1. Introduction

The Z-scan technique [1–5] has been largely used for measuring the nonlinear (NL) refractive index, n₂, and the NL absorption coefficient, α₂, of materials. The NL parameters are obtained analyzing the variation of the far-field intensity profile of a laser beam transmitted through a thin sample located at different positions with respect to the focusing lens [1–3] or reflected by the surface of a nontransparent sample [4, 5]. The technique is simple and elegant and has been used routinely to characterize optical materials. However, when the sample is a heterogeneous medium (e.g. a homogeneous solid or a liquid embedded with micro or nanoparticles (NPs), or molecule aggregates suspended in a liquid) the Z-scan technique does not distinguish if the variation of the transmitted light during the measurement is due the linear light scattering or to the effective nonlinearity of the sample. Furthermore, since it is necessary to move the sample to determine the NL parameters, the conventional Z-scan technique does not allow single-shot measurements. Indeed, this may be a serious limitation if one is dealing with fragile samples, e.g. organic and biological materials, that can be damaged by multiple laser pulses [6] or when photo-induced effects occurs in solids [7, 8]. Many attempts were made by several authors to determine n₂ in single-shot experiments [9–16] but the measurement of sign and magnitude of n₂ during one laser pulse remains a challenging task. On the other hand, the limitation of the Z-scan technique for the study of scattering media led to development of other techniques such as the spectral re-shaping [17] and the dual-arm Z-scan technique [18], that present good sensitivity but require laborious experimental setup.

In the present paper we report on a new method for measurements of n₂ that consists of recording with a camera the scattered light in the transverse direction to the beam propagating inside a scattering medium. The method used, labeled as Scattered Light Imaging Method (SLIM), was originally developed for laser beam characterization [19–21] using either continuous-wave or pulsed lasers. However, besides the already exploited features [19–23], one is capable of performing measurements to study the beam self-focusing (or self-defocusing)in NL experiments. In the present paper it is shown that the SLIM can be used to determine the effective NL refractive index of a composite system, $n_2^{\mathit{eff}}$, using lasers delivering pulses in a given repetition rate or in the single-shot regime. As a demonstration of the SLIM for measuring $n_2^{\mathit{eff}}$, we performed proof-of-principle experiments to characterize liquid colloids containing silica NPs.

2. Theory

To describe the laser beam propagation inside a cell filled with a liquid colloid with NPs we followed the approach described in [24] and consider the medium as a stack of thin NL lenses that depend on the incident beam intensity, I. The refractive behavior of the composite medium is described by $n^{\mathit{eff}}(I)=n_0^{\mathit{eff}}+n_2^{\mathit{eff}}I$, where $n_0^{\mathit{eff}}$ and $n_2^{\mathit{eff}}$ are the effective linear and NL refractive indices, respectively. Both effective refractive indices contain information related to the host liquid and the NPs. Since the SLIM allows accurate measurements of the beam waist and divergence angle, the theoretical approach described in [24] was adapted to take into account the experimentally accessible parameters obtained from images collected in the perpendicular direction with respect to the beam propagation direction (z axis).

Accordingly, for the theoretical interpretation of the data we considered that the laser beam has a Gaussian profile given by $I(r)=I_0{\left(\frac{w_0}{w_{\mathit{lin}}(z)}\right)}^2\text{exp}\left(-\frac{2r^2}{w_{\mathit{lin}}^2(z)}\right)$, where $w_{\mathit{lin}}(z)=w_0\sqrt{1+{\left(\frac{z}{z_R}\right)}^2}$ is the beam waist for light propagating linearly for a distance z, w₀ is the minimum beam waist, $z_R=\frac{n_0^{\mathit{eff}}\pi w_0^2}{\lambda M^2}$ is the Rayleigh length, λ is the light wavelength, M² is the beam quality (or propagation) factor [22, 23], I₀ is the on-axis intensity and r is the transverse distance from the optical axis. In the limit of an infinite number of thin NL lenses, and considering that the on-axis NL phase-shift, $\mathrm{\Delta }\mathrm{\Phi }=\frac{2\pi }{\lambda }{n}_2^{\mathit{eff}}{I}_{0}z$ is small when the light propagates through the sample for a distance z such that |ΔΦ| ≪ π, the propagation can be described by the ABCD matrix

The nonlinearity is taken into account through the effective focal distance f_eff that depends on a geometrical parameter a = 6.4 [24], and on the quantity $J={\int }_0^{\frac{z}{z_R}}du{\left(1+u^2\right)}^{-2}\approx \frac{\pi }{4}$, where the approximation is correct within 2% for z ≥ 3z_R. The a value originates from the fact that in the Z-scan technique, a = 6.4 is associated with an infinitesimal aperture positioned in the far-field, in front of the detector. In this case, the variation in the intensity transmitted by the aperture is only due to the NL change of the beam divergence angle. Since to implement the SLIM we detect the beam’s divergence angle, the measurements are equivalent to those obtained with very small aperture placed in the far-field region in the Z-scan method.

