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Abstract
We propose a feedback-based wavefront shaping with an annular phase mask to control the spatial characteristics of femtosecond laser filamentation in K9 glass. A closed-loop feedback driven by a genetic algorithm was used to search for the optimal phase profile for generating the specified filaments. We demonstrate the flexibility of this method to extend or shorten filaments, improve continuity, and simultaneously control the position of filaments with specified lengths. Our approach offers a flexible regulation of the spatial characteristics of femtosecond laser filamentation for its potential applications.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Femtosecond laser filamentation has attracted extensive attention because of its involvement of rich physics since its first observation [1,2]. Once formed, filaments can propagate over distances considerably larger than the Rayleigh length by the dynamic balance between nonlinear self-focusing and plasma defocusing [3]. Owing to the unique properties of filamentation, it can be used in several applications, such as guiding and triggering of electric discharges [4,5], channeling microwaves [6], spectral analyses [7], tumor treatment [8], pulse compression [9], and terahertz generation [10]. To satisfy different requirements in these practical applications, it is important to accurately control and optimize the properties of femtosecond laser filamentation, such as the quantity and spatial distribution of filaments, their initial position and length, and the plasma electron density.
To date, various methods have been proposed to manipulate femtosecond laser filamentation. Adjusting the initial parameters of a laser pulse, such as pulse width, divergence angle, and power, is a common way to control filaments [11,12]. In addition, controlling the filaments by regulating the optical field of the input femtosecond laser is feasible. For instance, the distribution of filaments was demonstrated to be changed by the phase control of the optical field with a phase plate or diffraction optical elements [13]. Special beams were also used to obtain controllable filaments [14–17]. Moreover, a combination of various lenses, such as an orthogonal astigmatic cylindrical mirror [18], axicon array [19], and microlens array [20], has been proposed to regulate filaments. Recently, with continuous research on the mechanism of filament formation, the double-pulse technique has been introduced into filament manipulation [21]. The above methods, which control femtosecond laser filaments, are in the spatial dimension. There are also some methods in the temporal dimension, such as temporal chirp compensation [22] and beam interference [23]. The control of filaments is multidimensional and diversified, which lays a foundation for more potential applications.
The current megapixel spatial light modulator (SLM) provides an effective way to control the laser wavefront. Previous studies have shown the feasibility of using SLM to control the characteristics of filaments. One approach is directly generating a desirable light field by using SLM. For instance, a multi-filamentation with designable quantity of filaments have been performed by patterned optical fields [24]. Furthermore, the use of wavefront shaping, especially with an appropriate optimization algorithm, has allowed controlling the onset of filaments [25] and producing filaments at user-defined locations [26]. In addition, the control of filament position and filament shortening have been achieved through the manipulation of the spectral phase [27]. However, merely a single attribute of filaments was regulated in these previous works. Recently, the filament-induced nonlinear spectroscopy (FINS) through femtosecond laser ionization and fragmentation of molecules has been widely employed in environmental sensing [28]. For example, the FINS has been applied to the diagnostics of high-temperature combustion fields [7]. A flexible control the length and position of filaments is beneficial for improving the detection sensitivity and the longitudinal resolution in this application.
In fact, a filament can survive as it is surrounded by the background reservoir, which contains most of the energy but low power density [29]. There is a dynamic energy exchange between the filament core and background reservoir [30]. Consequently, background reservoir plays an important role in the formation and propagation of filaments. According to this idea, a phase-nested beam that constructed by the central part and annular part has been proposed to extend the length of filament in BK7 glass [31]. A ‘‘multi-focal-length’’ beam with fixed annular phase masks using an SLM has also been reported to lengthen the filamentation. [32]. In this study, we demonstrated the control of the spatial characteristics of femtosecond laser filamentation in K9 glass via feedback-based wavefront shaping with an annular phase mask by using SLM. Our scheme divides the input femtosecond laser beam into two areas based on the idea of collinear femtosecond double pulses. The central region was operated as the formation of a filament. The annular region around it plays the role of a background reservoir that was wavefront-shaped by an SLM as an auxiliary dressing beam to control the filaments. A closed feedback loop driven by a genetic algorithm (GA) was used to search for the optimal phase profile for generating the specified filaments. GA is an adaptive learning method that is inspired by the process of biological evolution. It was selected for the iterative optimization owing to its high convergence speed and favorable stability. Unlike partly controlling a few spatial characteristics of the femtosecond laser filamentation using the above methods, the flexibility of the proposed method to extend or shorten filaments, improve continuity, and simultaneously control the position of filaments with specified lengths was demonstrated. The optimum energy ratios of the annular region for each characteristics optimization were investigated. Our approach offers a flexible regulation of the spatial characteristics of femtosecond laser filamentation for its potential applications, such as remote pollutant detection and combustion diagnosis.
