Abstract

The effect of a uniform grating on the evolution of the orthogonal polarization modes in a nonlinear birefringent waveguide is examined for the first time to our knowledge. The problem involves the interplay of the linear coupling that is due to the grating, the linear birefringence, and the nonlinear mode coupling. After examining the basic formulation and methods of solution, we present a series of results demonstrating a variety of nonlinear phenomena, such as bistability. Other two-mode systems are predicted to behave in a similar way. The possible implementation of three logic gates from one device configuration is discussed.

© 1994 Optical Society of America

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References

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  1. D. B. Ostrowsky and R. Reinisch, eds., Guided Wave Nonlinear Optics (Kluwer, Dordrecht, The Netherlands, 1992).
    [Crossref]
  2. A. C. Newel and J. V. Moloney, Nonlinear Optics (Addison-Wesley, Palo Alto, Calif., 1992).
  3. G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
    [Crossref]
  4. H. G. Winful, “Nonlinear optical phenomena in single-mode fibers,” in Optical Fiber Transmission, E. E. Basch, ed.-in-chief (Sams, Indianapolis, Ind., 1987), pp. 179–240.
  5. H. G. Winful, “Self-induced polarization changes in birefringent optical fibers,” Appl. Phys. Lett. 47, 213–215 (1985).
    [Crossref]
  6. H. G. Winful, “Polarization instabilities in birefringent nonlinear media: application to fiber-optic devices,” Opt. Lett. 11, 33–35 (1986).
    [Crossref] [PubMed]
  7. S. J. Garth and C. Pask, “Polarization behavior in lossy nonlinear birefringent optical fibers,” Opt. Quantum Electron. 22, 37–53 (1990).
    [Crossref]
  8. S. E. Feldman, D. A. Weinberger, and H. G. Winful, “Observation of polarization instabilities and modulational gain in a low-birefringent optical fiber,” Opt. Lett. 15, 311–313 (1990).
    [Crossref]
  9. H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
    [Crossref]
  10. A. Mecozzi, S. Trillo, and S. Wabnitz, “All-optical limiting, thresholding, and switching in a nonlinear DFB waveguide,” presented at the Thirteenth European Conference on Optical Communications, Finland, 1987.
  11. A. Mecozzi, S. Trillo, and S. Wabnitz, “Spatial instabilities, all-optical limiting, and thresholding in nonlinear distributed feedback devices,” Opt. Lett. 12, 1008–1010 (1987).
    [Crossref] [PubMed]
  12. B. Svenson, G. Assanto, and G. I. Stegeman, “Guided-wave optical bistability and limiting in zinc sulfide thin films,” J. Appl. Phys. 67, 3882–3885 (1990).
    [Crossref]
  13. G. Assanto, J. E. Ehrlich, and G. I. Stegeman, “Feedback-enhanced bistability in grating coupling into InSb waveguides,” Opt. Lett. 15, 411–413 (1990).
    [Crossref] [PubMed]
  14. J. E. Ehrlich, G. Assanto, G. I. Stegeman, and T. H. Chiu, “Guided-wave optical bistability in indium antimonide thin film,” IEEE J. Quantum Electron. 27, 809–816 (1991).
    [Crossref]
  15. Y. Chen, “Grating-assisted nonlinear couplers,” J. Mod. Opt. 38, 1731–1738 (1991).
    [Crossref]
  16. S. Trillo, S. Wabnitz, and G. I. Stegeman, “Nonlinear codirectional guided wave mode conversion in grating structures,” J. Lightwave Technol. 6, 971–976 (1988).
    [Crossref]
  17. R. A. Betts, T. Tjugiarto, Y. L. Xue, and P. L. Chu, “Nonlinear refractive index in erbium doped optical fiber: theory and experiment,” IEEE J. Quantum Electron. 27, 908–913 (1991).
    [Crossref]
  18. P. L. Chu and B. Wu, “Optical switching in twin-core erbium-doped fibers,” Opt. Lett. 17, 255–257 (1992).
    [Crossref] [PubMed]
  19. G. I. Stegeman, “Guided wave approaches to optical bistability,” IEEE J. Quantum Electron. QE-18, 1610–1619 (1982).
    [Crossref]
  20. G. Assanto and G. I. Stegeman, “Optical bistability in nonlocally nonlinear periodic structures,” Appl. Phys. Lett. 56, 2285–2287 (1990).
    [Crossref]

