Multichannel-Hadamard calibration of high-order adaptive optics systems

Youming Guo; Changhui Rao; Hua Bao; Ang Zhang; Xuejun Zhang; Kai Wei

doi:10.1364/OE.22.013792

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 [1,2]. 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 [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]. 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 [12]. 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 [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. 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 [16]. 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. [8]. 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 D_m, the calibration error here is defined as

J = t r a c e [{(D_{0} - D_{m})}^{T} (D_{0} - D_{m})] / t r a c e (D_{0}^{T} D_{0}) .

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. 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:

Fig. 1 Multichannel mode of the 595-actuator DM. The circles denote the DM actuators and the numbers in the circles represent which channel the actuators belong to. The squares mean the subapertures.

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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 3.4d_act, where d_act 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 as

C (i, j) = {\begin{matrix} \begin{matrix} = 1 & while the i^{th} actuator belongs to the j^{th} channel \end{matrix} \\ \begin{matrix} = 0 & otherwise \end{matrix} \end{matrix} .

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. The EIA can be described by an effective influence matrix (EIM) defined as

\begin{array}{l} S_{e f f e c t} (_{2 p - 1, q) 2 n_{s u b} \times n_{a c t}} = S_{e}_{f f e c t} (_{2 p, q) 2 n_{s u b} \times n_{a c t}} \\ = {\begin{matrix} \begin{array}{l} 1; while the center of the p^{th} subaperture is \\ in the E I A of the q^{th} actuator \end{array} \\ \begin{matrix} 0; & otherwise \end{matrix} \end{matrix}, \end{array}

where

n_{s u b}

is the subaperture number and

n_{a c t}

is the actuator number. In Eq. (3), the row number of S_effect is 2n_sub 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.

Fig. 2 Channel dividing matrix (C) of the 595-actuator DM. The 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.

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Fig. 3 The effective influence matrix S_effect. The 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.

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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 be

V_{M O} = v_{M O} I_{n_{M O}},

where v_MO is the amplitude of the driving voltage and

n_{M O}

is the channel number of the MO method. Different from that, our MH method employs a Hadamard matrix pattern for the channels to benefit from the noise rejection effect of the HO method [9]. Thus, the voltage matrix for the MH method can be described as

V_{M H} = v_{M H} H_{n_{M H}},

where v_MH is the amplitude of the driving voltage and n_MH is channel number of the MH method.

H_{n_{M H}}

is a Hadamard matrix of dimension n_MH. Depending on the value of n_MO, several channels without any actuators may be necessary to make the channel number n_MH satisfy the dimension requirement of a Hadamard matrix. In that case,

C_{M H} = [\begin{matrix} C & 0 \end{matrix}],

where the dimension of the block matrix 0 is

n_{a c t} \times (n_{M H} - n_{M O})

. When driving the channels, the WFS measurement matrix

G = D_{c} V + N,

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_{c m} = G V^{- 1} .

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_{t e m p} = D_{c m} C_{M H}^{T} .

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. This 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. This procedure can be expressed in mathematics as

D_{M H} = D_{t e m p} \times S_{e f f e c t} .

Thus, the matrix D_MH is the IM that we need which maps the DM actuators’ commands into the WFS measurements.

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 error

Δ D_{M H} = D_{M H} - D_{0} \approx - \frac{1}{v_{M H} n_{M H}} N_{M H} H_{n_{M H}}^{T} C_{M H}^{T} \times S_{e f f e c t},

where N_MH is the noise matrix of the MH method whose dimension is

2 n_{s u b} \times n_{M H}

. Similarly, the IM error of the MO method

Δ D_{M O} = D_{M O} - D_{0} \approx - \frac{1}{v_{M O}} N_{M O} C^{T} \times S_{e f f e c t},

where N_MO is the noise matrix of the MO method whose dimension is

2 n_{s u b} \times n_{M O}

. According to [9], the IM error of the HO method here is

Δ D_{H O} = D_{H O} - D_{0} = - \frac{1}{v_{H O} n_{H O}} N_{H O} H_{n_{H O}}^{T},

where N_HO is the noise matrix of the HO method whose dimension is

2 n_{s u b} \times n_{H O}

, v_HO is the driving voltage amplitude and

H_{n_{H O}}

is a Hadamard matrix of dimension n_HO. As the slope measurement noise of each subaperture can be considered as a Gaussian white noise, the noise matrix meets

N_{M H}^{T} N_{M H} = 2 n_{s u b} σ_{n}^{2} I_{n_{M H}},

where

I_{n_{M H}}

is the identity matrix of dimension

n_{M H}

and

σ_{n}^{2}

is the average variance . If one push-pull is applied when driving the actuators, this variance would be

σ_{n}^{2} = {〈 {[\frac{n o i s e (j, k) - n o i s e (j, k - 1)}{2}]}^{2} 〉}_{j, k}

where

n o i s e (j, k)

is the

j_{t h}

element of the slope noise vector at the

k_{t h}

time of driving the DM, and

< >_{j, k}

means the ensemble average. Besides, the Hadamard matrix meets

H_{n_{M H}} H_{n_{M H}}^{T} = n_{M H} I_{n_{M H}},

and the trace of the channel dividing matrix

t r a c e (C_{MH} C_{MH}^{T}) = n_{a c t} .

