lett. 25, 182–183 (1989).
    [CrossRef]
  7. B. Y. Kim, J. N. Blake, S. Y. Huang, H. J. Shaw, “Use of highly elliptically core fibers for two-mode fiber devices,” Opt. Lett. 12, 729–731 (1987).
    [CrossRef] [PubMed]
  8. A. W. Snyder, W. R. Young, “Modes of optical waveguides,” J. Opt. Soc. Am. 68, 297–309 (1978).
    [CrossRef]
  9. J. D. Love, C. D. Hussey, “Variational approximations for higher-order modes of weakly-guiding fibres,” Opt. Quantum Electron. 16, 41–48 (1984).
    [CrossRef]
  10. S. Sudo, H. Itoh, “Efficient non-linear optical fibers and their applications,” Opt. Quantum Electron. 22, 187–212 (1990).
    [CrossRef]
  11. S. Sudo, I. Yokohama, A. Cordova-Plaza, M. M. Fejer, R. L. Byer, “Uniform refractive index cladding for LiNbO3single-crystal fibers,” Appl. Phys. Lett. 56, 1931–1933 (1990).
    [CrossRef]
  12. S. J. Garth, C. Pask, “Polarization rotation in nonlinear bimodal optical fibers,” IEEE J. Lightwave Technol. 8, 129–137 (1990).
    [CrossRef]
  13. R. H. Stolen, J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
    [CrossRef]
  14. C. G. Lambe, C. J. Tranter, Differential Equations for Engineers and Scientists (English U. Press, London, 1961), p. 338.

1990 (3)

S. Sudo, H. Itoh, “Efficient non-linear optical fibers and their applications,” Opt. Quantum Electron. 22, 187–212 (1990).
[CrossRef]

S. Sudo, I. Yokohama, A. Cordova-Plaza, M. M. Fejer, R. L. Byer, “Uniform refractive index cladding for LiNbO3single-crystal fibers,” Appl. Phys. Lett. 56, 1931–1933 (1990).
[CrossRef]

S. J. Garth, C. Pask, “Polarization rotation in nonlinear bimodal optical fibers,” IEEE J. Lightwave Technol. 8, 129–137 (1990).
[CrossRef]

1989 (1)

S. J. Garth, C. Pask, “Nonlinear polarization behaviour on circular core bimodal optical fibres,” Electron. Lett. 25, 182–183 (1989).
[CrossRef]

1988 (2)

H. G. Park, C. C. Pohalski, B. Y. Kim, “Optical Kerr switch using elliptical core two-mode fiber,” Opt. Lett. 9, 776–778 (1988).
[CrossRef]

R. V. Plenty, I. H. White, A. R. L. Travis, “Non-linear, two-moded, single-fiber, interferometric switch,” Electron. Lett. 24, 1338–1339 (1988).
[CrossRef]

1987 (2)

S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, G. I. Stegeman, “Experimental observation of polarization instability in a birefringent optical fiber,” Appl. Phys. Lett. 49, 1224–1226 (1987).
[CrossRef]

B. Y. Kim, J. N. Blake, S. Y. Huang, H. J. Shaw, “Use of highly elliptically core fibers for two-mode fiber devices,” Opt. Lett. 12, 729–731 (1987).
[CrossRef] [PubMed]

1985 (1)

H. G. Winful, “Self-induced polarization changes in birefringent optical fibers,” Appl. Phys. Lett. 47, 213–215 (1985).
[CrossRef]

1984 (1)

J. D. Love, C. D. Hussey, “Variational approximations for higher-order modes of weakly-guiding fibres,” Opt. Quantum Electron. 16, 41–48 (1984).
[CrossRef]

1982 (1)

R. H. Stolen, J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
[CrossRef]

1978 (1)

Assanto, G.

S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, G. I. Stegeman, “Experimental observation of polarization instability in a birefringent optical fiber,” Appl. Phys. Lett. 49, 1224–1226 (1987).
[CrossRef]

Bjorkholm, J. E.

R. H. Stolen, J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
[CrossRef]

Black, R. J.

R. J. Black, A. Henault, L. Gagnon, F. Gonthier, S. Lacroix, “Intensity-dependent few-mode light guide interferometry,” in Nonlinear Guided Wave Phenomena: Physics and Applications, Vol. 2 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989) pp. 66–69.

Blake, J. N.

