1. Introduction

The third-order nonlinear optical materials have been receiving considerable interests because of their potential applications in optical communication, data storage, optical computing, optical switching and optical modulation. These materials should have large optical nonlinearities and fast response, especially in the femtosecond (fs) time domain [1–14]. To find proper nonlinear optical materials for use in photonic devices, a wide variety of novel materials, including semiconductors, polymers, nanomaterials, organometallic compounds and special glasses, have been studied for their third-order optical nonlinearities by femtosecond pulsed laser [1–15]. And a number of approaches, including the Z-scan [16], degenerate four-wave-mixing (DFWM) [1–5], optical Kerr gate (OKG) [6–10]/optical Kerr effect (OKE) [11,12] and optical heterodyne detection of optical Kerr effect (OHD-OKE) techniques [13–15], are widely being used to characterize the third-order optical nonlinearities of these novel materials. In the latter three methods, for simplifying the measurements the CS₂ is usually chosen as a reference and assumed no imaginary part for susceptibility tensor elements. However, these methods relate to the modulus of ${\chi }_{ijkl}^{(3)}$ of the reference, (within the references [1–14], $\left|{\chi }_{xxxx}^{(3)}\right|$ for DFWM, $\left|{\chi }_{xyyx}^{(3)}+{\chi }_{xyxy}^{(3)}\right|$ for OKG/OKE and OHD-OKE) [1–15]. Recently, several linearly polarized light open-aperture (OA) Z-scan experiments were carried out in CS₂ around 800 nm [17–21]. In these measurements there was obvious nonlinear absorption in CS₂, which means the no-imaginary assumption is challenged and the values from DFWM, OKG/OKE and OHD-OKE methods may be not reliable enough. Recently, the signal of the linearly polarized light OA Z-scan experiment was verified to arise mainly from nonlinear scattering, and the nonlinear absorption (or $\mathrm{Im}({\chi }_{xxxx}^{(3)})$) of linear polarized light is negligible around 800 nm [22]. But we could not conclude that $\mathrm{Im}({\chi }_{xyyx}^{(3)})$ or $\mathrm{Im}({\chi }_{xyxy}^{(3)})$ is negligible. So, it is better to check whether the two imaginary parts are negligible.

As listed in Table 1 , the reference values of CS₂ are not identical for each method. One reason for the non-uniform reference values is the lack of report on the third-order susceptibility tensor elements at fs time scale. To make that the nonlinearities values measured by the three methods could be explicitly compared among novel materials, firstly, the reference value should be identical for every method. So, it is very necessary to know fully the real and imaginary parts of susceptibility tensor elements of CS₂ before determining the reference values for each method. Minoshima et al have given the value of $\mathrm{Re}({\chi }_{zzzz}^{(3)})$ at 100-fs pulses [23]. However we cannot obtain the accurate values of real parts of other tensor components by using the relationship of $\mathrm{Re}({\chi }_{xyyx}^{(3)})/\mathrm{Re}({\chi }_{xyxy}^{(3)})=1$ as done by some researchers [11,13,14], because the Kleinman symmetry rule is no longer valid [24,25]. At this time scale of most commercial fs laser used for studying novel materials, there are fast non-instantaneous nuclear contribution ($\mathrm{Re}({\chi }_{xyyx}^{(3),nucl})/\mathrm{Re}({\chi }_{xyxy}^{(3),nucl})=6$ [26,27]) as well as nonresonant electron contribution ($\mathrm{Re}({\chi }_{xyyx}^{(3),elec})/\mathrm{Re}({\chi }_{xyxy}^{(3),elec})=1$ [25,26], ) to the nonlinear susceptibility of CS₂, which means that the ratio $\mathrm{Re}({\chi }_{xyyx}^{(3)})/\mathrm{Re}({\chi }_{xyxy}^{(3)})$ is between 1 and 6 [26–29]. To our best knowledge, the research on the CS₂ third-order nonlinear susceptibility tensor elements at fs time scale is scarce although the nonlinear refractive index, nonlinear absorption coefficient or single tensor element has been widely studied [17–23]. Thus, to accomplish the measurement of susceptibility components of CS₂ and supply proper reference values for the three methods, it is better to study systematically the real and imaginary parts of third-order nonlinear susceptibility tensor elements of CS₂ at the fs time scale.

Table 1. Referenced Value of CS₂ Used in Literature

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The well-known Z-scan method has been widely used to measure the nonlinear refractive index and nonlinear absorption coefficient [16]. Krauss et al showed that the independent components of χ⁽³⁾ can be specified in Z-scan method for isotropic medium by using both linearly and circularly polarized lights [30].

In this work, we report on the values of real parts of susceptibility tensor elements of CS₂ determined by polarized lights Z-scan method with common fs laser pulses (λ = 800 nm, τ_FWHM = 125 fs) [1–14], and check whether the imaginary parts are negligible. The motivations of this work are both to supplement the values of CS₂ susceptibility tensor elements and to provide the reference values for DFWM, OKG/OKE, OHD-OKE and the ellipse rotation [15] measurements.

