Experimental and theoretical study of
higher-order nonlinearities in chalcogenide glasses
G. Boudebs
a,*, S. Cherukulappurath
a, H. Leblond
a, J. Troles
b, F. Smektala
b, F. Sanchez
aaLaboratoire POMA, UMR 6136, UniversiteedÕAngers, 2 Bd. Lavoisier, 49045 Angers, France
bLaboratoire des Verres et Ceeramiques, UMR 6512, Universiteede Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France Received 6 December 2002; received in revised form 26 February 2003; accepted 10 March 2003
Abstract
We investigate experimentally and theoretically the nonlinear properties of two distinct chalcogenide glasses.
Nonlinearities are measured through a spatially resolved Mach–Zehnder interferometer. It is shown that the resulting nonlinear index coefficient cannot be correctly described with the usual cubic model. A more convenient theory is developed assuming a medium with two-photon nonlinear absorption and both cubic (n2) and quintic (n4) nonlinear index variations. The resulting closed-form expression for the effective nonlinear index coefficient allows to extract the cubic and quintic index coefficients.
Ó 2003 Elsevier Science B.V. All rights reserved.
PACS:42.70.Km; 42.65.-k; 42.65.Tg
Keywords:Chalcogenide glasses; Mach–Zehnder interferometer; Nonlinear refractive index; Fourier transform; Nonlinear absorption
1. Introduction
From the theoretical viewpoint, the nonlinear response of a medium to a high power light beam is commonly described by means of an expansion in a power series of the electric field wave ampli- tude, the coefficients of which are the so-called nonlinear susceptibilities. In a first stage, the
theory of light propagation in nonlinear media considers only the first nonzero nonlinear term in this expansion, which is the cubic one in centro- symmetrical media. As it is well-known, such a modeling of the response of the medium gives ac- count for self-focusing, and for collapse in more than ð1þ1Þ dimensions [1,2]. From both the fundamental and applicative viewpoints, propa- gating stable two- or three-dimensional light pul- ses, or light bullets, would be of major interest. A large amount of research has been done in that direction. All theoretical predictions of such structures involve some saturation of the nonlin- earity. A way to describe the latter is to take into
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*Corresponding author. Tel.: +33-02-41-73-54-26; fax: +33- 02-41-73-52-16.
E-mail address:georges.boudebs@univ-angers.fr (G.Boudebs).
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doi:10.1016/S0030-4018(03)01341-5
account the second term in the expansion of the material response function, which is the quintic nonlinear susceptibility. The so-called cubic–
quintic model obtained this way is able to support light bullets [3], spinning solitons [4], and so on.
However, experimental measurements of non- linear susceptibilities and nonlinear indices have shown that many real materials can be considered as pure Kerr media, i.e., media presenting only the cubic nonlinearity. Experiments in PTS and more recently in chalcogenide glasses [5] have shown the experimental relevance of more sophisticated models. In reference [5], the third-order nonlinear- ity related ton2 coefficient has been measured for As2S3 and As2Se3using the Z-scan method. It has been seen that this measured index is not a constant but decreases as the input intensity increases. It must be noted that the so-called model of the sat- urated nonlinearity, Dn¼n2I=ð1þI=IsÞ, cannot give account for this sign change, which seems to be a necessary condition for the formation of spinning solitons [4]. The experimental observation states thus the validity of the cubic–quintic model, and allows to expect the experimental realization of the theoretical predictions. Further, experimental val- ues of the fifth-order nonlinear index coefficient (n4) are needed in order to compare the theoretical predictions involving this quantity to experiment, and to conceive experiments in which light bullets or spinning solitons could be observed.
The paper is organized as follows. In Section 2 we consider the experimental results obtained using a powerful technique to measure the real and imaginary part of the third-order susceptibility [6–8]. This technique is based on a pump/probe experiment using a Mach–Zehnder interferometer combined with a CCD camera, which allows to obtain a great number of measurements with a single laser shot in the material. We then show that intensity dependent nonlinear index occurs in chalcogenide glasses characterized in this study.
This result cannot be explained in terms of third- order susceptibility. In Section 3 we develop a model for a medium exhibiting both linear and two- photon absorption and both cubic and quintic nonlinear index. Analytical solutions are obtained for the intensity and the nonlinear phase shift. This allows to obtain a closed-form expression for the
effective nonlinear index coefficient. A linear fit on the experimental data leads to the determination of the cubic and quintic nonlinear index coefficients.
2. Experimental results 2.1. Experimental setup
In this section we briefly explain the experi- mental procedure for the determination of the nonlinear index coefficient. Full details are given in [6–8]. The experimental setup is shown in Fig. 1.
