Multi-focus manipulation system based on separable natural evolution strategy aberration self-calibration

Wavefront shaping using digital micromirror devices (DMDs) allows inertia-free focus manipulation with numerous modulation modes and high refresh rates. However, the aberration caused by the curvature of DMDs affects the focusing performance. Here, we propose an aberration self-calibration method based on separable natural evolution strategies. This method searches optimal Zernike coefficients of aberration globally and completes compensation using super-pixel encoding. Compared to the genetic algorithm method, we improve the speed by 62% and achieve better-focused spots. The method enables simultaneous scanning of 25 independent focal spots. This advancement supports wavefront-shaping applications in optical imaging, industrial inspection, and laser processing.


I
nertia-free focus manipulation can scan multiple independent light beams along customized trajectories with high precision, efficiency, and flexibility, and is valuable in imaging and industrial processing. 1) The inertia-free focus manipulation methods include acousto-optic methods, electro-optic methods, 2) and wavefront shaping.Among them, wavefront shaping stands out for its ability to create high-resolution optical foci and flexibly change the direction of scanning beams using spatial light modulators. 3,4)][7] However, the quality of scanning focus is inevitably affected by imperfections in the system, such as the curvature of the DMD, inherent aberrations of optical elements, and mechanical misalignments. 8)Therefore, accurately and efficiently compensating for these aberrations is crucial for the performance of wavefront shaping in focus manipulation.
The traditional aberration correction methods are mainly based on interferometry. 9)An additional reference beam interferes with the working beam and interference patterns are obtained to quantify and eliminate aberrations of the working beam wavefront. 10,11)However, extra reference devices significantly increase the complexity and decrease the stability of the system.Thus, these interferometry methods are not suitable for aberration correction of integrated scanning systems. 12)By contrast, optimization methods based on iterative algorithms, such as the genetic algorithm (GA), 13,14) particle swarm optimization, and simulated annealing, are coming to the fore.][17] These methods successfully search and compensate for complex aberrations such as distortion caused by strong scattering 18) and achieve high-quality controllable multi-point focusing. 19)However, these state-of-the-art algorithms have limited global search capability and are sensitive to the initial condition and the local minima.In addition, the search is known to be time-consuming.Therefore, it is desirable to develop a new optimization method with a high speed and strong global search capabilities to improve the performance of aberration correction in integrated scanning systems.
Here, we propose an efficient and highly accurate online self-calibration method based on separable natural evolution strategies (SNES). 20,21)Using the central light intensity of the focus as feedback, SNES searches for the Zernike coefficients of the system aberrations globally. 22,23)In the process of iteration, we use super-pixel encoding 24) of DMDs to modulate the compensated wavefront for aberrations, and make the scanning focus gradually approach the ideal Airy spot.This method avoids the cumbersome interferometric setup.The gradient information-based SNES ensures global search capability.We construct an integrated high-speed aberration self-calibration beam scanning system.We demonstrate the advantages of our method by experimentally producing high-quality focus and multi-focus scanning patterns.The method is expected to contribute to the advancement of optical imaging, industrial inspection, and laser processing.
As shown in Fig. 1, SNES searches for the Zernike coefficients of system aberrations and continuously updates the compensation wavefront.The super-pixel encoding is employed to modulate this compensatory wavefront using a DMD.Finally, the self-calibration of aberrations is completed and high-quality multi-focus scanning can be achieved.
In super-pixel encoding, multiple adjacent micromirrors on the DMD are combined to form a super-pixel.The light emitted from this super-pixel, following the optical path shown in Fig. 1, can form a complex amplitude field E mod on the focal plane of lens L2 through an off-axis 4 f system with a low-pass filtering pinhole.By encoding the "ON" and "OFF" combinations of micromirrors within this super-pixel, modulation of the amplitude and phase of E mod can be achieved.If there are no aberrations, to generate p scanning foci behind the focusing lens L3, E mod is described as follows: where a p and b p represent the components along the x-axis and y-axis of the plane wave corresponding to the pth focus, respectively.In fact, aberrations j ext are inevitable, and the light field becomes In order to achieve high-quality scanning, SNES is used to compensate for system aberrations.In the iterative process, two Gaussian parameters m and s are updated to realize the search for actual aberrations.Specifically, during the jth iteration, N Gaussian random samples s n following ( ) m s N , j j 2 are generated.The size of each sample is Ḱ 1, where K represents the order of Zernike coefficient, so that one sample corresponds to one searched aberration j , with Z k being the terms in the Zernike polynomial.The conjugates j n ext , * of these researched aberrations are then transformed into binary patterns of DMD using the superpixel encoding.DMD displays these patterns sequentially, and N focused images can be captured by the camera.The central intensity f n of focus is served as the feedback, and the samples s n are reordered as s m according to the increasing order of f .n Subsequently, the gradients  m J and  s J are calculated using the formulas in Fig. 1, and then m + j 1 and s + j 1 are updated.In the calculation of gradients, incremental weight factors u m are set, giving higher weights to samples with higher f n to ensure the search is in the direction where feedback values increase.Through repeated iterations, m gradually approaches a global optimum, and s converges to a small value.Ultimately, m and s with the highest feedback values are the global optimal solution, and the corresponding sample s is the optimal Zernike coefficient that describes the system aberrations.This sophisticated approach highlights the intricate balance of precision and efficiency in optimizing optical systems for complex tasks like high-resolution scanning.
