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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 17 — Aug. 13, 2012
  • pp: 18827–18835
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Self-accelerating self-trapped nonlinear beams of Maxwell's equations

Ido Kaminer, Jonathan Nemirovsky, and Mordechai Segev  »View Author Affiliations


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

The research on accelerating beams has been growing rapidly since it was introduced into the domain of optics, by Demetri Christodoulides, in 2007 [1

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

,2

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]

]. The best known accelerating beam is the paraxial Airy beam, which propagates along a parabolic trajectory while preserving its amplitude structure indefinitely. The effect is caused by interference: the waves emitted from all points on the Airy profile maintain a propagation-invariant structure, which shifts laterally along a parabola. This beautiful phenomenon has led to many intriguing ideas such as guiding particles along a curve [3

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

], light-induced curved plasma channels [4

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]

], and to recent studies on shape-preserving accelerating beams in nonlinear optics [5

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

12

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).

]. The idea of nonlinear accelerating beams goes back to Giannini and Joseph [13

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]

] who found temporal self-accelerating solitons in Kerr media. However, that pioneering idea has been largely forgotten. Instead, scientists attempted to launch Airy beams into nonlinear media, and have found, in experiments and in simulations, that the beams do not propagate as an accelerating shape-preserving entity [9

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

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]

]. Except for cases where the nonlinear effects on the beam were weak [4

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]

], the beams deform and even breakup [9

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

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]

]. Thus, the prevailing opinion around 2010 considered strong nonlinearity as an obstacle for having accelerating beams. In spite of this general opinion, we have shown [5

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

] that a specific design of the beam profile can make it accelerate in a shape-preserving manner while propagating in the nonlinear Kerr [5

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

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]

] nonlocal [7

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

], and quadratic [5

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

,8

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]

] media. Moreover, a recent study [12

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).

] has shown, by analyzing the spectrum, how an initial linear Airy beam launched into the nonlinear medium is evolving into our nonlinear solution [5

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

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]

]. In fact [12

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).

], proves that our nonlinear accelerating solution is an attractor for the nonlinear dynamics. All of these studies, however, were in the paraxial domain.

Here we present the first nonlinear self-trapped accelerating beams of the full time harmonic Maxwell equations [23

23. This concept of nonlinear accelerating beams of Maxwell's equations was first proposed in Nonlinear Photonics Topical Meeting, paper NTu3C.7, submitted 27/02/2012.

]. These beams accelerate along a circular trajectory in a shape-preserving manner. We show that such nonlinear non-paraxial accelerating beams involve coupling between forward- and backward-propagating beams, giving rise to unique effects: Specifically, part of the beam is reflected due to the nonlinear change of the permittivity. However, the reflections are exactly compensated by the backward moving part of the wavepacket. We study the behavior of the beam in the absence of the backward part, showing that in the presence of a strong nonlinearity the trajectory of the shape-preserving beam is altered from being exactly circular acceleration. Moreover, at a very high nonlinearity, such nonlinear coupling between the forward and the backscattered waves leads to regime of instability, where unexpected highly nonlinear, possibly chaotic, dynamics arises. In addition, accelerating shape-preserving propagation occurs even when the beam is constructed of subwavelength features, such that a significant part of its power resides in the evanescent regime. In this context, this work proves that the nonlinearity takes a unique role in the propagation, preventing the evanescent spectrum from decaying.

Derivation of the solution

We begin from Maxwell's equations with nonlinearity of the form n2(E) = n02(1 + ε(|E|)) for a general saturable or Kerr-like nonlinearity that depends only on the intensity I = |E|2. In general, the nonlinear permittivity 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]

] that also depends on the electric field 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 = EY(x,z,t)ŷ, Maxwell's equation yield the nonlinear Helmholtz equation:
Exx+Ezz+k02E+k02ε(I)E=0.
(1)
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 r,θ by taking z = r sin(θ), x = r cos(θ), and seek shape-preserving solutions of the form E = U(r)eiαθ. Here, 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:
d2dr2U+1rddrUα2r2U+k02U+k02ε(U)U=0.
(2)
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]

], giving the nonlinear Bessel beam with zero angular momentum. In contrast to that, we are interested in high values of α (large beam bending). To find accelerating (α>0) solutions of Eq. (2), we notice that for some sufficiently small r = r0 value (typically α/k0), 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: U(r0) = Jα(r0k0), U'(r0) = J'α(r0k0), 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

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

]). Since we work at non-paraxial angles, diffraction effects are very strong; hence in order to display considerable nonlinear effects, the nonlinearities should be strong. This logic of having nonlinear and diffractive effects comparable is well known from solitons [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]

], where for a simple Kerr soliton the width is inversely proportional to its height. Consider for example the existence curve of the Kerr solitons, where the family of solutions forms a hyperbola in the height-width plane. This means that there is a tradeoff between high non-paraxial angles of bending and high nonlinear effects.