1992 (1)

1991 (3)

J. E. Ehrlich, G. Assanto, G. I. Stegeman, and T. H. Chiu, “Guided-wave optical bistability in indium antimonide thin film,” IEEE J. Quantum Electron. 27, 809–816 (1991).
[Crossref]

Y. Chen, “Grating-assisted nonlinear couplers,” J. Mod. Opt. 38, 1731–1738 (1991).
[Crossref]

R. A. Betts, T. Tjugiarto, Y. L. Xue, and P. L. Chu, “Nonlinear refractive index in erbium doped optical fiber: theory and experiment,” IEEE J. Quantum Electron. 27, 908–913 (1991).
[Crossref]

1990 (6)

G. Assanto and G. I. Stegeman, “Optical bistability in nonlocally nonlinear periodic structures,” Appl. Phys. Lett. 56, 2285–2287 (1990).
[Crossref]

G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
[Crossref]

S. J. Garth and C. Pask, “Polarization behavior in lossy nonlinear birefringent optical fibers,” Opt. Quantum Electron. 22, 37–53 (1990).
[Crossref]

S. E. Feldman, D. A. Weinberger, and H. G. Winful, “Observation of polarization instabilities and modulational gain in a low-birefringent optical fiber,” Opt. Lett. 15, 311–313 (1990).
[Crossref]

B. Svenson, G. Assanto, and G. I. Stegeman, “Guided-wave optical bistability and limiting in zinc sulfide thin films,” J. Appl. Phys. 67, 3882–3885 (1990).
[Crossref]

G. Assanto, J. E. Ehrlich, and G. I. Stegeman, “Feedback-enhanced bistability in grating coupling into InSb waveguides,” Opt. Lett. 15, 411–413 (1990).
[Crossref] [PubMed]

1988 (1)

S. Trillo, S. Wabnitz, and G. I. Stegeman, “Nonlinear codirectional guided wave mode conversion in grating structures,” J. Lightwave Technol. 6, 971–976 (1988).
[Crossref]

1987 (1)

1986 (1)

1985 (1)

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

1982 (1)

G. I. Stegeman, “Guided wave approaches to optical bistability,” IEEE J. Quantum Electron. QE-18, 1610–1619 (1982).
[Crossref]

1979 (1)

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[Crossref]

Assanto, G.

J. E. Ehrlich, G. Assanto, G. I. Stegeman, and T. H. Chiu, “Guided-wave optical bistability in indium antimonide thin film,” IEEE J. Quantum Electron. 27, 809–816 (1991).
[Crossref]

B. Svenson, G. Assanto, and G. I. Stegeman, “Guided-wave optical bistability and limiting in zinc sulfide thin films,” J. Appl. Phys. 67, 3882–3885 (1990).
[Crossref]

G. Assanto and G. I. Stegeman, “Optical bistability in nonlocally nonlinear periodic structures,” Appl. Phys. Lett. 56, 2285–2287 (1990).
[Crossref]

G. Assanto, J. E. Ehrlich, and G. I. Stegeman, “Feedback-enhanced bistability in grating coupling into InSb waveguides,” Opt. Lett. 15, 411–413 (1990).
[Crossref] [PubMed]

Betts, R. A.

R. A. Betts, T. Tjugiarto, Y. L. Xue, and P. L. Chu, “Nonlinear refractive index in erbium doped optical fiber: theory and experiment,” IEEE J. Quantum Electron. 27, 908–913 (1991).
[Crossref]

Chen, Y.

Y. Chen, “Grating-assisted nonlinear couplers,” J. Mod. Opt. 38, 1731–1738 (1991).
[Crossref]

Chiu, T. H.

J. E. Ehrlich, G. Assanto, G. I. Stegeman, and T. H. Chiu, “Guided-wave optical bistability in indium antimonide thin film,” IEEE J. Quantum Electron. 27, 809–816 (1991).
[Crossref]

Chu, P. L.

P. L. Chu and B. Wu, “Optical switching in twin-core erbium-doped fibers,” Opt. Lett. 17, 255–257 (1992).
[Crossref] [PubMed]

R. A. Betts, T. Tjugiarto, Y. L. Xue, and P. L. Chu, “Nonlinear refractive index in erbium doped optical fiber: theory and experiment,” IEEE J. Quantum Electron. 27, 908–913 (1991).
[Crossref]

Ehrlich, J. E.

