The Multiparameter Measurement Technique in a Large-Aperture Rectangular Laser Beam Aberration Correction System

Adaptive optics (AO) can effectively improve the beam quality of solid-state slab lasers. However, the aperture of the output beam increases as the output power of the laser increases, resulting in a larger measurement system. Ultimately, a more complex AO system needs to be designed. To meet the requirements of conjugate imaging in an AO system, it is of research significance to coordinate and optimize the system's structural dimensional parameters while enabling the detection of multiple parameters, such as the wave-front and beam quality. In this paper, a multiparameter measurement technique in a large-aperture rectangular laser beam aberration-correction system is proposed. The system layout is optimized with total dimensions of 400 mm × 150 mm × 246 mm (L × W × H). The AO system conforms to the requirements for conjugate detection and can perform wave-front detection, far-field evaluation, and near-field detection of a 160 mm × 120 mm rectangular beam produced from a solid-state slab laser. The findings reveal that the measured PV values of wave-fronts of the measurement system are less than 0.288 μm, the RMS is no more than 0.079 μm, the average far-field beam quality factor is 1.248 times the diffraction limit, and the average near-field beam uniformity is 0.533 at a temperature of 20 °C ± 10 °C; these results satisfy the technical parameters.

the power and improving the beam quality. Solid-state slab lasers have continually demonstrated their superiority with the rapid development of the gain medium [3], [4], cooling technology [5], [6], and resonant cavity optimization [7]. The output power has increased from the kilowatt scale to the megawatt scale; however, the beam quality declines as power increases, owing to thermal effects and other factors, resulting in a limited brightness increase. Benefiting from the increasingly mature adaptive optics (AO) technology, solid-state slab lasers have the potential to combine high power and high beam quality, and corresponding research results have been achieved. For example, in 2014, Yang et al. from the Institute of Optics and Electronics, Chinese Academy of Sciences, enhanced the beam quality of a solid-state slab laser with an output power of 1.3 kW from 13.1 to 2.3 times the diffraction limit using an AO system without wave-front detection [8]. In 2017, the Institute of Optics and Electronics at the Chinese Academy of Sciences reduced the output beam's PV values of wave-front from 57.26 to 1.87 μm with a low-order aberration autocorrection approach under limitations [9]. In 2018, Yang et al. determined the beam quality factor for a 750-MW solid-state slab laser to be 1.64 times the diffraction limit using an AO system combined with the related low-order aberration-correction approach [10]. In 2021, Wang et al. employed a completely closed-loop AO-controlled off-axis multirange amplification system to strengthen the beam quality of an 1178-J, 527-nm laser to almost the diffraction limit [11].
The gain medium size grows as the power of the solid-state slab laser increases, leading to a steady increase in the output beam aperture and increased difficulty in designing the AO system. The difficulty of designing the detection unit in an AO system stems from the multiparameter detection and evaluation of the rectangular large-aperture laser beam, which entails detecting wave-front aberrations and assessing the beam quality and uniformity while meeting the constraints of conjugate imaging, small volume, and environmental temperature change. Furthermore, the high energy of the laser being measured affects the environmental temperature, thereby introducing additional thermogenic [12] aberration into the detection unit.
Essential research has been conducted to address the abovementioned issues. To meet the requirement for large-aperture wave-front detection, Wang et al. [11] established a Keplerian system using two lenses to compress a 260 mm × 260 mm square aperture laser beam. Nonetheless, the total focal lengths of this This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ Keplerian system reached 2.05 m. Despite the employment of a reflector to fold the optical path, the detection unit was still huge and could only measure a single parameter. Zhang [13] achieved a single-parameter measurement of the far-field beam quality of a high-power fiber laser using large measurement equipment. The beam quality factor was measured to be 1.41 at 90.1 W and 1.81 at 3.04 kW. In Li's study on high-power laser systems with high-quality near-field beams, a 10× telescope system was adopted as the detection system to compress the beam to 6 mm to satisfy the target surface of the near-field camera. Nevertheless, the telescope system has a tube length up to 1.3 m and a single parameter measurement system [14]. Xiang et al. [15] adopted a nonfocal Keplerian telescope system with a compressional magnification of 11× to perform wave-front detection with a 150 mm × 150 mm square beam aperture, and the tube length reached 2.42 m, which was also excessive. Sylvain et al. [16] compressed a beam with an aperture of approximately 75 mm using a Keplerian telescope system in the detection system to meet the effective target surface of the wave-front sensor in a wave-front correction study of ultra-high intensity laser beams for the 200-TW laser system. However, the tube length was too long, nearly 1.04 m.
The current technical method is superior for detecting laser beam parameters, such as wave-front distortion, near-field (uniformity), and far-field (beam quality). However, it is difficult to realize miniaturization and simultaneous detection of various parameters, and the effect of temperature on the measurement system has not been considered. To address the existing problems, first, a scheme using large magnification beam compression followed by splitting detection was selected based on the technical requirements, i.e., a large detection aperture, conjugate imaging, and multiparameter detection, as well as the imaging principles of the Keplerian system. Second, simulation models were built using the technical indexes of the measurement system on the common-aperture telescope, far-field detection subunit, and near-field detection subunit to analyze the imaging quality and tolerance. The simulation results lay a foundation to establish the experimental platform. Third, the mechanical system was designed under the optical system, and the adaptability of the environment was analyzed. Finally, relevant experiments were performed to confirm the design outcomes. Fig. 1 illustrates the schematic diagram of the measurement system, which consists of a common-aperture telescope, wavefront detection subunit, far-field detection subunit, and near-field detection subunit. The common-aperture telescope primarily transforms the detection problem of a large aperture into that of a small aperture. Then, it cooperates with the wave-front detection subunit, far-field detection subunit, and near-field detection subunit to detect the wave-front distortion, far-field, and near-field of the corrected laser beam, respectively.

