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A novel method of measuring the interaction matrix for diffractive liquid crystal wavefront correctors
Optics Communications 380 (2016) 361–367
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
A novel method of measuring the interaction matrix for diffractive liquid crystal wavefront correctors Zhaoliang Cao a,n, Lina Shao a,b, Yukun Wang a, Quanquan Mu a, Lifa Hu a, Xingyun Zhang a, Li Xuan a a State Key Laboratory of Applied Optics, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun, Jilin 130033, China b University of Chinese Academy of Sciences, Beijing 100049, China
art ic l e i nf o
a b s t r a c t
Article history: Received 20 January 2016 Received in revised form 14 June 2016 Accepted 15 June 2016 Available online 21 June 2016
The interaction matrix significantly affects the correction accuracy of wavefront correctors. For diffractive liquid crystal wavefront correctors (LCWFCs), the diffraction wavefront error must be taken into account to improve the accuracy of the interaction matrix. In this paper, a tunable Zernike polynomial coefficient method is demonstrated that decreases the effect of the diffraction wavefront error on the interaction matrix. Moreover, to eliminate the effect of the random error, a least squares method is combined with the tunable coefficient method. Experimental results show that, with the combined method, the error of the interaction matrix is decreased by a factor of 2.5 compared to that when the least squares method is used alone. Furthermore, an open loop adaptive correction experiment was performed and a considerable improvement of the image resolution was obtained by the novel method. Therefore, this method is very useful for liquid crystal adaptive optics systems to acquire the high correction accuracy. & 2016 Elsevier B.V. All rights reserved.
Keywords: Adaptive optics Atmospheric correction Liquid-crystal devices
1. Introduction Adaptive optics systems (AOSs) are widely used in groundbased large-aperture telescopes to overcome the effects of the atmospheric turbulence [1,2]. With the advantages of high spatial resolution, low cost, low power consumption, and large phase modulation [3,4], the liquid crystal wavefront corrector (LCWFC) is one of the most attractive wavefront correction devices, and liquid crystal adaptive optics systems (LC AOSs) have been investigated by many researchers [5–7]. To avoid energy loss caused by dispersion of the LCWFC, open loop control is utilized for the LC AOSs [8]. More importantly, the open loop control is an essential technique for multi-object adaptive optics (MOAO), which is an advanced method to enlarge the field of view of the AOSs [9]. The interaction matrix describes the control relationship between the wavefront corrector and the wavefront sensor (WFS), is used to derive the wavefront corrector signal from the WFS measurements during AO correction [10]. If the interaction matrix was not exact, the control relationship will be nonlinear, resulting in a decrease of the correction accuracy. To facilitate the open loop control, the interaction matrix of the WFC, which determines the correction accuracy, must be precise. n
Corresponding author. E-mail address: [email protected] (Z. Cao).
http://dx.doi.org/10.1016/j.optcom.2016.06.044 0030-4018/& 2016 Elsevier B.V. All rights reserved.
The Zernike polynomial modal method has been applied to LC AOSs to measure the interaction matrix of the LCWFC [10]. A set of Zernike modes is sent to the LCWFC in turn and the responded wavefronts are measured by a Shack–Hartmann wavefront sensor (S-H WFS). The measured information is recorded in a matrix called the interaction matrix. To eliminate the measurement noise, a least squares (LS) method has been presented by Zhang et al. [11]; although they were able to eliminate the measurement noise, they did not consider the response ability of the LCWFC for different Zernike modes. Because the phase wrapping or kinoform technique has been used to extend the correction magnitude [12], the LCWFC is a diffractive type device. The response error of the LCWFC is generated in the phase wrapping due to quantization. With the mode number of the Zernike polynomial increasing, the Zernike wavefront is complicated and generates the high phase gradient where the quantization level drops. Thus, the response abilities of the LCWFC to different Zernike polynomial are different. As the identical coefficients are used in the conventional measurement method, the measurement accuracy of the interaction matrix will be decreased. Therefore, in this paper, the response ability of the LCWFC will be first investigated for different Zernike modes. Then, a tunable coefficient (TC) method will be presented to make the response abilities of the LCWFC identical and reduce the response error of the LCWFC. Finally, a novel method, which combines the TC method with the LS method, will be introduced to improve the measurement accuracy of the interaction matrix.
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2. Tunable coefficient method 2.1. Response ability of the LCWFC According to diffraction theory, the response error of the LCWFC can be expressed as [13]
ε=
λ 2 3N
(1)
where λ is the wavelength, and N is the quantization level. Therefore, the response ability of the LCWFC is determined by the quantization level. To correct the wavefront with large magnitude (Fig. 1(a)), the distorted wavefront (Fig. 1((b)) should be wrapped into 1λ, and then be quantified with corresponding LC pixels as shown in Fig. 1(c). It is shown that, for the 35th Zernike polynomial mode with 64 64 pixels, the quantization level is different at different pixel positions. The quantization level is low where the curve has a large gradient. Furthermore, the quantization level reaches a minimum at the position with the maximum gradient. The minimum quantization levels of each Zernike polynomial mode are different. According to Eq. (1), the minimum quantization level determines the response error of the LCWFC. Before calculating the minimum quantization levels of each Zernike polynomial mode, we first introduce the meaning of the quantization level. The quantization level means the number of the LC pixels to realize 1λ phase modulation. For a large gradient, the 1λ phase modulation contains a smaller number of the LC pixels. Thus, the essence of quantization level is to determine the number of pixels with different gradient wavefront in 1λ phase modulation. It may be expressed as
N ( x , y) =
2c j P
1 → × G (x, y)
(2)
where cj is the coefficient of the jth Zernike mode, P is the pixel → number of LCWFC along one dimension, and G (x, y ) is the gradient of the Zernike polynomial. Because a Zernike mode of the coefficient of 1 has a range of 1 to 1, the 2 in the denominator of Eq. (2) is the normalized factor to make the peak to valley (PV) value equal to 1. To calculate the quantization level conveniently, the Zernike polynomial is expressed in a rectangular coordinate system:
⎧ ⎪ Z (x, y) ⎪ j m /2 ⎪ m! t ⎪ = R nm( x, y) ∑ ( −1) xm − 2t y2t when j is even 2 t ( )! ⎪ t=0 ⎪ ⎨ ⎪ Zj(x, y) ⎪ m /2 − 1 ⎪ m! t = R nm( x, y) ∑ ⎪ ( −1) xm − 2t − 1y2t + 1when j 2t + 1)! ⎪ t=0 ( ⎪ ⎩ is odd
(3)
where
R nm(x, y) (n − m) /2
=
∑ s= 0
n − m − 2s ( − 1)s (n − s )! (x2 + y2 ) 2 s ![(n + m)/2 − s ]![(n − m)/2 − s ]!
