More Zernike modes&#x2019; open-loop measurement in the sub-aperture of the Shack&#x2013;Hartmann wavefront sensor

Zhaoyi Zhu; Quanquan Mu; Dayu Li; Chengliang Yang; Zhaoliang Cao; Lifa Hu; Li Xuan

doi:10.1364/OE.24.024611

1. Introduction

Adaptive optics (AO) is now widely applied in many research areas, such as astronomy observation through large aperture telescopes, optical microscopy and free space optical communications [1–3]. The wavefront sensor is one of the basic elements of an AO system (AOS). Shack-Hartmann Wavefront Sensor (SHWFS) is a type of conventional pupil plane wavefront sensor and its characteristics of simple setup, high accuracy and high processing speed make it popular in AOS [4–7].

The SHWFS senses the wavefront through a sub-aperture method. The turbulent wavefront is sampled by the lenslet array in front of the sensor. The sampled sub-wavefronts are considered as planes and the slopes are used to reconstruct the whole pupil wavefront. The sampled size of the sub-aperture determines the precision of the wavefront reconstruction. When aiming for atmospheric turbulence compensation in AOS, the diameter d of the sub-aperture is designed based on a statistic parameter, the atmospheric coherent length denoted as r₀ and d is typically set to be equal to r₀ [8]. In this case, the number of the sub-apertures is determined to be (D/r₀)² where D is the diameter of the telescope aperture, and the number of Zernike modes used in the wavefront reconstruction is also determined correspondently. Besides, the total energy collected in the telescope aperture is divided into sub-apertures, resulting in little energy contained in a focal spot which may be easily lost in the noise of the CCD detector. It means that the number of the sub-apertures limits the dim targets that can be sensed. Therefore, it needs more sub-apertures to achieve a better compensation, but the demand for sensing dim targets limits the number of the sub-apertures used. r₀ varies in real-time with the randomly changing atmospheric turbulence. When the real r₀ is smaller than the one applied in the design, the sub-wavefront cannot be seen as a plane that the higher order modes cannot be measured. The higher modes in the sub-aperture can also change the energy distribution of the focal spot, resulting in a wrong slope calculation. The above discussions present the factors affecting the efficiency of the SHWFS, and it has been confirmed that the plane sub-wavefront assumption is the main constraint of the sensing ability of the SHWFS.

Another type of wavefront sensing technique is the focal plane technique, such as the phase retrieval (PR) algorithm and the phase diversity (PD) algorithm by Gonsalves [9,10]. The focal plane technique derives the wavefront in the aperture from the focal plane spots or images. The intensity distribution of the focal spot is the square of the Fourier transform of the turbulent wavefront in the aperture and the image is degraded by the distorted spot. For large aperture telescopes, the aberrations are complex because of the atmospheric turbulence, so to achieve high sensing precision, the iterative process is needed for PR and PD, and they are limited by their slow data processing speed and still not used in the real-time AOS. Gonsalves proposed a small phase estimation method based on PD where the intensity distribution of the spot is described to be in a linear relation with the small aberration in the aperture [11]. Two measurements were needed in this method. Meimon et al [12], modified this method and proposed the linearized focal-plane technique (LIFT) in which single measurement was realized. They applied LIFT in the laser guide star based AOS (LGS-AOS) for the total tip/tilt and defocus sensing, but the iteration remains necessary to enlarge the linearity range [12–14].

The sampled wavefront in SHWFS is much simpler because of the division of sub-apertures, and for point-like astronomy target, the focal spot of each sub-aperture can be treated as diffraction spot only affected by the sub-wavefront. The focal plane technique then can be applied as the sub-wavefront estimation method, and then more information can be obtained in the sub-apertures, thus improving the efficiency of the SHWFS. Several problems still exist and limit the application of the focal plane technique in the SHWFS. First, considering all the methods discussed, the iterative method is applied to enlarge the sensing range, which is complicated and unsuitable for open-loop systems such as the liquid-crystal-on-silicon (LCOS) device-based AOS [15,16] and the multi-conjugation AOS. Second, the method of determining the introduced small phase remains unclear and it has been confirmed that a symmetrical phenomenon exists for the introduced mode, which may result in a wrong estimation. Third, the energy is finally collected by the discrete pixels, thus the sampling can affect the precision and the sensing range of the focal plane technique [17], which is not discussed yet.

