## Self-accelerating self-trapped nonlinear beams of Maxwell's equations |

Optics Express, Vol. 20, Issue 17, pp. 18827-18835 (2012)

http://dx.doi.org/10.1364/OE.20.018827

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### Abstract

We present shape-preserving self-accelerating beams of Maxwell's equations with optical nonlinearities. Such beams are exact solutions to Maxwell's equations with Kerr or saturable nonlinearity. The nonlinearity contributes to self-trapping and causes backscattering. Those effects, together with diffraction effects, work to maintain shape-preserving acceleration of the beam on a circular trajectory. The backscattered beam is found to be a key issue in the dynamics of such highly non-paraxial nonlinear beams. To study that, we develop two new techniques: projection operator separating the forward and backward waves, and reverse simulation. Finally, we discuss the possibility that such beams would reflect themselves through the nonlinear effect, to complete a 'U' shaped trajectory.

© 2012 OSA

## Introduction

1. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**(8), 979–981 (2007). [CrossRef] [PubMed]

2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. **99**(21), 213901 (2007). [CrossRef] [PubMed]

3. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics **2**(11), 675–678 (2008). [CrossRef]

4. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved Plasma Channel Generation Using Ultraintense Airy Beams,” Science **324**(5924), 229–232 (2009). [CrossRef] [PubMed]

5. I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-Accelerating Self-Trapped Optical Beams,” Phys. Rev. Lett. **106**(21), 213903 (2011). [CrossRef] [PubMed]

13. J. A. Giannini and R. I. Joseph, “The role of the second Painlevé transcendent in nonlinear optics,” Phys. Lett. A **141**(8-9), 417–419 (1989). [CrossRef]

9. Y. Hu, S. Huang, P. Zhang, C. Lou, J. Xu, and Z. Chen, “Persistence and breakdown of Airy beams driven by an initial nonlinearity,” Opt. Lett. **35**(23), 3952–3954 (2010). [CrossRef] [PubMed]

11. Y. Fattal, A. Rudnick, and D. M. Marom, “Soliton shedding from Airy pulses in Kerr media,” Opt. Express **19**(18), 17298–17307 (2011). [CrossRef] [PubMed]

4. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved Plasma Channel Generation Using Ultraintense Airy Beams,” Science **324**(5924), 229–232 (2009). [CrossRef] [PubMed]

9. Y. Hu, S. Huang, P. Zhang, C. Lou, J. Xu, and Z. Chen, “Persistence and breakdown of Airy beams driven by an initial nonlinearity,” Opt. Lett. **35**(23), 3952–3954 (2010). [CrossRef] [PubMed]

11. Y. Fattal, A. Rudnick, and D. M. Marom, “Soliton shedding from Airy pulses in Kerr media,” Opt. Express **19**(18), 17298–17307 (2011). [CrossRef] [PubMed]

5. I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-Accelerating Self-Trapped Optical Beams,” Phys. Rev. Lett. **106**(21), 213903 (2011). [CrossRef] [PubMed]

5. I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-Accelerating Self-Trapped Optical Beams,” Phys. Rev. Lett. **106**(21), 213903 (2011). [CrossRef] [PubMed]

6. A. Lotti, D. Faccio, A. Couairon, D. G. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, “Stationary nonlinear Airy beams,” Phys. Rev. A **84**(2), 021807 (2011). [CrossRef]

7. R. Bekenstein and M. Segev, “Self-accelerating optical beams in highly nonlocal nonlinear media,” Opt. Express **19**(24), 23706–23715 (2011). [CrossRef] [PubMed]

**106**(21), 213903 (2011). [CrossRef] [PubMed]

8. I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental Observation of Self-Accelerating Beams in Quadratic Nonlinear Media,” Phys. Rev. Lett. **108**(11), 113903 (2012). [CrossRef] [PubMed]

**106**(21), 213903 (2011). [CrossRef] [PubMed]

6. A. Lotti, D. Faccio, A. Couairon, D. G. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, “Stationary nonlinear Airy beams,” Phys. Rev. A **84**(2), 021807 (2011). [CrossRef]

