## Simulations of the nonlinear Helmholtz equation: arrest of beam collapse, nonparaxial solitons and counter-propagating beams

Optics Express, Vol. 16, Issue 17, pp. 13323-13329 (2008)

http://dx.doi.org/10.1364/OE.16.013323

Acrobat PDF (272 KB)

### 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

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

*D*NLS to predict the possibility of a catastrophic collapse of beams whose input power is above the

*critical power for collapse*. In the case of propagation through planar waveguides, the 1D NLS was used to predict the existence of spatial solitons [2]. Both beam collapse in bulk medium and spatial solitons in planar waveguides were observed in experiments [3

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]

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

*D*NLS predicts that the beam radius shrinks to zero at collapse, the paraxial approximation breaks down at this point. In the case of spatial 1

*D*solitons, the paraxial approximation sets a lower limit on the soliton radius.

*D*NLS solutions for collapsing beams is non-physical. Therefore, an important question is whether the singularity formation is already arrested by taking one step back in the aforementioned series of approximations and employing the scalar NLH model, or only in a more comprehensive model. Both the mathematical analysis and simulations of the scalar NLH have proved to be considerably more difficult than for the NLS, since for the NLH one solves a nonlinear boundary-value problem, whereas the NLS requires solving an initial value problem. An additional computational obstacle is that unlike the NLS, which governs the slowly varying envelope, the NLH has to be approximated with sub-wavelength resolution. For these reasons, the question of collapse in the scalar NLH model was not fully answered for over 40 years.

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]

*D*cubic NLS is extremely sensitive to small perturbations [10

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]

*P*, but diverged for input powers higher than

_{cr}*P*. It was unclear, however, whether the divergence above

_{cr}*P*was due to limitations of the numerical method, or because collapse is not arrested in the scalar NLH model. Subsequently, the method of [11

_{cr}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. Diff. Eq. **226**, 572 (2006). [CrossRef]

14. M. Sever, “An existence theorem for some semilinear elliptic systems,” J. Diff. Eq. **226**, 572 (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]

*D*NLH, which models the propagation of plane waves in a Kerr medium. In this case, the solution always exists, but becomes non-unique (bistable) above a certain input power threshold [16

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

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]

*D*NLH for input powers above

*P*. Hence, we obtain the first ever computational evidence that the collapse of the beam is indeed arrested in the scalar NLH model. We also calculate the field backscattered from the domain. Moreover, we solve the (1+1)

_{cr}*D*NLH for a “nonparaxial” soliton with radius equal to half a wavelength, and observe that it propagates virtually unchanged over 40 diffraction lengths. This indicates that such beams are still in the paraxial regime. Finally, we solve the (1+1)

*D*NLH for two counter-propagating beams and compare the results to those obtained using the coupled NLS model.

*E*is the electric field,

*k*

_{0}is the linear wavenumber,

*n*

_{0}is the linear index of refraction and

*n*

_{2}is the Kerr coefficient. In the bulk medium (2+1)

*D*case

**x**

_{⊥}=(

*x*,

*y*) and Δ

_{⊥}=∂

^{2}

_{x}+∂

^{2}

_{y}; in the planar waveguide (1+1)

*D*case

**x**

_{⊥}=

*x*and Δ

_{⊥}=∂

^{2}

_{x}. We consider an incoming beam traveling in the positive

*z*direction (henceforth “forward” or “right”) impinging on a finite-length Kerr material slab at the

*z*=0 interface and exiting the Kerr medium at the

*z*=

*Z*

_{max}interface, see Fig. 1(A). A portion of the field may be reflected by the interfaces at

*z*=0 or

*z*=

*Z*

_{max}, or backscattered inside the Kerr medium, because of the variations of the index of refraction induced by the forward-propagating beam. To derive the NLS, the standard approach is to represent the field as

*A*is assumed slowly varying. Using the standard rescaling

**x̃**

_{⊥}=

**x**

_{⊥}/

*r*

_{0},

*z̃*=

*z*/2

*L*and

_{DF}*r*

_{0}is the input beam radius and

*L*=

_{DF}*k*

_{0}

*r*

^{2}

_{0}is the diffraction length, the NLH can be written in the dimensionless form

*nonparaxiality parameter*. Typically

*λ*

_{0}≪

*r*

_{0}so that

*f*

^{2}≪1 and

*f*

^{2}

*Ã*

_{z̃z̃}≪

*Ã*. Therefore, the paraxial approximation, which consists of neglecting

_{z̃}*f*

^{2}

*Ã*, leads to the NLS

_{z̃z̃}*D*NLH with cylindrical symmetry, i.e.,

*E*=

*E*(

*z*,

*r*) where

*δ*≤

*z*≤Z

_{max}+

*δ*in order to implement the BCs in the linear regions. At the material interfaces

