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
Articles you may be interested in
Z-scan measurements of the nonlinear refractive index of a pumped semiconductor disk laser gain medium
Appl. Phys. Lett. 106, 011105 (2015); 10.1063/1.4905346
Kerr-driven nonlinear refractive index of air at 800 and 400 nm measured through femtosecond laser pulse
filamentation
Appl. Phys. Lett. 99, 181114 (2011); 10.1063/1.3657774
Direct measurements of the nonlinear index of refraction of water at 815 and 407 nm using single-shot
supercontinuum spectral interferometry
Appl. Phys. Lett. 94, 211102 (2009); 10.1063/1.3142384
Femtosecond investigation of the non-instantaneous third-order nonlinear suceptibility in liquids and glasses
Appl. Phys. Lett. 87, 211916 (2005); 10.1063/1.2136413
Antimony orthophosphate glasses with large nonlinear refractive indices, low two-photon absorption coefficients,
and ultrafast response
J. Appl. Phys. 97, 013505 (2005); 10.1063/1.1828216
Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 193.52.40.1 On: Tue, 03 May 2016
15:18:40
JOURNAL OF APPLIED PHYSICS 105, 103106 共2009兲
Absolute measurement of the nonlinear refractive indices
of reference materials
Georges Boudebsa兲 and Kamil Fedusb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 surprisingly n2 values 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 absolute n2 measurements 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.1,2 The 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 yta兲
Author to whom correspondence should be addressed. Electronic mail:
[email protected]. 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: [email protected]. FAX: 共33兲 共0兲2.41.73.52.16.
0021-8979/2009/105共10兲/103106/5/$25.00
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 , . . ..3 In 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
and 5 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 system7 while 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. in 1
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 / ␻2e 兴,
where r is the radial coordinate in the transverse plane and
E0共t兲 is the amplitude of the electric field containing the oscillating term and the temporal envelope of the laser pulse. In
the 4-f imaging system and for f 1 = f 2, 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 lens L1. By considering a spatial and temporal Gaussian profiles and after integration on these both coordinates, we obtain
105, 103106-1
© 2009 American Institute of Physics
Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 193.52.40.1 On: Tue, 03 May 2016
15:18:40
103106-2
J. Appl. Phys. 105, 103106 共2009兲
G. Boudebs and K. Fedus
f1
BS1
f1
f2
L1
Sample
f2
L2
CCD
BS2
(a)
z
L3
M1
y
M2
x
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M 1 , M 2兲;
beam splitters 共BS1 , BS2兲.
I0 = 4冑␲ ln共2兲
冉 冊
␻e
␭f 1
2
␧
,
␶
共1兲
where ␶ is the pulse duration of full width at half maximum
共FWHM兲, f 1 is the focal length of lens L1, and ␧ is the
energy given by the joulemeter. We already checked9 numerically 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 ⌬Tpv,
the difference between the normalized peak and valley transmittance by using6
⌬Tpv = 0,41共1 − S兲0,25兩␸NL0兩,
(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 CS2 at 1064 nm, 17
ps pulse duration, and 1% linear transmittance of the numerical aperture.
The central peak incident intensity is 23 GW/ cm2.
共2兲
where S is the linear transmittance of the aperture 共numerical
one in our case兲. In the case of negligible nonlinear absorption, this allows us to evaluate n2 with the following relationship: ␸NL0 = 2␲Ln2I0 / ␭ where L is the thickness of the material. It is clear that the measurement of ␸NL0 is quiet easy
and direct to determine. However, the problem in nonlinear
measurements comes from the uncertainty that we have in
measuring I0. 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.10 A beam splitter at the entry of the setup
allows to monitor fluctuations occurring in the incident laser
beam. The focal lengths of lenses L1 and L2 are 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 versus z, the sample position for a positive n2 material
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 CS2 at 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
Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 193.52.40.1 On: Tue, 03 May 2016
15:18:40
103106-3
J. Appl. Phys. 105, 103106 共2009兲
G. Boudebs and K. Fedus
M1
P1
L1
M2
BS
NL
delay
P
Time
L2
L3
P2 CCD
FIG. 3. Kerr shutter experimental setup: M i: mirrors, Li: lenses, Pi: polarizers, BS: beam splitter, P: prism, and NL: nonlinear material.
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.1,8,11 The 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 directions 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 polarizers 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 polarizers P1 and P2 have their axis
at ⫾45° with respect to polarization plane of the pump beam;
共ii兲 we used lens L3 to obtain a magnified image of the exit
surface of the sample; and 共iii兲 the probe beam is slightly
focused by lens L1 into 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 lens L2 where an intense 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 temporally averaged signal which is considered as proportional to a
third-order autocorrelation function G共3兲共td兲
G共3兲共td兲 =
冕
I2p共t兲Ipr共t − td兲dt,
共3兲
where td is the delay between incident pump and probe
pulses, I p and Ipr 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
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.
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 estimate 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 joulemeter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
Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 193.52.40.1 On: Tue, 03 May 2016
15:18:40
103106-4
J. Appl. Phys. 105, 103106 共2009兲
G. Boudebs and K. Fedus
TABLE I. Values of the experimentally measured nonlinear coefficients 共n2 and ␤兲 corresponding to the studied
reference materials at different wavelengths.
