Using the Z-scan technique with 532 nm 16 picosecond laser pulses, we observe reverse saturable absorption and positive nonlinear refraction of toluene solutions of both C60 and C70. By deducting the positive Kerr nonlinear refraction of the solvent, we notice that the solute molecules contribute to nonlinear refraction of opposite signs: positive for C60 and negative for C70. Attributing nonlinear absorption and refraction of both solutes to cascading one-photon excitations, we illustrate that they satisfy the Kramers-Kronig relation. Accordingly, we attest the signs and magnitudes of nonlinear refraction for both solutes at 532 nm by Kramers-Kronig transform of the corresponding nonlinear absorption spectra.
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The fullerenes C60 and C70 are expected to show large third order nonlinearities due to the extensive delocalization of their three-dimensional π-electron conjugated systems [1–3]. Therefore, nonlinear absorption and refraction of C60 and C70 solutions have been widely studied for their potential applications in power limiting, all optical switching, optical bistability, etc [4–15]. When multi-energy-band models were used to explain the observed nonlinearities as a result of sequential ground state and first excited singlet state excitations, Li et al. strictly derived the nonlinear optical susceptibility by using the density matrix formulation of quantum mechanics [14,15].
In this report we investigate nonlinear absorption and refraction of toluene solutions of both C60 and C70, dubbed as C60-toluene and C70-toluene respectively, using the Z-scan technique with 16 picosecond (ps) laser pulses at 532 nm. As a result, we verify that both solutions show reverse saturable absorption (RSA) and positive nonlinear refraction. After deducting the positive Kerr nonlinear refraction of the solvent from the experimental results, it is interesting to know that C60 and C70 themselves cause nonlinear refraction of opposite signs: positive for C60 and negative for C70. To elucidate the opposite signs of nonlinear refraction for both solutes, we first show that the nonlinear susceptibility χ (NL), in addition to the linear one χ (1), pertaining to the solutes satisfies the Kramers-Kronig relation (KKR). Next, we substitute the nonlinear absorptive coefficients of both solutions, in the wavelength range between ~440 and 900 nm, to the Kramers-Kronig transform (KKT) equation to extract the nonlinear refractive coefficients of the solutes in the wavelength range between 480 and 850 nm. Consequently, we find that the signs of nonlinear refraction determined by the KKT equation for both solutes at 532 nm agree with those determined by the Z-scan technique. Furthermore, the magnitudes of nonlinear refractive coefficients for both solutes at 532 nm are close to those determined by the Z-scan technique.
2. The experimental
Figure 1 shows the Z-scan apparatus . Briefly, a small portion of each incident laser pulse traveling along the positive z axis is split by the beam splitter BS1 and directed to the photodetector D1 that monitors the fluctuation in input pulse energy ε ng. The rest of this pulse, after being focused by the lens and transmitted through the sample at a certain position z relative to the beam waist, is split into two by the beam splitter BS2 and directed to the apertured detector D2 and the detector D3 that monitor the axial and total transmitted pulse energy, respectively. Readings of D1, D2 and D3 for 10 laser pulses are recorded at each sample position z. Since the laser tends to fluctuate from pulse to pulse, the D2 and D3 readings for each laser pulse are divided by the corresponding D1 reading to correct for the laser energy fluctuation. The (D2/D1) and (D3/D1) ratios averaged over 10 laser pulses at each z, after normalized with those at large where the incident intensity is low and thus linear response prevails, are named the normalized axial and normalized total transmittances and denoted by NTa and NT, respectively. When NTa involves nonlinear refraction and nonlinear absorption (if present), NT reflects nonlinear absorption alone. Plots of NTa and NT as functions of z are called the closed-aperture and open-aperture Z-scan curves. Division of NTa by NT, denoted by NTd, effectively eliminates nonlinear absorption and retains nonlinear refraction . A plot of NTd as a function of z, termed the divided Z-scan curve, tells the signs of nonlinear refraction and the corresponding lensing effect. Nonlinear refraction results from the change of a sample′s refractive index (Δn tot) with the intensity I or generalized fluence () and hence the lateral distance ρ associated with a TEM00 mode pulse. The positive lensing effect (Δn tot has a maximum at the beam center (ρ = 0)) of a sample renders a valley on the –z side and a peak on the + z side in a divided Z-scan curve. This is because it slows the light speed most at the beam center. Therefore, it augments the beam divergence at the aperture in front of D2 when the sample is before the beam waist and lessens the beam divergence there when the sample is after the beam waist. Oppositely, a negative lensing effect (Δn tot has a minimum at the beam center) causes a peak on the –z side and a valley on the + z side in a divided Z-scan curve.
