Diffraction-free beams in fractional Schrödinger equation

We investigate the propagation of one-dimensional and two-dimensional (1D, 2D) Gaussian beams in the fractional Schrödinger equation (FSE) without a potential, analytically and numerically. Without chirp, a 1D Gaussian beam splits into two nondiffracting Gaussian beams during propagation, while a 2D Gaussian beam undergoes conical diffraction. When a Gaussian beam carries linear chirp, the 1D beam deflects along the trajectories z = ±2(x − x0), which are independent of the chirp. In the case of 2D Gaussian beam, the propagation is also deflected, but the trajectories align along the diffraction cone and the direction is determined by the chirp. Both 1D and 2D Gaussian beams are diffractionless and display uniform propagation. The nondiffracting property discovered in this model applies to other beams as well. Based on the nondiffracting and splitting properties, we introduce the Talbot effect of diffractionless beams in FSE.


Results
Diffraction free beams. The 1D FSE without potential has the form 18 where α is the Lévy index (1 < α ≤ 2). When α = 2, one recovers the usual SE in free space 20,26,27 . We consider the limiting case α = 1 4 . By taking Fourier transform of Eq. (1), one obtains with k being the spatial frequency. Eq. (2) demonstrates that in the inverse space a beam propagates in a symmetric linear potential. Recall that the potential in the inverse space is parabolic for the standard SE without potential 28 . This seemingly minor difference between the equations brings a crucial change in the behavior of beams. The solution of Eq. (1) is easily obtained from Eq. (2) and can be written as a convolution , the solution will also be symmetric, which can be written as ψ(± |x|, z). Here, we would like to note that iz/{π[4x 2 + (iz) 2 ]} is a complex Lorentzian function. The convolution between a Lorentzian and a Gaussian function is the Voigt function. Because the complex Lorentzian is singular at x = ± z/2 and has two peaks, the gap between the peaks will increase linearly with propagation distance. We should note that Eq. (3) can be expressed in terms of Fox's H functions 23,25 . However, such a treatment does not depict the physical picture clearly, because of the complexity of mathematical expressions. Here, we utilize an approximate analytical method to display propagation dynamics of chirped Gaussian beams. Also, an equivalent method that provides a simple physical interpretation is presented in the "Methods" section.
For simplicity, let us assume that the input is a chirped Gaussian beam in 0 2 with x 0 being the transverse displacement, C being the linear chirp, and σ controlling the beam width. The corresponding Fourier transform of the input is in 2 0 which is also a Gaussian beam. We consider first the case C = 0. As is evident from the above, one cannot obtain an analytical result from the convolution directly. Therefore, we plug Eq. (5) into Eq. (3) and after some mathematical steps, obtain an analytical but approximate solution The detailed derivation of Eq. (6) is provided in the section on "Methods". From this solution one can infer that the Gaussian will split into two diffraction-free Gaussian beams -because the beam width is not affected -along the trajectories z = ± 2(x − x 0 ).
The propagation of the Gaussian beam with C = 0 is shown in Fig. 1(a,b), in which Fig. 1(a) depicts numerical simulation and Fig. 1(b) displays the corresponding analytical result. Clearly, they agree with each other rather well, although the numerical solution is a bit wider than the analytical approximation. As expected, the initial Gaussian beam splits into two diffraction-free Gaussian beams, which can be also called the one-dimensional conical diffraction. This is starkly different from the usual behavior of Gaussian beams in free space. The physical reason for such a splitting can be inferred from Eq. (3), which is a convolution between a Gaussian and a function with two peaks. Since the trajectories are linear and the beam widths are unaffected during propagation, the motion of two Gaussian beams is uniform.
The solution for the case C ≠ 0 is also provided in the "Methods" section. For C of small absolute value, one still observes the 1D conical diffraction, but the intensity of the two branches is not equal. However, when C is of high-enough absolute value, the Gaussian beam depicted in Eq. (5) will be mainly in the k > 0 or k < 0 region, so that the corresponding solution in Eq. (1) can be approximately written as Scientific RepoRts | 6:23645 | DOI: 10.1038/srep23645 0 0 2 in which ± corresponds to  C 0. Therefore, the beam will not split during propagation, because of the high chirp. Other than this, from Eq. (7) one can see that the chirp does not affect the trajectory or the "velocity" of uniform motion, nor the transverse displacement of the beam at the output. Thus, the chirped Gaussian beam remains diffraction-free and without acceleration during propagation. The corresponding trajectories from numerical and analytical results are shown in Fig. 1(c,d). We must emphasize that the solutions in Eqs (6) and (7) are only approximate analytical solutions and not the exact solutions. Still, they agree with numerical results quite well.
Since the motion of the Gaussian beam in such a model is uniform, the acceleration of caustics is missing and the beam cannot self-heal when it encounters an obstacle (the corresponding numerical simulations not shown). This is different from an Airy beam 29,30 , a Fresnel diffraction pattern 31 , and other nonparaxial accelerating beams 32 . It is worth mentioning that a beam always acquires a symmetric linear phase during propagation in the inverse space, according to Eq. (2), which will not affect the beam profile except for a transverse displacement in the real space. Thus, the nondiffracting property is feasible for all kinds of beams, including Airy beams, Bessel-Gaussian beams, and Hermite-Laguerre-Gaussian beams, to name a few. One should bear in mind that the quadratic phase obtained in the inverse space would change the beam profile in the real space; this is why the Gaussian beam broadens in free space (corresponding to Eq. (1) for α = 2, which is the usual SE). Now, it is clear that the beam splits and is diffraction-free for α = 1, while the beam diffracts but does not split during propagation for α = 2. Therefore, for the cases in-between, i.e., 1 < α < 2, the beam will both split and diffract.
Talbot effect. Considering the time-reversal symmetry of the system, the initial Gaussian beam in Fig. 1(a) can be viewed as a collision of two diffraction-free Gaussians. From this point of view, one can consider an input composed of a superposition of equally separated Gaussian beams without chirp, with c n being the amplitude of each component. The propagation of such a superposition is described by Let us assume that the coefficients c n are constant and independent of n. If z = 2x 0 , Eq. (9) can be written as   which is same as the input if we substitute l by n, except for a transverse displacement. In Fig. 2(a,b), the intensity and the corresponding phase of the input beam during propagation are shown numerically, from which one can conclude that the Talbot effect in the form of a Talbot net has formed -the beam recovers at the Talbot length and there is a π phase shift of the self-images at the half-Talbot length. One should note that this Talbot effect is not the result of the diffraction of beams -which is absent here -but that it comes from the transverse periodicity of colliding beams 33,34 . The generation of such a Talbot effect can be understood as follows: Each transverse component splits into two diffraction-free Gaussian beams, which collide with other diffraction-free Gaussian beams that split from other components, to form the self-imaging at the half-Talbot length. The self-imaging at half-Talbot length will then repeat itself for another half-period, to form self-images at the Talbot length and so the full cycle is closed; it repeats, to form the full Talbot net. Thus, the recurrence here is the consequence of peculiar input superposition and its propagation, not of the near-field diffraction.
When we choose the coefficients as =   c i i i [ , , 1, , 1, , ] n , the numerical results are depicted in Fig. 2(c,d). One sees that the Talbot effect is still there, but the Talbot length is doubled and at the half-Talbot length the beam recovers itself, which represents the fractional Talbot effect 34 . According to the analysis in previous literature 34 , the fractional Talbot effect for such a special choice of coefficients can be understood easily.
Two-dimensional case. The 2D FSE without a potential can be written as For α = 1, the 2D FSE in the inverse space is x y x y x y 2 2 The input 2D chirped Gaussian beam can be written as x y 2 where C x and C y are the chirp coefficients along the x and y directions, and = + r x y 2 2 . If C x = C y = 0, the Gaussian beam is not chirped, so it will undergo conical diffraction, which is equivalent to the continuous split around the full circle. The radius of the cone is r = z/2. Such a propagation is displayed in Fig. 3(a), in which the isosurface plot depicts the panoramic view of propagation. It is indeed a conical diffraction, after an adjustment over a short propagation distance. The intensity is normalized, to show the propagation more clearly. The cross sections of intensity distributions along x and y axes are also displayed in the horizontal x = − 50 and the vertical y = − 50 plane.
As shown in Fig. 3(b), the chirped Gaussian beams execute uniform motion, and the direction of the "velocity" depends on the sign of the chirp. One can see that no matter how the chirp coefficients change, the output Gaussian beam will always lie on the dashed circle -the location of the conical diffraction, as shown in Fig. 3(b). This fact confirms that the trajectory of the beam can only be along the diffraction cone. It can also be verified that the trajectory is given by from which one finds that the chirp can only affect the location of the output beam on the dashed circle. For eventual experimental observation of these theoretical and numerical findings, we propose a system composed of two convex lenses and a phase mask 18 , as shown in Fig. 4. The first lens transforms the input beam into the inverse space 35 , then a phase mask produces the phase change at certain propagation distance z, as required by Eq. (2), and at last the second lens transforms the output beam back into the real space. Thus, the treatment is executed in the inverse space and the management of the fractional Laplacian in Eq. (1) in the direct space is avoided.