## Derivation of the solution

*n*(

^{2}*E*)

*= n*(1 +

_{0}^{2}*ε*(|

*E*|)) for a general saturable or Kerr-like nonlinearity that depends only on the intensity

*I = |E|*. In general, the nonlinear permittivity

^{2}*ε*is a tensor [24

24. O. Peleg, M. Segev, G. Bartal, D. N. Christodoulides, and N. Moiseyev, “Nonlinear Waves in Subwavelength Waveguide Arrays: Evanescent Bands and the “Phoenix Soliton”,” Phys. Rev. Lett. **102**(16), 163902 (2009). [CrossRef] [PubMed]

*E*, but we analyze here the case of a TE-polarized beam which does not mix the polarization. For a y-polarized time-harmonic field propagating in the x-z plane,

*Ē = E*, Maxwell's equation yield the nonlinear Helmholtz equation:Equation (1) displays full symmetry between the x and z coordinates. Hence it is logical to seek a shape-preserving beam whose trajectory resides on a circle. We therefore transform to polar coordinates

_{Y}(x,z,t)ŷ*r,θ*by taking

*z = r sin(θ), x = r cos(θ)*, and seek shape-preserving solutions of the form

*E = U(r)e*. Here,

^{iαθ}*α*is some real number indicating the phase velocity of the beam in the azimuthal direction, which is the direction of propagation. The radial function

*U(r)*must satisfy:A special case of this equation (specific nonlinearity and no acceleration;

*α =*0) is solved in [25

25. M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Trapani, “Nonlinear Unbalanced Bessel Beams: Stationary Conical Waves Supported by Nonlinear Losses,” Phys. Rev. Lett. **93**(15), 153902 (2004). [CrossRef] [PubMed]

*α*(large beam bending). To find accelerating (

*α*>0) solutions of Eq. (2), we notice that for some sufficiently small

*r*=

*r*

_{0}value (typically

*α/k*), the amplitude is small enough such that the nonlinear term is negligible. Hence, we use the analytic Bessel solution of the linear equation to set the boundary conditions:

_{0}*U(r*, and with them we solve the eigenfunction Eq. (2) numerically. This method successfully finds solutions for the Kerr nonlinearity in both focusing and de-focusing cases, and also works for saturable nonlinearities. Other types of nonlinearity would also yield solutions, but in certain cases of strong nonlinearities will have a threshold for the existence of an accelerating solution. For example, above a certain value of defocusing Kerr, no solution exists (a similar effect is found in the paraxial regime; see [5

_{0}) = J_{α}(r_{0}k_{0}), U'(r_{0}) = J'_{α}(r_{0}k_{0})**106**(21), 213903 (2011). [CrossRef] [PubMed]

26. G. I. Stegeman and M. Segev, “Optical Spatial Solitons and Their Interactions: Universality and Diversity,” Science **286**(5444), 1518–1523 (1999). [CrossRef] [PubMed]

1. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**(8), 979–981 (2007). [CrossRef] [PubMed]

2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. **99**(21), 213901 (2007). [CrossRef] [PubMed]

20. I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting Accelerating Wave Packets of Maxwell’s Equations,” Phys. Rev. Lett. **108**(16), 163901 (2012). [CrossRef] [PubMed]

*k/α*([20

20. I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting Accelerating Wave Packets of Maxwell’s Equations,” Phys. Rev. Lett. **108**(16), 163901 (2012). [CrossRef] [PubMed]

*d*

^{3}

*k*

^{2})

^{−1}, where

*d*is the scale of the Airy beam and is typically the width of the main lobe. From such comparison we get that

*α =*2(

*dk*)

^{3}= 16π

^{3}(

*d/λ*)

^{3}~500 × (the number of wavelengths in the main lobe)

^{3}. To proceed, we obtain that needed number by considering the nonlinear effect on a single lobe. That is, from intuitive soliton considerations (the existence curve) [26

26. G. I. Stegeman and M. Segev, “Optical Spatial Solitons and Their Interactions: Universality and Diversity,” Science **286**(5444), 1518–1523 (1999). [CrossRef] [PubMed]

