We demonstrate the ability to position single and multiple filaments arbitrarily within the energy reservoir of a high power femtosecond laser pulse. A deformable mirror controlled by a genetic algorithm finds the optimal phase profile for producing filaments at user-defined locations within the energy reservoir to within a quarter of the nominal filament size, on average. This proof-of-principle experiment demonstrates a potential technique for fast control of the configuration of the filaments.
© 2016 Optical Society of America
A filamenting pulse maintains a high intensity over many Rayleigh lengths, and can even be made to persist after free propagation in air on the order of a kilometer . Above the critical power for self-focusing, the Kerr effect causes collapse of a pulse having an initial spatial intensity profile that is greater in its center than at the edges. This is balanced by diffraction and defocusing of the pulse in the plasma generated behind the pulse front , resulting in a long plasma channel and relatively constant pulse intensity. At a wavelength of 800 nm in air, the critical power for self-focusing, Pcr, ranges between 3-10 GW . When the pulse power is near but greater than Pcr a single filament forms, but at orders of magnitude greater than Pcr, multiple filaments appear. In either case, the energy needed to maintain the plasma generation within the hot core of the pulse where the filaments are localized comes from a surrounding reservoir of non-filamenting laser light [4–6].
Many envisioned applications of filamentation require manipulating filaments in air at some great distance from the laser that produces the pulses. Examples include white light LIDAR , guiding high voltage discharges , remote sensing , and remote generation of terahertz radiation . These depend on controlling the location, extent, and configuration of the plasma channels generated with the laser pulse. By configuration, we mean the spatial distribution of individual filaments with respect to the others in a multi-filamenting pulse in the plane transverse to propagation. Successful use of filamentation in technology strongly depends upon our understanding of and ability to control the evolution of the filamentation process.
In this paper, we demonstrate control over multiple filaments in a laser pulse at powers of 50 − 100Pcr by shaping the wavefront with a deformable mirror. The mirror figure is governed by an adaptive feedback loop – the desired configuration of filaments is sought by an algorithm that modulates the mirror and then learns from the history of mirror iterations to reach convergence with user-defined filament locations based on rapidly processing images of the exit mode at the termination of the filaments. The advantage of this approach is that there is no need to guess at the mirror shape that will accurately place the filaments in both transverse dimensions within the energy reservoir. Using this approach, we are able to position one filament at almost any location within the energy reservoir with high shot-to-shot repeatability, in addition to creating arbitrary configurations of multiple filaments. Given enough time, the algorithm can usually converge to the desired filament configuration. Further, the use of reflecting instead of transmitting optics to configure the filaments avoids energy losses, in addition to undesirable effects such as the rapid accumulation of B-integral, dispersion, and self-phase modulation of the laser light.
Adaptive optics should allow for fast manipulation of the number and position of the filaments if the corresponding mirror figures are stored. Other efforts to control multiple filamentation have succeeded in creating stationary configurations of multiple filaments. The ability to tune the onset and extent of single and multiple filaments has also been demonstrated. Beam ellipticity  and imposed astigmatism  can provide one dimensional control over the filament configuration. A circularly polarized input beam has been shown to create a pair of co-propagating filaments with high spatial stability . Amplitude masks have been employed to create two-dimensional filament arrays [14, 15], as have phase plates , although these approaches incur significant loss of pulse energy. Some researchers have indeed used deformable mirrors to fix the onset of filamentation [17, 18], and also to impose transverse phase profiles that seed multiple filaments the beam [19, 20] albeit without the ability to precisely choose the locations within the energy reservoir where the filaments appear.
Since filamentation is sensitive to so many aspects of a laser pulse in addition to atmospheric fluctuations, some researchers investigating remote applications of filamentation have turned to evolutionary algorithms similar to that discussed here in order to optimize the signal derived from their experiments. These include maximizing target fluorescence at a distance for remote sensing applications , maximizing the supercontinuum emissions from the filaments , and constraining their onset and termination . The work presented here focuses particularly upon showing the feasibility of using such an algorithm to configure the filaments within the energy reservoir.
