We study experimentally and theoretically the nonlocal response of a medium with thermal nonlinearity and show that despite its inherently infinite range it can be accurately characterized by a well-defined nonlocal response function. We retrieve the shape of this function and analyze its transformation with the change of boundaries

Dreischuh, Alexander

Kivshar, Yuri

Krolikowski, Wieslaw

Minovich, Aliaksandr

Neshev, Dragomir

Optics Letters

The Australian National University

June 15, 2007 / Vol. 32, No. 12 / OPTICS LETTERS 1599Experimental reconstruction of nonlocal responseof thermal nonlinear optical mediaAlexander Minovich,1,* Dragomir N. Neshev,1 Alexander Dreischuh,1,2 Wieslaw Krolikowski,1 andYuri S. Kivshar11Nonlinear Physics Centre and Laser Physics Centre, Research School of Physical Sciences and Engineering,Australian National University, Canberra ACT 0200, Australia2Permanent address: Department of Quantum Electronics, Faculty of Physics, Sofia University, Bulgaria*Corresponding author: min124@rsphysse.anu.edu.auReceived December 15, 2006; revised March 11, 2007; accepted April 1, 2007;posted April 11, 2007 (Doc. ID 78152); published June 5, 2007We study experimentally and theoretically the nonlocal response of a medium with thermal nonlinearity andshow that despite its inherently infinite range it can be accurately characterized by a well-defined nonlocalresponse function. We retrieve the shape of this function and analyze its transformation with the change ofboundaries. © 2007 Optical Society of AmericaOCIS codes: 190.4420, 190.4870.Optical nonlinearity is an important property of amaterial that characterizes its response to an exter-nal electromagnetic field. It is usually accepted thatthe material response at a certain point depends onlyon the value of the field at that point. However, theresponse of many materials is spatially nonlocal. In anonlocal medium the nonlinear response induced at acertain point is carried away to the surrounding re-gions. The size of this extended region determinesthe range of nonlocality. In materials where the non-linearity appears as a result of, e.g., diffusion pro-cesses, the nonlocal region expands to microscopic oreven macroscopic scales.The physical processes leading to a long-range non-local nonlinear response are rather diverse, includingplasma heating and ionization [1], atomic diffusion[2], and drift and/or diffusion of photoexcited carriersin photorefractive media [3,4]. Similarly, the heatconduction in materials with thermal nonlinearity[5,6] results in a spatially nonlocal change of the op-tical refractive index often termed a “response withan infinite range of nonlocality” [7]. The physics ofthese nonlocal mechanisms is very different; how-ever, the description of the nonlinear response can beunified, provided that the nonlinear change of the re-fractive index, nx ,y, is expressed by a nonlocal re-sponse function Rx ,y determining the degree ofnonlocality [8,9]:nx,y =  Rx − x,y − yIx,ydxdy, 1where Ix ,y is the light intensity at the point x ,yand  is a normalization coefficient, such thatRx ,ydxdy=1. Obviously, in this approach theshape and spatial extent of the nonlocal responsefunction are the only characteristics that determinethe light propagation in such materials. In this Let-ter, we study experimentally and theoretically thecharacteristics of the nonlocal response of a thermalnonlinear liquid as a medium with an infinite extentof nonlocality. We demonstrate that, despite being in-herently dependent on boundaries, the response of0146-9592/07/121599-3/$15.00 ©finite-size thermal medium can be accurately de-scribed by the generic spatially invariant nonlocal re-lation Eq. (1) with a well-defined, localized nonlocalresponse function dependent only on sample geom-etry.To determine experimentally the characteristics ofthe nonlocal response function we use a pump–probetechnique to induce a refractive index modulation ina weakly absorbing liquid. The intensity-dependentrefractive index change of such a medium resultsfrom heating of the liquid by the absorbed laser light,volume expansion, and subsequent index change viathe thermo-optic effect. To induce an index change,the wavelength of the pump light has to be partiallyabsorbed by the sample. On the other hand, thesample should not be affected by the probe beam. Toachieve such conditions, we use a pump laser at532 nm and probe at 633 nm (see Fig. 1), where theintensity of the probe beam is several orders of mag-nitude lower than the pump intensity. Both laserbeams pass through a 2 mm thick glass cell filledFig. 