Abstract

We investigate numerically the interaction between two copropagating Bragg solitons in a fiber grating. We find that, in the low-intensity limit, the interaction is reminiscent of the nonlinear Schrödinger solitons in that Bragg solitons attract or repel each other, depending on their relative phases. However, the relative phase between two Bragg solitons is found to depend on their initial separation. We discuss the implications of the numerical results for laboratory experiments.

© 1999 Optical Society of America

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References

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  1. J. P. Gordon, “Interaction forces among solitons in optical fibers,” Opt. Lett. 8, 596–598 (1983).
    [CrossRef] [PubMed]
  2. F. M. Mitschke and L. F. Mollenauer, “Experimental observation of interaction forces between solitons in optical fibers,” Opt. Lett. 12, 407–409 (1987).
    [CrossRef] [PubMed]
  3. J. S. Aitchison, A. M. Weiner, Y. Silberberg, D. E. Leaird, M. K. Oliver, J. L. Jackel, and P. W. E. Smith, “Experimental observation of spatial soliton interactions,” Opt. Lett. 16, 15–17 (1991).
    [CrossRef] [PubMed]
  4. M. Shalaby, F. Reynaud, and A. Barthelemy, “Experimental observation of spatial soliton interactions with a π/2 relative phase difference,” Opt. Lett. 17, 778–780 (1992).
    [CrossRef] [PubMed]
  5. W. Chen and D. L. Mills, “Gap solitons and nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
    [CrossRef] [PubMed]
  6. J. E. Sipe and H. G. Winful, “Nonlinear Schrödinger solitons in a periodic structure,” Opt. Lett. 13, 132–133 (1988).
    [CrossRef]
  7. D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
    [CrossRef] [PubMed]
  8. A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37–42 (1989).
    [CrossRef]
  9. C. M. de Sterke and J. E. Sipe, “Gap solitons,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1994), Vol. 33, pp. 203–260.
  10. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
    [CrossRef] [PubMed]
  11. C. M. de Sterke, B. J. Eggleton, and P. A. Krug, “High-intensity pulse propagation in uniform gratings and grating superstructures,” J. Lightwave Technol. 15, 1494–1502 (1997).
    [CrossRef]
  12. B. J. Eggleton, R. E. Slusher, T. A. Strasser, and C. M. de Sterke, “High intensity pulse propagation in fiber Bragg gratings,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper BMB1–1.
  13. B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
    [CrossRef]
  14. B. J. Eggleton, C. M. de Sterke, R. E. Slusher, A. Aceves, J. E. Sipe, and T. A. Strasser, “Modulational instabilities and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
    [CrossRef]
  15. D. Taverner, N. G. R. Broderick, D. T. Richardson, R. I. Laming, and M. Ibsen, “Nonlinear self-switching and multiple gap-soliton formation in a fiber Bragg grating,” Opt. Lett. 23, 328–330 (1998).
    [CrossRef]
  16. C. M. de Sterke and B. J. Eggleton, “Bragg solitons and the nonlinear Schrödinger equation,” Phys. Rev. E (to be published).
  17. T. Iizuka and M. Wadati, “Grating solitons in optical fibers,” J. Phys. Soc. Jpn. 66, 2308–2313 (1997).
    [CrossRef]
  18. B. J. Eggleton, R. E. Slusher, N. M. Litchinitser, G. P. Agrawal, and C. M. de Sterke, “Experimental observation of interaction of Bragg solitons,” in International Quantum Electronics Conference (IQEC), Vol. 7 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper QTuJ5.
  19. H. G. Winful, “Pulse compression in optical fiber filters,” Appl. Phys. Lett. 46, 527–529 (1985).
    [CrossRef]
  20. C. M. de Sterke, K. R. Jackson, and B. D. Robert, “Nonlinear coupled mode equations on a finite interval: a numerical procedure,” J. Opt. Soc. Am. B 8, 403–412 (1991).
    [CrossRef]
  21. C. M. de Sterke, N. G. R. Broderick, B. J. Eggleton, and M. J. Steel, “Nonlinear optics in fiber gratings,” Opt. Fiber Technol. 2, 253–268 (1996).
  22. G. P. Agrawal, Fiber-Optic Communication Systems, 2nd ed. (Wiley, New York, 1997).
  23. C. Desem and P. L. Chu, IEE Proc.-J: Optoelectron. 134, 145–151 (1987).
    [CrossRef]
  24. P. St. J. Russell, “Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,” J. Mod. Opt. 38, 1599–1619 (1991).
    [CrossRef]
  25. N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: Theoretical model and design criterion for nearly ideal pulse compression,” J. Lightwave Technol. 15, 1303–1313 (1997).
    [CrossRef]

