## Abstract

Linear dynamics of an accelerating wave packet, which is produced by adding shifted copies of the fundamental Airy beam, due to parabolic optical potentials are investigated. A new type of self-imaging phenomenon, referred to as period-reversal accelerating self-imaging, is demonstrated theoretically and numerically. Unlike ordinary Talbot effects, where optical field pattern reappears at constant intervals and follows a straight line, here, the field pattern of this new self-imaging effect propagating along a periodic oscillating trajectory, can self-reproduces itself at nonconstant intervals, and begins to invert after the phase transition points, where the superposition of fundamental Airy beams forms multi-beams interference fringes. A completely spatially reversal replica of the initial field distribution is observed at odd multiplies of the period halves. Moreover, the properties of the multi-beams interference fringes are discussed in detail and can be used for the measurement of the system parameter. The above results can be generalized in the case of two transverse dimensions, where it can be treated as a product of two independent one-dimensional cases. The theoretical calculations and numerical simulations verify each other completely.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Self-accelerating Airy beams, a class of non-diffracting optical beams, propagate along a parabolic self-accelerating trajectory, while maintaining a shape-preserving Airy profile. Another fundamental property of Airy beams is their ability to self-heal during propagation [1]. Such optical phenomena have been the focus of considerable attention during the last decade, since Airy beams were firstly predicted theoretically and observed experimentally by Siviloglou *et. al.* in 2007 [2,3]. Although, the field structure of a truncated Airy beam with finite intensity presents transverse self-accelerating behavior, the “center of mass” always travels instead in a straight line, so as to ensure the transverse momentum preservation. Actually, Airy beam has been proven to be the eigenmode in the accelerated frame, and the only nondispersive solution of the one-dimensional paraxial wave equation in free space [4]. The path manipulation of self-accelerating Airy beams is of significant importance in practice. It is shown that the field distribution and propagation trajectory of self-accelerating Airy beams can be controlled effectively, by employing various refractive index potentials, which can be physically realized by tailoring the transverse refractive index distribution of the medium. In transversely linear index potentials with a gradient that dynamically changes along the propagation direction, the exact solutions of Airy beams with infinite and finite intensity can be found. Besides the Airy beam remaining nondiffracting, it can propagate along any predefined trajectory by engineering the index gradient [5,6]. The presence of a parabolic index potential also modifies the dynamical behavior of Airy beams, which was proved to exhibit periodic oscillation and inversion of its profile, the self-Fourier transformation is also shown at the phase transition points [7,8]. Speaking in general, in the last decade, considerable research efforts have been devoted to self-accelerating beams, from one-dimensional pattern to two and three dimensions, and even the multiplexing of arbitrary structures, from theoretical predictions to experimental observations, from linear to nonlinear regimes, from fundamental aspects to demonstration of possible applications. For more details, interested readers can see the reviews [1,9–11] and the references therein.

Talbot effect, also referred to as self-imaging, is concerned with periodic wave in near-field diffraction. Replicas of the input intensity profile reappear after a specific propagation distance called Talbot length. Potential applications of Talbot effects in a variety of research fields made them considerably attraction in the last years [12]. At present, Talbot effect has been demonstrated in numerous optical systems, and several corresponding conceptual extensions of Talbot effect have been proposed, such as quantum Talbot effect [13], nonlinear Talbot effect [14], plasmon Talbot effect [15], angular Talbot effect [16], discrete Talbot effect [17], Talbot-Lau effect [18], and electromagnetically induced Talbot effect [19]. As regards Talbot effect based on accelerating Airy beams most significant is work of Lumer *et. al.* [20], who demonstrated the Airy-Talbot effect, i.e., the self-imaging of accelerating optical beams, by propagating a sum of Airy beams with successively changing transverse displacements. The important feature that distinguishes the traditional Talbot effect, where the input wave needs be periodic, instead is an asymmetric self-accelerating initial profile, as a result, the field is self-imaged along a parabolic trajectory [20,21]. Furthermore, the intensity distribution and trajectory manipulation of Airy-Talbot effect have also been demonstrated by selecting different input field and employing dynamic linear index potentials [22]. However, linear dynamics of self-accelerating wave packet constructed from superposition of fundamental Airy beams in an external parabolic potential have not been reported. As mentioned above, if one launches an Airy beam into such a potential, it exhibits periodic oscillation. Hence the question arises how both effects interplay.

