Abstract
The unique bending and shapepreserving properties of optical Airy beams offer a large range of applications in for example beam routing, optical waveguiding, particle manipulation and plasmonics. In these applications and others, the Airy beam may experience nonlinear lightmatter interactions which in turn modify the Airy beam properties and propagation. A wellknown example is light selffocusing that leads to the formation of spatial soliton. Here, we unveil experimentally the selffocusing properties of a 1DAiry beam in a photorefractive crystal under focusing conditions. The transient evolution involves both selfbending and acceleration of the initially launched Airy beam due to the onset of an offshooting soliton and the resulting nonlocal refractive index perturbation. Both the transient and stationary selffocusing properties can be tuned by varying the bias electric field, the injected Airy beam power and the background illumination.
Introduction
Although being a truncated solution of the ideal Airy waveform, the optical Airy beam keeps its accelerating, nonspreading and selfhealing properties over a finite distance^{1}. The propagation of Airy beams in nonlinear media has been first studied in an unbiased photorefractive medium, where an optical beam is mainly subject to the diffusion effect. It has been shown that the peculiar asymmetrical Airy field distribution enables the Airy beam to undergo selftrapping, i.e. the diffraction of an Airy beam can be annihilated via the carrier diffusion effect in a nonlinear medium over a longer distance^{2,3}. This observation also holds for Airy pulses^{4}.
Besides the shapepreserving propagation of an Airy beam in diffusive media, the propagation direction of the accelerating beam can also be altered via an externally applied electric field. By studying further the impact of the nonlinearity of the medium on the propagation of an Airy beam, various theoretical as well as experimental studies have demonstrated that the shape and trajectory and therefore the acceleration of the Airy beam can be engineered via a refractive index variation^{5,6,7}. The control of the Airy beam’s ballistic properties using either the medium nonlinearity or photonic lattices^{8,9} offers new possibilities in alloptical waveguiding or routing.
All these previous studies considered the nonlinear propagation of an Airy beam where the beam preserves its multilobe distribution and its curved trajectory (i.e. selftrapping). When applying a small bias electric field to the nonlinear medium, the Airy beam still undergoes selftrapping and the main lobe is narrower while still accelerating^{3}. However, recent works have suggested the possibility to induce spatial solitons through selffocusing (and not selftrapping) of optical Airy beams^{3,10,11,12,13}. When applying a larger bias electric field to the nonlinear medium, the Airy beam does not entirely turn into a soliton, but decomposes itself into a socalled offshooting soliton and an accelerating wave packet. However these selffocusing properties of an optical Airy beam have not been experimentally demonstrated.
In this Letter we experimentally demonstrate the existence of solitonic beam structures induced by an Airy beam under strong nonlinear selffocusing conditions. When applying a bias electric field in the direction of the caxis of a photorefractive medium, the Airy beam splits into a weak accelerating structure and an offshooting soliton propagating along the crystal without transverse acceleration. These results match the theoretical predictions in literature^{3,10,11,12,13}. By studying experimentally for the first time the transient selffocusing of the Airy beam, we unveil a twosteps dynamics. When applying an external bias voltage on the medium, the Airy beam first selffocuses into a solitonic structure as theoretically predicted. This solitonic beam coexists with an accelerating wave and, for longer times, relaxes into an Airylike accelerating beam. By analyzing the properties of this offshooting soliton buildup, we show that the onset of an offshooting soliton shed from the initial Airy beam involves both selfbending and acceleration of the initially launched Airy beam. In addition, we demonstrate that both the transient and stationary selffocusing properties can be tuned by varying the bias electric field, the injected Airy beam power and an external background illumination applied on top of the photo refractive crystal [Fig. 1].
Results
Our experiment consists of propagating a onedimensional Airy beam (λ = 532 nm) into a photorefractive SBNcrystal with dimensions 5 mm*5 mm*1 cm (n_{SBN} = 2.3) as depicted on Fig. 1. The onedimensional Airy beam is generated using a cubic phase modulation on a spatial light modulator and is defined by the initial conditions:
ψ(x)_{z=0,t=0} = Ai((x + x_{A})/x_{A})exp(a(x + x_{A})/x_{A}), where x_{A} = 10 μm is the main lobe’s waist and a = 0.04 the truncation parameter of the Airy beam. Under linear conditions, the beam propagates along the zaxis of the crystal with a transverse parabolic acceleration along the caxis (parallel to the xaxis) of the photorefractive crystal. In our experiment we set x = 0 as the transverse output position of the linear Airy beam. (where k = 2πn/λ is the wave vector) mathematically describes the transverse parabolic acceleration. As a consequence, the Airy main lobe has initially been launched at −x_{peak}(z = 0) = 34 μm.
