Adaptive optics (AO) technology can effectively correct the dynamic wavefront distortion in real time [1]. It has been widely used in astronomical observation [2], laser beam purification, retinal high-resolution imaging, laser communication and other fields [3–5]. However, all AO systems applied in these fields would be affected by a bright light from background, backscatter or straylight. Therefore, in order to avoid the effect of the daytime stray light in astronomical observation, so far, almost of astronomical observation are working at night or the other low background or stray light scene. Similarly, in the field of laser communication, the stray light generated from high-power laser transmission will be received by the high-sensitivity detector. During the measurement of ocular aberrations, the signal light scattered from fundus will be mixed with corneal reflex. The bright stray light or corneal reflex which would affect the sensitivity of AO system. Shack-Hartmann wavefront sensor (SHWFS) [6, 7] is one of the most important components of an AO system, which is used to detect the wavefront distortion from the target to observation [8]. Many applications show that the SHWFS can meet a high accuracy and achieve a stable property under the night or the other low background or stray light scene. However, under the bright background light scenes such as the above conditions of daytime stray light and corneal reflex, the signal-to-background ratio (SBR) [9] or signal-to-noise ratio (SNR) of each sub-aperture of SHWFS will be limited obviously, where SBR usually used to indicate the intensity level of the background light. Under ultra-worse condition, one may not find any signal light from the SHWFS, which limit its measurement ability of wavefront aberration. For solving this problem, some authors have proposed several kinds of methods. Beckers [10] applied the active wavefront detection technology based on ultra-narrow band filtering principle to daytime AO system. Gonglewski [11] used the field stop to eliminate the skylight, but this could not effectively solve the impact of bright skylight on wavefront detection in daytime. [12] proposed the method of field shift SHWFS, which could detect the mixed information and skylight separately through a tilt mirror or beam splitting prism. However, this field shift method requires a time-sharing detection or two-way wavefront detection set-up, which increased the complexity of control. On the anther hand, the method can only eliminate the evenly distributed skylight. [13] proposed using polarization filtering technology to suppress polarized background light, but the polarization of skylight is relatively weak, and the filtering method also weaken the target signal. Therefore, this polarization filtering technology only have a limited improvement in detection contrast of the target signal. Huang et al. [14] proposed a new daytime pyramid wavefront sensor (PyWFS) to separate the object signal from background signal to improve the SNR in daylight AO system. The results shown that the new PyWFS approach was practical when SNR is greater than 1.9 but less than 16.9. In fact, these proposed methods have limited improvement on the performance of SHWFS for daytime AO system in terms of hardware. In addition, there are some improvements come from the algorithm of data processing. Li et al. [15] proposed a SHWFS-Neural Network (SHNN) to calculate centroid in SHWFS from the extreme situations such as strong environmental light and noise pollution. The results shown that a False Rate with 50 hidden layer neurons was 6% under the peak SNR of 3. Xu et. al [16] propose a Gaussian modeling centroid extraction algorithm which performs real-time daylight AO closed-loop corrections. This method can provide a lower centroid estimation error under bright sky background conditions. Guo et al. [17] proposed a new wavefront detection method based on linear polarized modulation. The proposed method can reduce the influence of bright skylight, but the rotating polarizer limits the performance of real-time and accuracy. The mentioned methods can improve in some certain the SBR or SNR of the SHWFS, but the performance of the SHWFS cannot meet the requirement under bright stray light.

Based on above methods, we proposed a polarized SHWFS (p-SHWFS) in this paper, which utilize the difference of state of polarization (SoP) between signal and stray light to improve the image contrast. The p-SHWFS can be composed simply by a micro-lens array and a linear polarization camera. The camera uses four directional polarizing filters at 0°, 135°, 45° and 90° on every four pixels. Light passing through these four filters can then be linear interpolated to provide a signal pixel value in polarized dimension. Then, the unpolarized and polarized light can be identified and separated. Thus, the degree of linear polarization (DoLP) and angle of linear polarization (AoLP) for each sub-aperture can be analyzed, and the SBR can be improved in some certain depending on the difference of SoP. In this paper, the basic principle of the proposed p-SHWFS is described, and its performance is also validated by numerical simulation and practical experiments. The experimental results show that the p-SHWFS can improve obviously the measurement accuracy under strong stray light when the difference of SoP exists.

Parameters of p-SHWFS used in simulation are shown in Table 1.

In order to completely eliminated the effect of unpolarized light on wavefront aberrations measurement, the incident light intensity I (x, y) is generated by numerical method as, I x , y = I P x , y + I N x , y (10) where, (x, y) is the sampling point at the detector. The object and stray light may be partially light. Thus, I P x , y is the polarized parts of the total light and the direction is θ. I N x , y is the unpolarized parts of the total light. To quantify the relative intensity of the stray light, the SBR is defined as, SBR = 10 lg ∬ x , y ∈ D I N x , y ∬ x , y ∈ D I P x , y (11)

The performance of p-SHWFS is affected by the difference of SoP between the signal and the stray light. Thus, in order to quantify the polarization characteristics of I N x , y , the polarization degree P of I N x , y is defined as, P = I N m a x − I N m i n I N m a x + I N m i n (12) p-SHWFS can remove the I N x , y completely when the P is zero. By simulation, the total light intensity calculated from I ₀, I ₁₃₅, I ₄₅ and I ₉₀ are shown in Figure 2. It can be seen that the signal is completely submerged in the background (such as Figure 2C) with the increment of stray light intensity. Thus, it is difficult to measure wavefront aberrations accurately by the traditional method from the total light.