The laser used in the Z-scan measurements is a frequency doubled, Q-switched and mode-locked Nd:YAG laser operating in the TEM00 mode and delivering in each second (s) 10 pulses with a half-width at e −1 maximum of τ = 16 ps. Each output pulse is focused to a waist at z = 0 of radius w 0 ≡ w(0) = 21 μm half-width at e −2 maximum (HWe −2M). In the laboratory coordinate frame, the electric field strength of each pulse propagating along + z direction and irradiating the sample at z can be written as Eqs. (1)-(3), c denotes the speed of light in free space. t′ (t − z/c) and ρ refer to the temporal and lateral distributions of each pulse. w(z) = w 0 × [1 + (z/z 0)2]1/2 is the beam radius (HWe −2M) at z. R(z) = z × [1 + (z 0/z)2] is the curvature radius of the wave front at z and k = 2π/λ (λ = 532 nm being the central wavelength) is the central wave propagation number. is the diffraction length of the beam. is the on-axis peak electric field strength at the beam waist and I 00 is the corresponding intensity, related to asEqs. (1)-(4) pertain to free space. The integration of Eq. (2) over t′ (from −∞ to + ∞) and over the whole beam cross section relates I 00 to ε ng as
Results of experimental observed NT and NTa as functions of the sample position z are presented by the dots and crosses in Fig. 2(a) for C60-toluene and in Fig. 2(b) for C70-toluene. NTd as a function of z is presented by the dots in Fig. 3(a) for C60-toluene and in Fig. 3(b) for C70-toluene.
3. Theoretical model
An explanation that the KKR holds for the nonlinear susceptibility of C60 and C70 is made below based on the three-energy-band model simplified from a five-energy-band model. Given the nonlinear absorption spectra, a KKT equation is derived to calculate the nonlinear refraction spectra of C60 and C70.
In the laboratory coordinate frame () with absorption and refraction of a sample are governed by the electromagnetic wave equation Eq. (7) with a corresponding response function decomposed from R (1)(t). Similarly, can also be subdivided into components to denote different mechanisms.
According to Eq. (7), linear polarization in the frequency domain i.e., the temporal Fourier Transform (FT) of () , is related to the electric field strength in the frequency domain i.e., temporal FT of as
The causality relation between and demandsEq. (9) yields 
To ensure that R (1)(t) is real, has to be Hermitian, i.e., and Accordingly, we can separate the real and imaginary parts of Eq. (11) as
Since the terms before in Eq. (1) involves t, and hence are not monochromatic. Accordingly, temporal FT of is not a delta function of ω. However, when represents the electric field strength of a laser pulse with a width (τ = 16 ps) greatly longer than the reciprocal of the central angular frequency (ω −1 = λ/2πc = 2.8 × 10−16 s) and a beam waist radius (w 0 = 21 μm) much larger than the central wavelength (λ = 532 nm), can be regarded as a quasi monochromatic plane wave and expressed asEq. (8).Eq. (16), we ignore the first term on the left of Eq. (6) since divergence of , a quasi monochromatic plane wave, nearly equals zero. Also, we neglect but retain in the expansion of because our samples meet the thin sample condition (L < nz 0) so that beam broadening or narrowing induced by diffraction is negligible in the samples. In Eq. (16), the second order derivatives of with respect to both z and t are negligible compared with the corresponding first order derivatives multiplied by 2ik and This is because the slowly varying amplitude condition () is satisfied. For the same reason, the first two terms on the right are negligible compared with the third one. Thus, Eq. (16) can be approximated asEq. (17) more easily, we make use of the coordinate transformation: t′ = t − z/c and z′ = z. In this new coordinate system, Eq. (17) becomesEq. (18) into two, one for absorption and the other for refractionEqs. (19) and (20) govern linear absorption and refraction of the light at a central angular frequency ω. To incorporate nonlinear absorption and refraction, we simply need to substitute and by () and () in Eqs. (19) and (20) as well as replace α (1)(ω) and Δn (1)(ω) by the total absorptive coefficient α(ω) (α (1)(ω) + α (NL)(ω)) and total refractive index change Δn(ω) (Δn (1)(ω) + Δn (NL)(ω)). Here and denote the real and imaginary parts of the nonlinear ( dependent) susceptibility and represent the nonlinear absorptive coefficient and refractive index change.