2. Experiments and results

In our experiments, fs laser pulses at the wavelength of 800 nm with the duration of 120-fs (FWHM) were generated from a regenerated amplifier (mode-locked Ti: sapphire laser, Spitfire pro, Spectra Physics). The repetition rate was set to be 1 KHz. The details of the polarized fs pulses Z-scan experimental setup are described in Fig. 1 [31]. A spatial filter was placed before the setup to produce a near Gaussian spatial distributed beam, which was confirmed by a CCD camera (L230, LBA-USB) [22]. The beam was focused using a 250 mm focal-length lens preceded by a polarizer-waveplate (λ/4) combination to allow for a continuous change in its polarization state. After passing the polarizer-waveplate combination, the pulse width was about 125 fs. The beam waist radius w₀ of the focused laser beam was measured to be 33 ± 2 μm. The elliptically polarized light was produced by orienting the slow axis of the waveplate an angle (φ) 25 degrees to the polarizer [31]. The CS₂ sample was filled in a 1-mm thick quartz cell. The measurement system was calibrated with ZnSe (OA Z-Scan [32]) and toluene (CA Z-scan [20]).

 

Fig. 1 Scheme of polarized lights Z-scan measurement. CA: with an aperture before detector 2; OA: with lens 2 before detector 2.

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The real parts

The on-axis peak intensity I ₀ for closed-aperture (CA) Z-scan measurements was set to be 33-101 GW/cm². The normalized transmittance traces for different polarized lights at I ₀ = 55 GW/cm² are shown in Fig. 2(a) . Under this intensity, there was no observable nonlinear refraction signal from cell for these three type polarized lights in our measurements.

 

Fig. 2 (a) CA Z-scan traces of different polarized lights and theoretical fitting curves at 55 GW/cm². (b) The dependence of n₂ on I ₀ and polarization state.

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From the figure we find the nonlinear refractive index decreases with ellipticity (0 for linearly polarized light, 0.4663 for elliptically polarized light and 1 for circularly polarized light). From the fittings of these experimental data the nonlinear refractive index n ₂ can be obtained for all polarized lights and listed in the inset of the figure [31]. Furthermore, the values of $\mathrm{Re}({\chi }_{xxxx}^{(3)})$ and $\mathrm{Re}({\chi }_{xyxy}^{(3)})$ could be calculated from linearly and circularly polarized cases, respectively. Owing to ${\chi }_{xxxx}^{(3)}=2{\chi }_{xyxy}^{(3)}+{\chi }_{xyyx}^{(3)}$ for isotropic medium, the value of $\mathrm{Re}({\chi }_{xyyx}^{(3)})$ could be determined based on the values of $\mathrm{Re}({\chi }_{xxxx}^{(3)})$ and $\mathrm{Re}({\chi }_{xyxy}^{(3)})$. The ratio of $\mathrm{Re}({\chi }_{xyyx}^{(3)})$ to $\mathrm{Re}({\chi }_{xyxy}^{(3)})$ is about 2.1, which is consistent with Burgin’s result [26]. The n _2,lin value agrees well with others’ [17–21], and the value of $\mathrm{Re}({\chi }_{xxxx}^{(3)})$ is in good agreement with Gong and Minoshima’s results [10,23].

The nonlinear refractive indexes were also measured at other intensities. The dependence of nonlinear refractive index on intensity and polarization state is shown in Fig. 2(b). The lines are constant fittings, thus, there is no higher-order nonlinearity in the intensity region. By combining the constant fitting results of linearly and circularly polarized cases, the values of $\mathrm{Re}({\chi }_{xyxy}^{(3)})$ and $\mathrm{Re}({\chi }_{xyyx}^{(3)})$ were determined and listed in Table 2 . The ratio $\mathrm{Re}({\chi }_{xyyx}^{(3)})/\mathrm{Re}({\chi }_{xyxy}^{(3)})$ is about 1.8. So, there is fast non-instantaneous nuclear contribution at the time scale [26–29], and it is unsuitable to calculate the real part value of other components from $\mathrm{Re}({\chi }_{xxxx}^{(3)})$ by using the Kleinman approximation. Furthermore, the values of the real parts of susceptibility components from different contributions can be separated through the values of $\mathrm{Re}({\chi }_{xyxy}^{(3)})$ and $\mathrm{Re}({\chi }_{xyyx}^{(3)})$ because the ratio $\mathrm{Re}({\chi }_{xyyx}^{(3)})/\mathrm{Re}({\chi }_{xyxy}^{(3)})$ is different for electronic and nuclear contributions. The results are listed in Table 3 . These values of the real parts are relatively independent of wavelength over the visible and near infrared region [17,19]. When the pulse width is less than 1 ps, these values increase little with pulse width (about 20% from 110 fs to 475 fs) due to the small increase of nuclear contribution [17,26].