1998

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, A. Aceves, J. E. Sipe, and T. A. Strasser, “Modulational instabilities and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

D. Taverner, N. G. R. Broderick, D. T. Richardson, R. I. Laming, and M. Ibsen, “Nonlinear self-switching and multiple gap-soliton formation in a fiber Bragg grating,” Opt. Lett. 23, 328–330 (1998).
[CrossRef]

1997

T. Iizuka and M. Wadati, “Grating solitons in optical fibers,” J. Phys. Soc. Jpn. 66, 2308–2313 (1997).
[CrossRef]

C. M. de Sterke, B. J. Eggleton, and P. A. Krug, “High-intensity pulse propagation in uniform gratings and grating superstructures,” J. Lightwave Technol. 15, 1494–1502 (1997).
[CrossRef]

N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: Theoretical model and design criterion for nearly ideal pulse compression,” J. Lightwave Technol. 15, 1303–1313 (1997).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
[CrossRef]

1996

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

1992

1991

1989

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[CrossRef] [PubMed]

A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37–42 (1989).
[CrossRef]

1988

1987

F. M. Mitschke and L. F. Mollenauer, “Experimental observation of interaction forces between solitons in optical fibers,” Opt. Lett. 12, 407–409 (1987).
[CrossRef] [PubMed]

W. Chen and D. L. Mills, “Gap solitons and nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

C. Desem and P. L. Chu, IEE Proc.-J: Optoelectron. 134, 145–151 (1987).
[CrossRef]

1985

H. G. Winful, “Pulse compression in optical fiber filters,” Appl. Phys. Lett. 46, 527–529 (1985).
[CrossRef]

1983

Aceves, A.

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, A. Aceves, J. E. Sipe, and T. A. Strasser, “Modulational instabilities and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

Aceves, A. B.

A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37–42 (1989).
[CrossRef]

Aitchison, J. S.

Barthelemy, A.

Broderick, N. G. R.

Chen, W.

W. Chen and D. L. Mills, “Gap solitons and nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

Christodoulides, D. N.

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[CrossRef] [PubMed]

Chu, P. L.

C. Desem and P. L. Chu, IEE Proc.-J: Optoelectron. 134, 145–151 (1987).
[CrossRef]

de Sterke, C. M.

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, A. Aceves, J. E. Sipe, and T. A. Strasser, “Modulational instabilities and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

C. M. de Sterke, B. J. Eggleton, and P. A. Krug, “High-intensity pulse propagation in uniform gratings and grating superstructures,” J. Lightwave Technol. 15, 1494–1502 (1997).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

C. M. de Sterke, K. R. Jackson, and B. D. Robert, “Nonlinear coupled mode equations on a finite interval: a numerical procedure,” J. Opt. Soc. Am. B 8, 403–412 (1991).
[CrossRef]

Desem, C.

C. Desem and P. L. Chu, IEE Proc.-J: Optoelectron. 134, 145–151 (1987).
[CrossRef]

Eggleton, B. J.

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, A. Aceves, J. E. Sipe, and T. A. Strasser, “Modulational instabilities and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

C. M. de Sterke, B. J. Eggleton, and P. A. Krug, “High-intensity pulse propagation in uniform gratings and grating superstructures,” J. Lightwave Technol. 15, 1494–1502 (1997).
[CrossRef]

N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: Theoretical model and design criterion for nearly ideal pulse compression,” J. Lightwave Technol. 15, 1303–1313 (1997).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Gordon, J. P.

Ibsen, M.

Iizuka, T.

T. Iizuka and M. Wadati, “Grating solitons in optical fibers,” J. Phys. Soc. Jpn. 66, 2308–2313 (1997).
[CrossRef]

Jackel, J. L.