Using the standard way for describing the Gaussian beams propagation by an ABCD matrix and applying Eq. (1), considering that the beam at z = 0 (cell’s entry face) has the minimum waist, it is possible to obtain the beam waist, w_NL(z), for arbitrary positions inside the sample including the NL effects. Then defining the NL and linear local divergence angles respectively as ${\theta }_{\mathit{NL}}=\frac{d{w}_{\mathit{NL}}(z)}{dz}$ and ${\theta }_0(z)=\frac{d{w}_{\mathit{lin}}(z)}{dz}$, it is possible to show that for z_R ≪ z ≪ f_eff we have

3. Experimental details

The experiments for NL characterization of the scattering media were performed with samples having different concentrations of silica NPs suspended in a liquid. The samples were prepared by adding acetone in an ethanol solution containing spherical silica NPs with average diameter of (120 ±21) nm. The NPs concentration is expressed through the filling fraction, f, that is the ratio between the volume of the NPs and the total volume of the sample. The ethanol solution of 1 ml corresponds to f = 4.5 × 10⁻⁴, which provides a turbid liquid that presents strong light scattering behavior. The value of f was changed adding small quantities of acetone in the solution and the values $n_2^{\mathit{eff}}$ were determined using the values for acetone, ethanol and silica [25] and their respective volume fractions through the expression: $n_2^{\mathit{eff}}=\left(n_2^{\mathit{acetone}}\times V_{\mathit{acetone}}+n_2^{\mathit{ethanol}}+V_{\mathit{ethanol}}+n_2^{\mathit{silica}}\times V_{\mathit{silica}}\right)\times {\left(\mathit{total}\hspace{0.17em}\mathit{volume}\right)}^{-1}$. The parameters of the samples are given in Table 1.

Table 1. Characteristics of the samples.

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For the experiments it was used the second harmonic beam obtained from a linearly polarized Q-switched and mode-locked Nd:YAG laser (10 Hz, 80 ps, 532 nm). For the NL measurements the beam intensity on the samples was adjusted from 7 to 16 GW/cm² by using a λ/2 plate followed by a Glan-Laser prism, as indicated in Fig. 1(a); the beam was focused with a 50 mm focal length lens to the center of the cell. Figure 1(b) shows the imaging optical system that was positioned in a direction perpendicular to the beam direction. It consisted of two identical cylindrical lenses with 80 mm focal length, aligned orthogonally to each other. The first lens focusses the light in the y direction (vertical plane) and second lens focusses in the z direction (horizontal plane) of the object. The overall distance (from the object to the image) was the same for both lenses. The object distance was chosen such that the magnifications on the vertical and horizontal directions were 3 and 1/3, respectively. In order to obtain an optical resolution approximately equal to the pixel size of the CCD camera (6.45μm×6.45μm) it was used a 6 mm width rectangular slit in front of each lens leading to a f -number=13. More details on the SLIM setup are given in [20]. Figure 1(c) shows the scattered light image for a laser intensity of 5 GW/cm² propagating in a liquid scattering medium (soluble oil), corresponding to the linear regime of light propagation. NL behavior of soluble oil was no observed even for I = 16 GW/cm², which corresponds to the maximum laser intensity, and the evolution of the beam radius along propagation can be adequately described by w_lin(z). Notice that the beam can be visualized in the entire cell. The image is processed as a matrix where each element is an image pixel. Each column represents a specific z position and the elements of the row are the measured intensity along the y direction. To determine the beam radius as a function of the z position, the laser beam profile was numerically adjusted by a Gaussian function along the columns [22, 23]. The divergence angle, θ, was determined by the slope of a linear fit adjusted in the last positions along the z axis of the beam radius curve. The intensity value is determined simultaneously with the image capture using an external beam sampler. Since the laser beam intensity presented large fluctuations, we performed a post-filtering selection and recorded only images corresponding to intensities varying at most by ±5%. For each sample four values of intensity were taken. For each of these values we recorded 50 images (corresponding to 50 pulses). The results, shown in Fig. 1(d), lead to θ₀ = (10.7 ±0.6) mrad, determined from the slope of a linear fit measured for z > 7 mm. The beam waist measured was w₀ = (37 ± 7) μm at the center of the 10 mm quartz cell that contains the colloid. For these measurements we obtained M² = 2.34.

