1. Introduction

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 [16]. 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 $\left|z\right|,$ where the incident intensity is low and thus linear response prevails, are named the normalized axial and normalized total transmittances and denoted by NT_a and NT, respectively. When NT_a involves nonlinear refraction and nonlinear absorption (if present), NT reflects nonlinear absorption alone. Plots of NT_a and NT as functions of z are called the closed-aperture and open-aperture Z-scan curves. Division of NT_a by NT, denoted by NT_d, effectively eliminates nonlinear absorption and retains nonlinear refraction [16]. A plot of NT_d 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 ($F_G\equiv {\displaystyle {\int }_{-\infty }^{t}Id{t}^{″}}$) 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.

 

Fig. 1 The Z-scan apparatus which records the axial and total transmittances as functions of the sample position z.

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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 [17]

Results of experimental observed NT and NT_a 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. NT_d 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.

 

Fig. 2 Z-scan results of C₆₀-toluene (a) and C₇₀-toluene (b): dots for the experimental open-aperture curves, crosses for the experimental closed-aperture curves and solid-line curves for the corresponding theoretical fits.

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Fig. 3 The divided Z-scan results of C₆₀-toluene (a) and C₇₀-toluene (b): dots for the experimental curves and solid-line curves for the theoretical fits.

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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 ($\stackrel{\rightharpoonup }{r},t$) with $\stackrel{\rightharpoonup }{r}=\rho \widehat{\rho }+\stackrel{\rightharpoonup }{z},$ absorption and refraction of a sample are governed by the electromagnetic wave equation [18]

According to Eq. (7), linear polarization in the frequency domain ${{\stackrel{\rightharpoonup }{P}}^{\prime }}^{(1)}(\stackrel{\rightharpoonup }{r},\omega ),$ i.e., the temporal Fourier Transform (FT) of ${\stackrel{\rightharpoonup }{P}}^{(1)}(\stackrel{\rightharpoonup }{r},t)$ (${(2\pi )}^{-1}\times {\displaystyle {\int }_{-\infty }^{\infty }{\stackrel{\rightharpoonup }{P}}^{(1)}(\stackrel{\rightharpoonup }{r},t)\times e^{-i\omega t}dt}$) [19], is related to the electric field strength in the frequency domain ${\stackrel{\rightharpoonup }{E}}^{\prime }(\stackrel{\rightharpoonup }{r},\omega ),$ i.e., temporal FT of $\stackrel{\rightharpoonup }{E}(\stackrel{\rightharpoonup }{r},t),$ as

The causality relation between ${\stackrel{\rightharpoonup }{P}}^{(1)}(\stackrel{\rightharpoonup }{r},t)$ and $\stackrel{\rightharpoonup }{E}(\stackrel{\rightharpoonup }{r},t)$ demands

To ensure that R ⁽¹⁾(t) is real, ${\chi }^{(1)}(\omega )={\chi }_{\mathrm{Re}}^{\left(1\right)}(\omega )+i{\chi }_{\mathrm{Im}}^{\left(1\right)}(\omega )$ has to be Hermitian, i.e., ${\chi }_{\mathrm{Re}}^{(1)}(\omega )={\chi }_{\mathrm{Re}}^{(1)}(-\omega )$ and ${\chi }_{\mathrm{Im}}^{(1)}(\omega )=-{\chi }_{\mathrm{Im}}^{(1)}(-\omega ).$ Accordingly, we can separate the real and imaginary parts of Eq. (11) as

Since the terms before $e^{-i(kz-\omega t)}$ in Eq. (1) involves t, ${\stackrel{\rightharpoonup }{E}}_0$ and hence $\stackrel{\rightharpoonup }{E}(\stackrel{\rightharpoonup }{r},t)$ are not monochromatic. Accordingly, temporal FT of $\stackrel{\rightharpoonup }{E}(\stackrel{\rightharpoonup }{r},t)$ is not a delta function of ω. However, when $\stackrel{\rightharpoonup }{E}(\stackrel{\rightharpoonup }{r},t)$ 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), $\stackrel{\rightharpoonup }{E}(\stackrel{\rightharpoonup }{r},t)$ can be regarded as a quasi monochromatic plane wave and expressed as

Substituting Eqs. (14) and (15) into Eq. (6), we obtain the equation

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) [21]. Each state, including the associated zero-point level |0) and vibronic levels |ν ≠ 0), is conventionally named as S_m for the singlet manifold and T_m 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 [6]. 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].

 

Fig. 4 The scheme of the five-energy-band (S₀, S₁, S₂, T₁ and T₂) model. Upward-pointing arrows denote optical excitations; wiggly lines indicate nonradiative relaxations; and downward-pointing arrows show radiative relaxation.

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Although Eqs. (14)-(20) are all related to $\stackrel{\rightharpoonup }{E}(\stackrel{\rightharpoonup }{r},t)$ with a central wavelength of λ = 532 nm, they hold even if we tune the central wavelength of $\stackrel{\rightharpoonup }{E}(\stackrel{\rightharpoonup }{r},t)$ 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).

