COACH-based Shack&#x2013;Hartmann wavefront sensor with an array of phase coded masks

Nitin Dubey; Ravi Kumar; Joseph Rosen

doi:10.1364/OE.438379

1. Introduction

Shack-Hartmann wavefront sensor (SHWS) was developed for optical metrology purposes [1] and later has been used in various other fields, such as adaptive optics [2], microscopy [3], retinal imaging [4], and high-power laser systems [5]. Usually, in SHWS, a microlens array is used to measure the local gradients of the wavefront. For this, the incident wavefront is spatially sampled such that each lenslet focuses the local sub-aperture into an imaging sensor located at the focal plane of the lenslet. The position of the focused spot corresponds to the average slope of the local wavefront of each microlens. The local slope of the wavefront can be evaluated by calculating the spot displacement from the optical axis. Afterward, a wavefront reconstruction algorithm is utilized to estimate the complete wavefront using these estimated slopes [6].

Recently, an incoherent digital holography technique called coded aperture correlation holography (COACH) [7,8] has been developed. In this study, we present a new technique of COACH-based SHWS that increases the accuracy of wavefront sensing. Instead of using the regular lenslet array, a coded phase mask (CPM) is introduced in each sub-aperture of the array. Based on several recently published studies [9–11] about interferenceless COACH, we conclude that using a certain CPM can yield a sharper cross-correlation peak. The sharper peak enables detecting a more accurate displacement of the peak from its reference position, and the displacement is translated to a more accurate estimated tilt angle of the local wavefront than in the case of the regular SHWS [12]. Narrowing the focal spots can be achieved in a regular SHWS by changing the lenslet diameter and focal length, but this way of increasing the accuracy has various penalties in the other SHWS performances. According to [12], the dynamic range of the sensor is d/2f, where d and f are the lenslet diameter and its focal length, respectively. On the other hand, the sensitivity, defined as the minimal detectable tilt of the wavefront angle, is p/f, where p is the minimum detectable spot displacement determined by the pixel size of the detector. If the lenslet diameter is determined according to considerations of wavefront sampling, then decreasing the lenslet focal lengths improves the dynamic range on the one hand but harms the sensitivity on the other hand. Our goal in the present study is to increase the accuracy of the SHWS without changing neither the dynamic range nor the sensitivity. Improving the accuracy is achieved by narrowing the moving spot for the same values of lenslet diameter and focal length. Thus, by use of the CPM array instead of the conventional lenslet array, we show that one can achieve more accurate displacement of each correlation peak, which improves the overall accuracy of the measured wavefront. The main goal of this study is to achieve a high precision measurement of the local tilt-angle of wavefront across each sub-aperture of SHWS. To a large extent, this is possible due to various correlation techniques adapted from the field of pattern recognition. Coded aperture-based SHWS is, to the best of our knowledge, the first attempt to design a correlation technique-based wavefront sensor with a CPM array.

There are several applications of wavefront sensing which use the incoherent or partially coherent light source (for instance, ocular and astronomical). In the current analysis, we have performed all the experiments with a coherent light source (i.e., HeNe laser). However, we note that in many applications of wavefront sensing with incoherent light, the phase object is illuminated by a guidestar, which induces spatial coherence light over the aperture of the optical system. Adding a chromatic filter can convert the light to both spatial and temporal coherent. Therefore, our proposed wavefront sensor can be relevant also for these ‘incoherent’ cases.

2. Methodology

Modified Gerchberg-Saxton algorithm (GSA) is utilized for synthesizing the CPMs [9–11], as shown in Fig. 1. The CPMs displayed on a spatial light modulator (SLM) generate an ensemble of sparse randomly distributed intensity dots on the sensor plane. The ensemble of dots obtained from the measured wavefront is cross-correlated with a reference dot pattern created by illuminating a single CPM with a plane wave. The intensity response in each cell of the array is shifted according to the average slope of the wavefront in each corresponding cell. Composing the entire slopes yields a three-dimensional curvature used as an approximation to the wavefront incident on the SHWS.

Fig. 1. Flow chart of modified GSA for synthesizing the CPMs with a sparse dot pattern and the CPM intensity response of 3, 5, 10, and 12 dots patterns.

