Pattern formation and direct measurement of the spatial resolution in a Photorefractive Liquid-Crystal-Light-Valve

Transcription

1 Pattern formation and direct measurement of the spatial resolution in a Photorefractive Liquid-Crystal-Light-Valve U. Bortolozzo a,, S. Residori a, A. Petrosyan b and J.P. Huignard c a Institut Non Linéaire de Nice, 1361 Route des Lucioles, Valbonne, France b Laboratoire de Physique de l ENS-Lyon, 46 Allée ditalie Lyon, France c Thales Research & Technology, RD Palaiseau Cedex, France Abstract In a photorefractive liquid crystal light valve, acting as a Kerr-like nonlinear optical medium, we show the appearance of optical patterns induced by a single mirror feedback. The spatial wavelength of the patterns scales with the distance between the mirror and the valve and the contrast of the patterns decreases for decreasing this distance. We use these properties to setup a new optical scheme for the measurement of the spatial resolution of the nonlinear device. Key words: Nonlinear optics, liquid crystals, spatial light modulators, pattern formation. PACS: k; Kr; r 1 Introduction A photorefractive liquid crystal light valve (PLCLV) can be considered as a nonlinear optical component which exhibits very attractive capabilities for laser beam manipulation or spatial control of an incident wavefront. Motivated by the possibility of using such devices in optical pattern formation, we have set up a single mirror feedback experiment where a PLCLV acts as a Kerr-like nonlinear medium [1]. Another type of nonlinear behavior of the valve is the first demonstration in [2] of coherent image amplification through Corresponding author. Tel.: +33-(0) ; fax.: +33-(0) address: umberto.bortolozzo@inln.cnrs.fr (U. Bortolozzo). Preprint submitted to Elsevier Science 10 January 2006

2 two wave mixing and dynamic holography. Also the PLCLV is a very useful device for laser beam shaping and wavefront correction [3]. In other terms, it is viewed as a nonlinear component with an hybrid structure combining a piece of photoconductive crystal sandwiched to a liquid crystal layer. The excellent photosensitivity arises from the photoconductor crystal while a large electro-optic effect at low applied voltage is due to the large birefringence of the liquid crystals. Therefore, in a liquid crystal light valve both photoconductive and electro optic properties can be separately optimised. Note that in a conventional photorefractive crystal these two properties at the origin of the nonlinear effect are present in the same volume of the crystal. All these applications require a precise knowledge of the spatial frequency response of the optically addressed spatial phase modulator in order to quantify its performance for the processing and manipulation of laser wavefronts carrying spatial informations. The presented device works in transmission, hence allows much more simplified schemes with respect to classical types of liquid crystal light valves that work in reflections [4]. Here, we show that, when inserted in the optical feedback loop, the PLCLV gives rise to optical pattern formation. The characteristic size of the patterns scales with the free propagation length in the feedback loop. When this distance decreases, the pattern wavelength decreases and eventually approaches the diffusion length of the nonlinear medium, hence the contrast of the patterns is strongly diminished. We use these properties to perform a direct measurement of the spatial frequency response of the nonlinear device. Several other methods have been used to quantify the spatial resolution of a light valve, in particular: incoherent image projection of a binary chart and measurement of image contrast for each spatial frequency of the pattern [5]; holographic recording and probing the diffraction with an auxiliary laser beam. In comparison with these methods requiring more sophisticated optical set up the experiment based on nonlinear pattern formation in a transmission type light valve is easy to implement. The laser is only passing through the device with a retromirror, and it permits to access to the nonlinear parameters and to the spatial resolution of the device. The method is very general and it can be extended to other nonlinear optical media. 2 Experimental setup The PLCLV is realized by using a B 12 SiO 20 (BSO) photorefractive crystal as a photoconductor [6], cut in the form of a thin slice, 1 mm thickness, 20x30 mm lateral sizes. On one side the BSO is coated with an Indium-T in- Oxide (ITO) transparent electrode. The other wall is a glass window (BK7) coated with ITO. Both the BSO surface and ITO of the glass window are 2

