1. Introduction

The availability of high peak-power ultrashort pulses made it possible to generate high order nonlinear effects, even in media with very low nonlinearities, such as gas. Self focusing, for example, comes into play very often and it is an important mechanism to initiate light filamentation[1]. Filamentation accompanies a highly dynamic propagation regime for the pulse and it is proved to possess big potentials for many applications ranging from pulse compression[2–4] to terahertz radiation generation[5] and LIDAR[6].

One of the earliest and most comprehensive works on self focusing was done by Marburger in 1975 [7]. He showed that, when the power of a laser pulse exceeds a certain threshold called the critical power (P_cr= 3.77λ ²/8πn ₂, λ, being the laser center wavelength and n ₂ the nonlinear index of refraction), the self focusing effects overcome diffraction, resulting in collapse of the beam on itself. In real experiments, this catastrophic collapse is avoided due to the onset of ionization. The reduction of the beam size increases the pulse intensity to a level sufficient for multi-photon absorption. As a result, a weakly ionized plasma is generated, which defocuses the beam, stopping further decrease of the beam size[1]. Formation of this plasma usually sets the beginning of filamentation.

The terminology of critical power is very often used in the context of filamentation. It is important to note, however, that in practice, the power computed from the formula mentioned above is a reference number, rather than a strict threshold for beginning of filamentation. Due to ionization, filamentation may start at a slightly lower power than P_cr. The beam profile and group velocity dispersion also have significant effects in this regards[8, 9]. It is also well known that, if the power in the laser beam is too high, the beam may break into multiple filaments. The structure of the input spatial mode, as well as the focusing geometry, strongly affects both the formation and organization of the multiple filaments[10, 11]. While multi-filamentation may become apparent from the output spatial mode for the case of several filaments, the beginning of this regime is not always visually obvious. There is not a well determined relation to evaluate at exactly what power level the multi-filamentation begins.

The dependence of single and multi filamentation on the properties of the media, as well as on the input pulse and experimental geometry makes it very difficult to analytically evaluate at what pulse energies these processes begin. For many applications, such values would be very useful. For pulse compression by filamentation, for example, multi-filamentation limits the maximum power that can be extracted. In the presence of multi-filamentation, the output spatial and temporal mode becomes complicated and unstable; hence, this regime should be avoided for proper pulse compression.

Here we propose an experimental method to evaluate different nonlinear propagation and filamentation regimes in gases. Our method is based on scanning the input power of the laser pulse and measuring the relative transmittance of it through an aperture placed well after the geometrical beam focus. By scanning the power, we can cover various propagation regimes, from linear to filamentation, then to multi-filamentation. The turning points in such transmission curves give us a direct measure of the powers the filamentation and multi-filamentation begins. We call this method “P-scan” referring to our main approach of scanning the input power. We describe this new method in detail, along with earlier related works in section 2. In section 3, we describe the analysis of the P-scan data. We present our experimental and simulation results in section 4.

2. Characterization of nonlinear propagation in gases

Before going into the details of our scheme, we briefly describe some earlier related works. Shimoji et.al.[12] directly observed the effect of self focusing by measuring the beam size of an intense laser pulse, after propagation through the nonlinear gas. With this method, they were also able to extract the n ₂ values. In order to see significant change in the beam size and make a reliable measurements, very high pressures (~40atm) and long propagation lengths (25 m) were required.

More recently, Liu and Chin[13] presented a method for direct observation of moving focus due to self focusing. They used an intensified CCD to observe the luminescence from a beam focused into the gas under investigation. When the input power of the laser is low, the position of luminescence stays constant. However, when power approaches P_cr, the nonlinear focus, hence the position of the luminescence is shown to move towards the focusing lens. Using the lens transformation formula[7] for the position of the nonlinear focus, the critical power can be estimated.

Measuring self focusing effects and n ₂ has been much easier in solids and liquids, due to relative strengths of these effects. One of the most commonly used methods to find n ₂ in solids and liquids is the Z-scan[14]. Z-scan involves scanning a thin sample of material to be characterized, through the focal region of a beam. Due to the self focusing, the sample increases or decreases the divergence of the beam, depending on the scan position. Hence, by measuring transmission through a small aperture in the far field, the n ₂ of the sample can be measured.