To explain the absorption and refraction of both C60-toluene and C70-toluene, we invoke the five-energy-band model (Fig. 4 ) to interpret the 16 ps pulse-induced optical excitations and subsequent relaxations of C60 and C70 molecules dissolved in the solvent (toluene) . Each state, including the associated zero-point level |0) and vibronic levels |ν ≠ 0), is conventionally named as Sm for the singlet manifold and Tm for the triplet manifold, where the subscript m refers to the state formed from certain electronic configurations in molecular orbitals. When C60-toluene and C70-toluene are in thermodynamic equilibrium, all C60 and C70 molecules reside in S0 with a uniform population density of N S0 (−∞) equal to 1.2 × 1017 cm−3 for C60 and 6.0 × 1016 cm−3 for C70. After being pumped by a 16 ps laser pulse with λ = 532 nm (ω = 3.5 × 1015 s−1), some of the solute molecules are promoted via one-photon absorption to |ν)S1 and then relax to |0)S1 in sub-ps to ps . From |0)S1, they undergo the nanosecond (ns) order relaxations |0)S1ÎS0 or |0)S1Î|ν)T1Î|0)T1 [22–24] or get excited to |ν)S2, via one-photon absorption by the later portion of the same pulse. The solute molecules excited to |ν)S2 nonradiatively decay to |0)S1 in sub-ps to ps [6,25,26].
Although Eqs. (14)-(20) are all related to with a central wavelength of λ = 532 nm, they hold even if we tune the central wavelength of in the range between ~440 and 900 nm (correspondingly, in the angular frequency range between ~2.1 × 1015 and 4.3 × 1015 s−1) but maintain all the other parameters unchanged. This is because both the thin sample condition and the slowly varying amplitude condition hold in this wavelength range. Accordingly, Eqs. (19) and (20) will be used to interpret the white light pulse induced-nonlinear absorption and refraction in the wavelength range between ~440 and 900 nm (vide infra).
Here ℏ is the Planck′s constant. and are the dipole matrix elements associated with S0→|ν)S1 and |0)S1→|ν)S2 excitations respectively. N S0(t′) and N S1(t′) represent the population densities of S0 and |0)S1 at time t′ relative to the pulse peak. and are the damping rates associated with and The angular frequencies and correspond to the energy difference between |ν)S1 and S0 and that between |ν)S2 and |0)S1. Since the time intervals for relaxations |ν)S1Î|0)S1 and |ν)S2Î|0)S2 are considerably shorter than the pulse width (τ = 16 ps), we approximate the population densities of |ν)S1 and |ν)S2 as zero in the course of light-matter interaction. Accordingly, we neglect the contribution of stimulated emissions |ν)S1→S0 and |ν)S2→|0)S1 to the susceptibility in Eqs. (21) and (22).
By substituting and (the real and imaginary parts of the total susceptibilities of the solutes) in Eqs. (21) and (22) into Eqs. (19) and (20) for and and introducing the Kerr nonlinear refraction of the solvent into Eq. (20), we obtain the governing equations of absorption and refraction of the solutions asEqs. (23) and (24), Δn tot(ω) denotes nonlinear refractive index change of the solution. α(ω) and Δn(ω) represent the (linear plus nonlinear) absorptive coefficient and refractive index change of the solutes. Note that the term in Eq. (24) does not originate from χ(ω) of the solutes (Eqs. (21) and (22)). Instead, it arises from the Kerr nonlinear refraction of the solvent (toluene) with a coefficient of n 2 = 7.7 × 10−15 cm2W−1 at ω = 3.5 × 1015 s−1 (λ = 532 nm). σ (ω) and γ (ω) represent the absorptive and refractive cross sections of the states specified by the subscripts:
By solving the density matrix equation of motion, we obtain the population change rates of S0 and |0)S1 in the course of light-matter interaction as
The population density of |0)S2, in addition to those of |ν)S1 and |ν)S2, is taken to be zero because the time interval for relaxation |0)S2Î|ν)S1Î|0)S1 is considerably shorter than the pulse width (τ = 16 ps). The population densities of the triplet states are approximated as zero because the ns order time interval for |0)S1Î|ν)T1Î|0)T1 relaxation is much longer than τ = 16 ps. This facilitates the simplification of the five-energy-band model as the three-energy-band (S0, S1 and S2) model in this study. Similarly, population relaxation to S0 from |0)S1 is disregarded because the ns order time interval for |0)S1ÎS0 relaxation is much longer than τ = 16 ps. Sum of Eqs. (29) and (30) equals zero, leading to the population conservation relation N S0(−∞) = N S0(t′) + N S1(t′).