2. Experimental methods
2.1. Genetic algorithm
In experiments where the outcome is sensitive to a very large parameter space, it can be useful to use advanced optimization algorithms to search for a solution. Of these, evolutionary algorithms allow us to locate and refine a result by starting with a random subset of the parameter space, and construct a solution by iteratively mixing attributes of that initial subset. Our experiment relies on a genetic algorithm (GA), which is a class of evolutionary algorithm inspired by the process of natural selection where the fitness of each candidate solution can be expressed as a single number called a figure of merit (FOM). Each candidate solution, or child, consists of its attributes, or genes. Here each mirror figure is a child, and the position of each of the mirror actuators constitutes a gene. GAs are based upon biological sexual reproduction and execute inheritance, crossover, and mutation of genes to create new children in the search process .
To evolve the set of candidate solutions, the algorithm calculates an FOM for each child. A greater value indicates greater fitness. Our experiment is initialized with a population of 100 random mirror figures, and each subsequent iteration tests 100 different figures. The actuator positions of the figures that give the ten biggest FOMs are combined (that is, crossed over) to make the next 100 mirror figures. Evolution of the solution set is accomplished by the inheritance of the optimal actuator positions between iterations. However, if the algorithm happens to be initialized with no figures that improve the FOM, or if the inherited actuator positions between iterations decrease the FOM, failure of the algorithm may be avoided by allowing mutation. Each of the actuator positions in each of the 100 children has a user-defined probability of being reset to a random value. If a mutation increases the FOM enough it is retained in the next iteration, otherwise it is discarded. Determining the optimal mutation probability is not an exact process - too much mutation dominates inheritance and keeps the mirror figure from converging on a solution, but too little can lock the feedback loop to a highly suboptimal solution. The results presented in Section 3 use a mutation probability of 20%. This determines both the likelihood of a mutation occurring, and the amplitude of the maximum possible change in position that can be applied to the mutated actuator. The effect of the mutation probability on the solution is described in the Appendix.
The FOM used to determine the fitness of each child from the exit mode image is given by25,26], Eq. 1 encourages filaments to occur simultaneously at all coordinate pairs (xk,yk). If the pixel values at or near (xk,yk) are low, then the whole FOM is multiplied by a small number, even if there are filaments at the other desired coordinates. The image moment formulation – multiplying several times by the inverse of the distance in the image between each pixel and (xk,yk) – tends to draw regions of high pixel values toward (xk,yk). Since the filaments are typically the brightest features in the exit mode images, their locations in the energy reservoir are expected to settle on each (xk,yk) given enough iterations of the GA. Clearly, if N is greater than the number of filaments that can be produced with the available pulse energy, the algorithm will fail.
2.2. Imaging the filament exit mode
The λ3 laser  in the Center for Ultrafast Optical Science at the University of Michigan was used in the experiments, and delivers 800 nm pulses of up to 15 mJ with a full-width-at-half-maximum pulse duration of about 35 fs with a repetition rate of 500 Hz. For all tests, the pulse energy is maintained within 12.5–15 mJ. If Pcr ~ 5 GW, then each laser pulse achieves approximately 86Pcr. If we use a value of about 10Pcr per filament needed to form a multi-filamenting pulse  as a heuristic, there should be less than 10 filaments visible in the exit mode.
Figure 1 shows the experimental setup. The beam exits the laser after reflecting off of the deformable mirror (DM37PMNS4 by AOA Xinetics, Northrop Grumman) which has 37 independent actuators spaced 7 mm apart in a square grid. The aperture of the mirror is 47 mm, and each actuator has a maximum stroke of 5λ (4 μm at λ = 800 nm) with resolution better than λ/100. The actuator voltages are determined by the GA, which is a subroutine in a LabVIEW program that takes camera settings and user-defined filament locations as inputs, interfaces with the mirror to set the actuator voltages, and calculates the FOM corresponding to each tested mirror figure. The beam fills the deformable mirror and is subsequently transported about 7 m in air, where it is focused through a f/20 lens, for a total of 8 m of propagation between the deformable mirror and the filamentation region.
The image of the exit mode from the filament region is formed with minimal astigmatism by reflecting the beam off of a wedge placed far beyond the end of the filament, and then a concave f/7.5 mirror, which nearly retro-reflects the attenuated beam back off the wedge and on to a turning mirror. The light remaining after both wedge reflections is neutrally filtered, and then bandpass filtered at 800 ± 5 nm before illuminating a CCD camera (DMK21BU04 by Imaging Source). Even with self-phase modulation of the laser light, the filament spectrum peaks at 800 nm, so filaments appear as bright spots in the images. The CCD sensor has a resolution of 640 by 480 with 5.6 μm pixels and 8-bit depth. The camera is triggered by the laser, and integrates for a single shot, recording once every 33 shots.