2. Experiment
Our experimental apparatus is shown in Fig. 1(a). An amplified Ti:sapphire femtosecond laser system (Libra-USP-HE, Coherent Inc., USA) operating with a pulse duration of 50 fs and central wavelength of 800 nm at a repetition rate of 1 kHz with a maximum power of approximately 3 W was used in our experiment. The average power of the input laser pulse was adjusted using a neutral density filter (NDF) and fixed at approximately 10 mW for all the experiments. Subsequently, the laser was introduced onto an SLM (Holoeye Pluto HED6000-L), which operates in a reflective mode with controllable liquid crystal arrays of 1920 × 1080 pixels. Each pixel was 8 µm × 8 µm, and the phase can be set in a range from 0 to 2π. Then, the modulated laser beam was focused inside a piece of K9 glass to produce the filament by a convex lens with 75-mm focal length. The size of the K9 glass was 4 mm × 22 mm × 66 mm. The critical power of the K9 glass for self-focusing is 1.8 MW similar to the BK7 [33]. The average power of the input laser pulse corresponds to 100Pcr. The power density is approximately within a range of 1010−1011 W/cm2 in our experiment, which is lower than the breakdown threshold of the K9 glass∼1012 W/cm2 [34]. A plasma channel then generated due to filamentation. The plasma fluorescence from the side of the glass was capture using a charge-coupled device (CCD). Because the plasma fluorescence was accompanied with the filamentation, the length and the position of the plasma fluorescence were used to describe the length and the position of filament indirectly. Of course, the plasma inside of the filament might be still exited beyond the plasma emitting regions when the brightness of the plasma fluorescence was lower than the sensitivity of the CCD. This method was still a simple method to character the spatial properties of the filamentation in most cases. In our experiment, the intensity, length and position of the plasma fluorescence was observed to be stable with the SLM phase held constant. To ensure the image intensity was strong enough, the integration time of CCD was set to 50 milliseconds. The SLM and CCD were connected to a computer that was governed by the GA. The image of the filament was captured as the feedback signal. A closed-loop iterative optimization feedback driven by the GA was performed to search for the optimal phase profile for generating the specified filaments.
Fig. 1. Experiment device schematic diagram. (a) Experimental setup. (NDF = neutral density filters; SLM = spatial light modulator). (b) Annular region of the phase modulation. (c) Selection of the annular region.
The GA is a type of optimization algorithm that simulates natural selection and biological evolution to obtain the optimal solution [35]. It is independent of concrete models and is suitable for the global optimization problems of complex systems with multiple variables. At the beginning of the GA, an initial population was randomly generated that consists of 32 phase masks (individuals). The working elements of the SLM were combined with 10*10 pixels to reduce the computational load in our experiment. The phase of each of the pixels was set in a range from 0 to 2π with 256 levels. Then, a figure of merit (FOM) was used to evaluate every individual in the current population. The phase masks in the population would be ranked in descending order according to FOM and the top 16 of the phase masks were selected as new parents. Every new offspring had a percentage of mutation rate expressed as R= (R0-Rend)·exp(-n/λ)+Rend, where R0 and Rend were the initial and final mutation rate, respectively; λ was the decay factor; n was the iteration number. Here, R0 was set to 0.03, Rend was 0.0025, λ was 200. The mutation rate decreased with the iteration number in order to find the optimal value faster and retained the elite individuals better. The values of FOM as a function of iteration number were displayed in a live graph plotting, which helped the user to determine if a convergence has been reached. It should be noticed that the SLM was rectangle and the laser spot was round, there were some pixels without laser illuminations. These pixels without laser illuminations were excluded from the FOM calculation to prevent divergence of the GA. In our experiment, there was approximately 5024 total modulation units in the illuminated region of the SLM.
An annular region was chosen as the phase modulation range in our experiment, as shown in Fig. 1(c). The input pulse with a radius of 3.2 mm irradiates on the SLM. The center of the annular region coincides with the center of the laser beam. A discrete rectangular coordinate system is consisted of pixels on the SLM, as shown in Fig. 1(c). Firstly, the center of the annular region serves as the origin of the rectangular coordinate system. Then, a region with an inner radius of rinner was selected for the formation of the filament, the phases of which were not changed by the GA. The annular region within the radius that greater than rinner and less than router was selected for modulation. The proportion of the energy in the annular region to the energy of the whole beam was defined as the energy ratio. The dependence between various energy ratios and the inner radius of the ring is shown in Table 1. There were approximately 2300 total modulation units in the annular region, which varied with the energy ratio of the annular region.
