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Abstract
We propose a fast and robust method for calibrating Spatial Light Modulators (SLMs) based on polarization phase-shifting interferometry. Our method effectively calibrates the SLM by addressing both the static aberration and nonlinear phase response, utilizing specially designed gray images loaded sequentially onto the SLM. Notably, we introduce a novel kinoform that effectively eliminates the influence of tilt phase shift between two shots of the polarization camera. This results in a highly accurate phase aberration map and phase modulation curve with exceptional stability, making it an ideal method to calibrate the SLM with exceptional efficiency and precision in real applications.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Liquid-crystal spatial light modulators (LC-SLMs), as dynamic and reconfigurable devices, have the capability to manipulate the phase, amplitude, and polarization state of incident light. Among them, the phase-only reflective SLM stands out for its high diffraction efficiency and large phase modulation depth, making it indispensable in numerous fields such as laser processing [1,2], null measurement of aspheric surfaces [3,4], optical tweezing [5,6], microscopic imaging [7], and adaptive optics [8]. To ensure the high modulation precision required for these applications, two critical factors need to be addressed: the static aberration caused by the non-flatness of the SLM's backplane and the nonlinearity of the phase response resulting from the complex physical model of liquid crystal molecules in terms of voltage, deflection angle, and equivalent refractive index. Consequently, accurate calibration of the SLM becomes necessary.
Since the advent of SLM, numerous calibration methods have been proposed, which can be roughly categorized into three groups. The methods in first group [9–14] only achieve the compensation for the static aberration by using Shack-Hartmann sensor [9], phase-retrieval algorithm [11], interferometry [10,12–14], etc. The methods in second group [15–25] aim at correcting the nonlinear phase response. Among them, diffraction self-reference interferometric methods [15,18], shear interferometric methods [19,25], polarization-based methods [16,21,24] and diffraction intensity analysis method [23] are roust against experimental turbulences. However, the approaches in this group cannot measure the static aberration. The methods of last group are full calibration approaches which can address both the static aberration and nonlinear phase response of SLM simultaneously. These approaches stand out for their comprehensiveness and efficiencies, and therefore are detailly discussed in the following paragraph.
Samuel McDermott [26] employed ptychography to measure the phase response of the SLM. While this approach theoretically offers an unlimited field of view, the experimental recovered phase map exhibits noticeable boundary noise, which is attributed to the stitching algorithm. Haolin Zhang [27], based on the Shack-Hartmann principle, calibrated the SLM by loading several kinoforms composed of a lens array. However, the static aberration measured by this method provides only a rough estimation, and significant errors can be observed in the corners of the SLM. Both of these diffraction-based calibration methods suffer from limited accuracy. Aiming at calibrating SLM precisely, interferometry [28–34] are more widely used. Traditional interferometric methods [28,30] employ the non-common path Twyman-Green configuration combined with temporal phase-shifting technology, making them susceptible to environmental disturbances. Efforts to overcome this limitation can be classified into two types. On one hand, Minchol Lee [34] achieved rapid acquisition by simultaneously controlling the SLM and camera, acquiring 18 interferograms in just 4.5 seconds. This rapid acquisition method effectively reduces the impact of low-frequency environmental turbulence but struggles with high-frequency turbulences. On the other hand, Qiang Lu [31] and Zhen Zeng [33] utilized a Fizeau interferometer to calibrate the SLM. Although this semi-common path setup is more robust, it remains vulnerable to the vibration of SLM itself. To date, no full calibration approach exists that simultaneously guarantees high precision and strong robustness against experimental turbulence. Additionally, the aforementioned full calibration methods require the loading of multiple kinoforms on the SLM when correcting for the nonlinear phase response, resulting in time-consuming procedures.
To efficiently, accurately, and stably compensate for the SLM's static aberration and nonlinear phase response, this paper proposes a novel full calibration method based on polarization multiplexing. In comparison to existing full calibration approaches, the proposed method offers three distinct advantages. Firstly, by combining a polarization camera with polarization phase-shifting technology, four interferograms can be simultaneously acquired. This feature ensures that our method remains insensitive to environmental turbulence and achieves a precision as high as traditional interferometric methods, when the static aberration is measured. Secondly, the proposed method only requires two shots of the polarization camera to achieve full compensation of the SLM, significantly enhancing the efficiency of calibration process. Thirdly, a unique kinoform is designed to effectively eliminate the influence of tilt phase shift between two shots of camera. This innovative kinoform further enhances the robustness of the proposed method.