Jackson, K. R.

Joseph, R. I.

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[CrossRef] [PubMed]

Krug, P. A.

C. M. de Sterke, B. J. Eggleton, and P. A. Krug, “High-intensity pulse propagation in uniform gratings and grating superstructures,” J. Lightwave Technol. 15, 1494–1502 (1997).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Laming, R. I.

Leaird, D. E.

Litchinitser, N. M.

N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: Theoretical model and design criterion for nearly ideal pulse compression,” J. Lightwave Technol. 15, 1303–1313 (1997).
[CrossRef]

Mills, D. L.

W. Chen and D. L. Mills, “Gap solitons and nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

Mitschke, F. M.

Mollenauer, L. F.

Oliver, M. K.

Patterson, D. B.

N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: Theoretical model and design criterion for nearly ideal pulse compression,” J. Lightwave Technol. 15, 1303–1313 (1997).
[CrossRef]

Reynaud, F.

Richardson, D. T.

Robert, B. D.

Russell, P. St. J.

P. St. J. Russell, “Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,” J. Mod. Opt. 38, 1599–1619 (1991).
[CrossRef]

Shalaby, M.

Silberberg, Y.

Sipe, J. E.

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, A. Aceves, J. E. Sipe, and T. A. Strasser, “Modulational instabilities and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

J. E. Sipe and H. G. Winful, “Nonlinear Schrödinger solitons in a periodic structure,” Opt. Lett. 13, 132–133 (1988).
[CrossRef]

Slusher, R. E.

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, A. Aceves, J. E. Sipe, and T. A. Strasser, “Modulational instabilities and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

Smith, P. W. E.

Strasser, T. A.

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, A. Aceves, J. E. Sipe, and T. A. Strasser, “Modulational instabilities and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

Taverner, D.

Wabnitz, S.

A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37–42 (1989).
[CrossRef]

Wadati, M.

T. Iizuka and M. Wadati, “Grating solitons in optical fibers,” J. Phys. Soc. Jpn. 66, 2308–2313 (1997).
[CrossRef]

Weiner, A. M.

Winful, H. G.

J. E. Sipe and H. G. Winful, “Nonlinear Schrödinger solitons in a periodic structure,” Opt. Lett. 13, 132–133 (1988).
[CrossRef]

H. G. Winful, “Pulse compression in optical fiber filters,” Appl. Phys. Lett. 46, 527–529 (1985).
[CrossRef]

Appl. Phys. Lett.

H. G. Winful, “Pulse compression in optical fiber filters,” Appl. Phys. Lett. 46, 527–529 (1985).
[CrossRef]

IEE Proc.-J: Optoelectron.

C. Desem and P. L. Chu, IEE Proc.-J: Optoelectron. 134, 145–151 (1987).
[CrossRef]

J. Lightwave Technol.

N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: Theoretical model and design criterion for nearly ideal pulse compression,” J. Lightwave Technol. 15, 1303–1313 (1997).
[CrossRef]

C. M. de Sterke, B. J. Eggleton, and P. A. Krug, “High-intensity pulse propagation in uniform gratings and grating superstructures,” J. Lightwave Technol. 15, 1494–1502 (1997).
[CrossRef]

J. Mod. Opt.

P. St. J. Russell, “Bloch wave analysis of dispersion and pulse propagation in pure distributed feedback structures,” J. Mod. Opt. 38, 1599–1619 (1991).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. Soc. Jpn.

T. Iizuka and M. Wadati, “Grating solitons in optical fibers,” J. Phys. Soc. Jpn. 66, 2308–2313 (1997).
[CrossRef]

Opt. Commun.

B. J. Eggleton, C. M. de Sterke, R. E. Slusher, A. Aceves, J. E. Sipe, and T. A. Strasser, “Modulational instabilities and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998).
[CrossRef]

Opt. Lett.

Phys. Lett. A

A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37–42 (1989).
[CrossRef]

Phys. Rev. Lett.

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746–1749 (1989).
[CrossRef] [PubMed]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996).
[CrossRef] [PubMed]

W. Chen and D. L. Mills, “Gap solitons and nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

Other

C. M. de Sterke and J. E. Sipe, “Gap solitons,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1994), Vol. 33, pp. 203–260.

