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Abstract
A hybrid adaptive optics (AO) system with an influence function (IF) optimization method is presented for high precision wavefront correction of a traditional Shack-Hartmann AO system. The hybrid AO system consists of a Shack-Hartmann wavefront sensor (SHWFS) and a deflectometry system (DS) to measure the wavefront of the laser beam and the IF of the deformable mirror, respectively. An IF optimization method is used to generate a hybrid IF (H-IF) through a position-calibration algorithm and a resolution-conversion algorithm by use of the original IFs measured by the SHWFS (S-IF) and the DS (D-IF). Configuration of the hybrid AO system is introduced. Principles and calculation results of the IF optimization method are presented. Comparison of the wavefront correction ability between the H-IF and the original IF is carried out in simulation. Closed-loop performance of the hybrid AO system using the H-IF is investigated in experiment. Simulation and experiment results show that for a traditional Shack-Hartmann AO system, the H-IF has better correction ability than the original S-IF and the IF optimization method could help improve closed-loop performance without sacrificing the simplicity of the system structure and the rapidity of the closed-loop correction.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Adaptive optics (AO) is widely used to test and correct wavefront aberration occurring in different systems of various areas, including astronomical observation [1–3], laser systems [4–6], biomedical microscopy [7–9], vision science research [10–12] and inertial confinement fusion [13–15]. A traditional AO system contains a wavefront sensor, a data processor and a controller. Among various kinds of wavefront sensors used in AO, the Shack-Hartmann wavefront sensor (SHWFS) is a widely used one for its quick response and simple structure. For a SHWFS, the measurement precision is a limitation of the correction performance, and the performance could be improved with a higher precision SHWFS. However, for a SHWFS, a higher measurement precision means a lower dynamic range. A micro-lens array (MLA) is used in a SHWFS to segment the wavefront of incident beam into several sub-apertures. In a static measurement situation, to attain a wavefront with higher precision, the MLA needs to reach a higher resolution and segment the wavefront into more sub-apertures, which results in a narrower dynamic range of the SHWFS. In a dynamic measurement and real-time wavefront correction situation, a high dynamic range of the SHWFS is necessary, which results in a low-resolution MLA and low-precision measured wavefront. The dynamic range and precision of a SHWFS is limited by the number of sub-apertures of the MLA. The number of sub-apertures is insufficient on the premise of large dynamic range, which leads to low precision [16]. In a Shack-Hartmann AO system, the wavefront aberration is measured by a SHWFS and corrected by a deformable mirror (DM). As a core component of an AO system, the DM uses its actuators to generate a conjugate surface shape and correct the wavefront aberration. The influence function (IF) of a DM is defined as the phase differences brought by the mirror after the variation of each actuator. Therefore, the precision of the IF is consistent with that of the measured wavefront, which means that the IF has limited precision. Apparently, the limited precision of the wavefront and the IF restricts the wavefront correction ability of a Shack-Hartmann AO system.
Various studies have been implemented to improve AO system’s wavefront correction ability by adding measurement algorithms, changing system structure or optimizing the IF measurement. For example, several studies focused on measurement algorithms [16–19]. An adaptive spot search method enabled expanding the dynamic range of a SHWFS, measuring the wavefront correctly when the spot was beyond the detection area [16]. A two-step system identification was proposed containing an offline calibration and an iterative identification method to optimize the wavefront correction [17]. A sub-regional wavefront hybrid algorithm was proposed to cut the wavefront and the voltage matrix into several sub matrixes and execute global correction [18]. A sorting algorithm helped to expand the dynamic range of a SHWFS by tagging spots in special sequences [19]. With the help of these algorithms, the dynamic range of a SHWFS was enlarged, and the correction performance for low-order aberrations was improved. Nonetheless, the algorithms bring complexity in the real-time calculation process, and for high-order aberrations, the correction improvements are limited.
