Multichannel-Hadamard calibration of highorder adaptive optics systems

Multichannel-Hadamard calibration of highorder adaptive optics systems Youming Guo, 1,,3 Changhui Rao, 1,,* Hua Bao, 1, Ang Zhang, 1, Xuejun Zhang, 1, and Kai Wei 1, 1 Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu 61009, China Key Laboratory on Adaptive Optics, Chinese Academy of Sciences, Chengdu 61009, China 3 University of Chinese Academy of Sciences, Beijing, 100049, China * chrao@ioe.ac.cn Abstract: 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. aking the 595-element adaptive optics system as an example, the procedure is described in detail. he optimal channel dividing is discussed and tested by numerical simulation. he 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. 014 Optical Society of America OCIS codes: (010.1080) Active or adaptive optics; (090.1000) Aberration compensation; (010.185) Atmospheric correction. References and links 1. J. W. Hardy, Adaptive Optics for Astronomical elescopes (Oxford University, 1998), Chap. 1.. H. W. Babcock, Adaptive optics revisited, Science 49(4966), 53 57 (1990). 3. C. Boyer, V. Michau, and G. Rousset, Adaptive optics: Interaction matrix measurements and real time control algorithms for the COME-ON project, Proc. SPIE 137, 406 41 (1990). 4. W. Jiang and H. Li, Hartmann-Shack wavefront sensing and wavefront control algorithm, Proc. SPIE 171, 8 93 (1990). 5. A. Bouchez, R. Dekany, J. Angione, C. Baranec, M. Britton, K. Bui, R. Burruss, J. Cromer, S. Guiwits, J. Henning, J. Hickey, D. Mckenna, A. Moore, J. Roberts,. rinh, M. roy,. ruong, and V. Velur, he PALM-3000 high-order adaptive optics system for Palomar Observatory, Proc. SPIE 7015, 70150Z (008). 6. J.-F. Sauvage,. Fusco, C. Petit, S. Meimon, E. Fedrigo, M. Suarez Valles, M. Kasper, N. Hubin, J.-L. Beuzit, J. Charton, A. Costille, P. Rabou, D. Mouillet, P. Baudoz,. Buey, A. Sevin, F. Wildi, and K. Dohlen, SAXO, the extreme Adaptive Optics System of SPHERE. Overview and calibration procedure, Proc. SPIE 7736, 77360F (010). 7. S. J. homas, L. Poyneer, R. De Rosa, B. Macintosh, D. Dillon, J. K. Wallace, D. Palmer, D. Gavel, B. Bauman, L. Saddlemyer, and S. Goodsell, Integration and test of the Gemini Planet Imager, Proc. SPIE 8149, 814903 (011). 8. S. Oberti, F. Quirós-Pacheco, S. Esposito, R. Muradore, R. Arsenault, E. Fedrigo, M. Kasper, J. Kolb, E. Marchetti, A. Riccardi, C. Soenke, and S. Stroebele, Large DM AO systems: synthetic IM or calibration on sky? Proc. SPIE 67, 670 (006). 9. M. Kasper, E. Fedrigo, D. P. Looze, H. Bonnet, L. Ivanescu, and S. Oberti, Fast calibration of high-order adaptive optics systems, J. Opt. Soc. Am. A 1(6), 1004 1008 (004). 10. S. Oberti, H. Bonnet, E. Fedrigo, L. Ivanescu, M. Kasper, and J. Paufique, Calibration of a curvature sensor/bimorph mirror AO system: interaction matrix measurement on MACAO systems, Proc. SPIE 5490, 139 150 (004). 11. A. Riccardi, R. Briguglio, E. Pinna, G. Agapito, F. Quiros-Pacheco, and S. Esposito, Calibration strategy of the pyramid wavefront sensor module of ERIS with the VL deformable secondary mirror, Proc. SPIE 8447, 84475M (01). 1. S. Meimon,. Fusco and C. Petit, An optimized calibration strategy for high order adaptive optics systems: the Slope-Oriented Hadmard Actuation, in 1st AO4EL Conference (010), paper 07009. 13. W. Zou and S. A. Burns, High-accuracy wavefront control for retinal imaging with Adaptive-Influence-Matrix Adaptive Optics, Opt. Express 17(), 0167 0177 (009). 14. F. Wildi and G. Brusa, Determining the interaction matrix using starlight, Proc. SPIE 5490, 164 173 (004). #07731 - $15.00 USD Received 6 Mar 014; revised 19 May 014; accepted 1 May 014; published 30 May 014 (C) 014 OSA June 014 Vol., No. 11 DOI:10.1364/OE..01379 OPICS EXPRESS 1379

