Z-scan study of thermal nonlinearities in silicon naphthalocyanine-toluene solution with the excitations of the picosecond pulse train and nanosecond pulse Sidney S. Yang1 , Tai-Huei Wei2 , Tzer-Hsiang Huang3 , and Yun-Ching Chang1 1 Institute of Photonics Technologies, National Tsing Hua University, Hsinchu 300, Taiwan, R.O.C. [email protected] 2 Department of Physics, National Chung-Cheng University, Ming-Hsiung, Chia-Yi 621, Taiwan, R.O.C. [email protected] 3 Department of Electronic Engineering,Wu Feng Institute of Technology, Ming-Hsiung, Chia-Yi 621, Taiwan, R.O.C. Abstract: Using the Z-scan technique, we studied the nonlinear absorption and refraction behaviors of a dilute toluene solution of a silicon naphthalocyanine (Si(OSi(n-hexyl)3 )2 , SiNc) at 532 nanometer with both a 2.8-nanosecond pulse and a 21-nanosecond (HW1/eM) pulse train containing 11 18-picosecond pulses 7 nanosecond apart. A thermal acoustic model and its steady-state approximation account for the heat generated by the nonradiative relaxations subsequent to the absorption. We found that when the steady-state approximation satisfactorily explained the results obtained with a 21-nanosecond pulse train, only the thermal-acoustic model fit the 2.8-nanosecond experimental results, which supports the approximation criterion established by Kovsh et al. © 2007 Optical Society of America OCIS codes: (190.4870)Optically induced thermo-optical effects; (190.4710)Optical nonlinearities in organic materials; (000.6850)Thermodynamics References and links 1. J. W. Perry, L. R. Khundkar, D. L. Coulter, D. Alvarez, Jr., S. R. Marder, T. H. Wei, M. J. Sence, E. W. Van Stryland, and D. J. Hagan, in Organic Molecules for Nonlinear Optics and Photonics, NATO ASI Series E, J. Messier, F. Kajzar, and P. Prasad, eds, (Kluwer, Dordrecht, 1991), Vol. 194, pp. 369-382. 2. J. S. Shirk, J. R. Lindle, F. J. Bartoli, C. A. Hoffman, A. H. Kafafi, and A. W. Snow, ”Off-resonat third-order optical nonlinearities of meta-substituted phthalocyanines,” Appl. Phys. Lett. 55, 1287-1288 (1989). 3. T. H. Wei, D. J. Hagan, M. J. Sence, E. W. V. Stryland, J. W. Perry, and D. R. Coulter, ”Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines,” Appl. Phys. B 54, 46-51 (1992). 4. A. A. Said, T. Xia, D. J. Hagan, A. Wajsgrus. S. Yang, D. Kovsh and E. W. Van Stryland, in Conference on Nonlinear Optical Liquids, Proc. SPIE-2853, (1996). #77284 - $15.00 USD (C) 2007 OSA Received 21 November 2006; revised 26 January 2007; accepted 29 January 2007 19 February 2007 / Vol. 15, No. 4 / OPTICS EXPRESS 1718 5. J. W. Perry, K. Mansour, J. Y. S. Lee, X. L. Xu, P.V. Bedwhorth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, ”Organic optical limiter with a strong nonlinear absorptive response,” Science 23, 1533-1536 (1996). 6. S. R. Mishra, H. S. Rawat, M. P. Joshi and S. C. Mehendale, ”The role of non-linear scattering in optical limiting in C60 solution,” J. Phys. B:At. Mol. Phys. 27, 157-163, (1994). 7. T. Tomiyama, I. Watanabe, A. Kuwano, M. Habiro, N. Takane, and M. Yamada, ”Rewritable optical-disk fabrication with an optical recording material made of naphthalocyanine and polythiophene,” Appl. Opt. 34, 8201-8208 (1995). 8. J. Seto, S. Tamura, N. Asai, N. Kishii, Y. Kijima, and N. Matsuzawa, ”Macrocyclic functional dyes: Applications to optical disk media, photochemical hole burning and non-linear optics,” Pure and Appl. Chem. 68, 1429-1434 (1996). 9. D. I. Kovsh, S. Yang, D. J. Hagan, and E. W. Van Stryland, ”Nonlinear optical beam propagation for optical limiting,” Appl. Opt. 38, 5168-5180 (1999). 10. P. Brochard and V. Grolier-Mazza, ”Thermal nonlinear refraction in dye solution: a study of the transient regime,” J. Opt. Soc. Am. B 14, 405-414 (1997). 11. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, ”Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760-769 (1990). 12. D. G. Mclean, R. L. Sutherland, M. C. Brant, D. M. Brandelik, P. A. Fleitz, T. Pottenger. ”Nonlinear absorption study of a C60-toluene solution,” Opt. Lett. 18, 858-860 (1993). 13. T. H. Wei, T. H. Huang, S. Yang, D. Liu, J. K. Hu and C. W. Chen, ”Z-scan study of optical nonlinearity in C60-toluene solution,” Mol. Phys. 103, 1847-1857 (2005). 