The first arm of the Mach–Zehnder interferometer is used as the reference beam. LetI1ðrÞdenote the intensity of the reference beam. The second arm (I2ðrÞ) is used as the test area (the probe beam).
The nonlinear medium (NL) is illuminated by a pump beam, focused by means of the lens L. At the output of the setup, the interference pattern intensity INLðrÞ, is recorded on a CCD camera placed in theOxyplane perpendicular to the light propagation axis. The Mach–Zehnder interfer- ometer is adjusted to obtain rectilinear fringes. The intensity at the output of the setup is then given by INLðrÞ ¼I1ðrÞ þI2ðrÞTðrÞ2
þ2TðrÞ ffiffiffiffiffiffiffi I1I2
p cos½uLþuNLðrÞ: ð1Þ
Physically, local fringe alterations occur in a small region of the interference pattern. These local
Fig. 1. Spatially resolved Mach–Zehnder interferometer for nonlinear coefficient measurements.
alterations are a combination of a local displace- ment of the fringes attributed to the nonlinear dephasing uNLðrÞ and a local modification in the fringe visibility attributed to the amplitude non- linear transmission TðrÞ. In order to extract this information of interest, a numerical spatial Fou- rier transform (FT) is performed on the acquired image INLðrÞ. The first and the second terms of Eq. (1) are not of particular interest. Their FT is centered at the origin of the Fourier plane. Thus if the fringe spacing is sufficiently small, the FT of the cosine function in the third term of Eq. (1) consists of two Dirac delta functions far enough from the origin convoluting a complex function which con- tains the information about the nonlinearities.
Considering the part of the spectrum around one of these Dirac delta functions and performing an in- verse Fourier transform on this part, we extract from the interference pattern the sought quantities TanduNLas functions of the radial coordinater.
In the case where the sample under investiga- tion is inhomogeneous (phase and amplitude variations independent of the optical intensity), the procedure is slightly modified [7]. In order to extract the information related to the nonlineari- ties only, we acquire further a ‘‘linear fringe pat- tern’’, for which the sample is inserted in the interferometer but the pump beam is switched off.
In this situation, the nonlinearities are negligible, thus allowing to measure the ‘‘linear response’’ of the sample. Then, comparing the ‘‘linear’’ and the
‘‘nonlinear’’ fringe patterns, we are able to get spatially resolved information on the nonlineari- ties and to fully characterize the nonlinear re- sponse of the material.
The nonlinear dephasing allows finally to find an effective nonlinear index coefficient assuming a purely third-order nonlinearity [9]:
uNLðrÞ ¼kLn2IeffðrÞ; ð2aÞ with
IeffðrÞ ¼ 1
bLln½1þqðrÞ; ð2bÞ
whereIeffðrÞis the effective pump intensity inside the nonlinear medium,kthe modulus of the wave vector, L the length of the sample, b is the two- photon absorption coefficient andqðrÞ ¼bLeffI0ðrÞ
withLeff ¼ ð1eaLÞ=a.I0ðrÞis the incident pump intensity distribution andais the linear absorption coefficient.
In the case where n2 is constant with the effec- tive intensity, it means that the material exhibits a purely third-order susceptibility otherwise, it means that higher-order terms should be taken into account.
The two-photon absorption coefficient b is de- duced from the amplitude transmittance which writes
TðrÞ ¼ ðeaL½1þqðrÞÞ1=2: ð3Þ
2.2. Results for chalcogenide glasses
Let us first consider the As2Se3 chalcogenide glass. The pump intensity profile is shown on Fig.
2(a). It has been reconstructed from the spatially resolved sample transmission using relation (3) which relatesTðrÞtoI0ðrÞ. Surprisingly, the pump beam exhibits two maxima. We have easily checked that this pump profile results in fact from the interference of the two first beams transmitted through the nonlinear medium which acts as a poor quality Fabry–Perot interferometer. Indeed, the input/output faces are not anti-reflection coated. In Fig. 2(b) we have represented the spatially resolved measured nonlinear dephasing change. In agree- ment with the pump intensity profile, the nonlinear phase change has two extrema. The important point here is that the maximum corresponding to the secondary pump peak is positive while the maximum spatially located at the principal maxi- mum of the pump beam is negative. This experi- mental observation makes conspicuous, for the first time in our knowledge, simultaneous positive and negative index changes in a nonlinear material.
