Shack–Hartmann Wavefront Sensing Based on Four-Quadrant Binary Phase Modulation

Abstract

Aiming at the problem that it is difficult for the conventional Shack–Hartmann wavefront sensor to achieve high-precision wavefront reconstruction with low spatial sampling, a kind of Shack–Hartmann wavefront sensing technology based on four-quadrant binary phase modulation is proposed in this paper. By introducing four-quadrant binary phase modulation into each subaperture, the technology is able to use an optimization algorithm to reconstruct wavefronts with high precision. The feasibility and effectiveness of this method are verified at extreme low spatial frequency by a series of numerical simulations, which show that the proposed method can reliably reconstruct wavefronts with high accuracy with rather low spatial sampling. In addition, the experiment demonstrates that with a 2 × 2 microlens array, the four-quadrant binary phase-modulated Shack–Hartmann wavefront sensor is able to achieve approximately 54% reduction in wavefront reconstitution error over the conventional Shack–Hartmann wavefront sensor.

1. Introduction

The Shack–Hartmann Wavefront Sensor (SHWFS) is a common optical measuring device, which consists of a microlens array and a detector located at the focal plane of the microlens array. It has been widely used in astronomy observation [1,2], ophthalmology [3,4], biomedical imaging [5], and laser beam characterization [6,7] for its simple and compact structure, strong adaptability, and measurement speed. In principle, the SHWFS approximates the wavefront through subaperture as the tilted wavefront with an average x and y tilt component [8]. Then, the average slopes of the local wavefront are estimated by calculating the centroid displacement of the subspot [9]. Finally, the incident wavefront is reconstructed from the slope data with the help of the wavefront reconstruction algorithm [10,11,12]. It is known that the accuracy of the SHWFS is limited by the spatial frequency of the microlens array. As a result, an SHWFS is not able to obtain accurate wavefront information with low spatial frequency, which limits the detectable faint targets [13,14,15].

To improve the detection performance of the subaperture detail in low spatial resolution, a series of improvement methods have been proposed in recent years. Some researchers proposed a kind of defocus SHWFS wavefront sensing method to improve reconstruction precision [16,17]. However, defocused structures would reduce the dynamic range. Wang et al. used a binary filtered Shack–Hartmann wavefront sensing technology that can gain noise benefits from using single detectors [18]. However, the method increases the complexity of the system. Moreover, an estimation method based on the focal plane sensing technique has been proposed to detect more Zernike modes in each subaperture [13].

A kind of Shack–Hartmann wavefront sensing technology based on four-quadrant binary phase modulation is provided in this paper. The four-quadrant binary phase modulation is introduced into each subaperture; then, the stochastic parallel optimization algorithm is performed to retrieve Zernike coefficients from the spot array image. Numerical simulation and experimental results show that four-quadrant binary phase modulation is of great help with the detection of high-order wavefront information in subaperture, enhancing the wavefront detection accuracy and speed of the SHWFS with low spatial frequency.

The structure of this paper is arranged as follows: Section 2 introduces the principle of the SHWFS based on four-quadrant binary phase modulation and the stochastic parallel optimization algorithm. Section 3 introduces a series of numerical simulations, analyzing the influence of four-quadrant binary phase modulation on the reconstruction precision and the convergence rate of the optimization algorithm. In addition, the robustness of the proposed method is analyzed and compared with the conventional method for different subaperture resolution. Section 4 gives experimental results and discussion. Finally, Section 5 summarizes the paper.

