## Abstract

Considering nonlinear optical propagation through photorefractive crystals in which the bias voltage is periodically modulated along the propagation direction, we are able to identify the conditions in which a beam forms a soliton in a straight line down to micron-sized widths. The effect, which is numerically investigated considering the full (3+1)D spatio-temporal light-matter dynamics, emerges when the period of modulation of the bias is smaller than the beam diffraction length. In conditions in which the two scales are comparable, the soliton follows a characteristic wiggling trajectory, oscillating in response to the oscillating bias. The finding indicates a method to achieve highly miniaturized soliton-based photonic applications that do not require specific off-axis alignment.

©2008 Optical Society of America

## 1. Introduction

Self-action is a fundamental mechanism by which light propagating through a nonlinear material can manifest a variety of manifestations among which solitons are arguably the most known and significant entities. In a more general perspective, self-action can be viewed as a feasible means to achieve useful handling of optical radiation [1, 2], an essential requirement for modern photonic applications. A most suitable environment for light steering and beam handling are photorefractive media [3] since the underlying physical mechanism and associated nonlinear 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 bias voltage fixes the existence relation between soliton width and intensity [5], where for standard Kerr solitons, this relationship depends uniquely on the nonlinear properties of the material itself. In addition, photorefractives support electro-activation, whereby the sample is biased by a tunable voltage which, together with the stored charge, produces a fast reconfigurable refractive index pattern, able to control light propagation without a correspondingly slow charge redistribution [6, 7].

In most photorefractive schemes, the external bias voltage is delivered through a pair of electrodes in a standard capacitor-like configuration (i.e. a pair of homogeneous electrodes is positioned on two opposite crystal facets). In this condition, within the sample bulk there is an almost uniform electric field on which photorefractive charge migration and dynamics take 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 voltage is 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 with quasi-digital bias voltages in a specific region embedded beneath the electrode pair and showed how 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 this paper we investigate nonlinear optical propagation through a photorefractive crystal biased by a system of electrodes characterized by a spatial periodicity along the optical beam propagation direction. To probe the features of the novel setting we resort to numerical analysis by considering the full (3+1)D spatio-temporal light-matter coupled dynamics where the micron-sized optical beam is described within the paraxial approximation and charge dynamics is consistently dealt with by means of the full band transport model (an extension of the (2+1)D calculation of Ref([9])). We identify two main regimes, i.e., when the spatial period of the applied voltage is greater or smaller than the optical beam diffraction length. We show that in the first situation conditions can be found for which the optical beam does not diffractively spread and follows a spatial 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 diffraction length experiences a lateral deflection in the direction of the local electric field. These terms increase 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 impede soliton formation, the wiggling trajectory is a consequence of the electrode cascading beneath which the electric field is periodically reversed. In the second regime where the optical beam experiences the effect of a fast oscillating bias voltage we show that optical propagation occurs on a straight line and that self-focusing yields a steady state almost non-diffracting trapped optical beam.

## 2. Self-bending for two-dimensional beams

The physical features of the beam deflection can be explained by means of a simple
symmetry argument in the relevant situation *ρ*
≪ *qN _{a}* and

*χ*≫ 1 (unsaturated charge regime [9]) where

*N*≃ (

_{e}*βα*/

*γ*)

*Q*and Eq. (2) is $\nabla \xb7\left(Q\mathbf{E}+\frac{{K}_{B}T}{q}\nabla Q\right)$. The solution of this equation can be sought in the form

where **Φ** is a suitable potential satisfying the equation
∇∙(*Q*∇**Φ**)
= 0 and the same boundary conditions of the electrostatic potential
(**Φ** = *V*
_{0} and
**Φ** = -*V*
_{0} on the two
electrodes), since log*Q* = 0 at electrodes (the light never reaching
the boundary of the crystal). Consider now a plane *z* =
*z*
_{0} onto which the intensity distribution
*Q*(*x*, *y*,
*z*
_{0}) is invariant under the reflection
*y* → *L _{y}* -

