we present a novel technique of calibrating the interaction matrix for high-order adaptive optics systems, called the multichannel-Hadamard method. In this method, the deformable mirror actuators are firstly divided into a series of channels according to their coupling relationship, and then the voltage-oriented Hadamard method is applied to these channels. Taking the 595-element adaptive optics system as an example, the procedure is described in detail. The optimal channel dividing is discussed and tested by numerical simulation. The proposed method is also compared with the voltage-oriented Hadamard only method and the multichannel only method by experiments. Results show that the multichannel-Hadamard method can produce significant improvement on interaction matrix measurement.
© 2014 Optical Society of America
Adaptive optics (AO) systems can compensate for the turbulence-induced phase distortion in real time so as to improve the image quality of astronomical telescopes [1,2]. In order to get the proper commands to control the deformable mirror (DM), the gain matrix i.e., interaction matrix (IM) that maps DM commands into wavefront sensor (WFS) measurements has to be calibrated. The traditional zonal method is to drive the different actuators successively and measure the corresponding slope response vectors which compose the columns of the IM [3,4]. However, with the development of high-order AO systems [5–7], the traditional zonal method becomes difficult to implement since the procedure is unacceptably time consuming and sensitive to noise . Considering the saturation of DM actuators, a voltage-oriented Hadamard method was proposed by Kasper to reduce the impact of turbulence and photon noise . Nevertheless, in closed-loop systems based on the curvature WFS or the pyramid WFS, the linear range of the WFS is much smaller than the DM so the maximum voltage can hardly be achieved [10,11]. Thereby, Meimon suggested a slope-oriented Hadamard method to make the WFS response matrix satisfy a Hadamard matrix pattern. However, an initial estimate of the IM is necessary to calculate the optimal DM commands . Zou has developed an iterative technique for improving the IM calibration to overcome the non-linearity of the DM, in which the static wavefront error is firstly corrected with a generic IM . Both the slope-oriented method and the iterative method need an initial IM, which influences both the performance and the stability of these methods. Therefore, a good calibration of the initial IM is important for the convergence of these iterative methods. Other calibration methods like those using modulation techniques have also been introduced to reduce the turbulence induced error when calibrating the IM of an adaptive secondary AO system [14,15]. The sinusoidal calibration technique has already been tested on sky at the LBT . These modulation methods are particularly fit for adaptive secondary AO systems because the power spectral density (PSD) of the atmospheric turbulence decreases rapidly with frequency.
In this paper we describe a novel IM calibration method called the multichannel-Hadamard (MH) method for traditional high-order AO systems with the advantages of both the voltage-oriented Hadamard only (HO) method and a kind of multichannel only (MO) method. The IM acquired by this method can be used for other iterative or closed-loop calibration methods as the initial IM. The IM calibration metric is defined in section 2. The principle of the multichannel-Hadamard method is introduced in detail in section 3. Section 4 shows the experimental results.
2. Calibration error
In order to quantify the performance of different IM calibration methods, the definition of the calibration error is necessary. The calibration error used here is a minor modification of the one used by S. Oberti et. al. . To give the calibration error more meaning, the normalization is used here. Thus, the calibration error is defined as the normalization of the mean-square error between the exact IM and the measured IM. If the exact IM is D0 and the estimated IM is Dm, the calibration error here is defined as
Obviously, the smaller the IM error is, the smaller the calibration error will be. Owing to the normalization, if the estimated IM is a null matrix, J equals 1, while if the estimated IM equals the exact IM, i.e., the perfect calibration performance achieved, J equals 0. Although this metric is different to the one used in  which is the geometric mean of the standard deviation of the IM elements, both of them reflect the same physical meaning that is the deviation of the measured IM from the exact IM.