14. T. H. Wei and T. H. Huang, ”A study of photophysics using the Z-scan technique: lifetime determination for high-lying excited states,” Opt. and Quantum Electron. 28, 1495-1508 (1996). 15. A. Seilmeier and W. Kaiser, in Ultrashort Laser Pulses 2nd ed., W. Kaiser, eds. (Springer-Verlag, Berlin, 1993), pp. 305. 16. J. H. Brannon and D. Madge, ”Picosecond laser Photophysics. group 3A phthalocyanines,” J. Am. Chem. Soc. 102, 62-65 (1980). 17. C. Jensen, in High Power Dye Lasers, F. J. Durate, eds. (Springer-Verlag, Berlin, 1991), pp. 48. 18. C. Li, L. Zhang, M. Yang, H. Wang, and Y. Wang, ”Dynamic and steady-state behaviors of reverse satura absorption in metallophthalocyanines,” Phys. Rev. A 49, 1149-1157 (1994). 19. T. H. Wei, T. H. Huang, and M. S. Lin, ”Signs of nonlinear refraction in chloroaluminum phthalocyanine solution,” Appl. Phys. Lett. 72, 2505-2507 (1998). 20. D. R. Lide. in CRC Handbook of Chemistry and Physics, 77th ed., D. R. Lide and et al, eds. (CRC Press, Boca Raton, 1996), pp. 6-128. 21. D. I. Kovsh, D. J. Hagan, and E. W. Stryland, ”Numerical modeling of thermal refraction in liquids in the transient regime,” Opt. Express 4, 315-327 (1999). 22. D. Landau and E. M. Lifshitz, in Course of theoretical physics (Pergamon Press), Vol. 6. 23. J. -M. Heritier, ”Electrostrictive limit and focusing effects in pulsed photoacoustic detection,” Opt. Comm. 44, 267-272 (1983). 24. C. W. Chang, M. S. thesis, National Chung Cheng University, pp. 28, (1999). 1. Introduction Two-dimensional molecules with π -conjugated electron systems, such as porphyrins, phthalocyanines, and their derivatives, and their nonlinear optical properties have been widely investigated recently [1, 2]. These molecules show potential in optical-limiting applications due to their large excited-state absorption cross sections in both singlet and triplet manifolds within the visible spectrum [3-6]. They are also good candidates for optical recording materials because of their large nonlinear refraction in the infrared regime [7, 8]. Using the Z-scan technique, we characterized the nonlinear absorption and refraction properties of a silicon naphthalocyanine (Si(OSi(n-hexyl)3 )2 , dubbed SiNc)-toluene solution at 532 nanometer (nm). Using a laser pulse with a width of τ = 2.8 nanoseconds (ns) (HW1/eM) and a Gaussian distributed train, composed of 11 18-picosecond (ps) pulses 7 ns apart, with an envelope width of τenv = 21 ns, nonradiative relaxations induced a thermal lensing effect (∆ntherm ), in addition to internal nonlinearities, is expected to contribute to the nonlinearities. ∆ntherm results from a temperature rise (∆θ ), caused by nonradiative relaxation subsequent to optical excitation, and the solvent density change (∆ρ ) induced by a ∆θ -driven thermal acoustic wave. Strictly speaking, ∆ρ needs to be derived by solving the thermal acoustic wave; however, a steady-state approximation of the wave equation #77284 - $15.00 USD (C) 2007 OSA Received 21 November 2006; revised 26 January 2007; accepted 29 January 2007 19 February 2007 / Vol. 15, No. 4 / OPTICS EXPRESS 1719 BS1 BS2 Sample D6 -z +z D4 D5 Fig. 1. The Z-scan experimental setup. D4 , D5 , and D6 are photodetectors. BS1 and BS2 are beam splitters. A sample placed on a motion control stage can be moved from -z to +z. can be made to simplify the calculation of ∆ρ provided that the pulse duration is more than 1.5 times longer than the thermal transit time τac (time for the acoustic wave to propagate across the beam cross section) [9]. Given the acoustic wave speed of νs = 1170 m/s for the solvent (toluene)[10], we respectively focused a 2.8-ns pulse and a 21-ns pulse train to have a beam waist radius of w0 = 14.1 µ m and 18.9 µ m (HW1/e2 M for both) in this study. This resulted in τac = w0 /νs = 12.0 ns for the 2.8-ns pulse and τac = w0 /νs = 16.2 ns for the pulse train. The steady state approximation was relatively appropriate for a 21-ns pulse train (τenv /τac = 1.3) compared with a 2.8-ns pulse (τ /τac = 0.2). In this paper, we respectively derive ∆ρ by strictly solving the thermal acoustic wave equation and from the steady state approximation for both a 2.8-ns pulse and a 21-ns pulse train. As a result, ∆ρ obtained using both approaches yields close ∆ntherm ’s for a 21-ns pulse train but causes significantly different ∆ntherm ’s for a 2.8-ns pulse (vide infra). 