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In the modified GSA, an initial random phase mask is Fourier transformed from the CPM plane to the sensor plane. On the sensor plane, the magnitude distribution is replaced with the chosen pattern of a randomly scattered sparse dot pattern, whereas the phase distribution remains unchanged. The resulting complex amplitude is inversely Fourier transformed to the CPM plane, and the magnitude distribution is replaced with the uniform magnitude. This iterative process continues till the generated intensity profile converges to satisfy the constraints. Once the CPM is generated, it was displayed on the SLM with a diffractive lens to satisfy the Fourier relation between the CPM plane and sensor plane. The intensity responses of the CPM with 3, 5, 10, and 12 sparse dot patterns are also shown in Fig. 1, along with the schematic of the GSA.

The present study is based on previous researches on COACH. The response of multiple dots is a middle pattern between a single dot obtained by each lenslet and a chaotic continuous response over the entire sensor plane. The number of dots is determined by an optimization algorithm in which the number is increased from one until the optimum cost function (such as signal-to-noise ratio) is achieved. The locations of the dots are randomly chosen to avoid high side-lobes of the cross-correlation function.

The optical scheme of COACH-based SHWS is shown in Fig. 2(a). A plane wave passes through a phase object, and the emitted wavefront is sampled by an array of identical phase elements, each of which is a product of the CPM and a lens transmittance. The lens performs a Fourier transform which, according to the GSA, is needed for the response of the dots on the camera plane. As in conventional SHWS, in front of each CPM, the local wavefront emitted from the phase object is approximated to a tilted plane wave. The goal of the system is to measure the tilt angle of each local planar wavefront and to estimate the global shape by fusing all the local angles. As in ordinary SHWS, estimating the tilt angle is done by measuring the shift of the intensity response recorded on the camera. Since in COACH-based SHWS, this intensity response is the ensemble of dots, their common shift is measured by cross-correlation with a reference pattern. As much as the cross-correlation peak is sharper, the accuracy of the translation of tilt angle into the displacement is higher, and so the measurement of the local slope is more accurate [12]. Therefore, to narrow the peak as much as possible, the cross-correlation is done by a phase-only filter (POF) in some experiments or by a nonlinear process with two parameters optimized to yield the sharpest cross-correlation peak in other experiments.

Fig. 2. (a) Optical Schematic and (b) experimental setup for the COACH-based SHWS. PO - Phase object, BS - Beamsplitter. All distances in (b) are in centimeters.

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We start the formal analysis by calculating the distribution of the reference pattern. Each cell of the size a × a in the CPM array designed by the GSA has the distribution of t(x,y)=exp [iφ(x,y)]Rect(x/a,y/a), where φ(x,y) is a chaotic phase function and Rect(x/a,y/a) = 1 for all |x|,|y|≤a/2 and 0 otherwise. t(x,y) is attached to a diffractive microlens with f focal length. It is well-known that illuminating these combined phase elements at a single cell of the array with a plane wave yields the following intensity [13],

(1)$${I_R}(x,y) = {|{\nu [{{1 / {\lambda f}}} ]{\cal F}\{{t(x,y)} \}} |^2},$$

where ${\cal F}$ is two-dimensional Fourier transform and ν [·] is the scaling operator such that ν[b]g(x)=g(bx). I_R(x,y) plays the role of the reference pattern in the cross-correlation. As mentioned above, φ(x,y) is designed by the GSA to yield an ensemble of sparse randomly distributed dots as a result of a Fourier transform of exp[iφ(x,y)] [9]. Therefore,

(2)$$\begin{aligned} {I_R}(x,y) &= {|{\nu [{{1 / {\lambda f}}} ]{\cal F}\{{\textrm{exp}[{i\varphi ({x,y} )} ]\textrm{Rect}({{x / a},{y / a}} )} \}} |^2}\\ & \cong \sum\limits_{p = 1}^P {{C_p}\textrm{sin}{\textrm{c}^2}\left( {a\frac{{x - {x_p}}}{{\lambda f}},a\frac{{y - {y_p}}}{{\lambda f}}} \right),}\end{aligned}$$

where $\textrm{sinc}[{a \cdot ({x,y} )} ]= {\cal F}\{{\textrm{Rect}({{x / a},{y / a}} )} \},$C_p is a positive constant, P is the number of dots, and {x₁,…,x_P} is a set of random numbers determined by the GSA. I_R(x,y) is measured once by illuminating the central CPM cell in the array with a plane wave propagating in the z-direction and without the presence of the phase object.