Besides the algorithms, several studies investigated the improvement brought by the change of system structure. A DM group consisting of nine movable single-actuator DMs was constructed, and the DMs generated a virtual DM of different types with variable distances of adjacent actuators [20,21]. The IF of the virtual DM could be adjusted according to the real-time correction, while the precision’s limitation by the SHWFS was not removed. A liquid crystal display (LCD) panel was used in front of the MLA, controlling the number of lenses to provide an adjustable dynamic range and adjustable precision [22]. For this adjustable MLA, the precision of the wavefront in a static situation was improved when the MLA was adjusted for high precision measurement, while the precision of the IF was still limited by the low precision MLA since the IF had high dynamic range. Besides, the system structure loses its simplicity and the work procedure is more complicated.
In addition to the measurement algorithms and system structures, several studies investigated the optimization method of the IF. An IF fitting method was proposed, using a Gaussian function to fit the IF measured by a SHWFS [23]. After the fitting method, the IF had better ability in wavefront compensation. However, the precision of the IF was restricted by the resolution of the SHWFS since the measurement method was limited to only the SHWFS. A modified IF fitting method was implemented to use a Gaussian function to optimize the IF which is measured by an interferometer [24]. Through the optimization, the wavefront fitting ability for the IF could be improved, whereas the optimization was limited to an ideal IF and suitable for simulation with complex structure, which made it not applicable for a practical AO system. Although a variety of methods have been implemented, the SHWFS precision was not improved effectively, which limited the improvement of the wavefront correction ability of a Shack-Hartmann AO system.
In this paper, a hybrid AO system based on a Shack-Hartmann AO system is presented using an auxiliary deflectometry systems (DS) to measure the variation of a mirror’s surface shape, which has a simple structure and high precision [25–31]. Hybrid AO systems have advantages in some applications [32–35]. Some people combined two different phase-manipulation technologies for visual simulation and wavefront correction simultaneously [32]. Some people combined two DMs, one for low-order aberration correction, one for high-order aberration fine correction, which is so-called woofer-tweeter system [33]. Some people combined wavefront sensing-based and image-based AO into one system to account for difficult situations [34]. Some people combined computational and hardware AO together for speckle reduction in optical coherence microscopy [35]. In this paper, the hybrid AO system aims at the improvement of correction performance brought by the combination of a DS and a traditional Shack-Hartmann AO system. In Section 2, the configuration as well as the principle of the hybrid AO system is presented. In Section 3, an IF optimization method is introduced, containing a position-calibration algorithm and a resolution-conversion algorithm. Through the IF optimization method, a hybrid IF (H-IF) is generated. Comparison of the wavefront aberration correction ability of the H-IF and the original IF measured by the SHWFS (S-IF) is made through simulation, including the correction of 3rd to 10th Zernike mode aberrations and a random aberration. In Section 4, the ability of the IF optimization method in real-time wavefront correction and closed-loop correction of the hybrid AO system is investigated through experiment. In Section 5, the results of the simulation and the experiment are analyzed and discussed.
2. Configuration and principle of the hybrid AO system
In order to investigate the wavefront correction improvement brought by the hybrid AO system, a Shack-Hartmann AO system with an auxiliary measurement system (i.e. the DS) is proposed in this paper, as presented in Fig. 1.
Fig. 1. Configuration of the hybrid AO system. Fiber laser generates a laser beam. L1, L2 and L3 are collimating lenses. DM is a deformable mirror. The laser beam is split by a beam splitter and two split beams enter into power collector and SHWFS. The DS consists of an LCD and a charge-coupled device (CCD) with a lens (L4). The SHWFS consists of an MLA and a CCD. Wavefront and IF data measured by SHWFS and DS are transmitted into a personal computer (PC). After calculating, control signals are obtained and sent into high-voltage driver by the PC. Finally, the DM is driven by corresponding driving voltages and the compensation wavefront is generated.
