Abstract

We solve the (2+1)D nonlinear Helmholtz equation (NLH) for input beams that collapse in the simpler NLS model. Thereby, we provide the first ever numerical evidence that nonparaxiality and backscattering can arrest the collapse. We also solve the (1+1)D NLH and show that solitons with radius of only half the wavelength can propagate over forty diffraction lengths with no distortions. In both cases we calculate the backscattered field, which has not been done previously. Finally, we compute the dynamics of counter-propagating solitons using the NLH model, which is more comprehensive than the previously used coupled NLS model.

© 2008 Optical Society of America

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  1. P. L. Kelley, "Self-Focusing of Optical Beams," Phys. Rev. Lett. 15, 1005 (1965).
    [CrossRef]
  2. G. A. Askar�??yan, "Self-Focusing Effect," Sov. Phys. JETP 15, 1088 (1962).
  3. Y. Shen, "Self-focusing: Experimental," Prog. Quantum Electron. 4, 1 (1975).
    [CrossRef]
  4. A. Barthelemy, S. Maneuf, and C. Froehly, "Propagation soliton et auto-confinement de faisceaux laser par non linearit optique de kerr," Opt. Commun. 55, 201 (1985).
    [CrossRef]
  5. O. Cohen et al., "Collisions between Optical Spatial Solitons Propagating in Opposite Directions," Phys. Rev. Lett. 89, 133,901 (2002).
  6. G. Fibich and B. Ilan, "Vectorial and random effects in self-focusing and in multiple filamentation," Physica D 157, 112 (2001).
    [CrossRef]
  7. S. N. Vlasov, "Structure of the field of wave beams with circular polarization near a nonlinear focus in a cubic medium journal," Sov. J. Quantum Electron. 17, 1191 (1987).
    [CrossRef]
  8. M. Feit and J. Fleck, "Beam nonparaxiality, filament formation and beam breakup in the self-focusing of optical beams," J. Opt. Soc. Am. B 5, 633 (1988).
    [CrossRef]
  9. G. Fibich, "Small beam nonparaxiality arrests self-focussing of optical beams," Phys. Rev. Lett. 76, 4356 (1996).
    [CrossRef] [PubMed]
  10. G. Fibich and G. C. Papanicolaou, "Self-focusing in the perturbed and unperturbed nonlinear Schrodinger equation in critical dimension," SIAM J. Appl. Math. 60, 183 (1999).
    [CrossRef]
  11. G. Fibich and S. V. Tsynkov, "High-order two-way artificial boundary conditions for nonlinear wave propagation with backscattering," J. Comp. Phys. 171, 632 (2001).
    [CrossRef]
  12. G. Fibich and S. V. Tsynkov, "Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions," J. Comp. Phys. 210, 183 (2005).
    [CrossRef]
  13. G. Fibich, B. Ilan, and S. V. Tsynkov, "Backscattering and nonparaxiality arrest collapse of damped nonlinear waves," SIAM J. Appl. Math. 63, 1718 (2003).
    [CrossRef]
  14. M. Sever, "An existence theorem for some semilinear elliptic systems," J. Differ. Equations 226, 572-593 (2006).
    [CrossRef]
  15. G. Baruch, G. Fibich, and S. V. Tsynkov, "Self-focusing in the perturbed and unperturbed nonlinear Schrodinger equation in critical dimension," J. Comput. Phys. 227, 820 (2007).
    [CrossRef]
  16. W. Chen and D. L. Mills, "Optical response of a nonlinear dielectric film," Phys. Rev. B 35, 524 (1987).
    [CrossRef]
  17. P. Chamorro-Posada, G. McDonald, and G. New, "Non-paraxial solitons," J. Mod. Opt. 45, 1111 (1998).
    [CrossRef]
  18. P Chamorro-Posada, G. S. McDonald, and G. H. C. New, "Non-paraxial beam propagation methods," Opt. Commun. 192, 1-12 (2001).
    [CrossRef]
  19. J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, "Helmholtz bright and boundary solitons," J. Phys. A: Math. Theor. 40, 1545 (2007).
    [CrossRef]
  20. J. M. Christian, G. S. McDonald, R. Potton, and P. Chamorro-Posada, "Helmholtz solitons in power-law optical materials," Phys. Rev. A 76, 033,834 (2007).
  21. S. Chi and Q. Gou, "Vector theory of self-focusing of an optical beam in Kerr media," Opt. Lett. 20, 1598 (2001).
    [CrossRef]

