Light-induced interconnects using nonlinear Airy beam interactions

We analyze numerically optical waveguiding structures created in photorefractive media by two incoherent counter-propagating 1D Airy beams under nonlinear self-focusing conditions. We then inject a Gaussian probe beam to test our waveguiding structure. By using an anti-symmetric Airy beam configuration in stationary conditions, we find rich and complex waveguiding structures with multiple input to multiple output configurations and transverse input-to-output shifts up to 13 times the guided beam’s waist.


Introduction
In 1979, in the context of quantum mechanics, Berry and Balazs predicted a new solution to the Schrödinger equation [1], the Airy wave packets. In 2007, by truncating the ideal Airy waveform the first optical Airy beam was observed in free space [2]. This self-healing, shape-preserving, accelerating beam has multiple applications from manipulating microparticles [3] and plasma channel generation [4] to all-optical routing [5], or light-sheet microscopy [6]. Recently, the self trapping character of Airy beams in biased nonlinear media has suggested interesting dynamics such as soliton-like behaviors [7][8][9][10][11]. Co-propagating Airy beams and their interactions have been studied in different medias ranging from nematic liquid crystals [12] to photonic lattices [13] or threewave systems [14]. In Kerr or photorefractive (PR) media, the solitonic structures of two co-propagating coherent Airy beams will exhibit attraction when in-phase and repulsion when out-of-phase [15][16][17] whereas two incoherent Airy beams will always exhibit attraction [18]. Other research on the propagation dynamics have shown that by using external potentials and weakening or strengthening of the autofocussing effect, the accelerating trajectory of Airy beam like structures can be controlled [19,20]. The interaction of Airy beams and solitons in nonlinear media can even reproduce gravitational dynamics [21].
Similarly to studies with two co-propagating Airy beams, two counter-propagating Airy beams in a saturable nonlinear media will shed OSSs given enough nonlinearity strength [22], furthermore the OSSs tend to be attracted by the lobes of the counter-propagating Airy beam [23]. However, by contrast to studies with two copropagating Airy beams, a counter-propagating configuration combined with the large transverse dimension of the Airy shape and their curved trajectory allow several interaction schemes where one or several lobes of the counter-propagating beams can overlap [10,22,23]. The interaction of such counter-propagating Airy beams introduce new dynamics and open new interesting fields for optical interconnections. Interestingly, both multiplexing (combining light beams in a single waveguide) and demultiplexing (splitting light beams in different waveguides) can be achieved using Airy beams. Moreover, waveguiding with a greater output-to-input transverse shift can be obtained by using Airy beams rather than with conventional Gaussian beams.
Waveguiding possibilities in the case of counter-propagating Airy beams have been studied in a symmetric configuration [22], but to our knowledge no study exists on an anti-symmetric configuration.
In this letter we analyze numerically optical waveguiding structures created in PR media by two incoherent counter-propagating 1-D Airy beams under nonlinear self-focusing conditions. In order to create complex waveguides we chose to propagate the Airy beams anti-symmetrically i.e. with each Airy beam accelerating towards the counter-propagating beams. This allows a larger overlapping of the Airy beams' secondary lobes. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
The overlapping results in stronger waveguiding structures with multiple input to multiple output configurations and transverse input-to-output shifts up to 13 times the guided beam's waist.