2. Experiments The Z-scan technique (Fig. 1) is a simple yet sensitive technique for measuring the nonlinear absorption and refraction of materials. Its operation has been described in detail by Sheik-Bahae et al. [11]. Briefly, the beam splitter BS1 splits and directs a small portion of the incident pulse to the detector D4 , which monitors the fluctuation of the incident pulse energy. The rest of the pulse is tightly focused by a lens and transmitted through the sample at various positions (z) relative to the beam waist at z = 0. The beam splitter BS2 divides the transmitted pulse into two and directs them to detectors D6 and D5 . When D6 monitors the total transmitted pulse energy, D5 , which has an aperture in front, measures the energy of the axial portion of a transmitted pulse. With the sample in the linear regime, we carefully adjusted the aperture radius to allow 40% of the transmitted energy to reach D5 . We devided D6 and D5 by D4 and then normalized the values with the corresponding values obtained in the linear regime (at the starting z), which yielded the normalized transmittance (NT) and the normalized axial transmittance (NTa ) as a function of sample position z. Because D6 (which has no aperture) collected all the transmitted energy, NT involves nonlinear absorption alone. The partially obstructed D5 reflects beam broadening or narrowing at the aperture, a result of nonlinear refraction, in addition to nonlinear absorption. NTa reveals, therefore, not only the nonlinear absorption, but also the nonlinear refraction. If we divide NTa by NT, the resultant ratio (NTd ) retains only the information of nonlinear refraction. The 21-ns pulse trains and 2.8-ns pulses used in this study were generated using a Q-switched and mode-locked Nd:YAG laser and a seeding injected Q-switched Nd:YAG laser respectively. Both lasers were frequency doubled to have a wavelength of λ = 532 nm and were operated in #77284 - $15.00 USD (C) 2007 OSA Received 21 November 2006; revised 26 January 2007; accepted 29 January 2007 19 February 2007 / Vol. 15, No. 4 / OPTICS EXPRESS 1720 Fig. 2. The temporal profile of the full pulse envelope. The numbers above spikes mark their order. the TEM00 mode at 10 Hz. The incident intensity (Fig. 2) and phase of the nth pulse in a train are, respectively, " # 2 w0 −2r2 t − n × 7ns 2 (n) (1) I0 = I (z, r,t) = I00 × 2 × exp 2 × exp − w (z) w (z) τ and φ0 = φ (z, r,t) = − kr2 . 2R(z) (2) Here, w(z) = w0 [1 + (z/z0 )2 ]1/2 is the laser beam radius (HW1/e2 M) at z. w0 denotes w(0) and (n) equals 21 µ m. I00 is the on-axis peak intensity at z = 0. τ is the pulse width (HW1/eM) and equals 18 ps. t and r refer to the temporal distribution of the intensity relative to the peak of the 0th pulse and the lateral distribution of the laser beam, respectively. k = 2π /λ (λ = 532 nm) is the wave propagation number. R(z) = z[1 + (z0 /z)2 ] is the curvature radius of the wave front at z. z0 = π w20 /λ is the diffraction length. All the above-introduced parameters pertain to free space. Integration of Eq. (1) over the pulse width (from −∞ to ∞) and the beam cross section (n) relates I00 to the pulse energy ε (n) as (n) I00 = 2ε (n) . π 3/2 w20 τ (3) Because the envelope of the pulse train fits with a Gaussian function peaked at the 0th pulse with a HW1/eM width of τenv = 21 ns, the energy of the nth pulse is " 2 # n × 7ns . (4) ε (n) = ε (0) × exp − τenv #77284 - $15.00 USD (C) 2007 OSA Received 21 November 2006; revised 26 January 2007; accepted 29 January 2007 19 February 2007 / Vol. 15, No. 4 / OPTICS EXPRESS 1721 Summing up this equation from n = −5 to 5, we obtain the full pulse train energy εt , with which ε (n) is expressed as 2 n×7ns εt × exp − τenv " ε (n) = (5) #. 