*z*=0 and

*z*=

*Z*

_{max}where the index of refraction is discontinuous, Maxwell equations for a normal-incident field imply that

*E*and

*E*are continuous across the interfaces. At

_{z}*z*=

*Z*

_{max}+

*δ*we imposed the radiation BC that the field does not have any left-going component for

*z*>

*Z*

_{max}, see Fig. 1(C). Similarly, at

*z*=-

*δ*we implement the two-way radiation BC that for

*z*<0 the field does not have right-going components except for the prescribed incoming beam which impinges on the interface

*z*=0 with a transverse profile

*E*

_{inc}(

*r*), see Fig. 1(D). Because

*z*=-

*δ*and

*z*=

*Z*

_{max}+

*δ*are outside the Kerr slab, the field propagation there is linear, which simplifies the implementation of the radiation BCs, see [11

**171**, 632 (2001). [CrossRef]

**210**, 183 (2005). [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]

*n*

_{0}in the Kerr medium (0≤

*z*≤

*Z*

_{max}) and in the surrounding linear medium (

*z*<0 and at

*z*>

*Z*

_{max}) are chosen to be equal, so that to eliminate the reflections due to discontinuity of

*n*

_{0}at the interfaces. However, discontinuities in the nonlinear coefficient are not eliminated, and are a source of reflections at

*z*=0,

*Z*

_{max}. Our numerical method can be applied to the case of different

*n*

_{0}with no change [15

**227**, 820 (2007). [CrossRef]

*simulations in this Letter are valid for any physical value of k*that corresponds to the same dimensionless quantities

_{0}, n_{0}, n_{2}*f*

^{2}and

*P*/

*P*.

_{cr}*r*

_{0}=1.27

*λ*

_{0}, corresponding to nonparaxiality parameter of

*f*

^{2}=(

*k*

_{0}

*r*

_{0})

^{-2}=1/64, and input power of

*P*=1.29

*P*. The NLH solution initially self-focuses, until

_{cr}*z*≈0.8

*L*where the collapse is arrested, after which the solution defocuses, see Fig. 2(A). The corresponding NLS solution collapses at

_{DF}*z*=0.68

_{c}*L*, see Fig. 2(C). This comparison of the NLH and NLS provides a direct numerical evidence that

_{DF}*collapse is arrested in the scalar NLH model*.

*E*|

^{2}in the

*z*direction in Fig. 2(A) are not a numerical artifact, but rather account for the actual physics. Indeed, let us first note that a part of the forward-propagating wave is reflected backwards by the material interfaces at

*z*=0 and

*z*=

*Z*

_{max}. In addition, since the forward propagating beam induces changes in the refraction index, part of the beam may be backscattered inside the Kerr medium. The presence of both forward and backward traveling fields, i.e,

*E*|

^{2}should undergo oscillations with wavenumber ~2

*k*

_{0}. Note that the analytical solutions of the (0+1)

*D*NLH also exhibit these 2

*k*

_{0}intensity oscillations [16

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

*k*

_{0}oscillations implies that the index of refraction also oscillates. In other words, the backward traveling field induces a 2

*k*

_{0}Bragg grating. This prediction may be tested by pump-probe experiments. In order to find a smoother representation of the solution, recall that for the NLS (3) the conserved beam power is

*P*=∫|

_{NLS}*Ã*|

^{2}

*d*

**x̃**

_{⊥}. For the NLH (1), however, the conserved beam power is

*P*=∫

_{NLH}*S*

_{z}*d*

**x**

_{⊥}, where

**S**=

*k*

_{0}Im(

*E**∇

*E*) is the energy flux, or Poynting vector, and

*z*-component. Specifically, for the field (4) the value of

*S*reduces to the flux difference

_{z}*S*≈

_{z}*k*

^{2}

_{0}(|

*A*|

^{2}-|

*B*|

^{2}). It is therefore much smoother than |

*E*|

^{2}, and provides a “more natural” depiction of the NLH solution, as confirmed by comparing

*S*of Fig. 2(B) with |

_{z}*E*|

^{2}of Fig. 2(A). The energy flux

*S*shows the arrest of collapse and the focusing-defocusing dynamics more clearly, see also Fig. 2(C).