Byer, R. L.

S. Sudo, I. Yokohama, A. Cordova-Plaza, M. M. Fejer, R. L. Byer, “Uniform refractive index cladding for LiNbO3single-crystal fibers,” Appl. Phys. Lett. 56, 1931–1933 (1990).
[CrossRef]

Cordova-Plaza, A.

S. Sudo, I. Yokohama, A. Cordova-Plaza, M. M. Fejer, R. L. Byer, “Uniform refractive index cladding for LiNbO3single-crystal fibers,” Appl. Phys. Lett. 56, 1931–1933 (1990).
[CrossRef]

Fejer, M. M.

S. Sudo, I. Yokohama, A. Cordova-Plaza, M. M. Fejer, R. L. Byer, “Uniform refractive index cladding for LiNbO3single-crystal fibers,” Appl. Phys. Lett. 56, 1931–1933 (1990).
[CrossRef]

Gagnon, L.

R. J. Black, A. Henault, L. Gagnon, F. Gonthier, S. Lacroix, “Intensity-dependent few-mode light guide interferometry,” in Nonlinear Guided Wave Phenomena: Physics and Applications, Vol. 2 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989) pp. 66–69.

Garth, S. J.

S. J. Garth, C. Pask, “Polarization rotation in nonlinear bimodal optical fibers,” IEEE J. Lightwave Technol. 8, 129–137 (1990).
[CrossRef]

S. J. Garth, C. Pask, “Nonlinear polarization behaviour on circular core bimodal optical fibres,” Electron. Lett. 25, 182–183 (1989).
[CrossRef]

Gonthier, F.

R. J. Black, A. Henault, L. Gagnon, F. Gonthier, S. Lacroix, “Intensity-dependent few-mode light guide interferometry,” in Nonlinear Guided Wave Phenomena: Physics and Applications, Vol. 2 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989) pp. 66–69.

Henault, A.

R. J. Black, A. Henault, L. Gagnon, F. Gonthier, S. Lacroix, “Intensity-dependent few-mode light guide interferometry,” in Nonlinear Guided Wave Phenomena: Physics and Applications, Vol. 2 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989) pp. 66–69.

Huang, S. Y.

Hussey, C. D.

J. D. Love, C. D. Hussey, “Variational approximations for higher-order modes of weakly-guiding fibres,” Opt. Quantum Electron. 16, 41–48 (1984).
[CrossRef]

Itoh, H.

S. Sudo, H. Itoh, “Efficient non-linear optical fibers and their applications,” Opt. Quantum Electron. 22, 187–212 (1990).
[CrossRef]

Kim, B. Y.

H. G. Park, C. C. Pohalski, B. Y. Kim, “Optical Kerr switch using elliptical core two-mode fiber,” Opt. Lett. 9, 776–778 (1988).
[CrossRef]

B. Y. Kim, J. N. Blake, S. Y. Huang, H. J. Shaw, “Use of highly elliptically core fibers for two-mode fiber devices,” Opt. Lett. 12, 729–731 (1987).
[CrossRef] [PubMed]

Lacroix, S.

R. J. Black, A. Henault, L. Gagnon, F. Gonthier, S. Lacroix, “Intensity-dependent few-mode light guide interferometry,” in Nonlinear Guided Wave Phenomena: Physics and Applications, Vol. 2 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989) pp. 66–69.

Lambe, C. G.

C. G. Lambe, C. J. Tranter, Differential Equations for Engineers and Scientists (English U. Press, London, 1961), p. 338.

Love, J. D.

J. D. Love, C. D. Hussey, “Variational approximations for higher-order modes of weakly-guiding fibres,” Opt. Quantum Electron. 16, 41–48 (1984).
[CrossRef]

Park, H. G.

H. G. Park, C. C. Pohalski, B. Y. Kim, “Optical Kerr switch using elliptical core two-mode fiber,” Opt. Lett. 9, 776–778 (1988).
[CrossRef]

Pask, C.

S. J. Garth, C. Pask, “Polarization rotation in nonlinear bimodal optical fibers,” IEEE J. Lightwave Technol. 8, 129–137 (1990).
[CrossRef]

S. J. Garth, C. Pask, “Nonlinear polarization behaviour on circular core bimodal optical fibres,” Electron. Lett. 25, 182–183 (1989).
[CrossRef]

Plenty, R. V.

