We present the results of Z-scan studies carried out on fused silica at 1064nm and 532nm with two different nanosecond pulse durations. Such measurements in silica and in the nanosecond regime are possible thanks to a high sensitivity setting up of the Z-scan method and in-situ characterizations of the spatio-temporal parameters of the beam. Besides, with the use of a newly adapted numerical simulation only the calibration errors of the measurement devices are significant. In these conditions, we found a higher value of the nonlinear refractive index than in the femtosecond regime and we show that these values depend on pulse duration, which indicates the contribution of nanosecond mechanisms like electrostriction.
© 2004 Optical Society of America
When a high irradiance laser beam is propagated through any transparent medium, photo-induced refractive index variations may lead to self-focusing of the beam. These photo-induced refractive index variations are commonly described by the simple relationship :
where n 0 is the linear refractive index, I is the intensity of the incident light, expressed inW/m 2 and γ is defined as the nonlinear refractive index, expressed in m 2/W. In the case of multiple mechanisms leading to a refractive index variation, the coefficient γ is a global or an effective parameter that contains all the different contributions. It can be easily demonstrated that the self-focusing process is all the more important that the total power of the beam is high . Self-focusing compensates diffraction and thus may lead to a catastrophic collapse of the beam for a total power greater than a critical power Pc, which is inversely proportional to the nonlinear refractive index γ. Depending on the total power of the beam and its spatial repartition, this self-focusing process may distort the beam by enhancing hot spots inside the beam (small-scale self-focusing) or focus the whole beam (large-scale self-focusing). Even if the power is inferior to the critical power, these two types of self-focusing lead to a local enhancement of the electric field. Therefore, this mechanism plays an important role in the laser damage process. In the nanosecond regime, the laser damage process is very complicated, because fast electronic, as well as opto-mechanical and thermo-optical processes may be involved . In order to study the impact of self-focusing on laser damage, it is crucial to have a reliable measurement of the nonlinear refractive index.
For this purpose, we developed a metrology bench based on the Z-scan method . This method, based on beam distorsion measurements, is particularly suitable for the study of the self-focusing process. Moreover, it is quite sensitive and can be used with nanosecond pulses. In that regime, it then takes into account all kind of mechanisms that may lead to a photo-induced change of the refractive index. In the case of fused silica, the nonlinear refractive index is quite small, roughly 3×10-20 m 2/W , which is about two order of magnitude below the nonlinear refractive index of most of the materials usually studied with the Z-scan method. Moreover, in the nanosecond regime, the maximum intensity that can be reached is limited by the damage fluence of the sample, which is roughly 100J/cm 2 in the case of fused silica at 1064nm . Therefore, for a measurement of the nonlinear refractive index near damage threshold, the maximum intensity available is limited to only few GW/cm 2. In these conditions, it would be necessary to have an important beam diameter to reach a high total power and then induce an important self-focusing inside the sample. However, the interpretation of the measured Z-scan curves (normalized transmittance curves) would be difficult as the transmitted power on the photodiodes may not follow linearly the instantaneous output power of the laser. For an accurate metrology purpose, the Z-scan method should be used in its linearity regime, that is to say at a power far below the critical power. Therefore, we chose to work with a small beam radius at the waist position, roughly equal to 15µm at 1/e 2 in intensity. The power is then about 10 times below the critical power, but the on-axis fluence and peak intensity are maximum (roughly 100 to 200J/cm 2 and 5 to 10GW/cm 2 at 1064nm).
Our Z-scan measurement bench  is illustrated on Fig. 1. Two different Nd:YAG Q-switched nanosecond lasers are used in order to provide two different pulse durations. A half-wave plate as well as a polarizer are used to tune finely the power of the laser beam. An assembly of a KTP crystal and a half-wave plate is available for frequency conversion in the second harmonic (532nm). Two apertures have been set up, in order to set up a spatial filtering of the beam. A trimmed Airy beam is thus set up, which has numerous advantages. First, it is quite easier a bit easier to set up than a conventional spatial filtering composed of two lenses and a pinhole, especially in the case of high fluence lasers like Q-switched lasers. Secondly, the trimmed Airy beam configuration enhances the Z-scan setup sensitivity . Two wedges (W1,W2) are used for the different reflections, particularly for the reference arm in order to avoid distorsions of the beam due to interferences. Another wedge (W3) is used for the main arm in order to take into account the angular fluctuations of the beam. This contributes to have correlated signals on the two photodiodes. The use of a reference arm is crucial here. It must, however, be as similar as possible to the main arm in order to measure the Z-scan normalized transmittance exactly as it is defined in previous works  and in order to account for each type of beam fluctuations. The correlation between the two arms is tuned finely thanks to the X-Y micro-positioning mountings of the lenses. In these conditions, the fluctuations of the transmitted power are estimated to be around ±3% (standard deviation) on each photodiode. However, these signals are perfectly correlated and the ratio of the two signals gives less than ±0.5% fluctuations with that particular type of setup. This makes possible nanosecond regime measurements in fused silica. The sample is placed on a motorized translation stage controlled by a computer and the photodiodes are connected to the computer via a digital oscilloscope. The setup is thus automatized.