_{z}*f*

^{2}, in Fig. 3 we fix the wavelength and vary the input beam radius

*r*

_{0}(while keeping the power unchanged) so that

*f*

^{-2}=36, 64, and 144. All the NLH solutions initially follow the collapsing NLS solution, but later the collapse is arrested and the solution defocuses. As expected, for a wider input beam (lower nonparaxiality), the deviations from the NLS solution and the arrest of collapse occur later, and the maximum self-focusing is higher. Again we see that |

*E*|

^{2}has 2

*k*

_{0}oscillations (whose magnitude increases as the input beam becomes more nonparaxial), while the energy flux

*S*is smooth.

_{z}*z*=0. To the best of our knowledge,

*this is the first ever calculation of the backscattered field of collapsing beams*, which is due to backscattering from inside the Kerr medium and reflections from the nonlinear interface. As the input beam radius

*r*

_{0}decreases, the power of the backscattered field increases from 0.46% to 0.63%, to 2.1% of the incoming beam power. This, as well as a comparison of magnitudes of oscillations in Figure 3(A), shows that the backscattered field increases as the input beam becomes more nonparaxial.

*D*configurations, the NLS possesses stable soliton solutions. It is generally believed that the paraxial approximation breaks down when the beam width becomes comparable to λ

_{0}and that, therefore, no solitons of such narrow width exist. To see that this is not the case, the (1+1)

*D*NLH (1) is solved for the incoming NLS-soliton profile

*x*/

*r*

_{0}) with width

*r*

_{0}=λ

_{0}/2, impinging on a Kerr slab of finite length

*Z*

_{max}=40

*L*. As in the (2+1)

_{DF}*D*case, we impose continuity of

*E*and

*E*at the material interfaces

_{z}*z*=0 and

*z*=

*Z*

_{max}, and apply the radiation BCs in the linear regions at

*z*=-

*δ*and

*z*=

*Z*

_{max}+

*δ*. The solution inside the Kerr-slab resembles a “nonparaxial soliton” which propagates virtually unchanged, see Fig. 4(A). We note that even for such a narrow beam, the nonparaxiality parameter is still moderate, as

*f*

^{2}=1/

*π*

^{2}≈0.1, which may explain why there there still exist soliton-like solutions. Similarly to the (2+1)

*D*case, because a part of the forward propagating beam is backscattered, |

*E*|

^{2}exhibits the fast 2

*k*

_{0}oscillations (Fig. 4(B)), while

*S*is smooth. In this case, the backscattered field leads to 10% oscillations in |

_{z}*E*|

^{2}.

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

18. C.-P. P., M. G.S., and G. New, “Non-paraxial beam propagation methods,” Opt. Commun. **192**, 1 (2001). [CrossRef]

*D*Helmholtz equation over a semi-infinite Kerr medium, of the form

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

*z*=

*Z*

_{max}, and the full NLH as a boundary-value problem must be solved.

*D*configuration of recent interest is that of counter-propagating beams, when a right traveling soliton impinges at the left interface and a left traveling beam impinges at the right interface (Fig. 1(B)). This configuration was analyzed numerically by Cohen et al.[5] using a coupled NLS system, which is derived from the NLH by employing the paraxial approximation and further assuming that asynchronous terms of the Kerr nonlinearity can be neglected. In doing so, the BCs should simultaneously account for the coupled incoming and outgoing fields at each interface. As noted in [5], these BCs can only be approximately accommodated in the coupled NLS model. In contrast, they can be fully implemented in the NLH model, without any approximation. Figure 5 presents our solution of the NLH for counter-propagating beams of radius

*r*

_{0}=

*λ*

_{0}that enter a Kerr material slab at the opposite interfaces with a transverse displacement of

*d*=4.4

*λ*

_{0}, and propagate over 10

*L*. It shows that the beams are slightly attracted toward each other and also become wider as they propagate. The results are in close agreement with the coupled NLS model, see Fig. 5(B). Therefore, the more comprehensive NLH model confirms the validity of the coupled NLS model for counter-propagating beams even for the “extreme” parameters of

_{DF}*r*

_{0}=

*λ*

_{0}and

*d*=4.4

*λ*

_{0}.

*f*

^{2}, and hence are of the same order of magnitude as nonparaxiality. In bulk media, they have been shown to have the same effect as nonparaxiality, which is to arrest the collapse [21

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

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

*r*

_{0}=

*λ*

_{0}/2,

*f*

^{2}is small, and so we expect that both nonparaxial and vectorial effects are likely to have a secondary effect on the propagation dynamics. Therefore, the sub-wavelength solitons predicted in the Letter are likely to remain stable also in the more comprehensive vector NLH model.

