Wiggling and bending-free micron-sizedsolitons in periodically biased

photorefractives

A. Ciattoni1,2, E. DelRe3,4, A. Marini2, C. Rizza2,1

1 Consiglio Nazionale delle Ricerche - CASTI Regional Lab, 67100 L’Aquila, Italy2 Dipartimento di Fisica, Universita dell’Aquila, 67010 L’Aquila, Italy,

3 Dipartimento di Ingegneria Elettrica e dell’Informazione, Universita dell’Aquila, Montelucodi Roio 67100 L’Aquila, Italy

4 CRS SOFT, INFM-CNR, Universita di Roma ”La Sapienza”, 00185 Roma

Abstract: Considering nonlinear optical propagation through photore-fractive crystals in which the bias voltage is periodically modulated alongthe propagation direction, we are able to identify the conditions in whicha beam forms a soliton in a straight line down to micron-sized widths.The effect, which is numerically investigated considering the full (3+1)Dspatio-temporal light-matter dynamics, emerges when the period of modu-lation of the bias is smaller than the beam diffraction length. In conditionsin which the two scales are comparable, the soliton follows a characteristicwiggling trajectory, oscillating in response to the oscillating bias. Thefinding indicates a method to achieve highly miniaturized soliton-basedphotonic applications that do not require specific off-axis alignment.

© 2008 Optical Society of America

OCIS codes: (190.5330) Photorefractive optics; (190.6135) Spatial solitons

References and links1. D. N. Christodoulides, F. Lederer, and Y. Silberberg, ”Discretizing light behaviour in linear and nonlinear

waveguide lattices,” Nature (London) 424, 817-823 (2003)2. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, ”Soliton control in fading optical lattices,” Opt. Lett. 31, 2181-

2183 (2006)3. L. Solymar, D. J. Webb, and A. Grunnet-Jepsen, The Physics and Applications of Photorefractive Materials

(Oxford Press, 1996)4. A. Ciattoni, E. DelRe, C. Rizza and A. Marini, arXiv:0804.3687v1 (2008)5. S. Trillo and W. Torruellas, Spatial Solitons (Springer, Berlin, 2001)6. E. DelRe, M. Tamburrini, and A. J. Agranat, ”Soliton electro-optic effects in paraelectrics,” Opt. Lett. 25, 963-

965 (2000).7. E. DelRe, B. Crosignani, P. Di Porto, E. Palange, and A. J. Agranat, ”Electro-optic beam manipulation through

photorefractive needles,” Opt. Lett. 27, 2188-2190 (2002).8. A. DErcole, E. Palange, E. DelRe, A. Ciattoni, B. Crosignani, and A. J. Agranat, ”Miniaturization and embedding

of soliton-based electro-optically addressable photonic arrays,” Appl. Phys. Lett. 85, 2679-2681 (2004).9. E. DelRe, A. Ciattoni, and E. Palange, ”Role of charge saturation in photorefractive dynamics of micron-sized

beams and departure from soliton behavior,” Phys. Rev. E 73, 017601 (2006)10. M. I. Carvalho, S. R. Singh, and D. N. Christodoulides, ”Self-deflection of steady-state bright spatial solitons in

biased photorefractive crystals,” Opt. Commun. 120, 311-315 (1995)11. E. DelRe, G. De Masi, A. Ciattoni, and E. Palange, ”Pairing space-charge field conditions with self-guiding for

the attainment of circular symmetry in photorefractive solitons,” Appl. Phys. Lett. 85, 5499-5501 (2004).

#93999 - $15.00 USD Received 20 Mar 2008; revised 29 May 2008; accepted 29 May 2008; published 7 Jul 2008

(C) 2008 OSA 21 July 2008 / Vol. 16, No. 15 / OPTICS EXPRESS 10867

1. Introduction

Self-action is a fundamental mechanism by which light propagating through a nonlinear mater-ial can manifest a variety of manifestations among which solitons are arguably the most knownand significant entities. In a more general perspective, self-action can be viewed as a feasiblemeans to achieve useful handling of optical radiation [1, 2], an essential requirement for mod-ern photonic applications. A most suitable environment for light steering and beam handlingare photorefractive media [3] since the underlying physical mechanism and associated nonlin-ear optical response can be greatly affected by the use of externally applied voltage profiles,which thus form an additional control on optical behavior [4]. For example, the external biasvoltage fixes the existence relation between soliton width and intensity [5], where for standardKerr solitons, this relationship depends uniquely on the nonlinear properties of the materialitself. In addition, photorefractives support electro-activation, whereby the sample is biased bya tunable voltage which, together with the stored charge, produces a fast reconfigurable re-fractive index pattern, able to control light propagation without a correspondingly slow chargeredistribution [6, 7].