According to [14] and [15], cascading one-photon excitations S₀→|ν)S₁ and |0)S₁→|ν)S₂ for C₆₀ and C₇₀ give

Here ℏ is the Planck′s constant. ${\wp }_{01^{\prime }}$and ${\wp }_{12^{\prime }}$ 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. ${\Gamma }_{01^{\prime }}$ and ${\Gamma }_{12^{\prime }}$ are the damping rates associated with ${\wp }_{01^{\prime }}$ and ${\wp }_{12^{\prime }}.$ The angular frequencies ${\omega }_{01^{\prime }}$ and ${\omega }_{12^{\prime }}$ 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 ${\chi }_{\text{Re}}(\omega )$ and ${\chi }_{\text{Im}}(\omega )$ (the real and imaginary parts of the total susceptibilities of the solutes) in Eqs. (21) and (22) into Eqs. (19) and (20) for ${\chi }_{\mathrm{Re}}^{\left(1\right)}(\omega )$ and ${\chi }_{\mathrm{Im}}^{\left(1\right)}(\omega )$ and introducing the Kerr nonlinear refraction of the solvent into Eq. (20), we obtain the governing equations of absorption and refraction of the solutions as

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 ${{\stackrel{\rightharpoonup }{E}}^{\prime }}_t$ but N _S0(−∞) does not, we can use the population conservation relation to separate the linear and nonlinear components of ${\chi }_{\mathrm{Re}}(\omega )$ and ${\chi }_{\mathrm{Im}}(\omega )$ (see Eqs. (21) and (22)) as

4. Results and discussion

A computer fitting algorithm, as already described in detail in [11], 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 ${\varphi }_0$ 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 I_L and phase ${\varphi }_L$ (or equivalently, the electric field E_L) 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 [27], we deduce the intensity at the aperture (I_a) from E_L. When integration of I_L over the pulse width and the whole beam cross section gives an energy corresponding to that detected by D3, integration of I_a 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 NT_a to be compared with the experimental ones.

Since the contributions of the first terms on the right of Eqs. (23) and (24) (${\sigma }_{\text{S0}}(\omega )N_{\text{S0}}(-\infty )$ and ${\gamma }_{\text{S0}}(\omega )N_{\text{S0}}(-\infty )$) are normalized out in the open-aperture and closed-aperture Z-scan curves, the fitting parameters for both NT and NT_a as functions of z turn out to be (${\sigma }_{\text{S1}}(\omega )-{\sigma }_{\text{S0}}(\omega )$) and (${\gamma }_{\text{S1}}(\omega )-{\gamma }_{\text{S0}}(\omega )$) at ω = 3.5 × 10¹⁵ s⁻¹ (λ = 532 nm). To best fit the open-aperture Z-scan results, we use (${\sigma }_{\text{S1}}(\omega )-{\sigma }_{\text{S0}}(\omega )$) = 2.5 × 10⁻¹⁷ cm² for C₆₀-toluene and 8.0 × 10⁻¹⁸ cm² for C₇₀-toluene [11,28]. Positive (${\sigma }_{\text{S1}}(\omega )-{\sigma }_{\text{S0}}(\omega )$) 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 (${\gamma }_{\text{S1}}(\omega )-{\gamma }_{\text{S0}}(\omega )$) = 1.2 × 10⁻¹⁸ cm² for C₆₀ and −8.3 × 10⁻¹⁸ cm² for C₇₀ in combination with (${\sigma }_{\text{S1}}(\omega )-{\sigma }_{\text{S0}}(\omega )$) pertaining to both solutes [11,28]. Although C₆₀ causes a positive lensing effect with a positive (${\gamma }_{\text{S1}}(\omega )-{\gamma }_{\text{S0}}(\omega )$) and C₇₀ renders a negative lensing effect with a negative (${\gamma }_{\text{S1}}(\omega )-{\gamma }_{\text{S0}}(\omega )$), 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 (${\gamma }_{\text{S1}}(\omega )-{\gamma }_{\text{S0}}(\omega )$) 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 [23] and [29] 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 ${{\stackrel{\rightharpoonup }{E}}^{\prime }}_t$, 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 (${\sigma }_{\text{S1}}(\omega )-{\sigma }_{\text{S0}}(\omega )$) 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, ${\sigma }_{\text{S1}}$ 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).

 

Fig. 5 Absorptive cross sections as functions of λ for C₆₀-toluene (a) and C₇₀-toluene (b): solid-line curves for the |0)S₁ state and dashes for S₀. λ indicates the wavelength.

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In [23] and [29], 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 [23] and [29], 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 $({\gamma }_{\text{S1}}(\omega )-{\gamma }_{\text{S0}}(\omega )),$ 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 $({\gamma }_{\text{S1}}-{\gamma }_{\text{S0}})$ 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, $({\gamma }_{\text{S1}}-{\gamma }_{\text{S0}})$ = 9.0 × 10⁻¹⁹ cm² for C₆₀ and −9.4 × 10⁻¹⁸ cm² for C₇₀. Signs of $({\gamma }_{\text{S1}}-{\gamma }_{\text{S0}})$ agree with those determined by the Z-scan technique for both solutes. In addition, magnitudes of $({\gamma }_{\text{S1}}-{\gamma }_{\text{S0}})$ 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).

 

Fig. 6 Nonlinear refractive coefficient $({\gamma }_{\text{S1}}-{\gamma }_{\text{S0}})$ calculated as a function of λ for C₆₀-toluene (a) and C₇₀-toluene (b).

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5. Conclusion

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 $({\gamma }_{\text{S1}}-{\gamma }_{\text{S0}})$ of C₆₀ and C₇₀ molecules in the wavelength range between 480 and 850 nm from the spectra of $({\sigma }_{\text{S1}}-{\sigma }_{\text{S0}})$ measured in the wavelength range between ~440 and 900 nm. The calculated $({\gamma }_{\text{S1}}-{\gamma }_{\text{S0}})$ 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.