^{−4}, the typical width of a beam which would be significantly affected by the nonlinearity is around 20 wavelengths. Substituting this estimate forces us to take

*α*to be at least 4 × 10

^{6}. From

*α*, it is easy to find the needed transverse width of the beam: to achieve significant bending, the initial aperture should be at least half the radius of the circular trajectory ([20

20. I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting Accelerating Wave Packets of Maxwell’s Equations,” Phys. Rev. Lett. **108**(16), 163901 (2012). [CrossRef] [PubMed]

*α*/

*k*, which is more than 4 meters, for the above parameters in the visible wavelength regime. The important consequence of this calculation is that one needs a strong nonlinearity to observe the nonlinear self-accelerating Maxwell beam. In particular, for the Kerr nonlinearity, the dimensionless nonlinear refractive index change is

*n*= 2(

_{2}I*dk*)

^{−2}. Hence we can write

*α*directly using the refractive index change:

*α*= 2

^{(5/2)}(

*n*)

_{2}I^{(−3/2)}. A strong nonlinearity of

*n*~10

_{2}I^{−2}, which is nevertheless accessible in organic materials, yields

*α*~5,500, which would require an initial beam width of ~1cm to observe significant non-paraxial nonlinear bending.

*x-z*plane). One might be tempted to assume that this beam can propagate in circles endlessly, anti-clockwise, with its phase velocity

*α*. However, the boundary conditions here are still at plane z = 0: the incident beam is launched as

*E(x,z = 0)*and is propagating in the forward

*+ z*direction only. Therefore, the solution of (2) should be used only as the initial condition for

*x>0*in the

*z*= 0 plane, where

*α*>0 yields a beam bending anti-clockwise, or for

*x<0*in the z = 0 plane where

*α*<0 generates a clockwise-bending beam. This selection of initial conditions breaks the full symmetry of the polar coordinates, and limits the acceleration of the beam to one quarter of the circle, after which the trajectory should (according to the solution of Eq. (2)) bend back toward

*-z*. However, only a backward propagating beam can actually go back along the second quarter of the circle, and such a beam does not agree with the physical boundary conditions [20

**108**(16), 163901 (2012). [CrossRef] [PubMed]

*z = 0*, but instead it approximates the desired beam with increasing accuracy for larger

*α*. This peculiarity is discussed in more details in [20

**108**(16), 163901 (2012). [CrossRef] [PubMed]

27. I. Kaminer, Y. Lumer, M. Segev, and D. N. Christodoulides, “Causality effects on accelerating light pulses,” Opt. Express **19**(23), 23132–23139 (2011). [CrossRef] [PubMed]

28. G. Fibich and S. Tsynkov, “High-order two-way artificial boundary conditions for nonlinear wave propagation with backscattering,” J. Comput. Phys. **171**(2), 632–677 (2001). [CrossRef]

## Decoupling the forward and backward waves

*x→k*) which takes the form of

_{x}*z*= 0). However, in a nonlinear wave system, the task of separating the wavepakcet into the forward and backward propagating parts is more complicated, but possible; see Fig. 2(c,d) for the forward and backward beams, calculated by the method explained in this section. We suggest a general technique for this separation, that can actually be utilized for any wave system which is second-order in the propagation parameter

*z*. For each plane of constant

*z*, we substitute the field into the nonlinear term to define a linear operator

_{0}*M = ∂*; Eq. (1) is now

_{x}^{2}+ k_{0}^{2}+ k_{0}^{2}ε(I)*E*. Using the eigenvalues

_{zz}= -ME*λ*and the eigenfunctions

_{i}*u*of

_{i}= u_{i}(x)*M*, one can express any solution (close enough to the plane

*z = z*) as a sum of the forward and backward propagating parts (first and second sums respectively)Note that all the terms of the sums are not evanescent, since

_{0}*M*is a positive definite operator, hence all its eigenvalues are positive. This property of

*M*also causes the square root of the operator

*√M*to be well defined (

*(√M)*). One can prove that this yields a projection operator of the form

^{2}= M*z = z*separately, since our beam changes in different planes and vary the induced refractive index. Numerically, the calculation is done by algorithms for the square root of matrixes.