In most photorefractive schemes, the external bias voltage is delivered through a pair ofelectrodes in a standard capacitor-like configuration (i.e. a pair of homogeneous electrodesis positioned on two opposite crystal facets). In this condition, within the sample bulk thereis an almost uniform electric field on which photorefractive charge migration and dynamicstake place and eventually giving rise to the steady state configuration at equilibrium. Recently,more elaborate electrode geometries have been investigated in which the external bias voltageis delivered by means of a pair of parallel electrodes positioned on a single crystal facet (top-sided electrode geometry, see Fig. 1(a)) [8]. This allowed the excitation of soliton channels withquasi-digital bias voltages in a specific region embedded beneath the electrode pair and showedhow the nonlinear response could be significantly tailored in space through the voltage profile.

With the aim of expanding the possibilities of nonlinear effects and beam control, in thispaper we investigate nonlinear optical propagation through a photorefractive crystal biased bya system of electrodes characterized by a spatial periodicity along the optical beam propagationdirection. To probe the features of the novel setting we resort to numerical analysis by consider-ing the full (3+1)D spatio-temporal light-matter coupled dynamics where the micron-sized op-tical beam is described within the paraxial approximation and charge dynamics is consistentlydealt with by means of the full band transport model (an extension of the (2+1)D calculationof Ref([9])). We identify two main regimes, i.e., when the spatial period of the applied voltageis greater or smaller than the optical beam diffraction length. We show that in the first situationconditions can be found for which the optical beam does not diffractively spread and follows aspatial mean wiggling trajectory reproducing the oscillations of the applied voltage. Physically,this is a consequence of the asymmetric terms in the response that lead to the well-known self-bending effect, according to which a beam travelling through a region longer than its diffractionlength experiences a lateral deflection in the direction of the local electric field. These terms in-crease for decreasing soliton width and introduce a lower limit to the achievable soliton width,of the order of several micrometers [9]. In conditions in which self-bending does not impedesoliton formation, the wiggling trajectory is a consequence of the electrode cascading beneathwhich the electric field is periodically reversed. In the second regime where the optical beamexperiences the effect of a fast oscillating bias voltage we show that optical propagation occurson a straight line and that self-focusing yields a steady state almost non-diffracting trappedoptical beam.

#93999 - $15.00 USD Received 20 Mar 2008; revised 29 May 2008; accepted 29 May 2008; published 7 Jul 2008

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Fig. 1. Top-sided electrode settings in the standard (a) and in periodically biased (b) con-figuration. Electrodes are positioned on the x = Lx facet, their transverse spacing is dy andtheir voltages are −V0 and V0 for black and red electrodes, respectively. In configuration(b) longitudinal periodicity is dz whereas electrodes longitudinal length is dz/4.

2. Self-bending for two-dimensional beams

The physical features of the beam deflection can be explained by means of a simple symmetryargument in the relevant situation ρ << qNa and χ >> 1 (unsaturated charge regime [9]) where

Ne � (β α/γ)Q and Eq. (2) is ∇ ·(

QE+ KBTq ∇Q

). The solution of this equation can be sought

in the form

E = −KBTq

∇ logQ−∇Φ (1)

where Φ is a suitable potential satisfying the equation ∇ · (Q∇Φ) = 0 and the same boundaryconditions of the electrostatic potential (Φ = V0 and Φ = −V0 on the two electrodes), sincelogQ = 0 at electrodes (the light never reaching the boundary of the crystal). Consider nowa plane z = z0 onto which the intensity distribution Q(x,y,z0) is invariant under the reflectiony → Ly − y so that the function −Φ(x,Ly − y,z0) also satisfies ∇ · (Q∇Φ) = 0 and the sameboundary conditions as Φ, this implying that Φ(x,y,z0) = −Φ(x,Ly − y,z0). As a consequence,from Eq. (1), we deduce that the Ey component of the electric field is the superposition of aneven and an odd contributions (with respect to the plane y = L y/2) and therefore the overallnonlinear refractive index governing light dynamics (see the RHS of Eq. (3)) has no definitespatial reflection symmetry and it shows a hump located at the right or the left side of theintensity profile (according to the sign of the bias voltage) along the y- axis. This lateral guidingregion is the responsible for the self-bending deflection of the optical beam.

