Abstract

We give what we believe to be the first closed-form exact expression for the dynamic evolution of nonstationary beams of arbitrary intensity and width propagating in a uniform nonlinear medium and in both two and three dimensions. This shows that periodic and quasi-periodic (nonradiating) beams can exist in a non-Kerr nonlinear medium. The Schrödinger equation is solved for Gaussian beams in a saturable medium. For one critical (initial) beam width, the Gaussian is a stable stationary soliton or bullet, independent of its intensity; otherwise, it breathes. New quasi-periodic beams (mighty morphing solitons) and bullets (mighty morphs) of elliptical cross section also exist whose ellipticity changes with propagation.

© 1997 Optical Society of America

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  1. The inverse scattering technique does not give a closed-form expression for the evolution of sech profile beam of arbitrary amplitude and width, only for special values. It provides an implicit description at asymptotic distances for 1+1-D beams of a cubic (Kerr) nonlinearity.
  2. M. Segev, B. Crosignani, A. Yariv, and B. Fischer, Phys. Rev. Lett. 68, 923 (1992); Phys. Rev. Lett. 74, 1978 (1995); M. Shih, P. Leach, M. Segev, M. H. Garrett, G. Salamar, and G. C. Valley, Opt. Lett. 21, 324 (1996).
    [Crossref] [PubMed]
  3. V. Tikhonenko, J. Christour, and B. Luther-Davies, J. Opt. Soc. Am. B 12, 2046 (1995).
    [Crossref]
  4. Y. Silberberg, Opt. Lett. 15, 1282 (1990). Here the usual ∇t2 operator is generalized to ∂2/∂x2+∂2/∂y2+∂2/∂t¯2, where t¯=d2ω/dk2t, with t the retarded time.
    [Crossref] [PubMed]
  5. Periodic solitons of a saturating medium have been reported previously in a non-Kerr medium, but they suffer a small radiation loss. See A. W. Snyder, S. Hewlett, and D. J. Mitchell, Phys. Rev. E 51, 6297 (1995).
    [Crossref]
  6. Saturation ensures that beams of circular cross section are highly stable and can be steered in 3D, and their collisions lead to versatile devices, as shown by A. W. Snyder, D. J. Mitchell, L. Poladian, and F. Ladouceur, Opt. Lett. 16, 27 (1991); L. Poladian, A. W. Snyder, and D. J. Mitchell, Opt. Commun. 85, 59 (1991); and A. W. Snyder and A. P. Sheppard, Opt. Lett. 18, 499 (1993).
    [Crossref]
  7. A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, Mod. Phys. Lett. B 9, 1479 (1995). Section 8 shows that Eq. (2) approximates Maxwell’s equations provided that n≅n0.
    [Crossref]
  8. This is seen from Eq. (4) with the identification b=n0 Imq˙/q/kΔ.
  9. Solitons are stable to arbitrary perturbations when dI0/dP is positive. Mighty morphing solitons have I0=γP, where γ is a positive constant and P is the beam power. See A. W. Snyder, D. J. Mitchell, and A. Buryak, J. Opt. Soc. Am. B 13, 1146 (1996).
    [Crossref]
  10. This is in homage to the creators of “Mighty Morphin Power Ranges.” Morphing is used because the beams can change dramatically, and mighty because they are impervious to changes in intensity and can be propagated numerically, even with asymmetric perturbations and even in an ln1+I/It medium.
  11. The solution of the Schrödinger equation for a (linear) parabolic index medium n2=n12z-η2zx2, which changes in the axial z direction, is generalized from Arnaud12 to have the Gaussian form ψ of Eq. (3) but with q obeying q¨+η2zq˙=0. By inverting this expression for Ix=ψ2 to find x2I and substituting this into x2 in the expression for n2x,z above, we find that the Gaussian must be a soliton of a In I nonlinearity as in Eq. (1). Substituting Ix into Eq. (1) shows that η2z=Δkn0 Imq˙/q for self-consistency. This leads to Eq. (4) and hence to our general solution.
  12. J. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), p. 52.
  13. A. Buryak, Optical Sciences Centre, Australian National University, Canberra, Australia (personal communication, August5, 1996).

1996 (1)

1995 (3)

A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, Mod. Phys. Lett. B 9, 1479 (1995). Section 8 shows that Eq. (2) approximates Maxwell’s equations provided that n≅n0.
[Crossref]

V. Tikhonenko, J. Christour, and B. Luther-Davies, J. Opt. Soc. Am. B 12, 2046 (1995).
[Crossref]

Periodic solitons of a saturating medium have been reported previously in a non-Kerr medium, but they suffer a small radiation loss. See A. W. Snyder, S. Hewlett, and D. J. Mitchell, Phys. Rev. E 51, 6297 (1995).
[Crossref]

1992 (1)

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, Phys. Rev. Lett. 68, 923 (1992); Phys. Rev. Lett. 74, 1978 (1995); M. Shih, P. Leach, M. Segev, M. H. Garrett, G. Salamar, and G. C. Valley, Opt. Lett. 21, 324 (1996).
[Crossref] [PubMed]

1991 (1)

1990 (1)

Arnaud, J.

J. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), p. 52.

Buryak, A.