2. Principle

2.1. Four-Quadrant Binary Phase-Modulated Shack–Hartmann Wavefront Sensor

The four-quadrant binary phase-modulated Shack–Hartmann wavefront sensor (FBPM-SHWFS) proposed in this paper consists of a four-quadrant binary phase-modulated microlens array and a CCD located at the focal plane of the microlens array, as shown in Figure 1a. The four-quadrant binary phase microlens array consists of a microlens array and a four-quadrant binary phase array. It may be two independent static elements or a composite static element inscribed together. As an example, a 2 × 2 four-quadrant binary phase microlens array is shown in Figure 1b, where the left is a 2 × 2 microlens array, and the right is a 2 × 2 four-quadrant binary phase array and a single four-quadrant binary phase. This is able to help the SHWFS provide more wavefront information and reduce the possibility of multiple solutions of the input wavefront for phase retrieval. The four-quadrant binary phase array is expressed as follows:

$${\Phi }_{\mathrm{mask}}\left(x,y\right)=\mathrm{comb}\left(\frac{x}{a},\frac{y}{b}\right)\ast \left[\mathrm{rect}\left(\frac{x}{a},\frac{y}{b}\right)\cdot {\Phi }_{\mathrm{submask}}\right],$$

(1)

where a and b represent the dimensions of the microlens in the x-axis and y-axis directions, respectively. The comb and rect represent the comb function and rectangular function, respectively. The ${\Phi }_{\mathrm{submask}}$ is the phase distribution of four-quadrant binary phase in each subaperture. It can be written as:

$${\Phi }_{\mathrm{submask}}=\{\begin{array}{c}\frac{\pi }{2}, x_0{y}_0\ge 0 \\ 0, x_0{y}_0<0\end{array},$$

(2)

where $(x_0,y_0)$ represents the coordinates of the subaperture. Moreover, $x_0$ and $y_0$ take values in the range of [−a/2, a/2] and [−b/2, b/2], respectively. The four-quadrant binary phase modulation is one of the simplest phase modulations, while not causing excessive dispersion of the light spot [19].

The incident beam before the microlens array is:

$$E_{in}\left(x,y\right)=A_{near}\left(x,y\right)\cdot \exp \left[i{\Phi }_{in}\left(x,y\right)\right],$$

(3)

where $A_{\mathit{near}}$ and ${\Phi }_{\mathit{in}}$ represent the amplitude and phase distribution of the incident beam, respectively. ${\Phi }_{\mathit{in}}$ can be represented as a linear combination of M order Zernike polynomials as:

$${\Phi }_{\mathit{in}}\left(x,y\right)={\displaystyle \sum }_{i=1}^M{a}_i{Z}_i(x,y),$$

(4)

where $Z_i(x,y)$ represents the ith Zernike polynomial and $a_i$ represents the corresponding Zernike coefficients.

The light is then focused and imaged by the modulated microlens array. According to Fresnel diffraction, when the light propagates to the focal plane, the result is:

$$E_{\mathit{far}}\left(x^{\prime },y^{\prime }\right)=F^{-1}\left\{F\left\{E_{\mathit{in}}\left(x,y\right)\cdot T_{\mathit{tf}}\left(x,y\right)\right\}\cdot H\left(u,v\right)\right\},$$

(5)

where $F^{-1}\{\cdot \}$ represents the inverse Fourier transform, $F\{\cdot \}$ represents the Fourier transform, and $H\left(u,v\right)$ is the free-space optical transfer function. The $\left(x^{\prime },y^{\prime }\right)$ and $\left(u,v\right)$ represent the corresponding coordinates of diffraction field and frequency domain, respectively. $T_{\mathit{tf}}\left(x,y\right)$ is complex amplitude transmittance of the microlens array, which can be written as:

$$T_{\mathit{tf}}\left(x,y\right)=\mathrm{comb}\left(\frac{x}{a},\frac{y}{b}\right)\ast \{\mathrm{rect}\left(\frac{x}{a},\frac{y}{b}\right)\cdot \exp \left[-\mathrm{j}\frac{k}{2f}\left(x^2+y^2\right)+{\Phi }_{\mathit{mask}}\left(x,y\right)\right]\},$$

(6)

where k is wave number and f is the focal length of the microlens array.