We have numerically integrated the full (3+1)D photorefractive model describing light-matter spatio-temporal coupled dynamics. At each instant of time, we evaluate the change ofthe space charge density ρ due to optical photo-excitation and spatial redistribution by meansof the charge continuity equation

∂ρ∂ t

= −μq

[∇ · (NeE)+

KBTq

∇2Ne

], (2)

where E is the electrostatic field (space charge field), μ and q are the electron mobility andcharge and Ne is the electron density which, from the full band transport model [9], is given by

#93999 - $15.00 USD Received 20 Mar 2008; revised 29 May 2008; accepted 29 May 2008; published 7 Jul 2008

(C) 2008 OSA 21 July 2008 / Vol. 16, No. 15 / OPTICS EXPRESS 10869

Ne =− β2γ

[Q+ χ

(1+ ρ

qNa

)]+ β

2γ

√[Q+ χ

(1+ ρ

qNa

)]2+4χQ

(α − ρ

qNa

)where β is the rate

of thermal excitation of electrons, γ is the electron-ionized trap recombination rate, N a is theacceptor impurity density, χ = γNa/β , α = (Nd −Na)/Na (Nd being the donor impurity den-sity) and Q = 1 + I/Ib, I and Ib being the optical beam intensity distribution and the uniformbackground illumination intensity. Subsequently, we calculate the electrostatic potential dis-tribution resulting from the boundary non-uniform bias voltage and the overall charge density(ρ(r,t + dt) as obtained from Eq. (2)) by solving the 3D Poisson equation within the sample.Finally, assuming a boundary (at z = 0) Gaussian beam profile, we evaluate the optical field dis-tribution by solving the parabolic equation with the standard quadratic electro-optic responseor

i∂A∂ z

+12k

(∂ 2A∂x2 +

∂ 2A∂y2

)= − k

n0

[−1

2n3

0gε20 (εr −1)2E2

y

]A (3)

where n0 is the uniform background refractive index, k = ωn 0/c, g is the effective electro-opticcoefficient for the geometry considered (see Fig. 1(a)), where the dominant response for the y-polarized beam is driven by the y- component of the electrostatic field E.

Consider the rectangular sample of Fig. 1(a), whose sides are Lx,Ly and Lz. Since Eqs. (2)and (3) are left invariant by the reflection transformation y → L y − y and Ey →−Ey, swappingthe electrode voltages implies that the underlying resulting electric field, and hence the opticalfield driven by it, are the mirror images (with respect to the plane y = L y/2) of the formerones. Hence the effects on the beam that depend on the orientation of the bias field, i.e., theself-bending mechanisms which amount to a beam deflection along the y− axis, allow us totailor the beam trajectory by suitably reversing the bias along the propagation direction [10].Along these lines, in Fig. 1(b) we consider a photorefractive setting where a rectangular gridof electrodes is positioned onto the facet x = Lx. To maximize the effect, the optical beamis launched in the z direction at a distance Lx/2 from electrode facet, comparable with bothdy and dz/4, i.e., the y− and z− electrode grid spacings, where the electrostatic field is mostappropriate.

We have performed all our numerical simulations choosing a crystal sample of potassiumlithium tantalate niobate (KLTN) (εr = 3 ·104, g = 0.13m4C−2, n0 = 2.4) at room temperatureof length Lz = 2000 μm through which an optical beam (whose wavelength is λ = 0.5 μm)propagates from z = 0 where its Gaussian distribution (centered at x = Lx/2 and y = Ly/2)has an Half Width at Half Maximum (HWHM) w0 = 3.3 μm (with a diffraction length of �240 μm) and a peak intensity 25 Ib. In Fig. 2(a1) (and related movie) we report, for comparisonpurposes, the temporal evolution of the optical beam in the standard configuration of Fig. 1(a)with dy = 80 μm, Lx = 100 μm and the bias voltage V0 = 12V has been chosen so as to observea self-trapped optical beam with an almost z-independent width. Note that, at steady state, self-bending is particularly evident since it yields a beam deviation of � 25 μm after a propagationlength approximately amounting to 8 diffraction lengths.

Our argument agrees with the numerical simulations (based on the full photorefractivemodel) as shown in Fig. 2(a2) where the nonlinear refractive index supporting propagation ofthe optical beam of Fig. 2(a1) is reported (in the form of level plots sliced at different z-planes)and it shows a light-off-axis guiding region (red curves).