The hybrid AO system shown in Fig. 1 is based on a traditional Shack-Hartmann AO system, adding a DS measurement part to improve the wavefront correction ability. In the system, lenses L1 (1050 mm focal length), L2 (1050 mm focal length) and L3 (75 mm focal length) are used to collimate the laser beam generated by a fiber laser. After the beam splitter, a laser beam enters into SHWFS and wavefront of the laser beam is measured. The other reflected laser beam is for the use of laser applications, and in this configuration, is collected by a power collector. The SHWFS consists of an MLA (20 × 20 sub-apertures) and a CCD (1004 × 1004 pixels, 7.4µm × 7.4µm pixel size). The MLA is used to separate the wavefront into sub-apertures, while the CCD records the light intensity data. Independent of the SHWFS, the DS directly measures the surface shape variation of the DM (84mm × 84mm size and 49 square-distribution actuators). The DS consists of an LCD and a CCD (1384 × 1036 pixels and 6.45µm × 6.45µm pixel size) with a lens, an LCD generating standard fringe patterns and a CCD recording the fringe patterns reflected by the DS. The measured data of the SHWFS and the DS is transmitted to a PC and the wavefront of the laser beam as well as the surface shape of the DM are calculated. After calculation by the PC, control signals are obtained and transmitted into a high-voltage driver, which provides corresponding driving voltages to the DM to generate the compensation wavefront.
Unlike traditional Shack-Hartmann AO systems, the hybrid AO system is based on the addition of the DS measurement. The process of wavefront correction is calculating the value of the driving voltage to be applied to the DM by the use of the wavefront data and the IF data. In a traditional Shack-Hartmann AO system, the wavefront and the IF are both measured by a SHWFS. In the hybrid AO system presented in this paper, the wavefront is measured by a SHWFS, while the IF of the DM measured by a DS instead. Based on the SHWFS-measured wavefront and the DS-measured IF, subsequent calculations are carried out and the wavefront aberration is corrected.
In the wavefront measurement part shown in red in Fig. 1, the beam passing through L1 is reflected by a DM (84 mm × 84 mm size). Through the collimation of L2 and L3, the beam passes through the MLA and covers the CCD image plane, resulting in a measured wavefront matrix with a resolution of 20 × 20.
For the IF measurement part shown in blue in Fig. 1, the detailed configuration is displayed in Fig. 2. Shown in Fig. 2(a) is a simplified configuration of the DS. Note that, in order to explain more concisely the transformation of the image size, the CCD with a lens is modeled as an ideal pinhole camera. To implement high accuracy IF measurement, the DS is calibrated first. During the IF measurement process, the fringes generated by the LCD are reflected by the DM and recorded by the CCD, before and after the deformation of the DM’s surface shape. Before the deformation, two sets of fringe patterns with sinusoidal intensity profile along x-direction and y-direction are generated by the LCD and recorded by the CCD. Figure 2(b) shows the fringe patterns along y-direction as an example. Then, an actuator is driven and the DM’s surface shape is deformed. The phase change happens on the reflected beam and the distorted fringes are recorded by the CCD [Figs. 2(c)–2(d)], accordingly. A phase-shifting method is implemented to calculate the slopes of the phase differences caused by the deformation of the DM, and a reconstruction algorithm is carried out to derive phase differences [36,37]. Finally, the IF of the driven actuator of the DM is measured.
Fig. 2. Configuration and fringe patterns of the DS. (a) Simplified configuration of the DS. (b) Reference fringes along y-direction on the CCD. (c) Deformed fringes on the CCD. (d) Deformed fringes on the CCD, local enlarged.
The effective measured areas of the LCD and the CCD are shown in Fig. 3. For the LCD, the area of the generated fringe patterns reflected by the DM is effective in the measurement, which is encircled by the red rectangle as shown in Fig. 3(a), while the corresponding highlighted area of the CCD image plane is shown in Fig. 3(b). Note that the CCD is modeled as an ideal pinhole camera, based on the geometrical relationship of the distances between the LCD, the DM and the CCD in the implemented experiment. In our simulation and experiment, the DM size is 84 mm × 84 mm, the effective area of the LCD is 168 mm × 168 mm and the corresponding highlighted area of the CCD is 2 mm × 2 mm in the image plane. As the pixel size of the CCD is 6.45µm × 6.45µm, the surface map captured by the CCD covers 310 × 310 pixels, in which all pixels are used to measure the distortion and calculate the slopes of the phase change.