2.2. Wavefront Reconstruction Algorithm

The conventional Shack–Hartmann wavefront reconstruction algorithm is unable to recover the wavefront details inside the subaperture at low spatial sampling. In order to retrieve wavefront details from the spot array image with more information, the incident wavefront is reconstructed using the stochastic parallel gradient descent (SPGD) algorithm, which has been successfully applied to the field of wavefront correction [20,21,22]. The Zernike coefficients are the control parameters of the SPGD algorithm. The correlation function ($C_{orr}$) between the theoretical and measured light intensities is used as the cost function:

$$C_{orr}(E_{\mathrm{far}},I_{\mathrm{far}})=\frac{{\displaystyle \sum }{\displaystyle \sum }\left({\left|E_{\mathrm{far}}\right|}^2-\overline{{\left|E_{\mathrm{far}}\right|}^2}\right)\cdot \left(I_{\mathrm{far}}-\overline{I_{\mathrm{far}}}\right)}{\sqrt{\left[{\displaystyle \sum }{\displaystyle \sum }{\left({\left|E_{far}\right|}^2-\overline{{\left|E_{far}\right|}^2}\right)}^2\right]\cdot \left[{\displaystyle \sum }{\displaystyle \sum }{\left(I_{far}-\overline{I_{far}}\right)}^2\right]}},$$

(7)

where ${\left|E_{\mathit{far}}\right|}^2$ and $\overline{{\left|E_{\mathit{far}}\right|}^2}$ represent the calculated far-field intensity distributions and its statistical averages, respectively, and both $I_{\mathit{far}}$ and $\overline{I_{\mathit{far}}}$ represent the corresponding measured far-field intensity.

The random perturbation vectors are generated to perform positive and negative perturbations on the Zernike coefficients. Then, the gradient of the correlation function is estimated based on the variation amount of the coefficients and the cost function. Finally, the Zernike coefficients of the incident wavefront are estimated accurately by searching the minimal value of the cost function in the gradient direction through continuous iterations and updates. The phase iteration equation can be written as:

$${\Phi }^{\left(n+1\right)}={\Phi }^{\left(n\right)}+\gamma \Delta {\Phi }^{\left(n\right)}\Delta c_{\mathit{orr}}{}^{\left(n\right)},$$

(8)

where ${\Phi }^{\left(n\right)}$ represents the phase of the nth iteration, γ represents the gain factor, and $\Delta c_{\mathit{orr}}{}^{\left(n\right)}$ represents the variation amount of the cost function after applying current perturbation. Figure 2 and Algorithm 1 illustrate the process of wavefront reconstructed by the SPGD algorithm. The termination condition is generally selected to reach a certain number (N) of iterations or reach a certain value (T) of the cost function.

Algorithm 1. The procedure of the Stochastic Parallel Gradient Descent (SPGD) method

Require: the cost function $c_{\mathit{orr}}$, the gain factor γ, the certain value of ${\Phi }_{in}$, the measured far-field intensity $I_{\mathit{far}}$, and the maximal number of iterations N.

Ensure: The optimal parameters γ.

Initialize the indexes $c_{\mathit{orr}}$ = 0, ${\Phi }_{in}$ = 0, n = 0;

Initialize the coefficients a(0) = [a1, a2, …, aM].

Whilen ≥ N or Corr ≥ T do

1:  Randomly generated disturbance vectors Δa(n) = [Δa1(n), Δa2(n), …, ΔaM(n)];

2:  Calculate the corresponding perturbation phase $\Delta {\Phi }^{\left(n\right)}=\Delta a^{(n)}\cdot Z$;

3:  Calculate the far-field distribution after forward perturbation

$E_{\mathit{far}+}{}^{\left(\mathrm{n}\right)}=E_{\mathit{far}}({\Phi }^{\left(\mathrm{n}\right)}+\Delta {\Phi }^{\left(\mathrm{n}\right)})$;

