1. Introduction

Since 1990s [1], carbon disulfide (CS₂) has been applied in nonlinear refraction coefficient calibration [2] as an inexpensive and frequently-used material [3, 4]. Nonlinear absorption (corresponding to the imaginary part of nonlinear susceptibility) in CS₂ had been studied later since the refraction process in it was investigated widely. In early studies, the nonlinear absorption process at 800 nm is assumed to be two-photon absorption (2PA) [5–8]. Ganeev et al. [5,6] showed that the 2PA coefficient of CS₂ becomes smaller with increase of incident pulse intensity, which conflicts with the common view that with increase of input pulse intensity in Z-scan experiment, the fitted absorption coefficient is supposed to be stable [1, 9]. Therefore, they presumed that both 2PA and 3PA process took place in CS₂. However, in recent research [10–12], it has been proved that the 2PA process in fresh CS₂ can be neglected. For CS₂, many studies show that a variety of nonlinearity including filamentation [13] and nonlinear scattering [11, 14] can take place together with nonlinear absorption process. Besse et al. [13] investigated filamentation at relatively low intensity in CS₂ (5.54 GW/cm²). They presumed that self-focusing and defocusing in the filament can lead to inaccurate peak intensity and nonlinear absorption coefficient extracted. Yan et al. [11] and Gnoli et al. [14] carried out open aperture (OA) Z-scan experiments with pulse width of about 120 fs at intensity up to 183 GW/cm² and did not find absorption signal, so they claimed that the nonlinear absorption signal shown in previous Z-scan experiments for CS₂ at 800 nm might be caused by nonlinear scattering, i.e., Stokes stimulated Raman scattering and Stokes stimulated Rayleigh-wing scattering around 800 nm. Nevertheless, Kong et al. [15] reported good fitting results of CS₂ based on 3PA model at 790 nm with pulses of 27 ps at intensity of 137 GW/cm². It is wondering why two OA Z-scan results that seems conflicting were reported at similar intensity, what nonlinearities occur in this process, and how they affects nonlinear absorption in OA Z-scan. It is also confusing that if filamentation occurs at such low intensity (5.54 GW/cm²), why no effect on nonlinear refraction was observed in the both studies since the filaments generated should have affected the results.

In order to separate absorption from nonlinearity including filamentation and nonlinear scattering inside CS₂, parametric studies should be introduced for Z-scan setup. Spectral resolved studies were applied, in which work we replaced the CCD, which is the detector of a traditional Z-scan system, by two spectrometers. Moreover, we took pictures of Z-scan spots with a digital camera, which can reveal spatial properties of filament generated in CS₂. Using this system, it was found that for CS₂, the major part of effect on 3PA at 800 nm came from plasma generated in filaments, which can be eliminated by system introduced in our work. Therefore, it would be feasible to apply 3PA model at 800 nm even if filaments were generated in CS₂.

2. Experimental setup

A regenerative amplified femtosecond laser system is applied to generate pulses at central wavelength of 790nm. The repetition rate, pulse energy, and pulse duration are 1kHz, ∼4 mJ, and ∼50 fs, respectively. A half-wave plate and a Glan prism are placed to attenuate the pulse energy to ∼30 nJ. The iris in Fig. 1 is used to limit cross section of the beam.

 

Fig. 1 Experimental setup diagram. In part of Z-scan experiments, the red lines are optical paths; the blues lines are control lines accessing PC, while the black ones are optical fibers. “IS” is an abbreviation for integrating sphere. After completing Z-scan experiments, photographs of the spots were captured. In this part, the optical elements following the screen does not work, so the optical path is presented as dashed line. “ET” is exposure time of the digital camera. Narrow-band filters of various wavelength are replaced to obtain spot photographs at 585, 633, and 800 nm.

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Our experiment includes 2 parts. In the first part, an Z-scan system referring to white light continuum (WLC) Z-scan setup [16] was established. The distinction between our Z-scan system and a traditional one is that we placed two spectrometers as the detectors in which one is taken as signal and the other is reference. The advantage of this setup is that it can separate different spectral components instead of integrating them (like CCD camera) in an open-aperture (OA) Z-scan experiment if filamentation or nonlinear scattering occurs. After repeating Z-scan 3 times, a screen is placed to show the Z-scan spots in which part we called it photographic part. In order to realize spectral resolution for this part, 3 narrow-band filters at wavelength of 585 nm, 633 nm, and 800 nm are successively placed in front of the camera. The exposure time of camera is set to be 5 seconds to obtain higher clarity of pictures. The stage and spectrometers are controlled by a computer automatically to ensure the acquisition of z-values is the same in two experimental parts.