In this paper, a sub-wavefront estimation method based the focal plane technique is proposed and applied in the SHWFS, in which five Zernike modes from Z2 to Z6 following Noll’s ordering [18] are used for estimation. This paper focuses on the analysis of the influence of the related parameters including the size of the diffraction spot, the introduced phase with its set amplitude and the pixel number for sampling in each sub-aperture, and the optimization of these parameters in order to improve the efficiency of the proposed method. After the analysis and the optimization, a modified SHWFS is simulated to calibrate the improvements brought by the new method. As discussed above, open-loop measurement is to be realized. More Zernike modes will be used in the whole pupil wavefront reconstruction and lower residual errors can be achieved.

2. Sub-wavefront sensing in SHWFS

2.1 Theoretical description

In Fourier optics, the energy distribution of the diffraction spot is the square of the Fourier transform of the generalized pupil function that includes the aberrations in the pupil. The distribution is not in a linear relationship with the aberrations, and a phase indetermination appears because of the second power function [9–14,19]. Based on the small phase estimation method proposed by Gonsalves and LIFT by Meimon, a small even phase is introduced, and the theoretical descriptions of the focal plane technique are as follows:

I (x, y) = I_{0} \cdot {| ℱ {p (ξ, η) \cdot \exp [j \cdot (W (ξ, η) + ϕ_{o f f})]} |}^{2}

where I(x, y) is the intensity distribution function in the focal plane of coordinates (x, y), I₀ is the total energy collected in the spot,

ℱ

stands for the Fourier transform, p(ξ, η) is the pupil function which consists of 0 and 1, W(ξ, η) is the unknown phase in the pupil plane of coordinates (ξ, η) and is decomposed by the Zernike modes as

W (ξ, η) = \sum_{k} a_{k} \cdot Z_{k} (ξ, η)

, and

ϕ_{o f f}

is the known even phase offset which helps solve the phase indetermination.

Then, I(x, y) can be rewritten and linearized by the first order Taylor expansion as follows:

I (x, y) / I_{0} = I_{o f f} + \sum_{k} a_{k} \cdot I^{'} (Z_{k} + ϕ_{o f f}),

where I(x, y)/I₀ is the normalized intensity distribution and I_off is the normalized intensity distribution with only the phase offset.

I^{'} (Z_{k} + ϕ_{o f f})

is defined as

I^{'} (Z_{k} + ϕ_{o f f}) = \frac{I (Δ a_{k} \cdot Z_{k} + ϕ_{o f f}) - I (- Δ a_{k} \cdot Z_{k} + ϕ_{o f f})}{2 \cdot Δ a_{k}},

where

I (Δ a_{k} \cdot Z_{k} + ϕ_{o f f})

is the intensity distribution of the phase consisting of the kth Zernike mode at

Δ a_{k}

and the phase offset, and so is

I (- Δ a_{k} \cdot Z_{k} + ϕ_{o f f})

with the kth Zernike mode at

- Δ a_{k}

.

Δ a_{k}

is as small as 0.1 rad RMS and k is from 2 to 6.

In practice, the energy is collected by the discrete pixels of the focal plane CCD, and it’s convenient to treat the discrete intensity pattern as a vector consisting of the pixel signals. Let I(x, y)/I₀-I_off be the vector ΔI[p], H(A_off) be the matrix whose element at [p, k] is the pth pixel signal $I^{'} (Z_{k} + ϕ_{o f f})$ of the kth Zernike mode, and A is the vector of the decomposed coefficient a_k. Thus, Eq. (3) can be rewritten as:

Δ I = H (A_{o f f}) \cdot A,

where H can be defined as the sub-aperture interaction matrix (IM) for the coefficients estimating of the unknown phase, and the least square solution of A is:

A = H^{†} (A_{o f f}) \cdot Δ I,

where

H^{†} (A_{o f f})

is the pseudo-inverse matrix of

H (A_{o f f})

.