4:  Calculate the value of correlation function after forward perturbation

$C_{orr+}{}^{\left(\mathrm{n}\right)}$ = $C_{orr}(E_{\mathit{far}+},I_{\mathit{far}})$;

5:  Calculate the far-field distribution after negative perturbation

$E_{\mathit{far}-}{}^{\left(\mathrm{n}\right)}=E_{\mathit{far}}({\Phi }^{\left(\mathrm{n}\right)}-\Delta {\Phi }^{\left(\mathrm{n}\right)})$;

6:  Calculate the value of correlation function after negative perturbation

$C_{orr-}{}^{\left(\mathrm{n}\right)}$ = $C_{orr}(E_{\mathit{far}-},I_{\mathit{far}})$;

7:  Calculate the variation amount of correlation function $\Delta c_{\mathit{orr}}{}^{\left(n\right)}=C_{orr+}-C_{orr-}$;

8:  Calculate new phase distribution ${\Phi }^{\left(n+1\right)}={\Phi }^{\left(n\right)}+\gamma \cdot \Delta {\Phi }^{\left(n\right)}\cdot \Delta c_{\mathit{orr}}{}^{\left(n\right)}$;

9:  n = n + 1;

end while

3. Numerical Simulation

To investigate the performance of the FBPM-SHWFS, a series of simulations are executed. The key parameters of the Shack–Hartmann wavefront sensor in the simulation are shown in

Table 1. Key parameters of the FBPM-SHWFS.

Parameters	Values
wavelength (λ)	1064 nm
microlens size	960 µm
focal length of the microlens	27 mm
pixel size	15 µm × 15 µm

.

3.1. Analysis on 2 × 2 Subapertures

Firstly, the performance of the FBPM-SHWFS is evaluated using the single Zernike polynomials (excluding piston) with a coefficient of 1 as the incident wavefront. The root mean square (RMS) of the incident wavefronts is about 0.16λ. Figure 3 compares the residual wavefront RMS of three methods with 2 × 2 subapertures. The modal algorithm [11] can only effectively recover the first four orders of Zernike polynomials. The SPGD algorithm based on SHWFS is able to effectively recover the first 35 orders of Zernike polynomials. However, with the help of the four-quadrant binary phase modulation, the first 80 orders of Zernike polynomials can be recovered with high precision by the SPGD algorithm. The results indicate that the modulated spot array image is able to provide higher-order aberration wavefront information.

To further validate the performance of the FBPM-SHWFS, a random incident wavefront is used to evaluate the FBPM-SHWFS. As shown in Figure 4, the wavefront consists of the first 35 orders of Zernike polynomials (excluding piston and tilt), whose coefficients are generated randomly based on the Kolmogorov turbulence model [23,24]. The wavefront’s RMS and peak-to-valley (PV) are 0.5320λ and 3.1169λ, respectively. The corresponding spot intensity distributions of the 2 × 2 SHWFS and FBPM-SHWFS are shown in Figure 5a1,b1, respectively. The sub-spot in the upper right corner of spot arrays of the SHWFS and FBPM-SHWFS are shown individually in Figure 5a2,b2, respectively. It is clear that the four-quadrant binary phase modulation changes the intensity and shape of the spot.

Figure 6 illustrates the wavefront reconstruction results of the three methods. The 2 × 2 SHWFS can only obtain the average slope of the four sub-wavefronts, which is insufficient to describe the incident wavefront. As can be seen, the modal algorithm cannot effectively reconstruct the incident wavefront. However, the SPGD algorithm retrieves the phase directly from the spot intensity distribution. This is beneficial to obtain more wavefront information. The RMS of the residual wavefront is 0.0718lλ based on the SHWFS, which is 16.37% of that of the incident wavefront. However, with the help of the four-quadrant binary phase modulation, the RMS of the residual wavefront base on the FBPM-SHWFS is reduced to 0.0243lλ, which is only 5.54% of the input value. Figure 7 compares the evolution curves of the residual wavefronts’ RMS for both structures. Obviously, the time cost of the FBPM-SHWFS is greatly reduced compared to the SHWFS, and the number of iterations is reduced from 1500 to 440 for the same reconstruction precision. These results indicate that four-quadrant binary phase modulation is helpful for the SHWFS to improve wavefront reconstruction precision and speed up wavefront retrieval.