(4)
Thus, the gradient of the each Zernike polynomial mode may be expressed as
∂Zj → ∂Zj→ → G (x, y) = ∇Zj = x + y = ∂x ∂y
⎛ ∂Zj ⎞2 ⎛ ∂Zj ⎞2 ⎜ ⎟ +⎜ ⎟ ⎝ ∂x ⎠ ⎝ ∂y ⎠
(5)
To calculate the gradient numerically, the unit circle is discretized with P P points. Hence, the values of x and y will be in the range from 1 to 1, and then, the step size is 2/P. The gradient of Zj at a certain point can be computed by substituting Eq. (3) and the values of x and y at this point into Eq. (5). Thus, the distribution of the gradient of each Zernike mode can be obtained. Based on Eq. (3) to Eq. (5), the quantization level can be computed for different Zernike modes. Fig. 2 shows the calculated results for the 4th, 11th, and 28th modes. It can be seen that the maximum quantization level decreases when the Zernike mode is increased. Moreover, as shown in the enlarged inset, the trend of the minimum quantization level is similar to that of the maximum quantization level. In the conventional Zernike polynomial method, the coefficient is selected to be 1; hence, we also use the coefficient of 1 to perform the analysis. As the response error is limited by the minimum quantization level, the maximum response error and the minimum quantization level are calculated according to Eqs. (1) and (2) for different Zernike modes as shown in Fig. 3. It can be seen that the quantization level drops drastically from 128 to 6 in the first 25 modes, and then decreases slowly (Fig. 3(a)). The corresponding maximum response error is shown in Fig. 3(b), and the maximum value reaches to 0.25λ. This response error is large, and must be eliminated to obtain high correction accuracy.
Fig. 1. Phase distribution for the 35th mode Zernike: (a) Wavefront; (b) Phase profile along the x dimension; (c) Wrapped into 1λ and quantified.
Z. Cao et al. / Optics Communications 380 (2016) 361–367
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Fig. 4. Tunable coefficient as a function of the Zernike mode.
Fig. 2. Quantization level distributions along the x-dimension for different Zernike modes.
2.2. TC method From the above analysis, we can see that, because the minimum quantization level is different for different Zernike modes, the response error must also be different. Therefore, to achieve the same response ability, the minimum quantization level must be the same for different Zernike modes. Furthermore, to achieve high response accuracy, the minimum quantization level should exceed a certain threshold. According to Eq. (2), to increase the minimum quantization level, the number of pixels of the LCWFC should be increased, or else the coefficients of the Zernike polynomials should be decreased. For a given LCWFC, the coefficients of the Zernike polynomials have to be tuned to obtain a suitable minimum quantization level: we call this the tunable coefficients (TC) method. To adjust the coefficient, the minimum quantization level should be first calculated for different Zernike modes by using the coefficient of 1. Then, the coefficient is tuned by:
⎧ P ⎪ → ⎪ 2 N cj = ⎨ t ar Gmax ⎪ ⎪ ⎩1
Nmin < Nt ar Nmin ≥ Nt ar
(6)
where Gmax is the maximum gradient of the Zernike polynomial, Nmin is the minimum quantization level, and Ntar is the expected
quantization level. For Ntar ¼10, P ¼256, and 200 Zernike modes, the tuned coefficients are calculated as shown in Fig. 4. With these tuned coefficients, we assume that the response ability of the LCWFC is the same for different Zernike modes. Furthermore, as Ntar ¼10, the response error is as small as 0.029λ, and then the measurement accuracy of the interaction matrix will be high. 2.3. Validation experiment To evaluate the validity of the TC method, a single Zernike mode correction experiment was performed for 65 Zernike modes. Assuming simulating the conditions of a 2 m telescope and an atmospheric coherence length of 10 cm, the coefficients of each Zernike mode in atmospheric turbulence are calculated according Noll's atmospheric turbulence model [14] as shown in Fig. 5. Each coefficient is utilized as the magnitude of the distorted wavefront for a single Zernike mode correction experiment. Consequently, the compensated wavefront for different Zernike modes can be calculated with the produced coefficients, and then sent to the LCWFC. The responded wavefront can be measured with an S-H WFS. For example, a defocus term with the coefficient of 1.1 was selected as the distorted wavefront as shown in Fig. 6(a). The compensated wavefront (Fig. 6(b)) calculated based on the measured interaction matrix was sent to the LCWFC and the responded wavefront (Fig. 6(c)) was measured by the S-H WFS. Thus, the correction error could be computed with the difference between Fig. 6(a) and (c). Therefore, the coefficient relative error may be defined as
Fig. 3. Response errors for different Zernike modes: (a) The minimum quantization levels; (b) The maximum response errors.
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Fig. 7. Optical setup of the validation experiment.
Fig. 5. Zernike coefficients calculated with Noll's model under the conditions of a 2 m telescope and an atmospheric coherence length of 10 cm.