B. Technical Indicators and Analysis
Tables I-III list the optical indexes of the multiparameter measurement system, whereas Table IV shows the mechanical indexes.
The transmission peak-to-valley (PV) value must be less than or equal to λ w /3, where λ w denotes the wavelength of the laser, with the development of beam quality and load capacity requirements for high-power laser devices [17]. To ensure a correct result, the wave-front detected by the wave-front detection subunit and that corrected by the deformable mirror should have the same physical value, which can be achieved by conjugating the detection surface of the wave-front detection subunit and the deformable mirror. It can be understood that the deformable mirror is at the diaphragm position, and images are at the exit pupil position of the telescope system through the common-aperture telescope system. Therefore, the wave-front detection subunit is positioned at the exit pupil to ensure conjugation relation. Therefore, the entrance pupil position of the common-aperture telescope is 0.5 m. Because the input energy of the laser is too high, many beam splitters are required to output majority of the energy, leaving just a small amount of energy for weak light detection. To ensure the smooth entry of the compressed beam into the subsequent system, the optical path is required to use beam splitting and folding. Owing to the limitations of the installation position of the wave-front detection subunit, an exit pupil position of 40 mm or greater is needed to achieve optical path folding and installation.
The more pixels a spot occupies, the more sensitive the calculation will be. According to John E. Greivenlamp [18], the minimum criterion for the accuracy of the centroid calculation is to cover at least 8 pixels in the diameter of the Airy disk. The far-field detection unit is made up of a far-field detection subunit and common-aperture telescope, which have a combined focal length of f z = 5500 mm. The theoretical Airy spot radius is 1.22·λ·f z /D = 35.7 μm, where λ is the main wavelength, f z is the combined focal length, and D is the equivalent circular aperture. Because the pixel size of the selected camera is 6.9 μm, the theoretical radius of the image spot is 35.70/6.9 = 5.17 pixels, and the aperture of the diffracted spot is about 10 pixels. These meet the technical index requiring that the aperture of the image spot be greater than 8 pixels, which indicates that the combined focal length of the common-aperture telescope and the far-field detection subunit is reasonably designed.
The following two factors are primarily considered in nearfield detection, that is, the coverage of the camera, and the ability to guarantee the integrity of the light spot in the case of an off-axis field of view. If the coverage area is too large, the offaxis field of view cannot be assured, while calculation accuracy will be insufficient if it is too narrow. The near-field detection subunit and common-aperture telescope work together to create a near-field detection unit with a combined shrinkage ratio of 49.5 times and a beam size of 3.23 mm × 2.42 mm. The pixel size of the detection camera is 6.9 μm, and the number of pixels that can be covered is 3.23/0.0069 × 2.42/0.0069 = 468.1 × 350.7 pixels. The system meets the minimum coverage of 360 × 270 pixels, which indicates good magnification distribution. It is undeniable that the AO system's aberration control is quite crucial. When designing the optical system, especially the common-aperture telescope system, the double Gaussian initial structure is employed to get rid of distortion, astigmatism, coma, etc., and the spherical aberration caused by the large-aperture system is eliminated via the aspheric technology. Simultaneously, the aberration in the system can further be eliminated by calibration.

