Absolute measurement of the nonlinear refractive indices of reference materials

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Absolute measurement of the nonlinear refractive indices of reference materials

Georges Boudebs and Kamil Fedus

Citation: Journal of Applied Physics 105, 103106 (2009); doi: 10.1063/1.3129680 View online: http://dx.doi.org/10.1063/1.3129680

View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/105/10?ver=pdfcov Published by the AIP Publishing

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Absolute measurement of the nonlinear refractive indices of reference materials

Georges Boudebs^a兲 and Kamil Fedus^b兲

Laboratoire des Propriétés Optiques des Matériaux et Applications, FRE CNRS 2988, Université d’Angers, 2 Boulevard Lavoisier, 49045 Angers Cedex 01, France

共Received 23 January 2009; accepted 8 April 2009; published online 21 May 2009兲

We report absolute measurements of the nonlinear refractive index on carbon disulfide共CS₂兲 and fused silica. These materials are commonly used as standard references in nonlinear optical experiments. To obtain more accurate values than those usually used, we have combined the z-scan method inside a 4-f imaging system共in order to analyze the spatial distortion of the diffracted pump beam兲with the “Kerr shutter” experiment共to evaluate the temporal pulse width durations for three different wavelengths such as 1064, 532, and 355 nm兲. We obtained surprisinglyn₂values one order of magnitude less than the one usually taken into account in the picosecond regime and a more significant dispersion of the nonlinear refraction index. Experimental and simulated Z-scan transmittance profiles as well as acquired autocorrelation functions in the Kerr-gating experiments are presented here in order to validate our measurements. © 2009 American Institute of Physics.

关DOI:10.1063/1.3129680兴

I. INTRODUCTION

Carbone disulfide共CS₂兲 and fused silica are two standard references materials in nonlinear experiments. Both of these optical-Kerr media have negligible nonlinear absorption in the range of the low third order optical nonlinearities.

CS₂ is generally used to calibrate the incident intensities in the experimental setup for characterizing materials having relatively high nonlinear refraction index 共n₂兲共of about 10⁻¹⁸ m²/W兲. Fused silica is used for materials with nonlinearities of two orders of magnitude less. Nowadays, the ab- soluten₂measurements are rare. We understand that the order of magnitude is sufficient sometimes to compare nonlinearities together, but more precise and accurate absolute values will allow us to better understand the physical properties of nonlinear phenomena. The absolute values in glasses are also of interest for high-power laser because of their wide use as optical components inside the system.

When a physical quantity is investigated it is necessary to make as many measurements as possible with different experimental parameters and different operators in order to obtain a final reliable mean value. Accurate measurements need a good characterization of the spatiotemporal profiles related to the laser pump beam.^1,2The far-field diffraction pattern in Z-scan experiment is an excellent way to obtain the spatial parameters of the focused laser beam inducing nonlinearities.

On the contrary, it is more difficult to obtain information about the temporal profile of the beam. Generally, one relies on the pulse duration values given by the laser manufacturers. Rarely, even though the second order autocorrelation technique is used to determine the pulse duration in the fundamental wavelength 关1064 nm with neodymium doped yt-

trium aluminum garnet共Nd:YAG兲laser兴, the pulse durations of generated harmonics共532 nm, 355 nm…兲are theoretically deduced by dividing the obtained result at 1.064 ␮^{m with}

冑

2 ,

冑

3 , . . ..³In this paper, we will show that such theoretical prediction can be one of the sources of important errors.

The aim of this communication is to add absolute values to the already existing n₂ values 共see, for example, Refs. 4 and5 and references therein兲. To do this, we combined the classical Z-scan analyses⁶ with a third-order autocorrelator technique, as in Ref.1. The spatial shape of the incident and diffracted beam is characterized inside a nonlinear 4-f imaging system⁷while the pulse duration of the temporal profile 共supposed to be Gaussian兲is measured using the well known optical Kerr shutter experiment 共see, for example, Ref. in1 or 8兲.