2007 (3)

G. Baruch, G. Fibich, and S. V. Tsynkov, "Self-focusing in the perturbed and unperturbed nonlinear Schrodinger equation in critical dimension," J. Comput. Phys. 227, 820 (2007).
[CrossRef]

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, "Helmholtz bright and boundary solitons," J. Phys. A: Math. Theor. 40, 1545 (2007).
[CrossRef]

J. M. Christian, G. S. McDonald, R. Potton, and P. Chamorro-Posada, "Helmholtz solitons in power-law optical materials," Phys. Rev. A 76, 033,834 (2007).

2006 (1)

M. Sever, "An existence theorem for some semilinear elliptic systems," J. Differ. Equations 226, 572-593 (2006).
[CrossRef]

2005 (1)

G. Fibich and S. V. Tsynkov, "Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions," J. Comp. Phys. 210, 183 (2005).
[CrossRef]

2003 (1)

G. Fibich, B. Ilan, and S. V. Tsynkov, "Backscattering and nonparaxiality arrest collapse of damped nonlinear waves," SIAM J. Appl. Math. 63, 1718 (2003).
[CrossRef]

2002 (1)

O. Cohen et al., "Collisions between Optical Spatial Solitons Propagating in Opposite Directions," Phys. Rev. Lett. 89, 133,901 (2002).

2001 (4)

G. Fibich and B. Ilan, "Vectorial and random effects in self-focusing and in multiple filamentation," Physica D 157, 112 (2001).
[CrossRef]

G. Fibich and S. V. Tsynkov, "High-order two-way artificial boundary conditions for nonlinear wave propagation with backscattering," J. Comp. Phys. 171, 632 (2001).
[CrossRef]

S. Chi and Q. Gou, "Vector theory of self-focusing of an optical beam in Kerr media," Opt. Lett. 20, 1598 (2001).
[CrossRef]

P Chamorro-Posada, G. S. McDonald, and G. H. C. New, "Non-paraxial beam propagation methods," Opt. Commun. 192, 1-12 (2001).
[CrossRef]

1999 (1)

G. Fibich and G. C. Papanicolaou, "Self-focusing in the perturbed and unperturbed nonlinear Schrodinger equation in critical dimension," SIAM J. Appl. Math. 60, 183 (1999).
[CrossRef]

1998 (1)

P. Chamorro-Posada, G. McDonald, and G. New, "Non-paraxial solitons," J. Mod. Opt. 45, 1111 (1998).
[CrossRef]

1996 (1)

G. Fibich, "Small beam nonparaxiality arrests self-focussing of optical beams," Phys. Rev. Lett. 76, 4356 (1996).
[CrossRef] [PubMed]

1988 (1)

1987 (2)

S. N. Vlasov, "Structure of the field of wave beams with circular polarization near a nonlinear focus in a cubic medium journal," Sov. J. Quantum Electron. 17, 1191 (1987).
[CrossRef]

W. Chen and D. L. Mills, "Optical response of a nonlinear dielectric film," Phys. Rev. B 35, 524 (1987).
[CrossRef]

1985 (1)

A. Barthelemy, S. Maneuf, and C. Froehly, "Propagation soliton et auto-confinement de faisceaux laser par non linearit optique de kerr," Opt. Commun. 55, 201 (1985).
[CrossRef]

1975 (1)

Y. Shen, "Self-focusing: Experimental," Prog. Quantum Electron. 4, 1 (1975).
[CrossRef]

1965 (1)

P. L. Kelley, "Self-Focusing of Optical Beams," Phys. Rev. Lett. 15, 1005 (1965).
[CrossRef]

1962 (1)

G. A. Askar�??yan, "Self-Focusing Effect," Sov. Phys. JETP 15, 1088 (1962).

Askar???yan, G. A.

G. A. Askar�??yan, "Self-Focusing Effect," Sov. Phys. JETP 15, 1088 (1962).

Barthelemy, A.

A. Barthelemy, S. Maneuf, and C. Froehly, "Propagation soliton et auto-confinement de faisceaux laser par non linearit optique de kerr," Opt. Commun. 55, 201 (1985).
[CrossRef]

Baruch, G.