A calibrated pyrometer is used to measure the total energy of the pulses after the main lens. A removable mirror can be set on the main arm, allowing repetitive characterizations of the beam spatial profile at any position. This is done thanks to a CCD camera with an objective lens. The camera has been calibrated and the resolution of the camera is 0.7µm per pixel. This allows us to perform accurate spatial characterizations of the beam. Figure 2 represents a snapshot of the beam at the waist position and the evolution of the on-axis intensity along the optical axis. As it can be seen, the shape of the beam at the waist position is close to a gaussian shape, but as soon as we look at the on-axis intensity evolution, we find out that the depth of focus is greater in the case of a trimmed Airy beam, which is at the origin of the sensitivity enhancement. The measurements are compared to a simulation of the evolution of the beam profile, using a propagation algorithm based on the Fresnel integral. It must be pointed out that the measurements are in good agreement with the simulations. The simulations have been carried out without any fitting on the spatial parameters of the setup, but by using distances and aperture radii that have been measured on the setup.
In order to study now the temporal profile of the laser pulses, we used a 25GHz InGaAs photodiode. It is connected to an oscilloscope with a 2GHz bandwidth via a high frequency wiring. This allows us to perform in-situ temporal characterizations of the instantaneous output power of the laser. As it is shown on Fig. 3, the temporal profile p(t) is complex because the laser is multimode longitudinally. As we are able to measure every temporal profiles of the pulses, this complex behavior will be taken into account in the estimation of the temporal parameters of the beam. Indeed, when the power of the beam is far from the critical power, we checked numerically that the instantaneous normalized transmittance can be decomposed into the following form, as soon as the refractive index change is typically inferior to 10-5:
where Δn(t) is the instantaneous refractive index variation on the optical axis. F(z) represents a function of z, which also depends on the beam shape and dimensions, the wavelength, the sample thickness and the linear refractive index. Actually, for sensitivity reasons, it is more convenient to measure the transmitted energy on the photodiodes. In these conditions, the time-integrated normalized transmittance is thus related to the spatio-temporal parameters of the beam by the following relationship:
where Ae is the effective area, which is evaluated from the 2D-profile as the summation of all the pixels of the image divided by the maximum pixel value and multiplied by the equivalent area of one pixel, that is to say (0.7µm)2 in the case of Fig. 2. E stands for the total energy of the beam inside the sample, and the temporal parameter τe is then defined by:
We chose this definition (with the introduction of √2π) because in the case of a gaussian profile having a half-width at 1/e equal to τ, the right-hand term of relation 4 is exactly equal to . This definition thus simplify comparisons with other works. In our case, this temporal parameter τe will be evaluated exactly as it is defined by using the temporal profile characterization setup that has been described earlier.
Besides, numerical simulations of the propagation of the trimmed Airy beam inside the nonlinear sample have been carried out. Indeed, in the case of a trimmed Airy beam combined with a sample thickness greater than the depth of focus, no analytical model is available. Hence, it is crucial to provide an accurate description of the propagation of any beam shape inside the nonlinear sample, in order to provide a reliable value of the nonlinear refractive index. We developed a general simulation tool based on the split-step fast Fourier transform method  and we adapted this method to the case of the circular symmetry by using Fast Hankel Transforms . This simulation tool allows us to take into account any beam shape (with circular symmetry), any sample thickness and any type and strength of the refractive index variation.
In these conditions, accurate measurements have been carried out on different types of fused silica (herasil and suprasil). An example of the measurements is represented on Fig. 4 with the corresponding simulations. The experimental curves are really close to the simulation curves and this, without any fit on the spatial parameters of the beam. The only parameter that is fitted to obtain the correspondence between experimental and simulation curves is the value of the nonlinear refractive index. This good correlation has been made possible because of our accurate characterization of the beam and the use of our adapted simulations that makes no approximations on the beam shape.
As the real spatio-temporal parameters of the beam are taken into account in our simulations and interpretations, the remaining absolute error on the nonlinear refractive index estimation is the summation of the calibration errors of the characterization tools, which is evaluated to be 12% at 1064nm and 16% at 532nm. Seven samples made of different fused silica types and with different thicknesses have been characterized by this method. No change greater than 5% has been noticed from one sample to the other. However, these samples have been tested with two different pulse durations (7ns and 20ns) and it has been noticed that the nonlinear refractive index decreases with the pulse duration.
As a conclusion, the nonlinear refractive index of fused silica has been evaluated to be (4.9±0.6)×10-20 m 2/W at 1064nm and (3.4±0.5)×10-20 m 2/W at 532nm for a 20ns pulse duration. Besides, for a 7ns pulse duration, (3.9±0.5)×10-20 m 2/W has been found at 1064nm and (2.6±0.4)×10-20 m 2/W at 532nm. This great dependence upon the pulse duration indicates that nanosecond mechanisms like electrostriction play an important role in the self-focusing process in the nanosecond regime and can not be neglected. Thermal effects, that could also occur when absorption is present are not significative with our samples which have an absorption coefficient inferior to 10-5 cm -1. These results are consistent with previous measurements made with shorter pulse durations (between 100 and 200 ps) and using time-resolved interferometry at 1064nm  and 355nm . In these works, the value of the nonlinear refractive index has been found equal to (2.73±0.27)×10-20 m 2/W at 1064nm and (2.5±1.2)×10-20 m 2/W at 355nm. These values give a limit of the nonlinear refractive index for very short pulses and are also in good agreement with our observation of a decrease of the nonlinear refractive index of fused silica when the wavelength is reduced.
We thus developed a very sensitive metrology setup based on the Z-scan method that can measure nonlinear refractive indexes as small as the one of fused silica, and this in the nanosecond regime and close to the damage threshold of the sample. Besides, accurate characterizations of the real spatio-temporal parameters of the beam and the use of a versatile simulation tool allow us to provide a very reliable value of the nonlinear refractive index. In these conditions, the dependence of the nonlinear refractive index of fused silica on the pulse duration have been noticed, as could be expected.
We would like to thank CEA/CESTA, Bordeaux, France, for financial support.
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