II. PRINCIPLE OF THE MEASUREMENT

We reported a nonlinear-imaging technique using a 4-f coherent imaging system to characterize the value of the nonlinear refractive index 共n2兲 of materials placed in the Fourier plane of the setup.⁹ Here, the scheme of the experimental setup, as shown in Fig. 1, is similar but without any object phase at the entry. Only linearly polarized Gaussian beam of beam-waist ␻e is used: E共r,t兲=E₀共t兲exp关−r²/␻e2兴, wherer is the radial coordinate in the transverse plane and E₀共t兲is the amplitude of the electric field containing the os- cillating term and the temporal envelope of the laser pulse. In the 4-f imaging system and for f₁=f₂, the magnification is equal to one. It is important to note that in this configuration one can measure directly and precisely the beam waist of the Gaussian beam. Thus we can obtain I₀, the central peak intensity in the focal plane of the focusing lensL₁. By considering a spatial and temporal Gaussian profiles and after integration on these both coordinates, we obtain

a兲Author to whom correspondence should be addressed. Electronic mail:

georges.boudebs@univ-angers.fr. Tel.: 共33兲共0兲2.41.73.54.26. FAX: 共33兲共0兲2.41.73.52.16.

b兲Electronic mail: kamil.fedus@univ-angers.fr. FAX:共33兲共0兲2.41.73.52.16.

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I₀= 4

冑

␲^ln共2兲

冉

␭^␻f^e₁

冊

²^␧␶^, ^共¹^兲 where␶is the pulse duration of full width at half maximum 共FWHM兲, f₁ is the focal length of lens L₁, and ␧ is the energy given by the joulemeter. We already checked⁹ nu- merically and experimentally that the addition of lens L₂ 共necessary to obtain the image of the Gaussian beam at the entry in the linear regime兲, does not affect the results of Z-scan. In fact, this lens contributes to produce the Fourier transform of the field at the exit surface of the sample, which is physically similar to the far field diffraction pattern obtained with the original Z-scan method. By applying this method, one can determine ␸NL0, the on-axis nonlinear dephasing in the focal plane from the measurement of⌬T_pv, the difference between the normalized peak and valley transmittance by using⁶

⌬Tpv= 0,41共1 −S兲^0,25兩␸NL0兩, 共2兲 whereSis the linear transmittance of the aperture共numerical one in our case兲. In the case of negligible nonlinear absorption, this allows us to evaluaten₂with the following relationship: ␸NL0= 2␲^Ln2I₀/␭ whereL is the thickness of the material. It is clear that the measurement of␸NL0is quiet easy and direct to determine. However, the problem in nonlinear measurements comes from the uncertainty that we have in measuringI₀. One can see from Eq.共1兲that if generally␧is given by a calibrated joulemeter, a significant source of errors can originate from the measurement of␻e共especially for experiments using photodiodes兲and␶共especially for the harmonics of the Nd:YAG laser兲. Thereby, we will focus our attention on how to evaluate these two parameters with more accuracy.

III. EXPERIMENTS A. Z scan

The excitation is provided by a Nd:YAG laser delivering linearly polarized pulse at 1064 nm and its harmonics 共532 and 355 nm兲. The nominal pulse duration at 1064 nm is given to be ␶= 17 ps at FWHM. The image receiver is a 1000⫻1018 pixels, 共12⫻12 ␮m²兲 cooled charge couple device共CCD camera兲共−30 ° C兲 with a fixed gain placed in

the image plane of the 4-f setup共see Fig.1兲. Thermo-optical effects are negligible in the picosecond range and low repetition rate共10 Hz兲.¹⁰A beam splitter at the entry of the setup allows to monitor fluctuations occurring in the incident laser beam. The focal lengths of lensesL₁andL₂are both equal to 20 cm. In Fig. 2共a兲, we show the variation of the measured value of the beam waist at the output of the experimental setup versusz, the sample position for a positiven₂material without nonlinear absorption 共CS2兲. This figure clearly shows the broadening and the narrowing of the output beam for prefocal and postfocal positions, respectively. In the linear regime, we obtain a more constant value and generally, we calculate the mean value by averaging the linear and the nonlinear regime acquisitions. It has to be added that the minimum in the variation of the beam waist shown in Fig.