Once the reference function is known, the phase object is introduced into the setup, and wavefront sensing is performed. The wavefront coming from the phase object is divided into M×N subareas as the number of phase elements at the array. At each subarea, the wavefront is approximated to a tilted planar wavefront with a tilt angle that should be measured to estimate the shape of the overall wavefront. Since each phase element in the array is multiplied by approximated linear phase, the dot response in the Fourier plane is shifted a distance proportional to the tilt angle. The intensity response at the (m,n)-th subarea on the camera for the tilted planar wavefront with a tilt angle of (θ_m,θ_n) is [14],

(3)$$\begin{aligned} I_{m,n}(x,y) &= \left| {\nu \left[ {{1 / {\lambda f}}} \right]{\cal F}\left\{ {{\exp}} \right.} \right.\left[ {\left( {{{i2\pi } / \lambda }} \right)\left( {x\tan \theta _m + y\tan \theta _n} \right)} \right. + \left. {\left. {\left. {i\varphi \left( {x,y} \right)} \right]{\textrm{Rect}}\left( {{x / a},{y / a}} \right)} \right\}} \right|^2 \\ &\cong \sum\limits_{p = 1}^P {C_p{\textrm{sin}}{\textrm c}^2\left( {a\displaystyle{{x-x_p-f\tan \theta _m} \over {\lambda f}},a\displaystyle{{y-y_p-f\tan \theta _n} \over {\lambda f}}} \right).} \end{aligned} $$

Cross-correlation between I_m,n(x,y) and I_R(x,y) yields a correlation peak at a distance (f·tanθ_m,f·tanθ_n) from the center of the (m,n)-th subarea. Measuring this shift of the peak gives an estimation of the value of the tilt angle of (θ_m,θ_n), and collecting all the local angles of the entire M×N subareas enables the restoration of the wavefront emitted from the phase object. The accuracy of the peak shift is dependent on the peak width, and therefore we propose to cross-correlate the signals in a nonlinear way with two parameters optimized to produce the narrowest peak [10]. To understand the relation of the peak width with the parameters of the nonlinear cross-correlation, we consider the Fourier transforms of the two correlated functions as follows,

(4)$$\begin{array}{l} {{\tilde{I}}_{m,n}}(u,v) = {\cal F}\{{{I_{m,n}}(x,y)} \}= |{{{\tilde{I}}_{m,n}}(u,v)} |\times \exp [{i\Phi (u,v) + i \cdot f({u\tan {\theta_m} + v\tan {\theta_n}} )} ]\\ {{\tilde{I}}_R}(u,v) = {\cal F}\{{{I_R}(x,y)} \}= |{{{\tilde{I}}_R}(u,v)} |\exp [{i{\Phi _R}(u,v)} ],\end{array}$$

where $\Phi (u,v)$ and ${\Phi _R}(u,v)$ are the phase of ${\tilde{I}_{m,n}}(u,v)$ [for (θ_m,θ_n)=(0,0)] and ${\tilde{I}_R}(u,v),$ respectively. Since $|{{{\tilde{I}}_{m,n}}(u,v)} |= |{{{\tilde{I}}_R}(u,v)} |$ and $\Phi (u,v) = {\Phi _R}(u,v),$the nonlinear cross-correlation (NCC) with the optimized parameters α and β becomes [10],

(5)$$\begin{aligned}C({x,y} )&= |{{{\cal F}^{ - 1}}\{{{{|{{{\tilde{I}}_{m,n}}(u,v)} |}^\alpha }\exp [{i\Phi (u,v)} } } { { { + i \cdot f({u\tan {\theta_m} + v\tan {\theta_n}} )} ]{{|{{{\tilde{I}}_R}(u,v)} |}^\beta }\exp [{ - i{\Phi _R}(u,v)} ]} \}} |\\ &= {|{{{\tilde{I}}_R}(u,v)} |^{\alpha + \beta }}\exp [{i \cdot f({u\tan {\theta_m} + v\tan {\theta_n}} )} ]= \Lambda ({x - f\tan {\theta_m},y - f\tan {\theta_n}} ),\end{aligned}$$

where Λ(·) is the peak function which should be as narrow as possible to increase the accuracy of the tilt angle measurement. Theoretically, the values of α and β that yield the sharpest peak are the values that satisfy the equation α + β = 0. However, the noisy experimental environment usually yields different optimal parameters and may not follow this equation. In this study, the optimal values of the parameters α and β of the NCC are found once by minimizing the entropy and then by minimizing the mean square error (MSE). Entropy is a blind figure-of-merit that minimizing it yields sharps peaks regardless of the shape of the measured wavefront [10]. In MSE, on the other hand, the measured wavefront is compared to some specific ideal wavefront. However, minimizing the MSE is actually maximizing the accuracy of the measurement, which is the purpose of this study. In other words, the MSE assesses the reconstruction error by calculating the deviation of the reconstructed wavefront from the digitally simulated wavefront, and minimum MSE is the measure of accuracy which we aim to increase. The MSE is calculated as,