The remainder of this paper is organized as follows. Section 2 presents the principle and process of the proposed calibration method, including the principle of polarization phase shift interferometry, the process of compensating static aberration, and the correction flow for nonlinear phase response. In Section 3, a commercially available SLM (Upololabs HDSLM80R) with a resolution of 1200*1920 is calibrated using the proposed method. Additionally, the performance of the proposed method is compared with three representative calibration methods: the Twyman-Green based method to compensate for static aberration, the diffraction grating self-interferometry method and the polarimetric method to correct nonlinear phase response. Finally, the main conclusions are summarized in Section 4.
2. Principle of calibration
2.1 Optical setup and principle of polarization phase-shifting interferometry
The proposed calibration method is based on Twyman-Green interferometry, as is shown in Fig. 1(a). Passing through a polarizer (0°, that is the modulation direction of SLM) and collimated by a beam-expanding system (composed of an objective, a pinhole and a collimation lens), the wavefront can be formulated by Jones matrix as:
where ${A_0}$ is the amplitude of wavefront. Next, ${E_1}$ is divided to the reference beam ${E_R}$ and the test beam ${E_{SLM}}$ by BS. The reference beam ${E_R}$, which is reflected by a mirror and passes through a quart wave plate (QWP, at 45°) twice, becomes:
Fig. 1. Schematic diagram of the proposed calibration method. (a) The optical setup. (b) The polarization channel distribution of polarization camera. (c1) and (c2) Two kinoforms loaded on SLM. (d1) and (d2) Two sets of interferograms extracted from polarization camera’s four channels when (c1) and (c2) are loaded respectively. (e1) and (e2) The restored phase maps of (c1) and (c2) respectively, and (e1) is the static aberration of SLM. (f) Equals to (e2) minus (e1), and is used to correction the nonlinear phase response of SLM.
It should be noted that ${E_{SLM}}^{\prime}$ cannot interfere with ${E_R}^{\prime}$ because their polarization directions are perpendicular to each other. Subsequently, the merged beam ${E_2}$ passes through a 4f-system and a QWP at 45°, becoming:
The 4f- system makes the SLM plane and the CCD plane conjugate, so the edge effect [35], which is caused by the diffractive artifacts at the edges, can be avoided.
If using a polarizer to modulate ${E_2}^{\prime}$, the obtained intensity map can be formulated as:
As is depicted in Fig. 1(b), the polarization camera has four channels whose polarization angles are $\theta $=0°, 45°, 90° and 135° respectively. Hence, the intensity maps captured by polarization camera’s four channels are:
Utilizing four-step phase-shifting algorithm, the demodulated phase map can be calculated as:
where ${\phi _{SLM}}$ represents the wrapped phase map of ${\varphi _{SLM}}$, and its value range is limited in $[{ - \pi ,\pi } ]$ by the function $arc\tan ({\bullet} )$. Using the phase unwrapping algorithm [36–38], true phase map ${\varphi _{SLM}}$ can be retrieved from ${\phi _{SLM}}$.2.2 Compensation of static aberration
The calibration process for static aberration is straightforward. By loading a plane kinoform (as shown in Fig. 1(c1)) onto the SLM, four interferograms (as seen in Fig. 1(d1)) can be synchronously captured by the polarization camera. After the demodulation and unwrapping processes, the resulting phase map, denoted as ${\varphi _{SA}}$, represents the static aberration of SLM.
It is worth mentioning that the phase unwrapping algorithm in Ref. [36] is applied here. This algorithm, based on Zernike polynomial fitting, effectively reduces errors caused by the fringing-field effect [39] in two ways. Firstly, it ensures that the restored phase map is continuous, resulting in a smooth compensation kinoform that avoids fringing-field errors caused by outlier noise points. Secondly, a tilt-free phase map ${\varphi _{SA}}^{\prime}$ can be easily obtained by setting the first three coefficients of the Zernike polynomial to zero. This operation significantly decreases the fringe number of compensating kinoform and therefore the corresponding fringing-field error is reduced.