Since both N S0(t′) and N S1(t′) depend on the intensity I and hence on but N S0(−∞) does not, we can use the population conservation relation to separate the linear and nonlinear components of and (see Eqs. (21) and (22)) asEqs. (31) and (32) is independent. It fulfills the KKR (Eqs. (11)-(13)) because its corresponding linear response function R (1)(t) satisfies Eq. (9). On the other hand, is the nonlinear susceptibility because included in Eqs. (33) and (34) is dependent (see Eq. (30)). Since (i) each term in the bracket on the right of Eq. (33) replicates the term on the right of Eq. (31), (ii) each term in the bracket on the right of Eq. (34) imitates the term on the right of Eq. (32) and (iii) appears on the right of both Eqs. (33) and (34) in the same manner, nonlinear susceptibility and thus the total susceptibility of the solutes satisfy the KKR (Eqs. (11)-(13)). By substituting and into Eq. (12) for and we obtain the KKR between Δn (NL)(ω) and α (NL)(ω) asEq. (24)) and (the part of α(ω) dependent on in Eq. (23)), Eq. (35) leads to the KKT equation
4. Results and discussion
A computer fitting algorithm, as already described in detail in , is used to simulate the measured Z-scan data of both C60-toluene and C70-toluene in this study. By best fitting the experimental results, we obtain both (σ S1(ω)−σ S0(ω)) and (γ S1(ω)−γ S0(ω)) at ω = 3.5 × 1015 s−1 (λ = 532 nm). In addition, Eq. (36) is employed to extract (γ S1(ω)−γ S0(ω)) from (σ S1(ω)−σ S0(ω)) measured in the angular frequency range between ~2.1 × 1015 and 4.3 × 1015 s−1.
Given the input intensity I 0 and phase at a certain sample position z by Eqs. (2), (3) and (5) with ε ng = 1.0 μJ for C60-toluene and 0.9 μJ for C70-toluene, we numerically integrate Eqs. (23) and (24) over the sample thickness L = 0.1 cm to obtain the transmitted irradiance IL and phase (or equivalently, the electric field EL) at the exit surface of the sample. At each penetration depth z' and lateral distance ρ, the population densities N(t′)s used in Eqs. (23) and (24) are obtained from time integration of Eqs. (29) and (30) using N S0(−∞) = 1.2 × 1017 cm−3 for C60-toluene and 6.0 × 1016 cm−3 for C70-toluene and N S1(−∞) = 0 for both solutions as the initial conditions.
Using the Huygens-Fresnel propagation formalism , we deduce the intensity at the aperture (Ia) from EL. When integration of IL over the pulse width and the whole beam cross section gives an energy corresponding to that detected by D3, integration of Ia over the pulse width and the aperture cross section yields an energy corresponding to that detected by D2. Division of both D3 and D2 by D1 yields the total and axial transmittances at a certain sample position z. By repeating the calculation at all sample positions, we obtain the total and axial transmittances as functions of z which, after normalization with the corresponding transmittances in the linear regime, yield the simulated NT and NTa to be compared with the experimental ones.
Since the contributions of the first terms on the right of Eqs. (23) and (24) ( and ) are normalized out in the open-aperture and closed-aperture Z-scan curves, the fitting parameters for both NT and NTa as functions of z turn out to be () and () at ω = 3.5 × 1015 s−1 (λ = 532 nm). To best fit the open-aperture Z-scan results, we use () = 2.5 × 10−17 cm2 for C60-toluene and 8.0 × 10−18 cm2 for C70-toluene [11,28]. Positive () signifies RSA (increase of α with I or F G) and results in a symmetrical open-aperture Z-scan curve with a valley at the beam waist. To best fit the closed-aperture Z-scan results, we use () = 1.2 × 10−18 cm2 for C60 and −8.3 × 10−18 cm2 for C70 in combination with () pertaining to both solutes [11,28]. Although C60 causes a positive lensing effect with a positive () and C70 renders a negative lensing effect with a negative (), the divided Z-scan curves for both C60-toluene and C70-toluene show positive lensing effects. This is understandable because the positive Kerr nonlinear refraction of the solvent (the last term on the right of Eq. (24) with n 2 = 7.7 × 10−15 cm2W−1 at ω = 3.5 × 1015 s−1) enhances the positive lensing effect of C60 molecules and surpasses the negative lensing effect of C70 molecules.