Figure 1 also shows an example of the air plasma left in the wake of the filaments viewed in the sagittal plane. For this particular test case, a few filaments appear to converge and fuse near the geometrical focus of the f/20 lens. This image is included because it is representative of the appearance of the filaments when the deformable mirror is not flat. The aberrations introduced by the deformable mirror lengthen the filaments relative to those observed when the mirror is flat, and create the branching filament structure. The focusing lens constrains the pulse’s energy deposition in space and forces the filaments to come together.
The subsequent sections discuss three related experimental results. The first involves using the GA to create an apparent intensification of a single filament by setting N = 1 and (xk,yk) to the center of the pulse in Eq. 1. The second demonstrates that the GA can be used to reposition a single filament arbitrarily within the energy reservoir reliably and with reasonable shot-to-shot repetition. These are discussed in Section 3.1. Test cases where N ≠ 1 are discussed in Section 3.2.
3.1. Single filament control
The intensity clamping phenomenon inherent to filamentation  prevents greater intensity in a filament to be achieved simply by tighter focusing of the pulse. Figure 2 compares the exit mode observed when the mirror is flat with one observed after the GA has optimized the exit mode for a single filament (N = 1) with the target filament location set to the initial filament location, that is, the center of the pulse.
Break-up of the beam caused by multiple filamentation is responsible for the poor beam quality shown in Figs. 2(a) and 2(b). The bottom inset in Fig. 2(a), whose intensity values do not correspond to the top insets and are therefore normalized, shows that the beam quality of the laser is quite good in the absence of filamentation. The structure of the energy reservoir for a filamenting pulse evolves randomly as it propagates. In comparing Figs. 2(a) and 2(b), it seems that not only is the brightness of the filament is increased, but also that of the energy reservoir. The lineouts in each of the top insets include the maxima in both images, and indicate a nearly three-fold increase in the apparent brightness for the optimized case. With respect to the energy reservoir, we can only speculate that the deformable mirror may shape the wavefront so as to redistribute energy in the tails of the pulse toward the center. Confirmation of such an effect will need further study.
The increased brightness in the single observed filament is likely due to overlapping of multiple filaments at the pulse center such that the overlapping filaments are more tightly arranged than when the mirror is flat. Another effect possibly contributing to the increase in brightness observed in Fig. 2(b) is that the filament may be closer to the optimal object plane of the concave mirror imaging the exit mode. The image in Fig. 2(a) was recorded with the imaging plane manually optimized such that the pulse diameter seen on the camera was minimized. We have observed that running the GA continuously while moving the imaging plane causes the downrange end of the filament to move in lockstep with the camera in order to keep the end of the filament in focus. While the layout of the imaging optics is unchanged between Figs. 2(a) and 2(b), some part of the brightness increase in Fig. 2(b) may be due to the GA performing a minor adjustment to the length of the filament in order to optimize its longitudinal termination at the object plane. However, this does not explain necessarily the increased brightness of the reservoir.
The bottom inset in Fig. 2(b) shows the typical evolution of the FOM as the GA iterations advance. In most cases, the envelope traced by the 10 best children in each iteration exhibits a logarithmic rise, making it clear when the algorithm has converged, and it is acceptable to stop its function.
In order to demonstrate that our approach permits arbitrary positioning of a filament within the energy reservoir, we ran successive cases in which the target filament location was moved in increments of about 50 μm along each of the four cardinal directions (θ = 0,π/2,π,3π/2) defined in the sense of a unit circle. Figure 3 shows one case in which the filament is positioned along θ = 3π/2. It is clear that the location of the filament moves within the energy reservoir, and that the trivial solution to Eq. 1 with N = 1–steering the whole beam so that its center coincides with (xk,yk) – is not observed.