This last issue is very important, namely, making sure there are physical parameters that support a nonlinear Maxwell accelerating beam. This is not so simple because, as we explained above, taking higher nonparaxial bending makes the nonlinear effects smaller in comparison, and vice versa. Hence, higher beam bending inevitably involves larger nonlinearities. To explain the tradeoff, consider an accelerating beam of high acceleration. As known from the paraxial [1

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

,2

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]

] and the non-paraxial [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]

] regimes, this requires narrow lobes, which therefore experience faster diffraction broadening. This implies that in order to observe significant nonlinear effects – the nonlinear index change must be very strong. On the other hand, for a beam with wider lobes diffraction effects are weaker, hence weaker nonlinearities will affect the propagation dynamics. To give a qualitative explanation, we first find a relation between the width of the lobes to the bending of the beam, for the linear case: compare the acceleration of the linear non-paraxial beam, 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]

]), to the acceleration of an Airy beam, (2d3k2)−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 = 16π3(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]

], for a nonlinear index change of ~10−4, 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 × 106. 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]

]), 0.5α/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 n2I = 2(dk)−2. Hence we can write α directly using the refractive index change: α = 2(5/2)(n2I)(−3/2). A strong nonlinearity of n2I~10−2, 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.

Decoupling the forward and backward waves

The coupling of the forward and backward waves is crucial for finding the correct physical solution. Had the system been linear, we could have used the projection in Fourier plane (x→kx) which takes the form of i2(k02kx2)12z+12to exactly eliminates the backward propagating waves and gets the solution which satisfies the physical boundary conditions (i.e. beam is launched at 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 z0, we substitute the field into the nonlinear term to define a linear operator M = ∂x2 + k02 + k02ε(I); Eq. (1) is now Ezz = -ME. Using the eigenvalues λi and the eigenfunctions ui = ui(x) of M, one can express any solution (close enough to the plane z = z0) as a sum of the forward and backward propagating parts (first and second sums respectively)
E(x,zz0)=ici+ui(x)eiλiz+iciui(x)eiλiz.
(3)
Note that all the terms of the sums are not evanescent, since 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)2 = M). One can prove that this yields a projection operator of the form i2(M)1z+12 that eliminates the backward propagating waves and leaves just the exact forward beam. Note that this projection must be evaluated for each plane z = z0 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.

Unfortunately, however, splitting the solution of Eq. (2) only at 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

Before proceeding to the full numerical solution, we suggest a simplified scenario. Assume that two beams are launched at opposite azimuthal directions. The first is launched counter-clockwise from θ = 0 and the second is launched clockwise from some other angle θ1<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 θ = 0. Second, the clockwise propagating part of the beam is chosen as the boundary condition at θ1 (due to azimuthal symmetry we can use the values of the backward part of Fig. 2(d) on any 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]

], which studied the coupled nonlinear interaction between solitons propagating in opposite direction. Instead, we prefer to make use here of another approach, reverse simulation, which numerically simulates the propagation arising from a single beam launched at z = 0, with no pre-assumptions.

The reverse simulation

At an infinite distance z from the launch plane z = 0, we expect the beam to have only a forward propagating part, since no backward propagating beam would be back reflected from infinity, where the nonlinear index change is negligible so no reflections would occur. This means that at some large enough z1 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 z = z1 and use the nonlinear projection of it onto the forward propagating wavepacket (Fig. 2(c)). Second, we propagate this beam in reverse until 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).

Our method of nonlinear reverse propagation naturally assumes that the propagation dynamics of the nonlinear Helmholtz equation is reversible in 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?

