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Abstract
A new technique is suggested for the generation of laser beams with an intensity profile specified. The technique is based on the coherent combining of radiation of a fiber laser array with adaptive control of the power and phase of Gaussian subbeams with plane wavefronts. The power and phase of the subbeams are determined for each intensity profile specified in the far field based on the inverse problem solution, for example, by the Gershberg–Saxton method. To form a required phase profile, the stochastic parallel gradient descent (SPGD) method is used along with the inversion of a required phase distribution with a phase corrector. The main advantages of the technique are the adaptive control of the intensity profile and a possibility of generating high-power laser beams. The results of numerical and field experiments are described.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The problem of beam shaping is urgent in the laser processing of materials, where the radiation intensity profile of lasers used (Gaussian in most cases) does not ensure the proper execution of necessary technological operations [1–4]. A uniform or asymmetric intensity profile is often required, which cannot be formed without special techniques and tools. In addition, a possibility of rapid beam shaping can significantly expand the range of operations without retuning. Leading manufacturers (Coherent, Trumpf, NLight, SPI Lasers, IPG, and Amada) of process lasers [5] show an active interest in a possibility of dynamical beam shaping [6], which emphasizes the relevance of the problem.
Most of the known techniques for intensity profiling involve intracavity or extracavity beam conversion with the use of static optical elements [7–10]. The disadvantages of these devices include impossibility of rapid intensity profiling and of elimination of dynamic aberrations and phase noise in the optics system and distortions along a propagation path. These disadvantages can be removed by using dynamic devices, such as deformable mirrors [11] or liquid-crystal spatial light modulators (LCSLM) [12], in combination with adaptive optics methods [6,13], which allow quick wavefront restructuring according to changing conditions. However, the use of such devices in active channel when positioning them directly between a laser and a target assumes limitations to the optical power of laser beams because of limited damage threshold of these elements.
Phase-controlled laser arrays suggest broad prospects in problems of radiation intensity profiling in a plane specified. These arrays are 2D structures of lasers arranged in a certain order; they generate beams with a Gaussian intensity profile, which propagate parallel to each other. The presence of a phase control unit in each channel makes it possible to control the phase distribution in a beam, which is formed as a result of overlapping and interference of individual subbeams in the far optical field or in the focal plane of a focusing lens. Power amplifiers with controllable gain factor mounted in each channel allow controlling the amplitude distribution in a beam synthesized. The idea of using phased laser arrays was first associated with a possibility of increasing the irradiance by coherent combining of laser beams [14–19]. However, the capability of controlling the phase and amplitude of subbeams, inherent in coherent beam-combining schemes, stimulates interest in the synthesis of beams with different intensity profiles on this basis. A possibility of forming different intensity profiles based on a piston-phase laser array was theoretically and experimentally studied in a number of works [20–22]. For this, some previously calculated ratios between the subbeam amplitude and phase were suggested to use.
The situation is complicated by the fact that the subbeam phase randomly varies with time due to acoustic and temperature fluctuations in the fiber channels and is needed to be stabilized [23]. However, several techniques have been developed for subbeam phase synchronization in problems of coherent laser beam combining, when the phases of all subbeams vary with time but are kept equal [24,25]. In [26], simple relations between the amplitudes and phases of central and peripheral beams of a hexagonal array of seven subbeams were derived for the generation of a dark hollow beam. In that work, the stabilization of phases of individual subbeams relative to a reference phase-modulated beam was used to maintain those relations. A low energy efficiency of the technique suggested for the generation of a dark hollow beam (∼ 8%) should be noted (according to the relations between the amplitudes), as well as the difficulty in using that technique for the phase stabilization in multi-aperture laser systems.
An approach to generation of combined laser beams with complex spatiotemporal characteristics was suggested in [27]. The so-called exotic beams with periodic, quasi-periodic, and random spatiotemporal modulations were considered. Many photodetectors were suggested to be used for the control of spatiotemporal beam characteristics. They record the result of interference of a central beam in the hexagonal subaperture array with each peripheral beam separately. In that case, the peripheral sections (tails) of Gaussian beams which participate in the beam synthesis were used for organization of a feedback channel. An advantage of the approach suggested is the absence of any additional optical elements at the system exit used for removing a part of the radiation and forming a feedback loop, which is to allow scaling the output optical power of a beam synthesized. However, the use of this approach to generation of beams with stable intensity profiles is a difficult task.