A. Wave-Front Detection Unit Design
The wave-front detection subunit is the wave-front detector in the measurement system and is situated at the exit pupil location of the common-aperture telescope; hence, this section concentrates on the optical design and analysis of the commonaperture telescope. Because of the size constraint of the target surface, high-magnification beam compression is required. The common-aperture telescope is actually a system that compresses the beam aperture. The optical system for beam compression [16] is a nonfocal telescope system [19] with a 160 mm × 120 mm aperture and a 200-mm-equivalent circular aperture. Although the detected beam has a rectangular aperture, we design the optical system based on the diameter of the outer circle of the rectangular aperture. Compared with the Galilean telescope system, the Keplerian telescope system can achieve image transmission, with an object-image conjugate relationship, thereby allowing for adaptive correction. Therefore, the Keplerian telescope system is selected for this application. In addition, a band-pass filter can be utilized to suppress stray light in the relevant experiment since the laser correction is mostly a single wavelength.
The focal lengths f 1 and f 2 of the objective and eyepiece groups were calculated as f 1 = 444.994 mm and f 2 = 40.454 mm using the design parameters of the common-aperture telescope, visual magnification formula, Gauss formula, and turning surface formula of the Keplerian telescope system. In theory, the tube length of the Keplerian telescope system is equal to the sum of the focal lengths of the objective and eyepiece groups, 485.448 mm, which is larger than 320 mm. The telephoto structure can further decrease the tube length.  The telephoto structure specifies that the tube length coefficient k is the ratio of the tube length to the focal length, and 0 < k < 1. The telephoto structure's parameters using the required two-piece lens are related as follows when combined with Gaussian optics: where f α and f β are the focal lengths of the objective group and eyepiece group of the telephoto structure, respectively; a is an introduced coefficient, 0 < a < k; l 2 is the flange back, expressed as the distance from the center of the rear surface of the last lens of the optical system to the image-space focus of the system; and W is the tube length of the system. The objective group of the common-aperture telescope is designed as a telephoto structure when the focal length is greater than the tube length and the initial selection of the tube length coefficient k 1 = 0. The theoretically determined ideal lens is replaced with an actual lens, and the structure of the wave-front detection unit is shown in Fig. 2. Because of the large aperture of the system and the common-aperture telescope tube length constraint of 320 mm, if all spherical lenses are employed to provide acceptable image quality, numerous lenses will be required to correct the aberration, resulting in a huge, complicated structure. Furthermore, the f-number of the first lens is rather high. To improve the optical performance, simplify the system structure, and minimize the system size [20], the front surface of the first lens is constructed as an aspherical surface with an 8th-order correction term. This surface has coefficients of −4.718 × 10 −9 , −4.476 × 10 −15 , and −4.635 × 10 −19 for the 4th-, 6th-, and 8th-order terms, respectively, through using optical design software, optimization computation, and analysis. The telephoto structure of the eyepiece also serves to expand the exit pupil distance to accommodate the installation area of the wave-front detector. The exit pupil distance is 42.1 mm, and the tube length is 319.5 mm, meeting the design specifications.
A telephoto structure is employed in both the far-field detection subunit and the near-field detection subunit to shorten the optical length and decrease the tube length of the subunit to reduce the volume and weight of the measurement system. Simultaneously, both the far-field detection subunit and nearfield detection subunit are connected with the exit pupil of the common-aperture telescope through a long entrance pupil distance, and the system volume is compressed through splitters and prisms.