The result of Z-scan part and photographic part (partially) are shown, respectively, in Fig. 2 and Fig. 3.

 

Fig. 2 Transmittance of CS₂ at wavelength from 700 nm to 900 nm along z-axis. (a)–(c) is transmittance for CS₂ in cuvette with optical path length of 1 mm, and (d)–(f) is that for 2 mm. The incident pulse energy increases from left to right whose corresponding peak intensity is given in Table. 1.

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Fig. 3 Z-scan spot photographs at wavelength of 585 nm ((a) and (a′)), 633 nm ((b) and (b′)), and 800 nm ((c) and (c′)) obtained by replacing narrow-band filters of different specifications. The photographs are captured along z-axis in which case the pulse energy is of ∼30 nJ, corresponding to the two groups of intensity at maximum (the third row of ∼40 GW/cm² in Table. 1). There are totally 22 photographs in each group of wavelength, which is ordered in two columns from top to bottom (1–11 (left) and 12–22 (right)). For clarity, data of two specifications are divided into two parts where the red part is for 1 mm cuvette ((a)–(c)) and the green one is for 2 mm cuvette ((a′)–(c′)).

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3. Data processing

3.1. Spectral data processing

As shown in Fig. 2, Z-scan traces are dispersed spectrally because we used spectrometers as detectors. Therefore, for validity, Z-scan trace at 790 nm (the central wavelength of incident pulses) is chosen to extract the 3PA coefficient β⁽³⁾ by assuming a 3PA model [9,15]. The fitted β⁽³⁾ at different peak intensity, which can be calculated using the formula $I_0=\frac{2{∊}_0}{{\pi }^{3/2}{\omega }_0^2{\tau }_{\text{p}}}$ [9,17], are listed in Table. 1 for optical path length of 1 mm and 2 mm. Due to the fact that filamentation takes place, we fitted the beam waist as a parameter to obtain an “effective” one, and the value measured is ∼20μm. The fitted curve are shown in Fig. 4 and Table. 1. Additionally, error information is also given in the figure and the table to show the stability of data in repetitive experiments considering the turbulence in filaments.

 

Fig. 4 Z-scan traces and error bars versus fitting lines. (a)–(c) are traces for CS₂ in cuvette with optical length of 1 mm, and (d)–(f) are those of 2 mm. The vertical axes are transmittance of Z-scan, which is labeled as “Tr”. The peak intensity of incident beam at geometric focus is given in the legend at bottom of each figure.

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Table 1. Averaged 3PA Coefficients Obtained by Fitting z-scan Traces at 790 nm. Averaged Peak Intensity is Also Calculated for Each Group of Different Pulse Energy and Optical Path.

View Table

3.2. Photographic processing

There was an intersection angle between the laser beam and the imaging path of camera. Hence, the captured photographs show projections of the spot images in side view, and they must be restored to front view. In order to calibrate the spot surface, we used a standard circular reference. Elliptical line model is applied to fit the edge of the reference. By constructing an affine matrix, the surface can be restored (see Figs. 5(a) and 5(b)).

 

Fig. 5 Image adjustment and analysis of captured photographs. (a) - projection analysis by calibrating a circular reference. Elliptical line model was applied to fit its edge. The center coordinates in pixel, semi-major axis “a”, semi-short axis “b”, and reflection angle “theta” are marked. (b) - an adjusted interference photograph (see Fig. 3(b19)) and the cropped area (green rectangle). X_min and Y_min represent the coordinates (in pixel) of lower-left corner of the cropped area. In addition, the width and height (in mm) of this area are also marked here. (c) - rotated (for convenience of processing) image and interference image cropped in order to analyse the fringe spacing. (d) - interference analysis. The red crosses are the sample data. The green solid lines are the fitted skeletons of fringes, while the dotted lines are the spaces between adjacent fringes. This space was calculated to be 4.3 mm.

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In some photographs, clear interference fringes can be observed, for example, as shown in Fig. 3(b19). Particularly, linear equidistant fringes usually indicate dual light source. We presume they are generated by dual filaments. Given three parameters: the distance between the sample and the screen surface (135.1 mm); averaged fringe spacing (corresponding to Figs. 5(c) and 5(d))), which is calculated to be 4.302 mm; and the wavelength of the captured light (633 nm), the distance between dual filaments is estimated to be 0.02 mm, which is similar to the observation in [13]. Moreover, the intensity where a second filament appears is at the same magnitude (15 GW/cm² in [13] and ≤26 GW/cm² in our work) as the previous work.