The controllability of subbeam phases allows generating beams with special properties. Thus, the conditions for generation of optical vortex beams with an orbital angular momentum (OAM) via coherent superposition of a radially symmetric array of phase-stabilized laser beams with a certain phase distribution over the array were theoretically considered in [20]. The idea is that N phase-stabilized beams equispaced on a circle with the phase distribution φj=πm(2j-1)/N (j = 1, 2…, N is the number of a beam in the array) form a scalar vortex beam with the topological charge m in the focal plane of a focusing lens as a result of interference in the far optical field [28]. The advantage of this scheme is that it allows scaling the vortex beam power. From the viewpoint of intensity distribution, in the far optical field, vortex beams correspond to a dark hollow beam. Therefore, the above phase relation between subbeams arranged on a circle can also be used to form such distributions.
Similar beams were first experimentally generated [29,30] in an array of six coherent Gaussian beams spaced on a circle by setting a fixed phase shift between neighboring subbeams after ensuring the phase synchronization of all beams, which remains “frozen” for a certain time after a synchronization controller is turned off. The details of the experiment are described in [31]. However, the problem of long-term support of the phase stability, characteristic of such beams, was not solved.
In [32], a technique is suggested for adaptive stabilization of a vortex beam with a preset OAM sign on the basis of allocation of a “price” function (metric) in the out-of-focus area of the system, in contrast to specifying a metric in the form of power within a limited aperture (PIB metric) used in problems of coherent beam combining [33]. This approach is based on the high-speed analysis of the images of a beam synthesized or signals of a photodetector array, which requires additional engineering and computational resources. An array of phase-modulated fiber lasers with a LCSLM was used to generate a vortex beam with m = 1 in recent work [34,35].
However, all these works are devoted to the generation of vortex beams. A possibility of forming and controlling an arbitrary specified intensity profile in the far optical field for beams synthesized using a fiber laser array has not yet been considered.
2. New technique for beam intensity profiling in a phase-modulated fiber laser array
Let consider a problem of generating radiation with a preset intensity profile using a fiber laser array (Fig. 1) of Nsub Gaussian emitters (subbeams). An algorithm for the problem solution is as follows [36]. Let radiation amplitudes and phases Ak0 and φk0 be formed in the initial plane z = 0 in the channels centered at points (xk, yk). Then the field for each subbeam is representable as
where k is the number of subbeams (k = 1, 2, …, Nsub). The field of the kth subbeam at the distance L can be written as the complex amplitude:
Fig. 1. Experimental setup: narrow-band laser (1); fiber optic amplifier (2); fiber splitter 1×8 (3); phase modulators (4); fiber collimators (5); long-focus lens (6); beam splitter (7); collimator 5× (8); beam profiler (9); computer (10); LCSLM (11); pinhole (12); broad-band photodetector (13); optimizing multichannel SPGD processor (14) [30]; control computer (15); power amplifiers (16); power amplifier controller (17).
The intensity distribution at the distance L is defined by the superposition of subbeam fields:
Many works are devoted to solution of the inverse problem of retrieving the initial beam amplitude and phase (Ak0 and φk0 in our case) from an intensity profile specified over the distance L. They are mainly aimed at calculation of the corresponding diffractive optical elements (DOEs). Most algorithms are based on the iterative error minimization procedure, which has been first suggested in [37] and is known as the Gershberg–Saxton algorithm. In some works, attempts have been made to improve the Gershberg–Saxton algorithm to increase the convergence rate and reduce the intensity profile retrieval error when solving one of the most difficult problems of converting a Gaussian beam into a beam with a flat-top intensity distribution, as well as of generating gradient-phase beams, line-shaped beams, and vortex beams [10,38–43]. Other common algorithms are the stimulated annealing algorithm suggested in [44] and developed in [45,46] and the stochastic parallel gradient descent (SPGD) algorithm [47,48]. Thus, using these approaches, we can determine Ak0 and φk0 values, which allow required beam intensity profiling along the distance L.
