Abstract

The paraxial wave theory is known to lead to inaccurate predictions in self-focusing of optical beams. The nonlinear Helmholtz equation describes more accurately wave propagation in dispersive, spatially local, Kerr-type media. We derive rigorous bright and dark solutions to the nonlinear Helmholtz equation in a full three-dimensional form. These expressions are new and unique. The solutions are obtained with a multidimensional extension of the (paraxial) nonlinear Schrödinger equation. We also establish energy conservation laws for both nonlinear wave equations in terms of spatial currents. Our results give insight, for example, into the diffraction and breakup of tightly confined nonlinear fields.

© 2000 Optical Society of America

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References

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  1. R. W. Boyd, Nonlinear Optics (London, Academic, 1992).
  2. J. S. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. M. Vogel, and P. W. E. Smith, “Observation of spatial optical solitons in a nonlinear glass waveguide,” Opt. Lett. 15, 471–473 (1990).
    [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. R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964); E. Garmire, R. Y. Chiao, and C. H. Townes, “Dynamics and characteristics of the self-trapping of intense light beams,” Phys. Rev. Lett. 16, 347–349 (1966).
    [CrossRef]
  5. V. I. Talanov, “Focusing of light in cubic media,” JETP Lett. 2, 138–141 (1965); P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965); Y. R. Shen, “Self-focusing: experimental,” Prog. Quantum Electron. PQUEAH 4, 1–34 (1975).
    [CrossRef]
  6. D. Anderson, M. Bonneal, and M. Lisak, “Self-trapped cylindrical laser beams,” Phys. Fluids 22, 1838–1840 (1979); D. Anderson and M. Bonneal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids 22, 105–109 (1979).
    [CrossRef]
  7. M. Karlsson, D. Anderson, and M. Desaix, “Dynamics of self-focusing and self-phase modulation in a parabolic index optical fiber,” Opt. Lett. 17, 22–24 (1992).
    [CrossRef] [PubMed]
  8. M. Karlsson, D. Anderson, M. Desaix, and M. Lisak, “Dynamic effects of Kerr nonlinearity and spatial diffraction on self-phase modulation of optical pulses,” Opt. Lett. 16, 1373–1375 (1991).
    [CrossRef] [PubMed]
  9. A. L. Dyshko, V. N. Lugovoi, and A. M. Prokhorov, “Self-focusing of intense light beams,” JETP Lett. 6, 146–148 (1967).
  10. M. D. Feit and J. A. Fleck, Jr., “Beam nonparaxiality, filament formation, and beam breakup in the selffocusing of optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988).
    [CrossRef]
  11. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Eq. (5.2-14).
  12. L. Gagnon, “Exact solutions for optical wave propagation including transverse effects,” J. Opt. Soc. Am. B 7, 1098–1102 (1990).
    [CrossRef]
  13. See Ref. 11, Eqs. (19.1–6) and (19.1–7) at frequency ω. Since the linear refractive index (n0) is constant and the nonlinearity is weak (γ≪1), Eq. (9) remains valid as long as the vector property of the electromagnetic field is not dominant.
  14. S. Maneuf, R. Desailly, and C. Froehly, “Stable self-trapping of laser beams: observation in a nonlinear planar waveguide,” Opt. Commun. 65, 193–198 (1988).
    [CrossRef]

1992 (1)

1991 (2)

1990 (2)

1988 (2)

S. Maneuf, R. Desailly, and C. Froehly, “Stable self-trapping of laser beams: observation in a nonlinear planar waveguide,” Opt. Commun. 65, 193–198 (1988).
[CrossRef]

M. D. Feit and J. A. Fleck, Jr., “Beam nonparaxiality, filament formation, and beam breakup in the selffocusing of optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988).
[CrossRef]

1967 (1)

A. L. Dyshko, V. N. Lugovoi, and A. M. Prokhorov, “Self-focusing of intense light beams,” JETP Lett. 6, 146–148 (1967).

Aitchison, J. S.

Anderson, D.

Desailly, R.

S. Maneuf, R. Desailly, and C. Froehly, “Stable self-trapping of laser beams: observation in a nonlinear planar waveguide,” Opt. Commun. 65, 193–198 (1988).
[CrossRef]

Desaix, M.

Dyshko, A. L.

A. L. Dyshko, V. N. Lugovoi, and A. M. Prokhorov, “Self-focusing of intense light beams,” JETP Lett. 6, 146–148 (1967).

Feit, M. D.

Fleck Jr., J. A.

Froehly, C.

S. Maneuf, R. Desailly, and C. Froehly, “Stable self-trapping of laser beams: observation in a nonlinear planar waveguide,” Opt. Commun. 65, 193–198 (1988).
[CrossRef]

Gagnon, L.

Jackel, J. L.

Karlsson, M.

Leaird, D. E.

Lisak, M.

Lugovoi, V. N.

A. L. Dyshko, V. N. Lugovoi, and A. M. Prokhorov, “Self-focusing of intense light beams,” JETP Lett. 6, 146–148 (1967).

Maneuf, S.

S. Maneuf, R. Desailly, and C. Froehly, “Stable self-trapping of laser beams: observation in a nonlinear planar waveguide,” Opt. Commun. 65, 193–198 (1988).
[CrossRef]

Oliver, M. K.

Prokhorov, A. M.

A. L. Dyshko, V. N. Lugovoi, and A. M. Prokhorov, “Self-focusing of intense light beams,” JETP Lett. 6, 146–148 (1967).

Silberberg, Y.