(6)$$\textrm{MSE} = \frac{1}{{L \cdot K}}\sum\limits_{l = 1}^L {\sum\limits_{k = 1}^K {{{({{{\hat{W}}_{k,l}} - {W_{k,l}}} )}^2}} } ,$$

where Ŵ_k,l, and W_k,l are the digital and experimental reconstructed wavefronts, respectively. K and L are the numbers of data points in the wavefront. In case the measured wavefront is a priory unknown, two strategies can be followed. First, the values of α and β can be calibrated with a phase object that its wavefront is a priory known, and then the same α and β can be used for the unknown wavefront. Alternatively, α and β can be found directly for the unknown wavefront, using an optimization function that does not need any reference wavefront [10,11,15], such as the entropy mentioned above. Following the procedure of the NCC, the center of the correlation peak for each cell is calculated by the center-of-mass method. The average slope of the tested wavefront in each cell is directly related to the displacement of the correlation peak from the center of the cell. Once the slope for each sub-aperture is calculated, then with the help of the zonal reconstruction technique [16,17], these slope values are fused to reconstruct the full wavefront. Apart from the NCC technique, the POF technique is also used for cross-correlation. The advantage of the POF technique is that it does not require any optimization like the NCC technique, but the reconstruction results usually suffer from the background noise present in the images. Note that in the NCC technique, the optimization of parameters is a one-time process for a given COACH-based SHWS. No more adjustments are required, and the system works with the same level of accuracy for any incoming wavefront.

Based on our previous investigations [8,9,11], it is well-known that cross-correlation between two bipolar functions reduces the noise and increases the sharpness of the correlation peaks. To see that cross-correlation between two bipolar functions is better than between two positive functions, we look at the spectral domain in which the spectrum of the cross-correlation is a product of the two spectrums of the corresponding functions. So, in the case of two positive functions, the product of the two spectrums yields a relatively high zero-order, whereas, in the case of two bipolar functions, the zero-order is relatively low. High zero order in the spectrum is translated to high background signal in the reconstruction domain, and hence bipolar dot pattern is preferred over positive dot pattern. Therefore, to further increase the accuracy of the tilt angle measurement, we tested the option of using bipolar functions by recording two camera shots with two independent CPMs and with two different dot responses, where one response is subtracted from the other. Note that recording two camera shots slow down the wavefront acquisition and limit its application to the case where accuracy is more important than measurement speed. In this scheme, the bipolar reference pattern is,

(7)$${I_R}(x,y) = {|{\nu [{{1 / {\lambda f}}} ]{\cal F}\{{{t_1}(x,y)} \}} |^2} - {|{\nu [{{1 / {\lambda f}}} ]{\cal F}\{{{t_2}(x,y)} \}} |^2},$$

where t_j(x,y)=exp[iφ_j(x,y)]Rect(x/a,y/a), and φ_j(x,y) is the phase distribution of the j-th CPM (j=1,2). I_m,n(x,y) is also obtained by two exposures with the same two CPMs so that its distribution is,

(8)$$ \begin{aligned} &I_{m, n}(x, y)=\left|v\left[\frac{1}{\lambda f}\right] \mathcal{F}\left\{\exp \left[\frac{i 2 \pi}{\lambda}\left(x \sin \theta_{m}+y \sin \theta_{n}\right)\right] t_{1}(x, y)\right\}\right|^{2} \\ &-\left|v\left[\frac{1}{\lambda f}\right] \mathcal{F}\left\{\exp \left[\frac{i 2 \pi}{\lambda}\left(x \sin \theta_{m}+y \sin \theta_{n}\right)\right] t_{2}(x, y)\right\}\right|^{2} \\ &\cong \sum_{p=1}^{P} C_{1, p} \operatorname{sinc}^{2}\left(\frac{x-x_{1, p}-f \tan \theta_{m}}{\lambda f / a}, \frac{y-y_{1, p}-f \tan \theta_{n}}{\lambda f / a}\right) \\ &-\sum_{p=1}^{P} C_{2, p} \operatorname{sinc}^{2}\left(\frac{x-x_{2, p}-f \tan \theta_{m}}{\lambda f / a}, \frac{y-y_{2, p}-f \tan \theta_{n}}{\lambda f / a}\right) \end{aligned} $$

where the two random series {x_1,1,…x_1,P} and {x_2,1,…x_2,P} are independent. The rest of the process is identical to the above-mentioned description of the process of the unipolar patterns.