5 m × 7ns 2 ∑ exp − τenv m=−5 (n) I00 needed in our theoretical analysis is derived from εt , experimentally measured by D4 , via Eqs. (3)-(5). The incident energy of a 2.8-ns pulse is expressed as 2 2 w t −2r2 × exp − I0 = I (z, r,t) = I00 × 2 0 × exp 2 , (6) w (z) w (z) τ and its phase is expressed by Eq. (2). In Eq. (6), we maintain the same notations used in Eq. (n) (1) except that I00 replaces I00 and denotes the on-axis peak intensity at z = 0. When I00 is needed in our theoretical simulation, it is related to the pulse energy ε , measured by D4 , as I00 = 2ε /π 3/2 w20 τ with w0 =14.1 µ m and τ = 2.8 ns. Using a 21-ns pulse train with a full train energy of εt = 0.8 µ J and 1.4 µ J, and a 2.8-ns pulse with a pulse energy of ε = 1.4 µ J and 2.5 µ J, we performed, at room temperature θe = 25 ◦ C, Z-scan measurements on a SiNc-toluene solution with a concentration of 6.1×1017 cm−3 and contained in a 1-mm-thick quartz cell. 3. Theoretical model Based on the 5-energy-band model [12], we interpret optical excitation and the associated population redistributions among various energy bands as well as the subsequent intramolecular conversion of absorbed photo energy as intramolecular heat [13]. We also explain the following intermolecular (solute-solvent) energy transfer, which leads to ∆θ and ∆ρ of the solution in sequence. Each energy band, including the associated zero-point level | 0) and vibronic level | ν 6= 0), is conventionally named Si for the singlet manifold and Ti for the triplet manifold (Fig. 3). The subscript i refers to the ordering of the electronic states. At thermodynamic equilibrium, all SiNc molecules reside in S0 and the solution has an equilibrium temperature of θe = 25 ◦ C and a solvent density of ρe = 0.79 g·cm−3 throughout the solution. The equations governing the intensity attenuation and phase change with the penetration depth z′ into the sample can be written as[14] dI (7) = −[(σ a)S0 NS0 + (σ a)S1 NS1 + (σ a)T 1 ]I − β NS0 I 2 dz′ and dφ = [(σ r)S0 NS0 + (σ r)S1 NS1 + (σ r)T 1 NT 1 ]I dz′ + γ NS0 I + kn2 I + k∆ntherm , (8) where I and φ are the intensity and phase, respectively, of a 2.8-ns pulse or an individual 18ps pulse within each train. α is the absorption coefficient and ∆n denotes the refractive index change. In Eqs. (7) and (8), I and φ changes are contributed to the one-photon excitations S0 →| ν )S1 , S1 →| ν )S2 , and T1 →| ν )T2 represented by their first three terms. σ a in Eq. (7) denotes the absorption cross section of the states specified by the subscripts. The σ r, the refractive cross section, of a band can be derived from σ a associated with the same band according to the Kramers-Kr¨onig relation. The 4th terms in Eqs. (7) and (8) pertain to the two-photon #77284 - $15.00 USD (C) 2007 OSA Received 21 November 2006; revised 26 January 2007; accepted 29 January 2007 19 February 2007 / Vol. 15, No. 4 / OPTICS EXPRESS 1722 |3) |3) |0) S 2 |1) |0) T 2 |6) W IC |2) |1) |0) S 1 W ISC |4) |3) |2) |1) W ISC |0) T1 Wf |0) S 0 Fig. 3. A five-energy-band model for SiNc-toluene: upward-pointing arrows, wiggly lines, and downward-pointing arrows indicate optical excitation, non-radiative relaxation and radiative relaxation, respectively. | ν ) refers to vibrational eigenstate. τ denotes lifetime (ISC ≡intersystem crossing, IC ≡internal conversion, and f ≡fluorescence). excitation S0 →| ν )S2 . The 5th term in Eq. (8) denotes the Kerr effect of the solvent (toluene), n2 = 5.5 × 10−15 cm2 /W, and the 6th term of Eq. (8) represents the thermal effect where ∆ntherm will be respectively estimated via Eq. (18) in combination with Eq. (16) or via Eq. (19) alone in this study (vide infra). Combining optical excitation with a 532-nm pulse and the subsequent relaxation, the population redistributes in various states with time rates of [13] (σ a)S0 NS0 I β NS0 I 2 NS1 NT 1 dNS0 =− − + + , dt 2¯hω τf τT 1 h¯ ω dNS1 (σ a)S0 NS0 I β NS0 I 2 NS1 NS1 = − , + − h¯ ω dt 2¯hω τf τISC (9) (10) and dNT 1 NS1 NT 1 . = − dt τISC τT 1 (11) N and τ are respectively the population density and relaxation time constant of the band specified by the subscripts. ω stands for the angular frequency of the laser. One-photon-excitation S0 →| ν )S1 and two-photon-absorption S0 →| ν )S2 induced population-density redistributions between S0 and | ν )S1 are denoted by the first two terms of