B. Far-Field Detection Unit Design
The far-field detection unit's design parameters are listed in Table II. The combined focal length of the far-field detection subunit and the common-aperture telescope is f z = 5500 mm, and the subunit focal length of far-field detection is f a = 5500/η = 500 mm, where η is the magnification of the common-aperture telescope. The far-field detection subunit uses a three-piece structure to focus the beam on the image surface, thereby reducing system space and weight. Fig. 2 depicts the far-field unit. The final tube length of the far-field detection subunit is 75 mm.

C. Near-Field Detection Unit Design
The near-field detection subunit requires that the exit pupil distance be greater than zero to satisfy the detection camera's reception. To decrease the space and weight of the system, the near-field detection subunit, combined with the parameters listed in Table III, uses a Keplerian structure system with two pieces in the front group and three pieces in the rear group to transfer the beam image and accomplish secondary beam compression. Fig. 2 illustrates the near-field detection unit. The final near-field detection subunit has a length of 146.4 mm and an entrance pupil distance of 34.3 mm.

IV. SIMULATION ANALYSIS
Athermalization design [21] and image-quality analysis of the system are carried out based on a sensible layout and volume compression of the optical system to eliminate thermal aberrations. Fig. 3(a) demonstrates the wave-front PV values of various FOV in the wave-front detection unit at various temperatures, with PV values greater than 0.1442λ. Fig. 3(b) depicts those in the far-field detection unit at various temperatures. The system shows good imaging quality with a PV value greater than 0.1223λ. Fig. 3(c) represents those in the near-field detection unit at various temperatures, with PV values greater than 0.1729λ, indicating that each subsystem has high imaging quality.
To further analyze the variation in the beam quality of the combined system, the ratio of the system's imaging spot to the diffraction-limited spot size at the 83.6% energy point in the surrounding energy curve is used to approximate the equivalent beam quality EBQ. Fig. 4 depicts the variation in the surrounding energy radius under different temperature conditions. The black curve in the figure depicts the energy diffraction limit in the far-field detecting unit, while the color one reflects the energy curve for each field of view. They almost coincide with the diffraction limit curve. With the small field of view, the curves of each field of view are assumed to coincide. The equivalent beam quality at 10°C, 20°C, and 30°C was obtained through the equivalent beam quality calculation approach. At 20°C, the far-field beam quality is 1.17 times the diffraction limit. At 10°C and 30°C, the far-field beam quality decreases somewhat, but EBQ values of 1.21 and 1.22 times the diffraction limit are achieved, respectively. The results reveal that temperature has little influence on the far-field beam quality, with the maximum fluctuation value being around ΔEBQ = 0.05 times the diffraction limit.
The distortion of the near-field detection unit at 10°C, 20°C, and 30°C, which are 0.2273, 0.2272, and 0.2274%, respectively. The system has high image quality and complies with index standards.

V. TOLERANCE
The variation in the manufacture and assembly of the optical system is analyzed, and the RMS wave-front is chosen as the reference for the tolerance of the measurement system, combined with the function of tolerance analysis in optical design software. The Monte Carlo approach is used with 1000 iterations to examine the tolerances of each system, as well as the most sensitive elements. The purpose of the Monte Carlo simulation in Zemax is to assess the global effect of tolerances. During the simulation, a series of random lenses that meet the tolerance requirements are generated and then evaluated against the criteria. Any number of designs can be generated during the process by employing the statistical approaches of Normal, Uniform Parabolic, and Parabolic Distribution. Table V shows the tolerance data range for the optical systems. Tables VI and VII show the results of their statistical analyses, respectively.
For both units, the first lens of the common-aperture telescope is the most sensitive element discovered through Monte Carlo analysis. The surface and element tilts are approximately ±0.6', and the decentering is approximately ± 0.015 mm. The tolerances of the far-field detection and near-field detection units are achievable.
The RMS values of wave-fronts of the far-field detection unit have a 98% probability of being less than 0.1723λ, which satisfies the wave-front aberration design standards. That of the near-field detection unit also meets the design standard with a 98% probability of being less than 0.1819λ.