To confirm the signs and magnitudes of nonlinear refractive coefficients () determined by the Z-scan technique for C60 and C70 molecules at ω = 3.5 × 1015 s−1 (λ = 532 nm), we substitute σ S0(ω) measured by a dual-beam spectra photometer and σ S1(ω) adopted from  and  into the KKT equation (Eq. (36)). Using a dual-beam spectra photometer, we measure the linear absorptive coefficients α (1)(ω) = σ S0(ω) × N S0(−∞) (the part of α independent of , see Eq. (23)) as a function of ω, ranging from ~2.1 × 1015 to 4.3 × 1015 s−1, for both C60-toluene and C70-toluene. Dividing the measured α (1)(ω)s by the corresponding N S0(−∞)s, we obtain σ S0(ω) as shown by the dashes in Fig. 5(a) for C60-toluene and in Fig. 5(b) for C70-toluene. Note that the variable ω has been converted into the wavelength λ = 2πc/ω. At λ = 532 nm, σ S0 equals 3.2 × 10−18 cm2 for C60-toluene and 4.4 × 10−17 cm2 for C70-toluene. When () has been determined to be 2.5 × 10−17 cm2 for C60-toluene and 8.0 × 10−18 cm2 for C70-toluene by the Z-scan technique, is derived to be 2.8 × 10−17 cm2 for C60-toluene and 5.2 × 10−17 cm2 for C70-toluene at ω = 3.5 × 1015 s−1 (λ = 532 nm).
In  and , Tanigaki, Ebbesen et al. used a pump-probe technique to measure the absorptive cross section of the |0)S1 state as a function of ω. First, they used an energetic 20 ps laser pulse at 354 nm to promote all the solute molecules from S0 to |0)S1 via the midway state |ν)S1, making N S1 = N S0(−∞). Next, they probed the |0)S1 state absorptive coefficient σ S1(ω) × N S0(−∞) as a function of ω (ranging from ~2.1 × 1015 to 4.3 × 1015 s−1 for both C60-toluene and C70-toluene) using a white light pulse. The widths of both the 354 nm pump pulse and the probe white light pulse and the separation between them were prepared much shorter than the ns order time intervals for relaxations |0)S1ÎS0 and |0)S1Î|ν)T1Î|0)T1. This ensured the absorption at each component wavelength of the white light pulse pertained to |0)S1→|ν)S2 excitation only. To extract σ S1(ω), we divide the observed σ S1(ω) × N S0(−∞) for C60-toluene and C70-toluene by the corresponding N S0(−∞)s. Since σ S1(ω) × N S0(−∞) was given in an arbitrary unit in  and , we rescale the resultant σ S1(ω) to yield σ S1 equal to that determined by the Z-scan technique in combination with the dual-beam spectra photometer at ω = 3.5 × 1015 s−1 (λ = 532 nm): 2.8 × 10−17 cm2 for C60-toluene and 5.2 × 10−17 cm2 for C70-toluene. The solid-line curves in Figs. 5(a) and 5(b) represent σ S1 as a function of λ = 2πc/ω for C60-toluene and C70-toluene respectively.
When dealing with P.V. integral in Eq. (36) to obtain we vary ω′ from ~2.1 × 1015 to 4.3 × 1015 s−1 excluding a tiny segment of 1.1 × 1012 s−1 centered at each observation angular frequency ω. Figures 6(a) and 6(b) show the calculated as a function of λ = 2πc/ω in the range between 480 and 850 nm for both C60 and C70 molecules. At λ = 532 nm, = 9.0 × 10−19 cm2 for C60 and −9.4 × 10−18 cm2 for C70. Signs of agree with those determined by the Z-scan technique for both solutes. In addition, magnitudes of closely meet those obtained by the Z-scan technique (1.2 × 10−18 cm2 for C60 molecules and −8.3 × 10−18 cm2 for C70 molecules).
Using the Z-scan technique with 16 ps laser pulses at 532 nm, we investigate nonlinear absorption and refraction of both C60-toluene and C70-toluene. After deducting the positive Kerr nonlinear refraction of the solvent (toluene) from the experimental results, we find that C60 and C70 contribute to nonlinear refraction of opposite signs: positive for C60 and negative for C70.
A KKR analysis based on the three-energy-band model is developed and used to calculate of C60 and C70 molecules in the wavelength range between 480 and 850 nm from the spectra of measured in the wavelength range between ~440 and 900 nm. The calculated shows good agreement with both signs and magnitudes of nonlinear refraction determined by the Z-scan technique with 16 ps laser pulses at 532 nm. Accordingly, we expect the KKR method has the potential to routinely extract nonlinear optical parameters of various materials with a simple three-energy-band system.
T. H. Wei gratefully acknowledges financial support from National Science Council grant NSC 96-2112-M-194-003-MY3 and that from Ministry of Economic Affairs grant 94-EC-17-A-08-S1-0006, Taiwan. J. L. Tang gratefully acknowledges financial support from National Science Council grant NSC 96-2112-M-194-004-MY3. T. M. Liu is presently at Department of Physics, University of Cincinnati OH, 45221, USA.
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