Eqs. (9) and (10). Nonradiative relaxations | ν )S1 | 0)S1 and | ν )S2 | 0)S1 are assumed to follow the above mentioned excitations well within the pulse width (18 ps or 2.8 ns) [14, 15]. The 3rd terms on the right-hand side #77284 - $15.00 USD (C) 2007 OSA Received 21 November 2006; revised 26 January 2007; accepted 29 January 2007 19 February 2007 / Vol. 15, No. 4 / OPTICS EXPRESS 1723 of Eqs. (9) and (10) denote fluorescent-decay | 0)S1 S0 induced population redistributions between | 0)S1 and S0 , τ f being the lifetime and equal to 3.1 ns [1]. Population redistribution between | 0)S1 and | 0)T1 via intersystem crossing (ISC) | 0)S1 | ν )T1 | 0)T1 is expressed by the 4th term of Eq. (10) and the 1st term of Eq. (11), respectively, τISC being the lifetime and equal to 16 ns [16]. Population redistribution between | 0)T1 and S0 as a result of intersystem crossing (ISC) | 0)T1 S1 , is expressed by the 4th term of Eq. (9) and the 2nd term of Eq. (11), respectively, τT 1 being the lifetime and falling in the µ s regime [17]. Since | 0)T1 →| ν )T2 absorption is verified in this paper (vide infra), we neglect the population redistributions between | 0)T1 and | ν )T2 because the | ν )T2 | 0)T2 | ν )T1 | 0)T1 relaxation subsequent to the excitation is believed to be much shorter than the pulse widths (18 ps or 2.8 ns) [14, 15, 18]. Accompanying the rapid nonradiative decays | ν )S1 | 0)S1 , | ν )S2 | 0)S1 , and | ν )T1 | 0)T1 [18], excess energy conceivably redistributes among various vibrations in the solute molecules, presumably via vibronic interaction, and turns into heat within the molecule at the speed of [13] dQ (σ a)S0 NS0 I = × h¯ (ω − ωS1 ) + (σ a)S1 NS1 I dt h¯ ω β NS0 I 2 × h¯ (2ω − ωS1 ) + (σ a)T 1 NT 1 I, + 2¯hω (12) where Q denotes the thermal energy accumulated within the solute molecules per unit volume. The first term on the right-hand side represents the heat generated via | ν )S1 | 0)S1 relaxation subsequent to the S0 →| ν )S1 excitation. ωS1 (λS1 = 780 nm) corresponds to the energy of | 0)S1 relative to S0 . The second and third terms describe the contributions of the sequential | ν )S2 | 0)S2 | ν )S1 | 0)S1 relaxations following the one-photon | 0)S1 →| ν )S2 excitation and the two-photon S0 →| ν )S2 excitation, respectively. The last term describes the contribution of the sequential relaxations, | ν )T2 | 0)T2 | ν )T1 | 0)T1 , following the one-photon excitation | 0)T1 →| ν )T2 . ∆θ occurs after the intramolecular heat (Q) dissipates throughout the surrounding solvent molecules in a local thermal equilibrium time τtherm . For the concentration (6.1 × 1017 cm−3 ) of present interest, τtherm is estimated to be 65 ps [19]. Since τtherm is significantly shorter than its pulse width, ∆θ is considered to increase simultaneously with Q when the sample is interacting with a 2.8-ns pulse. As a result, ∆θ at time t can be obtained as ∆θ = 1 ρCp Z t dQ −∞ dt ′ dt ′ . (13) When the leading edge of a 2.8-ns pulse (t = −∞) encounters the sample, ∆θ = 0. In Eq. (13) Cp denotes the isobaric specific heat and equals 1.71 J/g◦ C for toluene [20]. On the other hand, because τtherm is considerably longer than its width, an individual 18-ps pulse, say the nth, in a 21-ns train does not experience the thermal lensing effect induced by itself, but yields a temperature rise of Z ∞ dQn ′ 1 ∆θn = dt , (14) ρ Cp −∞ dt ′ for the following pulses to experience. The subscript n is introduced to single out ∆θ and Q caused by the nth pulse. Denoting the leading pulse in a train as the −5th one, the nth pulse in a train encounters the sample with a temperature rise of ∆θ = n−1 ∑ ∆θn′ . (15) −5 When the −5th pulse interacts with the sample, ∆θ = 0. Since the intersystem crossing time constant (τISC =16 ns) and µ s order for τT 1 are greatly longer than the relaxation time constants #77284 - $15.00 USD (C) 2007 OSA Received 21 November 2006; revised 26 January 2007; accepted 29 January 2007 19 February 2007 / Vol. 15, No. 4 / OPTICS EXPRESS 1724 within the singlet and triplet manifolds, heat generated via | 0)S1 | ν )T1 | 0)T1 and | 0)T1 S0 relaxations are ignored. How ∆θ drives an acoustic wave equation and thus induces ∆ρ can be understood via the thermal-acoustic wave equation. Based on three main equations of hydrodynamics: continuity (mass conservation), Navier-Stokes (momentum conservation), and energy transport equation (energy conservation), the wave equation has been derived [21-23] as ∂ 2 (∆ρ ) γe 2 − νs2 ∇2 (∆ρ ) = bρνs2 ∇2 (∆θ ) − ∇ I, ∂ t2 2nc (16) where νs is the velocity of the acoustic wave and equals 1170 m/s, b = −ρ (∂ ρ /∂ θ ) p , with the subscript p denoting the pressure, is the volume expansivity and equals 1.25×10−3 (◦ C−1 )[20], and γe = ρ (∂ n2 /∂ ρ )θ is the electrostrictive coupling constant. All the parameters pertain to the solvent (toluene). According to the Lorenz-Lorenz law, γe can be expressed as (n2 − 1)(n2 + 2)/3 with n denoting the refractive index [21]. Given the linear refractive index n0 = 1.49 for toluene [20], γe is estimated to be 1.71. As will be shown later, the second term on the righthand side of Eq. (16) (the electrostrictive effect) does not play a significant role compared with the first term (thermal effect) for our absorptive solution [21]. For the interaction of a 2.8-ns pulse with the sample, we substitute ∆θ derived from Eq. (13), in combination with I, into Eq. (16) to solve for ∆ρ with the initial condition of ∆ρ = 0 and ∂ (∆ρ )/∂ t = 0 at t = −∞. Regarding the interaction of the nth 18-ps pulse in a train with the solution, we substitute ∆θ derived from Eq. (15), in combination with I, into Eq. (16) to derive ∆ρ for the nth pulse to experience. Time integrations of ∂ 2 (∆ρ )/∂ t 2 and ∂ (∆ρ )/∂ t over the pulse separation of 7 ns are involved in solving the differential Eq. (16). Accompanying these integrations, ∆ρ and ∂ (∆ρ )/∂ t experienced by the (n − 1)th pulse are used as the initial conditions. Given ∆ρ = ∂ (∆ρ )/∂ t = 0 for the leading (−5th) pulse, ∆ρ and ∂ (∆ρ )/∂ t for each later pulse in a train can be obtained one by one. Once after ∆θ and ∆ρ are obtained for a 2.8-ns pulse or a 21-ns pulse train, thermally induced refractive index change can be deduced as ∂n ∂n ∆θ + ∆ρ . (17) ∆ntherm = ∂θ ρ ∂ρ θ Since the 1st term on the right-hand side is considerably smaller than the 2nd one [21], and (∂ n/∂ ρ )θ = γe /2nρ , as derived from γe = ρ (∂ n2 /∂ ρ )θ , Eq. (17) can be approximated as γe ∆ntherm ∼ ∆ρ . = 2nρ (18) Eq. (18) in combination Eq. (16) is suitable for analyzing the thermal effect of an absorptive solution. When τ or τenv is considerably longer than τac , the second-order time derivative of ∆ρ , i.e., the 1st term on the left-hand side of Eq. (16), can be ignored. This simplifies Eq. (16) as ∆ρ = −bρ ∆θ + γe I/2ncνs2 , which in turn approximates Eq. (18) as γ2 bγe ∆θ + 2e 2 I. ∆ntherm ∼ =− 2n 4n cνs (19) The Z-scan experiments are numerically fitted by calculating the normalized transmittance (NT) and the normalized axial transmittance (NTa ). Via Eqs. (7) and (8), we integrate through the thickness of the sample to obtain the intensity and the phase at the exit surface of the sample considering the initial input intensities given by Eqs. (1) to (5) and by Eq. (6) for ps pulse trains and ns pulses, respectively. Huygens-Fresnel formalism is thus applied to calculate the intensity distribution at the aperture. (σ a)S0 = 2.8 × 10−18 cm2 , (σ a)S1 = 5.0 × 10−17 cm2 , (σ r)S0 ∼ = #77284 - $15.00 USD (C) 2007 OSA Received 21 November 2006; revised 26 January 2007; accepted 29 January 2007 19 February 2007 / Vol. 15, No. 4 / OPTICS EXPRESS 1725 0, β = 0, (σ r)S1 = 1.2 × 10−18 cm2 , γ = 0, and n2 = 5.5 × 10−15 cm2 /W were previously determined in the study with single 18-ps pulses switched out of the pulse trains using a Pockels cell [24]. The triplet contributions and thermal effect were ignored in the fitting because the pulse duration is much shorter than both the intersystem crossing time and the thermal lensing formation time. The population densities required in Eqs. (7) and (8) are functions of both space and time. The dynamic behaviors can be obtained by calculating the rate equations (9) to (11). 