Several experimental acquisitions have been made in different impact regions of the nonlinear mate- rial. The resulting nonlinear dephasing always contains the negative response. We checked care- fully that the nonlinear negative dephasing is not a numerical artifact. Indeed, the deformation of the acquired rectilinear fringe pattern always presents displacement in both opposite directions, showing the simultaneous presence of responses with both signs. Moreover in these different experiments, the
condition of the nonlinear measurements (single pulse at 30 ps and repetition rate of 1 Hz maximum) does not lead to any photo-induced effect or any ablation in the material. In principle, the determi- nation of the nonlinear coefficients could be done from each laser shot. However, the low spatial resolution of the setup (36lm) [6] does not allow an accurate measure which requires a good separation between the positive and negative part of the non- linear dephasing. Therefore, the nonlinear coeffi- cients (n2;n4) are deduced from an averaging of the nonlinear dephasing over several laser shots (see Section 3.2).
Let us now consider the As2S3 chalcogenide glass. We have not observed a spatially resolved sign
change on the nonlinear dephasing. We expect that this is due to the lower value of the quintic nonlinear index coefficient. In addition, the experimental conditions to benefit from a two maxima pump in- tensity distribution were probably not fulfilled.
However, as we will see in the theoretical section, the effective nonlinear index coefficient will be correctly fitted assuming cubic and quintic nonlinearities.
Obviously our experimental data cannot be in- terpreted with purely third-order nonlinearity.
Higher order terms with opposite sign are required in order to compensate the cubic term and to generate a sign change on the dephasing term. In a first approach, we consider in the next section a cubic/quintic nonlinear medium. We will see that it will be sufficient to correctly explain the evolution of the effective nonlinear index coefficient as a function of the incident intensity.
3. Theoretical model 3.1. The model
We consider a beam propagating along the z- axis in a medium exhibiting: (i) linear absorption (a), (ii) two-photon absorption (b), (iii) third-order nonlinear index (n2) and (iv) fifth-order nonlinear index (n4). Under the slowly varying envelope ap- proximation and neglecting diffraction effects, the intensity (I) and the phase (u) of the beam verify the equations:
dI
dz¼ aIbI2; ð4Þ
du
dz ¼kðn2Iþn4I2Þ; ð5Þ
wherek¼2p=kis the modulus of the wave vector.
The equation for the intensity (4) can be directly integrated, yielding
IðzÞ ¼ I0eaz
1þbI0ð1eaz=aÞ: ð6Þ
For a medium of length L, the nonlinear phase shift is defined as
D/NL¼kn2
Z L 0
IðzÞdzþkn4
Z L 0
I2ðzÞdz: ð7Þ
Fig. 2. Experimental results for the As2Se3sample. (a) Trans- verse profile of the pump beam in the nonlinear material. (b) The corresponding induced transverse profile of the nonlinear dephasing. Thexandycoordinates are in pixels, the dimension of a pixel is 6 6lm2.
Eq. (7) leads to
D/NLðrÞ ¼kL½n2IeffðrÞ þn4ðI2ÞeffðrÞ; ð8Þ where
IeffðrÞ ¼ 1
bL ln½1þqðrÞ;
ðI2ÞeffðrÞ ¼ a b2L 1
þqðrÞ aLeff
qðrÞ 1þqðrÞ
ln½1þqðrÞg:
ð9Þ
We define an effective nonlinear index coefficient neff2 by means of
D/NLðrÞ ¼kLneff2 IeffðrÞ: ð10Þ This leads to:
neff2 ¼n2þn4Im; Im¼ðI2Þeff
Ieff ; ð11Þ
where we introduceIm as the ratio of the effective squared intensity to the effective intensity. An ad- vantage of the form (11) is its linearity with respect to Im, allowing the nonlinear index coefficients to be easily determined. The coefficients can be ac- curately computed using the least squares method.
Fig. 3 shows the simulation of Im versus the on- axis incident intensityI0ð0Þforb¼0,b¼0:4, and b¼7:45 cm/GW. The other parameters are those found experimentally characterizing the studied nonlinear materials: As2Se3 (open circles), a¼0:6 cm1, L¼1:44 mm; As2S3 (crosses), a¼0:3 cm1, L¼1:81 mm. The solid line repre- sents the theoretical case where a andb are both equal to zero (i.e., no linear or nonlinear absorp- tions). It is clear that in this particular case the intensity Im is simply equal toI0, the incident in- tensity (no losses in the material). One can see that Im is increasing with the incident intensity what- ever the value ofb, but the amplitude of the var- iation decreases whilebincreases.
3.2. Comparison with experimental results
In this section we use relations (11) to extractn2 andn4for As2Se3and As2S3 chalcogenide glasses.
Experimentally, we use the following procedure.
