1. Introduction

The Z-scan technique provides a highly sensitive and straightforward method for determining the nonlinear refractive index (γ) and nonlinear absorption coefficient (β) of a large variety of materials. Both the sign and the magnitude of the real and the imaginary part of the third order susceptibility can be deduced by this method. In the typical configuration the sample is moved along the propagation axis of the laser beam (z-axis) near the focal position. The transmitted signal is recorded by a detector placed in the far field behind a small aperture (closed-aperture Z-scan). A characteristic curve of collected intensity versus z-position of the sample is obtained, and the nonlinear refractive index is calculated following Sheik-Bahae formalism [1]. In the open-aperture configuration, another detector collects all the light transmitted through the sample yielding information on the nonlinear absorption coefficient.

To study third order nonlinearities of materials useful for photonic applications, where ultra-fast non-resonant electronic responses are desired, Z-scan experiments are typically carried out using femtosecond mode-locked lasers [2, 3]. Due to their usually high repetition rate, however, it has been recognized that cumulative effects, such as thermal lensing, may lead to erroneous interpretation of the origin and strength of the nonlinear response.

Thermal lensing induced by high repetition rate lasers in Z-scan experiments has been reported by various authors [4–8]. The light passing through the sample is partially absorbed and converted in heat. The temperature gradient results in a variation of the sample density and, hence, of the refractive index. Thermal heating induced by a single laser pulse persists over some characteristic time t_c. Thus, the thermal effect increases whenever the time interval between consecutive laser pulses is shorter than t_c. In this case, a stationary lens is formed when a steady state is reached between rate of heat generation and heat diffusion. If the laser repetition rate is much greater than 1/t_c, the time scale of this cumulative process is given by t_c = w²/4D, where D is the thermal diffusion coefficient (or diffusivity) of the material and w is the beam waist. In common experimental conditions, even a repetition rate as low as 1 kHz can be sufficient to generate thermal lensing.

Not only repetition rate but also pulse duration can lead to misleading results. Ganeev et al. [8] show how the γ value of CS₂ measured in Z-scan experiments grows by increasing the pulse duration as a result of the additional influence of physical effects with different time constants. Getting longer the pulse duration from 110 fs up to 75 ns, they observe the succeeding influence of electronic effects, molecular reorientational Kerr effects, and thermo-acoustic effects. It is a common assumption that in order to extract a nonlinear refractive index influenced by only electronic effects, Z-scan measurements should be made with a repetition rate of few tens of Hertz and pulses shorter than 1 ps.

In place of this approach, following a suggestion of Falconieri [9], we report a simple method to measure the nonlinear refractive index by Z-scan technique using high repetition rate lasers and managing cumulative thermal effects. Through this method, we are able to disentangle the cumulative thermal lens effect from other contributions to the nonlinear refractive index variation. We demonstrate the effectiveness of such method estimating the third order nonlinearities of CS₂ and toluene in excellent agreement with values reported by experiment performed in single-pulse configuration. Furthermore, measuring the Z-scan response of a sample of semiconductor nanocrystals embedded in SiO₂ matrix, we show that even for such materials the thermal contribution cannot be neglected.

2. Experimental

A schematic diagram of the experimental setup is shown in Fig. 1. The laser source is a mode-locked Ti:Sapphire laser (Coherent MIRA 900-F). Laser pulses are 120 fs wide, repetition rate is 76 MHz, and CW power is about 1 W. Closed-aperture and open-aperture detectors are two identical Si photodiodes with response time of 10 ns. In close similarity to the scheme reported by Falconieri and Salvetti [4], we acquire signals from the two photodiode detectors by means of a digital oscilloscope. The laser is modulated by a mechanical chopper placed in the focus of a 2× keplerian telescope. The chopper frequency is set at 20.8 Hz and its duty cycle is around 2% resulting in a sample illumination of 1 ms every 48 ms. Due to the finite size of the beam waist on the chopper wheel, the chopper opening risetime is about 18 μs. Liquid samples are held in spectroscopic-grade quartz cuvettes, with an optical path length of 1 mm, moved along the z-axis by a computer-controlled linear stage. The beam waist on the sample at the focal position is w ₀ = 16 μm. This setup allows us to simultaneously measure temporal evolution of open-aperture and closed-aperture transmittances for each position z of the sample. Oscilloscope traces are recorded and a normalized trace, obtained by standard closed-aperture to open-aperture ratio, is evaluated as a function of time and z. All the experiments reported here have been performed at wavelength of 770 nm.

 

Fig. 1. Experimental apparatus. L1, L2, L3 and L4 are lenses, Ch is a chopper, BS is a beam splitter, Pd1 and Pd2 are Si photodiodes for closed- and open-aperture Z-scan, respectively.