15. S. Esposito, R. ubbs, A. Puglisi, S. Oberti, A. ozzi, M. Xompero, and D. Zanotti, High SNR measurement of interaction matrix on-sky and in lab, Proc. SPIE 67, 671C (006). 16. E. Pinna, F. Quiros-Pacheco, A. Riccardi, R. Briguglio, A. Puglisi, L. Busoni, C. Arcidiacono, J. Argomedo, M. Xompero, E. Marchetti, and S. Esposito, First on-sky calibration of a high order adaptive optics system, Proc. SPIE 8447, 8447B (01). 17. X. Zhang, C. Arcidiacono, A. R. Conrad,. M. Herbst, W. Gaessler,. Bertram, R. Ragazzoni, L. Schreiber, E. Diolaiti, M. Kuerster, P. Bizenberger, D. Meschke, H. W. Rix, C. Rao, L. Mohr, F. Briegel, F. Kittmann, J. Berwein, and J. rowitzsch, Calibrating the interaction matrix for the LINC-NIRVANA high layer wavefront sensor, Opt. Express 0(7), 8078 809 (01). 1. Introduction 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,]. 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. he 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 [8]. Considering the saturation of DM actuators, a voltage-oriented Hadamard method was proposed by Kasper to reduce the impact of turbulence and photon noise [9]. 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]. hereby, 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 [1]. Zou has developed an iterative technique for improving the IM calibration to overcome the nonlinearity of the DM, in which the static wavefront error is firstly corrected with a generic IM [13]. Both the slope-oriented method and the iterative method need an initial IM, which influences both the performance and the stability of these methods. herefore, 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]. he sinusoidal calibration technique has already been tested on sky at the LB [16]. hese 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 () method for traditional high-order AO systems with the advantages of both the voltage-oriented Hadamard only () method and a kind of multichannel only (MO) method. he IM acquired by this method can be used for other iterative or closed-loop calibration methods as the initial IM. he IM calibration metric is defined in section. he principle of the multichannel-hadamard method is introduced in detail in section 3. Section 4 shows the experimental results.. Calibration error In order to quantify the performance of different IM calibration methods, the definition of the calibration error is necessary. he calibration error used here is a minor modification of the one used by S. Oberti et. al. [8]. o give the calibration error more meaning, the normalization is used here. hus, 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 D 0 and the estimated IM is D m, the calibration error here is defined as J = trace[( D D ) ( D D )] / trace( D D ). (1) 0 m 0 m 0 0 #07731 - $15.00 USD Received 6 Mar 014; revised 19 May 014; accepted 1 May 014; published 30 May 014 (C) 014 OSA June 014 Vol., No. 11 DOI:10.1364/OE..01379 OPICS EXPRESS 13793

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 [9] 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. herefore, 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. hese actuators are treated as a channel and the whole actuators of a DM can be divided into a series of channels. hen, 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. his calibration method is called as the multichannel-hadamard method. aking the 595-element AO system as an example (he layout of the DM actuators and the WFS subapertures is shown in Fig. 1), the coupling of the DM is about 9.5%. he detailed description of this method is below. he whole procedure can be summarized as three steps: 1. he 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 3.4d act, where d act is the spacing of two adjacent actuators in the same line. his rule makes actuators in the same channel almost uncoupled with each other. he detailed description of the channel dividing result is also shown in Fig. 1. he channel dividing result can also be expressed by a channel dividing matrix (Fig. ) which is defined as th th = 1 while the i actuator belongs to the j channel Ci (, j) =. () = 0 otherwise Here r equaling 1.7d act is defined as the effective influence radius (EIR), and the circle with center located at the actuator s position and radius equaling r is defined to be the effective influence area (EIA) of the actuator. he EIA can be described by an effective influence matrix (EIM) defined as S (p 1, q) = S ( p, q) effect nsub nact e ffect nsub nact th 1; while the center of the p subaperture is (3) = EIA, th in the of the q actuator 0; otherwise where n sub is the subaperture number and n act is the actuator number. In Eq. (3), the row number of S effect is n sub because both x-slope and y-slope responses are contained in the IM. he 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.. he 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 be #07731 - $15.00 USD Received 6 Mar 014; revised 19 May 014; accepted 1 May 014; published 30 May 014 (C) 014 OSA June 014 Vol., No. 11 DOI:10.1364/OE..01379 OPICS EXPRESS 13794