Figure 1 shows a scheme for the formation of a required radiation intensity profile in the far zone (distance L corresponds to the position of beam profiler 9). Radiation with the initial amplitude Ak0 and phase φk0 (in the plane of lens 6) is generated as follows. The required intensities in the fiber channels are first set; they should be proportional to the squared Ak0. This is achieved by introducing power amplifiers 16 into each radiation channel. The power amplifiers are controlled by corresponding controller 17. To achieve the needed values of the phase in the channels, the phase corrector (LCSLM-11 in the diagram in Fig. 1) should be used. The phase corrector is installed in the low-intensity feedback channel in the region of the converging synthesized beam, where there is still no overlapping of subbeams. In this case, in each channel, the phase value φk0 in the plane of long-focus lens (6), the phase shift Δφk at corrector (11), and the phase value φkf at photodetector (13) are related by the obvious relation
in which some extra term for the phase, which is identical for all the channels, is omitted. Whence it follows that if the phasing condition is fulfilled for all the channels in the plane of photodetector (13), that is ${\varphi _{kf}} = 0$, then the following condition is true: It should be noted that the phasing of the channels in our experiment was accomplished based on the SPGD approach, which is a standard approach for problems of this kind. Thus, the inclusion of LC SLM into the feedback loop allows the phase shifts Δφk to be transferred from SLM to the fiber array ${\varphi _k}(z = 0) = {\varphi _{k0}}$.Thus, this technique allows one to control the subbeam amplitudes and phases thus forming the desired intensity profile. The technique provides for high stability in setting the radiation amplitudes and phases at the fiber array, which ensures the stability of the intensity profile in the focusing plane.
3. Potential capabilities of the technique
Let us now consider the numerical simulation results which show the potential capabilities of the technique suggested in the problem of laser beam intensity profiling. It is assumed in the theoretical calculations that the beam phasing is ideal. As is well-known, phase fluctuations are always present in fiber optic channels. The SPGD method was used in our experiments to compensate for these fluctuations. However, despite the high efficiency of this method, it, like other methods, does not allow achieving complete compensation of fluctuations. Therefore, the beam phasing condition is always fulfilled with some error, that is ${\varphi _{kf}} \approx 0$. Correspondingly, the condition ${\varphi _{k0}} \approx{-} \Delta {\varphi _k}$ is also fulfilled with the same error. The analysis of the experimental data shows that this error does not exceed the level of λ/8 [49,50]. This level of error makes it possible using the approximation of ideal beam phasing in theoretical calculations to estimate the potential of the proposed method. The calculations were carried out for the case of the densest, that is, hexagonally-packed elements in a laser array for different number Nsub of subbeams, from 7 to 127, and for different distances to the radiation focusing plane. The larger the number of subbeams, the more degrees of freedom of the control and the more complex beam can be reproduced. However, an increase in the number of subbeams complicates the experimental (and production) implementation of the system. Hence, it will always be necessary to search for a compromise between two conflicting requirements: shape complexity and technological simplicity.
Figure 2 shows the distributions of the fiber array radiation phase and intensity at the distance z/zd = 0.04 (zd = ka2/2, a = asubNc is the radius of beam synthesized, asub is the radius of subbeam with Gaussian amplitude Ak0(x,y) = exp[-(x2+y2]/a02] a0/asub=0.89 [18], Nc is the number of subbeams at the central row, k is the wave number) calculated to get the radiation intensity uniformly distributed within a rectangle (Fig. 2(c)). The inverse problem was solved on the basis of the Gershberg–Saxton algorithm on the assumption of Gaussian intensity profile within each subbeam (Fig. 2(a)) and flat and parallel wavefronts (Fig. 2(b)). One can see that seven subbeams (Nsub = 7) are insufficient for producing a rectangular intensity distribution by the technique suggested. However, it is possible to select the initial phase and intensity distributions so as to provide a rectangular beam in the radiation focusing plane (in the far field) already at Nsub = 19.
Fig. 2. Initial fiber array (a) phase and (b) intensity distributions and (c) intensity distribution in the focal plane corresponding to this field (the required beam shape in the focusing plane is shown in the top right corner). The distance z/zd = 0.04. The size of the rectangle at the focal plane is equal to 4 × 8 Airy disk radiuses.