J. E. Ehrlich, G. Assanto, G. I. Stegeman, and T. H. Chiu, “Guided-wave optical bistability in indium antimonide thin film,” IEEE J. Quantum Electron. 27, 809–816 (1991).
[Crossref]

G. Assanto, J. E. Ehrlich, and G. I. Stegeman, “Feedback-enhanced bistability in grating coupling into InSb waveguides,” Opt. Lett. 15, 411–413 (1990).
[Crossref] [PubMed]

Feldman, S. E.

Garmire, E.

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[Crossref]

Garth, S. J.

S. J. Garth and C. Pask, “Polarization behavior in lossy nonlinear birefringent optical fibers,” Opt. Quantum Electron. 22, 37–53 (1990).
[Crossref]

Marburger, J. H.

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[Crossref]

Mecozzi, A.

A. Mecozzi, S. Trillo, and S. Wabnitz, “Spatial instabilities, all-optical limiting, and thresholding in nonlinear distributed feedback devices,” Opt. Lett. 12, 1008–1010 (1987).
[Crossref] [PubMed]

A. Mecozzi, S. Trillo, and S. Wabnitz, “All-optical limiting, thresholding, and switching in a nonlinear DFB waveguide,” presented at the Thirteenth European Conference on Optical Communications, Finland, 1987.

Moloney, J. V.

A. C. Newel and J. V. Moloney, Nonlinear Optics (Addison-Wesley, Palo Alto, Calif., 1992).

Newel, A. C.

A. C. Newel and J. V. Moloney, Nonlinear Optics (Addison-Wesley, Palo Alto, Calif., 1992).

Pask, C.

S. J. Garth and C. Pask, “Polarization behavior in lossy nonlinear birefringent optical fibers,” Opt. Quantum Electron. 22, 37–53 (1990).
[Crossref]

Stegeman, G. I.

J. E. Ehrlich, G. Assanto, G. I. Stegeman, and T. H. Chiu, “Guided-wave optical bistability in indium antimonide thin film,” IEEE J. Quantum Electron. 27, 809–816 (1991).
[Crossref]

G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
[Crossref]

G. Assanto and G. I. Stegeman, “Optical bistability in nonlocally nonlinear periodic structures,” Appl. Phys. Lett. 56, 2285–2287 (1990).
[Crossref]

G. Assanto, J. E. Ehrlich, and G. I. Stegeman, “Feedback-enhanced bistability in grating coupling into InSb waveguides,” Opt. Lett. 15, 411–413 (1990).
[Crossref] [PubMed]

B. Svenson, G. Assanto, and G. I. Stegeman, “Guided-wave optical bistability and limiting in zinc sulfide thin films,” J. Appl. Phys. 67, 3882–3885 (1990).
[Crossref]

S. Trillo, S. Wabnitz, and G. I. Stegeman, “Nonlinear codirectional guided wave mode conversion in grating structures,” J. Lightwave Technol. 6, 971–976 (1988).
[Crossref]

G. I. Stegeman, “Guided wave approaches to optical bistability,” IEEE J. Quantum Electron. QE-18, 1610–1619 (1982).
[Crossref]

Svenson, B.

B. Svenson, G. Assanto, and G. I. Stegeman, “Guided-wave optical bistability and limiting in zinc sulfide thin films,” J. Appl. Phys. 67, 3882–3885 (1990).
[Crossref]

Tjugiarto, T.

R. A. Betts, T. Tjugiarto, Y. L. Xue, and P. L. Chu, “Nonlinear refractive index in erbium doped optical fiber: theory and experiment,” IEEE J. Quantum Electron. 27, 908–913 (1991).
[Crossref]

Trillo, S.

S. Trillo, S. Wabnitz, and G. I. Stegeman, “Nonlinear codirectional guided wave mode conversion in grating structures,” J. Lightwave Technol. 6, 971–976 (1988).
[Crossref]

A. Mecozzi, S. Trillo, and S. Wabnitz, “Spatial instabilities, all-optical limiting, and thresholding in nonlinear distributed feedback devices,” Opt. Lett. 12, 1008–1010 (1987).
[Crossref] [PubMed]

A. Mecozzi, S. Trillo, and S. Wabnitz, “All-optical limiting, thresholding, and switching in a nonlinear DFB waveguide,” presented at the Thirteenth European Conference on Optical Communications, Finland, 1987.