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3. Results and discussion

In our data, the time evolution of traces recorded by the oscilloscope provides evidence of the thermal lensing effect. In Fig. 2(a) we report two normalized traces of the transmittance curve for a CS₂ sample. Traces are measured at z values representing prefocal and postfocal positions. It can be seen that after about 150 μs there is a crossing of the experimental data, disclosing role inversion of peak and valley. In Z-scan experiments the sign of γ is given by the shape of the normalized closed-aperture curve. A prefocal peak and a postfocal valley reveal a self-defocusing Kerr effect, i.e. γ < 0, while a valley coming before a peak implies a self-focusing Kerr effect, i.e. γ > 0. The crossing in Fig. 2(a) thus clearly demonstrates that electronic and thermal nonlinear refractive indexes have opposite signs in CS₂. This is true for the majority of liquids.

In the literature, there exist various models which predict both the z-dependence and the time evolution of the normalized transmittance. Such models allow a thorough understanding of the physical processes underlying the thermo-optical effects. In particular, the model presented in [9] gives a complete description of the thermal issue in Z-scan measurement, taking the general case of multiphoton absorption and the aberrated nature of the thermal lens into account. Within such an approach, the normalized signal intensity in a closed-aperture measurement takes the form

where q is the order of the multiphoton process leading to the heating effect and ζ = z/z ₀ is the normalized coordinate position, being z ₀ the Rayleigh range of the laser beam. The parameter ϑ is called the thermal lens strength.

The basic idea behind our method is to fit time evolution of normalized trace using Eq. (1) to reduce the noise, to improve the sensibility, and, above all, to extrapolate the Z-scan curve at t = 0, namely the very curve representative of the electronic nonlinearity. This approach provides a sizable improvement with respect to the method suggested in [4] where the temporal traces acquired at some tens of μs are taken as representative of the instantaneous electronic nonlinearity.

Fig. 2(b) reports the reconstruction of Z-scan profiles at each different time and the extrapolation of the profile at t = 0 obtained by fitting the normalized traces. It is worth noting that this extrapolation is meaningful only if there are no other physical processes hidden in the blind interval of the measurement, namely processes with a characteristic time shorter than the chopper opening risetime.

 

Fig. 2. (a) Normalized traces measured at prefocal and postfocal z positions of the transmittance curve for a CS₂ sample. Open symbols are experimental data. Red and black curves are fits obtained using Eq. (1) and a single exponential, respectively. Filled symbols are extrapolated values. (b) Z-scan profiles at different times, including that at t = 0 (thick black line), as reconstructed by the fitting procedure. In the inset, the open-aperture profile (dots) and its fit (red line) are shown.

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Besides what happens in the very first part of the time evolution, care must be taken also for times much greater than the thermal characteristic time, because in this case one has to consider the finiteness of the sample under study. In fact, after a first transient where the process dynamics is ruled by the beam size, the heat diffusion becomes dependent on the sample dimensions and, hence, the time evolution of the measurement deviates from the behavior predicted by Eq. (1). Therefore, it is crucial to choose an accurate illumination period. On the other hand, the dark time (i.e. the period in which the sample is not illuminated) has to be properly selected, in order to let the sample practically recover its initial temperature between two periods of illumination. For these reasons, in our experiments we have fixed the illumination time to 1 ms, not far from the thermal characteristic times of our samples, and the dark time to 47 ms. In Fig. 3 the temperature evolution of a typical liquid sample in these conditions is presented. This result is the numerical solution of the dynamical heat equation with a gaussian source heating the sample for 1 ms out of 48 ms.

 

Fig. 3. Temperature evolution of a liquid sample calculated at the center of the heating source.

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To test the proposed technique, we have measured CS₂ which is the most frequently used reference material for third order nonlinear measurements due to its large nonlinear response. For reasons to become apparent later on, in all fits we have used q = 2. In this way, we found a value γ = 2.1∙10^-19 m²/W which is in good agreement with values previously reported for femtosecond excitation and low laser repetition rates (see Tab. 1). This result confirms that the procedure illustrated here gives the nonlinear response of the sample as if it was probed by a single-shot pulse of 120 fs. This is precisely because there are not cumulative effects other than the thermal one. The potential contribution coming from molecular reorientation is known to develop over the time scale of picoseconds (1.5 ps for the CS₂ [8]) or longer. Hence, our pulses of about 120 fs are too fast to excite significant molecular reorientation. Anyhow, our repetition rate results in 13.3 ns delay between successive pulses and, then, the molecular reorientation cannot provide a cumulative effect.