σj =
aˊ j − a j aj
(7)
where αˊj is the calculated coefficient of the jth Zernike mode; and αj is the coefficient of the responded wavefront. The optical setup of the experiment is shown in Fig. 7. The beam emitted from a point light source is collimated by lens1, and then goes to a polarizer. The polarized beam is zoomed in by the combination of lens2 and lens3, and then is reflected by the LCWFC. The reflected light is zoomed out by the combination of lens3 and lens4, and is then detected by the S-H WFS. The polarizer ensures that the polarization direction of the light is parallel to the alignment direction of the LC molecules. The LCWFC was purchased from Boulder Nonlinear Systems Company; it has 256 256 pixels, a 24 μm pixel size, and a 200 Hz frame rate. The S-H WFS was fabricated in our laboratory; it has a 15 15 microlens array, 4.3 mm aperture size, and 2.5 kHz frame rate. To perform a comparison, the correction can also be performed with the interaction matrix measured with the conventional Zernike polynomial method. Fig. 8 shows the relative coefficient correction error for the TC and conventional methods. The upper ■ line denotes the tuned coefficients. Because the conventional interaction matrix measuring method utilizes the identical coefficient of 1, the coefficient relative error is nearly the same for two methods when the tuned coefficients equal to 1, but the coefficient relative error of the TC method decreases significantly for the coefficients less than 1. Furthermore, it can be seen that the coefficient relative error of
Fig. 8. Coefficient relative error as a function of the Zernike mode.
the TC method is nearly the same for different Zernike modes and the standard error is 0.005. This illustrates that, with TC method, the same response accuracy may be obtained for different Zernike modes. On the other hand, the response accuracy is very different with different Zernike modes for the conventional method. Therefore, the TC method can improve the measurement accuracy of the interaction matrix. Furthermore, the mean coefficient relative errors of the TC and conventional method are 0.08λ and 0.03λ, respectively. Consequently, with the TC method, the measuring error of the interaction matrix is decreased by a factor of 2.7 as compared with the conventional method. 3. Tunable coefficient least squares method 3.1. Principle We have mentioned that the LS method can improve the measurement accuracy of the interaction matrix [10] by
Fig. 6. Simulation of the correction process: (a) Distorted wavefront; (b) Compensated wavefront; (c) Responded wavefront.
Z. Cao et al. / Optics Communications 380 (2016) 361–367
eliminating the random noise. However, the response ability of the LCWFC is not considered for this method. Firstly, the coefficient of each Zernike mode is varied from 0.5 to þ0.5. Secondly, every wavefront sent to the LCWFC is randomly produced with the combination of certain Zernike modes. Thus, the magnitude of the random wavefront possibly exceeds the response ability of the LCWFC and the response error of the interaction matrix will be increased. The TC method can improve the response accuracy of the LCWFC, but cannot eliminate the random noise as shown in Fig. 9. It demonstrates that, for the measured slope of the x-tilt mode, the results of the LS method are much closer to the theoretical curve while those measured by the TC method deviate significantly from the theoretical values. Consequently, to improve the measurement accuracy of the interaction matrix, the TC method should be combined with the LS method; we call this the TCLS method. To implement the TCLS method, it is necessary to tune the coefficient of each Zernike mode and eliminate the random noise simultaneously. To achieve this, the response of each Zernike mode is first measured with a random coefficient, whose range is constrained by the calculated tuning result. Then different Zernike modes are measured in turn by the same method. Theoretically, assuming that the measured slope vector of the jth Zernike mode is sj and the corresponding measurement noise is snj, the interaction vector of the jth Zernike mode may be expressed as
sj = cj × Mj + snj .
(8)
here sj = [ sjx1 sjx2 ... sjxp sjy1 sjy2 ... sjyp ] and p is the number of the microlens. Because the LS method is used, the coefficient of Zernike mode cj should be a variable cj(i) that is produced randomly. Assuming that the number of random variables to be produced is k, cj(i) will be in the range from cj to cj (i ¼1,2,…, k); here, cj is obtained by Eq. (6). Assuming the random noise of the jth Zernike mode is snj, the following equations may be obtained:
⎧ s (1) = c (1) × M + s (1) j j nj ⎪ j ⎪ s (2) = c (2) × M + s (2) j ⎨ j j nj . ⎪ ... ⎪ s (k ) = c (k ) × M + s (k ) j j nj ⎩ j Eq. (9) may be expressed in a matrix form:
Fig. 9. Measured slopes by TC method and LS method.
(9)
365
Sj = Cj × Mj + Snj,
(10)
(1) (1) (1) (1) ⎡ s(1) ⎤ ⎡ s(1) jx1 s jx1 ... s jxp s jy1 s jy2 ⎢ j ⎥ ⎢ ⎢ (2) ⎥ ⎢ (2) (2) (2) (2) (2) where Sj = ⎢ s j ⎥ = ⎢ s jx1 s jx1 ... s jxp s jy1 s jy2 ... ⎢ ⎥ ⎢ ... ... ⎢ s (k ) ⎥ ⎢ s ( k ) (k ) (k ) (k ) (k ) ⎣ j ⎦ ⎢⎣ jx1 s jx1 ... s jxp s jy1 s jy2 ⎡ c (1) ⎤ ⎡ s (1) ⎤ ⎢ j ⎥ ⎢ nj ⎥ ⎢ c (2) ⎥ ⎢ (2) ⎥ j Cj = ⎢ ⎥, Snj = ⎢ snj ⎥. ... ⎢ ⎥ ⎢ ... ⎥ ⎢ c (k ) ⎥ ⎢ s (k ) ⎥ ⎣ j ⎦ ⎣ nj ⎦ The objective is to achieve Mj, and its least squares value may be solved by [15]:
(
˜ = C+ × S = CT × CT × C M j j j j j j
−1
)
⎤ ... s(1) jyp ⎥ ⎥ ... s(2) jyp ⎥, ⎥ ⎥ ... s(k ) jyp ⎥⎦
estimated
× Sj.