Wabnitz, S.

S. Trillo, S. Wabnitz, and G. I. Stegeman, “Nonlinear codirectional guided wave mode conversion in grating structures,” J. Lightwave Technol. 6, 971–976 (1988).
[Crossref]

A. Mecozzi, S. Trillo, and S. Wabnitz, “Spatial instabilities, all-optical limiting, and thresholding in nonlinear distributed feedback devices,” Opt. Lett. 12, 1008–1010 (1987).
[Crossref] [PubMed]

A. Mecozzi, S. Trillo, and S. Wabnitz, “All-optical limiting, thresholding, and switching in a nonlinear DFB waveguide,” presented at the Thirteenth European Conference on Optical Communications, Finland, 1987.

Weinberger, D. A.

Winful, H. G.

S. E. Feldman, D. A. Weinberger, and H. G. Winful, “Observation of polarization instabilities and modulational gain in a low-birefringent optical fiber,” Opt. Lett. 15, 311–313 (1990).
[Crossref]

H. G. Winful, “Polarization instabilities in birefringent nonlinear media: application to fiber-optic devices,” Opt. Lett. 11, 33–35 (1986).
[Crossref] [PubMed]

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

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[Crossref]

H. G. Winful, “Nonlinear optical phenomena in single-mode fibers,” in Optical Fiber Transmission, E. E. Basch, ed.-in-chief (Sams, Indianapolis, Ind., 1987), pp. 179–240.

Wright, E. M.

G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
[Crossref]

Wu, B.

Xue, Y. L.

R. A. Betts, T. Tjugiarto, Y. L. Xue, and P. L. Chu, “Nonlinear refractive index in erbium doped optical fiber: theory and experiment,” IEEE J. Quantum Electron. 27, 908–913 (1991).
[Crossref]

Appl. Phys. Lett. (3)

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

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[Crossref]

G. Assanto and G. I. Stegeman, “Optical bistability in nonlocally nonlinear periodic structures,” Appl. Phys. Lett. 56, 2285–2287 (1990).
[Crossref]

IEEE J. Quantum Electron. (3)

J. E. Ehrlich, G. Assanto, G. I. Stegeman, and T. H. Chiu, “Guided-wave optical bistability in indium antimonide thin film,” IEEE J. Quantum Electron. 27, 809–816 (1991).
[Crossref]

R. A. Betts, T. Tjugiarto, Y. L. Xue, and P. L. Chu, “Nonlinear refractive index in erbium doped optical fiber: theory and experiment,” IEEE J. Quantum Electron. 27, 908–913 (1991).
[Crossref]

G. I. Stegeman, “Guided wave approaches to optical bistability,” IEEE J. Quantum Electron. QE-18, 1610–1619 (1982).
[Crossref]

J. Appl. Phys. (1)

B. Svenson, G. Assanto, and G. I. Stegeman, “Guided-wave optical bistability and limiting in zinc sulfide thin films,” J. Appl. Phys. 67, 3882–3885 (1990).
[Crossref]

J. Lightwave Technol. (1)

S. Trillo, S. Wabnitz, and G. I. Stegeman, “Nonlinear codirectional guided wave mode conversion in grating structures,” J. Lightwave Technol. 6, 971–976 (1988).
[Crossref]

J. Mod. Opt. (1)

Y. Chen, “Grating-assisted nonlinear couplers,” J. Mod. Opt. 38, 1731–1738 (1991).
[Crossref]

Opt. Lett. (5)

Opt. Quantum Electron. (2)

S. J. Garth and C. Pask, “Polarization behavior in lossy nonlinear birefringent optical fibers,” Opt. Quantum Electron. 22, 37–53 (1990).
[Crossref]

G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
[Crossref]

Other (4)

H. G. Winful, “Nonlinear optical phenomena in single-mode fibers,” in Optical Fiber Transmission, E. E. Basch, ed.-in-chief (Sams, Indianapolis, Ind., 1987), pp. 179–240.