2共a兲, corresponds to a maximum of the normalized transmittance shown in Fig.2共b兲. In the latter, we can see the Z-scan trace for 1 mm thick cell of CS₂at 1064 nm and 17 ps pulse duration with 1% linear transmittance of the numerical aperture. Using Eq. 共2兲, we evaluate the nonlinear dephasing,

␸NL0= 0.77 rad. The 23 GW/cm² central peak incident intensity was obtained using relation 共1兲 taking into account the measured energy 6.5 ␮J共with a PE10 pyroelectric head

CCD

f1 f1 f2

L1 L2

L3

M1

Sample BS1

BS2

f2

z y x

M2

z

FIG. 1. Schematic of the 4-f coherent system imager. The sample is located in the focal region. The labels refer to: lenses共L₁-L₃兲; mirrors共M₁,M₂兲; beam splitters共BS₁, BS₂兲.

(a)

(b)

FIG. 2.共a兲Measurement of the beam waist at the output of the experimental setup vs z, the sample position at 1064 nm in the nonlinear regime.共b兲 Z-scan normalized transmittance for 1 mm thick cell of CS₂at 1064 nm, 17 ps pulse duration, and 1% linear transmittance of the numerical aperture.

The central peak incident intensity is 23 GW/cm².

103106-2 G. Boudebs and K. Fedus J. Appl. Phys.105, 103106共2009兲

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OPHIR joulemeter兲. The pulse duration is estimated using the Kerr shutter experiment that we will briefly describe hereafter.

B. Kerr shutter

The ultrafast optical Kerr shutter setup shown on Fig.3 is a well-known experimental configuration for determining pulse width in picosecond time scale.^1,8,11The beam splitter 共BS兲at the entry of the setup divides linearly polarized beam into two parts: the high intensity pump beam and low intensity probe beam. Both beams propagate in nonparallel direc- tions and intersect each other inside the sample 共NL兲 at a small angle共6°兲. This noncollinear configuration of the setup does not require using expensive polarizing elements and gives great flexibility to work with beams of the same polarization. The sample is inserted between two crossed polariz- ers 共P1 and P₂兲; thus, generally in absence of nonlinearity, the probe beam cannot reach the CCD camera.

In order to improve the efficiency of the signal obtained by the optical shutter:共i兲polarizersP₁andP₂have their axis at⫾45° with respect to polarization plane of the pump beam;

共ii兲 we used lensL₃ to obtain a magnified image of the exit surface of the sample; and 共iii兲 the probe beam is slightly focused by lensL₁into the impact region of the pump beam 共being careful to have always an area at least ten times greater than the focused pump beam area兲. Generally the sample is placed in the focal plane of lensL₂ where an in- tense pulse induces nonlinear change of the refractive index inside the material. During the presence of the pump beam, the induced birefringence causes the rotation of the probe beam polarization and consequently, a fraction of light can pass through the analyzer 共P2兲 and reaches the CCD.¹² The spatial integration on the acquired images provides a tempo- rally averaged signal which is considered as proportional to a third-order autocorrelation functionG^共³^兲共td兲

G^共3兲共td兲=

冕

^I^p²^共t兲I^pr^共t⁻^t^d^兲dt, ^共3兲

where td is the delay between incident pump and probe pulses, Ip andI_pr are the pump and probe beam intensities, respectively. Equation共3兲is valid for material with ultrafast response time共much less than the pulse duration兲and pump induced small phase shift. We have used two different nonlinear liquid materials: CS₂ in the infrared and the green共at 355 nm the transmittance is null兲and chlorobenzene for all

considered wavelengths. The comparison of the measured values shows that the response time of the materials is negligible. Figure 4 presents the autocorrelation functions G^共3兲 fitted to the experimental data at 1064 nm共the stars兲and 532 nm共the points兲. A deconvolution program was performed to obtain the assumed Gaussian temporal profile of the pump beam and its pulse width. The measured pulse durations 共FWHM兲are given to be 17 ps at 1064 nm, 7 ps at 532 nm, and 12 ps at 355 nm. The value at 1064 nm validates our measurement procedure because our laser has been already tested using the second order autocorrelation technique and the result was in excellent agreement with the obtained value here. At 532 nm, the 7 ps pulse that we obtain is very far from the expected one which is 12 ps共17/