VI. MECHANICAL SYSTEM DESIGN AND ANALYSIS
The mechanical structure of the multiparameter measurement system is designed based on the optical system. Under multiparameter measurement, the installation location is efficiently modified and planned, resulting in a compact optical-mechanical system structure. The system is composed of a common-aperture telescope, wave-front detection subunit, far-field detection subunit, and near-field detection subunit. The benefits of titanium alloy (TC4) include high strength and excellent heat resistance (lower thermal expansion coefficient). Therefore, TC4 is chosen as the mechanical system material to limit the impact of external temperature variations on the measurement system. The mechanical structure of the common-aperture telescope serves as the mainframe of the measurement system. As shown in Fig. 5, other subunits are attached to the mainframe. The main assembly and alignment principles for the measurement system follow. First, the mechanical supports and optics are constructed and tested to verify that the design tolerances are met. Then, the common-aperture telescope, wave-front detection subunit, far-field detection subunit, and near-field detection subunit are sequentially assembled and aligned, and the angular deviation of the lens is calibrated interferometrically. Finally, the overall structure is installed and debugged for good performance. The size of the system is 400 mm × 150 mm × 246 mm (L × W × H), and its weight is 23.84 kg, which meets the design requirements.
Finite-element analysis (FEA) is used to examine the structure of the measurement system. Considering that the main operating environment of the measurement system is 20°C ± 10°C, it is determined whether the deformation of the system induced by gravity and temperature is within the tolerance of the optical system.

A. Gravity Analysis of Common-Aperture Telescope Optical System
The surface shape of the lens and the imaging of the system are affected by gravity. As a result, a gravitational deformation analysis is performed on the common-aperture telescope's four larger-aperture lenses. The largest deformation in the radial direction occurs in the first lens with a deformation amount of 2.309 nm. Much smaller than the eccentricity tolerance, its effect is negligible.

B. Mechanical System Gravity Analysis
The four larger-aperture lenses are designed with independent support structures to enable slight adjustments of the lenses to improve image quality. Because the weight of the lens deforms the mechanical structure and affects the image quality, the equivalent mass of the larger-aperture lens and its independent support structure act on the corresponding position of the measurement system in the form of a distributed mass. The influence of this distributed mass is analyzed. Both the far-field and near-field detectors have a mass of 0.12 kg and also apply a distributed mass to their respective locations within the system. To optimize the analytical performance, these structures were excluded from the FEA. The measurement system showed the most deformation on the top of the front group and the bottom of the middle group of the mainframe with respective deformations of 55 nm and 27 nm. These maximum deformation values are smaller than the most sensitive decentering tolerance of ± 15 μm, indicating that the impact of gravity does not affect its ability to meet the tolerance requirement.

C. Influence of the Mechanical System on the Optical System Under Varying Temperature
The change in temperature causes thermal expansion and contraction of the structural materials, resulting in varying degrees of decentering and tilt in the optical elements or systems, which can influence the image quality of the optical system. If the mechanical structure is severely deformed, the optical system suffers irreversible image-quality degradation, making it   incapable of accomplishing normal measurement tasks. Using FEA software, a thermal study of the mechanical structure at 10°C and 30°C was performed.
Comprehensive analysis of gravity and temperature factors and the impact of temperature on the structure of the measurement system mainly occurs in the front and middle groups of the common-aperture telescope and HR1. The maximum relative decentering in the X and Y directions are 1.173 μm and 6.091 μm, respectively; the overall maximum tilt is 0.00129°; and the maximum tilt of HR1 is 0.0035°. The most sensitive tolerances obtained via analysis were ± 0.6 (± 0.01°) tilt and ± 15 μm decentering; thus, the maximum values obtained herein fulfill the tolerance standards.
The FEA results of optical element deformation, decentering, and tilt are imported into Zemax software to examine the wavefront aberration of the far-field detection unit and the near-field detection unit.
At 10°C and 30°C, Fig. 6 depicts the PV values of wave-fronts of the far-field detection unit and near-field detection unit following decentering and tilt. Although it meets the system tolerance standards, structural materials with lower expansion coefficients can be employed to improve the system's performance.