Compared to the IF measurement using the traditional SHWFS, which also measures the IF as the phase change after the deformation of the DM’s surface shape, the DS could achieve high precision IF measurement based on the same deformation with a much higher resolution. For the S-IF measured by the SHWFS, it is calculated using the slope values of each sub-aperture. In each sub-aperture, the slope value is calculated using the centroid deviation of the corresponding sub-beam. The resolution of the S-IF depends on the number of slope values used in the calculation [38]. For the IF measured by the DS (D-IF), the phase change of the deformation of the DM is measured using the change of the detected fringe patterns. The resolution of the D-IF depends on the interval of the phase shift and the number of the effective pixels of the LCD and CCD. As a result, in our simulation and experiment, the resolutions of the D-IF and the S-IF are 310 × 310 (N1) and 20 × 20 (N2), respectively.
In the hybrid AO system, the wavefront to be corrected is measured by the SHWFS and has the same resolution of the S-IF (20 × 20), which depends on the calculations of the centroid deviations of sub-beams. According to the analysis above, in the IF measurement, the raw D-IF data depends on the calculation of the fringes distortion and has a resolution as high as 310 × 310. In order to achieve high precision wavefront correction of the hybrid system, a hybrid IF (H-IF) matching the resolution of the measured wavefront is generated through an IF optimization method using the S-IF and D-IF as input parameters. The IF optimization method consists of a position-calibration algorithm and a resolution-conversion algorithm. In the following sections, the calculation process and relative results of the algorithms are described and the wavefront correction abilities of the S-IF and H-IF are investigated. The resolutions of different IFs and measured wavefront are listed in Table 1.
Table 1. Resolutions of different IFs and the measured wavefront
3. Simulation of the IF optimization method
3.1 IF optimization method
Figure 4 shows the comparison of the D-IFs and S-IFs of two typical actuators of the 49-actuator DM as a demonstration. As shown in Fig. 4(a), the DM is lab-manufactured, containing 49 square-distributed actuators with an interval d of 12 mm between each actuator [39].
Fig. 4. Comparison of S-IF and D-IF. (a) Distribution of 49 actuators of the DM. d is the distance between each actuator. (b) 10th S-IF; (c) 10th D-IF; (d) 39th S-IF; (e) 39th D-IF.
Shown in Fig. 4 is the original IF data measured by the DS and the SHWFS. From the information in the diagram, in a higher resolution, the D-IF is smoother and has less jagged edges than the S-IF, with more surface shape information contained. Considering that the wavefront aberration measured by the SHWFS is in a low resolution, to complete the wavefront correction, the D-IF is transformed into a new H-IF which has the same resolution with the S-IF. Besides, the position characteristics should be preserved. To meet the requirements of position characteristics preservation and resolution conversion, the IF optimization method is composed of two steps, of which the first step uses a position-calibration algorithm and the second step uses a resolution-conversion algorithm.
The position-calibration algorithm matches the effective area of the D-IF based on the S-IF. Because of different measurement principles and structures, the distributions of the S-IF and the D-IF are different from each other. In the position-calibration algorithm, the S-IF and the D-IF are measured at first, and the separate IFs of three actuators (i.e. the 9th, 13th and 37th actuators) are regarded as calibration references that are sufficient to provide the size comparison information, as shown in Fig. 5.
Fig. 5. Position-calibration for the D-IF. (a) 9th D-IF; (b) 9th S-IF; (c) 9th, 13th and 37th D-IF; (d) 9th, 13th and 37th S-IF; (e) Spliced matrix for D-IF; (f) Spliced matrix for S-IF; (g) 9th calibrated D-IF.
