Abstract

The analysis of Bespalov and Talanov concerning small scale self-focusing or filamentation effects for plane waves is generalized to a tapered beam case in this paper. A model of nonlinear phase and amplitude perturbations different from Siegman’s is suggested. The previous conclusion that self-focusing is very ineffectual for strongly tapered, both divergent and convergent, beams in optical Kerr media as compared to small-scale self-focusing in plane waves is found to be inappropriate. Moreover, the different conditions of phase matching in cases of cylindrical and spherical waves are explicated, which could throw some light on the issue.

© 2012 Optical Society of America

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References

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  1. V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307 (1966).
  2. R. Y. Chiao and P. L. Kelly, “Stimulated four-photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17, 1158–1161 (1966).
    [CrossRef]
  3. R. J. Lang, A. Hardy, R. Parke, D. Mehuys, S. O’Brien, J. Major, and D. Welch, “Numerical analysis of flared semiconductor laser amplifiers,” IEEE J. Quantum Electron. 29, 2044–2051 (1993).
    [CrossRef]
  4. S. Ramanujan and H. G. Winful, “Spontaneous emission induced filamentation in flared amplifiers,” IEEE J. Quantum Electron. 32, 784–789 (1996).
    [CrossRef]
  5. G. Levy, and A. A. Hardy, “Chaotic effects in flared lasers: a numerical analysis,” IEEE J. Quantum Electron. 33, 26–32 (1997).
    [CrossRef]
  6. A. E. Siegman, “Small-scale self-focusing effects in tapered optical beams,” www.stanford.edu/~siegman/self_focusing_memo.pdf (2002), pp. 1–13.
  7. R. W. Boyd, S. G. LuKishova, and Y. R. Shen, eds., Self-focusing: Past and Present (Springer, 2009), pp. 168, 215–216.
  8. A. E. Siegman, Lasers, 1st ed. (University Science, 1986), p. 630.
  9. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2003), p. 340.

1997 (1)

G. Levy, and A. A. Hardy, “Chaotic effects in flared lasers: a numerical analysis,” IEEE J. Quantum Electron. 33, 26–32 (1997).
[CrossRef]

1996 (1)

S. Ramanujan and H. G. Winful, “Spontaneous emission induced filamentation in flared amplifiers,” IEEE J. Quantum Electron. 32, 784–789 (1996).
[CrossRef]

1993 (1)

R. J. Lang, A. Hardy, R. Parke, D. Mehuys, S. O’Brien, J. Major, and D. Welch, “Numerical analysis of flared semiconductor laser amplifiers,” IEEE J. Quantum Electron. 29, 2044–2051 (1993).
[CrossRef]

1966 (2)

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307 (1966).

R. Y. Chiao and P. L. Kelly, “Stimulated four-photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17, 1158–1161 (1966).
[CrossRef]

Bespalov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307 (1966).

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2003), p. 340.

Chiao, R. Y.

R. Y. Chiao and P. L. Kelly, “Stimulated four-photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17, 1158–1161 (1966).
[CrossRef]

Hardy, A.

R. J. Lang, A. Hardy, R. Parke, D. Mehuys, S. O’Brien, J. Major, and D. Welch, “Numerical analysis of flared semiconductor laser amplifiers,” IEEE J. Quantum Electron. 29, 2044–2051 (1993).
[CrossRef]

Hardy, A. A.

G. Levy, and A. A. Hardy, “Chaotic effects in flared lasers: a numerical analysis,” IEEE J. Quantum Electron. 33, 26–32 (1997).
[CrossRef]

Kelly, P. L.

R. Y. Chiao and P. L. Kelly, “Stimulated four-photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17, 1158–1161 (1966).
[CrossRef]

Lang, R. J.

R. J. Lang, A. Hardy, R. Parke, D. Mehuys, S. O’Brien, J. Major, and D. Welch, “Numerical analysis of flared semiconductor laser amplifiers,” IEEE J. Quantum Electron. 29, 2044–2051 (1993).
[CrossRef]

Levy, G.

G. Levy, and A. A. Hardy, “Chaotic effects in flared lasers: a numerical analysis,” IEEE J. Quantum Electron. 33, 26–32 (1997).
[CrossRef]

Major, J.

R. J. Lang, A. Hardy, R. Parke, D. Mehuys, S. O’Brien, J. Major, and D. Welch, “Numerical analysis of flared semiconductor laser amplifiers,” IEEE J. Quantum Electron. 29, 2044–2051 (1993).
[CrossRef]

Mehuys, D.

R. J. Lang, A. Hardy, R. Parke, D. Mehuys, S. O’Brien, J. Major, and D. Welch, “Numerical analysis of flared semiconductor laser amplifiers,” IEEE J. Quantum Electron. 29, 2044–2051 (1993).
[CrossRef]

O’Brien, S.