## Acknowledgments

## References and links

1. | P. L. Kelley, “Self-Focusing of Optical Beams,” Phys. Rev. Lett. |

2. | G. A. Askar’yan, “Self-Focusing Effect,” Sov. Phys. JETP |

3. | Y. Shen, “Self-focusing: Experimental,” Prog. Quantum Electron. |

4. | A. Barthelemy, S. Maneuf, and C. Froehly, “Propagation soliton et auto-confinement de faisceaux laser par non linearit optique de kerr,” Opt. Commun. |

5. | O. Cohen et al., “Collisions between Optical Spatial Solitons Propagating in Opposite Directions,” Phys. Rev. Lett. |

6. | G. Fibich and B. Ilan, “Vectorial and random effects in self-focusing and in multiple filamentation,” Physica D |

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. |

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 |

9. | G. Fibich, “Small beam nonparaxiality arrests self-focussing of optical beams,” Phys. Rev. Lett. |

10. | G. Fibich and G.C. Papanicolaou, “Self-focusing in the perturbed and unperturbed nonlinear Schrodinger equation in critical dimension,” SIAM J. Appl. Math. |

11. | G. Fibich and S. V. Tsynkov, “High-order two-way artificial boundary conditions for nonlinear wave propagation with backscattering,” J. Comp. Phys. |

12. | G. Fibich and S. V. Tsynkov, “Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions,” J. Comp. Phys. |

13. | G. Fibich, B. Ilan, and S. V. Tsynkov, “Backscattering and nonparaxiality arrest collapse of damped nonlinear waves,” SIAM J. Appl. Math. |

14. | M. Sever, “An existence theorem for some semilinear elliptic systems,” J. Diff. Eq. |

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. |

16. | W. Chen and D. L. Mills, “Optical response of a nonlinear dielectric film,” Phys. Rev. B |

17. | P. Chamorro-Posada, G. McDonald, and G. New, “Non-paraxial solitons,” J. Mod. Opt. |

18. | C.-P. P., M. G.S., and G. New, “Non-paraxial beam propagation methods,” Opt. Commun. |

19. | J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, “Helmholtz bright and boundary solitons,” J. of Phys. A: Math. Theor. |

20. | J. M. Christian, G. S. McDonald, R. Potton, and P. Chamorro-Posada, “Helmholtz solitons in power-law optical materials,” Phys. Rev. A |

21. | S. Chi and Q. Gou, “Vector theory of self-focusing of an optical beam in Kerr media,” Opt. Lett. |

**OCIS Codes**

(190.3270) Nonlinear optics : Kerr effect

(190.5940) Nonlinear optics : Self-action effects

(190.6135) Nonlinear optics : Spatial solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: April 15, 2008

Revised Manuscript: June 23, 2008

Manuscript Accepted: July 25, 2008

Published: August 14, 2008

**Citation**

G. Baruch, G. Fibich, and Semyon Tsynkov, "Simulations of the nonlinear Helmholtz equation: arrest of beam collapse, nonparaxial solitons and counter-propagating beams," Opt. Express **16**, 13323-13329 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-13323

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### References

- P. L. Kelley, "Self-Focusing of Optical Beams," Phys. Rev. Lett. 15, 1005 (1965). [CrossRef]
- G. A. Askar�??yan, "Self-Focusing Effect," Sov. Phys. JETP 15, 1088 (1962).
- Y. Shen, "Self-focusing: Experimental," Prog. Quantum Electron. 4, 1 (1975). [CrossRef]
- 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]
- O. Cohen et al., "Collisions between Optical Spatial Solitons Propagating in Opposite Directions," Phys. Rev. Lett. 89, 133,901 (2002).
- G. Fibich and B. Ilan, "Vectorial and random effects in self-focusing and in multiple filamentation," Physica D 157, 112 (2001). [CrossRef]
- 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]
- 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]
- G. Fibich, "Small beam nonparaxiality arrests self-focussing of optical beams," Phys. Rev. Lett. 76, 4356 (1996). [CrossRef] [PubMed]
- 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 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]
- 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]
- M. Sever, "An existence theorem for some semilinear elliptic systems," J. Differ. Equations 226, 572-593 (2006). [CrossRef]
- 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]
- W. Chen and D. L. Mills, "Optical response of a nonlinear dielectric film," Phys. Rev. B 35, 524 (1987). [CrossRef]
- P. Chamorro-Posada, G. McDonald, and G. New, "Non-paraxial solitons," J. Mod. Opt. 45, 1111 (1998). [CrossRef]
- P Chamorro-Posada, G. S. McDonald, and G. H. C. New, "Non-paraxial beam propagation methods," Opt. Commun. 192, 1-12 (2001). [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).
- S. Chi and Q. Gou, "Vector theory of self-focusing of an optical beam in Kerr media," Opt. Lett. 20, 1598 (2001). [CrossRef]

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