R. V. Plenty, I. H. White, A. R. L. Travis, “Non-linear, two-moded, single-fiber, interferometric switch,” Electron. Lett. 24, 1338–1339 (1988).
[CrossRef]

Pohalski, C. C.

H. G. Park, C. C. Pohalski, B. Y. Kim, “Optical Kerr switch using elliptical core two-mode fiber,” Opt. Lett. 9, 776–778 (1988).
[CrossRef]

Seaton, C. T.

S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, G. I. Stegeman, “Experimental observation of polarization instability in a birefringent optical fiber,” Appl. Phys. Lett. 49, 1224–1226 (1987).
[CrossRef]

Shaw, H. J.

Snyder, A. W.

Stegeman, G. I.

S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, G. I. Stegeman, “Experimental observation of polarization instability in a birefringent optical fiber,” Appl. Phys. Lett. 49, 1224–1226 (1987).
[CrossRef]

Stolen, R. H.

S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, G. I. Stegeman, “Experimental observation of polarization instability in a birefringent optical fiber,” Appl. Phys. Lett. 49, 1224–1226 (1987).
[CrossRef]

R. H. Stolen, J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
[CrossRef]

Sudo, S.

S. Sudo, I. Yokohama, A. Cordova-Plaza, M. M. Fejer, R. L. Byer, “Uniform refractive index cladding for LiNbO3single-crystal fibers,” Appl. Phys. Lett. 56, 1931–1933 (1990).
[CrossRef]

S. Sudo, H. Itoh, “Efficient non-linear optical fibers and their applications,” Opt. Quantum Electron. 22, 187–212 (1990).
[CrossRef]

Tranter, C. J.

C. G. Lambe, C. J. Tranter, Differential Equations for Engineers and Scientists (English U. Press, London, 1961), p. 338.

Travis, A. R. L.

R. V. Plenty, I. H. White, A. R. L. Travis, “Non-linear, two-moded, single-fiber, interferometric switch,” Electron. Lett. 24, 1338–1339 (1988).
[CrossRef]

Trillo, S.

S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, G. I. Stegeman, “Experimental observation of polarization instability in a birefringent optical fiber,” Appl. Phys. Lett. 49, 1224–1226 (1987).
[CrossRef]

Wabnitz, S.

S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, G. I. Stegeman, “Experimental observation of polarization instability in a birefringent optical fiber,” Appl. Phys. Lett. 49, 1224–1226 (1987).
[CrossRef]

White, I. H.

R. V. Plenty, I. H. White, A. R. L. Travis, “Non-linear, two-moded, single-fiber, interferometric switch,” Electron. Lett. 24, 1338–1339 (1988).
[CrossRef]

Winful, H. G.

H. G. Winful, “Self-induced polarization changes in birefringent optical fibers,” Appl. Phys. Lett. 47, 213–215 (1985).
[CrossRef]

Yokohama, I.

S. Sudo, I. Yokohama, A. Cordova-Plaza, M. M. Fejer, R. L. Byer, “Uniform refractive index cladding for LiNbO3single-crystal fibers,” Appl. Phys. Lett. 56, 1931–1933 (1990).
[CrossRef]

Young, W. R.

Appl. Phys. Lett. (3)

H. G. Winful, “Self-induced polarization changes in birefringent optical fibers,” Appl. Phys. Lett. 47, 213–215 (1985).
[CrossRef]

S. Trillo, S. Wabnitz, R. H. Stolen, G. Assanto, C. T. Seaton, G. I. Stegeman, “Experimental observation of polarization instability in a birefringent optical fiber,” Appl. Phys. Lett. 49, 1224–1226 (1987).
[CrossRef]

S. Sudo, I. Yokohama, A. Cordova-Plaza, M. M. Fejer, R. L. Byer, “Uniform refractive index cladding for LiNbO3single-crystal fibers,” Appl. Phys. Lett. 56, 1931–1933 (1990).
[CrossRef]

Electron. Lett. (2)

R. V. Plenty, I. H. White, A. R. L. Travis, “Non-linear, two-moded, single-fiber, interferometric switch,” Electron. Lett. 24, 1338–1339 (1988).
[CrossRef]

S. J. Garth, C. Pask, “Nonlinear polarization behaviour on circular core bimodal optical fibres,” Electron. Lett. 25, 182–183 (1989).
[CrossRef]

