Third-order susceptibility measurements by ... - OSA Publishing

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J. Opt. Soc. Am. B / Vol. 13, No. 7 / July 1996

Boudebs et al.

Third-order susceptibility measurements by nonlinear image processing G. Boudebs*, M. Chis, and J. P. Bourdin Laboratoire des Proprietes ´ Optiques des Materiaux et Applications, Universite´ d’Angers, 2, boulevard Lavoisier, 49045 Angers Cedex 01, France Received July 21, 1995; revised manuscript received January 5, 1996 We present a two-wave mixing configuration associated with a coherent optical processor (4-f system) to characterize the third-order susceptibility coefficient. An accurate measurement of the spatial structure of the intensity becomes possible with a CCD camera, which simplifies and improves the accuracy of third-order susceptibility measurements. To verify the validity of the method, measurements of reference materials illuminated by linearly polarized light are carried out. Good agreement with other measurements by various authors is obtained.  1996 Optical Society of America

1.

INTRODUCTION 1

In Zernike spatial filtering experiments, an absorbing or dephasing filter placed at the common focus of a 4-f system (Fig. 1) is used to produce image processing. For instance, small defects of a phase object can be transformed into large variation of the intensity in the image. This principle has found many applications, particularly in phase-contrast microscopy. It has been shown2,3 that filtering by a nonlinear medium as in the setup described in Fig. 1 provides a simple method for all-optical image processing: realtime cross correlation, contrast inversion, edge enhancement, and phase object visualization. Here a new application of this nonlinear filtering setup is proposed for the measurement of third-order nonlinear susceptibilities. It is shown that a comparison of experimental filtered image of an object with its numerical simulation leads to knowledge of the nonlinear characteristics of the filtering material. The principle of the measurement method is described first, in Section 2. Then experiments and results are presented in Section 3 and 4, and finally the advantages of this method are discussed in Section 5.

2.

PRINCIPLE OF THE MEASUREMENT

A two-dimensional object (Fig. 1) is illuminated at normal incidence by a monochromatic linearly polarized plane wave hdefined by E expf2j svt 2 kzdg 1 c.c., where v and k are, respectively, the angular frequency and the wave vector of the plane wavej. If the transmittance of the object is tsx, yd, the electric field in the focal plane of the first lens is the spatial Fourier transform (FT) of Osx, yd Esx, ydtsx, yd (Ref. 4): 1 FT fOsx, ydg lf 1 ZZ Osx, ydexpf22pj sux 1 vydgdxdy , lf

Ssu, vd

where u xylf and v yylf denote the spatial frequencies in the focal plane, f is the focal length of lens L1, l is the wavelength of the exciting wave. This electric field can excite different phenomena, such as nonlinear absorption, nonlinear change of the refractive index, and stimulated scattering, in the nonlinear medium. After going through the nonlinear filter, the electric field becomes S 0 su, vd Ssu, vdTsu, vd ;

this relation defines the nonlinear transmittance T su, vd, which is defined in Eq. (6) below. In the image plane the electric field amplitude is U sx, yd lf FT 21 fSsu, vdTsu, vdg ,

0740-3224/96/071450-07$10.00

(3)

where FT21 denotes the inverse Fourier transform. For a given object Osx, yd, the spectrum Ssu, vd can be calculated. Using Eq. (3), one can deduce information on T su, vd from the measurements of the image intensity Iim sx, ydajU sx, ydj2 , particularly some characteristic constants of the nonlinear medium. A. Nonlinear Transmittance T(u, v) In what follows, nonlinear effects of orders higher than 3 are neglected, and stimulated scatterings (Brillouin, Raman, and Rayleigh) are not taken into account. It is assumed that self-focusing or self-defocusing phenomena are small and play a negligible role in image processing. These approximations turn out to be valid because the exciting laser intensity remains under a limit value (the corresponding threshold values). Thermo-optical effects are not significant when one is using ultrashort pulses in the picosecond range. In the third-order approximation of the nonlinear polarization, the electric field in the medium obeys the equation5 ≠S 2faS 1 s b 2 j gdjSj2 Sg , ≠z

(1)

(2)

(4)

where linear and nonlinear absorptions are characterized by coefficients a and b, respectively, and g is connected  1996 Optical Society of America

Boudebs et al.

