1. Introduction

In recent years, femtosecond laser filamentation has attracted considerable research interests because of its unique properties such as self-transformation to a so-called ‘white light laser’, intensity clamping, efficient power delivery without intensity decrease, and creation of a stable long plasma channel, etc [1, 2]. Many potential applications, including remote sensing [3, 4], lightening control [3], few-cycle laser science [5–7], virtual antenna [1] and remote laser induced breakdown spectroscopy (LIBS) [8, 9], have been highly expected for the filamentation. Therefore, it could be seen that the filamentation nonlinear optics will bring forth extensive impacts in wide range of fields [1, 10]. Without doubt, the intensity clamping phenomenon [11–13] is likely to be one of the most profound effects that will directly affect the light-matter interaction researches.

We have known that during the filamentation process, the intensity clamping is resulted from the subtle balance between the self- focusing of the laser beam induced by the optical Kerr effect and the defocusing effect of the self-generated plasma. It sets an up-limit for the highest light intensity that one could possibly achieve in an optical Kerr medium. In this case, any intensity sensitive light-matter interaction process would be constrained by this fundamental phenomenon. In this short communication, we report that in the course of femtosecond laser ablation of a metallic sample, intensity clamping in either ambient air or argon gas bath leads to an invariable ablation rate even when the laser pulse energy increases continuously.

2. Experiments and results

Our experimental setup is schematically shown in Fig. 1. A commercial femtosecond laser amplifier system (HP-Spitfire, Spectra-Physics Inc.) operating in single-shot mode was employed. The laser system generated 50 fs laser pulses with a single pulse energy up to 830 µJ and a central wavelength around 800 nm. The diameter of the laser beam was 8 mm at 1/e ². The laser pulses were focused by a 111 cm focal length lens in ambient air.

 

Fig. 1. Experimental setup. (SDG: synchronization delay generator.)

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During the experiment, a plasma channel with a length up to a few centimeters was formed. At the maximum laser pulse energy, we recorded the side image of the filament plasma channel using a CCD camera. As the filament in this case was about 5 cm long and the whole filament could not be imaged at one time due to the limited size of the field of view of the CCD camera, we imaged one section of the filament at a time. Then the strength of the plasma emission, which mainly consists of nitrogen fluorescence [14], was deduced from the CCD images taken at every distance. The corresponding results are presented in Fig. 2. Figure 2 indicates that the filament starts at the distance of 109 cm and ends at 114 cm.

On the other hand, it is known that in the case of external focusing, the filamentation will start at a distance of d = f·z_sf/(f+z_sf), where f denotes the focal length of the external focusing lens and z_sf represents the self-focusing distance of the parallel Gaussian beam given by the famous Margburger formula [15]: $Z_{\mathrm{sf}}=\frac{0.367k{a}^2}{{\left\{{\left[{\left(\frac{P}{P_{\mathrm{cr}}}\right)}^{\frac{1}{2}}-0.852\right]}^2-0.0219\right\}}^{\frac{1}{2}}}$ Here ka ² indicates the diffraction length with k being the wave number and a, the radius at 1/e level. P and P_cr correspond to the peak power of the laser pulse and the critical power for self-focusing, respectively. Bringing all the experimental parameters into the above equations and assuming P _cr = 8 GW [16], we found d≈1.09 m. This value is in consistent with the experimentally measured one. This implies that the self-focusing and filamentation dynamics played dominant role in our experiments.

 

Fig. 2. The normalized fluorescence signal intensity of the filament along the propagation axis taken by CCD camera.

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A piece of 14 µm thick aluminum foil was chosen as the ablation sample and inserted at the vicinity of the geometrical focus. To monitor exactly when the aluminum foil is pierced through by the laser pulses, a photodiode (Newport 818-SL) was used to detect the light power transmitted through the aluminum foil. The number of shots required to penetrate the foil was counted automatically by a PC based program which also sent trigger signals to both the laser amplification system and an oscilloscope. Note that the photodiode signal was monitored by the oscilloscope and analyzed by the PC. Once a clear signal (amplitude is higher than three times the standard deviation of the background noise) was received by the photodiode, a through hole was considered to be drilled on the aluminum foil. Then the trigger signal was discontinued and the laser amplifier system stopped firing laser pulses. Afterwards, the number of pulses required to drill through a hole was obtained. The interaction spot was changed to a new position after each datum.

Figure 3 plots the minimum number of pulses needed to drill a through hole on the aluminum sample as a function of the pulse energy in air. When the pulse energy equals to 150 µJ, this number is as high as 200 shots. With the pulse energy increasing, it decreases sharply to 60 shots when the pulse energy reaches 360 µJ (indicated by the blue arrow in Fig. 3). After this characteristic point, the number of pulses remains almost constant up to 830 µJ, the maximum pulse energy available in our experiment. A red line is indicated in Fig. 3 to help the visualization of this trend. In particular, we can find that the minimum number of pulses required for piercing through the Al sample slightly increase by 10 % for the pulse energy range of 360–830 µJ.

 

Fig. 3. In air, number of laser pulses required to penetrate the aluminum foil as a function of pulse energy.