3. Simulation study

We started the study with computer simulations of the unipolar COACH-based SHWS in comparison to the regular SHWS. MSE and Strehl ratio are used as parameters for comparison. Figure 2(a) shows the optical schematic of the wavefront sensing system used for the MATLAB simulation. For regular SHWS, a 30×30 array of microlenses is used, and this lenslet array is replaced with the same size of CPM array for COACH-based SHWS. The CPM with intensity response of 3 sparse dots pattern was chosen.

Figures 3(A, B, and C) are the ideal wavefronts generated by three low orders of Zernike polynomials. Figure 3(A₁, B₁, C₁) are the wavefront reconstructed from regular SHWS, whereas the corresponding reconstruction error maps $\Delta \phi ({x,y,z} )$ are shown in Fig. 3(A₄, B₄, C₄). $\Delta \phi ({x,y,z} )$ is calculated by the formula [18],

(9)$$\Delta \phi ({x,y,z} )= {\phi _{ideal}}({x,y,z} )- {\phi _{measured}}({x,y,z} ),$$

where ${\phi _{ideal}}({x,y,z} )$ is the ideal wavefront and ${\phi _{measured}}({x,y,z} )$ is the measured wavefront, which is either reconstructed by regular SHWS or by COACH-based SHWS. Similarly, Fig. 3(A₂, B₂, C₂) and Fig. 3(A₃, B₃, C₃) are the reconstructed wavefronts from COACH-based SHWS with the POF and NCC techniques, respectively. The parameter value α=0.1 and β=0.9 are used for the NCC technique. The error maps of COACH-based SHWS with POF and NCC technique are shown in Fig. 3(A₅, B₅, C₅) and Fig. 3(A₆, B₆, C₆), respectively.

Fig. 3. Simulated reconstruction results of three low orders of Zernike polynomials by regular SHWS and unipolar COACH-based SHWS system with POF and NCC technique.

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Another parameter commonly used to estimate the performance of wavefront sensors is the Strehl ratio. Strehl ratio is a measure used to compare the real performance of an optical system with its diffraction-limited performance. The Strehl ratio is related to root mean square error (RMSE) of the transmitted wavefront using the Maréchal approximation [19,20],

(10)$$S = \exp [{ - {{({2\pi {\sigma_\phi }} )}^2}} ]$$

Here σ_ϕ is the RMSE of the measured wavefront, calculated by taking the square root of MSE from Eq. (6).

The MSE and the Strehl ratio for regular and COACH-based SHWS are calculated and shown in Table 1. From Table 1, it is clear that COACH-based SHWS with a 3-dot pattern shows better performance than regular SHWS. For all the tested cases of Zernike polynomial wavefront reconstruction, COACH-based SHWS has a lower MSE value which indicates the better measurement accuracy, and it also shows a higher Strehl ratio value which confirms the better optical performance in comparison to the regular SHWS. From the simulation study, it is verified that COACH-based SHWS with POF and NCC techniques gives better reconstruction than the regular SHWS. The POF technique doesn’t require any optimization, and the NCC technique with optimal parameters has better background noise control in the reconstruction process.

Table 1. MSE and Strehl ratio values of regular SHWS and single-shot COACH-based SHWS for Zernike polynomial wavefronts.