Smith, P. W. E.

Vogel, E. M.

Weiner, A. M.

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

JETP Lett. (1)

A. L. Dyshko, V. N. Lugovoi, and A. M. Prokhorov, “Self-focusing of intense light beams,” JETP Lett. 6, 146–148 (1967).

Opt. Commun. (1)

S. Maneuf, R. Desailly, and C. Froehly, “Stable self-trapping of laser beams: observation in a nonlinear planar waveguide,” Opt. Commun. 65, 193–198 (1988).
[CrossRef]

Opt. Lett. (4)

Other (6)

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964); E. Garmire, R. Y. Chiao, and C. H. Townes, “Dynamics and characteristics of the self-trapping of intense light beams,” Phys. Rev. Lett. 16, 347–349 (1966).
[CrossRef]

V. I. Talanov, “Focusing of light in cubic media,” JETP Lett. 2, 138–141 (1965); P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965); Y. R. Shen, “Self-focusing: experimental,” Prog. Quantum Electron. PQUEAH 4, 1–34 (1975).
[CrossRef]

D. Anderson, M. Bonneal, and M. Lisak, “Self-trapped cylindrical laser beams,” Phys. Fluids 22, 1838–1840 (1979); D. Anderson and M. Bonneal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids 22, 105–109 (1979).
[CrossRef]

R. W. Boyd, Nonlinear Optics (London, Academic, 1992).

See Ref. 11, Eqs. (19.1–6) and (19.1–7) at frequency ω. Since the linear refractive index (n0) is constant and the nonlinearity is weak (γ≪1), Eq. (9) remains valid as long as the vector property of the electromagnetic field is not dominant.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Eq. (5.2-14).

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Figures (2)

Fig. 1
Fig. 1

The intensity distribution of bright solution (14) with three different nonlinearities: (a) γ=0.0001, (b) γ=0.0002, and (c) γ=0.0003.

Fig. 2
Fig. 2

The intensity distribution of dark solution (20) with three different nonlinearities: (a) γ=0.0001, (b) γ=0.0002, and (c) γ=0.0003.

Equations (35)

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2x2E+2y2E+k2E+γ|E|2E=0,
iτu+2x2u+γ|u|2u=0,
u(τ, x)=a2γ1/2 sech[a(x-bτ)]×expi12bx-14(b2-4a2)τ,
u(τ, x)=±a2|γ|1/2 tanh[a(x-bτ)]×expi12bx-14(b2+8a2)τ,
ihtF+2x02F+2x12F++2xN-12F
+γ|F|2F=0,
F(t, x¯)=A(v0x0+v1x1++vN-1xN-1+vNt+c0)×exp[iφ(k0x0+k1x1++kN-1xN-1+kNt+c1)],
F(t, x¯)=A0 sechγA022C11/2×v0x0++vN-1xN-1-2hC3t+c0×expik0x0++kN-1xN-1-1hC2-γA022t+ic1,
F(t, x¯)=±A0 tanh|γ|A022D11/2(v0x0++vN-1xN-1-2hD3t)+c0×expik0x0++kN-1xN-1-1h(D2+|γ|A02)t+ic1
2x2E+2y2E+2z2E+k2E+γ|E|2E=0,
EB(x, y, z)=A0 sechγA022C11/2(v0x+v1 y+v2z)+c0×exp[i(k0x+k1y+k2z)+ic1],
C1=v02+v12+v220,
C2=k02+k12+k22k2+γA022,
C3=v0k0+v1k1+v2k20.
EB(x, y)=A0 sechγA02γA02+2k21/2×k2-q2+γA0221/2x-q y×expiq x+ik2-q2+γA0221/2y.
ENSB(x, y)=A0 sechγA0221/2x-qky×expiq x+ik-q22k+γA024ky.
ED(x, y, z)=±A0 tanh|γ|A022D11/2×(v0x+v1y+v2z)+c0×exp[i(k0x+k1y+k2z)+ic1],
D1=v02+v12+v220,
D2=k02+k12+k22k2-|γ|A02,
D3=v0k0+v1k1+v2k20.
ED(x, y)=±A0 tanhγA022(k2-|γ|A02)1/2×[(k2-q2-|γ|A02)1/2x-q y]×exp[iq x+i(k2-q2-|γ|A02)1/2y].
ENSD(x, y)=±A0 tanh|γ|A0221/2x-qky×expiq x+ik-q22k-|γ|A022ky.
EB(r, θ, z)=A0 sechγA0221/2r cos(θ+θ0)×exp±ik2+γA0221/2z,
F(t, x¯)=4A0exp(iT/2)[cosh(3X)+3 exp(i4T)cosh(X)]cosh(4X)+4 cosh(2X)+3 cos(4T),
E(x, y)=A(x, y)exp(iky)=E(x, y)exp[iϕ(x, y)],
i2kyA+2x2A+2y2A+γ|A|2A=0.
i2kA*yA+A*2x2A+A*2y2A+γ|A|4=0.
kyρ2+xj1+yj2=0,
j1=ImA*xA,j2=ImA*yA.
kyρ2+xj1=0.
xj˜1+yj˜2=0,
j˜1=j1=E2xϕ,j˜2=E2yϕ.
j˜1=k0A02 sech2γA022(v02+v12)1/2(v0x+v1y),
j˜2=k1A02 sech2γA022(v02+v12)1/2(v0x+v1y).
xj˜1+yj˜2=C3E2,

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