It is shown in Fig. 5 that the IF distributions of the D-IF and the S-IF are different [Figs. 5(a)–5(b)]. Three actuators’ IFs are generated separately to get the calibration references as shown in Figs. 5(c)–5(d). Effective areas of these three separate IFs are spliced together to complete the calibration as shown in Figs. 5(e)–5(f). The calibration points are defined as the maximum points of the IFs of these three actuators. ${p_{D1}}$ (210), ${p_{D2}}$ (210), ${p_{S1}}$ (14) and ${p_{S2}}$ (14) are the intervals between the calibration points. Based on the calibration points of two calibration matrices, noting that ${p_{D1}} = {p_{D2}}$ and ${p_{S1}} = {p_{S2}}$, the calibrated IF matrix’s size is calculated to be R1 (300) [Eq. (1)].
After the position-calibration algorithm, the calibrated IF for one actuator (i.e. 9th actuator) is shown in Fig. 5(g), in which the part circled by the red line is the matched area cut from the original D-IF, and the resolution of the D-IF is calibrated to be R1 (300 × 300). The renewed D-IF of each actuator is cut based on the S-IF, which ensures the renewed D-IF and S-IF have the same edges and central points, reducing the mismatching of the IF. The calibration ensures that the resolution-conversion algorithm is able to be implemented precisely.The resolution-conversion algorithm is carried out to converse the calibrated D-IF into the H-IF with the resolution transformed from R1 to N2 (i.e. the resolution of S-IF). As shown in Fig. 6, in the resolution-conversion algorithm, the IF data is presented in a matrix form. The original matrix is the calibrated D-IF with a resolution of R1, while the computed matrix is the H-IF with a resolution of N2.
In the algorithm, the original matrix is divided into 400 (20 × 20, N2) sub-blocks, each sub-block containing a submatrix, from which a target value is derived. To maintain the characteristics of the D-IF, the target value of each sub-block is calculated using all the submatrix values [Eq. (2)].
Where b is the computed target value, ${a_{i,j}}$ the original submatrix value, and ${k_{i,j}}$ the coefficient of each value. The computed matrix is composed of the target values that are computed from each sub-block, having the same resolution with the S-IF and preserving the characteristics of the D-IF.Considering the distribution matrix of the coefficient ${k_{i,j}}$ to the resolution-conversion algorithm, three types of resolution-conversion algorithms with different coefficient distributions are analyzed in the simulation as shown in Fig. 7. As shown in Fig. 7(a), for Type 1, the coefficients are distributed in the uniform distribution [Eq. (3)] and the target value is derived as the mean value of the submatrix values, as each value has equal contribution. Parameter h in Eq. (3) means the sum of the number of the sample submatrix values involved in the calculation, which is 225 in total. For Type 2, as shown in Fig. 7(b), the coefficients are distributed in a linear distribution [Eq. (4)]. The center value accounts for the largest weight and the weight gets smaller linearly with the distance from the center increasing. Based on Fig. 6, the sampled submatrix has 15 × 15 values. In Eq. (4), m means the coordinate of the center sample submatrix value, which is 8, and p1 is used to conserve the mean intensity value after the conversion. For Type 3, as shown in Fig. 7(c), the coefficients are distributed in a normal distribution [Eq. (5)]. Compared to Type 2, the center value still accounts for the largest weight, and the weight gets smaller with the distance increasing. Nonetheless, the values near the center account for larger weights, and the values far from the center account for smaller weights on the contrary. In Eq. (5), $\sigma$ is the standard deviation that limits the calculating domain of the normal distribution, and p2 is used to conserve the mean intensity value after the conversion. The value of $\sigma$ is set to be $\frac{7}{3}$ to include all the submatrix values in the calculation.
Fig. 7. Three types of resolution-conversion algorithms with different coefficient distributions. (a) Type 1: Uniform distribution; (b) Type 2: Linear distribution; (c) Type 3: Normal distribution.
Figure 8 shows the conversion results of these three types’ algorithms, using the separate IF of the 17th actuator of the DM as a demonstration. The D-IF has a higher resolution (R1). The H-IF1, H-IF2 and H-IF3 are converted H-IFs using Type 1, Type 2 and Type 3 resolution-conversion algorithms corresponding with the resolution of N2. As shown in Fig. 8, the three H-IFs are similar to each other and the major characteristics as the location and the influence region of the raised point are consistent with the original S-IF.
