APPLIED PHYSICS LETTERS 93, 021118 共2008兲
Degenerate multiwave mixing using Z-scan technique G. Boudebs,a兲 K. Fedus, C. Cassagne, and H. Leblond Laboratoire des Propriétés Optiques des Matériaux et Applications-FRE CNRS 2988, Université d’Angers, 2 Boulevard Lavoisier, 49045 Angers Cedex 01, France
共Received 5 June 2008; accepted 24 June 2008; published online 17 July 2008兲 This study deals with a multiwave mixing experiment using Z-scan technique. We show that it is possible to simplify the complex degenerate four-wave mixing 共DFWM兲 experimental setup in order to characterize nonlinear materials. One object composed of three circular apertures is used as an input in a 4f coherent imaging system. When the nonlinear material is moved around the focus, we observe the variations of the forward fourth wave at the output as well as other diffracted waves. Experimental and simulated images are presented here to validate our approach. We show also that this method increases significantly the sensitivity of the measurement compared to DFWM. We provide a simple quadratic relation that allows the characterization of the cubic optical nonlinearity. © 2008 American Institute of Physics. 关DOI: 10.1063/1.2960336兴 The degenerate four-wave mixing is a well known method to characterize nonlinear optical properties. Generally, the phase conjugation properties of an incident wave on a counterpropagating two pump beam configuration are exploited.1 It is potentially a sensitive method, but requires a relatively complex experimental apparatus. To simplify the experimental procedure for nonlinear characterization, Sheik-Bahae et al.2 proposed a single beam Z-scan method Later, eclipsed Z-scan 共EZ-scan兲 was developed due to its more sensitive strioscopic properties.3 On the other hand, we have reported a one laser shot measurement technique using different objects 共circular and phase objects兲 at the entry of a 4f coherent imaging system to characterize the nonlinear refractive index of materials placed in the Fourier plane of the setup.4,5 In this method 共see Fig. 1兲, called nonlinear imaging technique 共NIT兲, we studied the Fraunhofer diffracted image intensity profiles. This 4f configuration has been combined6 with Z-scan technique to compare directly the sensitivity of both methods and to obtain the characterization of the nonlinear refraction in the presence of a relatively high nonlinear absorption. In this paper, we propose a three-circularaperture object eclipsed by an adapted spatial filter in the image plane allowing the combination of three different methods: 共i兲 Z-scan and its derivative EZ-scan, 共ii兲 NIT in a 4f setup, and 共iii兲 the forward degenerate four-wave mixing 共DFWM兲. We will show that such configuration offers simplicity of alignment as well as a relatively high sensitivity. The experimental acquisitions were fitted by a simple theoretical model based on Fourier optics. It is assumed that scalar diffraction theory is sufficient to describe image formation using 4f system. We briefly recall the theoretical model we use 共see Refs. 4 and 7–9 for more details兲. A two-dimensional object 关Fig. 2共a兲兴 is illuminated at normal incidence by a linearly polarized monochromatic plane wave 关defined by E = E0共t兲exp关−j共t − kz兲兴 + c.c., where is the angular frequency, k is the wave vector, and E0共t兲 is the amplitude of the electric field containing the temporal envelope of the laser pulse兴 delivered by a pulsed laser. Using the slowly varying envelope approximation to describe the propagation of the electric field in the nonlinear
medium10 and since we are concerned with the image intensity, the temporal terms will be omitted. Moreover, thermooptical effects are not significant when one is using ultrashort pulses in the picosecond range 共the full width at halfmaximum time is 17 ps兲 and low repetition rate 共10 Hz兲. For an object with a circular symmetry, one can use the Fresnel–Bessel transformation as in Ref. 6 to propagate the beam. In our case the object has no circular symmetry: we use three circular apertures located at three corners of a square 关see Fig. 2共a兲兴. The general scheme of this propagation is summarized hereafter. If the transmittance of the object 共see Fig. 2兲 is t共x , y兲 = CR共x + d , y − d兲 + CR共x − d , y − d兲 + CR共x − d , y + d兲, the amplitude of the field just behind the plane where the apertures are placed is O共x , y兲 = E · t共x , y兲. Here, the circular function CR共x , y兲 is defined as equal to 1 if the radius 冑x2 + y 2 is less than R and zero elsewhere 共R is the radius of the circular apertures and 2d is the distance between their centers兲. Let S共u , v兲 be the spatial spectrum of O共x , y兲:
Author to whom correspondence should be addressed. Tel.: 共33兲 共0兲2.41.73.54.26. FAX: 共33兲 共0兲2.41.73.52.16. Electronic mail:
[email protected].