IEEE J. Lightwave Technol. (1)

S. J. Garth, C. Pask, “Polarization rotation in nonlinear bimodal optical fibers,” IEEE J. Lightwave Technol. 8, 129–137 (1990).
[CrossRef]

IEEE J. Quantum Electron. (1)

R. H. Stolen, J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1072 (1982).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Lett. (2)

H. G. Park, C. C. Pohalski, B. Y. Kim, “Optical Kerr switch using elliptical core two-mode fiber,” Opt. Lett. 9, 776–778 (1988).
[CrossRef]

B. Y. Kim, J. N. Blake, S. Y. Huang, H. J. Shaw, “Use of highly elliptically core fibers for two-mode fiber devices,” Opt. Lett. 12, 729–731 (1987).
[CrossRef] [PubMed]

Opt. Quantum Electron. (2)

J. D. Love, C. D. Hussey, “Variational approximations for higher-order modes of weakly-guiding fibres,” Opt. Quantum Electron. 16, 41–48 (1984).
[CrossRef]

S. Sudo, H. Itoh, “Efficient non-linear optical fibers and their applications,” Opt. Quantum Electron. 22, 187–212 (1990).
[CrossRef]

Other (2)

R. J. Black, A. Henault, L. Gagnon, F. Gonthier, S. Lacroix, “Intensity-dependent few-mode light guide interferometry,” in Nonlinear Guided Wave Phenomena: Physics and Applications, Vol. 2 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989) pp. 66–69.

C. G. Lambe, C. J. Tranter, Differential Equations for Engineers and Scientists (English U. Press, London, 1961), p. 338.

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Figures (3)

Fig. 1
Fig. 1

Schematic of the output intensity pattern of an elliptical-core two-mode fiber.

Fig. 2
Fig. 2

Normalized power c22P/B versus normalized distance Bz for the three-mode interaction. The three modes are the fundamental mode (solid curves), the LP11 (odd) mode (dashed curves), and the LP11 (even) mode (dotted curves). δβ = 104 (m−1), and fiber parameters are given in Subsection 2.B. (a) c22p0 = B = 104 (m−1), (b) c22p0 = B = 500 (m−1), (c) c22p0 = B = 1 (m−1).

Fig. 3
Fig. 3

Normalized power versus normalized distance for the same interaction as in Fig. 2: (a) c22p0 = 1 (m−1) and B = 10 (m−1), (b) c22p0 = 10 (m−1) and B = 1 (m−1).

Tables (1)

Tables Icon

Table 1 Transverse Field Distribution, Propagation Constant, and Intensity Pattern for the Three Modes on a Bimodal Noncircular-Core Optical Fibera

Equations (69)