Vol. 13, No. 7 / July 1996/ J. Opt. Soc. Am. B

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B. Object O(x, y) The object is composed of two slits with their large dimension parallel to y (Fig. 2). This large dimension is greater than the beam diameter; the object transmittance can be written as √ ! √ ! x 1 x0 x 2 x0 , 0 (11) 1 rect tsx, yd t sxd rect 2d 2d Fig. 1. 4-f System used as a coherent optical processor. A nonlinear material (NL) is placed in the Fourier-transform plane of this setup. L’s, lenses.

to nonlinear changes of the refractive index. Eq. (4) gives

Solution of

where 2x0 denotes the distance between the slits, 2d being their width. Thus, if Esx, yd is the electric field incident upon the input plane of the object, the electric field Osx, yd in the output plane is Osx, yd Esx, ydtsx, yd .

exps2aLd

T su, vd "

# 1/2 1 2 exps22aLd jSsu, vdj2 a #) " ( 1 2 exps22aLd g , ln 1 1 b jSsu, vdj2 3 exp j 2b a 11b

(5) where L is the nonlinear material thickness. Thus the transmittance T su, vd is a function not only of the medium parameters a, b, and g but also of the intensity jSsu, vdj2 entering the sample. When nonlinear absorption is sufficiently small, i.e., s byadjSsu, vdj2 f1 2 exps22aLdg ,, 1, Eq. (5) yields " # 1 2 exps22aLd 2 T su, vd exps2aLdexp j g jSsu, vdj . 2a

In the remainder of Section 2, calculation of the theoretical image will take account of the real intensity profile of the laser beam. To predict and explain the main features of the image, we assume in this section that, within the slits, the laser profile along y is a Gaussian function, but this condition is not necessary for the measurement: ! √ y2 Esx, yd E 0 s yd E00 exp 2 2 , (13) 2s where E00 is the value of electric field in x 6x0 , y 0, and s is the full width at half-maximum. The Fourier transform of Osx, yd is then p 2p s E00 4d coss2px0ud Ssu, vd lf 3 sincs2dudexps22p 2 s 2v2 d ,

(6) The dephasing term in Eq. (6) describes a nonlinear variation Dn of the refractive index: Dnsu, vd

l 1 2 exps22aLd g jSsu, vdj2 . 2pL 2a

(7)

For 2aL ,, 1, Eq. (7) yields Dnsu, vd

l gjSsu, vdj2. 2p

(8)

The electric field Ssu, vd is connected to the intensity I su, vd: I su, vd 2e0 ncjSsu, vdj2 ,

(9)

(12)

(14)

where sincsud sinspudypu. We see from Eqs. (14) and (9) that Ssu, vd induces a phase sinusoidal diffraction grating. We expect the appearance of diffracted orders in the image. The intensities of the diffracted orders become important with increasing input intensity I su, vd in the plane of the Fourier spectrum [Eq. (9)]. Thus, by measuring the diffraction efficiency of this grating, we can find the values of Dnsu, vd and n2 [see Eq. (10)]. The choice of a two-slit object with 2x0 .. 2d gives rise to a sinusoidal excitation in the material [with a sinc(2ud) envelope] of the same type as that used in two-wave mixing experiments. The choice of the full width at half-

where n is the usual (i.e., linear or low-intensity) refractive index. Thus the variation of the refractive index can be written as follows: Dnsu, vd n2 Isu, vd ,

(10)

with n2 slgdys4pe0 cnd (I is expressed in watts per square meter and n2 in square meters per watt). The method remains valid for absorbing materials: The use of Eq. (5) in transformation (2) leads to a value Iim sx, yd of the image intensity that depends on a, b, and g. If a (describing linear absorption) is known, the determination of b and g is possible by the method described below.

Fig. 2. beam.

Two-slit object.