FIG. 1. Schematic of the 4f coherent system imager. The sample is located in the focal region. The labels refer to the object O共x , y兲, lenses 共L1 – L3兲, mirrors 共M1 , M2兲, beam splitters 共BS1, BS2兲, and spatial filters 共sf兲.
a兲
冕 冕 −⬁
˜ 关O共x,y兲兴 = S共u, v兲 = F
−⬁
−⬁
O共x,y兲
−⬁
⫻exp关− j2共ux + vy兲兴dxdy, ˜ denotes the Fourier transform operation, while u where F and v are the normalized spatial frequencies. Instead of
0003-6951/2008/93共2兲/021118/3/$23.00 93, 021118-1 © 2008 American Institute of Physics Author complimentary copy. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
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FIG. 2. 共a兲 The three circular apertures object. 共b兲 and 共c兲 Two different spatial filters placed in the image plane stopping the geometrical images of the circular apertures.
propagating the field in spatial domain and in order to reduce the computing time, we chose to propagate the spectrum of the object over a distance z⬘ by taking into account the transfer function of the wave propagation phenomenon 共see Chap. 3 in Ref. 7兲 H共u , v兲 = exp共j2z⬘冑1 − 共u兲2 − 共v兲2 / 兲, where is the wavelength. The field amplitude after the free propagation is obtained by computing the inverse Fourier trans˜ −1关S共u , v兲H共u , v兲兴. To calculate the output form O共x , y , z⬘兲 = F beam after passing through a lens of focal f, we apply the phase transformation related to its thickness variation: tL共x , y兲 = exp关−j共x2 + y 2兲 / f兴. The first propagation is performed on a distance z⬘ = f 1, where the beam enter lens L1 and tL is applied with f = f 1. Then we propagate the beam up to the sample located at z using z⬘ = f 1 + z in H, which is the optical transfer function 共z = 0 at the focus of the lens L1兲. The nonlinear response of the material is taken into account using T共u , v , z兲 given below in Eq. 共1兲. Next, we perform propagation on a distance z⬘ = f 2 − z, a phase transformation due to lens L2, and the final diffraction is calculated with z⬘ = f 2 at the output of the 4f system. The image intensity Iim is calculated in this plane taking into account the transmittance of the spatial filter. For the filter in Fig. 2共b兲 composed of a circular aperture situated at the position where a diffracted fourth wave should appear 共induced by the nonlinear regime兲, this transmittance is given by sfb共x , y兲 = C2R共x + d , y + d兲. The radius of this filter is two times larger than the radius of the object in order to receive all the energy diffracted through the circular aperture, while the filter in Fig. 2共c兲 is defined by three disks 共circular stop functions兲 with a radius two times larger than the geometrical images of the three apertures composing the object. Such a large opaque filter allows an easy alignment in the image plane: sfc = 1 − C2R共x + d , y − d兲 − C2R共x − d , y − d兲 − C2R共x − d , y + d兲. To simplify the problem we consider the particular case of a lossless Kerr material 共CS2兲 characterized by a cubic nonlinearity defined by n2, the nonlinear index coefficient. For samples considered as thin, the complex field at the exit face of the sample SL, can be written11 T共u, v,z兲 =
SL共u, v, f 1 + z兲 = exp关j2n2LI共u, v,z兲/兴, S共u, v, f 1 + z兲
共1兲
where S is the amplitude field at the entry, L is the sample thickness, and I共u , v , z兲 denotes the intensity of the laser
FIG. 3. Comparison between 共a兲 the experimental acquisition and 共b兲 the numerically calculated one of the images obtained at z = 0 for NL0 = 0.34. The “L” pattern appearing in 共a兲 is the spatial filter defined in Fig. 2共c兲. The coordinates x and y are expressed in number of pixels, and the natural logarithm is used to enhance lower intensity.