1. (Color online) Experimental setup: F, filter; SF,spatial filter; M, mirrors; L, lenses; BC, beam-splitter cube;P, polarizer; T, telescope;  /2, half-wave plate; A, attenua-tor. Inset, ethylene glycol transmission spectrum.2007 Optical Society of America1600 OPTICS LETTERS / Vol. 32, No. 12 / June 15, 2007with liquid (ethylene glycol or ethanol) dyed with io-dine. The iodine allows for higher absorption of thegreen pump beam than for the red probe (Fig. 1, in-set).For reconstruction of the induced refractive indexdistribution the cell is placed in the object arm of anadaptive interferometer [10]. It contains a program-mable phase modulator as a controllable-phase mir-ror in a diffraction-grating regime. The modulator in-troduces a desired phase shift in the first diffractionorder, which is selected by a spatial filter. The pumpbeam is combined with the probe by a pair of polar-izing beam-splitter cubes. The cell is subsequentlyimaged onto a high-resolution CCD camera.This interferometric technique allows one to quan-titatively determine the spatial distribution of the in-duced refractive index change, nx ,y, providingthat the probe beam does not change its shape insidethe cell. This condition is met in the thin cell. Thesubsequent phase reconstruction procedure is a par-ticular realization of the four-frame interferometrictechnique [11]. In our experiment a set of four inter-ferograms Ijx ,y j=1. . .4 is recorded (Fig. 2, rightinset) for phase shifts of j−1 /2 between the objectand the reference beams. The phase profile in the ob-ject arm of the interferometer can then be recon-structed according to the relation x ,y=tan−1I4−I2 / I1−I3. The experiments were performed atdifferent beam positions inside the sample away fromthe transverse boundaries.The measured maximum change of the refractiveindex versus the absorbed laser power, shown in Fig.2(a), is a linear dependence and agrees with earliertheoretical predictions [6]. The retrieved two-dimensional (2D) nonlinear refractive index nx ,yis shown in Figs. 2(b) and 2(c) as both 2D pseudocolorand 3D plots for ethylene glycol and ethanol, respec-tively. For comparison, in Fig. 2(a), left inset, weshow the intensity profile of the pump laser I0x ,y,which is clearly much narrower than the induced re-Fig. 2. (Color online) (a) Dependence of peak refractive in-dex change on absorbed power. Right inset, measured in-terference pattern (ethylene glycol, 2 mW); left inset, laserbeam profile. (b), (c) Reconstructed refractive index changeprofiles in ethylene glycol and ethanol at 2 mW.sponse. In the case of ethanol, the induced refractiveindex is strongly asymmetric, indicating the presenceof convection in the sample [6]. The convection effec-tively carries upward the absorbed heat, reducing thesize of the nonlocal response in the horizontal direc-tion.Having the index distribution and the pump-beamintensity profile, we determine the nonlocal responsefunction Rx ,y. This is essentially an inverse prob-lem [12], which is solved iteratively by Fourier decon-volution as Rx ,y=F−1Fnx ,y /FIx ,y. Thedependence of the full width at half-maximum(FWHM) of the response function in the horizontal xdirection for both liquids is shown in Fig. 3(a). In thecase of ethylene glycol the width of the response func-tion is independent of power, which is expected, asthe response function should not depend on the widthor intensity of the input beam. On the other hand,the width of the response function is reduced withpower in the case of ethanol. This is a consequence ofthe convection process, being power dependent. Atlow powers where no convection is present the re-sponse function has the same profile for both liquids.To understand why these two significantly differentmaterials have the same response function (in theabsence of convection) we consider the heat equationin a medium with thermal conductivity k and linearabsorption ,k2Tx,y,z = − Ix,y, 2where Tx ,y ,z is the temperature distribution in thesample and 2 is the Laplacian. The temperature dis-tribution inside the sample determines the inducedrefractive index change by nx ,y ,z= dn /dTT,where dn /dT is the thermo-optic coefficient of thematerial. For boundaries with constant temperaturethe solution of heat equation (2) can be written asTx ,y ,z= /kIx ,yGx ,x ,y ,y ,z ,zdxdydz,Fig. 3. (Color online) (a) Dependence of the response