Z-scan study of thermal nonlinearities in silicon

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
`