The coefficients of the sub-wavefront can be estimated by the proposed method, and the whole pupil wavefront is then reconstructed from these estimated coefficients of all the sub-apertures [19,20]. The linear relation of the whole pupil wavefront and the coefficients measured by SHWFS is obtained by the reconstruction matrix. Each column vector of the reconstruction matrix is the SHWFS measurement of each Zernike mode. The elements of each vector are the local measurements of five Zernike coefficients as [a_2,i, a_3,i, a_4,i, a_5,i, a_6,i] calculated in each sub-aperture. For the nth Zernike mode, the measured vector Y with np sub-apertures can be expressed as:

Y_{A 5}^{n} = {[a_{2, 1}, a_{3, 1}, a_{4, 1}, a_{5, 1}, a_{6, 1}, \dots, a_{2, n p}, a_{3, n p}, a_{4, n p}, a_{5, n p}, a_{6, n p}]}^{T} .

Then the reconstruction matrix can be

[Y_{A 5}^{2}, Y_{A 5}^{3}, \dots Y_{A 5}^{n z}]

denoted by M for the modified SHWFS with nz as the number of Zernike modes used for reconstruction. Finally the coefficients of the whole pupil wavefront can be calculated by M as:

A = M^{†} \cdot Y,

where A is the vector of the pupil wavefront coefficients, Y is the measurement of the estimating wavefront and

M^{†}

is the pseudo-inverse matrix of M.

2.2 Related parameters analysis

As discussed in the theoretical description, the parameters affecting the intensity distribution of the focal spot will have an effect on the proposed sub-wavefront estimation method. The phase offset should be even, such as the fourth focus, the fifth and sixth astigmatisms of the Zernike modes [18]. The size of the spot collected by the CCD is enlarged for the introduction of the phase offset. Considering that the crosstalk between the nearby sub-apertures must be avoided, the amplitude of the phase offset has a limit. In this paper, the Z4 focus is chosen as the phase offset because it keeps the spot round, which provides a larger limit for the set amplitude. In addition, the focus is easy to introduce by setting the CCD off the focal plane of the lenslets. Different pixel numbers in a sub-aperture can alter the discrete intensity distribution of the diffraction spot, which changes the relation between the sub-wavefront and the discrete intensity distribution. The size of the diffraction spot, the set amplitude of the chosen Z4 offset and the pixel number in each sub-aperture affect the accuracy of the proposed method. In addition, the noise of the detector is another influence factor, which should be considered. However, we focus on the optimization of the related parameters in this paper, and high intensity flux is provided in the experiments to decrease the influence of the noises.

3. Parameter analysis and optimization

We have just qualitatively described the influence of the related parameters in Section 2.2. In this section, we realize a numerical simulation of the sub-wavefront estimation method for the analysis and optimization of the related parameters. The accuracy of calculation and the linearity range are analyzed with each group of parameters. The error defined as $E = \sum_{k} {(c_{k, c a l} - c_{k, r e a l})}^{2}$ is used to estimate the accuracy of the coefficients calculation. $c_{k, c a l}$ is the kth coefficient calculated, and $c_{k, r e a l}$ is the kth real coefficient which is the decomposed coefficients of the sub-wavefront with the five Zernike modes.

3.1 Numerical simulation

The numerical simulation is based on the discrete Fourier transform (DFT). In the simulation, the sub-wavefront is expressed in 48 × 48 pixels, and the sub-wavefront is zero-padded into different pixel number to generate different sizes of diffraction spots. The intensity distribution of the spot is then computed from the zero-padded sub-wavefront through DFT. We pick out the center 48 × 48 pixels as the CCD collected signals corresponding to a sub-aperture because of the structure of the lenslet array, and then re-bin those signals into 24 × 24, 16 × 16, 12 × 12, 8 × 8, 6 × 6 and 4 × 4 pixels to simulate the practical sampling in the sub-window. The size of the diffraction spot is described by the ratio of the lenslet diameter over the size of the airy spot denoted as k_d. From the custom parameters of the lenslet array, we list a number of k_d from 6.0 to 20.0. For each k_d, the maximum permitted amplitude of the Z4 offset is pre-determined and shown in Table 1, in which the crosstalk has been avoided.