func-tion FWHM wx on absorbed power. (b), (c) Reconstructedresponse function profile (b) and comparison of the mea-sured (solid line) and calculated (dashed) response functionprofile (c) for ethylene glycol.June 15, 2007 / Vol. 32, No. 12 / OPTICS LETTERS 1601where Gx ,x ,y ,y ,z ,z is a Green function that de-pends on the geometry and boundary conditions only.Now one can remove the z dependence from the re-fractive index change nx ,y ,z by integrating italong the sample length L. Then we obtain nx ,y= dn /dT /L0LTx ,y ,zdz. Therefore the responsefunction calculated by inverting relation Eq. (1) in-deed depends only on the geometry of the sample andthe boundary conditions, but not on the properties ofthe medium, confirming our experimental observa-tions. In Fig. 3(b) and 3(c) we show the spatial profileof the retrieved nonlocal response function Rx ,y.The agreement with the profile calculated by usingthe heat equation (2) is excellent, as shown in Fig.3(c). The above representation also applies for thecase of an asymmetric boundary condition leading toan asymmetric response function Rx ,y.The dependence of thermal nonlinearity on bound-aries has serious consequences for the index mea-surements. Typically, in the context of beam propaga-tion in a thick thermal sample only the transversegeometry is considered. In a thick sample the slowvariation of the beam intensity along the propagationdirection does not lead to any appreciable tempera-ture gradient or heat flow in this direction. As a re-sult, the nonlocal response is a function of the trans-verse coordinates only. The situation becomes morecomplex when a thin sample is used (as in our experi-ments). In this case the front and back facets of thecell induce longitudinal heat flow, affecting the finaltemperature and refractive index distributions,which become a function of three spatial coordinates.To assess the impact of this effect on the nonlocal re-sponse function we solved numerically Eq. (2) assum-ing a Gaussian pump beam as a heat source. In thesimulations the transverse dimensions of the samplewere kept constant while the thickness was varied.Figure 4 shows the calculated width of the responseFig. 4. (Color online) Calculated width of the responsefunction (top) and corresponding maximum of the refrac-tive index change (bottom) as a function of the cell thick-ness. (a)–(c) Radical profiles of the response function forthicknesses of 2, 4, and 50 mm, respectively.function (top) and corresponding maximum refractiveindex change (bottom) versus the sample thickness.It is clear that for thin samples the heat flow towardthe back and front boundaries strongly affects themeasured index distribution, thus decreasing therange of nonlocality in transverse direction.On the other hand, when the thickness of thesample increases, the effect of longitudinal heat flowbecomes less pronounced until the response is com-pletely determined by the transverse boundaries.However, in all cases the nonlocality can still be welldescribed by the generic convolution model, Eq. (1).In fact this approximation is more accurate for a thinsample when the influence of transverse boundarieson the heat flow is weaker. The insets in Fig. 4 depictprofiles of the response function calculated for thick-nesses of 2, 4, and 50 mm. The increase of the extentof nonlocality and change of its shape with samplethickness is clearly visible.In conclusion, we have demonstrated that nonlocalnonlinearity of a thermal medium can be describedaccurately by a finite-extent nonlocal response func-tion even though the thermal effect is of an infiniterange. The response function is independent of thematerial parameters and is determined by the geom-etry of the sample. Therefore control of the remoteboundaries provides an efficient tool for affecting thepropagation of beams in thermal nonlinear media[13].The work was supported by the Australian Re-search Council and the National Science Fund (Bul-garia, F-1303).References1. A. Litvak, V. Mironov, G. Fraiman, and A. Yunakovskii,Sov. J. Plasma Phys. 1, 31 (1975).2. D. Suter and T. Blasberg, Phys. Rev. A 48, 4583 (1993).3. S. Gatz and J. Herrmann, Opt. Lett. 23, 1176 (1998).4. B. Crosignani, A. Degasperis, E. DelRe, P. Di Porto,and A. J. Agranat, Phys. Rev. Lett. 82, 1664 (1999).5. 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Experimental reconstruction of nonlocal response of thermal nonlinear optical media

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