beam within the sample 关proportional to 兩S共u , v , f 1 + z兲兩2兴. It is generally assumed2 that Eq. 共1兲 remains valid up to a maximum induced nonlinear dephasing NL0 = 2n2LI0 / less than 共I0 being the on-axis peak intensity at the focus兲. Excitation is provided by a Nd:YAG laser 共Continuum兲 delivering 17 ps single pulses at = 1.064 m with 10 Hz repetition rate. The input intensity is varied by means of a half-wave plate and a Glan prism in order to maintain linear polarization. A beam splitter at the entry of the setup 共Fig. 1兲 permits to monitor any fluctuation occurring in the incident laser beam. Other experimental parameters are f 1 = f 2 = 20 cm 共focal length of lens L1 and L2兲. The object with three circular apertures 关see Fig. 2共a兲兴 is placed in the front focal plane of lens L1. The radius of the apertures is R = 0.45 mm and the half of the distance between their central points is d = 1 mm. The latter is small compared to the beam waist of the incident laser beam 共1 cm兲. The image receiver is a cooled charge coupled device camera 共−30 ° C兲 with 1000⫻ 1018 pixels, each of which is 12⫻ 12 m2. The camera pixels have 4095 gray levels. The comparison between the experimental nonlinearly filtered image and its numerical simulation is shown in Fig. 3. Physically, in the focal plane region of lens L1, the intensity distribution pattern creates a combination of circular and sinusoidal gratings in the nonlinear material. For an instantaneous response of the medium, the self-diffracted spectrum on this induced pattern will generate diffracted beams at the output of the sample. Figure 3共a兲 is the natural logarithm of the acquisition in the image plane in the presence of the
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12兲. Note that the diffraction efficiency related to all the diffracted beams is five times higher than for one intense diffrated wave. Therefore, the signal to noise ratio will be enhanced in n2 measurement experiments. We checked by numerical simulation that is independent of the geometrical parameters characterizing the object 共R and d兲. It is natural to take into account this parameter to determine n2. Based on numerical fitting and assuming a relatively low nonlinearity 共NL0 ⬍ 1兲, we found a simple quadratic relationship relating the efficiency of all the diffracted waves to the maximum of the nonlinear dephasing: 2 = 2.41 ⫻ 共NL0 ⫻ 10−2兲.
FIG. 4. Experimental 共stars: all the diffracted waves, points: only one diffracted beam兲 and theoretical 共solid line兲 efficiency of the diffracted energy versus z; the sample position for I0 = 1.9 GW/ cm2 in 1 mm CS2 at 1064 nm 共NL0 = 0.34兲. The inset shows the energy diffraction efficiency vs I共z兲, the on-axis incident intensity for z, where z is a given sample position.
spatial filter 关Fig. 2共c兲兴 stopping the geometrical image of the apertures. A 1 m thick fused silica cell filled with CS2 was placed in the back focal plane of L1 at z = 0. The on-axis peak intensity of these three interfering beams was I0 = 1.7GW/ cm2, giving NL0 = 0.34 共I0 is measured after calibration using classical Z-scan and by considering the value of n2 = 3 ⫻ 10−18m2 / W for CS2 given in Ref. 2兲. Each circular beam has an Airy radius 0f = 288 m large enough to consider the corresponding Rayleigh range as much higher than the sample thickness. Figure 3共b兲 shows the simulation image 共natural logarithm of the intensity兲 obtained with the same experimental parameters. The good agreement between these two images validates our model and the corresponding numerical simulation. We distinguish nine intense diffracted waves in the acquisition. Typically, for NL0 = 0.34, the efficiency of the diffracted energy is about 6 ⫻ 10−4 for three of them 关see Fig. 3共b兲兴 and six times less for the other six 共four times less considering the intensity efficiency兲. This efficiency is comparable with the classical DFWM obtained with the fourth conjugated wave. Two sets of Z-scan acquisitions are carried out for two kinds of spatial filters placed just in front of the camera: the first one stops all of the beams except the forward fourth diffracted wave, as shown in Fig. 2共b兲, and the second one blocks just the geometrical image of the object 关see Fig. 2共c兲兴 in order to acquire the image of all the diffracted beams. We can see in Fig. 4 the result of the experimental acquisitions as well as the related numerical simulation. For each z position of the sample we have calculated the diffracted energy by summing the intensity diffracted through the spatial filters over all the pixels:Ed = /Iim共x , y兲sfm共x , y兲dxdy, where m defines the filter shape 共sfb or sfc兲. This was done after calibration using one of the diffracted beams. This diffracted energy was divided by the incident one calculating , the diffraction efficiency. We can see a perfect agreement between theory and experimental data. The inset of Fig. 4 shows the numerical evolution of the diffracted energy versus I共z兲, the on-axis incident intensity at z, where z is a given sample position. A similar variation can be found in the classical DFWM with the conjugated diffracted fourth wave 共see, for example, Ref.