Table 1. Maximum permitted amplitude of the Z4 offset for the listed k_d

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The sub-aperture IM is first built with the five Zernike modes based on a group of parameters as k_d, the set amplitude of the Z4 offset and the pixels number in each sub-aperture. Then both the diffraction spot of the estimated phase with the phase offset and the spot with only the phase offset are calculated. The coefficients of the estimated phase are calculated by the sub-aperture IM and the difference of the normalized diffraction spots based on Eq. (5). The estimated phases in the simulations are generated with a single mode of the five at different amplitudes or with the five modes at random amplitudes.

3.2 Linearity range changing with the pixel number in each sub-aperture

We first analyze how the pixel number in each sub-aperture effects on the linearity range and precision of coefficient calculation. The results are shown in Fig. 1, in which the sub-wavefront is generated with the Zernike mode Z2 and the size of the diffraction spot is kept the same. The range of linearity changes with the pixels number applied in the sub-window. Fewer pixels lead to a larger linearity range, but an aliasing phenomenon appears with fewer pixels, increasing the coefficients of the Zernike modes Z4 and Z6. Therefore, we choose the pixel number in each sub-aperture as 4 × 4, 6 × 6 or 8 × 8, which also lessens processing time and noise influence. The aliasing is considered to be decreased by optimizing the set amplitude of the Z4 offset.

Fig. 1 Linearity range changes with the pixel number in a sub-aperture, analyzed with Z2.

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3.3 Linearity range changing with the set amplitude of the Z4 offset

The influences of the amplitude of the Z4 offset are analyzed in this section. Different coefficients for the single testing mode, 1.0, 2.0, 3.0 rad RMS for the Zernike modes Z2 and Z3, 0.5, 1.0, 1.5 rad RMS for the Zernike modes Z4 to Z6, are analyzed and compared. The calculated coefficients vary with the set amplitude and the aliasing also appears. Figure 2 shows the results with that k_d is 13.0 and 4 × 4 pixels are set in each sub-aperture. For each k_d, an optimum amplitude of the Z4 offset exists to achieve the largest linearity range, and for k_d = 13.0, it is 5.5 rad RMS. The aliasing coefficients also change with the set amplitude and higher set amplitude leads to lower aliasing. In Fig. 3, we plot the linear relation of the set and calculated coefficients in the case of k_d = 13.0 and $ϕ_{o f f}$ = 5.5 rad RMS.

Fig. 2 Calculated coefficients change with the amplitude of the Z4 offset with k_d = 13.0 and 4 × 4 pixels in each sub-aperture, (a), (b), (c), (d) and (e) are for Z2, Z3, Z4, Z5 and Z6 respectively.

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Fig. 3 Linear relations of the set and calculated coefficients in case of k_d = 13.0, $ϕ_{o f f}$ = 5.5 rad RMS and 4 × 4 pixels in each sub-aperture, (a), (b), (c), (d) and (e) are for Z2, Z3, Z4, Z5 and Z6 respectively.

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3.4 Sensing error comparison with different parameters groups

In practice, the sub-wavefront randomly changes with the atmospheric turbulence. Here, we apply the random sub-wavefronts to optimize the discussed parameters. There is optimum amplitude of the phase offset for each k_d with different numbers of pixels in a sub-aperture as 4 × 4, 6 × 6 and 8 × 8 respectively. In Fig. 4, we conpare the local five modes sensing errors of the sub-wavefronts with different pixel numbers applied in the sub-aperture. For each situation we determine the k_d and the amplitude of the phase offset with the minimum sensing error for further analysis. Table 2 lists the chosen parameter groups. Figure 4 has just shown the comparisons in the situation of d/r₀ = 1.0. The linearity range has been proved to be as large as 3.0 rad RMS for tip/tilt and 2.0 rad RMS for the focus and astigmatism modes. The situations where d as 1.5 r₀ or 2 r₀ are analyzed as well, and results indicate that there are several groups of parameters making the sensing error less than 0.01 rad².

Fig. 4 Sensing errors of the local five modes by the new method, (a), (b) and (c) are the comparisons for 4 × 4, 6 × 6 and 8 × 8 pixels in a sub-aperture for d/r₀ = 1.0, respectively.