D. B. Ostrowsky and R. Reinisch, eds., Guided Wave Nonlinear Optics (Kluwer, Dordrecht, The Netherlands, 1992).
[Crossref]

A. C. Newel and J. V. Moloney, Nonlinear Optics (Addison-Wesley, Palo Alto, Calif., 1992).

A. Mecozzi, S. Trillo, and S. Wabnitz, “All-optical limiting, thresholding, and switching in a nonlinear DFB waveguide,” presented at the Thirteenth European Conference on Optical Communications, Finland, 1987.

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Figures (8)

Fig. 1
Fig. 1

Plots of Pxout, Pyout versus Pin (= Pxin + Pyin) for an LP input of 45° with initial linear birefringence δβyx = −2 and initial Δδxg = 0 and various values of the linear coupling coefficient K: (a) K = 0, (b) K = 0.1, (c) K = 0.5, (d) K = 1.

Fig. 2
Fig. 2

Pxout, Pyout versus Pin for an LP input of 45° for K = 2, Δβxg = 0, and (a) δβyx = −2, (b) Δβyx = +2. The no-coupling curve (dashed line) would be followed by Pxout and Pyout in the absence of a grating and nonlinear coupling.

Fig. 3
Fig. 3

Pxout, Pyout versus Pin for an LP input of 45° for K = 2, Δβxg = −2, and δβyx = −2. Also shown is the no-coupling curve (dashed line).

Fig. 4
Fig. 4

Pxout, versus Pxin for K = 2 and Δβxg = −2: a, NLDF case9; b, NLDF case with XPM term only; c, NLDF case with SPM term only; d, no-grating case (i.e., K = 0); e, linear case (i.e., neither XPM nor SPM terms).

Fig. 5
Fig. 5

Detuning curve for a linear Bragg reflector with the linear coupling coefficient K = 2.

Fig. 6
Fig. 6

Pxout, Pyout versus Pin for an LP input of 45° for K = 2, Δβxg = 0, and δβyx = 0 with the nonlinear coupling power terms artificially dropped from Eqs. (5).

Fig. 7
Fig. 7

(a) Pxout versus Pxin with K = 2 for the NLDFB (Δβxg = 0, δβyx = −2) and the NLDF (Δβxg = 0) structures.9 (b) Pyout versus Pyin with K = 2 for the NLDFB (Δβxg = 0, Δβyx = −2) and the NLDF (Δβyg = −2) structures.9

Fig. 8
Fig. 8

Plots of Pxout, Pyout versus Pin for an LP input at 45° for K = 2, Δβxg = 0, and δβyx = −2 [as in Fig. 2(a)]. Shown are the power levels Pth, Pin1, and Pin2 used for the implementation of logic gates.

Equations (21)