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4. Experimental results

The schematic of the experimental setup for COACH-based SHWS is shown in Fig. 2(b). A HeNe laser with a beam size of 8mm is used as the source, where a pair of lenses (f₀=10cm; f₁=48cm) is used to expand the beam diameter from 8mm to about 38mm. A pinhole of 20μm diameter used as a lowpass filter is placed at the common focal plane of the two lenses. The lowpass filter makes the wavefront after the beam expander as close as possible to a plane wave. This plane wave propagates through a phase object such that its phase distribution is represented by the wavefront emitted from the object. A positive spherical lens of 30cm focal length and a cylindrical lens of 30cm focal length are the phase objects in the experiment. The beam is polarized to the active orientation of the SLM (Holoeye PLUTO, 1920×1080 pixels, 8μm pixel pitch). The array of coded phase patterns displayed on the SLM is obtained by modulo-2π phase addition of the CPMs with the lenslet array of f=6.5mm focal length. The reflective SLM and the focal length enforce using a beamsplitter to reflect the modulated light coming from the SLM toward a digital camera (PCO.Edge 5.5 CMOS, pixel pitch=6.5μm, 2560×2160 pixel). An optical relay system is used to translate the focal plane of the microlens array onto the sensor plane. For comparison with our proposed method, we measured the wavefront with a regular SHWS implemented on the same setup by changing the phase pattern on the SLM to a lenslet array of f=6.5mm focal length. The central 1020×1020 pixels of the SLM were used to display an array of 30×30 CPMs attached to microlenses. The size of each cell is 34×34 pixels. The experimental results were compared with the digital wavefront. The digital wavefront is created by numerical Fresnel propagation from the phase object to the SLM plane using MATLAB with the same parameters as the experimental setup. The MSE was calculated between the digital wavefront and the tested wavefront reconstructed by the regular and by the COACH-based SHWS systems. For COACH-based SHWS, both techniques, unipolar and bipolar, were tested. Numbers of 3, 5, 10, and 12 dots were tested to find the optimal number of dots of the sparse response.

Figures 4(a₁) and 4(a₂) present the intensity response for two different CPMs when only the central cell is activated in the array. The dot structure of 5 dots is clearly seen in both figures and in the bipolar pattern of Fig. 4(a₃) obtained as the subtraction result of the intensities of Fig. 4(a₁) minus Fig. 4(a₂). Figures 4(b₁) and 4(b₂) and 4(c₁) and 4(c₂) show the intensity response in the center of the COACH-based array equipped with two different CPMs, where the tested and reference wavefront is introduced into the system. The corresponding bipolar responses are shown in Figs. 4(b₃) and 4(c₃).

Fig. 4. An intensity response of a single cell recoded by the sensor for (a₁) CPM1 and (a₂) CPM2 displayed on the SLM. The bipolar pattern in (a₃) is obtained as the subtraction result of the intensities of Fig. (a₁) minus (a₂); (b₁, b₂) and (c₁, c₂) are the central part of two intensity patterns on the sensor for the two CPM arrays and for the input of the test and reference wavefront, respectively; (b₃) and (c₃) are their corresponding bipolar pattern obtained as the result of b₁-b₂ and c₁ -c₂, respectively.

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To find the optimum parameter values for the NCC technique, one of the sub-apertures from the boundary of the CPM array was chosen, and the entropy and MSE values of the bipolar correlation for all the values α and β ranges from -1.0 to +1.0 are calculated. The NCC result of a single sub-aperture CPM with intensity response of 5 dots pattern is shown in Fig. 5 for parameters α in the range -0.2 to +0.7 and β in the range +0.1 to +1.0. Both the entropy and MSE are minimal in the range of β=0.8 to 1.0 and α=0.1 to 0.3, as emphasized in Fig. 5 by a red box. The product of MSE and entropy is minimal at (α=0.1; β=0.8) and (α=0.1; β=0.9). The reconstruction results of NCC reconstruction with (α=0.1; β=0.9) (green box) for a single camera shot and two cameras shot and regular SHWS reconstruction are all compared with a digital wavefront simulated numerically in the computer.

Fig. 5. Cross-correlation patterns of one sub-aperture of the CPM array located at the boundary of the array with the reference CPM. Entropy (in orange) and MSE (in yellow) values for the range of α and β parameter [α= -0.2 to +0.7; β= +0.1 to +1.0] are shown on each pattern. The MSE values are multiplied by 10⁻⁵ for all cases.

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Figure 6 shows the experimental results of the NCC in comparison with the focal spot of a single microlens, as shown in Fig. 6(a). The cross-correlation of Fig. 4(a₁) with its POF version and under the NCC process (α=0.1; β=0.9) are shown in Figs. 6(b) and 6(c), respectively. The cross-correlation of Fig. 4(a₃) with its POF version and under the NCC process (α=0.1; β=0.9) are shown in Figs. 6(d) and 6(e), respectively. Figure 6(f) shows the intensity response of regular SHWS for the test wave. Figure 6(h) shows the NCC of Fig. 4(a₃) with Fig. 4(b₃) [when the scale of 4(a₃) is the same as one cell of 4(b₃)] with the parameters α=0.1 and β=0.9. The cross-section plots of correlation peaks with POF and NCC with optimized values of α and β are shown in Fig. 6(g) in comparison to the curve of the microlens focal spot (blue). Based on the plots of Fig. 6(g), apparently, the bipolar NCC technique is narrower by 27% in comparison with the microlens focal spot of the regular SHWS.