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Table 2. The k_d and the amplitude of the Z4 offset pairs with the minimum sensing error

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4. Experimental validation of the parameters optimization

We perform a series of experiments in laboratory to verify the conclusions of the theoretical analysis. The optical setup of the experiments is shown in Fig. 5 in which a single lenslet of the modified SHWFS is simulated. The Zygo laser interferometer works as the source and it can generate a plane wavefront. The light passes through the beam-splitter and the polar beam-splitter, then falls on the LCOS device vertically [14, 15]. The plane wavefront is modulated by the LCOS, which changes the intensity distribution of the diffraction spot. The diffraction spot is collected by the CCD which is located at the focal plane of the focal lens. In the experiments, the Z4 offset is introduced in two ways. One is generated by the LCOS accompanied with the turbulent wavefront, and the other is to move the CCD detector off the focal plane. In the second method, the introduced amplitude of the Z4 offset can be calculated from the distance of the CCD from the focal plane by $Δ z / 8 / {(f / d)}^{2}$ in peak-to-valley (PV) value. Δz is set from 1 to 8 cm, and the same amplitude from 0.72 rad RMS to 5.76 rad RMS of the Z4 mode is generated. A circle stop of 2 mm is set in front of the LCOS in the optical setup to control the beam-width. The focal length of the focal lens is 200 mm. The size of the airy spot is 0.1544 mm at the wavelength of the Zygo interferometer as 632.8 nm and k_d is computed to be 2 mm/0.1544 mm≈13.0.

Fig. 5 Optical setup for a single lenslet test.

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In the test, we record the diffraction spots with or without the analyzed distorted wavefront and the difference of the normalized diffraction spots is used to calculate the coefficients based on the sub-wavefront estimation method. The test results are all compared with the simulated ones with the same parameters.

4.1 Linearity range test for a signal mode

We test the linearity range with the experimental parameters. The linearity range is from –3.0 to 3.0 rad RMS for the Zernike modes Z2 and Z3 and from –2.0 to 2.0 rad RMS for the Zernike modes Z5 and Z6, as shown in Figs. 6 and 7, respectively. The symmetric phenomenon of the Z4 offset appears in the test as seen in the simulation and with the experimental parameters, the linearity range for the Z4 mode is approximately from –2.0 to 2.0 rad RMS.

Fig. 6 Linearity range for Z2 with the experimental parameters ranging from –3.0 to 3.0 rad RMS when the Z4 offset is 5.0 rad RMS

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Fig. 7 Linearity range for Z5 with the experimental parameters ranging from −2.0 to 2.0 rad RMS when the Z4 offset is 5.0 rad RMS

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4.2 Wavefront sensing at the optimum amplitude of the Z4 offset

Several wavefronts with the Zernike modes Z2 to Z6 at different coefficients are tested in the experiments in this section. These wavefronts are generated by the LCOS accompanied with the Z4 offset at the determined optimum amplitude. Then the set coefficients are calculated by the proposed method in the situations where the energy is collected by 4 × 4, 6 × 6 or 8 × 8 CCD pixels. The sensing error of the calculated coefficients compared with the real coefficients is computed and compared with the sensing error calculated in the simulations. Five random wavefronts are tested, and the results are listed in Table 3. Considering the noise of the CCD detector, the sensing errors in the tests and simulations for each wavefront are approximately the same and adequately small for sensing, thus confirming that the conclusions are reliable.

Table 3. Sensing errors in the five modes sensing test with the optimum parameters

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5. Whole pupil wavefront reconstruction with the optimum parameter group

In this section, we realize the whole pupil wavefront reconstruction based on Eqs. (6) and (7) to analyze the improvements brought by the optimized sub-wavefront estimation method. More Zernike modes could be used in the wavefront reconstruction because more information can be achieved in the sub-apertures. The reconstruction with the five modes estimated per sub-aperture is compared with that with only local slopes are estimated per sub-aperture. The number of Zernike modes used and the residual wavefront errors are analyzed for comparison. The proposed two reconstruction methods are realized and compared with a SHWFS of 7 × 7 sub-apertures.