In order to distinguish filaments in the exit mode, the raw image files starting with the cases such as that in Fig. 3(a) are thresholded so that the filament optimized in the center of the reservoir appears on a dark background. The characteristic size and brightness of this center filament is used to determine which features in all other images, once thresholded, are filaments. The typical observed diameter for the filaments in our experiments is about 40 μm. The pulse energy is not attenuated to the single filament regime in these tests, so it is likely that at least in some of the test cases, multiple filaments have been made to overlap. Figure 3(e) shows a filament which has an apparent size approximately double that of the other cases, which based on repetitive images of this test case indicate possibly two filaments in close proximity at the target filament location.
Overlapping of filaments when the target location is off-center is rare. What is much more likely to happen is that a single filament is optimally imaged at the target location so that it appears very bright in the exit mode. If the filament is within about its characteristic size of the user-specified (xk,yk) then it is considered to be on target. Additional filaments commonly appear elsewhere in the reservoir. The phase profile imposed by the mirror tends to create these satellite filaments which are usually not quite as bright as the filament being controlled. They appear at other places within the reservoir which are clearly distinct from the target location. Many of the satellite filaments repeat shot-to-shot with moderate probability along with the controlled filament. This implies that they are unintended artifacts of the mirror shape, as opposed to the result of random fluctuations in the pulse profile.
Figure 4 quantifies the shot-to-shot reliability of the GA-optimized mirror figures, including the likelihood of creating satellite filaments along with the controlled filament. Twenty exit mode images are recorded for each target filament location with the deformable mirror fixed in its optimized shape. The position of the controlled filament is fairly stationary, however Fig. 4 demonstrates a few cases in which the GA fails (twice in Fig. 4(b) and twice in Fig. 4(e)) near the periphery of the energy reservoir. Averaging over each of the twenty four test cases shown in Figs. 4(a), 4(b), 4(d), and 4(e), the average difference between the target locations (xk,yk) and the mean location of the on-target filaments is less than two pixels (with an uncertainty of one pixel) or about a quarter of the nominal filament size. During experiments, the beam location tended to drift downward as viewed on the camera. This is a result of the heating of the air by the filament plasma. Because of the high pulse repetition rate (500 Hz) the air in the filamentation region cannot dissipate the heat the plasma generates between shots, creating a density depression that is subject to hydrodynamic forces from the surrounding air . In Fig. 4 the target filament location is adjusted to compensate for steering of the beam over time but maintain the 50 μm interval between targets, which is reflected in the downward drift of the diamond target location markers as the distance from the beam center increases.
As the target location is moved further away from the center of the reservoir, the shot-to-shot scatter of the filaments tends to increase, as Fig. 4(c) shows. Further, in Fig. 4(f) the aggregated data from Figs. 4(a), 4(b), 4(d), and 4(e) indicate the probability that the brightest (i.e. most in focus) filament is on target tends to decrease with distance. The likelihood that the mirror figure seeds one or more satellite filaments increases, as does the probability that no filaments are visible in the exit mode. Taken as a whole, the statistics of the test case samples lead us to conclude that the efficacy of the GA and the deformable mirror in controlling the filament is diminished with distance from the center of the energy reservoir. Two factors contributing to the decrease in the quality of the optimization may be: i) simply that there is less available laser energy around the periphery of the reservoir to ensure that the filament propagates into the optimal object plane, and ii) that the phase profile is subject to the action of fewer actuators at the edge of the mirror, so that the changes to the wavefront there are limited.
3.2. Multiple filament control
The GA can also manipulate the deformable mirror to place multiple filaments at different target locations. In Eq. 1, this corresponds to N > 1 and multiple target coordinate pairs (xk,yk). Figure 5 shows examples of two test cases where multiple filaments are configured within the energy reservoir. Unlike the test cases in Section 3.1, the FOM does not converge nicely. The stop condition for the GA when controlling multiple filaments is to observe the exit mode corresponding to the best child from an iteration, and stop after the exit mode appears to not change significantly for about ten iterations. The shot-to-shot variations taken together become more severe with the control of more filaments.
Table 1 quantifies the repeatability of the filament configurations shown in Fig. 5. The probability that one, two, three, or four, P(1 − 4), filaments are visible and at the target location are given in percentages. Samples where only satellite filaments are observed are accounted for in calculating P(1 − 4). The average scatter of the filaments from shot-to-shot are calculated for each filament f1−4, and are approximately a factor of 2 larger than that observed for single filament control. For the 2 filament case, the sample size is n = 61, while for the 4 filament case, it is n = 56. The 2 filament case is successful 46% of the time, while all filaments in the 4 filament case are on target in only about one of ten shots. However, the four filaments use up the available energy in the pulse, so there are few satellite filaments.