The open-aperture profile (see inset of Fig. 2(b)) is indicative of nonlinear absorption in CS₂. It is commonly recognized that two-photon absorption is the main source of absorption in CS₂ at wavelengths around 800 nm. In fact, linear absorption spectrum shows an intense band in the 290–410 nm region [10]. Using standard analysis for open-aperture experiments [11], we have measured β = 1.6∙10^-13 m/W in accordance with previously reported values obtained at 800 nm and 110 fs pulsewidth, as shown in Tab. 1. The model developed by Falconieri permits us to calculate the β value also in a different manner, using the time evolution of the closed-aperture traces. The fact that the absorption in CS₂ is a two-photon absorption process is confirmed by the peak-to-valley distance in our transmittance curves which is around 1.2 z ₀. This value is close to the theoretical one (1.13 z ₀) predicted in [9] for two photon absorption, well below the case of linear absorption (1.7 z ₀). Indeed, for multiphoton absorption the peak-to-valley separation becomes smaller as the order of absorption increases. For a two photon absorption process (q = 2), the thermal lens strength ϑ is related to β by [9]

where P is the CW laser power, d_c is the laser duty cycle, L is the sample thickness, λ is the wavelength, κ is the thermal conductivity, and dn/dT is the temperature derivative of the refractive index. ϑ(2) is calculated from the fit of the normalized transmittance and, once dn/dT and κ are known, the value of β can be deduced. With this method, we have obtained β = 1.2∙10^-15 m/W. This value is close to the one found in analogous manner by Falconieri (see Tab. 1). Our measurements, thus, underline a discrepancy of about two orders of magnitude between the β coefficient of CS₂ calculated by the open-aperture Z-scan measurement and the one deduced from the sample thermal dynamics. This difference has been already reported in the literature, but for the first time here it arises from the same set of measurements. In our opinion, β evaluated by the open-aperture profiles is overestimated because of the presence of stimulated Raman scattering processes generating radiation at angles wider than the collection angle [12]. Thus, the best estimation of β would come from the thermal-related procedure.

Table 1. Summary of the results and comparison with values taken from the literature

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In Tab. 1 we report our measurements of γ and β of CS₂ and toluene, together with the literature values. Toluene is a frequently used solvent for a large variety of materials, especially organic, and its nonlinear response has to be well-known in order to be subtracted from the response of the sample under investigation. As can be seen, our γ value for toluene is very close to that reported in the literature; as far as we know its β coefficient is here reported for the first time. The diffusivity D of the samples is reported in Tab. 1 as well. As mentioned in the introduction, the diffusivity is an important parameter related directly to the characteristic time of the heat diffusion. From the temporal behavior of the traces we have measured directly the diffusivity of our samples.

The usage of Eq. (1) to fit data can imply a quite heavy computational effort. For this reason, we have verified the possibility to use other simpler equations. It can be demonstrated that a biased single exponential can mimic the temporal trend of Eq. (1) up to the third order of expansion. Thus, at least within a time slot close to t_c, one can safely speed up data analysis using an exponential function. A comparison between Eq. (1) and a biased single exponential is shown in Fig. 2(a) confirming their equivalence in data fitting.

The method adopted here is suitable for a very large class of samples. The parameter stating the applicability of this method is the characteristic time constant t_c. It has to be larger than the time resolution of the technique which is essentially determined by the opening rise time of the laser modulator. For most liquids and optical glasses, the diffusivity ranges from 1∙10^-7 m²s^-1 to 6∙10^-7 m²s^-1. This implies that for typical beam waists used in Z-scan setups (10 μm or above) the resulting characteristic times are greater than about 40 μs. Hence, in most practical experimental conditions, this technique can be used successfully.

We have also applied this method to the study of semiconductor nanocrystals embedded in a dielectric matrix. The nonlinear optical properties of these materials are extremely interesting for possible photonics applications. Even if the thermal lensing effect in Z-scan measurements is usually neglected for these materials, we observe that this assumption should not be accepted as a general rule. Fig. 4 shows time evolution of normalized traces at prefocal and postfocal z positions for a sample of Ge nanocrystals (Ge-nc) in a film of SiO₂ and for an equivalent SiO₂ film without nanocrystals. As can be seen, the Ge-nc absorption induces a strong thermal effect. Therefore, the method proposed in this work, is essential for this kind of materials too. Detailed results on this subject will be published elsewhere.

 

Fig. 4. Time evolution of prefocal and postfocal traces for a sample of Ge nanocrystals and a bare SiO₂ layer.

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4. Conclusions

We have presented Z-scan measurements carried out with a high repetition rate laser which are representative of the actual instantaneous nonlinear optical response of the sample. The instantaneous response and the cumulative thermal effect are figured out from experimental data and separately analyzed. The effectiveness of the procedure is confirmed by the consistent results obtained on CS₂ and toluene. We show that this method provides a reliable alternative to single-shot type measurements in which a large number of averages are needed to obtain data with little experimental noise.