When an external electrical bias field E_{e} is applied along the xaxis at t = 0 s, the optical Airy beam photoinduces a refractive index variation in the crystal through the Pockels effect. The photorefractive effect in the SBNcrystal induces both a focusing and a shift of the optical energy along the transverse caxis thanks to mainly two contributions: the drift effect induced by the electrical bias field and the thermal diffusion effect^{14}. To optimize the nonlinear photorefractive and solitonic effect of our system, the external electrical bias field is set at E_{e} = 4 kV/cm. In this nonlinear regime, the propagating 1D Airy beam turns into a socalled “offshooting soliton” along the tangential direction (zaxis) and an accelerating beam^{13,15}. The theoretical output position of the offshooting soliton then matches with the input position of the main lobe of the Airy beam, i.e. no transverse shift along the xaxis. To analyze the dynamics of this nonlinear interaction, we image the output face of the crystal through a CCD camera. Figure 1(b–f) show the evolution of the intensity profile of the output beam versus time. Starting at t = 0 s, the intensity shifts towards the position of the linear second lobe, further towards the higher lobe’s orders [Fig. 1(b,c)] and finally reaches a maximum transverse shift x = −34 μm at t = 640 ms [Fig. 1(d)]. We will further refer to this position as the offshooting soliton’s position (red dashed line). Then, on a longer timescale a relaxationtype dynamics is observed towards a redistributed Airylike profile similar to the input beam at t = 0 s [Fig. 1(e,f)]. The spatiotemporal dynamics of the nonlinearly propagating Airy beam can therefore be summarized in three stages. (i) First the output beam focuses towards the red dashed line of Fig. 1(g,h). (ii) Then we observe two coexisting beam’s structures [Fig. 1(g,h)]: the socalled offshooting soliton at x/x_{A} = −3.7 and an accelerating structure at x/x_{A} = 0.5 with similar intensities. (iii) Finally the two previous solutions merge and form a new Airylike structure on a longer time scale. Similar to the relaxation dynamics of a spatial soliton formed by a selffocused Gaussian beam^{14,16}, the accelerating beam therefore relaxes for longer times into a less focused multilobe beam with a peak intensity that shifts back towards the +xaxis.
As we will now detail, the nonlinear interactions that emerge from the transient behavior of a single selffocused Airy beam can be characterized as an attraction of the accelerating structure towards the solitonic beam. Simultaneously the accelerating beam presents a tightening effect of the lobes. We show that both attraction and tightening effects can be tuned via the optical power of the initial Airy beam and a background illumination.
In order to characterize the attraction and deflection, we plot in Fig. 2 the nonlinear transverse intensity profile of the output beam for increasing times compared to the linear case at t = 0 s [Fig. 2(a)]. x_{d} corresponds to the shift of the accelerating wave packet induced by the attraction of the offshooting soliton with respect to the initial launched Airy beam. The position of the initial Airy main lobe defines the zero attraction position. Thus x_{d} < 0 illustrates the attraction of the Airy beam towards the offshooting soliton’s position (x/x_{A} = −3.7, see Fig. 2(c,d)). It is worth mentioning that on Fig. 2(c–e) the output profile of the accelerating structure does not match with an Airy distribution anymore, but the output beam still presents secondary lobes. This is due to the multichannel waveguiding structure photoinduced by the multilobe structure of the Airy beam at t = 0 s. After t = 1.9 s, the solitonic structure vanishes and the intensity redistributes into an Airylike profile [Fig. 2(f)]. Figure 2(g) details the temporal evolution of the transverse intensity peak’s position at the output of the crystal during the transient buildup regime of the offshooting soliton (t < 800 ms, stages (i)–(ii)). Initially the Airydistributed energy is mainly concentrated in the first lobe at x = 0. After the electrical switchon of the focusing nonlinearity of the system at t = 0 s, the position corresponding to the peak intensity shifts towards the position of the higher lobes’ orders along the −xaxis (bending effect). Then, around t = 500 ms, the position of the peak intensity reaches a quasisteady position corresponding to the location of the offshooting soliton until, as depicted on Fig. 1(g), for longer times beyond t = 1 s, the position of the peak intensity shifts back into the position of the main lobe of the accelerating beam.