Fig. 8. Results of three types of resolution-conversion algorithms for the IF of the 17th actuator. (a) D-IF; (b) H-IF1 using Type 1 conversion algorithm; (c) H-IF2 using Type 2 conversion algorithm; (d) H-IF3 using Type 3 conversion algorithm.
3.2 Wavefront aberration correction result
In the simulation, the wavefront aberration correction abilities of the converted H-IF and the original S-IF are investigated. Based on the wavefront aberration and the IF data, the voltage values of the actuators are calculated first, while after which the conjugate wavefront is derived to compensate the wavefront aberration and generate residual errors.
In the following simulation, to compare the aberration correction ability of H-IF and S-IF, the target wavefront aberrations are set composing of the 3rd to 10th Zernike mode aberrations and a random wavefront aberration. The Zernike mode aberrations are generated using the Zernike polynomials and the random wavefront aberration is generated as the sum of Zernike mode aberrations with random coefficients. The original data of H-IF and S-IF are measured in the experiment. The compensative wavefront is calculated using the IF and the voltage values as shown in Eq. (6), where ${S_c}$ is the conjugate wavefront, V the voltage-value matrix and I the IF matrix containing separate IFs of all the actuators. The voltage-value matrix V is calculated according to Eq. (7), where ${S_o}$ is the initial aberration that needs to be corrected. The residual error ${S_e}$ is calculated according to Eq. (8).
As to the 3rd to 10th Zernike mode aberration, the compensative wavefront is reconstructed and the residual error calculated for each mode. The initial aberration and the corrected residual errors of the 3rd Zernike mode aberration are shown in Fig. 9 as an example. According to Fig. 9, for the residual errors corrected by H-IFs, the surface in the center area is smoother than that using S-IF. The peak to valley (PV) value and root mean square (RMS) value of the initial Zernike mode aberrations and the corrected residual errors are shown in Fig. 10 to compare the calculation results in detail.Fig. 9. Simulation result of the 3rd Zernike mode aberration correction. (a) Initial aberration. Residual errors corrected by (b) S-IF, (c) H-IF1, (d) H-IF2, and (e) H-IF3.
Fig. 10. (a) PV and (b) RMS result comparison of the 3rd to 10th Zernike mode aberration correction.
As shown in Fig. 10, the PV values of the residual errors by using the three H-IFs are similar to each other, and are smaller than those using S-IF. It can be seen that the PV and RMS values of the residual error using H-IF could be up to 61% and 64% smaller than those using S-IF. The results indicate that H-IF could reach better correction than S-IF. For example, the initial PV value of the 5th Zernike mode aberration is 4µm, and is reduced to 0.669µm after correction by using S-IF. For H-IF1, H-IF2 and H-IF3, the corrected PV values are as small as 0.260µm, 0.268µm and 0.271µm, respectively. The correction results of H-IFs are better than those of S-IF with an approximate 60% reduction. It could be seen that in terms of the correction ability of the 3rd to 10th Zernike mode aberrations, H-IF performs better than S-IF.
In addition to the previous tests for the aberrations of different Zernike modes, the wavefront aberration correction ability of different IF requires an evaluation for a random aberration correction. As shown in Fig. 11(a), a random aberration is generated in the simulation as the correction target. According to the 3rd to 10th Zernike mode coefficients shown in Fig. 11(b), the initial aberration mainly consists of the 3rd, 6th and 8th Zernike modes. Residual errors using different IFs are calculated and shown in Figs. 11–12.
Fig. 11. Correction result of the random aberration in simulation. (a) Initial aberration and (b) Zernike mode decomposition coefficients. Residual errors corrected by using (c) S-IF, (d) H-IF1, (e) H-IF2 and (f) H-IF3.