Filaments on the corners of a square, as presented in Fig. 5(b), can also be produced by other means of multiple filament control, such as a periodic mesh . However since the pulse is partially blocked by the mesh this method significantly diminishes the pulse energy that is delivered at a distance.
There are many competing factors that influence the ability of the GA and deformable mirror to control multiple filaments. Some have to do with the parameters of the mirror, such as actuator size and spacing. Others, such as interference among filaments  are physical features of multi-filamentation itself. Our setup was not deliberately optimized to control multiple filaments at a distance. It is likely that improvement in the reliability of the control of multiple filaments could be accomplished by obtaining detailed understanding of how the wavefront as altered by the mirror changes as the pulse propagates, and modifying the layout of the experiment accordingly.
By reshaping the wavefront to manipulate the energy reservoir of a filamenting laser pulse, it is possible to precisely control the locations where filaments appear. The approach we have described demonstrates the feasibility of controlling a single filament within the energy reservoir, and can be extended to the simultaneous, independent control of multiple filaments, albeit with diminished reliability. The phase profiles which give the target filament configurations can be found quickly and accurately by exploiting a genetic algorithm to optimize the wavefront.
Absent from our measurements are phase profiles recorded in close proximity to the filamentation region with, for example, a Shack-Hartmann sensor. These measurements are part of our ongoing work, and would give definitive information about how the GA manipulates the deformable mirror to reorganize the energy reservoir of the pulses at a distance.
Our results demonstrate the principle of controlling filaments in this manner, but are far from indicating a mature approach for application. Possible improvements may be sought by using a different form for the FOM – perhaps one that actively discourages the formation of satellite filaments. More rapid, higher quality mirror optimizations might be achieved by a different, more natural basis set for the genes that the GA operates upon. For example, instead of each actuator position being its own completely independent gene, the set of genes may instead be coefficients of several superimposed Zernike polynomials. Also, it should be noted that the deformable mirror used in these experiments was relatively coarse, having only 37 actuators. Significantly improved control of the filaments could likely be achieved with a larger mirror having more actuators.
This experiment has several potential uses going forward. It provides opportunities to carefully characterize interactions between multiple filaments, and the effects of general beam aberrations on the evolution of the filaments. The effect of the precise configuration on established standoff applications of filamentation may be explored. With further development, such as the ability to learn from stored mirror figures for many target configurations, it is a potential technique for fast, even real-time control of filament position and structure in applications.
It is unclear from our experiments whether or not inputting the same number and target locations for the filaments into the GA cause it to always converge to the same (or even to a similar) mirror figure. Figure 6 shows results from two sets of three trials for two different input values of the mutation probability applied to the actuators. In all the trials, the target filament location is set to the center of the energy reservoir. Yet each set of trials produces mirror figures that have arguably no common features, as shown by comparing each of Figs. 6(d)–6(f) to each other, and also to each of Figs. 6(j)–6(l). The dominating effect leading to this observation is the noisy nature of the experiment. The fluctuations in the pulses and the atmosphere ensure that optimized mirror figures will always exhibit differences, even if all else remains unaltered.
In Figs. 6(a)–6(f) the mutation amplitude and probability is set to 2%, while it is set to 20% in Figs. 6(g)–6(l). It is clear that the latter gives better results, even though the positions of the center actuators in Figs. 6(j)–6(l) are not obviously much different from those actuators closer to the edge. Whereas in Fig. 6(e), the center actuator is significantly displaced from the others, and yet there is no filament on target in Fig. 6(b). When the mutation probability is small, it is likely that the small subset of the solution space to which the GA is initialized does not contain any significant maxima. Increasing the mutation probability allows the GA to access a larger portion of the solution space without having to test every mirror figure within it. However, too much mutation precludes convergence of the GA. Advanced versions of GAs such as ours can dynamically adjust the mutation probability based on the FOM history over the iterations.
A.C.E. is grateful to Jennifer Elle for helpful discussions. Funding for this research was provided by the Air Force Office of Scientific Research.
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