5.1 Determining the number of Zernike modes for reconstructed

In this section, the coefficients of local five modes or the local slopes are calculated by the phase IM, which are the decomposed coefficients of the Z2 to Z6 Zernike modes, in order to avoid the sensing error of the proposed new method. In Fig. 8, (a) is the random phase analyzed to determine the number of Zernike modes used for reconstruction and (b) (c) (d) (e) are the residual phases under different conditions. In addition, (b) (c) are reconstructed with 35 modes and (d) (e) used 65 modes. Five modes are sensed in (b) (d) and only the slopes are sensed in (c) (e). As for (e), a SHWFS of 10 × 10 sub-apertures is simulated, which confirms that when only the slopes are sensed in the sub-aperture, at least 10 × 10 sub-apertures are needed to reconstruct 65 Zernike modes. The residual error of the reconstruction with different numbers of modes is calculated and compared in Fig. 9. As can be seen from the figure in the 7 × 7 apertures situation, 65 Zernike modes are reconstructed with high accuracy when the five modes are sensed per sub-aperture.

Fig. 8 Random phase analyzed (a) and the residual phases, (b) for 35 modes and five modes sensed, (c) for 35 modes and only slopes sensed (d) for 65 modes and five modes sensed, (e) for 65 modes and only slopes sensed, and a SHWFS of 10 × 10 sub-apertures is used in this case

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Fig. 9 Residual errors of the phase reconstruction with the 7 × 7 sub-apertures SHWFS

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5.2 Residual error analysis of the reconstruction with the modified SHWFS

The above analysis verifies that using 65 Zernike modes in the reconstruction can achieve the lowest residual error when the chosen five modes are measured in each sub-aperture. Then we build the new method IM for sub-wavefront sensing and realize the whole pupil wavefront reconstruction in which 65 Zernike modes are used and the residual error is calculated. The reconstruction with the sub-phase IM is realized which is the same with that in Section 5.1 and the residual error is calculated for comparisons. The results are shown in Fig. 10. The pupil wavefront reconstruction is also analyzed when the size of the sub-aperture is equal to 1.5 × r₀ or 2 × r₀, because the new method has been confirmed to sense random sub-wavefronts with high accuracy when d is equal to 1.5 × r₀ or 2 × r₀. The applied parameters are the optimum parameters, and the experiments with 4 × 4, 6 × 6 or 8 × 8 pixels per sub-aperture are compared in Table 4. The residual errors of 100 frames of random wavefronts calculated using the new method IM and phase IM are almost the same, and the mean residual error is lower than 0.2 rad², which is more efficient than the wavefront reconstruction using the traditional centroid method.

Fig. 10 The residual error of the reconstruction with the new method compared with the reconstruction with the phase IM, (a) (b) (c) are the situations in which d/r₀ = 1.0, 1.5 and 2.0 respectively

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Table 4. Residual error of the reconstruction using the phase IM compared with that of the reconstruction using the new method IM built with 4 × 4, 6 × 6 or 8 × 8 pixels per sub-aperture

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6. Conclusions

In this paper, we presented a sub-wavefront estimation method to sense five Zernike modes in each sub-aperture, increasing the efficiency of the wavefront sensing when applied in the SHWFS. The theory of the proposed method is presented and its related parameters are analyzed and optimized. After optimization, open loop measurement is realized.

For the sub-wavefront sensing, setting 4 × 4, 6 × 6 or 8 × 8 pixels per sub-aperture is feasible, which signifies low data processing burden and noise propagation. The linearity ranges of the Zernike modes Z2 and Z3 can reach 3.0 rad RMS, and those of Z4, Z5 and Z6 are 2.0 rad RMS after optimization. It’s concluded that our new method can sense the atmospheric turbulent phase of diameter larger than r₀ with good accuracy and for d = 2 × r₀, the sensing error can be as low as 0.01 rad². Results of the experiments are almost the same with the simulation having the same parameters, indicating that the optimization is reasonable and that the optimum parameter groups can be used in the SHWFS design.

Improvements brought by the new method for the whole pupil wavefront reconstruction are analyzed based on a 7 × 7 sub-apertures modified SHWFS. Sixty-five Zernike modes rather than only 35 modes are used in the whole pupil wavefront reconstruction and lower residual error is achieved as lower than 0.2 rad².

Overall, two benefits can be achieved. One advantage is that more Zernike modes can be sensed for the whole pupil wavefront, leading to a better compensation. The other advantage is that the energy is divided into fewer sub-apertures in which case the dimmer stars can be observed. Further analysis about the influences of the noises and the estimation algorithm under noises are needed for the realization and applications of this modified type of SHWFS. Besides, the analysis about the influence of the misalignment is also needed.

