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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 4 — Feb. 19, 2007
  • pp: 1718–1731
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Z-scan study of thermal nonlinearities in silicon naphthalocyanine-toluene solution with the excitations of the picosecond pulse train and nanosecond pulse

Sidney S. Yang, Tai-Huei Wei, Tzer-Hsiang Huang, and Yun-Ching Chang  »View Author Affiliations


Optics Express, Vol. 15, Issue 4, pp. 1718-1731 (2007)
http://dx.doi.org/10.1364/OE.15.001718


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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

1. Introduction

Two-dimensional molecules with π-conjugated electron systems, such as porphyrins, phthalo-cyanines, and their derivatives, and their nonlinear optical properties have been widely investigated recently [1

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 SeriesE, J. Messier, F. Kajzar, and P. PrasadKluwer, Dordrecht, 1991),Vol. 194 pp.369–382.

, 2

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). [CrossRef]

]. 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

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). [CrossRef]

]. They are also good candidates for optical recording materials because of their large nonlinear refraction in the infrared regime [7

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). [CrossRef] [PubMed]

, 8

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). [CrossRef]

]. 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 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

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). [CrossRef]

]. Given the acoustic wave speed of νs = 1170 m/s for the solvent (toluene)[10

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). [CrossRef]

], we respectively focused a 2.8-ns pulse and a 21-ns pulse train to have a beam waist radius of w 0 = 14.1 μm and 18.9 μm (HW1/e2M for both) in this study. This resulted in τac = w 0/νs = 12.0 ns for the 2.8-ns pulse and τac = w 0/νs = 16.2 ns for the pulse train. The steady state approximation was relatively appropriate for a 21-ns pulse train (τenvac = 1.3) compared with a 2.8-ns pulse (t/tac = 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 Δntherm’s for a 21-ns pulse train but causes significantly different ∆ntherm’s for a 2.8-ns pulse (νide infra).

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.

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

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). [CrossRef]

]. 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 00 mode at 10 Hz. The incident intensity (Fig. 2) and phase of the nth pulse in a train are, respectively,

Fig. 2. The temporal profile of the full pulse envelope. The numbers above spikes mark their order.
I0=Izr,t=I00(n)×[w02w2(z)]×exp[2r2w2(z)]×exp[(tn×7nsτ)2]
(1)
ϕ0=ϕ(z,r,t)=kr22R(z).
(2)

and

Here, w(z) = w 0[1+(z/z 0)2]1/2 is the laser beam radius (HW1/e2M) at z. w 0 denotes w(0) and equals 21 μm. I 00 (n) 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+(z 0/z)2] is the curvature radius of the wave front at z. z 0 = πw 2 0/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 relates I 00 (n) to the pulse energy ε(n) as

I00(n)=2ε(n)π32w02τ.
(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

ε(n)=ε(0)×exp[(n×7nsτenv)2].
(4)

Summing up this equation from n= -5 to 5, we obtain the full pulse train energy εt, with which ε(n) is expressed as

ε(n)=εt×exp[(n×7nsτenv)2]m=55exp[(m×7nsτenv)].
(5)

I 00 (n) 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

I0=Izr,t=I00×[w02w2(z)]×exp[2r2w2(z)]×exp[(tτ)2]
(6)

Using a 21-ns pulse train with a full train energy of εt = 0.8 μJ and 1.4 μJ, and a 2.8-ns pulse with a pulse energy of ε = 1.4 μJ and 2.5 μJ, we performed, at room temperature θe = 25 °C, Z-scan measurements on a SiN c-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

12. D. G. Mclean, R. L. Sutherland, M. C. Brant, D. M. Brandelik, P. A. Fleitz, and T. Pottenger. “Nonlinear absorption study of a C60-toluene solution,” Opt. Lett 18,858–860 (1993). [CrossRef] [PubMed]

], 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

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).