共2兲
For instance, one intense forward diffracted beam gives 2 = 5.13共NL0 ⫻ 10−3兲. The sensitivity2 is generally defined as the slope of the curve giving the signal 共兲 vs NL0. By using all the diffracted waves, one can see that the sensitivity is enhanced by a factor 2.41/ 0.513⬇ 5. One of the most important advantages of this technique is the simplicity of the optical alignment compared to the classical DFWM. For nonlinear characterization, it is not necessary to obtain the conjugate fourth wave anymore. Any diffracted wave can be used to obtain the same result. It should be added that, as with the classical DFWM, the inconvenience of this method is that we are not able to separate measurement of the nonlinear absorption and the nonlinear refraction. Indeed, the signal is sensitive only to the modulus of the third order susceptibility.12 When compared to EZscan, an advantage of this method is that it does not require a perfect Gaussian incident beam. As for EZ-scan, the enhancement of the sensitivity will come at the expense of a reduction in accuracy even if we do not have to measure here the linear transmittance of the spatial filter. The use of this method with a reference material for calibration is recommended. For a specimen having a very good optical quality, NL0 is obtained using Eq. 共2兲 with high precision 共less than 1%兲. The other uncertainties 共thickness and wavelength兲 are negligible. If one considers that the value of n2 for CS2 given by Ref. 2 is correct, the main source of uncertainty comes from the joulemeter energy measurement 共⫾10% 兲. In summary, we have demonstrated that in one optical 4f setup it is possible to combine NIT,5 Z-scan,2 EZ-scan,3 and DFWM.1 By matching the spatial filter at the output with the object at the entry 共to stop the geometrical image兲, the multiwave mixing considered here and EZ-scan can be seen as two particular cases of the same nonlinear imaging strioscopic effect.
A. Yariv and D. M. Pepper, Opt. Lett. 1, 16 共1977兲. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. Hagan, and E. W. Stryland, IEEE J. Quantum Electron. 26, 760 共1990兲. 3 T. Xia, D. I. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, Opt. Lett. 19, 317 共1994兲. 4 S. Cherukulappurath, G. Boudebs, and A. Monteil, J. Opt. Soc. Am. B 21, 273 共2004兲. 5 G. Boudebs and S. Cherukulappurath, Phys. Rev. A 69, 053813 共2004兲. 6 G. Boudebs and S. Cherukulappurath, Opt. Commun. 250, 416 共2005兲. 7 J. W. Goodman, Introduction to Fourier Optics, 2nd ed. 共McGraw-Hill, New York, 1996兲. 8 G. Boudebs, M. Chis, and J. P. Bourdin, Opt. Commun. 13, 1450 共1996兲. 9 G. Boudebs, M. Chis, and A. Monteil, Opt. Commun. 150, 287 共1998兲. 10 Y. R. Shen, Nonlinear Optics 共Wiley, New York, 1984兲, Chap. 3, pp. 42–50. 11 J. A. Hermann, J. Opt. Soc. Am. B 1, 729 共1984兲. 12 R. W. Boyd, Nonlinear Optics 共Academic, New York, 1992兲, Chap. 6, p. 251. Author complimentary copy. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp 1 2