In order to show the effect of filaments intuitively, the normalized integral gray values for each photograph are shown in Fig. 6. We also showed beam intensity at incident plane of the sample using different colors as the background. By comparing the incident intensity and normalized integral gray value, emission intensity distribution along the optic axis (z-axis) and the diverse of its symmetric axis are presented.

 

Fig. 6 Calculated incident beam intensity along z-axis (colored background) and normalized integral gray value (norm. g. v.) of pictures (blue, orange, and purple lines). The background color, which is labeled at the color bar, represents intensity (GW/cm²) at incidental plane of sample along z-axis. The central pink lines represent the geometric focus of the beam. The circles correspond to the pictures shown in Fig. 3.

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4. Discussion

4.1. Effect of optical path on filamentation

As shown in Fig. 2, the white light continuum (WLC) is generated more intensively as we increase the incident pulse energy. From z=−10 mm to 10 mm, we can see transmittance increasing and then decreasing at wavelength shorter than 800 nm and a decrease around z=0 mm at wavelength longer than 800 nm. The reason of the trend is that WLC has been generated and emission at wavelength longer than 800 nm could be absorbed by excited CS₂ molecules. We will further discuss it in section 4.2. Clear WLC emission can be observed in Figs. 2(c) and 2(f), and obvious interference pattern with nonlinear scattering rings is shown in spot photographs (see Figs. 3(a), 3(b), 3(a′), and 3(b′)). Therefore, we have reasons to believe that filamentation takes place in such cases. Filamentation is described as a dynamic balance between Kerr self-focusing effect and plasma defocusing effect [18,19]. In the case of a thinner cuvette, the filament ends before plasma defocusing effect plays a more effective part. As a result, as shown in Figs. 3(a), 3(b), 3(a′), and 3(b′)), the averaged intensity of the WLC is larger and the interference pattern in filamentation image is more irregular for thicker cuvette (L=2 mm). Moreover, due to the weaker plasma defocusing effect in the case of a thinner cuvette, the contribution of self-focusing overtakes that of defocusing. Hence, even if energy of incident pulse is the same for both cases of L=1 mm and 2 mm, the calculated peak intensity is larger for the case of thinner cuvette (shown in Table. 1). Meanwhile, decrease of β⁽³⁾ with thinner cuvette can be seen in Table. 1. We drew a diagram of this process (see Fig. 7). The beam self-focuses and generate filaments, then plasma is generated near the exit surface of 1 mm cuvette. Due to the shorter optical path length, plasma in CS₂ maintains shorter length in 1 mm cuvette than that of 2 mm. This phenomenon leads to a “collective effect” that longer optical path length increases effective length of the focus. In order to describe this effect, which can be observed in Fig. 3 and Fig. 6, we define an “effective” focus located at geometric center of filamentation area. As shown in Fig. 7, the effective focus in the case of L=2 mm is in front (positive direction of z-axis) of that for L=1 mm. Meanwhile, self-focusing leads to a backward (negative direction of z-axis) shift of real focuses compared with the geometric focus of lens. In summary, effective focus moves backward due to self-focusing effect, while a longer filament moves it (geometric center of filament) forward (as shown in Fig. 7). Therefore, the effective focus corresponding to the symmetric axes of the normalized gray value curves in Fig. 6(a) shift to the negative direction of z-axis for L=1 mm, but in the case of L=2 mm, this kind of shift is less obvious as shown in Fig. 6(b).

 

Fig. 7 Diagram of filamentation and optical path in cuvette of different thickness.

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4.2. Effect of filamentation on nonlinear absorption and refraction