Finally, the optimal system aberration j ext is obtained according to the best Zernike coefficients searched by SNES.By modulating the compensation of this aberration by the DMD using super-pixel encoding, the scanning focus can approach to the ideal Airy spot, thereby achieving high-quality multi-beam scanning without aberrations.
To verify the performance of the proposed method, we first carried out a simulation.As shown in Fig. 2, a random aberration was added to the scanning plane wave.We set the same parameters of SNES and GA, including a population size of 30 and iteration steps of 200.Both methods searched only the first nine Zernike coefficients.The aberrations searched by GA and SNES are shown in Figs.2(b) and 2(c), respectively.The aberration obtained by SNES was more similar to the true aberration, while GA fell into local optimum.The comparison of the iteration curves [Fig.2(d)] shows that SNES significantly outperforms GA within the same number of iterations.The simulation demonstrated that SNES had better global search capabilities and enhanced the precision of aberration correction.Additionally, the time taken for the search process was evaluated: SNES took 702.58 s, while GA took 1125.97s, indicating that SNES also had the advantage in terms of speed.
Our high-speed light manipulation system based on SNES aberration self-calibration is shown in Fig. 3.The entire system consists of a scanning spot modulation module, an 032001-2 © 2024 The Author(s).Published on behalf of The Japan Society of Applied Physics by IOP Publishing Ltd image acquisition module, and a main control module.In the scanning spot modulation module, a light beam from a 532 nm laser was introduced into the optical path by a single-mode fiber with a numerical aperture of 0.12.After passing through a collimating lens (L1) with a focal length of 150 mm, a collimated parallel light beam is formed.The mirrors M1 and M2 adjusted the incident angle of light onto the DMD (DLP9500, JUOPT Technology Co., Ltd) to 24°.SNES was used to search for the Zernike coefficients and the compensation of aberrations.The DMD displayed the superpixel encoding pattern of aberration compensation.The firstorder diffracted light emitted from the DMD was filtered by a 4 f system composed of lenses L2 and L3 and a low-pass filtering pinhole.To increase the system's integration level, two mirrors were introduced between L2 and the filtering pinhole to fold the optical path.Subsequently, lens L4 converged the modulated beam, and the camera (aCA1920-155 nm, Basler) captured the scanning light spot.In the image acquisition module, an electric translation stage was used to adjust the camera to the focal plane.
We experimentally tested the scanning focal spot before and after compensating for aberrations.Figure 4(a) shows the focal spot before aberration correction, with diverged energy and irregular shape.Figure 4(b) shows the optimized spot using SNES, close to an ideal Airy spot.To quantify the improvement in scanning performance before and after aberration correction, we plotted the light intensity distribution in the central region of spots as shown in Fig. 4(c).The Then, we tested the lateral scanning range of our system.We modulated the inclination factor of the scanning light field and moved the focal spot outward along a spiral line with increasing radius.Figure 5(a) shows the intensity distribution of scanning focal spots.It was observed that as the radius increased, the intensity of spots decreased, and when the scanning radius exceeded a certain range the intensity rapidly diminished.To quantify this range, we calculated the critical position where the peak intensity of the spot fell to half.From the experimental curve in Fig. 5(b), we determined that the maximum lateral scanning radius of our system was 2.95 mm.Furthermore, we tested some complex patterns with multiple focal spot scanning to verify the performance of our aberration correction method.We manipulated 25 scanning focal spots simultaneously to draw a pattern containing 10,475 spots, as shown in Figs.6(a) and 6(b), and a pattern containing 22,400 spots, as shown in Figs.6(c) and 6(d).It could be seen that the scanning patterns became much clearer after aberration correction using our method.The results convincingly proved that our aberration selfcalibration scanning system based on SNES effectively corrected aberrations and substantially enhanced the scanning quality of complex patterns.
In conclusion, we proposed an aberration self-calibration method based on SNES.Super-pixel encoding of DMDs was used to modulate complex amplitude information of scanning light fields and create multiple independent focal spots.SNES was used to search for the optimal Zernike coefficients of system aberrations, which were compensated in the DMD's modulation field to correct the aberrations of the scanning spots.Our method's advantage is that it can achieve aberration self-calibration in small systems by avoiding the introduction of a reference arm, and ensures the quality of the light spot as well as the speed of aberration correction.Compared with the GA method, our method avoided local optima and converged to the global true value of aberrations, and SNES was 62%  032001-4 © 2024 The Author(s).Published on behalf of The Japan Society of Applied Physics by IOP Publishing Ltd faster than GA.The experimental results demonstrate that our method can optimize multiple focal spot quality in integrated systems for high-speed, high-quality scanning.
The resolution of focal spots can be improved by changing lenses with higher numerical aperture.Since the super-pixel encoding method creates focal spots using first-order diffracted light, the energy efficiency is severely limited. 25,26)hus, it is necessary to improve the efficiency of the scanning system for practical applications.In addition, our SNESbased aberration self-calibration method will also have significant application value in fields such as deep tissue imaging in biology and phototherapy.032001-5 © 2024 The Author(s).Published on behalf of The Japan Society of Applied Physics by IOP Publishing Ltd

Fig. 1 .
Fig. 1.Principle of the aberration self-calibration method based on SNES.

Fig. 4 .
Fig. 4. Experimental results.(a) The focal spot without aberration calibration.(b) The focal spot after aberration calibration.(c) The intensity distribution of the focal point.Scale bar, 50 μm.

Fig. 5 .
Fig. 5. Test of the lateral scanning range.(a) Scan along a spiral line.Scale bar, 1 mm.(b) Plot of the intensity of focus versus the radial distance.