In this paper, we present a detailed investigation on the evolution of an accelerating wave packet, which is designed by adding shifted copies of the fundamental Airy beam with a constant transverse interval in parabolic optical potentials. It is shown that, with the periodic inversion of its profile and periodic self-Fourier transformation during propagation, a period-reversal accelerating self-imaging in the self-accelerating regime is demonstrated theoretically and numerically. One should note that this self-imaging effect is not the result of the diffraction of beams, but it results from the interference of self-accelerating Airy beams with successively changing transverse displacement. In fact, the diffraction of single Airy beam is absent in a parabolic potential, and it exhibits periodic inversion and phase transition [7]. Unlike the ordinary Talbot effects, the optical field pattern of this new self-imaging effect self-reproduces itself at nonconstant intervals, and begins to invert after the phase transition points, where the superposition of fundamental Airy beams forms multi-beams interference fringes. A completely spatially reversal replica of the initial field distribution is formed at odd multiplies of the period halves. It is shown that the nonconstant intervals and the characteristics of interference fringes depend considerably on the depth of the potential and initial optical field parameters. These properties can be used to measure the system parameter. Furthermore, the generalization to two dimensions is discussed briefly. The two-dimensional self-imaging effect can be treated as a product of two independent one dimensional cases. Our results provide significant extensions of the ordinary Airy-Talbot effect in free space.

## 2. Theoretical model and numerical results

Let us consider an optical beam that propagates along the *z* axis in a medium with parabolic optical potentials and is permitted to diffract only along the *x* direction. The optical field of the incident beam can be expressed in terms of slowly varying envelope *ψ* (*x*, *z*), which can be described by the following Schrödinger-like equation,

*α*is the depth of the potential. Transverse and longitudinal coordinates have been normalized to the characteristic beam’s width and the diffraction length, respectively. It is well known that the evolution of the wave function

*ψ*(

*x*,

*z*) in a parabolic potential is fully determined for any input

*ψ*(

*x*, 0) by means of the following integral [23]

*ψ*(

*x*, 0)exp(

*ibx*

^{2}) with

*K*being the spatial frequency. By quoting the convolution theorem of Fourier transform, the analytical solution of Eq. (1) can be expressed as the convolution of the Fourier transforms of

*ψ*(

*x*, 0) and exp(

*ibx*

^{2}).

In order to find the wave packets that self-image, we set the initial field distribution whose spatial spectrum is constructed from a linear superposition of fundamental Airy beams, shifted at constant intervals *δ* in the following form:

*c*

_{n}represents the amplitude of the

*n*th component,

*a*<<1 is truncation constant for ensuring containment of infinite Airy trail. Of interest is the Fourier spectrum of the input wave, which in the normalized

*k*-space can be written as

*b*≠0 can be obtained after some cumbersome calculations and reads

*A*(

*z*) = 1/cos(

*αz*) is the modulation function of intensity amplitude. Accordingly, the beam propagation path is traced via the above equation

*T*=2π/

*α*. It is obvious that at

*z*=

*mT*with

*m*being an arbitrary integer, the optical field pattern would self-reproduce itself, as shown in Fig. 1(a). However, its profile is shown to exhibit periodic inversion at the propagation distance of

*z*=(2

*m*+1)

*T*/2, where we have

*ψ*(

*x*, 0)=

*ψ*(-

*x*,

*z*). At

*z*=(2

*m*+1)

*T*/4, i.e.,

*b*=0, Eq. (5) is invalid. In this case, the solution of Eq. (1) can be directly obtained from Eq. (2) and read as

*z*=(2

*m*+1)

*T*/4.

In the limit *a* = 0, corresponding to the infinite intensity distribution case, one obtains from Eq. (6)

*l*being an arbitrary integer, the phase difference between different components is 0, and the wave packet preserves its initial intensity distribution, shifted along the periodic oscillating trajectory given by Eq. (7), as shown in Figs. 1(b) and 1(c), which correspond to the numerical and analytical results, respectively. A comparison of numerical results with analytical predictions shows good agreement each other. Figures 1(d)–1(j) show the intensity profiles at some typical positon in the propagation direction. An important feature that distinguishes our solutions from the others reported in the literature [20–22], is the appearance of spatially periodic inversion of the self-imaging field at the propagation distance of

*z*=(2

*m*+1)

*T*/2, as shown in Figs. 2(a) and 2(b). Such periodic self-imaging effect based on Airy beams in parabolic optical potentials is referred to as