3 treated with polyvinyl-alcohol (PVA) for planar alignment. Teflon spacers of 14 µm are inserted between the two walls and the cell is sealed with UV photo-polymerizing glue. The assembled cell is filled with the nematic liquid crystal E48, which has a positive dielectric anisotropy, ε = 15.1 at 1.0 khz, and optical birefringence n = n e n o = , the extraordinary and the ordinary index being, respectively, n e = and n o = at λ = nm and T = 20 C. The final quality of the planar alignment is tested under the optical microscope. A schematic drawing of the finished PLCLV is shown in Fig.1a. electrode glass (ITO) window x BSO crystal electrode (ITO) ϕ/π [rad] 10 y z I pump 5 a) b) V 0 n BS Ipump [mw/cm2] I 0 c) L/2 V 0 CCD screen Fig. 1. a) Schematic representation of the PLCLV and b) its typical response for an applied voltage V 0 = 20 V olts, frequency f = 2 KHz. c) The single mirror feedback setup for pattern formation: L/2 is half the free propagation length. The BSO crystal is well-known for his photorefractive response [6]. Here, we make use of his large photoconductivity in the visible range, from λ = 400 to λ = 550 nm [2]. An AC voltage V 0 is applied across the device through the ITO electrodes. Under the application of an electric field, liquid crystals tend to realign in such a way to become parallel to the direction of the applied field [7]. When the light intensity on the BSO increases, its impedance decreases due to the photoconductivity and hence, the voltage drop across the liquid crystal layer increases, inducing liquid crystal reorientation. As a consequence, a light beam passing through the liquid crystal layer will experience a refractive index change and, thus, a phase change ϕ, which is a function of the light intensity on the BSO. 3

4 The response of the PLCLV is measured by inserting the component between two crossed polarizers, with the liquid crystal director making an angle of 45 with the polarizer axes. A diode pumped solid state laser DPSS, λ = 532 nm, incident on the BSO side acts as the pump beam whereas a low power He-Ne laser beam, λ = nm is used to probe the liquid crystal reorientation. A typical response is displayed in Fig.1b, where the phase change ϕ experienced by the probe beam is plotted as a function of the light intensity incoming on the photoconductor. The saturation of the response is attained when the liquid crystals are aligned along the direction of the applied field, which correspond to zero birefringence. The full range of phase variation is ϕ = k nd 11π, which corresponds to the maximum birefringence n = n e n o 0.2, realized for the initial planar alignment of the liquid crystals. For convenience, we denote as ϕ the modulus of the phase change. In Fig.1c it is shown the experimental setup for the optical feedback. The DPSS laser beam is enlarged and collimated, so that the beam diameter on the PLCLV is 30 mm. The beam is sent onto the BSO side of the PLCLV, pass through the liquid crystal layer and then is reflected back by a mirror. When the light pass for the second times trough the photorefractive crystal, it induces a local electric field modulation that is proportional to the feedback light intensity and phase shift. Because of its optical activity, the BSO crystal induces on the input beam a polarization rotation ψ 44 /mm [6]. To compensate this effect, a half-wave plate is inserted before the entrance face of the BSO and the input beam polarization is rotated of an angle ψ, so that it becomes parallel to the liquid crystal director n after its passage through the BSO. Under this condition, when the liquid crystal reorientation takes places only a phase shift occurs in the pump beam without any change in the intensity. 3 Spontaneous pattern formation According to the response of the PLCLV, for low pump intensity the device can be considered as a Kerr-like nonlinear medium, inducing on the input beam a phase variation which is proportional to the intensity of the pump beam. A model for the liquid crystal reorientation can be derived by considering a simple approximation that neglects the spatial dependence on the cell thickness. This is justified by the fact that the measured phase change is integrated along z, z being the longitudinal coordinate along the cell thickness. Under this approximation, and for n e, n 0 << n, the phase change can be expressed as ϕ = βcos 2 θ, where β = 2π n/λ and θ is the tilt angle averaged along z. We can write a local relaxation equation for ϕ τ t ϕ = (ϕ ϕ 0 ) + l 2 D 2 ϕ + f(i) (1) 4