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n ( z ) = n 0 + n 1 cos ( 2 β g z ) ,
P = 0 n 0 n 1 [ exp ( i 2 β g z ) + exp ( i 2 β g z ) ] E + 0 χ ( 3 ) [ 2 ( E · E * ) E + ( E · E ) E * ] ,
E = 1 2 [ A x F ( z ) exp ( i β x z ) + A x B ( z ) exp ( i β x z ) ] ψ ( r ) × exp ( i ω t ) x ˆ + 1 2 [ A x F ( z ) exp ( i β y z ) + A y B ( z ) exp ( i β y z ) ] × ψ ( r ) exp ( i ω t ) y ˆ ,
j = x , y β j d A j F d z ψ ( r ) e ˆ j exp [ i ( β j z ω t ) ] j = x , y β j d A j B d z ψ ( r ) e ˆ j exp [ i ( β j z ω t ) ] = i μ 0 d 2 P d t 2 ,
d a x F d Z = i K a x B exp ( i 2 Δ β x g Z ) + i { | a x F | 2 + 2 | a x B | 2 + 2 3 | a y F | 2 + 2 3 | a y B | 2 } a x F + i { 1 3 a y F 2 a x F * exp ( i 2 δ β y x Z ) + 2 3 a x B a y F a y B * exp ( i 2 δ β y x Z ) + 2 3 a x B * a y F a y B } ,
d a x B d Z = i K a x F exp ( i 2 Δ β x g Z ) i { | a x B | 2 + 2 | a x F | 2 + 2 3 | a y F | 2 + 2 3 | a y B | 2 } a x B + i { 1 3 a y B 2 a x B * exp ( i 2 δ β y x Z ) + 2 3 a x F a y F * a y B × exp ( i 2 δ β y x Z ) + 2 3 a x F * a y F a y B } ,
d a y F d Z = i K a y B exp [ i 2 ( Δ β x g + δ β y x ) Z ] + i { | a y F | 2 + 2 | a y B | 2 + 2 3 | a x F | 2 + 2 3 | a x B | 2 } a y F + i { 1 3 a x F 2 a y F * exp ( i 2 δ β y x Z ) + 2 3 a y B a x F a x B * exp ( i 2 δ β y x Z ) + 2 3 a y B * a x F a x B } ,
d a y B d Z = i K a y F exp [ i 2 ( Δ β x g + δ β y x ) Z ] i { | a y B | 2 + 2 | a y F | 2 + 2 3 | a x F | 2 + 2 3 | a x B | 2 } a y B i { 1 3 a x B 2 a y B * exp ( i 2 δ β y x Z ) + 2 3 a y F a x F * a x B exp ( i 2 δ β y x Z ) + 2 3 a y F * a x F a x B } .
Δ β x g = ( β x β g ) L , δ β y x = ( β y β x ) L , K = 1 2 β g n 1 n 0 L ,
a x , y F , B = C NL A ¯ x , y F , B ,
C NL = 3 ω χ ( 3 ) 4 c 2 n 0 2 0 A eff L ( W 1 ) , A eff = 2 π [ 0 ψ 2 ( r ) r d r ] 2 0 ψ 4 ( r ) r d r , A ¯ x , y F , B = A x , y F , B [ π c 0 n 0 0 ψ 2 ( r ) r d r ] 1 / 2 .
P x , y F , B = | a x , y F , B | 2 = C NL | A ¯ x , y F , B | 2 .
P x F + P y F ( P x B + P y B ) = P t ,
H a ¯ j F * = i a ¯ j F d Z , H a ¯ j B * = + i a ¯ j B d Z ,
a ¯ x F = a x F exp ( i Δ β x g Z ), a ¯ x B = a x B exp ( i Δ β x g Z ) , a ¯ y F = a y F exp [ i ( Δ β x g + δ β y x ) Z ] , a ¯ y B = a y B exp [ i ( Δ β x g + δ β y x ) Z ] .
H = K { a x B a x F * exp ( i 2 Δ β x g Z ) + a x F a x B * exp ( i 2 Δ β x g Z ) + a y B a y F * exp [ i 2 ( Δ β x g + δ β y x ) Z ] + a y F a y B * exp [ i 2 ( Δ β x g + δ β y x ) Z ] } + Δ β x g ( | a x F | 2 + | a x B | 2 + | a y F | 2 + | a y B | 2 ) + δ β y x ( | a y F | 2 + | a y B | 2 ) + 1 2 ( | a x F | 4 + | a x B | 4 + | a y F | 4 + | a y B | 4 ) + 2 ( | a x F | 2 | a x B | 2 + | a y F | 2 | a y B | 2 ) + 2 3 ( | a x F | 2 | a y F | 2 + | a x F | 2 | a y B | 2 + | a y F | 2 | a x B | 2 + | a y B | 2 | a x B | 2 ) + 1 6 [ a x F 2 a y F * 2 exp ( i 2 δ β y x Z ) + a x F * 2 a y F 2 exp ( i 2 δ β y x Z ) + a x B 2 a y B * 2 exp ( i 2 δ β y x Z ) + a x B * 2 a y B 2 exp ( i 2 δ β y x Z ] + 2 3 [ a x F a x B * a y F * a y B exp ( i 2 δ β y x Z ) + a x F * a x B a y F a y B * exp ( i 2 δ β y x Z ) + a x F a x B a y F * a y B * + a x F * a x B * a y F a y B ] .
ψ ( r ) = exp ( r 2 / 2 r 0 2 ) ,
C NL = 3 ω χ ( 3 ) 4 c 2 n 0 2 0 2 π r 0 2 L ( W 1 ) .
n 2 = 3 χ ( 3 ) 4 c 0 n 0 2 ( m 2 / W ) .
C NL = n 2 L λ r 0 2 .
Power = 10 8 λ 4 n 2 ( W ) .

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