To further evaluate the performance of the FBPM-SHWFS, we generate 80 sets of random incident wavefronts, with the mean RMS of 0.5320λ and the mean PV of 3.1169λ, and obtain the corresponding spot array images of the SHWFS and the FBPM-SHWFS, respectively. Figure 8a is the RMS of residual wavefronts after 1500 iterations. The RMS of the residual wavefront for the SHWFS is distributed between 0.0664λ and 0.1756λ. The mean and standard deviation are 0.1183λ and 0.0224λ, respectively. The RMS of residual wavefronts for the FBPM-SHWFS is distributed between 0.0168λ and 0.0829λ, with a mean and standard deviation of 0.0516λ and 0.0133λ, respectively. This means that the reconstruction accuracy of the FBPM-SHWFS is 56% better than that of the SHWFS. Figure 8b shows the number of iterations when the residual wavefront RMS reaches 0.1λ. The average number of iterations for the SHWFS is 1903. For the FBPM-SHWFS, the average number of iterations is 739. The results reveal that the FBPM-SHWFS takes less time compared with the conventional SHWFS for the same reconstruction accuracy.

3.2. Analysis on Different Subaperture Numbers

To analyze the effect of the subaperture number on the reconstruction precision of the FBPM-SHWFS, a test wavefront is reconstructed by the FBPM-SHWFS and SHWFS with subaperture numbers from 2 × 2 to 10 × 10, respectively.

Figure 9 provides the comparison results of the test wavefront for three methods, which intuitively illustrate the characteristics of each method. The wavefront reconstruction accuracy of all three methods tends to increase as the number of subapertures increases. As shown in Figure 9, the number of subapertures has a great influence on the reconstruction precision of the modal algorithm, while the method proposed in this paper has almost no effect. With low spatial sampling, the proposed method is still able to reconstruct the wavefront with high precision.

Figure 10 shows the statistical results of the RMS of 80 sets of residual wavefronts. When the number of subaperture is less than 8 × 8, the average slope information within subapertures cannot accurately represent the wavefront. This leads to the inability of the modal algorithm to effectively reconstruct the wavefront. However, the accuracy of the modal algorithm improves rapidly as the spatial frequency increases. As expected, the RMS of residual wavefront of the FBPM-SHWFS remains lower than those of the two methods when the subaperture is lower than 6 × 6. The above results demonstrate that modulated spot array intensity distribution is beneficial to extract more information and improve the wavefront detection precision. The FBPM-SHWFS demonstrates a high degree of robustness with different subaperture numbers.

4. Experiment

4.1. Experimental Setup

As shown in Figure 11, an optical experiment was designed to verify the validity of the proposed method. The light source of this optical system is a collimated 635 nm laser beam. The beam sequentially passes through a neutral optical attenuator (OA) that controls the beam energy, an aperture (AP) with a diameter of 1.6 mm, a linear polarizer (POL), and a non-polarizing beam splitter (BS) before reaching the spatial light modulator (SLM). The beam is vertically reflected by the SLM; at the same time, the phase is modulated and then reflected by the BS before being captured by the camera. The parameters of the main optical devices in this experiment are given in

Table 2. The parameters of the main optical devices in the experiment.

Parameters	Values
wavelength (λ)	635 nm
microlens size	800 µm
microlens numbers	2 × 2
focal length of the microlens	8 cm
diameter of aperture	1.6 mm
the hosting optical system field of view	0.57°
pixel size of CCD	6.4 µm × 6.4 µm
pixel size of SLM	12.5 µm × 12.5 µm

.