5 where ϕ 0 is the equilibrium value fixed by the external voltage V 0 in the absence of optical feedback, τ is the local relaxation time given by the response of the photorefractive crystal [2,6] and l D is a lateral diffusion constant, resuming the transverse diffusion of the charge carriers in the BSO and the elastic constants of the liquid crystals. Note that in our case l D practically coincides with the photocarriers diffusion length in the photoconductor, this length being much larger than the electrical coherence lengths of the liquid crystals [9]. Indeed, in the same way as for the homeotropic case [10], the electrical coherence lengths can be calculated in the planar geometry of the PLCLV and these are given by l x = K 22 /2 ε/e, l y = K 33 /2 ε/e and l z = K 11 /2 ε/e, corresponding to the elastic deformations occurring along x, y and z, respectively, when E > E F T, E F T being the Fréedericksz transition field. Note that when the applied field is slightly above E F T, the expressions for electrical coherence lengths reduce to l x = K 22 /K 11 d/π, l y = K 33 /K 11 d/π and l z = d/π [10]. In our case, by taking a typical value for the applied field, E = 1.4x10 5 V/m, and a typical value for the elastic constant in the single constant approximation, K 11 = K 22 = K 33 = 10 pn, we have that a typical electrical coherence length is of the order of 2 µm. Since this length is much smaller than the typical diffusion length of charges in a thick sample of BSO, we neglect the diffusion due to the elastic forces of the liquid crystals and we consider l D as an isotropic and macroscopic scale dominated by charge diffusion in the BSO. In other words, l D is the average width of the point spread function in the transverse plane of the spatial phase modulator. At variance with a recent work where the diffusion length in liquid crystals is measured in the absence of any electric or magnetic field [11], the diffusion length l D that we measure in the PLCLV results from a a process dominated by the applied electric field. The source term in Eq.1 is proportional to the sum of the pump beam and of the light intensity reflected by the mirror after a free propagation of length L, more precisely : f(i) = αi 0 ( 1+ e i L 2k 2 e iϕ 2), (2) where α is a coefficient accounting for the linear part of the PLCLV response, in our case α 4.4 rad cm 2 /mw, k is the optical wave number of the laser light and 2 is the transverse Laplacian. The sign of α gives the sign of the nonlinearity, α > 0 corresponding to a focusing and α < 0 to a defocusing medium. For the PLCLV the nonlinearity is of a defocusing type, since the liquid crystal refractive index changes from the extraordinary to the ordinary value, with n e > n 0 [7]. From the linear stability of the two coupled equations, Eq.(1) and Eq.(2), we know that the homogeneous stationary solution ϕ (0) = ϕ 0 + 2αI 0 becomes un- 5

6 stable at a critical pump power and at a critical wave number q c = 3kπ/L, leading to the appearance of a spatially modulated pattern [1]. In the experiment, we have applied a voltage V 0 = 20 V, frequency f = 2 KHz, and we have observed pattern formation for an input intensity I 0 ranging from 0.2 to 0.5 mw/cm 2. For these values of I 0 the working point of the PLCLV is in the linear part of its response, as it can be seen from Fig.1b. a) b) c) d) e) f) Fig. 2. Hexagonal patterns (near-field images) observed for decreasing the free propagation length L : a) L = 21 cm, b) L = 16 cm, c) L = 12 cm and d) L = 4 cm, e)l = 3 cm f) L = 1.8 cm. In Fig.2 are shown different hexagonal pattern states, as observed for different values of the free propagation length L. The patterns are near-field images recorded by a CCD camera imaging the glass window of the PLCLV. For each value of L, the input intensity has been set in such a way that patterns are close to their instability threshold. We can notice that the pattern size scales with L, as predicted by the linear stability analysis. As concerning the hexagonal symmetry, this is a consequence of the cubic nature of the light-matter interaction (Kerr-like effect), so that any spatial wave vector q i results from the quadratic coupling between two other modes q j and q k in such a way that q i = q j + q k, the three wave vectors having the same length and thus being angularly spaced of 120 [1]. The quadratic coupling is indicated on the far-field image displayed in Fig.3, where the six 6