An intriguing phenomenon was also observed that interference pattern and scattering rings were not observed at 800 nm on the screen for either L=1 mm or 2 mm, yet those at 585 nm and 633 nm were very clear as shown in Fig. 3. This phenomenon indicates that effect of filamentation on nonlinear absorption and refraction can be neglected, because if plasma generation, which is a part of filamentation, took place together with absorption, plasma would affect the beam so that shape change of the beam spot could be observed at 800 nm. It was considered to be caused by a poor sensitivity of the digital camera or small screen (Φ 29 mm), which cannot contain all emission, until we replaced the screen by a much bigger one (210 mm×150 mm, Daheng Optics, GCL-2002). Even using the bigger screen, we still found no visible interference pattern or scattering ring at 800 nm, while those could be observed clearly at 585 nm and 633 nm. To further explore this phenomenon, image entropy [20], which shows gray range of images, is calculated to show existence of possible interference pattern and scattering rings. The image entropy of rectangular areas (285×429 pix) at left upper corner of the 11th photographs in each of the 6 groups is calculated. The image entropy calculated for L=1 mm at wavelength of 585 nm, 633 nm, and 800 nm are respectively, 1.7366, 2.8792, and 1.3686, and for L=2 mm, the values are 2.8488, 2.7341 and 1.4174. As a reference, the image entropy of the controlling group (photograph captured in dark) is 1.3486, which means that light emission from plasma can be ignored at 800 nm. Therefore, we presume the reason why spots shown on the screen at 800 nm were not affected by plasma generated in filamentation is that they take place successively. Since our experiments are based on ultra-short pulses (50 fs) and the response time of plasma generation is much longer than the pulse duration, pulses exit from the sample before plasma is generated and the spot of the transmitted pulse is not affected by WLC. However, the response time of nonlinear absorption and refraction is short enough so that spot change due to them can be seen on the images in Figs. 2(c) and 2(c′). However, considering the rather long exposure time, generated plasma would still produce similar interference pattern and scattering rings on the screen. A reasonable explanation is that WLC emission at 800 nm was re-absorbed by CS₂. As shown in Fig. 2, WLC emission generated by plasma around 800 nm is supposed to increase progressively with an increase of intensity, but actually, it becomes weaker around focus (see Figs. 2(c) and 2(f)). It means that at higher intensity, the WLC is absorbed more than that at lower intensity around 800 nm. Therefore, we presume that the excited state of CS₂ maintains longer (∼ps) than the pulse width because it holds on till plasma are generated so that WLC emission around 800 nm is absorbed by excited molecules immediately.

We depicted the nonlinear process within a diagram (see Fig. 8). As shown in Fig. 8, linear and two-photon absorption at 800 nm (1.55 eV) are both prohibited by selecting rules, while linear absorption at 267 nm (4.65 eV) and 3-photon absorption process at 800 nm are allowed [11,15]. Due to self-focusing, excited molecules absorbed more energy and generated plasma, which produced WLC. Meanwhile, due to the absorption of excited molecules, most of the emission around 800 nm was re-absorbed in an ultrafast process, and this mechanism ensures that the spots (transmittance) are not affected by WLC. However, self-focusing, defocusing, and maybe some other nonlinearities, which participate in filamentation, still play important roles in the nonlinear absorption and refraction process before the plasma is generated, and that is why filament lengths still affect the results (see Table. 1). However, as shown in Figs. 3(c) and 3(c′), we found their effects on nonlinear absorption and refraction are not obvious in ultrafast process if we consider this part of filamentation process (without plasma) as a part of absorption process using an “effective” focus. From the view of result, nonlinear absorption and refraction play dominant roles at 800 nm, and the Z-scan spots were not affected by WLC. Overall, the major impact of filamentation on nonlinear absorption and refraction is due to the plasma generated. Therefore, if any setup could disperse Z-scan data spectrally and eliminate the impact of plasma generated in filamentation, and the optical path length is appropriate, extracting accurate nonlinear absorption coefficient at specific wavelength could be possible. Since the emission of the plasma takes part in impact of filamentation, to avoid this emission, one must block or absorb it around incident central wavelength. In this case, it is the excited CS₂ molecules that achieve this process, while in the cases of those close-aperture (CA) Z-scan experiments, of which the correctness have been verified in previous reports [11,15], most of the emission is blocked by the small far field aperture so that the effects of filamentation can be neglected.

 

Fig. 8 Energy levels diagram. The dashed levels represent virtual levels. The red, blue, purple arrows are respectively the transitions from ground state to virtual state S_im, S_in, and real state S₁. Light around wavelength of 800 nm (the red double arrow) within light generated by filamentation (the rainbow arrow) is re-absorbed after a delay.

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5. Conclusion

We established a Z-scan system that introduced spectral resolution to separate nonlinear absorption at 800 nm out. Moreover, effect of filaments generated in CS₂ are further investigated by Z-scan spot photography. The results show that nonlinear absorption as well as refraction and WLC generation are two sequential processes in CS₂ under femtosecond pulses. Besides, WLC emission from plasma around 800 nm can be re-absorbed by excited molecules of CS₂. Therefore, the effect of plasma generated, which is the main effect of filamentation on nonlinear absorption and refraction, is negligible. As a result, fitting Z-scan traces with three-photon absorption model is reasonable if the Z-scan trace at central wavelength is used. Since filamentation process are affected by optical path length inside CS₂, accuracy of extracted nonlinear absorption coefficient can be influenced.