3. Wiggling and Bending-free propagation

From Eq. (1) we note that the first contribution Ed = −(KBT/q)∇ logQ to the electric fieldis independent on the external bias voltage whereas the second one E D = −∇Φ physicallystems from the presence of electrodes biasing the sample. As discussed in the previous section,self-bending is due to the interplay between these two contributions. Modulating the external

#93999 - $15.00 USD Received 20 Mar 2008; revised 29 May 2008; accepted 29 May 2008; published 7 Jul 2008

(C) 2008 OSA 21 July 2008 / Vol. 16, No. 15 / OPTICS EXPRESS 10870

Fig. 2. Optical beam photorefractive time dynamics (equal intensity surfaces) (a1)(movie:Fig2a1-Mov.avi, 1371 Kb), (b1) (movie:Fig2b1-Mov.avi, 1658 Kb) and (c1)(movie:Fig2c1-Mov.avi, 1618 Kb) and refractive index patterns supporting the correspond-ing steady state optical propagation (a2) (b2) (c2). Configuration (a) corresponds to thewell-known setting of a longitudinally uniform biased top-sided electrode geometry of Fig.1(a) whose steady state is a bent optical beam. Configurations (b) and (c) correspond to aperiodically biased top-sided electrode geometry as in Fig. 1(b) with electrode periodicitygreater (b) and smaller (c) than the optical diffraction length, respectively. Note the wig-gling and straight steady state optical beam of settings (b) and (c), respectively. The HalfWidth at Half Maximum (HWHM) of the input Gaussian beam is, in all the simulations,w0 = 3.3 μm.

#93999 - $15.00 USD Received 20 Mar 2008; revised 29 May 2008; accepted 29 May 2008; published 7 Jul 2008

(C) 2008 OSA 21 July 2008 / Vol. 16, No. 15 / OPTICS EXPRESS 10871

bias voltage yields a local alteration of this interplay and allows a more complex and exter-nally driven optical beam spatial trajectory. We have numerically investigated this conjecturein a number of different settings and the main result is shown in Fig. 2(b1) where the temporaldynamics and steady state of the optical beam are reported in the case of periodically biased top-sided electrode geometry as in Fig. 1(b) with dz = 1000μm, dy = 80μm and V0 = 12 V. In Fig.2(b2) we report the refractive index pattern of steady state optical propagation which shows, ateach transverse plane, a profile resembling the well-known soliton-supporting structure with acentral guiding hump and two lateral anti-guiding regions (lobes) that, in the considered situ-ation, have different depth [11]. Note from Fig. 2(b1) that, in this situation, the optical beamis almost self-trapped and travels along a curved wiggling trajectory which follows the appliedvoltage oscillations, and this over a distance of eight diffraction lengths. This characteristicwiggling is a consequence of the fact that the period of the voltage modulation (approximatelyfour times the beam diffraction length) is such that each longitudinal step is sufficiently long toallow the bending mechanism to produce a noticeable deflection. This is placed further in evi-dence by the refractive index pattern of Fig. 2(b2), which shows, at each considered transverseslice within a single biased slab, that the guiding hump is not superimposed to the optical beamshape, being shifted (along the y-direction) toward the negative (−V 0) electrode. If the voltagespatial period is made comparable or smaller than the beam diffraction length, results indicatethat the optical beam travels along a straight line, and conditions can be found for the formationof a bending-free beam. This is shown in Fig. 2(c1), where we report the temporal dynamics andsteady state for dz = 200μm, dy = 80μm and V0 = 25 V. Although the corresponding refractiveindex pattern shown in Fig. 2(c2) still manifests, at each transverse plane, an alternating shift inthe guiding hump, this does not lead to observable wiggling, as the reversal occurs on a shorterscale. We note that here the bias voltage required to achieve self-trapped propagation is higherthan in previous cases (Fig. 2(a1) and Fig. 2(b1)), and this is a consequence of the fact that thesmaller longitudinal electrode spacing noticeably decreases the magnitude of electrostatic fieldwhere propagation takes place.

4. Conclusions

We have shown that an optical beam propagating through a periodically biased photorefractivecrystal generally follows a nontrivial oscillating trajectory due to a propagation dependent self-bending. We thus argue that the optical beam can be made to propagate along a prescribedtrajectory by means of a suitable choice of the externally applied, even not periodic, voltageprofile. When the period of the electrode grid is smaller than the optical diffraction length,quasi-soliton optical propagation takes place along a straight line. Therefore we have proposeda feasible way to produce straight photorefractive solitons whose width is of the order of fewmicrons, a possibility which, in the standard electrode geometry, is generally ruled out by thebeam deflection due to self-bending.

#93999 - $15.00 USD Received 20 Mar 2008; revised 29 May 2008; accepted 29 May 2008; published 7 Jul 2008

(C) 2008 OSA 21 July 2008 / Vol. 16, No. 15 / OPTICS EXPRESS 10872