Christour, J.

Crosignani, B.

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, Phys. Rev. Lett. 68, 923 (1992); Phys. Rev. Lett. 74, 1978 (1995); M. Shih, P. Leach, M. Segev, M. H. Garrett, G. Salamar, and G. C. Valley, Opt. Lett. 21, 324 (1996).
[Crossref] [PubMed]

Fischer, B.

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, Phys. Rev. Lett. 68, 923 (1992); Phys. Rev. Lett. 74, 1978 (1995); M. Shih, P. Leach, M. Segev, M. H. Garrett, G. Salamar, and G. C. Valley, Opt. Lett. 21, 324 (1996).
[Crossref] [PubMed]

Hewlett, S.

Periodic solitons of a saturating medium have been reported previously in a non-Kerr medium, but they suffer a small radiation loss. See A. W. Snyder, S. Hewlett, and D. J. Mitchell, Phys. Rev. E 51, 6297 (1995).
[Crossref]

Kivshar, Y. S.

A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, Mod. Phys. Lett. B 9, 1479 (1995). Section 8 shows that Eq. (2) approximates Maxwell’s equations provided that n≅n0.
[Crossref]

Ladouceur, F.

Luther-Davies, B.

Mitchell, D. J.

Poladian, L.

Segev, M.

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, Phys. Rev. Lett. 68, 923 (1992); Phys. Rev. Lett. 74, 1978 (1995); M. Shih, P. Leach, M. Segev, M. H. Garrett, G. Salamar, and G. C. Valley, Opt. Lett. 21, 324 (1996).
[Crossref] [PubMed]

Silberberg, Y.

Snyder, A. W.

Tikhonenko, V.

Yariv, A.

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, Phys. Rev. Lett. 68, 923 (1992); Phys. Rev. Lett. 74, 1978 (1995); M. Shih, P. Leach, M. Segev, M. H. Garrett, G. Salamar, and G. C. Valley, Opt. Lett. 21, 324 (1996).
[Crossref] [PubMed]

J. Opt. Soc. Am. B (2)

Mod. Phys. Lett. B (1)

A. W. Snyder, D. J. Mitchell, and Y. S. Kivshar, Mod. Phys. Lett. B 9, 1479 (1995). Section 8 shows that Eq. (2) approximates Maxwell’s equations provided that n≅n0.
[Crossref]

Opt. Lett. (2)

Phys. Rev. E (1)

Periodic solitons of a saturating medium have been reported previously in a non-Kerr medium, but they suffer a small radiation loss. See A. W. Snyder, S. Hewlett, and D. J. Mitchell, Phys. Rev. E 51, 6297 (1995).
[Crossref]

Phys. Rev. Lett. (1)

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, Phys. Rev. Lett. 68, 923 (1992); Phys. Rev. Lett. 74, 1978 (1995); M. Shih, P. Leach, M. Segev, M. H. Garrett, G. Salamar, and G. C. Valley, Opt. Lett. 21, 324 (1996).
[Crossref] [PubMed]

Other (6)

The inverse scattering technique does not give a closed-form expression for the evolution of sech profile beam of arbitrary amplitude and width, only for special values. It provides an implicit description at asymptotic distances for 1+1-D beams of a cubic (Kerr) nonlinearity.

This is in homage to the creators of “Mighty Morphin Power Ranges.” Morphing is used because the beams can change dramatically, and mighty because they are impervious to changes in intensity and can be propagated numerically, even with asymmetric perturbations and even in an ln1+I/It medium.

The solution of the Schrödinger equation for a (linear) parabolic index medium n2=n12z-η2zx2, which changes in the axial z direction, is generalized from Arnaud12 to have the Gaussian form ψ of Eq. (3) but with q obeying q¨+η2zq˙=0. By inverting this expression for Ix=ψ2 to find x2I and substituting this into x2 in the expression for n2x,z above, we find that the Gaussian must be a soliton of a In I nonlinearity as in Eq. (1). Substituting Ix into Eq. (1) shows that η2z=Δkn0 Imq˙/q for self-consistency. This leads to Eq. (4) and hence to our general solution.

J. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), p. 52.

A. Buryak, Optical Sciences Centre, Australian National University, Canberra, Australia (personal communication, August5, 1996).

This is seen from Eq. (4) with the identification b=n0 Imq˙/q/kΔ.

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Figures (1)

Fig. 1
Fig. 1

Salient characteristics of the periodic beam width ρz (a) The extremum values of ρz as found from Eq. (7), and (b) the dimensionalized period Zp=kzpΔ/n0, found by integration of Eq. (6). The minimum period is Zp=π2, occurring when ρzρ0. The dotted curves are the asymptotic approximation given in the text.

Equations (8)

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n2I=n02+Δ lnI/It,
2ikn0ψ/z+t2ψ+k2Δ lnI/Itψ=0.
ψ=γ expαx2expiϕ,
q¨+qkΔ/n0 Imq˙/q=0,
Iz=Imzexp-x2/ρ2z,
db/dZ=±2bHb-b2+b ln b1/2,
H=bmax-lnbmax=bmin-lnbmin.
I=I0ρx0ρy0ρxρyexp-x2ρx2-y2ρy2,

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