R. J. Lang, A. Hardy, R. Parke, D. Mehuys, S. O’Brien, J. Major, and D. Welch, “Numerical analysis of flared semiconductor laser amplifiers,” IEEE J. Quantum Electron. 29, 2044–2051 (1993).
[CrossRef]

Parke, R.

R. J. Lang, A. Hardy, R. Parke, D. Mehuys, S. O’Brien, J. Major, and D. Welch, “Numerical analysis of flared semiconductor laser amplifiers,” IEEE J. Quantum Electron. 29, 2044–2051 (1993).
[CrossRef]

Ramanujan, S.

S. Ramanujan and H. G. Winful, “Spontaneous emission induced filamentation in flared amplifiers,” IEEE J. Quantum Electron. 32, 784–789 (1996).
[CrossRef]

Siegman, A. E.

A. E. Siegman, “Small-scale self-focusing effects in tapered optical beams,” www.stanford.edu/~siegman/self_focusing_memo.pdf (2002), pp. 1–13.

A. E. Siegman, Lasers, 1st ed. (University Science, 1986), p. 630.

Talanov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307 (1966).

Welch, D.

R. J. Lang, A. Hardy, R. Parke, D. Mehuys, S. O’Brien, J. Major, and D. Welch, “Numerical analysis of flared semiconductor laser amplifiers,” IEEE J. Quantum Electron. 29, 2044–2051 (1993).
[CrossRef]

Winful, H. G.

S. Ramanujan and H. G. Winful, “Spontaneous emission induced filamentation in flared amplifiers,” IEEE J. Quantum Electron. 32, 784–789 (1996).
[CrossRef]

IEEE J. Quantum Electron. (3)

R. J. Lang, A. Hardy, R. Parke, D. Mehuys, S. O’Brien, J. Major, and D. Welch, “Numerical analysis of flared semiconductor laser amplifiers,” IEEE J. Quantum Electron. 29, 2044–2051 (1993).
[CrossRef]

S. Ramanujan and H. G. Winful, “Spontaneous emission induced filamentation in flared amplifiers,” IEEE J. Quantum Electron. 32, 784–789 (1996).
[CrossRef]

G. Levy, and A. A. Hardy, “Chaotic effects in flared lasers: a numerical analysis,” IEEE J. Quantum Electron. 33, 26–32 (1997).
[CrossRef]

JETP Lett. (1)

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307 (1966).

Phys. Rev. Lett. (1)

R. Y. Chiao and P. L. Kelly, “Stimulated four-photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17, 1158–1161 (1966).
[CrossRef]

Other (4)

A. E. Siegman, “Small-scale self-focusing effects in tapered optical beams,” www.stanford.edu/~siegman/self_focusing_memo.pdf (2002), pp. 1–13.

R. W. Boyd, S. G. LuKishova, and Y. R. Shen, eds., Self-focusing: Past and Present (Springer, 2009), pp. 168, 215–216.

A. E. Siegman, Lasers, 1st ed. (University Science, 1986), p. 630.

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2003), p. 340.

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Figures (4)

Fig. 1.
Fig. 1.

Perturbation model for divergent wave.

Fig. 2.
Fig. 2.

Evolution of the amplitude and phase ripples in plane, divergent spherical and cylindrical waves. (a) and (b) Growth of amplitude ripples; (c) and (d) growth of phase ripples.

Fig. 3.
Fig. 3.

Evolution of the amplitude and phase ripples in plane, convergent spherical, and cylindrical waves. (a) and (b) Growth of amplitude ripples; (c) and (d) growth of phase ripples.

Fig. 4.
Fig. 4.

Role of phase matching in a tapered beam.

Equations (12)

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(xy2+i2k0z+2n2nk02I)u=0,
u=u0|z|s/2exp(ik0x2+y22z)exp[iϕ(z)][1+c+1(z)exp(ik0p22z)exp(ik02px2z)+c1(z)exp(ik0p22z)exp(ik02px2z)].
c+1=c1=c2exp(ik0p2/2z).
u=u0|z|s/2exp(ik0x2+y22z)exp[iϕ(z)][1+c(z)cos(k0pxz)].
dc(z)dz=ik0p22z2c(z)+in2k0u02n|z|s[c(z)+c*(z)].
dcrdz=k0p22z2ci(z),dcidz=[2n2k0nu02|z|sk0p22z2]cr(z).
dcrdw=k0p22ci(1w),dcidw=[k0p222n2k0u02n]cr(1w).
Λ2=k0p22[2n2k0u02nk0p22].
γ(z)=1cr,idcr,idz1exp(Λz)ddzexp(Λz).
γ(z)=Λz2,
γ(z)2=k0p22z2[2n2k0nu02z2k0p22z2].
γ2=kx22k0[2n2k0Inkx22k0],

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