To compensate the static aberration of SLM, the corresponding kinoform can be calculated as:
where function $\bmod ({\bullet} )$ returns the modulus after division. It should be mentioned that the compensation kinoform is effective only when the nonlinear phase response of SLM has been calibrated to range $[{0,2\pi } ]$. This calibration process of phase response is illustrated in the next section.2.3 Correction of nonlinear phase response
To efficiently calibrate the nonlinear phase response of SLM, the kinoform with tilt phase map (equals to the upper part of Fig. 1(c2)) is commonly employed. Loading this kinoform only, Praveen Kumar [21] measured the phase response of SLM under different gray values. This method is based on polarimetry, rendering it immune to inherent static aberration and environmental disturbances. However, this kinoform is not a robust option for our optical setup, which relies on non-common-path interferometry. This is due to the tilt phase introduced between two camera shots, resulting from environmental vibrations, which significantly compromises the calibration precision. The forthcoming experimental results in Section 3.1 will further validate this observation.
To avoid the influence of tilt phase, we have designed a novel kinoform in which the tilt of the upper part is opposite to the tilt of the lower part. This innovative kinoform is depicted in Fig. 1(c2) and can be mathematically represented as:
When the kinoform $G(x,y)$ is loaded onto SLM, four interferograms can be extracted from a single shot of the polarization camera, as depicted in Fig. 1(d2). These four interferograms are demodulated by Eq. (9) and then unwrapped by the TIE based method in Ref. [37,38]. It should be noted that the Zernike-based method described in Ref. [36] is not suitable for this scenario, as the phase response of $G(x,y)$ cannot be fitted by Zernike polynomials well. The unwrapped phase map, shown in Fig. 1(e2), can be expressed as:
where the function $f(g)$ is the phase response of SLM under the gray value g; ${\varphi _{tilt}}$ is the tilt phase map caused by the environmental vibration, and it can be formulated as: where $\alpha$ represents the x-axis tilt phase coefficient, $\beta$ represents the y-axis tilt phase coefficient, and $\delta$ represents the spatially uniform phase shift.After obtaining ${\varphi _{SA}}$ and ${\varphi _2}$, the flow chart of calibrating the nonlinear phase response of SLM is presented in Fig. 2 and described as follows. First of all, the subtraction of ${\varphi _{SA}}$ from ${\varphi _2}$ yields the remaining phase map:
As is shown in Fig. 1(f) or Fig. 2(a), ${\varphi _3}(x,y)$ can also be divided into three parts. Secondly, the upper part ($0 \le y \le 0.4$) of ${\varphi _3}(x,y)$ can be formulated as:
Fig. 2. The process of calibrating SLM’s nonlinear phase response. (a) The larger vision of Fig. 1(f) (${\varphi _3}$). (b) The plot calculated by averaging the upper part of (a) along y axis (${C_1}(x)$). (c) The averaging of lower part (${C_2}(x)$). (d) Obtained by flipping (c) along x axis (${C_3}(x)$). (e) The final nonlinear phase response of SLM (${C_5}(x)$).
By averaging the upper part along the y-direction, a one-dimensional curve (depicted in Fig. 2(b)) is obtained:
In fact, ${C_1}(x)$ is the measured phase response of SLM when the traditional tilt phase kinoform in Ref. [21] is applied. According to Eq. (17), this phase response curve is susceptible to the x-axis tilt coefficient $\alpha$. Thirdly, the lower part ($0.6 \le y \le 1$) can be represented as:
By averaging the lower part along the y-direction, another one-dimensional curve (illustrated in Fig. 2(c)) is calculated:
Fourth, ${C_2}(x)$ is flipped along the x-axis, resulting in:
Finally, a tilt-free curve is obtained by combining ${C_1}(x)$ and ${C_3}(x)$:
In general, SLM’s phase response under the gray of 0 defaults to 0 rad ($f(0 )= 0$), thus Eq. (21) can be reformulated as:
Essentially, the curve ${C_5}(x)$ represents the plot of SLM’s phase response under different gray levels.
3. Experimental results
To validate the effectiveness of the proposed method, a SLM (Upololabs HDSLM80R) with a resolution of 1200*1920 was calibrated at the wavelength of 633 nm.