*period-reversal accelerating self-imaging*. Unlike the accelerating self-imaging in free space and dynamic linear potentials with a gradient, where the Airy-Talbot effect follows a parabolic trajectory or a predefined trajectory by engineering the index gradient, here the field pattern of this new self-imaging effect propagates along a periodic oscillating trajectory, and begins to invert after the phase transition points, where the superposition of fundamental Airy beams forms multi-beams interference fringes. Furthermore, different from other Talbot effects, the second important feature of this period-reversal self-imaging effect is that optical field pattern self-reproduces itself at nonconstant intervals, determined by Eq. (10). Another important feature is the

*z*modulation of intensity amplitude

*A*(

*z*) at these certain self-imaging recurrent planes. Recalled that infinite intensity distribution is not physically realistic. As done earlier [2,3], an apodization is needed. Based on Eq. (6), the evolution of wave packet still exhibits all above self-imaging features, in spite of its truncation (necessary for its realization).

An example of the period-reversal accelerating self-imaging effect obtained from a superposition of 11 Airy beams is shown in Figs. 1 and 2 for the parameters *α*=0.1, *a*=0.05, *δ*=2 *c*_{n}={···, 1, 1, 1, 1, 1, ···} and *n*ɛ [−5, 5]. Figure 1(a) clearly shows that the wave packet undergoes an oscillatory motion with period *T*=20π. Figure 2(a) shows the numerically calculated revival positons of self-imaging effect. From it, one can see that the self-imaging interval is suppressed with *z* approaching to the phase transition point. Recalled that accelerating self-imaging effect only persists in the accelerating range [20]. The acceleration of such finite Airy-like wave will stop when *z*→(2m+1)*T*/4. As a result, period-reversal accelerating self-imaging is fundamentally limited to a finite number of times, depending on the truncation constant *a*. The propagation of the finite energy input in the first quarter period is displayed in Figs. 1(b) and 1(c). The ideal trajectory for *n*=0 is denoted with white solid curve in Figs. 1(b) and 1(c). An inspection of Figs. 1(b) and 1(c) shows that the onset of self-imaging is clearly observed, where the optical wave field undergoes revivals during the accelerating propagation, which occur at the positions driven by Eq. (10). The three red dashed lines in Fig. 1(b) at *z *= 5.61, 8.99 and 10.83 represent first, second and third self-imaging recurrent positions shown in Fig. 2(c), which have been computed using Eq. (10). The corresponding intensity profiles are shown in Figs. 1(f), 1(h) and 1(j). One can see that the initial intensity profile can repeat itself along the propagation direction except for the *z* modulation of intensity amplitude, and the transverse displacement resulting from the accelerating trajectory. Figure 2(b) also shows that, after the phase transition point *z*=5π, the optical field pattern begins to invert, and a completely spatially reversal replica of the initial field distribution is formed at *z*=10π, as shown in Fig. 2(b4).

To further characterize this period-reversal self-imaging effect, cross-correlation coefficient *C* is introduced to estimate the degree of recurrence or similarity between the input intensity *I*(*x*, 0) and evolution intensity *I*(*x*, *z*), and reads [24]

*I*(

*x*, 0)=

*I*(

*x*,

*z*). Figure 2(d) shows the dependence of the cross-correlation coefficient

*C*with the propagation distance in the accelerating coordinates before the first phase transition points

*z*<

*T*/4. As is expected,

*C*exhibits periodic evolution behavior. Maximum of cross-correlation coefficient

*C*is reached at those positions marked by red crosses, where the self-imaging effect occurs. Close to phase transition point, the dependence of the cross-correlation coefficient

*C*displays an unorderly changing trend due to its intensity apodization and non-accelerating property. Moreover, another place of interest is the minimum of cross-correlation coefficient

*C*, which is denoted by green crosses, which correspond to green dash lines in Figs. 1(b) and 1(c). These locations can be obtained by setting

*δn*/(4

*b*)=(2

*l*+1)π in Eq. (6), where there is a π phase shift between two adjacent beam components, and the intensity profile is distorted and phase-shifted relative to the input. This phenomenon is called dual Airy

*–*Talbot effect [21]. In this case, the recurrent locations of dual Airy

*–*Talbot effect depend on the input parameters according to

*z*=(1/

*α*)arctan[2(2

*l*+1)

*απ*/(

*δn*)] shown in Fig. 2(c). From this relation, the first, second and third self-imaging recurrent planes are calculated at

*z*=3.04 7.56 and 10.04, the corresponding intensity profiles are shown in Figs. 1(e), 1(g), and 1(i), respectively.