*y*so that the function -

**Φ**(

*x*,

*L*-

_{y}*y*,

*z*

_{0}) also satisfies ∇∙(

*Q*∇

**Φ**) = 0 and the same boundary conditions as

**Φ**, this implying that

**Φ**(

*x*,

*y*,

*z*

_{0}) = -

**Φ**(

*x*,

*L*-

_{y}*y*,

*z*

_{0}). As a consequence, from Eq. (1), we deduce that the

*E*component of the electric field is the superposition of an even and an odd contributions (with respect to the plane

_{y}*y*=

*L*/2) and therefore the overall nonlinear refractive index governing light dynamics (see the RHS of Eq. (3)) has no definite spatial reflection symmetry and it shows a hump located at the right or the left side of the intensity profile (according to the sign of the bias voltage) along the

_{y}*y*-axis. This lateral guiding region 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 of the space charge density *ρ* due
to optical photo-excitation and spatial redistribution by means of the charge
continuity equation

where **E** is the electrostatic field (space charge field),
*μ* and *q* are the electron mobility
and charge and *N _{e}* is the electron density which, from
the full band transport model [9], is given by ${N}_{e}=-\frac{\beta}{2\gamma}\left[Q+\chi \left(1+\frac{\rho}{q{N}_{a}}\right)\right]+\frac{\beta}{2\gamma}\sqrt{{\left[Q+\chi \left(1+\frac{\rho}{q{N}_{a}}\right)\right]}^{2}+4\chi Q\left(\alpha -\frac{\rho}{q{N}_{a}}\right)}$ where

*β*is the rate of thermal excitation of electrons,

*γ*is the electron-ionized trap recombination rate,

*N*is the acceptor impurity density,

_{a}*χ*=

*γN*/

_{a}*β*,

*α*= (

*N*-

_{d}*N*)/

_{a}*N*(

_{a}*N*being the donor impurity density) and

_{d}*Q*= 1 +

*I*/

*I*,

_{b}*I*and

*I*being the optical beam intensity distribution and the uniform background illumination intensity. Subsequently, we calculate the electrostatic potential distribution resulting from the boundary non-uniform bias voltage and the overall charge density (

_{b}*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 distribution by solving the parabolic equation with the standard quadratic electro-optic response or

where *n*
_{0} is the uniform background refractive index,
*k* =
*ωn*
_{0}/*c*, *g* is
the effective electro-optic coefficient 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 *L _{x}*,

*L*and

_{y}*L*. Since Eqs. (2) and (3) are left invariant by the reflection transformation

_{z}*y*→

*L*-

_{y}*y*and

*E*→ -

_{y}*E*, swapping the electrode voltages implies that the underlying resulting electric field, and hence the optical field driven by it, are the mirror images (with respect to the plane

_{y}*y*=

*L*/2) of the former ones. Hence the effects on the beam that depend on the orientation of the bias field, i.e., the self-bending mechanisms which amount to a beam deflection along the

_{y}*y*- axis, allow us to tailor 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 grid of electrodes is positioned onto the facet

*x*=

*L*. To maximize the effect, the optical beam is launched in the

_{x}*z*direction at a distance

*L*/2 from electrode facet, comparable with both

_{x}*d*and

_{y}*d*/4, i.e., the

_{z}*y*- and

*z*- electrode grid spacings, where the electrostatic field is most appropriate.

We have performed all our numerical simulations choosing a crystal sample of
potassium lithium tantalate niobate (KLTN)
(*ε _{r}* = 3 ∙ 10

^{4},

*g*= 0.13

*m*

^{4}

*C*

^{-2},

*n*

_{0}= 2.4) at room temperature of length

*L*= 2000

_{z}*μm*through which an optical beam (whose wavelength is

*λ*= 0.5

*μm*) propagates from

*z*= 0 where its Gaussian distribution (centered at

*x*=

*L*/2 and

_{x}*y*=

*L*/2) has an Half Width at Half Maximum (HWHM)

_{y}*w*

_{0}= 3.3

*μm*(with a diffraction length of ≃ 240

*μm*) and a peak intensity 25

*I*. In Fig. 2(a1) (and related movie) we report, for comparison purposes, the temporal evolution of the optical beam in the standard configuration of Fig. 1(a) with

_{b}*d*= 80

_{y}*μm*,

*L*= 100

_{x}*μm*and the bias voltage

*V*

_{0}= 12

*V*has been chosen so as to observe a 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 propagation length approximately amounting to 8 diffraction lengths.