 

Fig. 1 (a) Laser pulse energy control and focusing system. (b) Experimental setup of the SLIM and ray diagram for scattered light (d₁ = 10.7 cm and d₂ = 21.3 cm). (c) A scattered light image obtained by SLIM and the definition of the beam waist 2 w₀ (vertical bar) and the divergence angle θ, between the arrow and the horizontal dashed line. (d) The beam radii (black open circles) measured for z >7 mm (it corresponds to 3 mm of the end of the cell) and the slope (red solid line) was equal to 0.0107. The linear fit determines the angle θ₀ of the laser beam propagation in the linear scattering medium with I = 5 GW/cm².

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Z-scan measurements were also performed in order to compare the SLIM results with a well-established technique. The Z-scan setup used is similar to the one described in [26]. It was used a 1 mm thick sample cell mounted on a translation stage to be moved in the focus region along the z axis. The transmitted beam spatial profile was monitored using detectors placed in the far-field region. A reference channel was used to achieve a better signal-to-noise (S/N) ratio in the Z-scan measurements as in [3, 26]. The closed- and open-aperture Z-scan schemes were used. The NL absorption coefficient of the samples was smaller than the minimum value that our apparatus allows to detect (0.25 cm/GW for 10 GW/cm²). Hence, only closed-aperture Z-scan profiles could be obtained.

4. Results

Figure 2 shows images obtained by SLIM of the laser beam propagation in the samples A, B, C and D with I = (9.7± 0.5) GW/cm². The laser beam waist was positioned in the entrance face of the cell and it is clear from Figs. 2(a)–2(d) that the scattered light increases with the concentration of scatterers. Contrarily to the images shown in Figs. 2(a)–2(c), in the Fig. 2(d) the laser beam propagation cannot be visualized with good S/N ratio along the whole sample owing to the strong light scattering. However, the θ value may be determined in a z region where the beam intensity spatial profiles were visualized with better S/N ratio, for example, in the middle of the cell. A good S/N ratio was obtained when the peak intensity is at least twice the baseline of the background (noise signal). It is important to remark that even analyzing the beam propagation along a region smaller than the entire cell, it was possible to determine $n_2^{\mathit{eff}}$ with accuracy.

 

Fig. 2 Images obtained by SLIM for I = (9.7 ±0.5) GW/cm²: (a) sample A; (b) sample B; (c) sample C and (d) sample D.

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Figure 3 shows the divergence angle, θ, for the samples A, B, C and D with I varying from 7 to 16 GW/cm². The θ value was determined as described in the section 3. Notice that θ varies linearly with the laser intensity and from the results in Fig. 3 the $n_2^{\mathit{eff}}$ values were determined using the Eq. (3), being presented in Fig. 4 for samples A, B, C and D. It is important to observe that the measured $n_2^{\mathit{eff}}$ values do not change as a function of laser intensity. This is an expected result since the nonlinearity of the colloid is mainly due to the third-order susceptibility. High-order nonlinearities of the liquid and the silica NPs are negligible.

 

Fig. 3 Divergence angle determined from an average of 50 images (or pulses) for I varying from 7 to 16 GW/cm². Black squares represent the experimental data and the red solid line corresponds to the line fit for (a) sample A: θ = 10.80 − 0.15I; (b) sample B: θ = 10.91 − 0.11I; (c) sample C: θ = 10.65 − 0.08I and (d) sample D: θ = 10.71 − 0.06I.

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Fig. 4 Values of $n_2^{\mathit{eff}}$ from the θ values determined for I varying from 7 to 16 GW/cm². Black squares represent the experimental data and the red dotted line corresponds to the average value for (a) sample A: 2.35×10⁻¹⁵ cm²/W ; (b) sample B: 1.53×10⁻¹⁵ cm²/W; (c) sample C: 1.46×10⁻¹⁵ cm²/W and (d) sample D: 0.94×10⁻¹⁵ cm²/W.

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The results of Fig. 3 and Fig. 4 were obtained performing an average for 50 acquired images by overlapping the data images (we refer to this procedure as P_av). However the experiments were also analyzed for single-shot pulses (procedure referred as P_ss). Figure 5 shows the single-shot NL refractive index behavior versus I, for the sample B. The $n_2^{\mathit{eff}}$ average values obtained by P_av (Fig. 4(b)) or by P_ss are similar. However, in the case P_ss, the larger standard deviation (about 20%) shows discrepancies in relation to P_av results, that are attributed to the laser fluctuations, mechanical vibrations and thermal instability of the laser cavity. These effects induce a variation of the laser beam pointing-stability and therefore it is observed when the P_ss is performed. These systematic errors make evident the importance of single-shot measurements even when no degradation of the sample is observed.