Funding

National Natural Science Foundation of China (NSFC) (61205021, 61475152, 61405194).

Acknowledgments

This work is supported by State Key Laboratory of Applied Optics, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences.

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k_d	6.0	7.0	8.0	9.0	10.0	11.0	12.0	13.0
amplitude (rad)	3.0	3.5	4.0	4.5	5.0	5.0	5.5	6.0
k_d	14.0	15.0	16.0	17.0	18.0	19.0	20.0
amplitude (rad)	6.5	7.0	7.5	7.5	8.0	9.0	9.5

k_d			6.0	7.0	8.0	9.0	10.0	11.0	12.0	13.0
$ϕ_{o f f}$	4 × 4 pixels		3.0	3.0	3.3	4.5	4.9	4.3	4.7	4.9
	6 × 6 pixels		2.4	2.7	3.9	3.4	3.7	3.9	4.2	4.6
	8 × 8 pixels		2.7	2.8	3.2	2.6	2.9	4.8	3.4	3.7
k_d			14.0	15.0	16.0	17.0	18.0	19.0	20.0
$ϕ_{o f f}$		4 × 4 pixels	5.3	6.9	7.1	7.5	8.0	7.9	8.4
		6 × 6 pixels	4.6	4.9	5.1	5.5	5.7	7.7	6.3
		8 × 8 pixels	5.9	6.4	6.5	7.1	4.6	7.4	5.0

Set coefficients (rad)	Test 1 (1.12 rad²) (0.5, −0.7, 0.3, 0.5, 0.2)		Test 2 (0.78 rad²) (−0.6, 0.1, −0.4, −0.4, 0.3)		Test 3 (5.12 rad²) (1.5, −0.7, 0.3, 1.5, 0.2)
	test	simulation	test	simulation	test	simulation
4 × 4 pixels	0.0223	0.0142	0.0240	0.0040	0.3048	0.5153
6 × 6 pixels	0.0171	0.0116	0.0130	0.0019	0.2087	0.2143
8 × 8 pixels	0.0081	0.0017	0.0063	0.0004	0.1431	0.0937
Set coefficients (rad)	Test 4 (5.78 rad²) (−1.6, 1.1, −0.4, −0.4, 1.3)		Test 5 (11.15 rad²) (−2.1, 1.8, 1.0, 0.9, −1.3)
	test	simulation	test	simulation
4 × 4 pixels	0.3190	0.1579	0.3020	0.5166
6 × 6 pixels	0.0466	0.0393	0.1239	0.2562
8 × 8 pixels	0.0503	0.0423	0.0979	0.0948

	d/r₀ = 1.0	d/r₀ = 1.5	d/r₀ = 2.0
Phase	0.0562	0.1794	0.1831
4 × 4 pixels	0.0617	0.2075	0.2209
6 × 6 pixels	0.0741	0.2929	0.3321
8 × 8 pixels	0.0657	0.2667	0.2856

k_d	6.0	7.0	8.0	9.0	10.0	11.0	12.0	13.0
amplitude (rad)	3.0	3.5	4.0	4.5	5.0	5.0	5.5	6.0
k_d	14.0	15.0	16.0	17.0	18.0	19.0	20.0
amplitude (rad)	6.5	7.0	7.5	7.5	8.0	9.0	9.5

More Zernike modes’ open-loop measurement in the sub-aperture of the Shack–Hartmann wavefront sensor

Abstract

1. Introduction

2. Sub-wavefront sensing in SHWFS

2.1 Theoretical description

2.2 Related parameters analysis

3. Parameter analysis and optimization

3.1 Numerical simulation

3.2 Linearity range changing with the pixel number in each sub-aperture

3.3 Linearity range changing with the set amplitude of the Z4 offset

3.4 Sensing error comparison with different parameters groups

4. Experimental validation of the parameters optimization

4.1 Linearity range test for a signal mode

4.2 Wavefront sensing at the optimum amplitude of the Z4 offset

5. Whole pupil wavefront reconstruction with the optimum parameter group

5.1 Determining the number of Zernike modes for reconstructed

5.2 Residual error analysis of the reconstruction with the modified SHWFS

6. Conclusions

Funding

Acknowledgments

References and links

Cited By

Figures (10)

Tables (4)

Equations (7)

Optics Express