Figure 1(b) displays the Fourier transform of the beams of Fig. 1(a). The dashed line indicates the edge of the evanescent regime, highlighting that a considerable part of the spatial spectrum of the nonlinear beam is evanescent when the nonlinearity is absent. As such, this accelerating nonlinear beam heavily relies on the nonlinearity to propagate without decaying, otherwise an essential part of its spectrum would decay and the accelerating propagation would stop. This finding is manifested by the thin sub-wavelength lobes shown in Fig. 1(a), where even the main lobe is about half a wavelength wide. As a consequence, when the beam completes near 90° bending, it breaks up, causing the nonlinear effect to drop quickly, causing most of the beam to decay. For stronger nonlinearities, we expect an even more abrupt decay, causing a larger portion of the power to backscatter. This breaks a fundamental rule of linear optics: linear propagation prohibits the backward radiation to be created from an incident beam in a homogenous medium; the essential reason is simple: forward propagating waves cannot change to become backward propagating waves, so nothing that was launched forward can turn backwards. Indeed, linear wave dynamics do not couple the forward and backward waves. And yet, this rule breaks down in the presence of nonlinear effects, because nonlinearity couples all waves, in particular – in this case it couples the forward and the backward propagating waves. Nonlinear optics allows the nonlinear beam to reflect itself (or part of it) – which is expressed in an effective bend of more than 90°. This beautiful phenomenon is at its best in the nonlinear dynamics of our accelerating beams, since the back-reflection changes from zero - along the propagation, to its maximum - right after the nonlinearity drops. This dynamics suggests the exciting possibility of making a beam that would somehow fully reflect on itself. Such a notion is known in nonlinear optics in the context of phase conjugation; however, here the beams are accelerating (bending), hence the trajectory of such a beam would more resemble a boomerang. Such an envisioned beam will have a 'U' trajectory: it will start propagating along one branch of the 'U' and then bend to propagate back along the other branch of the 'U'. It seems that such a beam is indeed in principle physically possible. However, very high nonlinearities are required to cause enough power to backscatter so that an actual “boomerang soliton” would form. Further study of this phenomenon is left for future research.

Conclusions

We presented shape-preserving nonlinear self-accelerating beams of Maxwell's equations, and analyzed their physical features.

Acknowledgments

This work was funded by an Advanced Grant from the European Research Council, by the Israel Science Foundation, and by the Binational US-Israel Science Foundation.

References and links

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]

6.

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7.

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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]

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E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106(21), 213902 (2011). [CrossRef] [PubMed]

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 19(17), 16455–16465 (2011). [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]

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. 37(10), 1736–1738 (2012). [CrossRef] [PubMed]

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 NTu3C.7, submitted 27/02/2012.

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]

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]

26.

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

29.

G. Fibich, “Small Beam Nonparaxiality Arrests Self-Focusing of Optical Beams,” Phys. Rev. Lett. 76(23), 4356–4359 (1996). [CrossRef] [PubMed]

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 5(3), 633–640 (1988). [CrossRef]

31.

H. E. Hemandez-Figueroa, “Nonlinear nonparaxial beam-propagation method,” Electron. Lett. 30(4), 352–353 (1994). [CrossRef]

32.

N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Does the nonlinear Schrödinger equation correctly describe beam propagation?” Opt. Lett. 18(6), 411–413 (1993). [CrossRef] [PubMed]

33.

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Propagation properties of non-paraxial spatial solitons,” J. Mod. Opt. 47, 1877–1886 (2000).

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]

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

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  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. A84(2), 021807 (2011). [CrossRef]
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  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. A141(8-9), 417–419 (1989). [CrossRef]
  14. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47(3), 264–267 (1979). [CrossRef]
  15. A. V. Novitsky and D. V. Novitsky, “Nonparaxial Airy beams: role of evanescent waves,” Opt. Lett.34(21), 3430–3432 (2009). [CrossRef] [PubMed]
  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. Express17(25), 22432–22441 (2009). [CrossRef] [PubMed]
  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.107(11), 116802 (2011). [CrossRef] [PubMed]
  18. E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett.106(21), 213902 (2011). [CrossRef] [PubMed]
  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. Express19(17), 16455–16465 (2011). [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]
  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.37(10), 1736–1738 (2012). [CrossRef] [PubMed]
  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 NTu3C.7, submitted 27/02/2012.
  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]
  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]
  26. G. I. Stegeman and M. Segev, “Optical Spatial Solitons and Their Interactions: Universality and Diversity,” Science286(5444), 1518–1523 (1999). [CrossRef] [PubMed]
  27. I. Kaminer, Y. Lumer, M. Segev, and D. N. Christodoulides, “Causality effects on accelerating light pulses,” Opt. Express19(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]
  29. G. Fibich, “Small Beam Nonparaxiality Arrests Self-Focusing of Optical Beams,” Phys. Rev. Lett.76(23), 4356–4359 (1996). [CrossRef] [PubMed]
  30. M. D. Feit and I. A. Fleck., “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B5(3), 633–640 (1988). [CrossRef]
  31. H. E. Hemandez-Figueroa, “Nonlinear nonparaxial beam-propagation method,” Electron. Lett.30(4), 352–353 (1994). [CrossRef]
  32. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Does the nonlinear Schrödinger equation correctly describe beam propagation?” Opt. Lett.18(6), 411–413 (1993). [CrossRef] [PubMed]
  33. P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, “Propagation properties of non-paraxial spatial solitons,” J. Mod. Opt.47, 1877–1886 (2000).
  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]

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