VMO = vmoin MO, (4) where v MO is the amplitude of the driving voltage and n MO is the channel number of the MO method. Different from that, our method employs a Hadamard matrix pattern for the channels to benefit from the noise rejection effect of the method [9]. hus, the voltage matrix for the method can be described as V = v Hn, (5) where v is the amplitude of the driving voltage and n is channel number of the method. H n is a Hadamard matrix of dimension n. Depending on the value of n MO, several channels without any actuators may be necessary to make the channel number n satisfy the dimension requirement of a Hadamard matrix. In that case, C = [ C 0 ], (6) where the dimension of the block matrix 0 is nact ( n nmo ). When driving the channels, the WFS measurement matrix G = DV + N, (7) c where V is the voltage matrix, N is the measurement noise matrix and D c is the relationship matrix that maps the channels commands into the WFS measurements which can be estimated by D 3. In order to translate D cm 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 D 1 cm = GV. (8) = D C. (9) temp cm Secondly, as the slope response of a channel is composed of the slope response of each actuator in that channel, elements in the slope response that belong to other actuators in the same channel should be set to zeros. his process depends on the EIA of the actuators. As the centre of a subaperture can only be in the EIA of one actuator in the same channel, one can easily determine which subaperture slope response is caused by which actuator in that channel. his procedure can be expressed in mathematics as D = Dtemp Seffect. (10) hus, the matrix D is the IM that we need which maps the DM actuators commands into the WFS measurements. #07731 - $15.00 USD Received 6 Mar 014; revised 19 May 014; accepted 1 May 014; published 30 May 014 (C) 014 OSA June 014 Vol., No. 11 DOI:10.1364/OE..01379 OPICS EXPRESS 13795