The circle represents the incident light

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Boudebs et al.

maximum s ø 2x0 .. 2d permits the confinement of the spectrum in one direction. This situation is useful when one is investigating integrated planar optical waveguides and for a line-by-line inspection of inhomogeneous nonlinear solid materials. Another technique for creating a diffracting grating consists in taking two-point input sources.6 The Fourier transforms of these points are two plane waves that interfere to produce a grating in the nonlinear medium. One inconvenience of this method is that the spectrum is spread out on a large area, and then the optical power delivered by the laser is not focused to provide the largest possible intensity. C. Image Intensity Iim (x, y) As was shown in Subsection 2.A, Eqs. (2) and (3) for a transparent medium yield Iim sx, ydajFT

21

hSsu, vdexpf j wsu, vdgjj , 2

(15)

where the function w is defined as wsu, vd gLjSsu, vdj2 .

(16)

The determination of the nonlinear constant n2 slgdys4pe0 cnd can be deduced from a comparison between the experimental image Iim sx, yd and that calculated theoretically by relation (15) and Eq. (16). Several ways of making this comparison are possible. Only two methods are detailed in what follows: The first is based on two-dimensional numerical simulation without any approximation in relation (15). The second supposes the input intensity is sufficiently weak to permit a first-order development of the phase exponential and an analytical one-dimensional calculation. In the two-dimensional method we calculate the twodimensional Fourier transform Ssu, vd and the phase exponential in relation (15) numerically without approximation. The g value is varied from an initial trial value to yield the coincidence between the measured and the calculated intensity profiles Iim sx, 0d. In the one-dimensional method we take into account the fact that the width of the slits is sufficiently small with respect to the transverse size of the laser beam that the electric field can be considered constant along the x direction of the slits: √ ! x 1 x0 Osx, yd > Es2x0, ydrect 2d √ ! x 2 x0 1 Esx0 , ydrect . (17) 2d In addition, the phase variation wsu, vd is assumed to be small enough to permit the approximation expf j wsu, vdg > 1 1 j wsu, vd. Thus, if we take into account our experimental conditions in Subsection 3.B sl 532 nm, L 1 mmd, the range of validity of this technique remains limited to Dns0, 0d ,, ly2pL ø 1024 ; if n2 ø 10218 m2yW, the corresponding limit value required for the intensity is I s0, 0d ,, 1014 Wym2 . With these approximations the following simple analytical relations are then obtained, from which the refraction coefficient g can be deduced [inclusion of the pulse’s temporal variation and temporal integration of Eqs. (18)– (20)

are nevertheless necessary to yield the spatial distribution of the energy density in the object and image planes]: Iobj s6x0 , yd E 2s6x0 , yd , 2e0 c Iim s6x0 , yd E 2s6x0 , yd , 2e0 c √ !4 Iim s63x0 , yd 9 2d 2 g 2L2 g667 s yd , 2e0 c 16 lf

(18) (19) (20)

with grst s yd Esrx0 , yd ≠ Essx0 , yd ≠ Estx0, 2yd ,

(21)

where r, s, t 1 or 2; ≠ in Eq. (21) denotes the convolution operation. In Eqs. (18)– (20) Iobj s6x0 , yd is the intensity profile along y, at x 6x0 in the object plane. Similarly, Iim s6x0 , yd and Iim s63x0 , yd are the intensity profiles along y, at x 6x0 and x 63x0 , respectively, of the diffracted orders 0 and 61 in the image plane. For materials that show simultaneously nonlinear refraction and linear and nonlinear absorption, Eqs. (19) and (20) are replaced by the following relevant equations: ( √ !2 Iim s6x0 , yd 3 2d exps22aLd E 2 s6x0 , yd 2 2e0 c 2 lf

)

3 bLeff Es6x0, ydf g666 s yd 1 2g677 s ydg , √

9 2d Iim s63x0 , yd 2e0 c 16 lf

(190 )

!4 exps22aLd

2 3 s b 2 1 g 2dL2eff g667 s yd ,

(200 )

with Leff

3.

