Optics Communications 245 (2005) 377–382
www.elsevier.com/locate/optcom
Characteristics of co-existence of third-order and
transient thermally induced optical nonlinearities
in nanosecond regime
Zhi-Bo Liu *, Wen-Yuan Zhou, Jian-Guo Tian *, Shu-Qi Chen, Wei-Ping Zang,
Feng Song, Chun-Ping Zhang
Photonics Center, College of Physics, Nankai University, Tianjin 300071, China
Received 4 June 2004; received in revised form 23 September 2004; accepted 28 September 2004
Abstract
We report the experimental results on characteristics of co-existence of third-order and transient thermally induced
optical nonlinearities using a nanosecond pulse laser and CS2 solutions of nigrosine. Results show that thermally
induced optical nonlinearity is very sensitive to linear absorption, beam waist size and pulsewidth. Numerical simulations obtained by solving simultaneously acoustic and electromagnetic wave equations, agrees basically with experimental results. Meanwhile, the transition of optical nonlinearity from self-focusing pattern to self-defocusing pattern was
observed when the contribution from thermally induced optical nonlinearity exceeds that from third-order optical
nonlinearity.
2004 Elsevier B.V. All rights reserved.
PACS: 46.25 k; 46.25.Jx; 78.20.Nv
Keywords: Thermally induced optical nonlinearity; Third-order nonlinearity; Z-scan
1. Introduction
In recent years thermally induced optical nonlinearities (TION) have been repaid attention
*
Corresponding authors. Tel.: +862223508379; fax:
+862223504856 (Z.-B. Liu).
E-mail addresses: rainingstar@eyou.com (Z.-B. Liu),
jjtian@nankai.edu.cn (J.-G. Tian).
because of their potential applications in optical
limiters in nanosecond regime [1–6]. Transient
TION arising from one-photon and two-photon
absorption have been studied by Z-scan method
and time resolution [7–9]. Brochard et al. [4] have
also reported the experimental and theoretical
study of TION in the transient regime using a 5 ns
pulse laser and Z-scan technique. But they only presented the experimental data about the dependence
0030-4018/$ - see front matter 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.optcom.2004.09.075
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Z.-B. Liu et al. / Optics Communications 245 (2005) 377–382
of TION on the beam radius. Since then there is almost no experimental data reported on thermal
refractive nonlinearity induced in the liquid absorptive medium by several ns laser pulses. For the theoretical investigation of TION, Kovsh et al. [10,11]
developed a numerical modeling of thermal refraction by solving simultaneously acoustic and electromagnetic wave equations.
In this paper, we report the study on the
dynamics of TION of the liquid absorptive medium with third-order nonlinearity in nanosecond
regime. The results show that the changes of liner
absorption a0, beam waist radius x0 and pulsewidth sp have an important influence on TION,
and a transition of optical nonlinearities from
self-focusing pattern to self-defocusing pattern will
occur when the contribution from TION exceeds
that from third-order nonlinearity. There is a basic
agreement between experimental data and numerical simulation. The co-existence of third-order
nonlinearity and TION provides us a better method
to study systematically the mechanism of TION
than the conventional method by using a pure
liquid absorptive medium without third-order
nonlinearity, such as solutions of nigrosine in
water, iodine in ethanol.
k = n0k0 = n0x/c. Dn(r, z, t) is nonlinear refractive
index distribution in the medium, which consists
of the refractive index change coming from optical
third-order nonlinearity and that from TION in
our discussion. Thus,
2
Dnðr; z; tÞ ¼ Dnh ðr; z; tÞ þ n2 jEj ;
ð2Þ
where n2 is third-order nonlinear coefficient,
Dnh(r, z, t) is thermally induced refractive index
change.
Assuming the density change and temperature
change in medium to be a small quantity, we can
obtain the acoustic wave propagation equation
as follows [4,11]:
o2 ðDnh Þ
c2S r2? ðDnh Þ
ot2
Z
ce c2S b t
¼
r2 ða0 Iðr; t0 ÞÞ dt0 ;
2nq cp 1 ?
ð3Þ
where b is the thermal expansion coefficient, q is
the density, n is the refractive index, cp is the constant pressure heat capacity, ce = q(on2/oq)T is the
electrostrictive coupling constant, I(r,t 0 ) is the laser
intensity.
By solving Eqs. (1)–(3) numerically, we can obtain the electric field distribution at the exit face of
medium and Z-scan characteristics for the coexistence of third-order and TION.
2. Theory
For several nanosecond laser pulse, TION
arises from acoustic wave propagation caused by
medium density change after local heating, and
its buildup time is determined by the time required
for a sound wave to propagate across beam size,
sac = x0/cS, where cS is the velocity of sound in
the medium. Applying paraxial approximation
and slowly varying envelope approximation, the
beam propagation in nonlinear medium can be described by following equation [11]:
~ z; tÞ þ 2ik
r2? Eðr;
þ
~ z; tÞ
oEðr;
~ z; tÞ
þ ika0 Eðr;
oz
2k 2
~ z; tÞ ¼ 0;
Dnðr; z; tÞEðr;
n
ð1Þ
~ z; tÞ is electric field envelope of optical
where Eðr;
beam, $^ is transverse Laplace operator,
3. Experiments and samples preparation
The Z-scan was used in our experiments [12]
since it can determine both the magnitude and sign
of nonlinearities simultaneously [13–15]. Our
Z-scan experimental arrangement is the same as
that in [12]. A frequency doubled Continuum Surelite Q-switched Nd:YAG pulse laser with a wavelength of 532 nm, a pulsewidth of 5.6 ns and a
repetition of 10 Hz, is used. A small part of the input beam obtained by a splitter (a glass plate) was
used to monitor the pulse-to-pulse energy fluctuation. At every position, we took the average of 50
data of transmittance (output pulse energy over
incident pulse energy) corresponding to 50 incident
pulses with less than 10% fluctuation of energy.
For all of Z-scan experiments, a sample of pure
CS2 was used as a standard for calibration.
Z.-B. Liu et al. / Optics Communications 245 (2005) 377–382
We choose the CS2 solutions of nigrosine as
samples in our experiments, based on that CS2
has a large thermal optical coefficient, its sign of
third-order nonlinearity is opposite to that of its
thermally induced optical nonlinearity, and it is always used as a conference standard for the determination of nonlinearity amplitude and beam
waist size. Nigrosine is chosen because it has very
little nonlinear response other than TION arising
from linear absorption for nanosecond input
pulses, and it has been used to study TION [4]
and thermal optical limiter [3]. The open-aperture
Z-scans for the CS2 solutions of nigrosine were
performed, and it was verified in our experiments
that the CS2 solutions of nigrosine do not exhibit
nonlinear absorption at 532 nm. CS2 solutions of
nigrosine with several different concentrations
were prepared, and were contained in 1 mm
path-length cells. For the solvent of CS2, the values of physical parameters are b = 1.21 · 103
K1, q = 1.26 kg m3, n = 1.6279, cp = 1.271.32
J cm3 K1, and cS = 1150 1170 m s1 [4].
4. Results and discussion
Because the refractive index change of TION
arises from acoustic wave propagation caused by
absorption-induced local heating, the influence of
TION will increase with the linear absorption if
sp and sac keep constant. Fig. 1 gives the experimental closed-aperture Z-scan curves of CS2 solutions of nigrosine with different linear absorption,
a0 = 0.054, 0.102, 0.152 and 0.206 cm1 at 532 nm,
where the beam waist radius x0 is 7 lm, and the
solid line represents the Z-scan curve of pure CS2
that can be considered as a typical medium with
pure third-order nonlinearity since its linear
absorption coefficient is so small (about 104
cm1) at 532 nm. In the case of x0 = 7 lm, the
buildup time of TION sac is 6.1 ns that is very close
to the pulsewidth sp = 5.6 ns, and the value of sp/
sac is about 0.92. It is seen in Fig. 1 that a transition from self-focusing pattern to self-defocusing
pattern happens as a0 increases. This illustrates
that the contribution to Z-scan curve coming from
TION increases with a0, and will surpass the contribution from third-order nonlinearity, and cause
379
Fig. 1. Closed-aperture Z-scan curves for different linear
absorption coefficient a0: 0.054, 0.102, 0.152 and 0.206 cm1.
The beam waist radius x0 is 7 lm, and pulsewidth sp is 5.6 ns.
The solid line is the Z-scan curve of pure CS2.
a conversion of Z-scan peak-valley pattern. In our
experiments, when the linear absorption coefficient
a0 is about 0.102, the third-order nonlinearity and
TION will counteract, and no obvious Z-scan
peak–valley feature appears.
We define that Z-scan curve difference DTp v
between extrema of the normalized transmittance
peak and valley is positive for a positive nonlinearity, and it is negative for a negative nonlinearity. In
Fig. 2. The ratio DTp v/DTp v(CS2) as a function of linear
absorption coefficient a0. Open circles: experimental results,
filled circles: numerical simulation. The beam waist radius x0 is
7 lm, and pulsewidth sp is 5.6 ns.
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Z.-B. Liu et al. / Optics Communications 245 (2005) 377–382
our experiments, the normalized transmittance
peak-to-valley difference of pure CS2, DTp v(CS2),
was adjusted to the value of 0.22 ± 0.01 by altering
the input intensity before the Z-scan measurements of CS2 solutions of nigrosine were performed. Because there is a positive nonlinearity
for pure CS2, the sign of DTp v(CS2) is positive.
In order to describe the change of TION explicitly,
we normalize DTp v of CS2 solutions of nigrosine
by DTp v(CS2). Fig. 2 gives the ratio of DTp v to
DTp v(CS2) for different linear absorption a0 of
CS2 solutions of nigrosine in the case of x0 = 7
lm (same as that in Fig. 1). The error bars for
experimental data are obtained from the errors
of Z-scan measurements, such as pulse-to-pulse
energy fluctuation, the error of linear absorption
determination. The filled circles are the data of
numerical simulation by solving simultaneously
Eqs. (1)–(3). Since physical parameters cS and cp
used for numerical solution are known with finite
precision, we also provide error bars for theoretical curves. Results show that experimental results
are in good agreement with those of numerical
simulations. The transition of DTp v/DTp v(CS2)
from positive to negative indicates that the contribution of TION to Z-scan curves will surpass that
of third-order nonlinearity as linear absorption
coefficient increases.
If we increase the beam waist size x0 of focused
beam and keep the pulsewidth sp unchanged, the
situation will significantly change. The acoustic
wave will not have enough time to grow within
the pulse duration, and the density and index cannot change significantly. Therefore no TION effect
appears during the pulse. By utilizing a lens with
larger focus length, we obtained a beam waist
radius x0 of 20 lm, which corresponds to sac of
18 ns and sp/sac of about 0.31. The ratios of DTp v
to DTp v(CS2) obtained from experimental data
and numerical simulation for different linear
absorption are given in Fig. 3. Results show that
the ratios of DTp v to DTp v(CS2) almost keeps
unchanged for different linear absorption. This
means that there is no influence coming from
TION. However, numerical simulations have a little dependence on linear absorption. Therefore,
further improvements of theoretical analysis
should be made.
In order to discuss further the influence of x0
on TION, we alter x0 by utilizing focus lenses with
different focus length in our Z-scan experimental
setup. The CS2 solution of nigrosine with a0 of
0.206 cm1 was chosen to perform Z-scan experiments for different x0. The dependence of
DTp v/DTp v(CS2) on x0 is given in Fig. 4.
Results show that the decrease of x0 can accelerate
the buildup of TION, and contrarily the increase
Fig. 3. The ratio DTp v/DTp v(CS2) as a function of linear
absorption coefficient a0: , experimental results; , numerical
simulations. The beam waist radius x0 is 20 lm, and pulsewidth
sp is 5.6 ns.
Fig. 4. The ratio DTp v/DTp v(CS2) as a function of beam
waist radius x0: , experimental results; , numerical simulations. The linear absorption coefficient a0 is 0.206 cm1, and
pulsewidth sp is 5.6 ns.
Z.-B. Liu et al. / Optics Communications 245 (2005) 377–382
of x0 can delay the buildup of TION. When x0 is
larger than a certain value, DTp v/DTp v(CS2)
will be close to 1 gradually. This is because that
TION has very little or no contribution to optical
nonlinearities. Therefore, the relationship between
x0 and TION is very important to the measurements of optical nonlinearity by Z-scan technique
using several nanosecond pulse laser. The Z-scan
characteristics of the co-existence of third-order
nonlinearity and TION in CS2 solution of nigrosine can give us the information that how much
x0 should be at least if the influence of TION
can be neglected when we measure the third-order
nonlinearity of medium by Z-scan technique.
Leaving sac unchanged, we can obtain different
values of sp/sac by altering sp and study further the
impact of sp/sac on TION. Altering sp/sac in this
way is convenient since it allows us to keep the
experimental geometry arrangement unchanged.
In our experiments, the change of sp can be realized by the adjustment of Q-delay of Continuum
Surelite Q-switched Nd:YAG laser. Fig. 5 gives
the pulse shape in time with different sp. The solid
lines in Fig. 5 are the Gaussian fittings. Results
show that the pulses still hold a good Gaussian
temporal profile after the adjustment of Q-delay.
We chose x0 of 9 lm and the CS2 solution of nigrosine with a0 = 0.102 cm1 to perform our Z-scan
measurements. In our experiments, DTp v(CS2)
was also controlled to be 0.22 ± 0.01 by adjusting
input intensities.
Fig. 5. The pulse shapes in time with different pulsewidth sp:
5.60, 5.93, 6.97, 9.17, 11.68 and 14.75 ns. The solid lines are the
Gaussian fittings.
381
Fig. 6. Closed-aperture Z-scan curves for different pulsewidth
sp: 5.60, 5.93, 6.97, 9.17, 11.68 and 14.75 ns. The beam waist
radius x0 is 9 lm and linear absorption coefficient a0 is 0.102
cm1. The solid line is the Z-scan curve of pure CS2.
Fig. 6 gives the Z-scan curves for different sp:
5.60, 5.93, 6.97, 9.17, 11.68 and 14.75 ns. The solid
line represents the Z-scan curve of pure CS2. Results show that the TION becomes strong gradually as pulsewidth increases. This is because that
as pulse broadens, the value of sp/sac increases,
and the buildup time of TION sac = 7.8 ns becomes short enough for the growth of TION and
TION will accumulate fast when pulses propagate
Fig. 7. The ratio DTp v/DTp v(CS2) as a function of pulsewidth sp. Open circles: experimental results, filled circles:
numerical simulations. The beam waist radius x0 is 9 lm, and
linear absorption coefficient a0 is 0.102 cm1.
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Z.-B. Liu et al. / Optics Communications 245 (2005) 377–382
through the medium. The experimental results and
numerical simulations of DTp v/DTp v(CS2) for
different sp are given in Fig. 7. Results show that
there is a basic agreement between them, and the
tendency of DTp v as a function of sp is also identical. The largest DTp v, corresponding to
sp = 14.75 ns (sp/sac is about 1.9), is about five
times more than DTp v(CS2). This implied that
the influence of sp on TION is of paramount
important. Therefore, one should consider the
contribution of TION to experimental results for
Z-scan measurements using several nanosecond
pulsed laser.
The above results illustrates that if the laser
pulsewidth is comparable with or larger than the
acoustic transient time, the contribution coming
from TION is important and even dominant, compared with other nonlinearity such as third-order
nonlinearity. Besides linear absorption, change of
sp/sac caused by different beam waist radius or pulsewidth can also affect TION greatly. If sp/sac is small
enough (<0.3 in our experiments), TION can be ignored even in a medium with large linear absorption
coefficient. Based on all of experiments and considerations, we conclude that TION is determined by
both linear absorption (if the medium do not exhibit
nonlinear absorption) and the ratio sp/sac.
5. Conclusion
In summary, the influences of linear absorption,
beam waist radius and pulsewidth on TION in
nanosecond regime have been analyzed experimentally and theoretically. It is found that the effect of
TION is enhanced by the increase of linear absorption, the pulse broadening or the diminution of
beam waist radius, and the transition of optical
nonlinearity from self-focusing pattern to selfdefocusing pattern was observed in CS2 solutions
of nigrosine. It may be the first time that these phenomena are observed experimentally in nanosecond regime when third-order nonlinearity and
TION co-exist. In most of cases, the experimental
results are well agreement with the theoretical
ones. However, when sp/sac is much smaller than
1 (e.g., Fig. 3), it appears that there is an obvious
disagreement between experimental and theoreti-
cal results. We think that in the processing of theoretical model derivation, some approximate
conditions will not be valid in the case of sp/
sac 1, e.g., the assumptions that the change of
density, temperature and pressure be small enough
will be not met. These will cause this disagreement.
Therefore, theoretical description of TION should
be improved further in order to have a better
agreement with the experimental results.
Acknowledgements
This research is supported by Project 60025512
supported by the National Natural Science Foundation of China, Fok Ying Tung Education Foundation (Grant 71008), Natural Science Foundation
of Tianjin (Grant 043601211), and Innovation
Foundation of Nankai University.
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