3. Principle of the multichannel-Hadamard calibration method
3.1. Basic principle
As each actuator of the DM can only influence local areas of the WFS, slopes measured in subapertures farther from the pushed actuator will have lower signal-to-noise ratios (SNR) so it is better to replace some of the slope responses with extremely low SNRs by zeros. Therefore, the IM of a large AO system is a sparse matrix and actuators far enough from each other can be driven at the same time with little coupling. These actuators are treated as a channel and the whole actuators of a DM can be divided into a series of channels. Then, the voltage-oriented Hadamard actuation is applied to these channels to reject the calibration noise. Finally, the IM mapping the multichannel pattern commands into the WFS measurements is translated into the one that maps the DM commands into the WFS measurements. This calibration method is called as the multichannel-Hadamard method. Taking the 595-element AO system as an example (The layout of the DM actuators and the WFS subapertures is shown in Fig. 1), the coupling of the DM is about 9.5%. The detailed description of this method is below. The whole procedure can be summarized as three steps:
1. The DM actuators are firstly divided into 19 channels by the rule that the distance between any of the two actuators in the same channel is not closer than 3.4dact, where dact is the spacing of two adjacent actuators in the same line. This rule makes actuators in the same channel almost uncoupled with each other. The detailed description of the channel dividing result is also shown in Fig. 1. The channel dividing result can also be expressed by a channel dividing matrix (Fig. 2) which is defined asEq. (3), the row number of Seffect is 2nsub because both x-slope and y-slope responses are contained in the IM. The EIM of the 595-element AO system is shown in Fig. 3, which indicates that this matrix is a sparse matrix. It is certain that the 9.5% coupling relationship in our system is a bit low, but one can easily adjust the EIR with different actuator coupling as required. If the coupling factor of the AO system increases, the EIR should increase too.
2. The 19 channels are treated as 19 “big” actuators and then a new DM with 19 “big” actuators is another form of the 595-actuator DM. Afterwards, if the MO method is used, it will do a push-pull on the channels and gets the average responses one by one. In that case, an identity matrix pattern is used, and the voltage matrix would be9]. Thus, the voltage matrix for the MH method can be described as
3. In order to translate Dcm into the IM mapping DM actuators’ commands into the WFS measurements, two steps have to be done. Firstly, the slope response of each channel is copied to the columns of the corresponding slope response of the actuators that belong to that channel by
3.2. Error analysis
To compare the performance of the MO, HO and MH methods, the calibration errors are analyzed in theory. For the MH method, if the EIR is large enough to make the elements that are going to be set to zeros small enough, the IM error9], the IM error of the HO method here isEqs. (1), (11), (14), (16) and (17), the calibration error of the MH method can be approximated byEqs. (18), (19) and (20)17].
3.3. Optimal EIR
When dividing the DM actuators, the EIR is one of the most important factors that influence the performance of the MH method. For different AO systems, the actuator couplings and the WFS measurement error may be different. During the IM calibration, the slope response’s SNR of a subaperture is in direct proportion to the actuator coupling while inversely proportional to the measurement noise. Therefore, the optimal EIR indeed exists.
The influence of the actuator coupling and the measurement noise on the EIR was analyzed by Monte Carlo simulations. Based on the influence function of the DM and the geometry relationship between the actuators and the subapertures, the ideal IM was firstly computed without adding any measurement noise, i.e., the slope response’s measurement SNR was infinite. Then, the IM errors were computed by simulating the IM calibration with different EIRs in different SNR conditions. During the simulation, the calibration SNR was increased from 5 to 15, and the actuator couplings were 9.5%, 20% and 30%, respectively. The calibration SNR here is defined as
During the simulation, in spite of the different channel numbers (all less than 64), an order 64 Hadamard matrix was used as the voltage matrix pattern to exclude the influence of different numbers of push-pulls required by different Hadamard matrices, so both n and nMH equaled 64 here. The results (Fig. 4) show that the optimal EIRs increase almost linearly with snr. Besides, the larger the coupling is, the larger the optimal EIR will be if snr is fixed. For the 595-element AO system we used (9.5% coupling), the optimal EIR is about 1.7~1.8dact when snr increases from 5 to 15. That’s why the EIR used in our experiment is 1.7dact.
In order to test the performance of the MO, HO and MH calibration methods, experiments have been done on the 595-element AO system. The 595-element AO system (located in the Institute of Optics and Electronics, Chinese Academy of Sciences) mainly contains a 595-actuator piezoelectric DM (9.5% actuator coupling), a 600-subaperture Shack-Hartmann WFS (28x28 subapertures). The optical schematic of this AO system is shown in Fig. 5. The source is a 600 nm semiconductor laser.