Table 1. Correspondence between the energy ratio of the annular region and the inner radius
3. Results and discussions
Long filaments are often required in many actual applications. We first demonstrated the extension of the filaments using our approach. The initial length of the filaments without wavefront shaping was approximately 3 mm, as shown in Fig. 2(a). The laser pulse was propagated from left to right in the image. Two rectangular regions outside of the initial filament were selected as the target region, which were surrounded by the dotted lines in Fig. 2(a). Then, we defined the image intensity of the target region as the FOM, which is used to evaluate the extension of the filaments. The various energy ratios of the annular region could be achieved by adjusting the radius of the central region, as shown in Table 1. When the energy ratio of the annular region was set to 0.5, the length was successfully increased by approximately 30% via the feedback-based wavefront shaping with the annular phase mask, as shown in Fig. 2(b).
Fig. 2. Extended filaments produced by the annular region modulation with (a) – (g) different energy ratios. The laser pulse was propagated from left to right in the image. (h) FOM measured as the function of iterations.
Because the length of the filaments might be affected by the energy ratios, we further studied the influence of different energy ratios on the extension of the filaments. Figure 2(b)–(e) show that the length of the filaments prolonged with increasing energy ratios in the range of 0.5 to 0.8. When the energy ratio was set to 0.8, the length of the filaments could reach up to 9 mm, as shown in Fig. 2(e), which indicated a threefold improvement compared with the initial filament length. However, when the energy ratio was further increased (higher than 0.8), the length of the filaments gradually decreased, as shown in Fig. 2(f)–(g). As the pulse energy was constant, the energy in the central region decreased with increasing energy of the annular region. We believe that when the energy of the central region is too weak, the length of the formed filament becomes extremely short. Therefore, the phase modulation of the annular region is difficult to lengthen the filament in this condition. The experimental results show that a suitable energy ratio of approximately 0.8 can make the best filament extension using our approach.
Furthermore, we quantitatively compared the optimization speed and optimization effect by using of the curves of the dependence of the FOM on the iteration times. Three typical energy ratios of 0.5, 0.8 and 1 were selected for comparison. The results were shown in Fig. 2(h). As depicted in Fig. 2(h), we can see that the optimization speed was relatively slow and the FOM was 3.6 × 104 at 100 iterations with the energy ratio set as 0.5. When the energy ratio was set as 0.8, the FOM increased rapidly to 5.5 × 104 at 100 iterations. And the FOM gradually reached to a stable value 6.8 × 104 at 500 iterations taking tens of minutes. When the energy ratio was set as 1, the optimization speed was the slowest and the FOM was only 1.1 × 104 at 100 iterations. Although the maximum FOM might be not less than that for the optimal energy ratio 0.8 when the whole laser-illuminated region was modulated, because a global optimal phase mask could be obtained by GA theoretically. It was impractical to use full-region modulation in this case due to its too slow optimization speed.
Long and continuous filaments are required in many practical applications. However, extended filaments always have poor continuity. Here, we further demonstrated that the continuity of extended filaments can be improved through our method. The energy ratio of the annular region was set to 0.8. A variance was used as the FOM to evaluate the continuity of the filaments, as shown in Eq. (1).
Fig. 3. Profiles of the filaments along the path in the case of (a) initial extended filaments and (b) after continuity optimization, (c) Image intensity profiles of the filaments in (a) – (b).
In order to obtain more information of filaments, the on-axis intensity distribution of filament image was shown in Fig. 3(c). The peaks and valleys represent the bright sections and dark regions separately. From the Fig. 3(c), we can see the valley intensity of initial extended filaments (black dotted line) was low but still greater than zero, which indicated that the filament still exist in the dark regions. The maximum intensity is approximately 30 times larger than the minimum intensity which confirms the poor uniformity of the filaments in Fig. 3(a). After optimization, the valley intensity was increased significantly. It means that the plasma density was well enhanced in dark regions in Fig. 3(b). Furthermore, all the peaks intensity was nearly the same after optimization. And the maximum intensity was only 6 times larger than the minimum intensity after optimization, which was only one fifth of that for initial filaments. The results showed that the continuity and uniformity of the filament was significantly improved.
Then, we demonstrated the shortening of the length of the filaments using our methods. An FOM was proposed for filament shortening, as shown in Eq. (2). $\bar{I}_{\textrm{tar}}$ is the average intensity of the target region, whereas $\bar{I}_{\textrm{out}}$ is the average intensity outside of the target region. The filament tends to have shorter length with the intensity decreasing outside the target region and increasing in the target region. In this experiment, the initial filament was approximately 3 mm, and the target region was delineated by a rectangle, as shown in Fig. 4(a). Meanwhile, to simultaneously determine the influence of the energy ratios on filament shortening, we implemented experiments under different energy ratios. The results are shown in Fig. 4(b)–(f). When the energy ratios were not less than 0.9, the filaments performed better shortening, as shown in Fig. 4(b) and (c). The length was reduced to one-third of the initial length, approximately 1 mm. However, the further decrease in the energy ratios to less than 0.8 was not ideally suited to the filament shortening, as illustrated in Fig. 4(d)–(f). We surmised that the modulation of the annular region primarily acts by decreasing the intensity of the filaments located outside of the target area. The central region has higher energy with decreasing energy ratio, which is beneficial for generating longer filaments. In this case, there was less energy and an adjustable area in the annular region, resulting in a weak shortening effect. Hence, our method provides an effective way to obtain shorter filaments without decreasing the input power.