Fig. 6. (a) intensity response of a single microlens focal spot. Correlation peaks by (b) unipolar POF, and (c) unipolar NCC with α=0.1, β=0.9, and (d) bipolar POF, (e) bipolar NCC with α=0.1, β=0.9. (f) Intensity response of microlens array for the test wavefront, (g) Horizontal cross-section of the microlens focal spot, unipolar and bipolar correlation peaks. (h) Correlation peaks with bipolar NCC (α=0.1, β=0.9) for the test wavefront.

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The reconstruction results of both the unipolar and bipolar SHWS are compared with the regular SHWS. For the experiments, 1020×1020 pixels of SLM were used to display a 30×30 array of microlenses and CPMs. The 30×30 data points of a reconstructed wavefront are resized into 1020×1020 data points, and the MSE is calculated in comparison to the simulated digital wavefront. Figures 7(A) and 7(B) show the 3D maps of the digitally simulated spherical and cylindrical wavefront. Figures 7 (A₁) and 7(B₁) show the reconstructed spherical and cylindrical wavefronts by the regular SHWS method. The 3D error maps between the reconstructed and digital wavefronts for both cases are shown in Figs. 7(A₂) and 7(B₂).

Fig. 7. (A, B) Simulated digital spherical and cylindrical wavefront. (A₁, B₁) Reconstructed experimental SHWS wavefront. (A₂, B₂) error maps.

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Figures 8(A) and 8(B) show the unipolar intensity response of CPM with 3 and 5 sparse dot patterns. Figures 8(a₁) and 8(c₁) show the corresponding reconstructed spherical wavefront by the NCC technique with parameters (α=0.1, β=0.9), and their corresponding error maps with MSE values are shown in Figs. 8(b₁) and 8(d₁), respectively. Figures 8(a₂) and 8(c₂) show the reconstruction of the spherical wavefront from 3 and 5 sparse dot patterns by POF technique, and the corresponding error maps are shown in Figs. 8(b₂) and 8(d₂), respectively. Similarly, POF and NCC techniques with parameters (α=0.1, β=0.9) were used for the reconstruction of a cylindrical wavefront. Figures 8(a₃) and 8(c₃) show the NCC reconstruction for 3 and 5 sparse dot patterns and Figs. 8(a₄) and 8(c₄) show POF reconstruction for both CPM patterns. From the error maps in Figs. 8(b₁), 8(d₁), 8(b₂), and 8(d₂), it is clear that the reconstructed wavefront of the unipolar method has a closer fit to the digital wavefront in comparison to the dome-shape error map of the regular SHWS shown in Fig. 7(A₂).

Fig. 8. Reconstruction results of spherical and cylindrical wavefront using unipolar COACH-based SHWS. (A) and (B) are the unipolar intensity response of 3 dots and 5 dots pattern, respectively. (a₁) and (c₁) show the spherical wavefront reconstructed by NCC technique with error maps given in (b₁) and (d₁). (a₂) and (c₂) show the spherical wavefront reconstructed by the POF technique with error maps shown in (b₂) and (d₂). Similarly, (a₃) and (c₃) show the cylindrical wavefront reconstructed by NCC technique with error maps in (b₃) and (d₃) and (a₄) and (c₄) show the cylindrical wavefront reconstructed by POF with error map shown in (b₄) and (d₄).

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The calculated MSE and Strehl ratio values of the regular SHWS and single-shot COACH-based SHWS for each number of sparse dots are given in the Table of Fig. 9. Along with the table, the graphs of MSE and Strehl ratio of the experimentally reconstructed wavefront are also shown in Fig. 9.

Fig. 9. Table and Graphs of MSE and Strehl ratio for the reconstruction from regular SHWS and unipolar COACH-based SHWS for various sparse dot patterns.

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The unipolar reconstructed image usually has strong background noise, which contributes to the error in the tilt-angle measurement. The sensor performance can be improved by synthesizing a bipolar dot pattern. The bipolar dot pattern for reference and test wavefront is synthesized by subtracting the respective intensity patterns generated from two different independent CPM arrays. Unlike a single-camera shot where image data has only positive values, in bipolar case data, values are varying from negative to positive, which makes the magnitudes of ${\tilde{I}_{m,n}}$and ${\tilde{I}_R}$more uniform than the cases of a single camera shot. Thus, each reconstructed point becomes sharper with less background noise.