9 8 6 11 18 8 6 7 1 6 7 17 1 4 5 10 11 4 5 6 5 8 1 4 5 15 9 1 3 16 9 1 10 4 10 4 3 14 16 1 3 13 14 7 15 8 3 17 11 8 3 9 13 1 10 11 8 6 10 11 1 6 13 7 14 9 5 1 10 8 15 11 7 6 13 9 7 4 8 5 10 4 16 15 1 13 9 6 14 5 1 8 4 15 1 5 1 16 9 3 1 11 6 10 4 7 15 8 1 10 4 17 9 3 1 16 14 3 18 15 7 1 5 8 3 11 6 5 4 13 3 1 13 10 5 17 10 11 1 13 6 14 9 7 13 1 14 10 3 16 9 8 11 7 6 15 11 7 6 13 9 4 8 10 16 4 15 9 1 13 1 6 7 6 15 14 1 8 16 14 1 8 15 7 5 1 11 17 3 1 6 5 4 7 15 8 1 5 4 9 3 1 4 9 3 1 16 6 3 9 1 5 8 11 10 3 8 11 5 1 10 3 1 10 5 13 10 5 4 1 10 14 15 6 13 9 1 14 17 6 9 14 3 9 13 11 8 6 7 11 15 6 7 11 13 17 16 18 10 19 15 16 13 18 10 1 13 6 6 7 15 1 16 17 8 1 16 14 8 15 7 1 4 7 1 4 7 1 4 7 4 5 9 4 1 3 9 4 1 3 6 9 11 3 5 8 11 3 5 8 11 3 5 8 1 3 1 10 13 5 10 13 5 1 10 4 14 6 9 1 14 6 9 1 14 6 9 11 9 8 11 14 6 7 11 14 6 7 11 13 8 16 10 13 15 16 10 13 15 16 10 1 6 7 15 3 8 1 15 17 8 9 14 1 5 1 7 1 4 7 1 4 7 13 3 1 4 5 16 9 4 1 3 10 15 6 3 9 15 11 3 5 8 11 3 5 8 1 14 13 1 10 13 5 1 16 4 7 10 13 4 14 6 9 1 14 6 9 11 15 9 17 11 14 6 7 11 13 8 11 17 8 16 10 13 15 16 10 1 6 1 15 3 8 9 14 1 5 1 14 1 5 1 7 1 4 7 13 3 8 7 4 1 10 15 6 3 9 15 6 3 9 15 11 3 5 8 1 14 10 5 1 16 4 7 10 13 4 7 10 13 4 14 6 9 11 15 9 6 11 13 8 11 16 8 11 16 8 17 10 1 16 8 7 3 5 1 1 14 5 1 1 6 5 7 1 1 6 9 17 15 6 9 3 14 15 9 3 4 7 3 10 4 13 7 10 4 Fig. 1. Multichannel mode of the 595-actuator DM. he circles denote the DM actuators and the numbers in the circles represent which channel the actuators belong to. he squares mean the subapertures. 50 100 150 00 Actuator 50 300 350 400 450 500 550 4 6 8 10 1 14 16 18 Channel Fig.. Channel dividing matrix (C) of the 595-actuator DM. he x axis represents the channel number and the y axis denotes the actuator number. Elements that equal 1 are black and those equal 0 are white. 00 400 Slope 600 800 1000 100 50 100 150 00 50 300 350 400 450 500 550 Actuator Fig. 3. he effective influence matrix S effect. he x axis represents the actuator number and the y axis denotes the slope number. Elements that equal 1 are black and those equal 0 are white. #07731 - $15.00 USD Received 6 Mar 014; revised 19 May 014; accepted 1 May 014; published 30 May 014 (C) 014 OSA June 014 Vol., No. 11 DOI:10.1364/OE..01379 OPICS EXPRESS 13796

3.. Error analysis o compare the performance of the MO, and methods, the calibration errors are analyzed in theory. For the method, if the EIR is large enough to make the elements that are going to be set to zeros small enough, the IM error Δ 1 D = D D 0 N Hn C, Seffect v n (11) where N is the noise matrix of the method whose dimension is n sub n. Similarly, the IM error of the MO method 1 Δ DMO = DMO D0 NMOC Seffect, (1) v where N MO is the noise matrix of the MO method whose dimension is n sub n MO. According to [9], the IM error of the method here is Δ 1 D = D D = 0 NHn, v n (13) where N is the noise matrix of the method whose dimension is n sub n, v is the driving voltage amplitude and H n is a Hadamard matrix of dimension n. As the slope measurement noise of each subaperture can be considered as a Gaussian white noise, the noise matrix meets MO N N = n σ I (14) sub n n, where I n is the identity matrix of dimension n and σ n is the average variance. If one push-pull is applied when driving the actuators, this variance would be σ n noise( j, k) noise( j, k 1) = j, k (15) where noise( j, k ) is the j th element of the slope noise vector at the k th time of driving the DM, and <> jk, means the ensemble average. Besides, the Hadamard matrix meets and the trace of the channel dividing matrix H H = n I, (16) n n n trace n act (C C ) =. (17) Because of the EIM S effect, lots of elements in the D with small SNRs are set to zeros. hus, with Eqs. (1), (11), (14), (16) and (17), the calibration error of the method can be approximated by J ( n n ) n n σ, (18) n n v trace( D D ) D 0 act sub n D 0 0 where n D is the total number of elements in S effect and n 0 is the number of the elements equaling 0. Similarly, it can be inferred that the calibration error of the MO method #07731 - $15.00 USD Received 6 Mar 014; revised 19 May 014; accepted 1 May 014; published 30 May 014 (C) 014 OSA June 014 Vol., No. 11 DOI:10.1364/OE..01379 OPICS EXPRESS 13797