G. Baruch, G. Fibich, and S. V. Tsynkov, "Self-focusing in the perturbed and unperturbed nonlinear Schrodinger equation in critical dimension," J. Comput. Phys. 227, 820 (2007).
[CrossRef]

Chamorro-Posada, P

P Chamorro-Posada, G. S. McDonald, and G. H. C. New, "Non-paraxial beam propagation methods," Opt. Commun. 192, 1-12 (2001).
[CrossRef]

Chamorro-Posada, P.

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, "Helmholtz bright and boundary solitons," J. Phys. A: Math. Theor. 40, 1545 (2007).
[CrossRef]

J. M. Christian, G. S. McDonald, R. Potton, and P. Chamorro-Posada, "Helmholtz solitons in power-law optical materials," Phys. Rev. A 76, 033,834 (2007).

P. Chamorro-Posada, G. McDonald, and G. New, "Non-paraxial solitons," J. Mod. Opt. 45, 1111 (1998).
[CrossRef]

Chen, W.

W. Chen and D. L. Mills, "Optical response of a nonlinear dielectric film," Phys. Rev. B 35, 524 (1987).
[CrossRef]

Chi, S.

Christian, J. M.

J. M. Christian, G. S. McDonald, R. Potton, and P. Chamorro-Posada, "Helmholtz solitons in power-law optical materials," Phys. Rev. A 76, 033,834 (2007).

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, "Helmholtz bright and boundary solitons," J. Phys. A: Math. Theor. 40, 1545 (2007).
[CrossRef]

Cohen, O.

O. Cohen et al., "Collisions between Optical Spatial Solitons Propagating in Opposite Directions," Phys. Rev. Lett. 89, 133,901 (2002).

Feit, M.

Fibich, G.

G. Baruch, G. Fibich, and S. V. Tsynkov, "Self-focusing in the perturbed and unperturbed nonlinear Schrodinger equation in critical dimension," J. Comput. Phys. 227, 820 (2007).
[CrossRef]

G. Fibich and S. V. Tsynkov, "Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions," J. Comp. Phys. 210, 183 (2005).
[CrossRef]

G. Fibich, B. Ilan, and S. V. Tsynkov, "Backscattering and nonparaxiality arrest collapse of damped nonlinear waves," SIAM J. Appl. Math. 63, 1718 (2003).
[CrossRef]

G. Fibich and S. V. Tsynkov, "High-order two-way artificial boundary conditions for nonlinear wave propagation with backscattering," J. Comp. Phys. 171, 632 (2001).
[CrossRef]

G. Fibich and B. Ilan, "Vectorial and random effects in self-focusing and in multiple filamentation," Physica D 157, 112 (2001).
[CrossRef]

G. Fibich and G. C. Papanicolaou, "Self-focusing in the perturbed and unperturbed nonlinear Schrodinger equation in critical dimension," SIAM J. Appl. Math. 60, 183 (1999).
[CrossRef]

G. Fibich, "Small beam nonparaxiality arrests self-focussing of optical beams," Phys. Rev. Lett. 76, 4356 (1996).
[CrossRef] [PubMed]

Fleck, J.

Froehly, C.

A. Barthelemy, S. Maneuf, and C. Froehly, "Propagation soliton et auto-confinement de faisceaux laser par non linearit optique de kerr," Opt. Commun. 55, 201 (1985).
[CrossRef]

Gou, Q.

Ilan, B.

G. Fibich, B. Ilan, and S. V. Tsynkov, "Backscattering and nonparaxiality arrest collapse of damped nonlinear waves," SIAM J. Appl. Math. 63, 1718 (2003).
[CrossRef]

G. Fibich and B. Ilan, "Vectorial and random effects in self-focusing and in multiple filamentation," Physica D 157, 112 (2001).
[CrossRef]

Kelley, P. L.

P. L. Kelley, "Self-Focusing of Optical Beams," Phys. Rev. Lett. 15, 1005 (1965).
[CrossRef]

Maneuf, S.

A. Barthelemy, S. Maneuf, and C. Froehly, "Propagation soliton et auto-confinement de faisceaux laser par non linearit optique de kerr," Opt. Commun. 55, 201 (1985).
[CrossRef]

McDonald, G.

P. Chamorro-Posada, G. McDonald, and G. New, "Non-paraxial solitons," J. Mod. Opt. 45, 1111 (1998).
[CrossRef]

McDonald, G. S.