]. 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 ∣ν 0) and vibronic level ν ≠ 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

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 281495–1508 (1996). [CrossRef]

]

dIdz'=[(σa)S0NS0+(σa)S1NS1+(σa)T1]IβNS0I2
(7)

and

dz'=[(σr)S0NS0+(σr)S1NS1+(σr)T1NT1]I+γNS0I+kn2I+kΔntherm,
(8)

where I and ϕ are the intensity and phase, respectively, of a 2.8-ns pulse or an individual 18-ps 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 S 0ν)S1, S 1 →∣ν)S 2, and T 1 →∣ ν)T 2 represented by their first three terms. σ a in Eq. (7) denotes the absorption cross section of the states specified by the subscripts. The σr, the refractive cross section, of a band can be derived from σa associated with the same band according to the Kramers-Krönig relation. The 4th terms in Eqs. (7) and (8) pertain to the two-photon excitation S 0 →∣ ν)S 2. The 5th term in Eq. (8) denotes the Kerr effect of the solvent (toluene), n 2 = 5.5 × 10-15 cm2/W, and the 6th term of Eq. (8) represents the thermal effect where Δntherm will be respectively estimated via Eq. (18) in combination with Eq. (16) or via Eq. (19) alone in this study (νide infra). Combining optical excitation with a 532-nm pulse and the subsequent relaxation, the population redistributes in various states with time rates of [13

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).

]

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).
dNS0dt=(σa)S0NS0IħωβNS0I22ħω+NS1τf+NT1τT1,
(9)
dNS1dt=(σa)S0NS0Iħω+βNS0I22ħωNS1τfNS1τISC,
(10)

and

dNT1dt=NS1τISCNT1τT1.
(11)

N and τ are respectively the population density and relaxation time S 0 -∣ ν)S 1 and two-photon-absorption S 0ν)S 2 induced population-density redistributions between S 0 and ∣ ν)S 1 are denoted by the first two terms of Eqs. (9) and (10). Nonradiative relaxations ∣ ν)S 1 ⇝∣ 0)S 1 and ∣ ν)S 2 ⇝∣ 0)S 1 are assumed to follow the above mentioned excitations well within the pulse width (18 ps or 2.8 ns) [14

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 281495–1508 (1996). [CrossRef]

, 15

15. A. Seilmeier and W. Kaiser, in Ultrashort Laser Pulses 2nd ed.,W. KaiseerSpringer-Verlag, Berlin, 1993), pp. 305.

]. The 3rd terms on the right-hand side of Eqs. (9) and (10) denote fluorescent-decay ∣ 0)S 1S 0 induced population redistributions between ∣ 0)S 1 and S 0, τf being the lifetime and equal to 3.1 ns [1

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 SeriesE, J. Messier, F. Kajzar, and P. PrasadKluwer, Dordrecht, 1991),Vol. 194 pp.369–382.

]. Population redistribution between ∣ 0)S 1 and ∣ 0)T 1 via intersystem crossing (ISC) ∣ 0)S 1 ⇝∣ ν)T 1 ⇝∣ 0)T 1 is expressed by the 4th term of Eq. (10) and the 1 st term of Eq. (11), respectively, τISC being the lifetime and equal to 16 ns [16

16. J. H. Brannon and D. Madge, “Picosecond laser Photophysics. group 3A phthalocyanines,” J. Am. Chem. Soc 102,62–65 (1980). [CrossRef]

]. Population redistribution between ∣ 0)T 1 and S 0 as a result of intersystem crossing (ISC) ∣ 0)T 1S 1, is expressed by the 4th term of Eq. (9) and the 2nd term of Eq. (11), respectively, τT1 being the lifetime and falling in the μs regime [17

17. C. JensenHigh Power Dye Lasers, F.J Durate, eds. (Springer-Verlag, Berlin, 1991), pp. 48.

]. Since ∣ 0)T 1 →∣ ν)T 2 absorption is verified in this paper (νide infra), we neglect the population redistributions between ∣0)T 1 and ∣ ν)T 2 because the ∣ ν)T 2 ⇝∣ 0)T 2 ⇝∣ ν)T 1 ⇝∣ 0)T 1 relaxation subsequent to the excitation is believed to be much shorter than the pulse widths (18 ps or 2.8 ns) [14

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 281495–1508 (1996). [CrossRef]

, 15

15. A. Seilmeier and W. Kaiser, in Ultrashort Laser Pulses 2nd ed.,W. KaiseerSpringer-Verlag, Berlin, 1993), pp. 305.

, 18

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). [CrossRef] [PubMed]

]. Accompanying the rapid nonradiative decays ∣ ν)S 1 ⇝∣ 0)S 1, ∣ ν)S 2 ⇝∣ 0)S 1, and ∣ ν)T 1 ⇝∣ 0)T 1 [18

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). [CrossRef] [PubMed]

], 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

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).