(11)
Assuming that the first q Zernike modes are used to measure the interaction matrix, the interaction matrix of the LCWFC may be expressed as
⎡M ˜ ⎤ ⎢ 1⎥ ⎢⋮ ⎥ ˜ ⎥ ˜ = ⎢M M ⎢ j⎥ ⎢⋮ ⎥ ⎢ ˜ ⎥ ⎣ Mq ⎦
(12)
˜ , the LCWFC can correct the With the interaction matrix M distortions accurately. 3.2. Validation experiment The single Zernike mode correction scheme was also utilized to validate the TCLS method. The experimental setup was the same as that of Fig. 6. Furthermore, the first 65 Zernike modes were selected, with the same coefficients as shown in Fig. 5. To enable the comparison, the correction must also be performed with the LS method. Fig. 10 shows the coefficient relative errors for both the LS and TCLS methods. The coefficient relative error of each Zernike mode is calculated by Eq. (7) as shown in Fig. 10. As can be seen, the coefficient relative errors of the two methods are nearly the same for the first 10 modes, but the errors of the TCLS method are smaller for larger Zernike modes; this illustrates that, while the LS method is effective for the lower Zernike modes, the mean coefficient relative errors of the LS, TCLS, conventional, and TC
Fig. 10. Coefficient relative error as a function of the Zernike mode.
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Fig. 11. Optical layout for open loop correction experiment.
methods are 0.05λ, 0.02λ, 0.08λ, and 0.03λ, respectively. Consequently, with the TCLS method, the measuring error of the interaction matrix is decreased by a factor of 2.5, compared to the LS method. 3.3. Open loop correction experiment with an atmospheric turbulence simulator To evaluate the improvement of the accuracy of the interaction matrix, simulated atmospheric turbulence was produced and an open loop correction was performed in the laboratory. The optical layout is shown in Fig. 11. The light emitted from the fiber bundle is reflected and collimated by M1 and L1 respectively. The collimated beam goes through an atmospheric turbulence simulator (ATS), and is then zoomed out by the combination of L2 and L3, L4, and L5. The reflected light from M3 is split into two beams by a dichroic filter (DF): The beam with the wavelengths in the 0.4– 0.7 μm range is reflected to the S-H WFS; the other beam, with wavelengths in the 0.7–0.9 μm range, is transmitted to the LCWFC. A polarized beam splitter (PBS) is utilized to obtain the linear
polarized light. The light reflected from the LCWFC goes to a rotatable mirror M6, which can switch the system to open loop for measuring the interaction matrix of the LCWFC. When M6 is rotated out, the light coming from the LCWFC goes to the S-H WFS and the interaction matrix can be measured. While M6 is rotated in, the light is reflected to a CCD camera for imaging. An ATS with an atmospheric coherence length of 0.78 mm was used to produce the atmospheric turbulence. The 2 m aperture of the telescope and the atmospheric coherence length of 10 cm were chosen to simulate the atmospheric turbulence. First, the distortion was measured by the S-H WFS; the wavefront is shown in Fig. 12 (a). Then, an open loop correction was performed with the interaction matrix measured by the conventional, TC, LS, and TCLS methods. The corrected images of the fiber bundle are shown in Fig. 12 (d)–(g). It can be seen that, after correction, the cores of the fiber can be resolved. However, the resolution of the image is different for different interaction matrices. To quantify the improvement of image quality, the radial average power spectra of the images were calculated as shown in Fig. 13. It can be seen that, with the correction, the power spectrum is closer to that of the
Fig. 12. Fiber bundle images: (a) Distorted wavefront; (b) Original image; (c) Before correction; (d) Correction with the conventional method; (e) Correction with the TC method; (f) Correction with the LS method; (g) Correction with the TCLS method.
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power spectra of the images, the degree of fidelity to the original image was ranked as TCLS4 LS4 TC4conventional method. Consequently, the TCLS is the best method to measure the interaction matrix of the diffractive LCWFC. Furthermore, although the response ability of the LCWFC was analyzed based on the effect of atmospheric turbulence on a large-aperture telescope, the TCLS method is also suitable for all LC AOSs that utilize a diffractive LCWFC to correct the distortions. Hence, the results of this work will be very helpful to improve the correction accuracy of all LC AOSs, especially the open-loop-controlled LC AOSs.
Acknowledgments
Fig. 13. Radially averaged power spectra of the images for different interaction matrices.
original image. Furthermore, the degree of the fidelity is ranked as TCLS4 LS4 TC4 conventional method. Meanwhile, the correction residual wavefront RMS errors are calculated with four methods: 0.221λ (conventional method), 0.106λ (TC method), 0.098λ (LS method), and 0.089λ (LSTC method). The Strehl Ratios (SRs) are 0.50, 0.71, 0.73, and 0.76, respectively. Hence, the TCLS method is the best method to measure the interaction matrix of the LCWFC.
4. Conclusions A novel method has been demonstrated to improve the accuracy of the interaction matrix of LCWFCs. Based on the response characteristics of diffractive LCWFCs, a TC method was first presented to decrease the response error of the LCWFC. By tuning the coefficients of the Zernike modes, the LCWFC can obtain an ideal response; then the error of the interaction matrix will be reduced. To validate this, a single Zernike mode correction experiment was performed and the results showed that the error of the interaction matrix of the TC method is reduced by a factor of 2.7, compared to the conventional method. Secondly, a TCLS method, which is a combination the TC and LS methods, was established to eliminate the random noise. A single Zernike mode correction experiment was performed and the results showed that the measuring error of the interaction matrix achieved with the TCLS method is reduced by a factor of 2.5, compared with the LS method by itself. Therefore, the TCLS method can not only improve the response accuracy of the LCWFC, but can also eliminate the random noise. Finally, the performance of each method was evaluated with an open loop correction experiment. Based on the radial average
The paper is supported by State Key Laboratory of Applied Optics, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences and National Natural Science Foundation of China (Nos. 11174274 and 11174279).