 

Fig. 5 Single-shot NL refractive index, $n_2^{\mathit{eff}}$, obtained by SLIM for sample B. Black circles: single-shot pulses; red dotted line: average value.

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As the SLIM is intrinsically a single-shot method of laser beam characterization [20], whereas the Z-scan technique requires averaging of each sample position (i.e., the Z-scan is not a single-shot method), therefore, the average of the results obtained by P_av are presented here to establish a basis for comparison between the SLIM and Z-scan.

In order to validate the SLIM results, we performed Z-scan measurements for all samples. The Z-scan measurements were performed averaging 10 pulses at 60 locations of the cell along the z axis, i.e.: processing 600 laser pulses; and the result for I = (9.4 ±1.9) GW/cm² is shown in Fig. 6. Notice that while the Z-scan profile of samples A and B, shown in Figs. 6(a) and 6(b), present good S/N ratio, the signal for sample C is degraded owing to the increase of the beam wavefront distortion due to the large linear light scattering. Indeed the S/N ratio becomes very poor and the Z-scan technique cannot be applied for samples with f > 3 ×10⁻⁴, as for the sample D plotted in Fig. 6(d). The solid curves in Figs. 6(a)–6(c) represent the best fit to the data. The value $n_2^{\mathit{eff}}=\left(2.40\pm 0.36\right)\times {10}^{-15}\hspace{0.17em}{\text{cm}}^2/\text{W}$ was obtained, in agreement with the $n_2^{\mathit{eff}}=2.38\times {10}^{-15}{\text{cm}}^2/\text{W}$ calculated for sample A. The measurements for the samples B and C also present similar agreement; for instance we obtained $n_2^{\mathit{eff}}$ (sample B)= (2.40 ±1.44) × 10⁻¹⁵ cm²/W and $n_2^{\mathit{eff}}$ (sample C)= (2.50 ± 1.89) × 10⁻¹⁵ cm²/W. The Z-scan technique cannot be applied for the sample D because of the large light scattering.

 

Fig. 6 Z-scan data for I = (9.4 ±1.9) GW/cm²: (a) sample A; (b) sample B; (c) sample C and (d) sample D. The open circles are experimental data and the solid lines are Z-scan fits.

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The $n_2^{\mathit{eff}}$ values were determined from the Z-scan data following the usual procedure introduced in [1, 2] by measuring the transmittance difference peak-to-valley and determining the NL phase-shift on the beam axis. Since the NL refractive index of ethanol and acetone are positive at 532 nm, the Z-scan profiles of Fig. 6 display a positive NL refractive index for the colloids. The NL contribution of the NPs is negligible.

Figure 7 summarizes the results obtained for $n_2^{\mathit{eff}}$ using the SLIM and the Z-scan technique as a function of f and V_acetone. It can be seen that with the Z-scan technique we could not detect the changes of $n_2^{\mathit{eff}}$ due to the decrease of V_acetone while with the SLIM the changes become clear. The solid line was drawn considering the NL refractive indices of acetone and ethanol, and their respective volume fractions. Notice that with the Z-scan technique, it was only possible to measure $n_2^{\mathit{eff}}$ for samples with f < 3 ×10⁻⁴ while using the SLIM we were able to measure $n_2^{\mathit{eff}}$ for f up to 4.5×10⁻⁴, which was limited by the maximum concentration of the samples available.

 

Fig. 7 Comparison of Z-scan results (filled blue circles) measured with I = (9.4 ±1.9) GW/cm² and SLIM (filled red squares) for I = (9.7 ±0.5) GW/cm² with the $n_2^{\mathit{eff}}$ versus volume fraction of acetone and filling fraction of silica NPs. The black solid line represents the theoretical predictions of our model.

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5. Summary

In summary the present experiments demonstrate the application of the Scattered Light Imaging Method (SLIM) for NL spectroscopy of scattering media. The effective NL refractive indices of colloids with large density of scatterers could be measured even when the Z-scan technique could not be applied. The experimental setup used has no movable parts, and the measurement of $n_2^{\mathit{eff}}$ requires, in principle, only one laser-shot. It is also important to notice that the single-shot results are not affected by laser beam pointing-stability. In conclusion, we evaluate SLIM as a powerful tool for NL studies of heterogeneous media which may present large elastic light scattering cross-sections.