1 2 exps22aLd . 2a

SIMULATION AND EXPERIMENT

A. Simulation In what follows, we present the numerical results given by the two-dimensional method. When Eqs. (1) and (16) and relation (15) are used, the theoretically calculated image of the two-slit object in the presence of the nonlinear filter presents the following characteristics (Fig. 3): • Diffracted orders appear at positions x 6s2m 1 1dx0 (m an integer). The number of diffracted orders increases with the input intensity. • Intensity Iim60 ; Iim s6x0 , yd at the central slits (defined by m 0) decreases when lateral orders sm fi 0d appear. • Intensity Iim61 ; Iim s63x0 , yd on the first lateral orders sm 1d as a function of the input intensity in the center of the spectrum I(0, 0) is seen in Fig. 3: Iim61 increases to a maximum and then decreases in correlation with the appearance of the second-order diffracted beams sm 2d.

Boudebs et al.

Fig. 3. Simulation of the central line in the image of a two-slit object filtered by a nonlinear medium: evolution of the intensities of the first diffracted orders with the exciting intensity I(0, 0) in the spectrum.

All these characteristics found by simulation turn out to be in good agreement with experimental data obtained with available intensities given by our laser fIs0, 0d ø 1012 – 1013 Wym2 g. However, when nonlinear effects are important, new high spatial frequencies appear in the nonlinear medium, and an important overlap is observed in the image plane, so the simulated image is largely affected [relation (15) and Eq. (16)]. The validity of our model using the two-dimensional method is estimated to be in the range of Dns0, 0d , 1023 as long as the object is sampled over 512 points, corresponding to the pixel number of the CCD camera. If n2 ø 10218 m2yW, the limit value required for the intensity is I s0, 0d , 1015 Wym2 . B. Experiment The setup is presented in Fig. 4. Excitation is provided by a Nd:YAG laser delivering 25-ps single pulses at l 532 nm. The input intensity is varied by means of a filtering system (te) made of a half-wave plate and a Glan prism to keep a linear polarization. The exciting intensity in the center of the spectrum I(0, 0) at the entrance of the cell is in the range 1012 – 1013 Wym2 . When necessary, circular or linear input polarization is obtained with the help of quarter-wave plate ly4. The dimensions of the object Osx, yd are 2d 0.15 mm and 2x0 2.5 mm, and the focal distance of the two lenses is f 20 cm. Here the nonlinear material is a liquid placed in a cell 1 mm thick. The image receiver is a CCD camera manufactured by I2S (model IAC 500) used with a fixed gain and a linear video signal. Because the characteristics of this camera are given for continuous excitation, calibration is necessary in the picosecond range. The camera is connected to a frame grabber manufactured by MATROX, inserted into a 386– 33-MHz computer. To monitor the entrance intensity distribution for each laser shot, we set prismatic plate P1 after the object to take a part of the exciting beam. An image of the slits before processing is thus directly acquired on the camera through lens L3; we call it the reference image. By a two-dimensional numerical Fourier transform of this image we can determine precisely the intensity for each point of the spectrum, particularly the intensity at the origin I(0, 0). With the help of an aperture (sf ), the


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diffracted order desired (the first in the present measurement) is selected. The diffracted-order image is synchronized with the reference image, and a comparison of their intensities is made. Neutral filters (to and tf ) are placed, respectively, before the diffracted order and before the reference image. They are necessary for the camera to be used in its linear range. Obtaining the absolute value of the nonlinear constants n2 and g [Eqs. (10)– (20)] requires absolute calibration of the camera. The absolute value is realized by comparison with a calibrated joulemeter. We take into account the defects of the camera that are due to the nonlinearity of its response. The pixels of the camera present 256 gray levels. For one image the sum of gray levels in all the pixels is proportional to the energy received by the camera. For a pulse duration of 25 ps the mean value of the measured response per pixel is 1.13 3 1025 Wygray level.

4.