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E ( r , ϕ , z , t ) = ½ ( E 1 + E 2 e + E 2 o ) = ½ [ a 1 ( z ) Ψ 1 ( r , ϕ ) exp ( i β 1 z ) + a 2 e ( z ) Ψ 2 e ( r , ϕ ) exp ( i β 2 e z ) + a 2 o ( z ) Ψ 2 o ( r , ϕ ) exp ( i β 2 o z ] exp ( - i ω t ) ,
d A 1 d z = i c 11 A 1 2 A 1 + 2 i c 12 ( A 2 e 2 + A 2 o 2 ) A 1 + i c 12 ( A 2 e 2 + A 2 o 2 ) A 1 * exp ( - 2 i δ β z ) ,
d A 2 e d z = i 3 2 c 22 A 2 e 2 A 2 e + i c 22 [ A 2 o 2 A 2 e + ½ A 2 o 2 A 2 e * exp ( - 2 i B z ) ] + i c 12 [ 2 A 1 2 A 2 e + A 1 2 A 2 e * exp ( 2 i δ β z ) ] ,
d A 2 o d z = i 3 2 c 22 A 2 o 2 A 2 o + i c 22 [ A 2 e 2 A 2 o + ½ A 2 e 2 A 2 o * exp ( 2 i B z ) ] + i c 12 [ 2 A 1 2 A 2 o + A 1 2 A 2 o * exp ( 2 i δ β z ) ] .
A j 2 = P j = 1 2 c 0 n co a j 2 0 Ψ j · Ψ j d r ,
c i j = 3 χ 4 λ n co 2 c o i i j j .
i i j j = 0 Ψ i 2 Ψ j 2 r d r 0 Ψ i 2 r d r 0 Ψ j 2 r d r .
A j = P j exp ( i φ j ) .
d P 1 d z = 2 c 12 P 2 e P 1 sin 2 θ e + 2 c 12 P 2 o P 1 sin 2 θ o ,
d P 2 e d z = - 2 c 12 P 2 e P 1 sin 2 θ e + 1 2 c 12 P 2 e P 2 o sin 2 θ ,
d P 2 o d z = - 2 c 12 P 2 o P 1 sin 2 θ o - 1 2 c 12 P 2 e P 2 o sin 2 θ .
d θ d z = B + d φ 2 e d z - d φ 2 o d z = B + c 12 P 1 ( cos 2 θ e - cos 2 θ o ) + c 22 ( P 2 e - P 2 o ) sin 2 θ ,
d θ e d z = δ β + d φ 1 d z - d φ 2 e d z = δ β + c 11 P 1 - 1 2 c 22 [ 3 P 2 e + P 2 o ( 2 + cos 2 θ ) ] + c 12 [ ( P 2 e - P 1 ) ( 2 + cos 2 θ e ) + 2 P 2 o cos 2 θ ] ,
d θ o d z = δ β + d φ 1 d z - d φ 2 o d z = δ β + c 11 P 1 - 1 2 c 22 [ 3 P 2 o + P 2 e ( 2 + cos 2 θ ) ] + c 12 [ ( P 2 o - P 1 ) ( 2 + cos 2 θ o ) + 2 P 2 e cos 2 θ ] .
P tot = P 1 + P 2 e + P 2 o const . ,
δ β = β 1 - β 2 ,             B = β 2 e - β 2 o .
ψ 1 ( r ) exp ( - r 2 / 2 r 0 2 ) ,             ψ 2 ( r ) ( r / r 1 ) exp ( - r 2 / 2 r 1 2 ) ,
111 2 1122 2 2222 = 1 r 0 2 = ln ( V 2 ) ρ 2 ,
V = ρ k n co ( 2 Δ ) 1 / 2 ,             Δ = ½ ( 1 - n cl 2 / n co 2 ) ,
c 11 ln ( V 2 ) λ ρ 2 ( 6.28 × 10 - 9 ) ( μ m W ) - 1 ,
δ β = β 1 - β 2 1 2 ρ ( U 2 2 - U 1 2 ) ρ k n co ( μ m - 1 ) ,
B = β 2 e - β 2 o = e 2 4 ρ U 2 2 ρ k n co K 1 2 ( W 2 ) K 0 ( W 2 ) K 2 ( W 2 ) ( μ m - 1 ) , W 2 = ( V 2 - U 2 2 ) 1 / 2 ,
c 11 P tot = 1.1 × 10 - 3 P tot ( m - 1 ) , B = 6.6 × 10 3 e 2 ( m - 1 ) , δ β = 1.3 × 10 4 ( m - 1 ) ,
d P 1 d z = - d P 2 d z = 2 c 12 P 1 P 2 sin 2 θ ,
d θ d z = δ β + c 11 P 1 - 3 2 c 22 P 2 - c 12 ( P 1 - P 2 ) ( 2 + 2 cos 2 θ ) ,
P 1 + P 2 = P , a constant .
P 1 ( z ) P 2 ( z ) = P 1 ( 0 ) P 2 ( 0 ) exp [ 2 c 12 P δ β sin 2 ( δ β z ) ] .
P 1 ( z ) P 2 ( z ) ½ P
d θ d z δ β + 1 2 ( c 11 - 3 2 c 22 ) P .