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, "Helmholtz bright and boundary solitons," J. Phys. A: Math. Theor. 40, 1545 (2007).
[CrossRef]

J. M. Christian, G. S. McDonald, R. Potton, and P. Chamorro-Posada, "Helmholtz solitons in power-law optical materials," Phys. Rev. A 76, 033,834 (2007).

P Chamorro-Posada, G. S. McDonald, and G. H. C. New, "Non-paraxial beam propagation methods," Opt. Commun. 192, 1-12 (2001).
[CrossRef]

Mills, D. L.

W. Chen and D. L. Mills, "Optical response of a nonlinear dielectric film," Phys. Rev. B 35, 524 (1987).
[CrossRef]

New, G.

P. Chamorro-Posada, G. McDonald, and G. New, "Non-paraxial solitons," J. Mod. Opt. 45, 1111 (1998).
[CrossRef]

New, G. H. C.

P Chamorro-Posada, G. S. McDonald, and G. H. C. New, "Non-paraxial beam propagation methods," Opt. Commun. 192, 1-12 (2001).
[CrossRef]

Papanicolaou, G. C.

G. Fibich and G. C. Papanicolaou, "Self-focusing in the perturbed and unperturbed nonlinear Schrodinger equation in critical dimension," SIAM J. Appl. Math. 60, 183 (1999).
[CrossRef]

Potton, R.

J. M. Christian, G. S. McDonald, R. Potton, and P. Chamorro-Posada, "Helmholtz solitons in power-law optical materials," Phys. Rev. A 76, 033,834 (2007).

Sever, M.

M. Sever, "An existence theorem for some semilinear elliptic systems," J. Differ. Equations 226, 572-593 (2006).
[CrossRef]

Shen, Y.

Y. Shen, "Self-focusing: Experimental," Prog. Quantum Electron. 4, 1 (1975).
[CrossRef]

Tsynkov, S. V.

G. Baruch, G. Fibich, and S. V. Tsynkov, "Self-focusing in the perturbed and unperturbed nonlinear Schrodinger equation in critical dimension," J. Comput. Phys. 227, 820 (2007).
[CrossRef]

G. Fibich and S. V. Tsynkov, "Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions," J. Comp. Phys. 210, 183 (2005).
[CrossRef]

G. Fibich, B. Ilan, and S. V. Tsynkov, "Backscattering and nonparaxiality arrest collapse of damped nonlinear waves," SIAM J. Appl. Math. 63, 1718 (2003).
[CrossRef]

G. Fibich and S. V. Tsynkov, "High-order two-way artificial boundary conditions for nonlinear wave propagation with backscattering," J. Comp. Phys. 171, 632 (2001).
[CrossRef]

Vlasov, S. N.

S. N. Vlasov, "Structure of the field of wave beams with circular polarization near a nonlinear focus in a cubic medium journal," Sov. J. Quantum Electron. 17, 1191 (1987).
[CrossRef]

J. Phys. A: Math. Theor. (1)

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, "Helmholtz bright and boundary solitons," J. Phys. A: Math. Theor. 40, 1545 (2007).
[CrossRef]

J. Comp. Phys. (2)

G. Fibich and S. V. Tsynkov, "High-order two-way artificial boundary conditions for nonlinear wave propagation with backscattering," J. Comp. Phys. 171, 632 (2001).
[CrossRef]

G. Fibich and S. V. Tsynkov, "Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions," J. Comp. Phys. 210, 183 (2005).
[CrossRef]

J. Comput. Phys. (1)

G. Baruch, G. Fibich, and S. V. Tsynkov, "Self-focusing in the perturbed and unperturbed nonlinear Schrodinger equation in critical dimension," J. Comput. Phys. 227, 820 (2007).
[CrossRef]

J. Differ. Equations (1)

M. Sever, "An existence theorem for some semilinear elliptic systems," J. Differ. Equations 226, 572-593 (2006).
[CrossRef]

J. Mod. Opt. (1)

P. Chamorro-Posada, G. McDonald, and G. New, "Non-paraxial solitons," J. Mod. Opt. 45, 1111 (1998).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (2)

A. Barthelemy, S. Maneuf, and C. Froehly, "Propagation soliton et auto-confinement de faisceaux laser par non linearit optique de kerr," Opt. Commun. 55, 201 (1985).
[CrossRef]

P Chamorro-Posada, G. S. McDonald, and G. H. C. New, "Non-paraxial beam propagation methods," Opt. Commun. 192, 1-12 (2001).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

J. M. Christian, G. S. McDonald, R. Potton, and P. Chamorro-Posada, "Helmholtz solitons in power-law optical materials," Phys. Rev. A 76, 033,834 (2007).