First of all, we perform the measurement of the
nonlinear absorption characterized by the coeffi- cient b using the relation for normalized power transmittance
Tt¼eaLlnð1þq0Þ
q0 ; ð12Þ
whereq0¼bI0ð0ÞLeff. Relation (12) gives the ratio of the total transmitted power to that of the incident one obtained by spatially integrating Eq. (3) overr for Gaussian incident intensity (I0ðrÞ). Results are given in Fig. 4. We obtain for As2Se3 b¼7:45 cm/
GW and for As2S3 b¼0:4 cm/GW. These values are of the same order of magnitude as those already reported in [10] by considering relation (3) that re- lates incident and transmitted intensities for one laser shot. Then, by considering a constant value for b(the previously calculated one), we deduce an ef- fective nonlinear index coefficient with the spatially resolved Mach–Zehnder interferometer using rela- tion (10). This is performed for several laser shot and for increasing pump input intensities. In the case of As2Se3 the on-axis pump input intensityI0 is limited to 3 GW=cm2 (photo-induced and/or
Fig. 3. Simulation givingIm versusI0forb¼0,b¼0:4, and b¼7:45 cm/GW fork¼1:064lm. The other parameters are those found experimentally characterizing the studied nonlinear materials: As2Se3 (circle points), a¼0:6 cm1, L¼1:44 mm;
As2S3 (crosses points), a¼0:3 cm1, L¼1:81 mm. The line represents the theoretical case whereaandbare both equal to zero (i.e., no linear or nonlinear absorptions).
destruction phenomena occur at higher intensities).
We obtain the evolution of neff2 as a function ofI0 (Fig. 5(a)). Note that the nonlinear evolution ofneff2 versusI0does not mean that we have necessarily a negative response in the nonlinear dephasing. In presence of relatively high nonlinear absorption and even ifn4is negligible the measuredneff2 should remain constant and equal ton2when plotted versus Ieffwhich is a nonlinear function ofI0(Eqs. (2a) and (2b)). A more convenient way to point out the im- portance and the presence of the coefficientn4in our experiment is to represent the evolution ofneff2 as a function ofIm(Fig. 5(b)). We then perform a linear fit using relation (11). For As2Se3 we obtain the following nonlinear index coefficients: n2¼2:2 1017m2=W and n4¼ 7:9 1031 m4=W2.For the As2S3sample we obtainn2¼5:8 1018 m2=W and n4¼ 6:3 1032m4=W2. Theoretical fits in Fig. 5(b) are in good agreement with the experi- mental data thus confirming the cubic/quintic model especially for the case of As2Se3 where the higher absolute value of the negative slope clearly confirms the existence of a negative quintic non- linear index coefficient. Moreover, the relatively lower value ofn4for As2S3could explain the reason
why it was not possible to obtain a negative part in the spatially resolved nonlinear dephasing such as for As2Se3in Fig. 2(b).
Finally, for simplicity we have considered here that there is no nonlinear absorption higher than the two photon one. In our materials, this is true in an approximate way only (roughly estimated at 30% frombmeasurements made for each laser shot as a function of the incident intensity). A more accurate approach taking into account higher
Fig. 5. (a) Evolution of the measured effective nonlinear index coefficient as a function of the incident intensity for As2Se3and As2S3 samples. (b) Evolution of the measured effective non- linear index coefficient as a function ofImðr¼0Þ. The points are the experimental data and the solid lines are the linear regres- sion fits using Eq. (11). Note the negative slope for both materials.
Fig. 4. Circle (ÔoÕ) and crosses points (Ô Õ) represent experi- mental data. The dashed lines represent the nonlinear fitting using Eq. (12). Thebvalues given by the fitting are 0.4 cm/GW for As2S3and 7.45 cm/GW for As2Se3characterizing the two- photon absorption in these infrared glasses at 1.064lm in the picosecond range.
orders will be considered in the future. However, simple analytical solutions are not possible when higher orders terms are taken into account for the absorption.
4. Conclusions
In summary, we have investigated both experi- mentally and theoretically the optical nonlinearities of two chalcogenide glasses. Experimental data clearly indicate that the samples used in our ex- periments cannot be described with the usual third- order nonlinear theory. Consequently, we have constructed a model based on the existence of both a cubic and quintic nonlinear index. The evolution of the resulting nonlinear index coefficient as a function of the intensity is in good agreement with the experimental data. A fit has allowed to deter- mine the values of the nonlinear index coefficients.
In particular, a negative quintic nonlinear index coefficient has been pointed out. Our results dem- onstrate that chalcogenide glasses are very good candidate to test some theoretical models con- cerning light bullets where a cubic/quintic nonlin- earity is required.
Acknowledgements
This work is supported by an interregional re- search program Bretagne-Pays de Loire: ‘‘Verres infrarouges pour fonctions optiques teeleecom’’.
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