Our calculations proved the diffraction effects on the deviation of the resulted beam intensity profile from the desired one. Figure 3 shows the resulted radiation intensity profile in the focusing plane when trying to generate a rectangular beam at different distances from the laser array. The effect of the laser array discreteness is negative (interference fringes appear) with a decrease in the focusing distance as compared to its optimal value. As the distance increases, the intensity distribution becomes diffuse due to diffraction spreading.
Fig. 3. The radiation intensity profile in the focal plane at different focusing distance z/zd, Nsub=19.
These results demonstrate that the quality of formation of the given beam shape depends essentially on the size ratio of the given intensity distribution and the subbeam diffraction size. The value of this size ratio can be controlled with a projection optical system with variable magnification. Figure 4 exemplifies complex-shape beams formed by 37 beams (Nsub = 37). The approach suggested evidently allows controlling the radiation intensity profile in a wide range at a sufficient number of the laser array components.
Fig. 4. Initial fiber array (a) intensity and (b) phase distributions and (c) intensity distribution in the focal plane corresponding to this field. The preset intensity distribution is given in the upper right corner.
4. Beam shaping in a phase-controlled fiber laser array: experiment
We believe that the main advantage of our approach is that it does not require a dynamic process to form beams with the preset intensity distribution. This stage of the process is accomplished with the methods of computational mathematics. As a result, we obtain the calculated values of initial phases and intensities for every channel. The necessary phase values in the channels are provided by setting phase LCSLM, while the intensity values are set by adjusting the signal amplification in the channels. The phase control is the most principal and newest point in the implementation of this method. The manuscript presents the results of experimental implementation of the method for two given intensity distributions. The results illustrate that when we used the calculated values of beam amplitude and phase, the resulted intensity profiles were close to those predicted in the numerical calculations. We treat the intensity control as additional advantages that allow extending the capabilities of this approach and that have found their solution in papers of other authors, for example, in [12,20–22].
Here, we present the results of experiments on generating beams of a shape specified by a phase-controlled fiber laser array. The experiments were carried out at a setup basic diagram of which is shown in Fig. 1. The feature of the scheme in Fig. 1 is the absence of power amplifiers in the fiber channels. A beam is synthesized by seven fiber channels.
Individual components of the experimental setup are shown in Fig. 5. A PLUTO-2-NIR-015 (HOLOEYE) phase LCSLM with the active region of 15.36 × 8.64 mm in size was used as dynamic phase corrector 11. The LCSLM was mounted so as subbeams falling on its active area did not overlap and were located at the center of the active area (Fig. 5(c)). Another advantage of the scheme with LCSLM is that it allows correction of static aberrations which are inevitably present in a real optical system. To eliminate aberrations of our optical system, we used a possibility of phase profiling with preset aberration parameters on the LCSLM display. For the control, we used the intensity profile formed in the coherent beam combining. The shape of a beam synthesized is controlled by the beam profiler.
Fig. 5. (a and b) Setup components (designations are like in Fig. 1); (c) subbeams on the LCSLM display
Figure 6(a) shows one of the realizations of the random intensity distribution under combining of beams with fluctuating nonsynchronized phases. The initial phase relief of the LCSLM was set in the form of a flat mirror. When phasing subbeams by SPGD method, the shape a beam synthesized corresponded to the coherent combining of equiphase beams (Fig. 6(b)). The effect of static aberration of the optical system was manifested in the asymmetric resulted intensity distribution and aberration peaks near the central peak of the distribution. To determine the phase distribution of LCSLM which compensates the aberrations caused by the optical elements, the SPGD method can also be used—to control the coefficients in the expansion in Zernike polynomials, which form the LCSLM surface. In our case, we limited ourselves to selection of a coefficient for the fourth-order Zernike polynomial Z40, which allowed us to increase the peak intensity of the central maximum by approximately 30% and form a more symmetric intensity distribution in the far optical field with minimal distortion (Fig. 6(d)). The phase screen Δφa introduced to compensate the aberrations of the optical system is then used as a background compensation map for applying the additional phase relief Δφk to produce the beam shape specified.
Fig. 6. Optical aberration compensation procedure: (a) random intensity distribution under beam combining without phase locking; (b) intensity distribution under coherent beam combining in the presence of aberration; (c) LCSLM phase profile for compensation of spherical aberration caused by plane-parallel beam splitter 7 (see Fig. 1); (d) intensity distribution under coherent beam combining with aberration compensation.