VII. MECHANICAL SYSTEM DESIGN AND ANALYSIS
The experimental system represented in Fig. 7 was built in the laboratory. The laboratory temperature has been kept at 22°C±5°C, with the lowest and maximum temperatures reaching 5°C and 40°C, respectively, and the relative humidity was 50%.
In an adaptive optics (AO) system, noise and detection error can produce errors in the slope measurement of a Hartmann-Shack (H-S) wave-front sensor and have further effects on the performance of the AO System. The noise in an AO system  can be divided into the readout noise and the photon noise. The detection error in an AO system results from the discrete sampling by using number-limited CCD pixels in the H-S sensor and the dead-space between the CCD pixels [22]. The adaptive optics technology's correction effect is essentially unaffected when the signal-noise ratio is high while the effect decreases dramatically compared to the noise-free state when the ratio declines. Therefore, it is vital to remove the impact of noise on the image. All in all, it is also a future research direction to probe into the influence of noise on the multiparameter measurement system under varying signal-noise ratios, even under the combined influence of noise and aberration. Fig. 8(a)-(c) show the three-dimensional (3D) image distribution of the wave-front detector at 10°C, 20°C, and 30°C, after noise removal, respectively. The corresponding PV values are 0.282 μm, 0.267 μm, and 0.288 μm, and the corresponding RMS values are 0.079, 0.072 μm, and 0.075 μm. Fig. 9(a)-(c) show beam-quality images at 10°C, 20°C, and 30°C with beamquality β factors of 1.26, 1.22, and 1.27 times the diffraction limit, respectively, meeting the technical requirements. Fig. 11 shows the 3D peak beam intensity. The β is defined as β = (A/A DL ) 1/2 [23], where A and A DL are the fractions of power contained with a far-field bucket with a radius of λ/D x · D y for the measured beam and a diffraction-limited beam with the same near-field dimensions and wavelength, respectively.D x and D y , respectively, are the width and height of the measured near-field beam profile [24]. The two-dimensional (2D) beam distributions obtained by the near-field detector at 10°C, 20°C, and 30°C are shown in Fig. 10(a)-(c), respectively. The near-field image changes slightly due to the temperature change, and the beam shape exhibits barrel distortion, which is primarily caused by machining and installation errors in the optical system, resulting in highorder aberrations in the system and diffraction deformation of the spot. Nevertheless, the imaging is improved. The Uniformity is defined as Uniformity = (I max − I min )/Ī [25]. Thus, the near-field depicted in Fig. 10(a)-(c) are 0.535, 0.474, and 0.502, respectively.
The multiparameter measurement system was tested 40 times in the laboratory under random temperature variations of 20°C ± 10°C. The measured data for far-field β are shown in Fig. 12. The maximum, minimum, and average values of β are β max = 1.284, β min = 1.226, and β ave = 1.248, respectively. Data regarding the near-field are shown in Fig. 13. Uni max = 0.570, Uni min = 0.472, and Uni ave = 0.533 are the maximum, minimum, and average values, respectively.

VIII. CONCLUSION
The dimensions of the proposed measurement system are 400 mm × 150 mm × 246 mm (L × W × H). It can concurrently measure the wave-front, far-field, and near-field of the 160 mm × 120 mm aperture laser beam emitted by the solid-state slab laser. The common-aperture telescope adopts the Keplerian system, which ensures the portability and compactness of the optical system while providing object-image conjugation capabilities to meet the AO requirements. Other detection subunits are constructed on the mainframe of the measurement system based on the mechanical structure of the common-aperture telescope. This effectively compresses the structural size of the measurement system, enhances its compactness, and reduces its volume. In 40 tests, β max = 1.284, β min = 1.226, β ave = 1.248, Uni max = 0.570, Uni min = 0.472, and Uni ave = 0.533. In the future, the optical system and mechanical system will be optimized to increase the index of the measurement system itself. The common-aperture telescope will be upgraded to reduce the tolerance sensitivity and simplify the installation and debugging of the system. Furthermore, the measurement system will be continuously evaluated.