Antisymmetric Airy beams interactions scheme
The typical scheme of antisymmetric Airy beams is presented in figure 1(a). Both beams propagate in a PR crystal along the z-axis. The forward beam F propagates from left to right and the backward beam B propagates from right to left. d is defined as the shift along the x-axis between the forward and backward Airy beam's main lobes upon injection at each side of the crystal. L is the length of the PR crystal. The x-axis is normalized by x 0 =10 μm i.e. the Airy beam's main lobe waist at 1/e of its maximum intensity. The z-axis is normalized by the diffraction length , with n b the unperturbed refractive index and λ the wavelength. The electric field is applied along the c-axis, parallel to the x-axis: this concerns nonlinear self-focusing conditions. The figures 1(b)-(d) show the intensity distribution of two counter-propagating anti-symmetrical Airy beams in linear conditions for different values of d. In a PR crystal due to the Pockels effect, the intensity distribution induces optically a refractive index distribution which can also be seen as a waveguiding structure. In the case d=1, the secondary lobes of the Airy beams greatly overlap resulting in a larger total intensity all along the secondary lobes' trajectory through the crystal and therefore corresponding to stronger variations of the refractive index. In what follows, we will mainly focus on this configuration which allows strong, numerous and complex waveguiding structures to appear.
The truncated Airy beams' input profiles are given by the following equations: Ai is the Airy function, a the truncation parameter of the Airy beam. To simulate the behavior of the propagation of the antisymmetric Airy beams, we use the same theoretical model as in [10,11,[22][23][24]. The nonlinear propagation of the two incoherent counter-propagating beams can be expressed as follows: eff e is the nonlinear PR coupling strength, r eff is the effective component of the electrooptic tensor and E e is the external electric field. E 0 is the homogeneous part of the x-component of the PR spacecharge field normalized by the external electric field. The temporal evolution of the space-charge field E 0 is considered with a saturable nonlinearity and calculated using a relaxation-type dynamic: where the relaxation time of the crystal τ is inversely proportional to the total intensity t = t + I 1 0 , and 2 is the intensity normalized by the effective background intensity. For small values of Γ (<7), the nonlinear effect is not high enough for producing waveguiding structures; when Γ>15 unstable dynamics are observed [22]. In this paper we present numerical simulations with a nonlinear PR coupling strength Γ=9 which allows the analysis of rich and complex stable waveguiding structures.

Optical interconnections for a transverse shift = d 1
In what follows, we focus on the configuration d=1 and Γ=9 because it has a rich variety of complex and interesting waveguiding structures. We are interested in the number of different outputs for a single input and in the transverse input-to-output shift of the waveguide. Figure 2 shows the intensity distribution of two counter-propagating anti-symmetrical Airy beams in nonlinear self-focusing conditions for d=1 and Γ=9. When the nonlinear self focusing effect compensates the diffractive nature of a light beam, a soliton-like structure appears [11]. The self focusing creates two new waveguiding structures of interest, an off-shooting beam (OSB) and an off-shooting soliton (OSS). The waveguiding structure shows different possible inputs, the OSB and OSS positions created by the backward Airy beam propagation, F 1 the forward Airy beams main lobe position, F 2 and F 3 the forward Airy beams secondary lobes position respectively. The waveguiding structure also shows different possible outputs, the OSB position and OSS position created by the forward Airy beam F, B 1 the backward Airy beam's main lobe position, B 2 , B 3 and B 4 the backward Airy beam's secondary lobes positions respectively, and finally F 1 the forward Airy beam's main lobe position after propagating through the crystal. The forward Airy beam's secondary lobes positions after propagating through the crystal are superimposed with the backward Airy beam's secondary lobe positions.
In order to analyze this waveguiding structure, we propagate a Gaussian probe beam (of waist x 0 = 10 μm) at different input positions of our crystal. The Gaussian probe beam's amplitude represents 10% of the Airy beam's amplitude in order for it's propagation to be linear. Figures 3(a)-(e) show the Gaussian beam's propagation in . It is worth mentioning that an output with less than 10% of the amplitude of the input beam is not of interest for optical interconnects applications. We therefore decided to take into account only the output peaks that are over 15% of the input probe beam's maximum intensity. Some guiding behaviors were already predicted in previous studies of our group in the case of the symmetric Airy beam configuration [22]. By comparison, in this anti-symmetrical configuration, we observe a larger number of outputs for all cases. In figure 3(a), for a probe beam injected in F 2 we observe simultaneously three outputs (OSS, B 1 and F 1 ) of equal peak intensity (around 20%). The OSS output is an expected output as it is observed even when the waveguide is obtained by injecting a single forward Airy beam [22]. However the two additional outputs called B 1 and F 1 are only obtained because of the interaction with the backward beam and thanks to the anti-symmetric configuration. For the symmetric counterpropagating beams interaction, the configuration with F 2 as an input did not yield interesting waveguiding. Similarly in figures 3(b)-(c), injecting in F 1 (respectively in the OSS) yields the expected OSS and B 1 outputs but we also observe an additional B 4 (respectively B 2 ) output. Other inputs give interesting results: F 3 and OSB (figures 3(d)-(e)). We obtain for both cases two outputs, for the first case in the B 1 area and for the second case in B 2 and F 1 positions. The fact that these inputs did not give any outputs in the previous study [22] shows the effectiveness of overlapping anti-symmetric secondary lobes.
The resulting photoinduced waveguiding structure yields more possibilities than what can be observed with counter-propagating Gaussian beams and even with Airy beams in a symmetrical configuration. The resulting waveguiding structure shows interesting features: we observe new optical interconnection schemes such as one input to three outputs with input-output shifts up to x 13 0 (figure 3(a)) (shifts of only x 6 0 were observed for symmetrically counter-propagating Airy beams [22]).
During our investigation we also found other interesting waveguiding possibilities when the probe beam is not injected in one of the inputs mentioned in figure 2 but in between them. In doing so, other configurations are obtained by a coupling of evanescent waves at the different input positions and they are reported in figure 4. By injecting the probe beam between F 1 and F 2 ( figure 4(a)) we combine outputs of the two cases presented in figures 3(a) and (b) (B 1 , B 4 and F 1 ) while suppressing others (OSS). Similarly by injecting between the OSS and F 1 ( figure 4(c)), we combine two outputs (OSS and B 2 ) but lose two outputs (B 1 and B 4 ). In figure 4(b), by placing the probe beam at the top of guide F 1 we can suppress the topmost output (OSS) while preserving the lobe outputs (B 1 and B 4 ). In figure 4(d) by placing the probe beam below the OSS input, we suppress the bottommost outputs (B 1 and B 2 ) and preserve the OSS output.
Such mechanics in Airy beam interactions can lead to the engineering of complex modular optical waveguides by varying different parameters such as the shift d between the beams, the nonlinearity of the medium and the size of the probe beam compared to the Airy beam.

