Degenerate multiwave mixing using Z-scan technique

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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 4fcoherent 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. Gener- ally, the phase conjugation properties of an incident wave on a counterpropagating two pump beam configuration are exploited.¹It is potentially a sensitive method, but requires a relatively complex experimental apparatus. To simplify the experimental procedure for nonlinear characterization, Sheik-Bahae et al.²proposed a single beam Z-scan method Later, eclipsed Z-scan 共EZ-scan兲 was developed due to its more sensitive strioscopic properties.³On 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 re- fractive index of materials placed in the Fourier plane of the setup.^4,5In 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-circular- aperture 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 and7–9for 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 byE=E₀共t兲exp关−j共␻^t−kz兲兴+ c.c., where

␻is the angular frequency,kis the wave vector, andE₀共t兲is the amplitude of the electric field containing the temporal envelope of the laser pulse兴delivered by a pulsed laser. Us- ing the slowly varying envelope approximation to describe the propagation of the electric field in the nonlinear

medium¹⁰and since we are concerned with the image intensity, the temporal terms will be omitted. Moreover, thermo- optical effects are not significant when one is using ultrashort pulses in the picosecond range 共the full width at half- maximum 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=C_R共x+d,y−d兲+C_R共x−d,y−d兲 +C_R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 functionC_R共x,y兲is defined as equal to 1 if the radius

冑

x²+y²is less thanRand zero elsewhere共Ris 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:

S共u,v兲=F˜关O共x,y兲兴=

冕

−⬁

冕

_−⬁^−⬁^O^共^x,y^兲

⫻exp关−j2␲共ux+vy兲兴dxdy,

where F˜ denotes the Fourier transform operation, while u and v are the normalized spatial frequencies. Instead of

a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:

georges.boudebs@univ-angers.fr.

FIG. 1. Schematic of the 4fcoherent system imager. The sample is located in the focal region. The labels refer to the objectO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. APPLIED PHYSICS LETTERS93, 021118共2008兲

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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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 distancez⬘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j2␲^z⬘

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- formO共x,y,z⬘^兲⁼^F^˜⁻¹^关S共u^,^v兲H共u^,v兲兴. To calculate the output beam after passing through a lens of focal f, we apply the phase transformation related to its thickness variation:

t_L共x,y兲= exp关−j␲共x²+y²兲/␭f兴.

The first propagation is performed on a distancez⬘⁼^f1, where the beam enter lens L₁ andt_L is applied with f=f₁. Then we propagate the beam up to the sample located at z using z⬘⁼^f1+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 usingT共u,v,z兲 given be- low in Eq.共1兲. Next, we perform propagation on a distance z⬘⁼^f2−z, a phase transformation due to lens L₂, and the final diffraction is calculated with z⬘⁼^f2 at the output of the 4f system. The image intensity I_im 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 sf_b共x,y兲=C_2R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 −C_2R共x+d,y−d兲−C_2R共x−d,y−d兲−C_2R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 sampleS_L, can be written¹¹

T共u,v,z兲=SL共u,v,f₁+z兲

S共u,v,f1+z兲 = exp关j2␲ⁿ2LI共u,v,z兲/␭兴, 共1兲 whereS 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

beam within the sample关proportional to 兩S共u,v,f₁+z兲兩²兴. It is generally assumed² that Eq. 共1兲 remains valid up to a maximum induced nonlinear dephasing ␸NL0= 2␲ⁿ2LI₀/␭

less than␲共I0being 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 f₁=f₂

= 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 isd= 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 instan- taneous response of the medium, the self-diffracted spectrum on this induced pattern will generate diffracted beams at the output of the sample. Figure3共a兲is the natural logarithm of the acquisition in the image plane in the presence of the

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.

FIG. 3. Comparison between共a兲the experimental acquisition and共b兲the numerically calculated one of the images obtained atz= 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.

021118-2 Boudebset al. Appl. Phys. Lett.93, 021118共2008兲

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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 CS₂was placed in the back focal plane of L₁atz= 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 ofn₂= 3⫻10⁻¹⁸m²/W for CS₂given 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. Figure3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⫻10⁻⁴for 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 eachzposi- tion of the sample we have calculated the diffracted energy by summing the intensity diffracted through the spatial filters over all the pixels:E_d=/I_im共x,y兲sf_m共x,y兲dxdy, wherem de- fines the filter shape共sfbor 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.4shows the numerical evolution of the diffracted energy versusI共z兲, the on-axis incident intensity atz, wherezis 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.

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Randd兲. It is natural to take into account this parameter to determinen₂. Based on numerical fitting and assuming a relatively low nonlinear- ity共␸NL0⬍1兲, we found a simple quadratic relationship re- lating the efficiency of all the diffracted waves to the maximum of the nonlinear dephasing:

␩^{= 2.41}⫻共␸NL02 ⫻10⁻²兲. 共2兲 For instance, one intense forward diffracted beam gives

␩= 5.13共␸NL02 ⫻10⁻³兲. The sensitivity²is 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 in- convenience 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 EZ- scan, an advantage of this method is that it does not require a perfect Gaussian incident beam. As for EZ-scan, the en- hancement 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 recom- mended. For a specimen having a very good optical quality,

␸NL0is 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 ofn₂for CS₂given 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,⁵Z-scan,²EZ-scan,³and DFWM.¹By 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.

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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 versusz; the sample position forI₀= 1.9 GW/cm²in 1 mm CS₂at 1064 nm 共␸NL0= 0.34兲. The inset shows the energy diffraction efficiency vsI共z兲, the on-axis incident intensity forz, wherezis a given sample position.

021118-3 Boudebset al. Appl. Phys. Lett.93, 021118共2008兲

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp