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.2004.09.075 378 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. 380 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. 382 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. 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