δ β eff = δ β + ½ ( c 11 - ³ / c 22 ) P δ β + c 11 P .
d P 1 d z 0 ,
d P 2 e d z = d P 2 o d z c 22 P 2 e P 2 o sin 2 θ ,
d θ d z B + c 22 ( P 2 e - P 2 o ) sin 2 θ .
P 2 ( z ) = P 2 ( 1 ± B c 22 P { [ 1 - 2 m + m sn 2 ( m ν B z ) ] ν 2 1 - m sn 2 ( m ν B z ) - 1 } ) ,
m = 1 2 ( 1 - [ 1 + ( c 22 P / B ) ( Q / P ) ] ν 2 ) ,
ν = [ 1 + 2 ( c 22 P / B ) ( Q / P ) + ( c 22 P / B ) 2 ] 1 / 4
P = P 2 e ( 0 ) + P 2 o ( 0 ) ,             Q = P 2 e ( 0 ) - P 2 o ( 0 ) .
Z p B z p 4 K ( m ) / ν ,
m = ¼ ( c 22 P / B ) 2 [ 1 - ( Q / P ) 2 ] 1 , ν 2 = 1 + ( c 22 P / B ) ( Q / P ) + ½ ( c 22 P / B ) 2 [ 1 - ( Q / P ) 2 ] .
P 2 e ( z ) ( P / 2 ) { 1 + ( Q / P ) + ½ ( c 22 P / B ) [ 1 - ( Q / P ) 2 ] sin 2 ( B z ) } .
m ½ [ 1 - ( Q / P ) ] ,             ν 2 c 22 P / B .
P 2 e ( z ) P 2 [ 1 + ( Q / P ) d n 2 ( m ν B z ) ] ,
P 2 e ( z ) P [ 1 - ( / P ) cos 2 ( ν B z ) ] .
P 2 e ( z ) d n 2 ( m ν B z ) .
P 2 e ( z ) P .
2 E = n 2 c 2 2 E t 2 + μ 0 2 P NL δ t 2 .
P NL = 0 χ [ 2 ( E · E * ) E + ( E · E ) E * ] .
j β j d a j d z Ψ j exp [ i ( β j z - ω t ) ] = - i μ 0 2 P NL t 2 .
Ψ j · Ψ k d r = δ j k N j 2 ,
P NL = 0 χ { 6 E 1 2 ( ½ E 1 + E 2 e + E 2 o ) + 6 E 2 e 2 × ( E 1 + ½ E 2 e + E 2 o ) + 6 E 2 o 2 ( E 1 + E 2 e + ½ E 2 o ) + 3 E 1 2 ( E 2 e * + E 2 o * ) + 3 E 2 e 2 ( E 1 * + E 2 o * ) + 3 E 2 o 2 ( E 1 * + E 2 e * ) + 6 E 1 * E 2 e E 2 o + 6 E 1 E 2 e * E 2 o + 6 E 1 E 2 e E 2 o * } x ^ .
u ˙ = - v ˙ = a u v sin θ ,
θ ˙ = b + ( c - a cos θ ) ( u - v ) ,
p = u + v = const . ,
Γ = b ( u - v ) - 2 u v ( c - a cos θ ) .
θ ¨ = - ½ a 2 p 2 sin 2 θ + a ( c p 2 + Γ ) sin θ ,
θ ˙ 2 = α cos 2 θ + β cos θ + γ ,
α = a 2 p 2 ,             β = - 2 a ( c p 2 + Γ ) ,             γ = b 2 + c 2 p 2 + 2 c Γ ,
x = ( α - β + γ α + β + γ ) 1 / 4 tan θ 2 = ν tan θ 2 .
x ˙ 2 = λ ( 1 + 2 μ x 2 + x 4 ) ,
λ = ¼ [ ( α + γ ) 2 - β 2 ] 1 / 2 ,             μ = ( γ - α ) / 4 λ .
x 2 = m sn 2 ( m ω z ) ,
x 2 = 1 - cn ( m ω z ) 1 + cn ( m ω z ) ,
x 2 = ( 1 - m ) 1 / 2 sn 2 ( m ω z ) cn 2 ( m ω z ) ,
2 u - p = θ ˙ - b c - a cos θ .
2 u - p = 2 ( x ˙ / ν ) - b [ 1 + ( x 2 / v 2 ) ( c - a ) + ( x 2 / ν 2 ) ( c + a ) ,
d u d z = - d v d z = u v sin θ , d θ d z = 2 B + ( 1 - cos θ ) ( u - v ) .
p = u 0 + v 0 , Γ = 2 B ( u 0 - v 0 ) = 2 B q ,
λ = B 2 [ 1 + ( p / B ) 2 + 2 ( p / B ) ( q / p ) ] 1 / 2 , μ = 1 + ( p / B ) ( q / p ) [ 1 + ( p / B ) 2 + 2 ( p / B ) ( q / p ) ] 1 / 2 .
x = ν tan ( θ / 2 ) = sn ( m ν B z ) d n ( m ν B z ) cn ( m ν B z ) ,

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