Two experiments were carried out to show the capability of the technique for intensity profiling. The aim of the first experiment was to demonstrate the potential of the technique even in the case of a small number of fiber channels. We set the task to form a stable intensity profile in the form of three peaks located at the vertices of an equilateral triangle by controlling only the phase of seven beams in the far optical field. Figure 7 shows the results of the experiment.
Fig. 7. Results of the experiment on the intensity profiling in the form of three peaks: (a) intensity distribution in the plane z = 0; (b) phase distribution in the plane z = 0: phase change from 0 to 2π is shown on the grayscale 2.4 (1), 0 (2), 0 (3), 2.5 (4), 2.4 (5), 2.4 (6), and 0 (7); (c) calculated intensity distribution in the focal plane; (d) LCSLM phase screen according to the calculation; (e) LCSLM phase screen with aberration pre-compensation; (f) intensity distribution formed.
To carry out the experiment in accordance with the calculations performed, the LCSLM phase screen was divided into sections so as the radiation of only one subbeam fell on each section (Fig. 7(b)). Thus, each subbeam received additional phase incursion according to the calculations. The resulted intensity distribution (Fig. 7(f)) is evidently in a good agreement with the distribution obtained in the numerical calculations (Fig. 7(c)). The second experiment was aimed at generation of a dark hollow beam. Six subbeams located around the circumference were used. The experimental results are shown in Fig. 8.
Fig. 8. Vortex beam generation: (a) phase screen with vortex phase distribution; (b) phase screen with pre-compensated aberration; (c) intensity distribution in the vortex beam synthesized (2D image); (d) intensity distribution in the vortex beam synthesized (3D image).
The easiest solution to this problem is to set a phase relief corresponding to the vortex beam with m = 1 (Fig. 8(a)), and this relief is superimposed on the phase screen which compensates aberration (Fig. 6(c)). The resulted LCSLM phase relief is shown in Fig. 8(b). An algorithm allowed generation of a vortex beam with dark hollow intensity distribution (2D and 3D images in Figs. 8(c) and 8(d)) in the focal plane of lens 6 on beam profiler 9 (see Fig. 1).
The experimental results show the effectiveness of the technique suggested for the intensity profiling in coherent beam combining. In the experiments, a dark hollow beam as well as three-peak intensity profile were generated. When we used the calculated values of beam amplitude and phase, the resulted intensity profiles were close to those predicted in the numerical calculations.
5. Conclusions
Thus, a new technique for generation of laser beams with a specified radiation intensity spatial profile is suggested. It is based on the coherent combining of radiation from a fiber laser array with adaptive control of the power and phase of the Gaussian subbeams with plane wavefronts. To form the required phase profile, the SPGD method is used in combination with inversion of a required phase distribution using a phase corrector. The main advantage of the technique is the adaptive control of parameters of an array emitter, which allows rapid beam shaping. In addition, the corrector is located in the feedback loop of the phased fiber laser array in our case, but not in the field of a high-power beam; the array uses only a small part of the radiation to stabilize the phase relations. Therefore, the approach suggested can be used to control high power radiation.
Numerical simulation allowed us to show the potential capabilities of the technique suggested in the problem of beam shaping. The calculations were performed for the case of hexagonally-packed components of the laser array, for the number of subbeams from 7 to 127. We have shown that the technique suggested allows controlling the radiation intensity profile in wide range at a sufficient number of matrix components. The radiation intensity profiles in the form of dark hollow (vortex) beam and three peaks at vertices of an equilateral triangle were produced in an experiment with seven-component hexagonal laser array.
The use of LCSLM in the experimental scheme provides another advantage, that is, it allows one to perform the preliminary correction of a static aberration in the system, and only then proceed to the beam generation. The effectiveness of the correction of static optical aberration of a phase-controlled laser array with LCSLM is shown for attainment of the maximal irradiance in the far optical field in the case of coherent beam combining.
Funding
Russian Science Foundation (18-19-00437); Russian Foundation for Basic Research (18-29-20115\18); Ministry of Science and Higher Education of the Russian Federation (budget funds for IAO SB RAS).
Disclosures
The authors declare no conflicts of interest.
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