In different methods, the DM actuators were divided into different number of groups, so different number of push-pulls were needed to drive the DM actuators during the whole calibration period. The first task was to define the calibration condition to exclude the impact of the different number of push-pulls. The calibration SNR defined in Eq. (22) was taken as the calibration condition. Generally, The RMS of a measured signal is inversely proportional to the square root of the measurements times when the measurement noise is white. If the numbers of push-pulls of driving the DM by using the MO, HO and MH methods in our system were denoted by nMO, nHO and nMH, the driving voltages of these methods should satisfyEqs. (21) and (23),Eq. (1). Although it couldn’t be really measured, we could use an IM calibrated in a high SNR condition to replace. In our experiment, an IM calibrated with snr equal to 35 was used as the real IM. Sets of ten IMs were taken by using each of the three methods mentioned above with different SNRs. The corresponding RMs were then calculated. The parameter during the experiment was about 1.99x10−5 rad. The simulation was also done on the computer using the real IM and the slope noise collected on the 595-element AO system with the calibration condition the same as the experiment. The experimental and simulation results are shown in Fig. 6. Because of the relatively small coupling value of the DM in our experiment, a great number of elements approaching zeros exist in the IM, which resulted in the calibration error of the MO method smaller than the HO method. However, the MO method isn’t always better than the HO method, especially if a larger coupling DM is used which makes less elements of the IM close to zeros. With the advantages of both the MO and HO methods, the MH method had the smallest error.
Since the IMs were used to calculate the RM and make the AO closed-loop system work, the most important way to judge the quality of them is through closed-loop performance. As the pseudo inverse of the IMs measured above, RMs were calculated and put to use in a kind of “one-shot” close, in which the compensation voltage vector calculated by the RM was directly applied to the DM. To simplify the closed-loop procedure, none iteration was used. It was a test of the wavefront reconstruction capability of different RMs. After “one-shot” close, the slope residual variance caused by the IM error can be given by Fig. 7. The similar trend of Fig. 6 and Fig. 7 demonstrates that the calibration error defined in Eq. (1) can indeed represent the slope residual variance caused by the IM error.
Although the slope error variance could indicate the quality of a zonal IM, the phase error variance would be a more convincing index. The phase error variances induced by the IM error after the “one-shot” close are shown in Fig. 8. Generally speaking, the phase error variances induced by the IM error decrease with the calibration SNR and the variances of the MH method are the smallest among the three methods. Even if the calibration SNRs of the other two methods reach 15, their variances are still larger than the variances of the MH method with SNR 5. These results agree with the slope variances in Fig. 7 very well.
In fact, almost every AO system works in real on-going closed-loop scene. Obviously, when correcting the same static aberration, a better IM should compensate the aberration faster and if the same closed-loop time is used, a larger Strehl ratio (SR) of the focal plane image should be achieved. In our experiment, the IMs and the corresponding RMs measured above were used to make the AO system closed-loop work and compensate the static aberration of the AO system. The sampling frequency was set at a small value about 10Hz for good compare of the closed-loop convergence process. Each close-loop procedure made 200 closed-loop iterations before the focal plane image was acquired. The controller was based on a PID control algorithm whose parameters were fixed all through. Some of the focal plane intensities and corresponding SRs are shown in Fig. 9. As it shows, the SRs acquired by using the MH method when the calibration SNRs equal 35 and 9 are approaching. The reason is that when snr is high, fitting error is the dominant factor that influences the focal plane intensity and the IM error becomes trivial. More importantly, although the SR acquired by using the MH method decreases with snr decreasing, the one acquired by the MH method with snr equivalent to 5 is still larger than those acquired by the MO and HO methods with snr equal to 10. These results can match the corresponding phase error variances shown in Fig. 8, which also indicts the superiority of the MH method. It means that if the aberration is static, the MH method can make a least phase residual variance in the three methods with the same closed-loop iterations and if the aberration is dynamic, the MH method will track the change of the turbulence best, too. The PSDs of the residual phases with snr 10 are shown in Fig. 10. As it shows, the MH method performs very well not only in high spatial frequencies but also in low spatial frequencies. Considering the calibration efficiency requirement of high-order AO systems, the MH method is particular useful for high-order AO systems.
In conclusion, a new multichannel-Hadamard method for IM calibration has been presented. The method is based on the sparse characteristic of the IM and the noise rejection effect of the voltage-oriented Hadamard method. The IM calibration error has been defined and validated by the experiments. The excellent slope error variances, phase error variances and focal plane intensity performance are also demonstrated by experiments. The MH calibration method is useful for high-order AO systems thanks to its strong anti-noise ability. It may either work independently or be used to provide an initial IM for other closed-loop or iteration calibration methods
This work has been supported by the National Natural Science Joint Foundation (NO. 11178004), the Hi-tech project of China and the Graduate Student Innovation Foundation of the Institute of Optics and Electronics, Chinese Academy of Sciences (2013.2).
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