J MO and the calibration error of the method J ( n n ) n n σ (19) n v D 0 act sub n D MOtrace( D0 D0) n n σ. (0) n v trace( D D ) act sub n 0 0 With Eqs. (18), (19) and (0) n n n n 1 J : JMO : J : : n n v n v n v. (1) D 0 D 0 D D MO For a zonal IM, the factor (n D -n 0 )/n D is from the EIA definition which makes the elements with low SNRs set to zeros. Obviously, this factor increases with the EIR which is partly determined by the DM coupling. If the EIR is set large enough, the factor will become 1 and n will equal n. In that case, the method is the same as the method. herefore, the method is a special case of the method which especially fits the AO system with a large coupling DM. What s more, the factors 1/ n and 1/ n are brought by the Hadamard matrix pattern of the driving voltage matrix because with a Hadamard matrix pattern, each actuator is driven up and down repeatedly to average the measurement noise. If the channel number n becomes larger, the calibration error will be smaller. However, the time used to finish the calibration will be longer like the principle of increasing the push-pull times to average the measurement noise [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 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. herefore, the optimal EIR indeed exists. he 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. hen, 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%, 0% and 30%, respectively. he calibration SNR here is defined as snr vmax( D ) n = 0 σ n n () where v is the amplitude of the driving voltage, max( D 0 ) is the maximum of the absolute value of the IM, n is the number of push-pulls added during the calibration, n is that of the method 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 n equaled 64 here. he 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 1.7~1.8d act when snr increases from 5 to 15. hat s why the EIR used in our experiment is 1.7d act. #07731 - $15.00 USD Received 6 Mar 014; revised 19 May 014; accepted 1 May 014; published 30 May 014 (C) 014 OSA June 014 Vol., No. 11 DOI:10.1364/OE..01379 OPICS EXPRESS 13798

.6.5.4.3 Coupling=9.5%:simulation Coupling=9.5%:fit Coupling=0%:simulation Coupling=0%:fit Coupling=30%:simulation Coupling=30%:fit Optimal EIR (d act )..1 1.9 1.8 1.7 4. Experiment 1.6 5 6 7 8 9 10 11 1 13 14 15 snr Fig. 4. he relationship between optimal EIR, coupling and the measurement SNR. he markers are the average of 10 times repeated results and the dashed lines are the fitted results. In order to test the performance of the MO, and calibration methods, experiments have been done on the 595-element AO system. he 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 (8x8 subapertures). he optical schematic of this AO system is shown in Fig. 5. he source is a 600 nm semiconductor laser. Fig. 5. Optical schematic of the 595-element AO system. 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. he first task was to define the calibration condition to exclude the impact of the different number of push-pulls. he calibration SNR defined in Eq. () was taken as the calibration condition. Generally, he 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, and methods in our #07731 - $15.00 USD Received 6 Mar 014; revised 19 May 014; accepted 1 May 014; published 30 May 014 (C) 014 OSA June 014 Vol., No. 11 DOI:10.1364/OE..01379 OPICS EXPRESS 13799

system were denoted by n MO, n and n, the driving voltages of these methods should satisfy v : v : v = 1: n / n : n / n (3) MO MO to keep these methods working in the same calibration condition. With Eqs. (1) and (3), nd J : JMO : J = 1: nmo : n n In our experiment, n MO is 19, n D is 714000 and n 0 is 69955 so J : JMO : J = 1:19:49.4 (5) In addition, the real IM was also required to calculate the calibration error J defined in 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. he corresponding RMs were then calculated. he parameter σ n during the experiment was about 1.99x10 5 rad. he 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. he 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 method. However, the MO method isn t always better than the 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 methods, the method had the smallest error. D 0 (4) 10 0 10-1 Sim: Sim:MO Sim: Exp: Exp:MO Exp: J 10-10 -3 4 6 8 10 1 14 16 snr Fig. 6. Calibration errors of the MO, and methods with calibration SNR from 5 to 15. hese errors are calculated by Eq. (1). he dash lines are simulation results by using the three methods and the markers are the experimental results of the average calibration error of ten IMs. 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. o simplify the closed-loop procedure, none iteration was used. It #07731 - $15.00 USD Received 6 Mar 014; revised 19 May 014; accepted 1 May 014; published 30 May 014 (C) 014 OSA June 014 Vol., No. 11 DOI:10.1364/OE..01379 OPICS EXPRESS 13800

