## Abstract

Anti-diffraction is a theoretically predicted nonlinear optical phenomenon that occurs when a light beam spontaneously focalizes independently of its intensity. We observe anti-diffracting beams supported by the peak-intensity-independent diffusive nonlinearity that are able to shrink below their diffraction-limited size in photorefractive lithium-enriched potassium-tantalate-niobate (KTN:Li).

© 2014 Optical Society of America

## 1. Introduction and motivation

Diffraction causes light beams to spread out, losing spatial definition and intensity [1, 2]. This forms a limit to the spatial resolution of optical imaging systems based on far-field optics, such as a standard wide-area microscope. In nonlinear materials, self-focusing can change this spreading, but the effect is intrinsically peak-intensity dependent [3]. When self-focusing exactly balances beam spreading caused by diffraction, something that imposes a precise peak-intensity beam-width relationship, stable non-spreading beams in the form of spatial solitons appear [4, 5].

Experiments in waveguide arrays and photonic crystals have shown that interference can cause beams in specific directions to suffer a cancelled diffraction [6–8]. In electro-magnetic-induced transparency experiments, interference can even lead to inverted (or negative) diffraction [9]. Based on interference, this modified diffraction occurs along specific directions and for beams with a small angular spectrum. A more general effect would be the observation of beams that literally ”anti-diffract” as they propagate in a substance. In such a system, beams will naturally converge instead of spreading, irrespective of direction of propagation and for a wide range of beam sizes, even with a considerable angular spectrum. In distinction to self-focusing, that depends on intensity and generally becomes stronger as beams shrink, anti-diffraction should be intensity-independent.

Studies in nanodisordered photorefractive crystals have shown that the diffusive nonlinearity in paraelectric samples [10–13] can strongly reduce natural diffraction, ultimately cancelling it, a phenomenon known as scale-free optics [14–17].

In this paper we theoretically predict anti-diffraction supported by the diffusive nonlinearity and report its first observation in lithium-enriched potassium-tantalate-niobate (KTN:Li).

## 2. Theoretical

In a photorefractive crystal, light absorbed by deep in-band impurities diffuses and gives rise to a static electric field **E*** _{dc}* = −(

*k*)∇

_{B}T/q*I/I*, where

*k*is the Boltzmann constant,

_{B}*T*the crystal temperature,

*q*the elementary charge,

*I*= |

*A*|

^{2}the optical intensity, and

*A*the optical field amplitude [10–13]. When the crystal is a disordered ferroelectric above its peak temperature

*T*[18], the electro-optic response of the mesoscopic dipoles (polar-nanoregions - PNRs) [19] gives rise to a scalar change $\mathrm{\Delta}n=-\left({n}_{0}^{3}/2\right)g{\epsilon}_{0}^{2}{\chi}_{\mathit{PNR}}^{2}{\left|{\mathbf{E}}_{dc}\right|}^{2}$ in the background index of refraction

_{m}*n*

_{0}[20], where

*χ*is the PNR low-frequency susceptibility,

_{PNR}*g*is the electro-optic coefficient, and $L=4\pi {n}_{0}^{2}{\epsilon}_{0}\sqrt{g}{\chi}_{\mathit{PNR}}({k}_{B}T/q)$ [14, 15]. In the paraxial approximation, the slowly varying optical amplitude

*A*obeys the equation

*k*=

*k*

_{0}

*n*

_{0},

*k*

_{0}= 2

*π/λ*,

*z*is the propagation axis, ∇

_{⊥}≡ (∂

*, ∂*

_{x}*), and*

_{y}*λ*is the optical wavelength. Separating the variables,

*A*(

*x*,

*y*,

*z*) =

*α*(

*x*,

*z*)

*β*(

*y*,

*z*),

*α*must obey

*β*replacing

*x*with

*y*. Eq. (2) is satisfied by the solution

*w*is the initial beam in the

_{ox}*x*–direction, and

*α*

_{0}is a constant. For a round launch beam with

*w*=

_{ox}*w*=

_{oy}*w*

_{0}, the waist in two transverse dimensions along the propagation direction

*z*is given by

*L*>

*λ*, Eq. (6) foresees beams that shrink into a point-like focus at a characteristic ”collapse length” independently of intensity.

## 3. Experimental

To experimentally demonstrate diffusive anti-diffraction described by Eq. (6) we use the setup illustrated in Fig. 1. A 0.8 mW (before L3) He-Ne laser operating at *λ* = 632.8nm is expanded and subsequently focused down to a spot with an *w*_{0} = 7.8*μ*m (intensity full-width-at-half-maximum of Δ*x* = Δ*y* ≃ 9.4*μ*m) at the input face of a sample of lithium-enriched potassium-tantalate-niobate (KTN:Li). The composite ferroelectric is grown through the top-seeded solution method so as to have a peak dielectric maximum *T _{m}* at room temperature and high optical quality [21]. Our specific crystal is a zero-cut 2.6 × 3.0 × 6.0 mm sample with a composition of K

_{1−x}Ta

_{1−y}Nb

*O*

_{y}_{3}:Li

*with*

_{x}*x*= 0.003,

*y*= 0.36. Cu impurities (approximately 0.001 atoms per mole) support photorefraction in the visible, whereas focusing and cross-polarizer experiments give

*n*

_{0}= 2.2 and

*g*= 0.14m

^{4}C

^{−2}. The beam is polarized in the

*x*direction and propagates inside the crystal for a distance of

*L*≃ 3.0mm. The crystal is rotated to a desired angle

_{z}*θ*in the

*x*,

*z*plane. The output intensity distribution of the beam is imaged by a CCD camera through an imaging lens (NA≃ 0.35). Light scattered in the vertical

*y*direction is captured by a second CCD camera placed above the sample in the

*y*direction through a high aperture microscope (NA≃ 0.8) positioned so as to image the plane of propagation.