_{0}*z*= 0 into the forward and the backward parts is not sufficient for identifying the physical solution arising from a beam launched at plane z = 0 and propagating in the forward ( + z) direction only. This is because the problem is nonlinear, hence even after the division; one cannot eliminate the backward waves, since they are part of the physical solution. Mathematically, this means that the correct boundary condition (e.g., at z = 0) has an unknown back-propagating part. Physically, any incoming accelerating beam is expected to backscatter part of it, since it has a wide

*k*-space spectrum. This creates a wave that propagates back which includes contributions from waves back-reflected from many planes, and might even reflect part of it forward again. In any case, the backward part of the beam at

*z*= 0 cannot be set to zero. Rather, the backward propagating part at z = 0 depends on the incoming beam in a complicated manner. This issue will be revisited below, and as we show there, one can resolve it numerically via “reverse simulation”.

## Coupled mode theory

*θ*= 0 and the second is launched clockwise from some other angle

*θ*<90°. An exact solution of (2), such as Fig. 2(a), would be achieved only under three terms: First, if the forward propagating part of the beam (Fig. 2(c)) is chosen as the boundary at

_{1}*θ*= 0. Second, the clockwise propagating part of the beam is chosen as the boundary condition at

*θ*(due to azimuthal symmetry we can use the values of the backward part of Fig. 2(d) on any

_{1}*θ*). Third, the two beams must be mutually coherent. We should highlight the fascinating underlying physical insight in this: the backscattered beam of the counterclockwise part destructively interferes with part of the clockwise propagating beam, and vise versa. All in all, the two nonlinear wavepackets (Fig. 2(c,d)) add up into the exact solutions presented in Fig. 2(a). Consequently, the nonlinear coupling of the backward (or backscattered) and to the forward waves is vital in understanding of the full nonlinear dynamics. Yet, simulating the nonlinear dynamics is essential for checking the stability of the solution which is by no means guaranteed, and the numerical simulation is the strongest method to verify stability. However, simulating this coupled dynamics of the two beams is very complicated. Such simulation can be carried out in an iterative manner similar to that of [34

34. O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, “Collisions between Optical Spatial Solitons Propagating in Opposite Directions,” Phys. Rev. Lett. **89**(13), 133901 (2002). [CrossRef] [PubMed]

*z*= 0, with no pre-assumptions.

## The reverse simulation

*z*the desired solution should consist of only a forward propagating part. Our simulation utilize this fact: First, we take the solution of Eq. (2) (Fig. 2(a)) at

_{1}*z = z*and use the nonlinear projection of it onto the forward propagating wavepacket (Fig. 2(c)). Second, we propagate this beam in reverse until

_{1}*z*= 0, by combining an explicit Runge-Kutta method with nonlinear beam propagation terms. In practice, of course we cannot take an infinite

*z*value, but still our numerical simulation yields precise accelerating beams (Fig. 2(b) presents the beam calculated by our reverse simulation method). Note that the solution is close to the full (forward + backward) solution of Eq. (2) (Fig. 2(a)). The reason is that large

*α*give a strongly anti-clockwise beam which has almost all the power in the forward part. Therefore, the initial beam can be chosen directly from the result of Fig. 2(a) with almost no change. The beam bends to an angle of 45°, while maintaining an almost shape-preserving profile. However, note that backscattering does play an important role – the trajectory is no longer an exact circle, and is instead stretched due to the absence of the balancing backward wave. The comparison to the exact circle is marked by the black dashed ark. For clarification, we emphasize that the beam of Fig. 2(b) is the physical solution of a beam launched at

*z*= 0, while the beam of Fig. 2(a) requires two counter propagating beams, and is otherwise only a mathematical solution of Eq. (2).