Because of the EIM

S_{e f f e c t}

, lots of elements in the D_MH with small SNRs are set to zeros. Thus, with Eqs. (1), (11), (14), (16) and (17), the calibration error of the MH method can be approximated by

J_{M H} \approx \frac{2 (n_{D} - n_{0}) n_{a c t} n_{s u b} σ_{n}^{2}}{n_{D} n_{M H} v_{M H}^{2} t r a c e (D_{0}^{T} D_{0})},

where n_D is the total number of elements in S_effect and n₀ is the number of the elements equaling 0. Similarly, it can be inferred that the calibration error of the MO method

J_{M O} \approx \frac{2 (n_{D} - n_{0}) n_{a c t} n_{s u b} σ_{n}^{2}}{n_{D} v_{M O}^{2} t r a c e (D_{0}^{T} D_{0})}

and the calibration error of the HO method

J_{H O} \approx \frac{2 n_{a c t} n_{s u b} σ_{n}^{2}}{n_{H O} v_{H O}^{2} t r a c e (D_{0}^{T} D_{0})} .

With Eqs. (18), (19) and (20)

J_{M H} : J_{M O} : J_{H O} \approx \frac{n_{D} - n_{0}}{n_{D} n_{M H} v_{M H}^{2}} : \frac{n_{D} - n_{0}}{n_{D} v_{M O}^{2}} : \frac{1}{n_{H O} v_{H O}^{2}} .

For a zonal IM, the factor (n_D-n₀)/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_MH will equal n_HO. In that case, the MH method is the same as the HO method. Therefore, the HO method is a special case of the MH method which especially fits the AO system with a large coupling DM. What’s more, the factors 1/ n_MH and 1/ n_HO 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_MH 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 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

s n r = \frac{v \max (| D_{0} |)}{σ_{n}} \sqrt{\frac{n}{n_{M H}}}

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_MH is that of the MH 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_MH 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 1.7~1.8d_act when snr increases from 5 to 15. That’s why the EIR used in our experiment is 1.7d_act.

Fig. 4 The relationship between optimal EIR, coupling and the measurement SNR. The markers are the average of 10 times repeated results and the dashed lines are the fitted results.

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4. Experiment

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.

Fig. 5 Optical schematic of the 595-element AO system.

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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 n_MO, n_HO and n_MH, the driving voltages of these methods should satisfy

v_{M H} : v_{M O} : v_{H O} = 1 : \sqrt{n_{M H} / n_{M O}} : \sqrt{n_{M H} / n_{H O}}

to keep these methods working in the same calibration condition. With Eqs. (21) and (23),

J_{M H} : J_{M O} : J_{H O} = 1 : n_{M O} : \frac{n_{D}}{n_{D} - n_{0}}

In our experiment, n_MO is 19, n_D is 714000 and n₀ is 699552 so

J_{M H} : J_{M O} : J_{H O} = 1 : 19 : 49.4

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. The corresponding RMs were then calculated. The parameter

σ_{n}

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.

Fig. 6 Calibration errors of the MO, HO and MH methods with calibration SNR from 5 to 15. These errors are calculated by Eq. (1). The 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.

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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 [12]

σ_{g I M}^{2} (D^{+}) = σ_{g}^{2} (D^{+}) - σ_{g}^{2} (D_{0}^{+}),

where

σ_{g}^{2} (D^{+})

,

σ_{g}^{2} (D_{0}^{+})

were the slope residual variances obtained by using

D^{+}

and

D_{0}^{+}

. Since D₀ was measured with a high SNR about 35 by using the MH method, it was considered as the real IM in our experiment. Therefore,

σ_{g}^{2} (D_{0}^{+})

was considered as the slope error variance which was only caused by the fitting error of the DM.

σ_{g I M}^{2} (D^{+})

was the slope residual variance caused by the IM error. The initial slope error in the experiment was caused by the static aberration of the AO system which scarcely varied. The initial slope error was then compensated by the ‘one-shot’ close. The slope variances induced by the IM error after the ‘one-shot’ close are shown in 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.

Fig. 7 Slope variances induced by the IM error after the “one-shot” close with calibration SNR from 5 to 15.

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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.

Fig. 8 Phase variances induced by the IM error after the “one-shot” close with calibration SNR from 5 to 15. These 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. The unit $λ$ denotes wavelength.

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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.

Fig. 9 Focal plane intensities of the imaging system by using MO, HO and MH methods for calibration.

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Fig. 10 PSD of the closed-loop residual phase error by using the IM calibrated with snr 10.

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5. Conclusion

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

Acknowledgments

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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Multichannel-Hadamard calibration of high-order adaptive optics systems

Abstract

1. Introduction

2. Calibration error

3. Principle of the multichannel-Hadamard calibration method

3.1. Basic principle

3.2. Error analysis

3.3. Optimal EIR

4. Experiment

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (10)

Equations (26)

Optics Express