RESULTS

The image obtained with a CCD camera representing the first diffracted order Iim61 sx, yd and the reference input image of the two slits is shown in Fig. 5. A comparison of the intensity Iim61 sx, y 0d in the first order with the reference image (see Subsection 3.B) is made for each

Fig. 4. Schematic of the experimental setup: NL, nonlinear material; L1 – L3, lenses; P1 , P2 , prismatic plates; M1 , M2 , mirrors; te, to, sf, tf, filters defined in text.

Fig. 5. Experimental images of a two-slit object filtered by a CS2 cell obtained in one laser shot. The reference input image ( between 100 and 250 on the abscissa) and the first lateral order sm 21, between 0 and 50 on the abscissa) are obtained with different values of filters tf and to.

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Table 1. Measured Value of the Nonlinear Refractive-Index Constant of CS2 in Linear Polarization Measurement Method

Experimental Conditions

One-wave method: self-focusing of the beamb One-wave method: Z scanb Two-wave method: interferometryc Two wave method: Jamin interferometryc Two-wave method: optical Kerr effectc Two-wave method: optical Kerr effectc This method: two-wave method

ps, 532 nm, L 1 cm, I 2.3 GWycm2 ps, 532 nm, L 1 mm, I 2.6 GWycm2 ps, 532 nm, probe 570 nm pump and probe, ps, 532 nm, L 2.4 cm, I 1 GWycm2 ns, 694 nm, probe 488 nm ps, 1060 nm, probe 532 nm L 1 cm ps, 532 nm, L 1 mm, I s0, 0d 2.5 GWycm2

CS2

n2

3 10218 m2yW

a

Reference

3.76

7

3.01

8

4.04

9

3.85

10

3.38

11

5.02

12

2.45

—

a When

necessary, a conversion to SI units has been made. One-wave method: Only one excitating wave is used. The refractive-index change is directly measured for this wave. Two-wave method: Excitation is provided by an intense beam. The refractive-index change induced on a probe wave is measured.

b c

Table 2. Absolute and Relative Values of n2 for Different Materials Absolute Value

Relative Values Referenced to CS2 Given by Different Authors Referencea

Materials

310218 m2yW (This Work)

This Work

10

11

12

CS2 C6 H5 NO2 C7 H8 CH3 COCH3

2.45 1.32 0.70 0.10

1 0.53 0.28 0.04

1 0.21 0.06 0.02

1 0.78 0.27 0.04

1 1.32 0.33 0.04

a The

experimental conditions for measurements in Refs. 10 – 12 are summarized in Table 1.

laser shot. The values of Iim61 sx, y 0d and Iobj (intensity of the reference image) are measured simultaneously. Diffracted orders have intensities much smaller than do central orders, usually Iim61 ø 1023 3 Iim60 . We vary the input intensity over a wide range to check the validity of the linear relation defined by Eq. (10). We examined the nonlinear refraction of carbon disulfide (CS2 ), which is often used as a standard reference nonlinear material, and the nonlinear refraction of several other nonlinear liquids (Tables 1 and 2). When CS2 is excited by a linearly polarized light, we check the linear dependence of the refractive-index change Dns0, 0d on the entrance intensity I s0, 0d (Fig. 6). The absolute value obtained for the constant n2 is 2.45 3 10218 m2yW (deduced from the slope of the linear regression line). Both of the two developed methods presented in Subsection 2.C lead to the same experimental results. The dominant mechanism for nonlinear refraction with picosecond pulses is the molecular reorientational Kerr effect; the slow effects, such as electrostriction and thermal nonlinearity, are not significant. A comparison with other measurements is made in Table 1. A large dispersion of the results is observed. Within intensity calibration errors, our value (rather smaller than others) is in agreement with the value obtained by the Z-scan technique.8 Measurement with a photodiode, which was done in almost all the studies cited so far in this paper, usually assumes use of a Gaussian spatial profile of the laser beam to yield the peak-on-axis intensity of the laser beam. It is clear that departures from this Gaussian profile lead to errors that can partly explain the dispersion observed in the reported values

of n2 . On the contrary, our method does not assume a particular form of the spatial profile and takes account of the real one, sampled by means of the CCD camera. In addition to these errors, the possible presence (in some experiments) of self-focusing and a strong large spectral-bandwidth-stimulated Rayleigh wing scattering,13 leading to a depletion of the pump, can also explain the dispersion of the results.