Discussion
In summary, we have introduced the diffraction-free beams in FSE without potential, taking chirped Gaussian beams as an example. The method is applicable to other beams as well. Without chirp, a 1D Gaussian beam splits into two diffractionless Gaussian beams whose motion is uniform. If the input is a superposition of equidistant 1D Gaussian beams, the fractional Talbot effect can be realized. We also find that conical diffraction is obtained for a 2D Gaussian beam without chirp, while for the chirped Gaussian beams, the motion of the beams is still uniform, but the propagation direction is determined by the chirp coefficients. Regardless of how the chirp coefficients change, the transverse displacement is not affected, but the position of beams on the diffraction cone is. In the end, an experimental optical implementation for such beam dynamics is proposed. This research may not only deepen the understanding of FSE and nondiffracting beams, but also help better control beams that show promise in various potential applications, such as producing beam splitters, beam combiners, and other.

Methods
Derivation of Eq. (6) for the input without chirp. We plug Eq. (5) with C = 0 into Eq. (3), and try to solve for an approximate but accurate solution, as follows:    The approximately-equal sign in Eq. (16) is there because we presume that x 0 − x ± z/2 is positive for  k 0 and negative for  k 0. We should mention that in Eq. (16), the expression in the second row is obtained by reversing the sign of k in the expression in the first row. Then we sum the first two rows and obtain the expression in the third row, based on the above assumption.
Solution for the input with C ≠ 0. For this case, we rewrite k + C = κ, so according to Eq. (16) one obtains    in Eq. (18) goes to 0, which leads to Eq. (7). Similar conclusions can be also obtained based on Eqs (19) and (20).

Factorization of the fractional Schrödinger equation in one-dimension.
We note that the analysis just described can be performed in a less rigorous manner by formally expressing the transverse fractional Laplacian in Eq. (1) for α = 1 as a first-order derivative,