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Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
A novel method of measuring the interaction matrix for diffractive liquid crystal wavefront correctors Zhaoliang Cao a,n, Lina Shao a,b, Yukun Wang a, Quanquan Mu a, Lifa Hu a, Xingyun Zhang a, Li Xuan a a State Key Laboratory of Applied Optics, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun, Jilin 130033, China b University of Chinese Academy of Sciences, Beijing 100049, China
art ic l e i nf o
a b s t r a c t
Article history: Received 20 January 2016 Received in revised form 14 June 2016 Accepted 15 June 2016 Available online 21 June 2016
The interaction matrix significantly affects the correction accuracy of wavefront correctors. For diffractive liquid crystal wavefront correctors (LCWFCs), the diffraction wavefront error must be taken into account to improve the accuracy of the interaction matrix. In this paper, a tunable Zernike polynomial coefficient method is demonstrated that decreases the effect of the diffraction wavefront error on the interaction matrix. Moreover, to eliminate the effect of the random error, a least squares method is combined with the tunable coefficient method. Experimental results show that, with the combined method, the error of the interaction matrix is decreased by a factor of 2.5 compared to that when the least squares method is used alone. Furthermore, an open loop adaptive correction experiment was performed and a considerable improvement of the image resolution was obtained by the novel method. Therefore, this method is very useful for liquid crystal adaptive optics systems to acquire the high correction accuracy. & 2016 Elsevier B.V. All rights reserved.
Keywords: Adaptive optics Atmospheric correction Liquid-crystal devices
1. Introduction Adaptive optics systems (AOSs) are widely used in groundbased large-aperture telescopes to overcome the effects of the atmospheric turbulence [1,2]. With the advantages of high spatial resolution, low cost, low power consumption, and large phase modulation [3,4], the liquid crystal wavefront corrector (LCWFC) is one of the most attractive wavefront correction devices, and liquid crystal adaptive optics systems (LC AOSs) have been investigated by many researchers [5–7]. To avoid energy loss caused by dispersion of the LCWFC, open loop control is utilized for the LC AOSs [8]. More importantly, the open loop control is an essential technique for multi-object adaptive optics (MOAO), which is an advanced method to enlarge the field of view of the AOSs [9]. The interaction matrix describes the control relationship between the wavefront corrector and the wavefront sensor (WFS), is used to derive the wavefront corrector signal from the WFS measurements during AO correction [10]. If the interaction matrix was not exact, the control relationship will be nonlinear, resulting in a decrease of the correction accuracy. To facilitate the open loop control, the interaction matrix of the WFC, which determines the correction accuracy, must be precise. n
Corresponding author. E-mail address: [email protected] (Z. Cao).
http://dx.doi.org/10.1016/j.optcom.2016.06.044 0030-4018/& 2016 Elsevier B.V. All rights reserved.
The Zernike polynomial modal method has been applied to LC AOSs to measure the interaction matrix of the LCWFC [10]. A set of Zernike modes is sent to the LCWFC in turn and the responded wavefronts are measured by a Shack–Hartmann wavefront sensor (S-H WFS). The measured information is recorded in a matrix called the interaction matrix. To eliminate the measurement noise, a least squares (LS) method has been presented by Zhang et al. [11]; although they were able to eliminate the measurement noise, they did not consider the response ability of the LCWFC for different Zernike modes. Because the phase wrapping or kinoform technique has been used to extend the correction magnitude [12], the LCWFC is a diffractive type device. The response error of the LCWFC is generated in the phase wrapping due to quantization. With the mode number of the Zernike polynomial increasing, the Zernike wavefront is complicated and generates the high phase gradient where the quantization level drops. Thus, the response abilities of the LCWFC to different Zernike polynomial are different. As the identical coefficients are used in the conventional measurement method, the measurement accuracy of the interaction matrix will be decreased. Therefore, in this paper, the response ability of the LCWFC will be first investigated for different Zernike modes. Then, a tunable coefficient (TC) method will be presented to make the response abilities of the LCWFC identical and reduce the response error of the LCWFC. Finally, a novel method, which combines the TC method with the LS method, will be introduced to improve the measurement accuracy of the interaction matrix.
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2. Tunable coefficient method 2.1. Response ability of the LCWFC According to diffraction theory, the response error of the LCWFC can be expressed as [13]
ε=
λ 2 3N
(1)
where λ is the wavelength, and N is the quantization level. Therefore, the response ability of the LCWFC is determined by the quantization level. To correct the wavefront with large magnitude (Fig. 1(a)), the distorted wavefront (Fig. 1((b)) should be wrapped into 1λ, and then be quantified with corresponding LC pixels as shown in Fig. 1(c). It is shown that, for the 35th Zernike polynomial mode with 64 64 pixels, the quantization level is different at different pixel positions. The quantization level is low where the curve has a large gradient. Furthermore, the quantization level reaches a minimum at the position with the maximum gradient. The minimum quantization levels of each Zernike polynomial mode are different. According to Eq. (1), the minimum quantization level determines the response error of the LCWFC. Before calculating the minimum quantization levels of each Zernike polynomial mode, we first introduce the meaning of the quantization level. The quantization level means the number of the LC pixels to realize 1λ phase modulation. For a large gradient, the 1λ phase modulation contains a smaller number of the LC pixels. Thus, the essence of quantization level is to determine the number of pixels with different gradient wavefront in 1λ phase modulation. It may be expressed as
N ( x , y) =
2c j P
1 → × G (x, y)
(2)
where cj is the coefficient of the jth Zernike mode, P is the pixel → number of LCWFC along one dimension, and G (x, y ) is the gradient of the Zernike polynomial. Because a Zernike mode of the coefficient of 1 has a range of 1 to 1, the 2 in the denominator of Eq. (2) is the normalized factor to make the peak to valley (PV) value equal to 1. To calculate the quantization level conveniently, the Zernike polynomial is expressed in a rectangular coordinate system:
⎧ ⎪ Z (x, y) ⎪ j m /2 ⎪ m! t ⎪ = R nm( x, y) ∑ ( −1) xm − 2t y2t when j is even 2 t ( )! ⎪ t=0 ⎪ ⎨ ⎪ Zj(x, y) ⎪ m /2 − 1 ⎪ m! t = R nm( x, y) ∑ ⎪ ( −1) xm − 2t − 1y2t + 1when j 2t + 1)! ⎪ t=0 ( ⎪ ⎩ is odd
(3)
where
R nm(x, y) (n − m) /2
=
∑ s= 0
n − m − 2s ( − 1)s (n − s )! (x2 + y2 ) 2 s ![(n + m)/2 − s ]![(n − m)/2 − s ]!