Fig. 6. Refractive-index change Dns0, 0d in the center of the Fourier spectrum in a 1-mm-long cell of CS2 , versus corresponding input intensity I s0, 0d.

Boudebs et al.

The calibration errors do not affect relative measurements. The relative values of n2 for different liquids have been measured and are listed in Table 2. They are compared with values obtained by other methods. The observed dispersion is relatively large. Besides the reasons given above, it can probably be explained by systematic errors that are due to the methods adopted; in particular, nonlinear absorption (in C6 H5 NO2 ) and Rayleigh wing scattering (in CS2 ), which may be strong in some experiments and are not taken into account in the models, can lead to errors.13,14 However, our experimental results for classical nonlinear liquid materials are consistent with those from previous measurements (Table 2).

5. COMPARISON OF THIS METHOD WITH OTHER TECHNIQUES Other, similar techniques of direct measurement of the nonlinear parameters exist7,8 ; it is therefore useful to make a careful comparison with a representative one, such as the Z-scan technique.8 The two methods are based on the principles of spatial distortions undergone by a laser beam focused into a nonlinear sample. The approximation that we used to model the propagation of light are the same: paraxial approximation of the transfer function representing the effects of the propagation outside the sample, the slowly varying envelope approximation, and neglect of diffraction inside a sample regarded as thin. The similarity between the models used to describe propagation inside the sample is clearly shown by the identity of the expression of their nonlinear transmittance, T su, vd [Eq. (5)]. The differences come from the use of the two-slit object, which leads to several interesting characteristics. In the Z-scan technique (ZST) the laser beam is directly focused into the sample. In the Zernike imaging technique (ZIT) the laser beam is focused after passing through a Young two-slit object; interference of the two beams inside the sample creates a phase or an amplitude grating or both in the nonlinear medium. The self-diffraction of the incident light creates new beams, resulting in lateral diffracted orders in the image plane. Thus one laser shot (of the correct intensity) is sufficient to permit us to observe nonlinearities and measure the nonlinear parameters from the characteristics of the lateral orders; it is also possible to follow the evolution of the value n2 during the processing of a nonlinear material to improve, in real time, the quality of the manufacturing. However, finding and measuring nonlinearities requires several laser shots, as the sample has to be moved along path z of the beam to yield the Z-scan transmittance curve. Besides its simplicity (no displacement of the sample, direct evidence of nonlinearities), the ZIT therefore has the advantage of lower sensitivity to the statistical fluctuation of the laser intensity because one laser shot is sufficient to yield one value of the nonlinear parameters. Another reason for the lower sensitivity of the ZIT to the laser fluctuations comes from the fact that each measurement takes account of the real spatial profile of each laser shot, whereas the validity of the results obtained with the ZST requires an excellent Gaussian mode.


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The Z-scan technique has the advantage of giving the sign of n2 . The choice of an amplitude object Osx, yd in this study has the inconvenience of leading to an image intensity that is independent of the sign of n2 . It follows from relation (15) and Eq. (16) that the image intensity becomes sensible to the sign of n2 if we choose an appropriate phase object. It must be added that there is a qualitatively simpler method to determine the sign of n2 by observing self-focusing sn2 . 0d or self-defocusing sn2 , 0d of the laser beam in the nonlinear material.15 Finally, we point out another interesting feature of the ZIT. As the Fourier spectrum of the two-slit object is confined along one axis — and is not circularly invariant as in the ZST—one can use this characteristic to inspect along different directions the nonlinear properties of anisotropic crystals. This feature, which is considered here for liquids excited by strong laser pulses, can also be used to study the anisotropic properties of photorefractive materials illuminated by cw low-intensity beams.16

6.