C. M. de Sterke and B. J. Eggleton, “Bragg solitons and the nonlinear Schrödinger equation,” Phys. Rev. E (to be published).

B. J. Eggleton, R. E. Slusher, T. A. Strasser, and C. M. de Sterke, “High intensity pulse propagation in fiber Bragg gratings,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper BMB1–1.

B. J. Eggleton, R. E. Slusher, N. M. Litchinitser, G. P. Agrawal, and C. M. de Sterke, “Experimental observation of interaction of Bragg solitons,” in International Quantum Electronics Conference (IQEC), Vol. 7 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper QTuJ5.

C. M. de Sterke, N. G. R. Broderick, B. J. Eggleton, and M. J. Steel, “Nonlinear optics in fiber gratings,” Opt. Fiber Technol. 2, 253–268 (1996).

G. P. Agrawal, Fiber-Optic Communication Systems, 2nd ed. (Wiley, New York, 1997).

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Figures (6)

Fig. 1
Fig. 1

Schematic of Bragg soliton interaction in (a) an infinitely long grating and in (b) a finite grating.

Fig. 2
Fig. 2

Illustration of the grating stop band in the parameter space (δ, ν), showing the region for which the soliton central frequency lies inside the stop band (shaded area). The filled point represents the parameters used in numerical simulations.

Fig. 3
Fig. 3

Interaction of two Bragg solitons in an infinite fiber grating for three different values of their initial separation LS: (a) LS=1.113 cm, (b) LS=1.252 cm, and (c) LS=1.391 cm. Other parameter values used are ν=0.745, δ=0.13, and κ=10 cm-1.

Fig. 4
Fig. 4

Evolution of a 60-ps sech-shaped pulse in a 50-cm-long fiber grating for an input pulse intensity of (a) I0=A02=3 GW/cm2, (b) I0=5.94 GW/cm2, and (c) I0=10 GW/cm2. A Bragg soliton is formed only in case (b).

Fig. 5
Fig. 5

Interaction of two Bragg solitons in a 50-cm-long fiber grating for the same initial soliton separations as in Fig. 3. The input pulses have a width of TFWHM=60 ps and a peak intensity I0=5.94 GW/cm2.

Fig. 6
Fig. 6

Interaction of two Bragg solitons in a 50-cm-long fiber grating for the same parameters as in Fig. 5, but with three different relative phase shifts: (a) ϕ=0, (b) ϕ=π/2, and (c) ϕ=π.

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

E(z, t)=[E+(z, t)exp(ikBz)+E-(z, t)×exp(-ikBz)]exp(-iωBt),
iE+z+incE+t+κE-+ΓS|E+|2E++2Γ×|E-|2E+
=0,
-iE-z+incE-t+κE++ΓS|E-|2E-+2Γ×|E+|2E-
=0.
E±=αE˜± exp[iη(θ)],
E˜+=κ2Γ× 1+ν1-ν1/4 sin δ exp(iσ)sechθ-iδ2,
E˜-=-κ2Γ× 1-ν1+ν1/4 sin δ exp(iσ)sechθ+iδ2,
1α2=1+ΓS2Γ×1+ν21-ν2,
exp[iη(θ)]=-exp(2θ)+exp(-iδ)exp(2θ)+exp(iδ) 2ΓSν2Γ×(1-ν2)+ΓS(1+ν2).
E±(z, 0)=E±(z-z0, 0)+E±(z-z0-LS, 0),
E+E-=a(z, t)A+A-exp[i(Qz-Ω±t)].
E+(z, 0)E-(z, 0)=a(z-z0, 0)A+A-exp[iQ(z-z0)]+a[z-(z0+LS), 0]A+A-×exp[iQ(z-z0)]exp(-iQLS).
ϕ=QLS.
E+(0, t)=A0 sech[(t-TS)/T0]+A0 sech(t/T0)exp(iϕ),
E-(L, t)=0,
E+(0, t)=A0 sech[(t-TS)/T0]+A0 sech(t/T0)exp(-iQLS),
Lc=π sinh(TS/T0)cosh(TS/2T0)TS/T0+sinh(TS/T0)T02|β2|,
β2=-1V2κ2[(Ω/V)2-κ2]3/2.

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