*z*. Is it also correct in the nonlinear case? At first glance, it appears that the Eq. (2) is symmetric to the exchange z↔-z so we expect it to be reversible. However, the answer is actually that the simulation is reversible only if also the noise is unchanged. That is, in order for the nonlinear dynamics to be completely reversible, all the boundary conditions must be the same for the forward and the backward propagation. This means that one should use the exact field amplitude at the output plane (including the weak noise, amplitude and phase). In practice, this means that our numerical method of reverse simulation could lead to errors in the presence of very strong nonlinearities, which might causes the dynamics to be unstable. That is, under very strong nonlinearities small values of noise could play a major role, hence the reverse simulation might be extremely sensitive to numerical errors. However, in practice, we find that as long as the maximum nonlinear index change is smaller than ~1, the reverse simulation method works well.

## Boomerang beam?

## Conclusions

## Acknowledgments

## References and links

1. | G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. |

2. | G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. |

3. | J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics |

4. | P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved Plasma Channel Generation Using Ultraintense Airy Beams,” Science |

5. | I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-Accelerating Self-Trapped Optical Beams,” Phys. Rev. Lett. |

6. | A. Lotti, D. Faccio, A. Couairon, D. G. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, “Stationary nonlinear Airy beams,” Phys. Rev. A |

7. | R. Bekenstein and M. Segev, “Self-accelerating optical beams in highly nonlocal nonlinear media,” Opt. Express |

8. | I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental Observation of Self-Accelerating Beams in Quadratic Nonlinear Media,” Phys. Rev. Lett. |

9. | Y. Hu, S. Huang, P. Zhang, C. Lou, J. Xu, and Z. Chen, “Persistence and breakdown of Airy beams driven by an initial nonlinearity,” Opt. Lett. |

10. | R. Chen, C. Yin, X. Chu, and H. Wang, “Effect of Kerr nonlinearity on an Airy beam,” Phys. Rev. A |

11. | Y. Fattal, A. Rudnick, and D. M. Marom, “Soliton shedding from Airy pulses in Kerr media,” Opt. Express |

12. | Y. Hu, Z. Sun, D. Bongiovanni, D. Song, C. Lou, J. Xu, Z. Chen, and R. Morandotti, “Reshaping the trajectory and spectrum of nonlinear Airy beams,” to appear in Opt. Lett. (2012). |

13. | J. A. Giannini and R. I. Joseph, “The role of the second Painlevé transcendent in nonlinear optics,” Phys. Lett. A |

14. | M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. |

15. | A. V. Novitsky and D. V. Novitsky, “Nonparaxial Airy beams: role of evanescent waves,” Opt. Lett. |

16. | L. Carretero, P. Acebal, S. Blaya, C. García, A. Fimia, R. Madrigal, and A. Murciano, “Nonparaxial diffraction analysis of Airy and SAiry beams,” Opt. Express |

17. | A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, “Generation and near-field imaging of airy surface plasmons,” Phys. Rev. Lett. |

18. | E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. |

19. | L. Froehly, F. Courvoisier, A. Mathis, M. Jacquot, L. Furfaro, R. Giust, P. A. Lacourt, and J. M. Dudley, “Arbitrary accelerating micron-scale caustic beams in two and three dimensions,” Opt. Express |

20. | I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting Accelerating Wave Packets of Maxwell’s Equations,” Phys. Rev. Lett. |

21. | F. Courvoisier, A. Mathis, L. Froehly, R. Giust, L. Furfaro, P. A. Lacourt, M. Jacquot, and J. M. Dudley, “Sending femtosecond pulses in circles: highly nonparaxial accelerating beams,” Opt. Lett. |

22. | I. Kaminer, R. Bekenstein, and M. Segev, CLEO: QELS-Fundamental Science, OSA Technical Digest (Optical Society of America, 2012), paper QM3E.3. The presentation included experimental results. |

23. | This concept of nonlinear accelerating beams of Maxwell's equations was first proposed in Nonlinear Photonics Topical Meeting, paper |

24. | O. Peleg, M. Segev, G. Bartal, D. N. Christodoulides, and N. Moiseyev, “Nonlinear Waves in Subwavelength Waveguide Arrays: Evanescent Bands and the “Phoenix Soliton”,” Phys. Rev. Lett. |