(4)
Thus, the gradient of the each Zernike polynomial mode may be expressed as
∂Zj → ∂Zj→ → G (x, y) = ∇Zj = x + y = ∂x ∂y
⎛ ∂Zj ⎞2 ⎛ ∂Zj ⎞2 ⎜ ⎟ +⎜ ⎟ ⎝ ∂x ⎠ ⎝ ∂y ⎠
(5)
To calculate the gradient numerically, the unit circle is discretized with P P points. Hence, the values of x and y will be in the range from 1 to 1, and then, the step size is 2/P. The gradient of Zj at a certain point can be computed by substituting Eq. (3) and the values of x and y at this point into Eq. (5). Thus, the distribution of the gradient of each Zernike mode can be obtained. Based on Eq. (3) to Eq. (5), the quantization level can be computed for different Zernike modes. Fig. 2 shows the calculated results for the 4th, 11th, and 28th modes. It can be seen that the maximum quantization level decreases when the Zernike mode is increased. Moreover, as shown in the enlarged inset, the trend of the minimum quantization level is similar to that of the maximum quantization level. In the conventional Zernike polynomial method, the coefficient is selected to be 1; hence, we also use the coefficient of 1 to perform the analysis. As the response error is limited by the minimum quantization level, the maximum response error and the minimum quantization level are calculated according to Eqs. (1) and (2) for different Zernike modes as shown in Fig. 3. It can be seen that the quantization level drops drastically from 128 to 6 in the first 25 modes, and then decreases slowly (Fig. 3(a)). The corresponding maximum response error is shown in Fig. 3(b), and the maximum value reaches to 0.25λ. This response error is large, and must be eliminated to obtain high correction accuracy.
Fig. 1. Phase distribution for the 35th mode Zernike: (a) Wavefront; (b) Phase profile along the x dimension; (c) Wrapped into 1λ and quantified.
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Fig. 4. Tunable coefficient as a function of the Zernike mode.
Fig. 2. Quantization level distributions along the x-dimension for different Zernike modes.
2.2. TC method From the above analysis, we can see that, because the minimum quantization level is different for different Zernike modes, the response error must also be different. Therefore, to achieve the same response ability, the minimum quantization level must be the same for different Zernike modes. Furthermore, to achieve high response accuracy, the minimum quantization level should exceed a certain threshold. According to Eq. (2), to increase the minimum quantization level, the number of pixels of the LCWFC should be increased, or else the coefficients of the Zernike polynomials should be decreased. For a given LCWFC, the coefficients of the Zernike polynomials have to be tuned to obtain a suitable minimum quantization level: we call this the tunable coefficients (TC) method. To adjust the coefficient, the minimum quantization level should be first calculated for different Zernike modes by using the coefficient of 1. Then, the coefficient is tuned by:
⎧ P ⎪ → ⎪ 2 N cj = ⎨ t ar Gmax ⎪ ⎪ ⎩1
Nmin < Nt ar Nmin ≥ Nt ar
(6)
where Gmax is the maximum gradient of the Zernike polynomial, Nmin is the minimum quantization level, and Ntar is the expected
quantization level. For Ntar ¼10, P ¼256, and 200 Zernike modes, the tuned coefficients are calculated as shown in Fig. 4. With these tuned coefficients, we assume that the response ability of the LCWFC is the same for different Zernike modes. Furthermore, as Ntar ¼10, the response error is as small as 0.029λ, and then the measurement accuracy of the interaction matrix will be high. 2.3. Validation experiment To evaluate the validity of the TC method, a single Zernike mode correction experiment was performed for 65 Zernike modes. Assuming simulating the conditions of a 2 m telescope and an atmospheric coherence length of 10 cm, the coefficients of each Zernike mode in atmospheric turbulence are calculated according Noll's atmospheric turbulence model [14] as shown in Fig. 5. Each coefficient is utilized as the magnitude of the distorted wavefront for a single Zernike mode correction experiment. Consequently, the compensated wavefront for different Zernike modes can be calculated with the produced coefficients, and then sent to the LCWFC. The responded wavefront can be measured with an S-H WFS. For example, a defocus term with the coefficient of 1.1 was selected as the distorted wavefront as shown in Fig. 6(a). The compensated wavefront (Fig. 6(b)) calculated based on the measured interaction matrix was sent to the LCWFC and the responded wavefront (Fig. 6(c)) was measured by the S-H WFS. Thus, the correction error could be computed with the difference between Fig. 6(a) and (c). Therefore, the coefficient relative error may be defined as
Fig. 3. Response errors for different Zernike modes: (a) The minimum quantization levels; (b) The maximum response errors.
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Fig. 7. Optical setup of the validation experiment.
Fig. 5. Zernike coefficients calculated with Noll's model under the conditions of a 2 m telescope and an atmospheric coherence length of 10 cm.
σj =
aˊ j − a j aj
(7)
where αˊj is the calculated coefficient of the jth Zernike mode; and αj is the coefficient of the responded wavefront. The optical setup of the experiment is shown in Fig. 7. The beam emitted from a point light source is collimated by lens1, and then goes to a polarizer. The polarized beam is zoomed in by the combination of lens2 and lens3, and then is reflected by the LCWFC. The reflected light is zoomed out by the combination of lens3 and lens4, and is then detected by the S-H WFS. The polarizer ensures that the polarization direction of the light is parallel to the alignment direction of the LC molecules. The LCWFC was purchased from Boulder Nonlinear Systems Company; it has 256 256 pixels, a 24 μm pixel size, and a 200 Hz frame rate. The S-H WFS was fabricated in our laboratory; it has a 15 15 microlens array, 4.3 mm aperture size, and 2.5 kHz frame rate. To perform a comparison, the correction can also be performed with the interaction matrix measured with the conventional Zernike polynomial method. Fig. 8 shows the relative coefficient correction error for the TC and conventional methods. The upper ■ line denotes the tuned coefficients. Because the conventional interaction matrix measuring method utilizes the identical coefficient of 1, the coefficient relative error is nearly the same for two methods when the tuned coefficients equal to 1, but the coefficient relative error of the TC method decreases significantly for the coefficients less than 1. Furthermore, it can be seen that the coefficient relative error of
Fig. 8. Coefficient relative error as a function of the Zernike mode.
the TC method is nearly the same for different Zernike modes and the standard error is 0.005. This illustrates that, with TC method, the same response accuracy may be obtained for different Zernike modes. On the other hand, the response accuracy is very different with different Zernike modes for the conventional method. Therefore, the TC method can improve the measurement accuracy of the interaction matrix. Furthermore, the mean coefficient relative errors of the TC and conventional method are 0.08λ and 0.03λ, respectively. Consequently, with the TC method, the measuring error of the interaction matrix is decreased by a factor of 2.7 as compared with the conventional method. 3. Tunable coefficient least squares method 3.1. Principle We have mentioned that the LS method can improve the measurement accuracy of the interaction matrix [10] by
Fig. 6. Simulation of the correction process: (a) Distorted wavefront; (b) Compensated wavefront; (c) Responded wavefront.
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eliminating the random noise. However, the response ability of the LCWFC is not considered for this method. Firstly, the coefficient of each Zernike mode is varied from 0.5 to þ0.5. Secondly, every wavefront sent to the LCWFC is randomly produced with the combination of certain Zernike modes. Thus, the magnitude of the random wavefront possibly exceeds the response ability of the LCWFC and the response error of the interaction matrix will be increased. The TC method can improve the response accuracy of the LCWFC, but cannot eliminate the random noise as shown in Fig. 9. It demonstrates that, for the measured slope of the x-tilt mode, the results of the LS method are much closer to the theoretical curve while those measured by the TC method deviate significantly from the theoretical values. Consequently, to improve the measurement accuracy of the interaction matrix, the TC method should be combined with the LS method; we call this the TCLS method. To implement the TCLS method, it is necessary to tune the coefficient of each Zernike mode and eliminate the random noise simultaneously. To achieve this, the response of each Zernike mode is first measured with a random coefficient, whose range is constrained by the calculated tuning result. Then different Zernike modes are measured in turn by the same method. Theoretically, assuming that the measured slope vector of the jth Zernike mode is sj and the corresponding measurement noise is snj, the interaction vector of the jth Zernike mode may be expressed as
sj = cj × Mj + snj .
(8)
here sj = [ sjx1 sjx2 ... sjxp sjy1 sjy2 ... sjyp ] and p is the number of the microlens. Because the LS method is used, the coefficient of Zernike mode cj should be a variable cj(i) that is produced randomly. Assuming that the number of random variables to be produced is k, cj(i) will be in the range from cj to cj (i ¼1,2,…, k); here, cj is obtained by Eq. (6). Assuming the random noise of the jth Zernike mode is snj, the following equations may be obtained:
⎧ s (1) = c (1) × M + s (1) j j nj ⎪ j ⎪ s (2) = c (2) × M + s (2) j ⎨ j j nj . ⎪ ... ⎪ s (k ) = c (k ) × M + s (k ) j j nj ⎩ j Eq. (9) may be expressed in a matrix form:
Fig. 9. Measured slopes by TC method and LS method.
(9)
365
Sj = Cj × Mj + Snj,
(10)
(1) (1) (1) (1) ⎡ s(1) ⎤ ⎡ s(1) jx1 s jx1 ... s jxp s jy1 s jy2 ⎢ j ⎥ ⎢ ⎢ (2) ⎥ ⎢ (2) (2) (2) (2) (2) where Sj = ⎢ s j ⎥ = ⎢ s jx1 s jx1 ... s jxp s jy1 s jy2 ... ⎢ ⎥ ⎢ ... ... ⎢ s (k ) ⎥ ⎢ s ( k ) (k ) (k ) (k ) (k ) ⎣ j ⎦ ⎢⎣ jx1 s jx1 ... s jxp s jy1 s jy2 ⎡ c (1) ⎤ ⎡ s (1) ⎤ ⎢ j ⎥ ⎢ nj ⎥ ⎢ c (2) ⎥ ⎢ (2) ⎥ j Cj = ⎢ ⎥, Snj = ⎢ snj ⎥. ... ⎢ ⎥ ⎢ ... ⎥ ⎢ c (k ) ⎥ ⎢ s (k ) ⎥ ⎣ j ⎦ ⎣ nj ⎦ The objective is to achieve Mj, and its least squares value may be solved by [15]:
(
˜ = C+ × S = CT × CT × C M j j j j j j
−1
)
⎤ ... s(1) jyp ⎥ ⎥ ... s(2) jyp ⎥, ⎥ ⎥ ... s(k ) jyp ⎥⎦
estimated
× Sj.