According to Figs. 11–12, the residual errors using three H-IFs have similar characteristics. The PV values and RMS values of the residual errors using H-IFs are smaller than that using S-IF. The wavefront aberration to be corrected has a PV value of 4.260µm and RMS value of 1.0689µm. After correction, the PV value of the residual error using S-IF is 0.792µm, and the largest PV value of the residual error using these three matched H-IF is 0.630µm. The RMS values of the residual error using H-IF1, H-IF2 and H-IF3 are 0.0828µm, 0.0788µm and 0.0786µm, respectively. The PV value of the residual error using H-IF is smaller than using S-IF for up to 21%. The results show that H-IF generated by the IF optimization method performs better than S-IF in the random wavefront aberration correction, which is consistent to that of the Zernike mode aberrations correction. This indicates that for a traditional Shack-Hartmann AO system, the correction ability could be improved by simply adding a DS measuring part and implementing the proposed IF optimization method, which generates a hybrid AO system with simple structure, good expansibility and easy implementation.
4. Experiment of the beam aberration correction
To investigate the closed-loop beam aberration correction ability of the H-IF, an actual hybrid AO system is built as shown in Fig. 13, which has a similar configuration to Fig. 1. In this experiment setup, the laser beam, generated by a fiber laser (Zweda Technology, 1053 nm and 0.14 numerical aperture fiber), is collimated by lens L1 (1050 mm focal length and 200 mm diameter) into a parallel beam and reflected by a lab-manufactured DM (84mm × 84mm size and 49 square-distribution actuators). Collimating lenses L1 and L2 both have a 1050 mm focal length and a 200 mm diameter, while collimating lens L3 has a 75 mm focal length and a 15 mm diameter. The SHWFS consists of an MLA (20 × 20 sub-apertures, 300µm sub-aperture interval and 6 mm × 6 mm size) and a CCD (Basler’s piA1000-48gm, 1004×1004 pixels, 7.4µm × 7.4µm pixel size and 7.4 mm × 7.4 mm sensor size), while the DS consists of an LCD (Lilliput’s UM-900, resolution 1920×1080, 0.1925mm × 0.1925mm pixel size) and a CCD (Point Grey’s GS3-U3-14S5M-C, 1384 × 1036 pixels and 6.45µm × 6.45µm pixel size) with a lens which has a focal length of 8 mm. In Fig. 13, the red lines and blue lines represent the optical routes of the SHWFS and DS measurement, respectively.
Fig. 13. Schematic diagram of actual structure of a hybrid AO system. The fiber laser generates a 1053 nm laser beam, and collimating lens L1 changes the beam into a parallel one. After the transmission of collimating lens L2 (1050 mm focal length) and L3 (75 mm focal length), the laser beam enters into the SHWFS and the wavefront aberration is measured. An LCD and a CCD with a lens form the DS part to measure the IF of the DM. The DM generates deformed surface shape to compensate the wavefront aberration of the laser beam. Red lines show the SHWFS measurement part, and the blue lines show the DS measurement part.
In the experiment, the IF of the DM is measured separately by the SHWFS and the DS, while the wavefront of the laser beam is measured by the SHWFS. The IF optimization method is implemented, as described in Section 3. Figure 14 shows the initial aberration measured by the SHWFS and the corrected residual errors using the original S-IF and the optimized H-IF. According to the 3rd to 10th Zernike mode coefficients shown in Fig. 14(b), the initial aberration mainly consists of the 3rd, 4th and 5th Zernike modes. The comparison of the PV and RMS values of the residual errors after the closed-loop correction is shown in Fig. 15.
Fig. 14. Correction result of the beam aberration in experiment. (a) Initial aberration and (b) Zernike mode decomposition coefficients. Residual errors corrected by using (c) S-IF, (d) H-IF1, (e) H-IF2 and (f) H-IF3.