As can be seen from Fig. 1(a), the linear superposition of fundamental accelerating beams forms the interference fringes around the phase transition points. The transverse intensity distribution at *z*=(2*m*+1)*T*/4 can be obtained from Eq. (8):

*β*=

*δnαx*/2,

*N*is the number of fundamental accelerating beams. The location of multiple-beam interference main maxima is determined by setting

*β*=

*M*π with

*M*standing for the order number of interference fringes. This equation means that the constructive interference between the beam components occurs and the resulting intensity is increased, as shown in Fig. 2(e). The intensity of the interference peaks is proportional to

*N*

^{2}, and modulated by the Gaussian envelope related to truncation constant

*a*. Moreover, between two adjacent interference peaks, there are

*N*-1 minimum with the locations given by

*Nβ*=

*M*π, where

*M*cannot be an integral multiple of

*N*. Then, there also are

*N*-2 secondary maxima. Obviously, the locations where the interference is constructive become more sharply with increasing the number of fundamental accelerating beams. Based on this dependence of interference fringes on the depth of the potential and initial optical field parameters, the system parameter

*α*can be easily obtained by measuring the spacing between two adjacent interference peaks.

Now, we choose different amplitude coefficient *c*_{n} to further investigate this period-reversal accelerating self-imaging and multi-beams interference fringes. Figure 3 summarizes our results with different amplitude coefficients. The top row [Figs. 3(a) and 3(c)] contains our results with *c*_{n}= {···, 1, *i*, 1, *i*, 1···}, while the corresponding numerical simulations with *c*_{n}= {···, 1, 0, 1, 0, 1, ···} are shown in the second row [Figs. 3(b) and 3(d)]. One can see that the optical wave follows the same trajectory described by Eq. (7) as the above case. In both cases, there are smaller self-imaging interval in comparison to that shown in Fig. 1. In fact, new self-imaging recurrent locations in both cases can be readily given by the replacement of *n* by 2*n* in Eq. (10), as shown in Fig. 2(c). For the case of *c*_{n}= {···, 1, *i*, 1, *i*, 1···}, although, there are complete same intensity distribution at input plane as that at the first self-imaging recurrent plane (*z*=3.04), the amplitude at *z*=3.04 is the conjugate of that at *z*=0 [21]. Figures 3(c) and 3(d) show that the spatial period of the interference fringe at the phase transition points is halved in comparison to that shown in Fig. 2(e), which can be easily explained by setting *β*=*δnαx*. Furthermore, compared with that in the case of *c*_{n}= {···, 1, *i*, 1, *i*, 1, ···}, the intensity amplitude is also halved, resulting from the π phase shift between adjacent components. For the input beam with *c*_{n}= {···, 1, 0, 1, 0, 1, ···}, the energy is approximately a quarter of that in Fig. 2(e), owing to the number reduced of fundamental accelerating components.

Finally, we proceed to a theoretical extension to two transverse dimensions, the extended normalized paraxial equation is of the following form

## 3. Conclusion

In conclusion, we have focused on the evolution dynamics of an accelerating wave packet, constructed from a superposition of fundamental Airy beams in parabolic optical potentials. With the periodic inversion of its profile and periodic self-Fourier transformation during propagation, a period-reversal accelerating self-imaging is demonstrated theoretically and numerically. Numerical simulations agree with the theoretical results very well. Unlike other Talbot effects, the optical field pattern of this new self-imaging effect self-reproduces itself at nonconstant intervals, and begins to invert after the phase transition points. A completely spatially reversal replica of the initial field distribution is observed at odd multiplies of the period halves. The linear superposition of fundamental accelerating beams forms the interference fringes around the phase transition points and the characteristics of interference fringes depend considerably on the depth of the potential and initial optical field parameters. Furthermore, the generalization to two dimensions is discussed briefly. It is shown that the two-dimensional case is equivalent to a product of two independent one dimensional cases. Our results not only provide significant extensions of the famous Airy-Talbot effect in free space, but also pave a promising way to find some potential applications in optical testing and florescence microscopy.

## Funding

National Natural Science Foundation of China (61605251); Fundamental Research Funds for the Central Universities (18CX05005A, 19CX02047A, 19CX05003A-10); Natural Science Foundation of Shandong Province (ZR2018MA027).

## Disclosures

The authors declare no conflicts of interest.

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