冑

2兲. Therefore one should be very careful to evaluate the incident intensity at the second and third harmonics of the Nd:YAG laser even if the fundamental wavelength was measured by the second order autocorrelation technique. The scale rule for pulse duration for higher harmonics has to hold only if we work in the linear part of the energy conversion efficiency with respect to the incident intensity. However, in order to increase the output energy, the laser manufacturers generally use the asymptotic values of this conversion. May be the beam is no more Gaussian as we can guess from the autocorrelation plots 共in the green and UV兲but it is more accurate to esti- mate the pulse width from experimental data by supposing so than to believe in theoretical prediction.

IV. RESULTS AND DISCUSSION

The measurements were performed more than ten times with different samples in different experimental conditions to check the reproducibility of the measurements. The incident energy was controlled with two different joulemeter. The overall experimental uncertainty was approximately ⫾20%

共principally originating from energy calibration of the joule- meter兲 and depending on the variation of the incident laser energy during the scan. We can see in Table I the absolute measurement of the nonlinear refractive indices for two of the most frequently used reference materials in nonlinear optical experiments. The beam waist at the entry of the setup BS NL

M1

P1

P2

M2

L1

L2

L3

CCD

Time delay

P

FIG. 3. Kerr shutter experimental setup:Mi: mirrors,Li: lenses,Pi: polar- izers, BS: beam splitter,P: prism, and NL: nonlinear material.

FIG. 4. Third-order autocorrelation function of 17 ps pulses at 1064 nm 共stars for experimental data and dashed line for numerical fitting兲and 7 ps pulses at 532 nm共points for experimental data and solid line for numerical fitting兲.

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共␻e兲was calculated in order to obtain a Rayleigh range larger than L, the thickness of the materials 共shown in the third column兲. This is necessary because the sample should be regarded as thin when applying Eq.共2兲to calculate␸NL0, the nonlinear dephasing. We have been careful not to use too high intensities in order to stay within the validity of the relationship共2兲共兩␸NL0兩⬍1兲and not to destroy the specimen.

We can notice the increasing of the measurement sensitivity of the system with the decreasing of the wavelength. Indeed, the required intensity共I0in column 6兲in the UV is six times less than the one used in the infrared. In column 5 appears␤^, the usual parameter that characterizes the nonlinear two- photon absorption共for more details concerning this parameter see Ref.6兲. It is clear that for both materials and for the considered wavelengths the nonlinear absorption is negligible at the corresponding intensities.

One of the main results of this paper is the obtainedn₂ absolute values for CS₂ and SiO₂ in the infrared and in the picosecond regime. Both of our values are one order of magnitude less than those found usually in the literature. We notice also a large dispersion of the results with respect to the wavelengths than the one considered up to now. For example, then₂value for CS₂in the green is found to be two times larger than the one obtained in the infrared. The same comment is valid also for the fused silica. The already existing dispersion theory for transparent glasses as fused silica proposed by Boling–Glass–Owyoung¹³was built on a simple model of harmonic oscillator at long wavelength 共far from the two-photon interband absorption edge兲providing a relation between linear and nonlinear refractive indices. Later, the dispersion predictions were improved by two other models. The first one employs the Kramers–Kronig transforma- tion of calculated two-photon absorption spectrum in the two-band configuration¹⁴in order to estimaten₂. This model provides better description rather in semiconductors than in wideband-gap glasses as fused silica.³The second one 共the PERT model¹⁵兲 is based on a fourth-order perturbation theory which assumes that the dispersion ofn₂in the visible and near IR can be attributed to a single resonance in the UV.

All these models indicate a normal dispersion, i.e.,n₂in UV is larger thann₂in IR which is the case in our experimental results. However, the ratio between nonlinear indices obtained in 355 and 1064 nm关n2共355兲/n₂共1064兲⬇1.5共Refs.4 and15兲兴is about three times less than the value found in this work 关n2共355兲/n₂共1064兲⬇4兴. All these theoretical models generally depend on a constant that the authors “adapt” 共to the material兲 in order to fit the experimental results or to

translate the theoretical curve to match some “important data.” In light of our results, we have to consider that the already existing theories should be improved. Note, that our n₂value for silica in UV is in a good agreement with values obtained in Refs. 3and16–18.