Then, we examined the feasibility of our method to control the locations of filaments. In contrast to the previous work, we demonstrated the simultaneous control of the locations and lengths of the filaments. First, we shortened the filaments at user-defined longitudinal positions using Eq. (2). The energy ratio of the annular region was set to 0.9. Three different areas were delineated by vertical dashed lines with a length of 1 mm, which were selected as target optimization regions, as shown in Fig. 5. The distance of each movement relative to the previous position was set to 1 and 2 mm, as shown in Fig. 5(c)–(d), respectively. After optimization, the filaments were successfully shortened to the target positions, as shown in Fig. 5(b)–(d). The shortening of the filaments and the change in the positions can be achieved at the same time. Moreover, we demonstrated the simultaneous control of the positions and extension of the filaments. The intensity of the target region was used as the FOM to evaluate the extension of the filaments. The energy ratio used in the optimization process was 0.8. As shown in the results presented in Fig. 6, the locations of the extended filaments also moved successfully. The distances of each movement of the extended filaments were approximately 3 mm. The simultaneous control of the lengths and locations of the filaments shown in our experiments has some potential applications, such as combustion diagnosis.
Fig. 5. Filaments shortened to different positions. The vertical dashed lines define the target position.
Fig. 6. Filaments extended to different positions. The vertical dashed lines define the target position.
Of note, the length of the filaments is related to the intensity of the incident pulse. Thus, the length of the filaments can also be controlled by varying the intensity of the input laser. However, precise location control could not be achieved using the laser intensity adjustment method. Moreover, the initial positions of the filaments could be controlled by varying the focal length of the incident pulse, but the length of the filament is not simultaneously controllable. The above methods can nearly control a single attribute of the filaments. Our method offers an efficient way to flexibly control various characteristics of filaments with the same input laser intensity. Our results demonstrate the feasibility of controlling filaments using this method. However, the significant bottleneck of this method is the runtime of the algorithm. The working elements of the SLM used in our experiments were combined to decrease the computational time, but 500 iterations and tens of minutes were required. In actual applications, this speed of the manipulation is far from the actual requirements. Further improvements should employ high-speed SLM and a high-frame-rate camera to provide high-efficiency control of the filament.
In addition, in the traditional genetic algorithm with full area modulation, with increasing the working elements of SLM, the computational time increase and the convergence speed decrease. Whereas, the optimization effect obtained by our method which is determined by the interaction of the modulated and unmodulated regions together. Hence, the optimal solution can be obtained more efficiently in an appropriate energy ratio. Besides, we also tried to reveal the mechanism of our method by analyzing a series of phase masks of the modulated femtosecond laser pulse corresponding to the converged optimization. But we failed due to the complexity and irregularity of these phase masks. That maybe when the wavefront of the femtosecond laser pulse was spatially shaped, its temporal intensity profile was also modified. The complex spatially and temporally shaped femtosecond laser pulse finally affected the nonlinear propagation of femtosecond laser pulses and filamentation. The experimental measurements of spatial and temporal characteristics of laser fields combined with the numerical simulations using nonlinear Schrodinger equation might figure out how the GA influences the wavefront to produce the optimal results. The related researches are right going on.
4. Conclusion
We demonstrated the control of the spatial characteristics of femtosecond laser filamentation via feedback-based wavefront shaping with an annular phase mask using an SLM. The length of filaments can be extended and shortened using different energy ratios of the annular region for optimization. The results show that an energy ratio of approximately 0.8 in the annular region is beneficial to the filament extension and an energy ratio relatively higher than or equal to 0.9 is advantageous to the filament shortening. Furthermore, our proposed method is scalable and can be used to improve the continuity of extended filaments. We have also shown that the simultaneous control of the locations and lengths of the filaments is possible. Our approach offers an efficient and flexible way to control filaments under the same initial conditions and thus has potential applications in combustion diagnosis and remote pollutant detection.
Funding
National Natural Science Foundation of China (61690221, 62027822); National Key Research and Development Program of China (2019YFA0706402); Natural Science Basic Research Program of Shaanxi Province (2018JM6012); Fundamental Research Funds for the Central Universities (xzy012019039).
Disclosures
The authors declare no conflicts of interest.
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