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Since when the PC program stopped the amplifier firing pulses in our experiment, only a tiny through hole is drilled by the central part of the laser beam, the curve shown in Fig. 3 is in fact inverse proportional to the laser ablation rate of the laser beam center. Since the laser ablation rate is directly governed by the laser peak intensity, the saturation behavior of the ablation rate measured in our experiment manifests the influence of the intensity clamping on the ablation process, just like those about the ionization rate of gases reported by Becker et al.[12]. It is well known that intensity clamping is an intrinsic concept of the filamentation phenomenon. In an optically transparent medium, filamentation takes place only when the peak power of a laser pulse is higher than the critical power for self-focusing given by P_cr = 3.77λ ²/(8π n ₀ n ₂) [15], where λ is the central wavelength of the laser beam, n ₀ denotes the linear refractive index, and n ₂ is related to the intensity dependent nonlinear refractive index defined by n = n ₀+n ₂ I. Based on the analysis of the drilling numbers under different energies, we infer that the 360 µJ characteristic pulse energy indicated in Fig. 3 corresponds to the self-focusing critical power. With the experimental data of pulse energy equal to 360 µJ and a pulse width of 50 fs, we obtain a peak power of 7.2 GW. This result is consistent with the one determined by the moving focus method (8 GW) [16].

It is necessary to mention that the weak increase of the data appeared after 360 µJ in Fig. 3 could be explained by the refined pulse length shortening dynamic during the filamentation process [17]. In other words, the experimental results shown in Fig. 3 confirm that when filamentation occurs for ultrashort laser pulses, the intensity claming phenomenon sets an up-limit for the laser ablation rate that a single laser pulse could possibly achieve.

At the same time, the results shown by Fig. 3 could leads to another potential application - determining the value of n ₂ of gas samples. We all know that light intensity dependent refractive index is a fundamental parameter in nonlinear optics. It results in numerous important nonlinear phenomena. Naturally, the measurement of the nonlinear refractive index coefficient n ₂ is a crucial issue for studying the nonlinear optics. In optically transparent condensed matters, the n ₂ measurement is often made by using the Z-scan method [18]. When implementing this technique, a finite aperture is fixed at some distance behind the geometrical focus of a focused Gaussian beam. Then, a thin block of the sample will be scanned along the propagation path (Z). At the same time, the light transmittance after the aperture is measured. By fitting the measured transmittance curve with the proper theoretical model, the sign and magnitude of the nonlinear refractive index could be deduced.

However, for gas samples, the values of n ₂ are typically 3 orders of magnitudes lower than those in condensed matters. Therefore, long gas cells are required to achieve accurate measurements in Z-scan technique. This makes Z-scan non-practical for gases. Another approach to measure n ₂ of gas sample is based on the self-phase modulation induced spectral broadening [19]. But, in order to retrieve the value of n ₂, considerable theoretical work is still needed to accurately analyze the experimental data. To accomplish the same task a simpler way is suggested in Ref. [16]. This method uses a rather expensive instrument –intensified CCD camera to directly record the focus movement when the laser power exceeds the critical power for self-focusing. More recently, Akturk et al. have demonstrated a so-called P-scan technique to classify distinct propagation regimes of the femtosecond laser pulse [20]. Our method is somewhat similar to P-scan method, but more relies on the intensity clamping phenomenon which is the most profound characteristic of the filamentation process.

 

Fig. 4. In argon gas, number of laser pulses required to penetrate the aluminum foil as a function of pulse energy.

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In order to further strengthen our discussions, we have repeated the experiment in argon gas. The experimental setup was essentially identical to that described above. In this case, the aluminum foil was placed inside a gas cell filled with argon gas at one atmospheric pressure. The gas cell was 1 meter in length and the foil was fixed at the middle of the cell, which was also about the location of the geometrical focus. Furthermore, the gas cell was installed on a translation stage allowing us to change ablation areas during the experiments. Like before, the minimum number of pulses needed to drill a through hole on the aluminum sample in argon gas is presented in Fig. 4 as a function of pulse energy. It can be seen that the general properties of the data shown in Fig. 4 are similar to those of Fig. 3. The laser ablation rate undergoes an initial relatively fast decrease and then arrives at a more constant value after the turning point of 190 µJ. In this case, a critical power of 3.8 GW could be deduced, which is again in good agreement with the result given in Ref. [20]. With λ = 800 nm, n ₀ = 1, we further get that n ₂ = 2.6×10^-19 cm²/W for argon gas at atmospheric pressure.

3. Summary

The laser ablation rate of an aluminum foil at various laser energies has been investigated in both air and argon gas bath. The experimental results demonstrate that once the laser peak power is higher than the critical power for self-focusing, i.e. when laser filamentation occurs, the ablation rate is nearly constant. This observation can be explained by the effect of intensity clamping. Since intensity clamping is a universal phenomenon for intense femtosecond laser pulse, our results should be helpful in bringing some greater attention to this effect for the expanding community of ultrashort laser pulse ablation. On the other hand, our experimental technique also provides a potentially simple way to determine the critical power for self-focusing, and thus the value of nonlinear refractive index n ₂. However, to make our technique practically validate, further investigation on the influence of the sample materials and the detailed laser ablation dynamics to the experimental results is under the way.