]

dQdt=(σa)S0NS0Iħω×ħ(ωωS1)+(σa)S1NS1I+βNS0I22ħω×ħ(2ωωS1)+(σa)T1NT1I,
(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)S 1 ⇝ ∣ 0)S 1 relaxation subsequent to the S 0 → ∣ ν)S 1 excitation. ωS1S1 = 780 nm) corresponds to the energy of ∣ 0)S1 relative to S 0. The second and third terms describe the contributions of the sequential ∣ ν)S 2 ⇝∣ 0)S 2 ⇝∣ ν)S 1 ⇝∣ 0)S 1 relaxations following the one-photon ∣ 0)S 1 →∣ ν)S 2 excitation and the two-photon S 0 →∣ ν)S 2 excitation, respectively. The last term describes the contribution of the sequential relaxations, ∣ ν)T 2 ⇝∣ 0)T 2 ⇝∣ ν)T 1 ⇝∣ 0)T 1, following the one-photon excitation ∣ 0)T 1 →∣ ν)T 2.

∆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

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). [CrossRef]

]. 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

Δθ=1ρCptdQdt'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

20. D. R. Lide. in CRC Handbook of Chemistry and Physics, 77th ed.,D. R. Lideet al, eds. (CRC Press, Boca Raton, 1996), pp. 6-128.

]. 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

Δθn=1ρCpdQndt'dt',
(14)

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

Δθ=5n1Δθn.
(15)

When the -5th pulse interacts with the sample, ∆θ = 0. Since the intersystem crossing time constant (τISC=16 ns) and μs order for τT1 are greatly longer than the relaxation time constants S1 ⇝∣ ν)T 1 ⇝∣ 0)T 1 and ∣ 0)T 1S 0 relaxations are ignored.

2(Δρ)t2vs22(Δρ)=vs22(Δθ)γe2nc2I,
(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

20. D. R. Lide. in CRC Handbook of Chemistry and Physics, 77th ed.,D. R. Lideet al, eds. (CRC Press, Boca Raton, 1996), pp. 6-128.

], and γe = ρ(∂n 2/∂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 (n 2 - 1){n 2 + 2)/3 with n denoting the refractive index [21

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). [CrossRef] [PubMed]

]. Given the linear refractive index n 0 = 1.49 for toluene [20

20. D. R. Lide. in CRC Handbook of Chemistry and Physics, 77th ed.,D. R. Lideet al, eds. (CRC Press, Boca Raton, 1996), pp. 6-128.

], γe is estimated to be 1.71. As will be shown later, the second term on the right-hand 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

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). [CrossRef] [PubMed]

]. 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

Δntherm=(nθ)ρΔθ+(nρ)θΔρ.
(17)

Since the 1st term on the right-hand side is considerably smaller than the 2nd one [21

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). [CrossRef] [PubMed]

], and (∂n/∂ρ)θ = γe/2nρ, as derived from γe = ρ(∂n 2/∂ρ)θ, Eq. (17) can be approximated as

Δnthermγe2Δρ.
(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/2ncv 2 s, which in turn approximates Eq. (18

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). [CrossRef] [PubMed]

) as

Δntherme2nΔθ+γe24n2cvs2I.
(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. (σa)S0 = 2.8 × 10-18 cm2, (σa)S1 = 5.0 × 10-17 cm2, (σr)S0 ≅ 0, β = 0, (σr)S1 = 1.2 × 10-18 cm2, γ=0, and n 2 = 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