Phys. Rev. B (1)

W. Chen and D. L. Mills, "Optical response of a nonlinear dielectric film," Phys. Rev. B 35, 524 (1987).
[CrossRef]

Phys. Rev. Lett. (3)

O. Cohen et al., "Collisions between Optical Spatial Solitons Propagating in Opposite Directions," Phys. Rev. Lett. 89, 133,901 (2002).

G. Fibich, "Small beam nonparaxiality arrests self-focussing of optical beams," Phys. Rev. Lett. 76, 4356 (1996).
[CrossRef] [PubMed]

P. L. Kelley, "Self-Focusing of Optical Beams," Phys. Rev. Lett. 15, 1005 (1965).
[CrossRef]

Physica D (1)

G. Fibich and B. Ilan, "Vectorial and random effects in self-focusing and in multiple filamentation," Physica D 157, 112 (2001).
[CrossRef]

Prog. Quantum Electron. (1)

Y. Shen, "Self-focusing: Experimental," Prog. Quantum Electron. 4, 1 (1975).
[CrossRef]

SIAM J. Appl. Math. (2)

G. Fibich, B. Ilan, and S. V. Tsynkov, "Backscattering and nonparaxiality arrest collapse of damped nonlinear waves," SIAM J. Appl. Math. 63, 1718 (2003).
[CrossRef]

G. Fibich and G. C. Papanicolaou, "Self-focusing in the perturbed and unperturbed nonlinear Schrodinger equation in critical dimension," SIAM J. Appl. Math. 60, 183 (1999).
[CrossRef]

Sov. J. Quantum Electron. (1)

S. N. Vlasov, "Structure of the field of wave beams with circular polarization near a nonlinear focus in a cubic medium journal," Sov. J. Quantum Electron. 17, 1191 (1987).
[CrossRef]

Sov. Phys. JETP (1)

G. A. Askar�??yan, "Self-Focusing Effect," Sov. Phys. JETP 15, 1088 (1962).

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Figures (5)

Fig. 1.
Fig. 1.

The physical setup: A: single beam, B: counter-propagating beams. C: A schematic of the upstream BC at z=Zmax+δ, which freely admits all forward propagating waves (red). D: A schematic of the downstream BC at z=-δ, which freely admits all the backward propagating waves (blue) to pass, and also specifies the (forward moving) incoming beam.

Fig. 2.
Fig. 2.

(color online) Arrest of collapse in the (2+1)D NLH. A: |E|2. B: Sz . C: comparison of normalized on-axis |E|2 (blue solid), Sz (red dashed), and NLS solution (black dotted)

Fig. 3.
Fig. 3.

(color online) NLH solutions with r 0 λ 0 = 3 π (blue, dots), 4 π (red, dash) and 6 π (green, solid). Solid black line is the NLS solution. A: Normalized on-axis intensity |E/E(z=0)|2. B: Normalized on-axis Poynting flux Sz /Sz (z=0). C: Transverse profile of the backward field at z=0-.

Fig. 4.
Fig. 4.

(1+1)D NLH soliton with r 0 = λ 0 2 propagating over 40LDF . A: Sz . B: On-axis |E|2.

Fig. 5.
Fig. 5.

(color online) Energy flux (Sz ) of the (1+1)D NLH with counter-propagating beams. A: Positive (forward) flux is red, negative (backward) flux is blue. B: The right-going beam at its incoming (blue dashed) and outgoing interface (red solid). Green dotted line is the coupled-NLS solution at the outgoing interface.

Equations (4)

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

E zz ( z , x ) + Δ E + k 0 2 ( 1 + ( 2 n 2 n 0 ) E 2 ) E = 0 ,
f 2 A ~ z ~ z ~ ( z ~ , x ~ ) + i A ~ z ~ + Δ A ~ + A ~ 2 A ~ = 0 ,
i A ~ z ~ ( z ~ , x ~ ) + Δ A ~ + A ~ 2 A ~ = 0 .
E A e ik 0 z + Be ik 0 z ,

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