CONCLUSION

Nonlinear Zernike imaging provides an interesting new method to measure nonlinear optical characteristics of materials. We have shown that the change in the nonlinear refractive index in a material can be measured by means of a simple setup, made of a two-lens system and a CCD camera linked to a computer. The alignment is simple, and the sensitivity of the method is good, as proved by the measurement performed in a cell filled with 1-mm thick acetone, which is a medium displaying a Kerr constant near 10219 m2yW. Besides being simple (no displacement of the nonlinear medium, direct evidence of nonlinearities, one single shot for estimating n2 , lower sensitivity to statistical fluctuations of the laser beam, and accurate measurement of the intensity by means of a CCD), the method permits us to choose an object so we can inspect along different directions the nonlinear properties of anisotropic crystals. Finally, we point out that this technique has been tested here only with Young two-slit object. It is clear that one can consider other objects to discover improvements of the method or to adjust this technique to particular characteristics of the nonlinear sample.

ACKNOWLEDGMENTS We are indebted to G. Rivoire for valuable and stimulating discussions and to R. Chevalier for his technical help. * G. Boudebs’ e-mail address is [email protected].

REFERENCES 1. G. Bruhat, Optique (Masson, Paris, France, 1992), Chap. XIV, p. 326. 2. L. Pichon and J.-P. Huignard, “Dynamic joint-Fouriertransform correlator by Bragg diffraction in photorefractive Bi12 SiO20 crystals,” Opt. Commun. 36, 277 – 287 (1981). 3. N. Phu Xuan, J. L. Ferrier, J. Gazengel, G. Rivoire, G. L. Brekhovskhikh, A. D. Kudriavtseva, A. I. Sokolovskaia, and N. V. Tcherniega, “Changes in the space structures of light beams induced by nonlinear optical phenomena: application to phase contrast and image processing,” Opt. Commun. 68, 244 – 250 (1990). 4. J. W. Goodman, Introduction to Fourier Optics (McGrawHill, New York, 1968), Chap. 5, pp. 77 – 89.

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5. Y. R. Shen, Nonlinear Optics (Wiley, New York, 1984), Chap. 3, pp. 42 – 50. 6. R. L. Townsend and J. T. LaMacchia, J. Appl. Phys. 41, 5188 – 5195 (1970). 7. W. E. Williams, M. J. Soileau, and E. W. Van Stryland, “Optical switching and n2 measurements in CS2 ,” Opt. Commun. 50, 256 – 260 (1984). 8. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. Styland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 – 769 (1990). 9. Y. A. Volkova, V. A. Zamkoff, and L. U. Nalbandov, Opt. Spectroscp. (USSR) 30, 300 – 311 (1971). 10. N. P. Xuan, J. L. Ferrier, J. Gazengel, and G. Rivoire, “Picosecond measurements of the third order susceptibility tensor in liquids,” Opt. Commun. 51, 433 – 437 (1984). 11. M. Paillette, “Recherche exp´erimentales sur les effets Kerr induits par une onde lumineuse,” Ann. Phys. 4, 617 – 712 (1969).

Boudebs et al. 12. P. P. Ho and R. R. Alfano, “Optical Kerr effect in liquids,” Phys. Rev. A 20, 2170 – 2187 (1979). 13. D. Wang and G. Rivoire, “Large spectral bandwidth stimulated Rayleigh-wing scattering in CS2 ,” Chem. Phys. 98, 9279 – 9283 (1993). 14. A. Fahmi, J. P. Bourdin, R. Chevalier, X. N. Phu, and G. Rivoire, “Influence of stimulated Rayleigh wing scattering in x 3 measurements and wave mixing experiments in CS2 ,” Nonlin. Opt. 12, 165 – 178 (1995). 15. B. Sahraoui, M. Sylla, J. P. Bourdin, G. Rivoire, and J. Zaremba, “Third order nonlinear optical properties of ehylenic tetrathiafulvalene derivaties,” J. Mod. Opt. 42, 2095 – 2107 (1995). 16. G. Boudebs, N. P. Xuan, J. Gazengrl, J. P. Lecoq, and G. Rivoire, “Etude du filtre spatiale non lin´eaire auto-induit dans Fe:NbLiO3 ,” in Opto 91 (Edition Scientifique Internationale, Paris, 1991), p. 80.