25. | M. A. Porras, A. Parola, D. Faccio, A. Dubietis, and P. Trapani, “Nonlinear Unbalanced Bessel Beams: Stationary Conical Waves Supported by Nonlinear Losses,” Phys. Rev. Lett. |

26. | G. I. Stegeman and M. Segev, “Optical Spatial Solitons and Their Interactions: Universality and Diversity,” Science |

27. | I. Kaminer, Y. Lumer, M. Segev, and D. N. Christodoulides, “Causality effects on accelerating light pulses,” Opt. Express |

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

29. | G. Fibich, “Small Beam Nonparaxiality Arrests Self-Focusing of Optical Beams,” Phys. Rev. Lett. |

30. | M. D. Feit and I. A. Fleck Jr., “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B |

31. | H. E. Hemandez-Figueroa, “Nonlinear nonparaxial beam-propagation method,” Electron. Lett. |

32. | N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Does the nonlinear Schrödinger equation correctly describe beam propagation?” Opt. Lett. |

33. | P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Propagation properties of non-paraxial spatial solitons,” J. Mod. Opt. |

34. | O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, “Collisions between Optical Spatial Solitons Propagating in Opposite Directions,” Phys. Rev. Lett. |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.3270) Nonlinear optics : Kerr effect

(260.2110) Physical optics : Electromagnetic optics

(350.7420) Other areas of optics : Waves

(190.6135) Nonlinear optics : Spatial solitons

(050.6624) Diffraction and gratings : Subwavelength structures

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: June 15, 2012

Revised Manuscript: July 16, 2012

Manuscript Accepted: July 16, 2012

Published: August 1, 2012

**Citation**

Ido Kaminer, Jonathan Nemirovsky, and Mordechai Segev, "Self-accelerating self-trapped nonlinear beams of Maxwell's equations," Opt. Express **20**, 18827-18835 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-17-18827

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### References

- G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett.32(8), 979–981 (2007). [CrossRef] [PubMed]
- G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett.99(21), 213901 (2007). [CrossRef] [PubMed]
- J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics2(11), 675–678 (2008). [CrossRef]
- P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved Plasma Channel Generation Using Ultraintense Airy Beams,” Science324(5924), 229–232 (2009). [CrossRef] [PubMed]
- I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-Accelerating Self-Trapped Optical Beams,” Phys. Rev. Lett.106(21), 213903 (2011). [CrossRef] [PubMed]
- A. Lotti, D. Faccio, A. Couairon, D. G. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, “Stationary nonlinear Airy beams,” Phys. Rev. A84(2), 021807 (2011). [CrossRef]
- R. Bekenstein and M. Segev, “Self-accelerating optical beams in highly nonlocal nonlinear media,” Opt. Express19(24), 23706–23715 (2011). [CrossRef] [PubMed]
- I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental Observation of Self-Accelerating Beams in Quadratic Nonlinear Media,” Phys. Rev. Lett.108(11), 113903 (2012). [CrossRef] [PubMed]
- Y. Hu, S. Huang, P. Zhang, C. Lou, J. Xu, and Z. Chen, “Persistence and breakdown of Airy beams driven by an initial nonlinearity,” Opt. Lett.35(23), 3952–3954 (2010). [CrossRef] [PubMed]
- R. Chen, C. Yin, X. Chu, and H. Wang, “Effect of Kerr nonlinearity on an Airy beam,” Phys. Rev. A82(4), 043832 (2010). [CrossRef]
- Y. Fattal, A. Rudnick, and D. M. Marom, “Soliton shedding from Airy pulses in Kerr media,” Opt. Express19(18), 17298–17307 (2011). [CrossRef] [PubMed]
- Y. Hu, Z. Sun, D. Bongiovanni, D. Song, C. Lou, J. Xu, Z. Chen, and R. Morandotti, “Reshaping the trajectory and spectrum of nonlinear Airy beams,” to appear in Opt. Lett. (2012).
- J. A. Giannini and R. I. Joseph, “The role of the second Painlevé transcendent in nonlinear optics,” Phys. Lett. A141(8-9), 417–419 (1989). [CrossRef]
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