(11)
Assuming that the first q Zernike modes are used to measure the interaction matrix, the interaction matrix of the LCWFC may be expressed as
⎡M ˜ ⎤ ⎢ 1⎥ ⎢⋮ ⎥ ˜ ⎥ ˜ = ⎢M M ⎢ j⎥ ⎢⋮ ⎥ ⎢ ˜ ⎥ ⎣ Mq ⎦
(12)
˜ , the LCWFC can correct the With the interaction matrix M distortions accurately. 3.2. Validation experiment The single Zernike mode correction scheme was also utilized to validate the TCLS method. The experimental setup was the same as that of Fig. 6. Furthermore, the first 65 Zernike modes were selected, with the same coefficients as shown in Fig. 5. To enable the comparison, the correction must also be performed with the LS method. Fig. 10 shows the coefficient relative errors for both the LS and TCLS methods. The coefficient relative error of each Zernike mode is calculated by Eq. (7) as shown in Fig. 10. As can be seen, the coefficient relative errors of the two methods are nearly the same for the first 10 modes, but the errors of the TCLS method are smaller for larger Zernike modes; this illustrates that, while the LS method is effective for the lower Zernike modes, the mean coefficient relative errors of the LS, TCLS, conventional, and TC
Fig. 10. Coefficient relative error as a function of the Zernike mode.
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Fig. 11. Optical layout for open loop correction experiment.
methods are 0.05λ, 0.02λ, 0.08λ, and 0.03λ, respectively. Consequently, with the TCLS method, the measuring error of the interaction matrix is decreased by a factor of 2.5, compared to the LS method. 3.3. Open loop correction experiment with an atmospheric turbulence simulator To evaluate the improvement of the accuracy of the interaction matrix, simulated atmospheric turbulence was produced and an open loop correction was performed in the laboratory. The optical layout is shown in Fig. 11. The light emitted from the fiber bundle is reflected and collimated by M1 and L1 respectively. The collimated beam goes through an atmospheric turbulence simulator (ATS), and is then zoomed out by the combination of L2 and L3, L4, and L5. The reflected light from M3 is split into two beams by a dichroic filter (DF): The beam with the wavelengths in the 0.4– 0.7 μm range is reflected to the S-H WFS; the other beam, with wavelengths in the 0.7–0.9 μm range, is transmitted to the LCWFC. A polarized beam splitter (PBS) is utilized to obtain the linear
polarized light. The light reflected from the LCWFC goes to a rotatable mirror M6, which can switch the system to open loop for measuring the interaction matrix of the LCWFC. When M6 is rotated out, the light coming from the LCWFC goes to the S-H WFS and the interaction matrix can be measured. While M6 is rotated in, the light is reflected to a CCD camera for imaging. An ATS with an atmospheric coherence length of 0.78 mm was used to produce the atmospheric turbulence. The 2 m aperture of the telescope and the atmospheric coherence length of 10 cm were chosen to simulate the atmospheric turbulence. First, the distortion was measured by the S-H WFS; the wavefront is shown in Fig. 12 (a). Then, an open loop correction was performed with the interaction matrix measured by the conventional, TC, LS, and TCLS methods. The corrected images of the fiber bundle are shown in Fig. 12 (d)–(g). It can be seen that, after correction, the cores of the fiber can be resolved. However, the resolution of the image is different for different interaction matrices. To quantify the improvement of image quality, the radial average power spectra of the images were calculated as shown in Fig. 13. It can be seen that, with the correction, the power spectrum is closer to that of the
Fig. 12. Fiber bundle images: (a) Distorted wavefront; (b) Original image; (c) Before correction; (d) Correction with the conventional method; (e) Correction with the TC method; (f) Correction with the LS method; (g) Correction with the TCLS method.
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power spectra of the images, the degree of fidelity to the original image was ranked as TCLS4 LS4 TC4conventional method. Consequently, the TCLS is the best method to measure the interaction matrix of the diffractive LCWFC. Furthermore, although the response ability of the LCWFC was analyzed based on the effect of atmospheric turbulence on a large-aperture telescope, the TCLS method is also suitable for all LC AOSs that utilize a diffractive LCWFC to correct the distortions. Hence, the results of this work will be very helpful to improve the correction accuracy of all LC AOSs, especially the open-loop-controlled LC AOSs.
Acknowledgments
Fig. 13. Radially averaged power spectra of the images for different interaction matrices.
original image. Furthermore, the degree of the fidelity is ranked as TCLS4 LS4 TC4 conventional method. Meanwhile, the correction residual wavefront RMS errors are calculated with four methods: 0.221λ (conventional method), 0.106λ (TC method), 0.098λ (LS method), and 0.089λ (LSTC method). The Strehl Ratios (SRs) are 0.50, 0.71, 0.73, and 0.76, respectively. Hence, the TCLS method is the best method to measure the interaction matrix of the LCWFC.
4. Conclusions A novel method has been demonstrated to improve the accuracy of the interaction matrix of LCWFCs. Based on the response characteristics of diffractive LCWFCs, a TC method was first presented to decrease the response error of the LCWFC. By tuning the coefficients of the Zernike modes, the LCWFC can obtain an ideal response; then the error of the interaction matrix will be reduced. To validate this, a single Zernike mode correction experiment was performed and the results showed that the error of the interaction matrix of the TC method is reduced by a factor of 2.7, compared to the conventional method. Secondly, a TCLS method, which is a combination the TC and LS methods, was established to eliminate the random noise. A single Zernike mode correction experiment was performed and the results showed that the measuring error of the interaction matrix achieved with the TCLS method is reduced by a factor of 2.5, compared with the LS method by itself. Therefore, the TCLS method can not only improve the response accuracy of the LCWFC, but can also eliminate the random noise. Finally, the performance of each method was evaluated with an open loop correction experiment. Based on the radial average
The paper is supported by State Key Laboratory of Applied Optics, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences and National Natural Science Foundation of China (Nos. 11174274 and 11174279).
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