Stability of the photoinduced waveguide
In order to get an overall picture of the different waveguiding possibilities, we have plotted a mapping of the output intensity distribution versus x for different input probe beam positions ( figure 5). For each input probe  figure 5(b). The most intense spots correspond to the best waveguiding conditions reported previously. The spots furthest from the topleft to downright diagonal (red) reveal the largest shifts from input to output positions. Circled in red is the F 1 output for the F 2 input (figure 3(a)) with an input to output shift equal to x 13 0 . Figures 5(a) and (c) show the waveguiding behavior of counter-propagating anti-symmetric Airy beams for d=−10 and d=10 respectively. The configuration differs only by the shift d between the forward and backward Airy beams, but the resulting waveguiding behaviors are quite different. As expected, most spots are close to the diagonal because it corresponds to an input-to-output shift of 0 but multiple outputs are still observed. For example in the case d=−10 ( figure 5(a)), the probe beam injected in the OSS position has two output positions B 1 and B 2 which is the same result obtained for a single forward beam [22]. In the case d=10 however ( figure 5(b)), the probe beam injected in the OSS position has three output positions F 1 , B 1 and B 2 . Therefore, figure 5 summarizes how changing parameters like the transverse shift d between forward and backward beam or the nonlinearity of the medium can give new waveguiding possibilities in comparison to the situation of photoinduced waveguiding from a single Airy beam or even two symmetric counter-propagating Airy beam.

Conclusion
We have analyzed the waveguides photoinduced by two counter-propagating anti-symmetric Airy beams. We have analyzed the propagation behavior of a Gaussian probe beam in such waveguides. We have found configurations giving multiple outputs for multiple input positions: up to three outputs whereas previous works had up to two outputs [22]. We have found greater input-to-output shifts (up to 13 beam waists) with counterpropagating anti-symmetric Airy beams compared to what was possible with counter-propagating Gaussian beams or even counter-propagating symmetric Airy beams configuration (typically maximum 6 beam waists [22]). Interactions of Airy beams in nonlinear self-focusing conditions yield much broader all-optical waveguiding possibilities than those observed so far with Gaussian beams. The situation discussed here with socalled anti-symmetric counter-propagating Airy beams yields to our knowledge the largest variety of all-optical waveguiding: either single input to single output but with possibly large transverse shifts, either single (multiple) inputs to multiple (single) outputs just by varying the initial transverse shift between the counter-propagating Airy beams or the nonlinearity strength.