Table 2. Values of Third-order Nonlinear Susceptibility Tensor Elements From Constant Fittings in Fig. 2(b)

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Table 3. Contribution of Nonresonant Electron and Fast Non-instantaneous Nuclear to Real Part of Susceptibility Tensor Elements

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The imaginary parts

According to Krauss’ work, the values of $\mathrm{Im}({\chi }_{xxxx}^{(3)})$ and $\mathrm{Im}({\chi }_{xyxy}^{(3)})$ could be obtained by linearly and circularly polarized lights OA Z-scans since the two-photon absorption coefficient $\beta =48{\pi }^2\omega [\mathrm{Im}({\chi }_{xyxy}^{(3)})+(1-{\sin }^{2}2\phi )\mathrm{Im}({\chi }_{xyyx}^{(3)})/2]/n_0^2{c}^2$ for the two polarized lights [30,33]. Furthermore, the value of $\mathrm{Im}({\chi }_{xyyx}^{(3)})$ could be calculated by subtracting $\mathrm{Im}(2{\chi }_{xyxy}^{(3)})$ from $\mathrm{Im}({\chi }_{xxxx}^{(3)})$. The normalized transmittance traces for I ₀ = 38 GW/cm² are shown in Fig. 3 . From the figure we could not find any observable nonlinear absorption for all the polarized lights within experimental error. We have checked the polarized lights OA Z-scan at other intensities, which are up to 183 GW/cm², and have not found any observable nonlinear absorption for the three polarized lights [22]. So, both $\mathrm{Im}({\chi }_{xyyx}^{(3)})$ and $\mathrm{Im}({\chi }_{xyxy}^{(3)})$ are negligible. The value of ${\beta }_{lin}$ from measurements of Falconieri and Gnoli et al is about 1.2 × 10⁻¹³ cm/W [34,35], which means $\mathrm{Im}({\chi }_{xxxx}^{(3)})$ = 5.1 × 10⁻¹⁷ esu. According to their results, $\mathrm{Im}({\chi }_{xxxx}^{(3)})$ is also negligible compared with $\mathrm{Re}({\chi }_{xxxx}^{(3)})$. The polarized lights OA Z-scan were also carried out at higher intensity, but it was found that the signal at the valley of Z-scan trace was mainly from nonlinear scattering for the three polarized lights. The polarized lights OA Z-scan was repeated at 780 nm, there was no observable nonlinear absorption signal before the nonlinear scattering appeared [22]. Therefore, the imaginary parts of susceptibility components are negligible around 800 nm, and the assumption of no-imaginary is reliable and available. But there is nonlinear absorption for CS₂ in the shorter wavelength region [36]. And, the aged CS₂ (yellow color) gives an absorption peak at 350 nm, with probable two-photon absorption around 700 nm [37]. So, $\mathrm{Im}({\chi }_{ijkl}^{(3)})$ may be non-negligible for aged CS₂ around 700 nm.

 

Fig. 3 OA Z-scan experimental data of different polarized lights.

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The reference value

Based on the values of susceptibility tensor elements, the reference value for non-aged CS₂ is suggested to be (9.8 ± 0.7) × 10⁻¹⁴ esu for $\left|{\chi }_{xxxx}^{(3)}\right|$ in DFWM, (7.2 ± 0.8) × 10⁻¹⁴ esu for $\left|{\chi }_{xyxy}^{(3)}+{\chi }_{xyyx}^{(3)}\right|$ in OKG/OKE and OHD-OKE, and (4.6 ± 0.8) × 10⁻¹⁴ esu for $\left|{\chi }_{xyyx}^{(3)}\right|$ in the ellipse rotation measurements when fs laser pulse is in near infrared range.

3. Conclusions

We have systematically studied the real and imaginary parts of third-order susceptibility independent components of CS₂ by using polarized lights Z-scan method. The real parts of tensor elements arising from different mechanisms have separately been determined, and the imaginary parts are negligible. The ratio of $\mathrm{Re}({\chi }_{xyyx}^{(3)})$ to $\mathrm{Re}({\chi }_{xyxy}^{(3)})$ is about 1.8, which means both nonresonant electron and fast non-instantaneous nuclear contribute to the third-order nonlinear susceptibility. Based on the values of these susceptibility elements, the reference value of non-aged CS₂ is suggested to be (9.8 ± 0.7) × 10⁻¹⁴ esu for .. in DFWM, (7.2 ± 0.8) × 10⁻¹⁴ esu for $\left|{\chi }_{xyxy}^{(3)}+{\chi }_{xyyx}^{(3)}\right|$ in OKG/OKE and (4.6 ± 0.8) × 10⁻¹⁴ esu for $\left|{\chi }_{xyyx}^{(3)}\right|$ in the ellipse rotation measurements respectively when experiments are carried out by using near infrared fs pulsed laser.