Our argument agrees with the numerical simulations (based on the full photorefractive
model) as shown in Fig. 2(a2) where the nonlinear refractive index supporting
propagation of the 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 **E**
_{d} =
-(*K _{B}T*/

*q*)∇log

*Q*to the electric field is independent on the external bias voltage whereas the second one

**E**

_{D}= -∇

**Φ**physically stems 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 bias voltage yields a local alteration of this interplay and allows a more complex and externally driven optical beam spatial trajectory. We have numerically investigated this conjecture in a number of different settings and the main result is shown in Fig. 2(b1) where the temporal dynamics 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

*d*= 1000

_{z}*μm*,

*d*= 80

_{y}*μm*and

*V*

_{0}= 12 V. In Fig. 2(b2) we report the refractive index pattern of steady state optical propagation which shows, at each transverse plane, a profile resembling the well-known soliton-supporting structure with a central guiding hump and two lateral anti-guiding regions (lobes) that, in the considered situation, have different depth [11]. Note from Fig. 2(b1) that, in this situation, the optical beam is almost self-trapped and travels along a curved wiggling trajectory which follows the applied voltage oscillations, and this over a distance of eight diffraction lengths. This characteristic wiggling is a consequence of the fact that the period of the voltage modulation (approximately four times the beam diffraction length) is such that each longitudinal step is sufficiently long to allow the bending mechanism to produce a noticeable deflection. This is placed further in evidence by the refractive index pattern of Fig. 2(b2), which shows, at each considered transverse slice within a single biased slab, that the guiding hump is not superimposed to the optical beam shape, being shifted (along the

*y*-direction) toward the negative (-

*V*

_{0}) electrode. If the voltage spatial period is made comparable or smaller than the beam diffraction length, results indicate that the optical beam travels along a straight line, and conditions can be found for the formation of a bending-free beam. This is shown in Fig. 2(c1), where we report the temporal dynamics and steady state for

*d*= 200

_{z}*μm*,

*d*= 80

_{y}*μm*and

*V*

_{0}= 25 V. Although the corresponding refractive index pattern shown in Fig. 2(c2) still manifests, at each transverse plane, an alternating shift in the guiding hump, this does not lead to observable wiggling, as the reversal occurs on a shorter scale. We note that here the bias voltage required to achieve self-trapped propagation is higher than in previous cases (Fig. 2(a1) and Fig. 2(b1)), and this is a consequence of the fact that the smaller longitudinal electrode spacing noticeably decreases the magnitude of electrostatic field where propagation takes place.

## 4. Conclusions

We have shown that an optical beam propagating through a periodically biased photorefractive crystal 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 prescribed trajectory by means of a suitable choice of the externally applied, even not periodic, voltage profile. 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 proposed a feasible way to produce straight photorefractive solitons whose width is of the order of few microns, a possibility which, in the standard electrode geometry, is generally ruled out by the beam deflection due to self-bending.

## References and links

**1. **D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in
linear and nonlinear waveguide lattices,”
Nature (London) **424**, 817–823
(2003) [CrossRef]

**2. **Y. V. Kartashov, V. A. Vysloukh, and L. Torner “Soliton control in fading optical
lattices,” Opt. Lett. **31**, 2181–2183
(2006) [CrossRef] [PubMed]

**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). [CrossRef]

**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). [CrossRef]

**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). [CrossRef]

**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) [CrossRef]

**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) [CrossRef]

**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). [CrossRef]