Figures 10(A) and 10(B) show the bipolar intensity response for 3 and 5 sparse dot patterns. Figures 10(a₁) and 10(c₁) show the reconstructed spherical wavefront by NCC technique with parameters (α=0.1, β=0.9). Their corresponding error maps with MSE values are shown in Figs. 10(b₁) and 10(d₁), respectively. Figures 10(a₂) and 10(c₂) show the reconstruction of the spherical wavefront from 3 and 5 sparse dot patterns by POF technique, and the corresponding error maps are shown in Figs. 10(b₂), and 10(d₂), respectively. Similarly, for cylindrical wavefront also POF and NCC techniques with parameters (α=0.1, β=0.9) were used for reconstruction. Figures 10(a₃) and 10(c₃) show the NCC reconstruction for 3 and 5 sparse dot patterns and Figs. 10(a₄) and 10(c₄) show POF reconstruction for both CPM patterns. The corresponding error maps of NCC and POF reconstruction in comparison with digitally simulated wavefront are shown in Figs. 10(b₃) and 10(d₃) and Figs. 10(b₄) and 10(d₄), respectively. From the flat error maps in Figs. 10(b₁) and 10(d₁) and 10(b₂) and 10(d₂), it is clear that the reconstructed wavefront of the bipolar method has a closer fit to the digital wavefront in comparison to the dome-shape error map of the regular SHWS shown in Fig. 7(A₂).

Fig. 10. Reconstruction result of spherical and cylindrical wavefronts using bipolar COACH-based SHWS. (A) and (B) are the bipolar response of 3 dots and 5 dots pattern, respectively. (a₁) and (c₁) show the spherical wavefront reconstructed by NCC technique with error maps given in (b₁) and (d₁). (a₂) and (c₂) show the spherical wavefront reconstructed by the POF technique with error maps shown in (b₂) and (d₂). Similarly, (a₃) and (c₃) show the cylindrical wavefront reconstructed by NCC technique with error maps in (b₃) and (d₃) and (a₄) and (c₄) show the cylindrical wavefront reconstructed by POF with error map shown in (b₄) and (d₄).

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The calculated MSE and Strehl ratio values of the regular SHWS and the bipolar COACH-based SHWS for each number of sparse dots are given in the Table of Fig. 11. Along with the table, the graphs of MSE and Strehl ratio of the experimentally reconstructed wavefront are also shown in Fig. 11. The lower MSE values confirm the improvement in the wavefront reconstruction accuracy of the bipolar COACH-based SHWS. The MSE values of spherical and cylindrical wavefronts from regular SHWS are 0.00705 and 0.0117, respectively. Note that COACH-based SHWS with the POF technique without any optimization gives MSE values of 0.00045 and 0.00540 for the case of 3 sparse dots, which is one order smaller than regular SHWS. Even for a case of 12 sparse dots, POF techniques give much better reconstruction accuracy than regular SHWS. Since the NCC technique reduces the background noise in the reconstruction due to the optimization process, in most cases, the accuracy of NCC is better than that of the POF technique. Therefore, NCC gives better MSE values than POF and regular SHWS for almost all cases. In the case of the Strehl ratio, for all the dot pattern cases, the optical performance of COACH-based SHWS is always above 85%, while the maximum optical performance of the regular SHWS is 75%.

Fig. 11. Table and Graphs of MSE and Strehl ratio for reconstruction from regular SHWS and bipolar COACH-based SHWS for various sparse dot patterns.

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5. Conclusions

In conclusion, we have presented a new SHWS technique with improved measurement accuracy. Coded phase masks are synthesized using the modified GSA algorithm to produce sparse dot patterns. Further, the bipolar sparse dot patterns are generated to reduce background noise. The experimental results with a different number of dot patterns are compared with the regular SHWS, and the results confirm the validity and effectiveness of our method. From the calculated MSE values, it can be noted that for the best case of 3-dot bipolar (overall 6 dots), the accuracy of COACH-based SHWS is improved by one order of magnitude in comparison with the regular SHWS. The cost of such improvement is the need for two camera shots. In case the speed of the measurement is more critical than the accuracy, the single-shot unipolar COACH-based SHWS can offer approximately double the accuracy of the regular SHWS.

Funding

Israel Science Foundation (1669/16).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this study are not publicly available at this time but may be obtained from the authors upon a reasonable request.

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COACH-based Shack–Hartmann wavefront sensor with an array of phase coded masks

Abstract

1. Introduction

2. Methodology

3. Simulation study

4. Experimental results

5. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (11)

Tables (1)

Equations (10)

Optics Express