24. C. W. Chang and M. S. thesis, National Chung Cheng University, pp. 28, (1999).

]. 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 Δntherm 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 μJ and 1.4 μJ. There are two parameters, (σa)T1 and (σr)T1, undetermined by the single 18-ps pulse Z-scan experiments. Using (σa)T1 = 6.0 × 10-17 cm2 and (σr)T1 = -5.5 × 10-17 cm2, we best fit the results with ∆n therm, determined using Eq. (18) in combination with Eq. (16). However, only a small deviation is generated when ∆n therm is determined using Eq. (19) alone given dn/ ≈ (∂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 τenvac =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

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). [CrossRef]

].

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 μJ) seems to be fitted better than the data obtained with lower level (0.8 μJ) (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.

Fig. 4. The Z-scan curves for 21-ns pulse trains with an energy level of 0.8 μJ. (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.
Fig. 5. The Z-scan curve for 21-ns pulse trains with an energy level of 1.4 μJ. (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.
Fig. 6. The Z-scan curve for 2.8-ns pulses with an energy level of 1.4 μJ. (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.
Fig. 7. The Z-scan curve for 2.8-ns pulses with an energy level of 2.5 μJ. (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.

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.

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 SeriesE, J. Messier, F. Kajzar, and P. PrasadKluwer, 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). [CrossRef]

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). [CrossRef]

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).

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,” Science23,1533–1536 (1996). [CrossRef]

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). [CrossRef]

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). [CrossRef] [PubMed]

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). [CrossRef]

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). [CrossRef]

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). [CrossRef]

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). [CrossRef]

12.

D. G. Mclean, R. L. Sutherland, M. C. Brant, D. M. Brandelik, P. A. Fleitz, and T. Pottenger. “Nonlinear absorption study of a C60-toluene solution,” Opt. Lett 18,858–860 (1993). [CrossRef] [PubMed]

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 281495–1508 (1996). [CrossRef]

15.

A. Seilmeier and W. Kaiser, in Ultrashort Laser Pulses 2nd ed.,W. KaiseerSpringer-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). [CrossRef]

17.

C. JensenHigh 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). [CrossRef] [PubMed]

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). [CrossRef]

20.

D. R. Lide. in CRC Handbook of Chemistry and Physics, 77th ed.,D. R. Lideet 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). [CrossRef] [PubMed]

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). [CrossRef]

24.

C. W. Chang and M. S. thesis, National Chung Cheng University, pp. 28, (1999).

OCIS Codes
(000.6850) General : Thermodynamics
(190.4710) Nonlinear optics : Optical nonlinearities in organic materials
(190.4870) Nonlinear optics : Photothermal effects

ToC Category:
Nonlinear Optics

History
Original Manuscript: November 21, 2006
Revised Manuscript: January 26, 2007
Manuscript Accepted: January 29, 2007
Published: February 19, 2007

Citation
Sidney S. Yang, Tai-Huei Wei, Tzer-Hsiang Huang, and Yun-Ching Chang, "Z-scan study of thermal nonlinearities in silicon naphthalocyanine-toluene solution with the excitations of the picosecond pulse train and nanosecond pulse," Opt. Express 15, 1718-1731 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-4-1718


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References

  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). [CrossRef]
  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). [CrossRef]
  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).
  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). [CrossRef]
  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). [CrossRef]
  7. T. Tomiyama, I. Watanabe, A. Kuwano,M. Habiro, N. Takane, andM. Yamada, "Rewritable optical-disk fabrication with an optical recording material made of naphthalocyanine and polythiophene," Appl. Opt. 34, 8201-8208 (1995). [CrossRef] [PubMed]
  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). [CrossRef]
  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). [CrossRef]
  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). [CrossRef]
  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). [CrossRef]
  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). [CrossRef] [PubMed]
  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). [CrossRef]
  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). [CrossRef]
  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). [CrossRef] [PubMed]
  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). [CrossRef]
  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). [CrossRef] [PubMed]
  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). [CrossRef]
  24. C. W. Chang, M. S. thesis, National Chung Cheng University, pp. 28, (1999).

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