We are able to achieve *L* > *λ* during a transient by operating near *T _{m}* = 287.5K, identified through dielectric constant measurements, and enacting a non-monotonic temperature trajectory

*T*(

*t*) [22–27]. In fact, considering the values of

*n*

_{0},

*g*, and

*k*≃ 25 mV,

_{B}T/q*L*∼

*λ*for ${\chi}_{\mathit{PNR}}~\lambda /\left(4\pi {n}_{0}^{2}{\epsilon}_{0}\sqrt{g}\left({k}_{B}T/q\right)\right)\simeq {10}^{5}$, i.e., an anomalously large value of susceptibility only observable in proximity of the dielectric peak. In each anti-diffraction experiment we enacted the following procedure: the crystal was first cleaned of photorefractive space-charge by illuminating it with a fully powered microscope illuminator placed at approximately 0.1 m above the crystal for over 10 minutes. Using a temperature controller that drives the current of a Peltier junction placed directly below the crystal in the

*y*direction, we brought the sample to thermalize at

*T*= 303K. The sample is then cooled from

_{A}*T*= 303K at the rate of 0.07 K/s to a temperature

_{A}*T*(that is fixed to different values in experiments, see below), where it is kept for 60 s. Then the sample is heated once again at a rate of 0.2 K/s to the operating temperature (>

_{D}*T*)

_{D}*T*= 290K. The strong transient response is observed to have a characteristic response time of 10–30 s, with measured values of collapse length

_{B}*z*= 3.9 − 6.8mm that depend on the actual value of

_{c}*T*used. This regime is not otherwise accessible with our apparatus by a standard rapid cooling (i.e., from

_{D}*T*directly to

_{A}*T*). Once

_{B}*T*is reached, the temperature cycle

_{B}*T*(

*t*) is complete and we switched on the laser beam, recording top-view and front view images of the captured intensity distribution. All intervals of time

*t*are indicated such that the laser is turned on at

*t*= 0.

## 4. Results

In Fig. 2 we show a condition of strong anti-diffraction observed when *T _{D}* = 283K. As shown in Fig. 2(a–c), the

*w*

_{0}= 7.8

*μ*m input beam diffracts to 38

*μ*m as it propagates to the output facet at the initial

*T*= 303K. After the cooling/heating cycle, the output beam shrinks to 5

_{A}*μ*m (

*L*≃ 0.643

*μ*m). Snapshots of the top-view scattered light illustrate the transition from the diffracting Fig. 2(d–f) to the shrinking beam condition Fig. 2(g), and ultimately to the once again spreading phase Fig. 2(h–i) with strongly reduced scattering. In this case, the crystal is rotated by

*θ*= 11°. The beam profiles of the input and output distributions (at

*t*= 15s) are compared in Fig. 2(j). From Eqs. (6–7) we deduce a value of

*z*= 3.9mm. To confirm the approximate intensity-independent and angle-independent nature of the effect, we repeated the experiment with different levels of beam power and propagation angles. We found same levels of anti-diffraction repeating experiments with 8, 30, 240, 800

_{c}*μ*W beams and for launch angles

*θ*= 5° − 11°. For example, at a fixed angle

*θ*= 11°, increasing the beam power from 30

*μ*W and 240

*μ*W, alters the minimum waist by less than 12%. In turn, at

*θ*= 5°, for beam powers from 30

*μ*W and 240

*μ*W, the minimum waist of the antidiffracting beams varies by less than 14 %. The only relevant systematic effect associated with different beam powers was a lenghtening of the anti-diffraction response time, as expected for the cumulative nature of the photorefractive response.

In Fig. 3 we show a condition of weaker anti-diffraction from 7.8 to 7 microns when *T _{D}* = 286K, (

*L*≃ 0.636

*μ*m). Here from Eqs. (6–7)

*z*= 6.8mm, and the maximum anti-diffraction occurs after 10 s from the end of the thermal cycle.

_{c}In Fig. 4 we show the time sequence for the two reported cases of Fig. 2 and Fig. 3. The transverse intensity distribution is shown for different intervals of time *t* from the completion of the temperature cycle and the launching of the laser beam, highlighting the transient nature of the anti-diffraction.

## 5. Conclusion

Anti-diffraction is a new nonlinear intensity-independent wave phenomenon that can possibily lead to new ideas in imaging techniques. From a purely fundamental perspective, we note that our paraxial theory will break down if *L _{z}* ≃

*z*, where the strong-focusing requires a fully nonparaxial treatment, so that future experiments with shorter

_{c}*z*or longer

_{c}*L*may hold further novel effects. Moreover, one phenomenological aspect that already at this stage of anti-diffraction merits discussion is the formation of transient patterns after the strong anti-diffraction stage, as reported in Fig. 4. The patterns are more evident as the value of

_{z}*z*decreases and are not strongly dependent on

_{c}*θ*for the range

*θ*= 5 − 22° we scanned. Since this excludes the possible influence of ferroelectric domains, which are pinned to the principal axes of the crystal in its nominal paraelectric m3m phase [28, 29], these patterns appear an effect of the nonlinear (but peak-intensity-independent) propagation itself.

## Acknowledgments

Funding from grants PRIN 2012BFNWZ2, Sapienza Ricerca 2012 and Award 2013 are acknowledged. A.J.A. acknowledges the support of the Peter Brojde Center for Innovative Engineering.

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