In the optical system, the SLM is employed to simulate the microlens array and also to generate the aberration wavefront. Therefore, the distance between the surface of the SLM and the camera is the focal length of the generated microlens array. Figure 12a,b exhibit the images used to generate the conventional microlens array and the four-quadrant binary phase-modulated microlens array, respectively. They are loaded on the SLM. Figure 12c,d show the acquired spot array images of the SHWFS and the FBPM-SHWFS with the plane waves, respectively. To demonstrate the intensity and shape of the spots more clearly, we display the single sub-spots with a hot colormap, as shown in Figure 12e,f, respectively.

4.2. Experimental Result and Disscussion

First, we calibrate the FBPM-SHWFS and SHWFS structure systems using Figure 12c,d acquired by the camera, respectively. As shown in Figure 13a, a set of test wavefronts is then generated randomly to evaluate the performance of the FBPM-SHWFS. The aberrated wavefront is a linear combination of the first 20 orders (except piston and tilt) of Zernike polynomials, which has an RMS of 0.2111λ and a PV of 1.0816λ. Figure 13b,c are the acquired spot array images of the FBPM-SHWFS and SHWFS, respectively. It is obvious that the modulated spot is more diffuse than the unmodulated spot, which helps to extract more wavefront information from the spot.

The comparison of the wavefront reconstruction results from the two structures is presented in Figure 14. The reconstructed wavefront and residual wavefront with the SHWFS are given in Figure 14a,b. The RMS and PV of the residual wavefront are 0.0552λ and 0.4132λ, respectively. Figure 14c,d are the reconstructed wavefront and residual wavefront with the FBPM-SHWFS. The RMS and PV of the residual wavefront are 0.0253λ and 0.1912λ, respectively. The results indicate that the FBPM-SHWFS is able to achieve approximately 54.16% and 58.27% reduction in RMS and PV of the residual wavefront over the SHWFS, respectively. Figure 15 illustrates the recovered Zernike coefficients. It is obvious that the FBPM-SHWFS structure has higher recovery accuracy than the conventional structure for each order of Zernike coefficients.

The FBPM-SHWFS proposed in this paper has a great advantage in improving the wavefront reconstruction precision. Further, we validate the impact of the FBPM-SHWFS on the SPGD algorithm. Figure 16 provides a comparison of the residual wavefront’s RMS as a function of iterations for both structures. The phase retrieval speed of the SPGD algorithm is significantly accelerated based on the FBPM-SHWFS. As expected, this phenomenon is displayed in Figure 16. Compared to the typical SHWFS, the FBPM-SHWFS structure requires fewer iterations for the same reconstruction precision. Thanks to the modulation of the four-quadrant binary phase, the number of iterations is reduced from 2835 to 450 for the wavefront residual RMS value of 0.05λ.

5. Conclusions

To improve the reconstruction precision of the SHWFS with extreme low spatial frequency, a kind of Shack–Hartmann wavefront sensing technology based on four-quadrant binary phase modulation is proposed. This technology introduces four-quadrant binary phase modulation in each subaperture and uses the optimization algorithm to retrieve wavefront detail. A large number of numerical simulations are conducted to validate the feasibility and performance of the proposed method. The simulation results show that the four-quadrant binary phase makes the spot more diffuse, which is more helpful to retrieve more wavefront information from the spot intensity distribution. This is a great help to improve the wavefront reconstruction precision and the convergence rate of the SPGD algorithm under the condition of extreme low spatial frequency. In addition, the method proposed in this paper demonstrates good robustness with different spatial sampling. The experimental results reveal that the FBPM-SHWFS is able to achieve approximately 54% and 58% reduction in the residual wavefront RMS and PV over the SHWFS, respectively. The number of iterations is reduced from 2835 to 450 for the residual wavefront RMS of 0.05λ. In future work, we will explore the influence of the incident beam energy, spot size, and other factors on reconstruction precision. The subject of temporal frequencies is also what we will focus on in the next step.