Moreover, our experimental results give a new look at the dependence of the nonlinear refraction on the pulse duration and its repetition rate. It is well known that only the instantaneous electronic processes and the molecular reorientation contribute to the nonlinear response under picosecond irradiation since the thermal effects and electrostriction are significantly slower processes.¹⁹ Hence, the picosecond range and low repetition rate 共10 Hz兲 in our work exclude thermal component. On one hand, considering fused silica one can neglect the molecular reorientation.²⁰ Therefore we can assume that our measurement gives only the electronic part of the cubic susceptibility and n₂ of the fused silica is weakly dependent on the pulse duration.²¹On the other hand, picosecond rotation of molecules gives an important contribution to Kerr susceptibility in CS₂共Refs.19and20兲and its n₂ should be dependant on pulse duration²² between pico- and femtosecond regimes. The obtained value in femtosecond regime 共due to the pure electronic response兲 is about 30⫻10⁻²⁰ m²/W in the infrared共800 nm兲.^23,24 Up to now we have been considering that then₂ of CS₂ in the picosecond range should be one order of magnitude larger due to the molecular contribution. Taking into account an extrapolated n₂value from our measurements at 1064 and 532 nm one can found a mean value of about 60⫻10⁻²⁰ m²/W at 800 nm, two times larger but still in the same order of magnitude as the one obtained in the femtosecond regime. This result shows that the electronic and the rotational contributions to then₂ value of CS₂are of the same order of magnitude.

Finally, we have performed²⁰systematic Z-scann₂deter- minations of a series of liquids after calibration of the setup with then₂value of CS₂given in Ref.6. The results obtained at 1064 nm were compared with the cubic susceptibilities obtained by the third harmonic generation 共THG兲 measurements performed on the same materials far from the resonance region. This paper shows that the absolute n₂ values for CS₂obtained by THG and Z-scan are approximately the same 共within the experimental errors兲 and the comparison would give less discrepancy in the results.

We think that the beam-waist and the pulse-width precise measurements are very critical parameters in alln₂measurement experimental procedure. This could explain the dis-

TABLE I. Values of the experimentally measured nonlinear coefficients共n₂and␤兲corresponding to the studied reference materials at different wavelengths.

Materials

共nm␭兲

L

共mm兲 n₂⫻10⁻²⁰ m²/W

␤⫻10⁻³ 共cm/GW兲

I₀ 共GW/cm²兲

SiO₂ 355 3.76 2.0⫾0.4 ⬍2 37

SiO₂ 532 3.76 0.9⫾0.3 ⬍0.2 76

SiO₂ 1064 3.76 0.4⫾0.07 ⬍0.2 221

CS₂ 532 1.0 80⫾17 ⬍17 5.5

CS₂ 1064 1.0 40⫾6 ⬍7 23

103106-4 G. Boudebs and K. Fedus J. Appl. Phys.105, 103106共2009兲

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crepancy of the absolute measurements obtained by different authors and worse with different experimental techniques.

V. CONCLUSION

We have reported on absolute measurements of the nonlinear refractive index for two standard reference materials 共CS₂and SiO₂兲. Inside a 4-f Z-scan imaging system we have analyzed the spatial distortion of the diffracted pump beam to measure precisely its beam waist at the entry of the setup.

The “Kerr shutter” technique was applied to evaluate the temporal pulse width for three harmonics delivered by a pulsed Nd:YAG laser 共1064, 532, and 355 nm兲 in the picosecond regime. In the infrared and the green wavelengths we obtainedn₂ values one order of magnitude less than the one usually taken into account. Another important result comes from the more significant dispersion of the nonlinear refraction index for both materials. Only in UV the n₂ value of fused silica is in good agreement with some values found in the literature. At the light of our results, values evaluated by comparison to one of these two reference materials should be corrected. It would be of interest to develop a model that could be used to predict the dispersion ofn₂more accurately taking into account our measured values.

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