7 spots corresponds to the three spatial modes (wave vectors indicated by a solid line) and their complex conjugates (wave vectors indicated by a dashed line). Fig. 3. Far-field image of the hexagonal pattern observed for L = 21 cm; arrows mark the wave vectors of the unstable modes. 4 Experimental results and discussion In order to perform a direct characterization of the spatial resolution of the PCLV, we have changed the light free propagation length from L = 21 cm to L = 10 mm and for each value of L we have measured the wavelength and the contrast of the spontaneously formed hexagonal pattern. For each value of L, the input intensity I 0 is set to a value for which the hexagons are just above their instability threshold, so that they are regular and stationary. Indeed, for increasing I 0 secondary instabilities may lead to space-time chaotic states [4]. 25 N [ lp/mm ] L [mm] Fig. 4. Pattern spatial frequency N as a function of the free propagation length L. In Fig.4, we plot in a log-log scale the measured spatial wavelength N as a function of the free propagation length L. N is expressed in line pairs/mm and has been evaluated by measuring the spatial period over the near-field images. The solid line is the theoretical prediction from the linear stability analysis in the case of a defocusing type Kerr nonlinearity, N = q c /π = 3k/Lπ. We can see that the experimental points follow rather well the predicted L 1/2 scaling. 7

8 M.T.F N [ lp/mm ] Fig. 5. MTF of the PLCLV as a function of the spatial frequency N. In order to measure the modulation transfer function (MTF) of the PLCLV, we have proceeded in the following way: for each value of L we have taken several (typically 20) linear intensity profiles on the near-field images, then we have found the maximum I max and the minimum I min value of the intensity for each profile, and calculated the normalized MTF as Imax I min MT F =, (3) I max + I min where <> indicates an ensemble average over all the profiles. The results are displayed in Fig.5, where the pattern visibility is plotted as a function of the pattern spatial frequency N. The curve has a 3 db bandpass at N = 30 lp/mm. The solid line is a best fit with the theoretical curve of Ref.[5], which express the MTF as a function of the physical parameters of the spatial light modulator (ε = 5.4, ε = 20.5). The sharp cutoff encountered at high spatial frequency is probably due to the fact that for short spatial period not all the photo-generated charges migrate to the BSO liquid crystal interface. This means that the spatial period has become of the order of the lateral diffusion length l D, which prevents pattern formation. As shown in [5] the value of the spatial frequency resolution is influenced by the respective thickness of BSO and liquid crystal layer, as well as by the optical wavelength. Using thinner BSO samples or shorter laser wavelength should lead to an increase of the spatial resolution, associated to a decreased value of the diffusion length l D. Further experimental verifications in this direction are in progress. 8

9 5 Conclusions We have shown pattern formation in a PLCLV with a single mirror feedback. We have characterized the spatial resolution of the device by measuring the pattern wavelength and the pattern contrast as a function of the free propagation length L. We have shown that the PLCLV acts as a Kerr-like nonlinear medium, for which a simple model can be used in comparison with the experimental results. When decreasing the light free propagation length, the pattern wavelength decreases together with the pattern visibility, which allows to perform a direct measurement of the spatial resolution of the PLCLV. The method is very general and could be extended to other Kerr-like nonlinear medium. 6 Acknowledgement U.B. acknowledges financial support from the FunFACS European project, n / References [1] G. Dlessandro and W.J. Firth, Hexagonal spatial patterns for a Kerr slice with a feedback mirror, Phys. Rev. A 46, (1992). [2] A. Brignon, I. Bongrand, B. Loiseaux and J.-P. Huignard, Signal-beam amplification by two-wave mixing in a liquid-crystal light valve, Opt. Lett. 22, (1997). [3] N. Sanner, N.Huot, E. Audouard, C.Larat, JP. Huignard,B.Loiseaux, Programmable focal spot shaping of amplified fs laser pulses, Opt. Lett. 30, 1479 (2005). [4] F.T. Arecchi, S. Boccaletti, S. Ducci, E. Pampaloni, P.L. Ramazza and S. Residori, The Liquid Crystal Light Valve with optical feedback: a case study in pattern formation, J. of Nonlinear Opt. Phys. & Materials 9, (2000); S. Residori, Patterns, fronts and structures in a Liquid-Crystal-Light-Valve with optical feedback, Physics Reports 416, (2005). [5] P. Aubourg, J.P. Huignard, M. Hareng and R.A. Mullen, Liquid crystal light valve using bulk monocrystalline Bi 12 SiO 20 as the photoconductive material, Appl. Opt. 21, 3706 (1982). [6] P. Gunter and J.P. Huignard, Photorefractive crystals and their applications, (Springer-Verlag, Berlin Heidelberg, 1989). 9

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