Unfortunately, it is not practical to apply Z-scan to measurements in a gas due to the requirement of scanning a thin sample. However, it would be straightforward to scan the input power instead. By doing so and, as done in Z-scan, by measuring the transmission through an aperture in the far field, we can cover different nonlinear propagation regimes and extract relevant information. As mentioned earlier, we call this method P-scan, due to its principle mechanism. We would like to note that, while we were inspired by the success of the Z-scan method, the P-scan is quite different in many aspects. In fact, the data in P-scan do not at all resemble that of Z-scan, and they require a completely different analysis, as we discussed below.

Figure 1 shows a schematic of P-scan. The laser beam first goes through a half wave plate and polarizer, which we use to adjust the laser power. We then focus the beam into the gas cell, which is long enough to avoid high intensities on the entrance and exit windows. After the exit cell, we use a small aperture, placed at the center of the beam, attenuate the intensity and measure the transmitted signal using a photodiode.

 

Fig. 1. The P-scan setup.

Download Full Size | PDF

While the setup of the P-Scan is relatively simple, by scanning a sufficient range of powers, we can extract a lot of information about the sample gas. If we start with low enough power to avoid nonlinear propagation, we expect a constant transmission. As the power increases gradually, self focusing first comes into play. It causes the focus of the beam to move towards the laser. As a result, compared to the linear regime, the beam size becomes slightly bigger in front the aperture, reducing the transmitted energy. Losses due to multi-photon absorption also contribute to this decrease. When the power increases further, the beam starts to collapse into a filament[1]. Due to the filament, the beam propagates over a relatively long distance with almost constant size, until the filamentation ends. In this case, the beam size at the aperture becomes relatively smaller. That is to say, formation of filament must increase the transmission. Finally, if the power exceeds several times that required for filamentation, the beam should start to break into multiple filaments. Formation of multiple filaments reduces both the portion of the energy in the beam center and consequently the transmission through the aperture.

3. The analysis of the P-Scan data

Figure 2 shows a typical experimental P-scan data obtained in air at normal pressure, showing power (normalized by P_cr defined in introduction) on the top x-axis, energy on the bottom x-axis and transmission through the aperture (transmitted average power, divided by input average power) on the y-axis. Note that the powers we use in the plots are always the input values. The instantaneous power may vary during the propagation of the pulse through the cell, due to the reshaping of the pulse profile.

 

Fig. 2 A typical experimental P-Scan data (for air). Regions I through IV indicate linear propagation, moving focus, filamentation and multi-filamentation, respectively.

Download Full Size | PDF

In the P-scan curve, we can identify four distinct propagation regimes. Going from low powers to higher, the region I indicates the linear propagation regime. For these energy levels, there are no significant nonlinear effects. Therefore, the transmission through the aperture stays constant. Yet, this region is still useful in the analysis: it provides a baseline for calibration of the absolute transmission.

In region II, the self focusing effects start to play a significant role. The nonlinear focus moves towards the laser, causing the slight drop in the transmission.

Region III is where filamentation happens. As can be seen in figure 2, there is a significant rise in the transmission, due to the enhancement of the propagation length of the beam in the form of a filament with near constant size, reducing the beam size at the aperture. In this region, higher input powers generate longer filaments and therefore increase the transmission. As a result, we can define the turning point at the beginning of region III as the beginning of filamentation. The corresponding power and energy values are very useful quantities, as they determine when filamentation begins.

Finally, the region IV of the P-scan data shows a sharp decrease in the transmission. It is not possible to concentrate powers much more than P_cr, into a single filament. For such high powers, the beam breaks up into multiple filaments[1]. When multi-filamentation occurs, the energy at the center of the original beam decreases, compared to the case of a single filament. For that reason, in the P-scan data, the significant drop of the transmission for high powers is a clear signature of the beginning of multi-filamentation.

The two principal turning points in the P-scan data enable us to evaluate the energy required to form a filament, and maximum energy that one can put into a single filament. In practice, these two quantities are of great importance since all the knowledge and interpretations of filamentation experiments are based on evaluation of the ratio of the power to its critical threshold.

4. P-Scan simulations and experiments

In the previous section, we showed a sample experimental data, which qualitatively agrees very well with what we expect. In order to test the method quantitatively, we performed simulations by numerically solving the nonlinear Schrödinger equation, using our filamentation code[15].

In our experiments, we used a Ti:Sapphire CPA laser system with one regenerative and one multi-pass amplification (Thales Alpha 100). The system can deliver pulses with up to 15 mJ energies and ~40fs widths, at 100 Hz repetition rate and 800 nm center wavelength. We measured the pulse durations and profiles using a commercial GRENOUILLE[16]. The beam size is 7.5 mm. After it goes through the half wave plate – polarizer, we use a 2 m lens and focus the beam into 1.5 m gas cell. At the output, we used a 0.5 mm diameter pinhole, placed at the center of the beam, and a photodiode behind the pinhole. For each input power, we average 500 shots at the photodiode. This averaging eliminates parasitic effects from random pointing instabilities of the input beam.