In a second step we now analyze whether the interactions between the solitonic and accelerating waves can be tuned by varying the nonlinearity. In what follows we vary the intensity of the input Airy beam to tune the refractive index modulation depth and analyze the corresponding attraction and tightening effects. Figure 2(g) shows the evolution of the deflection of the accelerating beam for increasing Airy beam powers. Similarly to the selffocusing properties of Gaussian beams^{17,18}, the transient time towards selffocusing is smaller when the input light intensity increases. In addition the increase of input power modifies the transient selffocusing properties. As depicted on Fig. 2(g), when the power increases from P_{A} = 250 μW to P_{A} = 300 μW, the maximum shift does not increase linearly, but jumps from the former second lobe’s position (x/x_{A} = −2.25) to the theoretical output position of the offshooting soliton. When further increasing the power (P_{A} > 700 μW), the maximal bending of the beam still deviates but saturates at the third lobe’s position (x/x_{A} = −3.7). By varying the optical power it is therefore possible to balance between diffraction and nonlinearity and tune our nonlinear system from a weak interaction (P ≤ 250 μW) to a strong attraction (P > 250 μW) between the accelerating wave packet and the solitonic structure.
In addition, during the transient selffocusing regime we also observe a tightening of the lobes of the accelerating beam. Figure 2(h) depicts the normalized peak tightening rate x_{0}/x_{A} of the lobes of the accelerating beam for increasing power, where x_{0} is computed by fitting the intensity profile with an Airy beam profile [Fig. 2(a,b)]. As the peak acceleration of an Airy beam is linked to the resulting lobes’ size x_{0} , the normalized peak acceleration for increasing input powers P_{A} is calculated as follows: (∂^{2}x_{peak}/∂z^{2})_{NL}/(∂^{2}x_{peak}/∂z^{2})_{lin} = (x_{A}/x_{0,NL})^{3}. The resulting evolution is also displayed on Fig. 2(h). As shown by the data on Fig. 2(h), the tightening of the interlobes’ distance induces an increase of the acceleration rate from three (P_{A} = 200 μW) to five times (P_{A} = 900 μW) the initial value.
In summary, the selfbending of the accelerating beam towards the offshooting soliton and its tightening effect are increased with the nonlinearity of the system during the transient selffocusing regime of an Airy beam. In particular the optical power enables to tune the effects of the selffocusing strength on the accelerating structure, hence offering an easytouse control parameter for interactions between the solitonic beam and the accelerating beam.
As already mentioned, the photorefractive nonlinearity of our system can be tuned by different physical parameters such as the external electric bias field (E_{e}), the intensity of the launched beam (with in the main lobe) but also the socalled dark intensity of the photorefractive crystal via an external background illumination [Fig. 1]. Such an illumination tends to artificially increase the dark conductivity of the photorefractive crystal which is initially very weak (I/I_{d} ≈ ∞). In photorefractive systems using Gaussian beams it has been shown that I_{d} plays a significant role during the selffocusing and solitonic regime^{17}. In our Airy beam system, we question how such a background illumination may influence the previous results. Figure 3(a,b) depict the influence of I/I_{d} on the transient and corresponding steady state (t > 8 s) peak values of x_{d} and x_{0}. On Fig. 3(a) for I/I_{d} = 15, the selfbending of the Airy beam observed previously is reduced in the transient regime from to . Contrary to the case without background illumination, where the Airylike structure of stage (iii) is superimposed with the initial Airy beam [Fig. 2(f)], adding I_{d} enables the accelerating beam to remain shifted even in the steadystate regime (maximum shift of −x_{A} for I/I_{d} = 30). As depicted on Fig. 3(b), the background illumination also influences the selffocusing effect both in the transient and the steadystate regime. In particular we still observe selffocusing of the accelerating beam after t > 8 s for I/I_{d} = 45.