␤ ⫻ 10−3
共cm/GW兲
Materials
␭
共nm兲
L
共mm兲
SiO2
SiO2
SiO2
355
532
1064
3.76
3.76
3.76
2.0⫾ 0.4
0.9⫾ 0.3
0.4⫾ 0.07
⬍2
⬍0.2
⬍0.2
CS2
CS2
532
1064
1.0
1.0
80⫾ 17
40⫾ 6
⬍17
⬍7
n2 ⫻ 10
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I0 in 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 twophoton 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 obtained n2
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, the n2 value for CS2 in 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–Owyoung13 was 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 transformation of calculated two-photon absorption spectrum in the
two-band configuration14 in order to estimate n2. This model
provides better description rather in semiconductors than in
wideband-gap glasses as fused silica.3 The second one 共the
PERT model15兲 is based on a fourth-order perturbation
theory which assumes that the dispersion of n2 in the visible
and near IR can be attributed to a single resonance in the UV.
All these models indicate a normal dispersion, i.e., n2 in UV
is larger than n2 in 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
and 15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
−20
m /W
2
I0
共GW/ cm2兲
37
76
221
5.5
23
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
n2 value for silica in UV is in a good agreement with values
obtained in Refs. 3 and 16–18.
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.21 On the other hand,
picosecond rotation of molecules gives an important contribution to Kerr susceptibility in CS2 共Refs. 19 and 20兲 and its
n2 should be dependant on pulse duration22 between picoand 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 the n2 of CS2 in the picosecond range should be one order of magnitude larger due to the
molecular contribution. Taking into account an extrapolated
n2 value 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
the n2 value of CS2 are of the same order of magnitude.
Finally, we have performed20 systematic Z-scan n2 determinations of a series of liquids after calibration of the setup
with the n2 value of CS2 given 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 CS2 obtained 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 all n2 measurement experimental procedure. This could explain the dis-
Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 193.52.40.1 On: Tue, 03 May 2016
15:18:40
103106-5
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CS2 and 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
obtained n2 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 of n2 more accurately
taking into account our measured values.
1
A. Major, F. Yoshino, J. S. Aitchison, and P. W. E. Smith, Opt. Lett. 29,
1945 共2004兲.
A. Major, F. Yoshino, J. S. Aitchison, and P. W. E. Smith, Opt. Lett. 30,
2653共2005兲.
3
R. DeSalvo, A. A. Said, D. J. Hagan, E. W. Van Stryland, and M. SheikBahae, IEEE J. Quantum Electron. 32, 1324 共1996兲.
4
D. Milam, Appl. Opt. 37, 546 共1998兲.
5
R. A. Ganeev, A. I. Ryasnyanski, and H. Kuroda, Opt. Spectrosc. 100, 108
2
J. Appl. Phys. 105, 103106 共2009兲
G. Boudebs and K. Fedus
共2006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.
7
G. Boudebs and S. Cherukulappurath, Opt. Commun. 250, 416 共2005兲.
8
M. A. Dugay, in Progress in Optics XIV, edited by E. Wolf 共NorthHolland, Amsterdam, 1976兲.
9
G. Boudebs and S. Cherukulappurath, Phys. Rev. A 69, 053813 共2004兲.
10
A. Gnoli, L. Razzari, and M. Righini, Opt. Express 13, 7976 共2005兲.
11
H.-St. Albrecht, P. Heist, J. Kleinschmidt, D. van Lap, and T. Schroder,
Appl. Phys. B B55, 362 共1992兲.
12
E. L. Falcão-Filho, C. B. de Araújo, C. A. C. Bosco, G. S. Maciel, L. H.
Acioli, M. Nalin, and Y. Messaddeq, J. Appl. Phys. 97, 013505 共2005兲.
13
N. L. Boling, A. J. Glass, and A. Owyoung, IEEE J. Quantum Electron.
14, 601 共1978兲.
14
M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland,
IEEE J. Quantum Electron. 27, 1296 共1991兲.
15
R. Adair, L. L. Chase, and S. A. Payne, Opt. Mater. 1, 185 共1992兲.
16
W. T. White III, W. L. Smith, and D. Milam, Opt. Lett. 9, 10 共1984兲.
17
W. L. Smith, J. H. Bechtel, and N. Bloembergen, Phys. Rev. B 12, 706
共1975兲.
18
T. Shimada, N. A. Kurnit, and M. Sheik-Bahae, in Laser-Induced Damage
in Optical Materials, edited by H. E. Bennett, A. H. Gunther, M. R.
Kozlowski, B. E. Newman, and M. J. Soileau 关Proc. SPIE 2714, 52
共1996兲兴 共Society of Photo-Optical Instrumentation Engineers, Bellingham,
WA, 1981兲;
19
R. W. Boyd, Nonlinear Optics 共Academic, New York, 1992兲.
20
I. Rau, F. Kajzar, J. Luc, B. Sahraoui, and G. Boudebs, J. Opt. Soc. Am. B
25, 1738 共2008兲.
21
F. Billard, “Metrology of the nonlinear refractive index of glasses in nanoseconde, picoseconde and sub-picoseconde regime,” Ph.D. thesis, Universite Paul Cezzanne Aix-Marseille III, 2005.
22
R. A. Ganeev, A. I. Ryasnyansky, M. Baba, M. Suzuki, N. Ishizawa, M.
Turu, S. Sakakibara, and H. Kuroda, Appl. Phys. B: Lasers Opt. 78, 433
共2004兲.
23
M. Falconieri and G. Salvetti, Appl. Phys. B: Lasers Opt. 69, 133 共1999兲.
24
S. Couris, M. Renard, O. Faucher, B. Lavorel, R. Chaux, E. Koudoumas,
and X. Michaut, Chem. Phys. Lett. 369, 318 共2003兲.
6
Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 193.52.40.1 On: Tue, 03 May 2016
15:18:40