4. Results and discussion In the presentation of the Z-scan data below, NT and NTa are marked with triangles and squares, respectively. NTd is marked with dots. Solid lines and dash lines represent the theoretical fitting with ∆ntherm from Eq. (18) in combination with Eq. (16) and that with Eq. (19) alone. We will discuss the results of the two different input excitations separately. 4.1. Pulse train results Figures 4 and 5 respectively show the experimental results obtained with εt =0.8 µ J and 1.4 µ J. There are two parameters, (σ a)T 1 and (σ r)T 1 , undetermined by the single 18-ps pulse Zscan experiments. Using (σ a)T 1 = 6.0 × 10−17 cm2 and (σ r)T 1 = −5.5 × 10−17 cm2 , we best fit the results with ∆ntherm determined using Eq. (18) in combination with Eq. (16). However, only a small deviation is generated when ∆ntherm is determined using Eq. (19) alone given dn/d θ ≈ (∂ n/∂ ρ )θ (∂ ρ /∂ θ ) p = −bγe /(2n), estimated to be −6.0× 10−4 (◦ C−1 ) using n = 1.49. Therefore, we claim that the ratio of τenv /τac = 1.3 can be considered large enough to satisfy the steady-state assumption. Another observation can be made. The thermal-lensing effect is obviously negative that can be easily confirmed by the fact that NTd is greater than 1 before, and less than 1 after, the beam waist in the Z-scan data (Figs. 4(c) and 5(c)). This type of Z-scan data indicates that the solution possesses negative nonlinear refraction, which we would expect from an absorptive liquid solution [11]. One interesting observation is that, although the steady-state equation, Eq. (19), can be applied to the data of both energy levels reasonably well, the data obtained with higher energy level (1.4 µ J) seems to be fitted better than the data obtained with lower level (0.8 µ J) (Figs. 4(b) and 5(b)). However, this is not generally true. When the thermal acoustic equation, Eq. (16), is used to emulate the thermal effect, the density variation ∆ρ is driven by the rising temperature ∆θ . Because the governing equation is an acoustic wave equation, we expect a wave-like profile of the density variation ∆ρ in the solution. So, too, should be the profile of the refractive index, since Eq. (18) reveals the proportionality. The approximation made in Eq. (19) smoothes the wiggling spatial feature and provides an averaged index refraction profile similar to the Gaussian profiles of the driving rising temperature, ∆θ , and the intensity I. Although the induced index refraction change is linearly proportional to the incident energy, the distortion of the refracted laser pulse in the far field (where the aperture is located) does not possess the proportionality when the induced thermal lens is strong. Therefore, the discrepancies (or errors) between the approximated simulation curve and the acquired Z-scan data are not expected to follow the variation of the incident pulse energies. We must also realize that Z-scan experimental data is obtained using an energy meter which neglects the fine spatial dependence even in a closed-aperture setup. Actually, the result obtained from the averaged steady-state equation, Eq. (19), can occasionally even out-fit the one obtained from the thermal-acoustic equation, Eq. (18), in combination with Eq. (16), because Eq. (16) is an approximation as well. #77284 - $15.00 USD (C) 2007 OSA Received 21 November 2006; revised 26 January 2007; accepted 29 January 2007 19 February 2007 / Vol. 15, No. 4 / OPTICS EXPRESS 1726 Fig. 4. The Z-scan curves for 21-ns pulse trains with an energy level of 0.8 µ J. (a) NT: triangles stand for the experimental result without an aperture, and the solid line for the theoretical simulation. (b) NTa : squares stand for the experimental result with an aperture, the solid line for the theoretical fit with Eqs. (16) and (18), and the dashed line for the theoretical fit with Eq. (19) alone. (c) NTd : dots stand for the experimental result with an aperture, the solid line for the theoretical fit with Eqs. (16) and (18), and the dashed line for the theoretical fit with Eq. (19) alone. #77284 - $15.00 USD (C) 2007 OSA Received 21 November 2006; revised 26 January 2007; accepted 29 January 2007 19 February 2007 / Vol. 15, No. 4 / OPTICS EXPRESS 1727 Fig. 5. The Z-scan curve for 21-ns pulse trains with an energy level of 1.4 µ J. (a) NT: triangles stand