In order to verify the wavefront measurement ability of p-SHWFS, the target wavefront and aberration components (the first 35 Zernike coefficients) are shown in Figure 3.

Figures 4A,B show the difference of wavefront measuring accuracy by traditional method and p-SHWFS under the polarization degree P of stray light is zero. It can be seen that the traditional method cannot accurately measure the target wavefront from the bright stray light. While the p-SHWFS can completely eliminate the stray light, as can be derived from Eq. 6. Therefore, under the P of stray light is zero, the RMSE (root mean square error) of Zernike coefficient restored by p-SHWFS is approximately zero (as in Figure 4A, the value of blue line is about 10^–14 μm). The RMSE and PV (peak valley) values of the residual wavefront are also zeros (Figure 4B, the values marked with “Δ” and hexagon are approximately 10^–13 μm). However, the ability of p-SHWFS to suppress stray light can be affected by uneven extinction ratio of micro polarizer array, detector quantum noise, readout noise and I _N(x,y). Limited to space, we only analyzed the influence of I _N(x,y) on the measurement accuracy of p-SHWFS.

Figures 4C,D show the wavefront measurement accuracy of the traditional method and p-SHWFS under different SBR when p = 20%. The results show that the traditional methods cannot effectively eliminate the stray light, which obviously restrict its wavefront measurement ability. However, the p-SHWFS can effectively eliminate the stray light (p = 20%, SBR ≤12.35 dB). RMSE of the residual wavefront is less than 0.0345 μm (≤0.0528 λ), PV is less than 0.2327 μm (≤0.3564 λ). RMSE of Zernike coefficient ≤0.0251 μm (0.0384 λ). From the results, we can know that the ability of p-SHWFS to eliminate the stray light will declined just when the SBR ≥13.10 dB (p = 20%).

Thus, the p-SHWFS can take advantage of the difference of SoP between the signal and stray light to effectively eliminate the influence of strong stray light on the wavefront detection accuracy.

In order to verify the wavefront detection ability of p-SHWFS, the optical path shown in Figure 5 was built. Where LD is the fiber laser with the wavelength λ= 653 nm. The lenslet array of p-SHWFS is 19 × 19 (lenslet spacing 0.36 mm, the number N of effective sub aperture is 253). The resolution of the detector is 2448 × 2048 (pixel size 3.45 μm, pixel format with polarized mono12, TRI050S-P, LICID Ltd.). The light beam emitted by LD passes through the collimator L and the polarizer Pr. The passing components will become the polarized light I _P. A strong stray light I _N from an incandescent lamp is transmitted to the p-SHWFS along the optical path. I _P and I _N are similar to the simulation and also can be explained as in Eq. 10. After passing through the linear polarizing filters of the p-SHWFS, the DoLP (one-quarter of the original 2448 × 2048 pixels) of the incident intensity will only remain the polarized components I _P and remove the unpolarized I _N. On the contrary, it is still true for the p-SHWFS to remain the unpolarized components I _N and remove the polarized components I _P, as I _{N =} I—I _P.

The results in our experiment and simulation have shown that the p-SHWFS would be a promising approach to improve the imaging contrast in AO systems to detect phase distortion characteristics. In order to facilitate the analysis, we used a mathematical model as shown in Eq. 10 to simulate the mixed incident light, where the difference of SoP between the object and the stray light can be modulated by changing the polarization degree of stray light. The simulated results shown that the performance measurement of p-SHWFS can be affected by the polarization degree of the stray light. In the future analysis and experiments, we will further focus on the performance measurement of p-SHWFS under greater polarization degree of partially polarized stray light.

In summary, a new p-SHWFS based on polarization imaging principle is proposed in this paper. The intensity information would be transformed into polarization dimension using micro linear polarizer array at the directions of 0°, 135°, 45° and 90°, so as to take advantage of the significant difference of SoP between stray and object light. The incident mixed light of each sub aperture of p-SHWFS is modulated in four directions to eliminate the interference of stray light. It can improve the SBR and wavefront detection ability under extreme conditions. The feasibility and accuracy of this method have been verified by simulation analysis and experimental system. The experiment results show that the RMSE and PV of wavefront measured by p-SHWFS are ≤0.0070 μm and ≤0.0694 μm under the stray light with SBR are 12.51 dB, 15.60 dB and 17.04 dB. RMSE of Zernike coefficient are less than 0.0394 μm. Compared to the traditional method, the ability of wavefront measurement by the p-SHWFS is obviously improved under the strong stray light when the difference of SoP exists. That may give us some initial reference to reduce the influence of background or stray light in laser communication, target detection, vision optics and other application fields.