We have done the simulations in accordance with the experimental parameters. Figure 3 shows the simulated intensity profiles for two input energies (pulse width 46 fs). For 10μJ, the propagation is almost linear (with a very weak ionization at the peak intensity), and we obtain a typical Gaussian beam propagation. On the other hand, for 250μJ input energy, the pulse leads to filamentation, which is evident from the almost flat peak-intensity region, which lasts for about 10cm. With the simulation results, we also calculated the transmission through the aperture, in order to reproduce the experimental results.

 

Fig. 3. The simulated pulse intensity vs propagation distance for two different input energies in air. With the higher input energy, the filamentation is manifest in the intensity plateau.

Download Full Size | PDF

Figure 4 shows the P-scan experimental data for air -same data as in figure 2- here plotted together with the simulation results. The input pulse width is 46 fs. The simulation agrees very well with the experimental data, proving the quantitative validity of P-scan. The simulations are not adjusted for multi-filamentation; therefore, the two curves start to deviate from each other when multi-filamentation begins. In other words, the simulation results should be considered as quantitatively valid for energies, in this case, up to about 0.5mJ. There is also a small shift between the two curves, which we believe is due to the retarded Kerr effect in air. The rotational contributions to the nonlinear index of refraction cause the effective value of n ₂ to vary slightly depending on the pulse width[17, 18]. In the simulations, we do take into computation both the instantaneous and retarded contributions to the n ₂. However, the relative strengths of these two contributions vary slightly with the pulse duration, changing the effective value of n ₂.

 

Fig. 4. P-scan experiment and simulation for air. The agreement is very well until the multi-filament regime.

Download Full Size | PDF

We performed several other P-Scan experiments. Figure 5 (left) shows P-Scan data for air, with a longer input pulse (125 fs). We increased the pulse duration not by chirping, but by limiting the bandwidth in the compressor stage of the CPA system. The plot on the right shows P-Scan for argon. We also did a measurement in Xenon. In all cases, the overall behavior of the transmission follows the same pattern, but with different quantitative values. As mentioned earlier, the turning points in the P-Scan data can be used to evaluate the input energies/powers required for single and multiple filaments. In order to increase the accuracy of the values of the turning points, we made polynomial fits on the experimental data and calculated the derivatives. In the fits, the residuals were typically less than 5 percent. In table 1, we tabulate the summary of our results. At this point, we would like to point out that Haiml et.al. [19] used a similar approach to characterize semiconductor saturable absorbers. Briefly, their method was based on scanning the laser fluence on the device to be characterized, and measuring the nonlinear reflectivity. They used a model function and fitted the experimental data to it. In our case, however, it is quite difficult to build an analytical function to model filamentation and multi-filamentation.

 

Fig. 5. Left: P-Scan in air at 1 bar, input pulse width 125fs. Right: P-Scan in argon at 1 bar, input pulse width 50fs.

Download Full Size | PDF

Several conclusions can be drawn from table 1. First, by comparing the two P-scan results for air, with two different input pulse durations, we can immediately see the dramatic effect of the retarded Kerr effect. The effective value of n ₂ is larger for longer pulses[17, 20], as a result of which we observe filamentation at lower powers. Secondly, since argon is used very commonly in pulse compression applications[2, 21], we see that the maximum input energy that can be used in the compression experiments is about 1mJ, to avoid multi filamentation. This threshold is lower for higher pressures and shorter input pulses. Low pressure is necessary in order to extract more energy from the compression. Lastly, the ratio of filamentation power to multi-filamentation power differs significantly for different gases. For argon, for example, this ratio is about 5, while for Xenon, it is only about 2.

Table 1. Summary of P-scan results for different gases. All measurements are done for atmospheric pressure. The power is calculated taking into account the measured intensity profile.

View Table

5. Conclusion

In conclusion, we introduced a novel method, P-scan, for the characterization of nonlinear propagation regimes in gases. By scanning the input power and measuring the transmission through an aperture after focus, we can cover linear, moving focus, filamentation and multi-filamentation regimes. We use the turning points to determine the thresholds for filamentation and multi-filamentation regimes. These quantities are very useful for the applications of filamentation.