Our experimental results can be reproduced qualitatively well by numerical simulations. See Fig. 3(c–g). The nonlinear propagation of the Airy beam in the photorefractive crystal can be simulated by the normalized nonlinear Schrödinger Eq. 1 suggested by Belić et al.^{19}, where an optical beam F(x, t) propagates following the nonlinear wave propagation equation in a photorefractive SBNcrystal:
where Γ = 10 is the nonlinear photorefractive coupling strength, r_{eff} = 235pm/V is the effective component of the electrooptic tensor, E_{e} the external electric field and E_{0} = E_{sc}/E_{e} is the homogeneous part of the xcomponent of the photorefractive space charge field. As the optical intensity modulates the space charge field, the steadystate E_{0} is equal to E_{0} = −I_{0}/(1 + I_{0}), I_{0} = F^{2}. The timedependency of the space charge field E_{0} is calculated from:
where τ = τ_{0}/(1 + I_{0}) is the relaxation time of the crystal, with τ_{0} the characteristic response time of the crystal. The numerical system is completely dimensionfree, in particular the propagation zaxis is normalized to the diffraction length and the transverse xaxis is normalized to the beam’s waist being the lobe’s waist x_{A} = 10 μm in the case of the Airy beam. We fix the truncation parameter a = 0.1 and vary the optical field amplitude F_{0}.
Similarly to the experiment, the Airy beam undergoes selffocusing into an offshooting soliton (at x/x_{A} = −5.5) and an accelerating beam (at x/x_{A} = 0) [Fig. 3(c–e)]. The selfbending and the acceleration effects induced by the offshooting soliton are also observed and can be enhanced by increasing the optical power and therefore the refractive index change [Fig. 3(f,g)]. The numerical results are in good qualitative agreement with the experimental observations of the nonlinear interactions between the offshooting soliton and the accelerating beam (stages (i) and (ii)). We note that the numerical simulations do not reproduce the relaxationtype dynamics of the beam (stage (iii)). Although the main focus here is the nonlinear interaction between the accelerating beam and the offshooting soliton well described by the model Eqs 1, 2, 3, 4, this suggests that a more complete description of the relaxation process in selffocusing requires to account for other transport mechanisms as for example carrier diffusion.
Conclusion and Discussion
In summary, this work is the first experimental analysis of the selffocusing properties of an Airy beam in a photorefractive nonlinear medium. The transient selffocusing behavior shows nonlinear interactions between a soliton and an accelerating beam. These interactions result in both (i) an attraction and deflection effect into the offshooting soliton’s position and (ii) a tightening of the interlobes distance which induces an increase of the acceleration rate. Both attraction and tightening effects can be tuned via the Airy beam intensity or the background illumination.
Very recently, the nonlinear interaction between an accelerating (Airy) beam and a spatial soliton  created by the selffocusing of a Gaussian beam through thermal nonlinearity^{20}  was analyzed in the context of optical gravitational lensing. Relying on the analogy with the NewtonSchrödinger model for quantum gravity, the authors relate the longrange interactions between the spatial soliton and the accelerating beam to the effects caused by a mass on light propagating in a gravitational field. In our configuration, where a soliton and an accelerating beam coexist during the transient selffocusing regime of an Airy beam, the offshooting soliton shed from the initial Airy beam similarly plays the role of a mass that attracts and deflects the remaining accelerating light from its own curved trajectory. The properties of this analogous gravitational lensing, i.e. deflection and acceleration, can be both controlled alloptically through the engineering of the optical photorefractive nonlinearity. Our observations being very similar to those of ref. 20 but in a different system, our conclusion is therefore that the analogy with gravitational lensing effects is not limited to the NewtonSchrödinger framework but applies more generally to a nonlinear Schrödinger equation that accounts for a nonlocal nonlinearity. Besides its interest for the analogy with gravitation, the twostages buildup dynamics of the focused beam provides a deeper insight into the subject of accelerating beams in nonlinear focusing media, and can be used to photoinduce multiple waveguide structures, as suggested in ref. 15.
Additional Information
How to cite this article: Wiersma, N. et al. Airy beam selffocusing in a photorefractive medium. Sci. Rep. 6, 35078; doi: 10.1038/srep35078 (2016).
References
 1.
Siviloglou, G. A. & Christodoulides, D. N. Accelerating finite energy Airy beams. Opt. Lett. 32, 979–981 (2007).
 2.
Jia, S., Lee, J., Fleischer, J. W., Siviloglou, G. A. & Christodoulides, D. N. DiffusionTrapped Airy Beams in Photorefractive Media. Phys. Rev. Lett. 104, 253904 (2010).
 3.
Kaminer, I., Segev, M. & Christodoulides, D. N. Selfaccelerating selftrapped optical beams. Phys. Rev. Lett. 106, 213903 (2011).
 4.