for the experimental result without an aperture, and the solid line for the theoretical simulation. (b) NTa : squares stand for the experimental result with an aperture, the solid line for the theoretical fit with Eqs. (16) and (18), and the dashed line for the theoretical fit with Eq. (19) alone. (c) NTd : dots stand for the experimental result with an aperture, the solid line for the theoretical fit with Eqs. (16) and (18), and the dashed line for the theoretical fit with Eq. (19) alone. #77284 - $15.00 USD (C) 2007 OSA Received 21 November 2006; revised 26 January 2007; accepted 29 January 2007 19 February 2007 / Vol. 15, No. 4 / OPTICS EXPRESS 1728 Fig. 6. The Z-scan curve for 2.8-ns pulses with an energy level of 1.4 µ J. (a) NT: triangles stand for the experimental result without an aperture, and the solid line for the theoretical simulation. (b) NTa : squares stand for the experimental result with an aperture, the solid line for the theoretical fit with Eqs. (16) and (18), and the dashed line for the theoretical fit with Eq. (19) alone. (c) NTd : dots stand for the experimental result with an aperture, the solid line for the theoretical fit with Eqs. (16) and (18), and the dashed line for the theoretical fit with Eq. (19) alone. #77284 - $15.00 USD (C) 2007 OSA Received 21 November 2006; revised 26 January 2007; accepted 29 January 2007 19 February 2007 / Vol. 15, No. 4 / OPTICS EXPRESS 1729 Fig. 7. The Z-scan curve for 2.8-ns pulses with an energy level of 2.5 µ J. (a) NT: triangles stand for the experimental result without an aperture, and the solid line for the theoretical simulation. (b) NTa : squares stand for the experimental result with an aperture, the solid line for the theoretical fit with Eqs. (16) and (18), and the dashed line for the theoretical fit with Eq. (19) alone. (c) NTd : dots stand for the experimental result with an aperture, the solid line for the theoretical fit with Eqs. (16) and (18), and the dashed line for the theoretical fit with Eq. (19) alone. #77284 - $15.00 USD (C) 2007 OSA Received 21 November 2006; revised 26 January 2007; accepted 29 January 2007 19 February 2007 / Vol. 15, No. 4 / OPTICS EXPRESS 1730 4.2. 2.8-ns pulse results We used parameters identical to those in the 18-ps pulse train data to emulate the 2.8-ns Z-scan results. While the steady state equation, Eq. (19), could not be applied to the Z-scan data, the thermal acoustic equation, Eq. (18), in combination with Eq. (16), still produced an excellent fit. Observed from a larger variation of the steady-state prediction than the experimental data in Figs. 6 and 7, it is clear that the thermal-induced negative-lensing effect is still building up within the duration of a 2.8-ns laser pulse, as the thermal acoustic equation successfully predicted. 5. Conclusion We successfully applied the Z-scan technique to the SiNc-toluene solution and quantitatively accounted for the energy transfer between the energy bands in the solute molecules and the heat generated from the non-radiative relaxations. The thermal-lensing effect due primarily to the density change in the solvent toluene was presented, and the resultant thermal-acoustic model was verified using two excitations: 2.8-ns pulses and 18-ps pulse trains. The validity of the simplified steady-state model was also examined. By introducing the thermal models, the internal nonlinearities of metallo-phthalocyanine molecules can be better characterized using the Z-scan technique, and the energy transfer from the molecules to the surrounding solvent can also be more accurately modeled. Acknowledgments The authors gratefully acknowledge financial support from National Science Council, Taiwan (NSC-94-2215-E-007-014, NSC 094-2811-M-194-002-, and NSC95-2112-M-194-007). We also thank E. W. Van Stryland , D. J. Hagan, M. Sheik-Bahae, and the researchers at CREOL, University of Central Florida, for the use of their facilities and for useful discussions. #77284 - $15.00 USD (C) 2007 OSA Received 21 November 2006; revised 26 January 2007; accepted 29 January 2007 19 February 2007 / Vol. 15, No. 4 / OPTICS EXPRESS 1731

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