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 [1] σ = σ σ (6) gim ( D + ) g ( D + ) g ( D + 0 ), where σ g ( D + ), σ ( D + g 0 ) were the slope residual variances obtained by using D + and D + 0. Since D 0 was measured with a high SNR about 35 by using the method, it was considered as the real IM in our experiment. herefore, σ ( D + g 0 ) was considered as the slope error variance which was only caused by the fitting error of the DM. σ gim ( D + ) was the slope residual variance caused by the IM error. he initial slope error in the experiment was caused by the static aberration of the AO system which scarcely varied. he initial slope error was then compensated by the one-shot close. he slope variances induced by the IM error after the one-shot close are shown in Fig. 7. he 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. he 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 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 method with SNR 5. hese results agree with the slope variances in Fig. 7 very well. Slope variance induced by IM error (mrad ) 1.8 1.6 1.4 1. 1 0.8 0.6 0.4 0. MO 0 4 6 8 10 1 14 16 snr Fig. 7. Slope variances induced by the IM error after the one-shot close with calibration SNR from 5 to 15. #07731 - $15.00 USD Received 6 Mar 014; revised 19 May 014; accepted 1 May 014; published 30 May 014 (C) 014 OSA June 014 Vol., No. 11 DOI:10.1364/OE..01379 OPICS EXPRESS 13801

Phase variance induced by the IM error ( λ ) 0.18 0.16 0.14 0.1 0.1 0.08 0.06 0.04 0.0 MO 0 4 6 8 10 1 14 16 snr Fig. 8. Phase variances induced by the IM error after the one-shot close with calibration SNR from 5 to 15. hese variances were calculated by subtracting the phase variance caused by the fitting error (this variance equaled the phase residual variance when snr equaled 35 in our experiments) from the phase residual variances. he unit λ denotes wavelength. 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. he sampling frequency was set at a small value about 10Hz for good compare of the closed-loop convergence process. Each close-loop procedure made 00 closed-loop iterations before the focal plane image was acquired. he 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 method when the calibration SNRs equal 35 and 9 are approaching. he 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 method decreases with snr decreasing, the one acquired by the method with snr equivalent to 5 is still larger than those acquired by the MO and methods with snr equal to 10. hese results can match the corresponding phase error variances shown in Fig. 8, which also indicts the superiority of the method. It means that if the aberration is static, the method can make a least phase residual variance in the three methods with the same closedloop iterations and if the aberration is dynamic, the method will track the change of the turbulence best, too. he PSDs of the residual phases with snr 10 are shown in Fig. 10. As it shows, the 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 method is particular useful for high-order AO systems. #07731 - $15.00 USD Received 6 Mar 014; revised 19 May 014; accepted 1 May 014; published 30 May 014 (C) 014 OSA June 014 Vol., No. 11 DOI:10.1364/OE..01379 OPICS EXPRESS 1380

Fig. 9. Focal plane intensities of the imaging system by using MO, and methods for calibration. 10-10 -3 :SNR=10 MO:SNR=10 :SNR=10 PSD(rad /m) 10-4 10-5 10-6 10-7 0 500 1000 1500 000 500 3000 3500 4000 4500 f(m -1 ) Fig. 10. PSD of the closed-loop residual phase error by using the IM calibrated with snr 10. 5. Conclusion In conclusion, a new multichannel-hadamard method for IM calibration has been presented. he method is based on the sparse characteristic of the IM and the noise rejection effect of the voltage-oriented Hadamard method. he IM calibration error has been defined and validated by the experiments. he excellent slope error variances, phase error variances and focal plane intensity performance are also demonstrated by experiments. he 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 Acknowledgments his 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 (013.). #07731 - $15.00 USD Received 6 Mar 014; revised 19 May 014; accepted 1 May 014; published 30 May 014 (C) 014 OSA June 014 Vol., No. 11 DOI:10.1364/OE..01379 OPICS EXPRESS 13803