Kaminer, I., Lumer, Y., Segev, M. & Christodoulides, D. N. Causality effects on accelerating light pulses. Opt. Express 19, 23132 (2011).
 5.
Hu, Y. et al. Optimal control of the ballistic motion of airy beams. Opt. Lett. 35, 2260 (2010).
 6.
Ye, Z. et al. Acceleration control of Airy beams with optically induced refractiveindex gradient. Opt. Lett. 36, 3230–3232 (2011).
 7.
Efremidis, N. K. Airy trajectory engineering in dynamic linear index potentials. Opt. Lett. 36, 3006 (2011).
 8.
Xiao, F. et al. Optical Bloch oscillations of an Airy beam in a photonic lattice with a linear transverse index gradient. Opt. Express 22, 22763 (2014).
 9.
Diebel, F. et al. Control of Airybeam selfacceleration by photonic lattices. Phys. Rev. A 90, 033802 (2014).
 10.
Bekenstein, R. & Segev, M. Selfaccelerating optical beams in highly nonlocal nonlinear media. Opt. Express19 19, 23706–23715 (2011).
 11.
Lotti, A. et al. Stationary nonlinear airy beams. Phys. Rev. A  At. Mol. Opt. Phys. 84, 021807 (2011).
 12.
Panagiotopoulos, P. et al. Nonlinear propagation dynamics of finiteenergy airy beams. Phys. Rev. A  At. Mol. Opt. Phys. 86, 1–15 (2012).
 13.
Hu, Y. et al. Reshaping the trajectory and spectrum of nonlinear airy beams. Opt. Lett. 37, 3201 (2012).
 14.
Maufoy, J., Fressengeas, N., Wolfersberger, D. & Kugel, G. Simulation of the temporal behavior of soliton propagation in photorefractive media. Phys. Rev. E 59, 6116–6121 (1999).
 15.
Wiersma, N., Marsal, N., Sciamanna, M. & Wolfersberger, D. Alloptical interconnects using airy beams. Opt. Lett. 39, 5997–6000 (2014).
 16.
Wolfersberger, D., Khelfaoui, N., Dan, C., Fressengeas, N. & Leblond, H. Fast photorefractive selffocusing in inp: fe semiconductor at infrared wavelengths. Appl. Phys. Lett. 92, 021106 (2008).
 17.
Chen, Z., Segev, M. & Christodoulides, D. N. Optical spatial solitons: historical overview and recent advances. Reports Prog. Phys. 75, 086401 (2012).
 18.
Petter, J., Weilnau, C., Denz, C., Stepken, A. & Kaiser, F. Selfbending of photorefractive solitons. Opt. Commun. 170, 291–297 (1999).
 19.
Belić, M. et al. Counterpropagating selftrapped beams in photorefractive crystals. J. Opt. B Quantum Semiclassical Opt. 6, S190 (2004).
 20.
Bekenstein, R., Schley, R., Mutzafi, M., Rotschild, C. & Segev, M. Optical simulations of gravitational effects in the newton–schrödinger system. Nat. Phys. 11, 872–878 (2015).
Acknowledgements
The authors acknowledge the support of Conseil Régional de Lorraine, Fondation Supélec, the IAP P7/35 (BELSPO) with the ‘Photonics@be’ project (2012–2017), Fonds européen de développement régional (FEDER) with the projects ‘PHOTON’ (20142015) and ‘APOLLO’ (2015), and Préfecture de Lorraine and Fonds National d’Aménagement et de Développement du Territoire (FNADT) with the project ‘APOLLO’.
Author information
Affiliations
LMOPS, CentraleSupélec, Université de Lorraine, 57070 METZ, France.
 Noémi Wiersma
 , Nicolas Marsal
 , Marc Sciamanna
 & Delphine Wolfersberger
LMOPS, CentraleSupélec, Université ParisSaclay, 57070 METZ, France.
 Nicolas Marsal
 , Marc Sciamanna
 & Delphine Wolfersberger
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Contributions
N.W. performed the experimental and numerical simulations under the supervision of D.W., N.M. and M.S. All the authors contributed to the data analysis and to the writing of the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Delphine Wolfersberger.
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Further reading

Counterpropagating interactions of selffocusing Airy beams
Scientific Reports (2019)

Propagation dynamics of Airy beams and nonlinear accelerating beams in biased photorefractive media with quadratic electrooptic effect
Applied Physics B (2018)
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