The initial aberration of the laser system has a PV value of 2.274µm, and after the correction, the PV value is reduced to 0.424µm by using S-IF. The closed-loop performance is limited as a result of S-IF’s low resolution. Compared to the correction ability of S-IF, the residual error using H-IF is at least 53% smaller (i.e. 0.200µm). The experiment results indicate that, by the implementation of the IF optimization method, a real-time Shack-Hartmann AO system could obtain the corrected wavefront with smaller PV and RMS values and realize better closed-loop performance than that of using the original S-IF. Note that S-IF and D-IF are directly measured and H-IF is calculated through the IF optimization method as prerequisites before the closed-loop process, which means that the rapidity of closed-loop correction is not affected by the IF optimization method. At the same time, the simple structure of the DS measuring part retains the AO system’s simplicity and expansibility.
5. Discussion
As illustrated above, the ability of the optimized H-IF to correct wavefront aberration is investigated through simulation and experiment. In the simulation, the 3rd to 10th Zernike mode aberrations and a random aberration are corrected, with the results shown in Figs. 10–12. According to Fig. 10, the PV and RMS values of the corrected Zernike mode aberrations by using the H-IF reduce by 36% and 22% respectively compared to those by the S-IF. As shown in Figs. 11–12, the random aberration in simulation mainly contains defocus, coma and spherical aberrations (i.e. the 3rd, 6th and 8th Zernike modes) and is corrected by using the S-IF and H-IF. The residual errors corrected by using the H-IF have a 21% smaller PV value and a 5% smaller RMS value on average than those corrected by using the S-IF. In the experiment, the beam aberration correction of an actual AO system is implemented using the S-IF and H-IF, and the results are shown in Figs. 14–15. Note that the beam aberration to be corrected in the experiment mainly includes defocus and astigmatism aberrations (i.e. the 3rd, 4th and 5th Zernike modes) as shown in Fig. 14 due to the stress and errors caused by the installation of optical components. As shown in Fig. 15, the comparison between the correction results by the H-IF and by the S-IF manifests that the residual errors after correction by the H-IF have a 53% smaller PV value and a 51% smaller RMS value on average, which correspond well with the simulation results in Fig. 10.
The PV value of the residual error is taken as the system performance value in the simulation and experiment. As can be seen clearly from Figs. 10–12 and 14–15, the aberration is well corrected and the PV value is effectively depressed. For various optical systems, different system performance values could be used to achieve corresponding control targets. For example, the RMS value could be adopted as the system performance value to control the flatness of the wavefront. Meanwhile, the least square method is used as the correction algorithm in our simulation and experiment. Other algorithms, including hill-climbing method, Newton method, simulated annealing algorithm and stochastic parallel gradient-descent algorithm, could also be employed and optimized to achieve better correction results by using the optimized IF proposed. It should be noticed that, compared with the simulation, in order to improve the precision of the measurement and calculation in the experiment, various practical factors should be taken into consideration and well controlled, such as air turbulence, environmental vibration, instability of laser power, hysteresis of actuators and high frequency distortion of the DM’s surface shape.
6. Conclusion
In this paper, a hybrid AO system is built using the DS as an auxiliary measurement to a traditional SHWFS AO system, and an influence function optimization method is implemented to realize high precision wavefront correction. Configuration of the hybrid AO system and principle of the IF optimization method are presented. The IF optimization method contains a position-calibration algorithm and a resolution-conversion algorithm to generate a hybrid IF (i.e. H-IF) for the hybrid AO system, using the S-IF and the D-IF as input IFs. To investigate the wavefront aberration correction ability of the H-IF, simulations and experiments are implemented. Simulation results show that the H-IF is more effective than the S-IF in the correction of the 3rd to 10th Zernike mode aberrations and a random wavefront aberration. Experiment results, coinciding well with the simulation results, indicate that the residual error could obtain smaller PV value and the hybrid AO system could achieve better closed-loop results by using the H-IF. In conclusion, for a traditional Shack-Hartmann AO system, the addition of the DS measuring part and the proposed IF optimization method could help to improve the closed-loop performance effectively and conveniently while maintaining the structure’s simplicity, the system’s expansibility and the closed-loop calculation’s rapidity.
Funding
National Natural Science Foundation of China (61775112); Laser Fusion Research Center, China Academy of Engineering Physics (6142A04180304).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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