3.1 Correction of nonlinear phase response
By successively loading the kinoforms depicted in Fig. 1(c1) and (c2), 8 interferograms (seen in Fig. 1(d1) and (d2)) can be extracted from the two shots of polarization camera. After demodulation and unwrapping, ${\varphi _{SA}}$ and ${\varphi _2}$ are obtained and depicted in Fig. 1(e1) and (e2) respectively. According to Eq. (15), the difference phase map ${\varphi _3}(x,y)$ is calculated, as shown in Fig. 2(a). It is observed that the dotted line in Fig. 2(a) exhibits a slight slant, indicating the presence of an x-axis tilted phase component in ${\varphi _3}(x,y)$. By performing averaging operations, ${C_1}(x)$ and ${C_2}(x)$ are derived and illustrated in Fig. 1(b) and (c) respectively. Notably, the peak-to-valley (PV) value of ${C_1}(x)$ (14.19 rad) is smaller than that of ${C_2}(x)$ (14.42 rad), indicating the existence of a y-axis tilted phase in ${\varphi _3}(x,y)$. Utilizing Eqs. (20–23), the phase response of the SLM under different gray values (${C_5}(x)$) is calculated, as depicted in Fig. 2(e). It is evident that this curve (PV = 14.31 rad) is smooth and free from outlier noise points, thus demonstrating the effectiveness of the proposed calibration method.
To demonstrate the stability of the proposed method, the nonlinear phase response of SLM was measured extra 8 times. Figure 3(a) displays the 9 curves of ${C_5}(x)$ obtained from these measurements, which exhibit a high degree of coincidence. This indicates the robustness and stability of our calibration method. Moreover, to illustrate the detrimental effects of tilt phase, the results of ${C_1}(x)$ are presented in Fig. 3(b), and the standard deviation of ${C_5}(x)$ and ${C_1}(x)$ under different gray values is plotted in Fig. 3(c). It can be observed from Fig. 3(b) and (c) that the fluctuations of ${C_1}(x)$ increase linearly with the gray values. This observation confirms the conclusion stated in Section 2.3, namely, that the stability of calibration is significantly impacted by the tilt phase map between the two shots of the camera when utilizing the traditional tilt phase kinoform. However, with the proposed novel kinoform, the fluctuations caused by the tilt phase map can be effectively eliminated, as demonstrated by the comparison in Fig. 3(c).
Fig. 3. The results of phase response with 9 times. (a) The result of our method (${C_5}(x)$). (b) The result of traditional tilt phase kinoform (${C_1}(x)$). (c) The standard deviation of ${C_1}(x)$ and ${C_5}(x)$.
To further demonstrate the robustness of the proposed method, we compare it with two representative calibration methods: the Stokes polarimetric method [21] and the diffraction grating self-interference method [18]. Both of these methods are based on the common-path interferometry and are known for their strong robustness against environmental disturbances. We calibrate the nonlinear phase response of the SLM using each of these methods, repeating the calibration process 9 times. The measured phase response curves obtained from the polarimetric method and the self-interference method are illustrated in Fig. 4(a) and Fig. 4(b) respectively. Additionally, the standard deviations of three methods (polarimetric method, self-interference method and the proposed method) are compared and plotted in Fig. 4(c). It can be concluded from Fig. 4(c) that the stability of the proposed method is comparable to the other two methods. The standard deviation values of the proposed method are consistently smaller than 1/100 $\lambda $, indicating its reliability and robustness in calibrating the nonlinear phase response of SLM.
Fig. 4. The comparison results of robustness. (a) The repeated results of self-reference method. (b) The repeated results of polarimetric method. (c) The standard deviations of three methods.
After obtaining the phase response of SLM under different gray values, the nonlinearity can be amended using a lookup table (LUT) or gamma correction. In the case of the SLM used in this paper, the manufacturer provides a gamma correction software that allows for modifying the mapping between voltage and gray values by loading a gamma curve interpolated from the measured phase response. By applying gamma correction, the SLM's phase response under the gray levels of 0 to 255 becomes theoretically linear and limited to the range of [0, 2π]. In this paper, three gamma plots are calculated based on the curves shown in Fig. 3(a), Fig. 4(a) and Fig. 4(b) respectively, and only one curve from each figure is utilized. Figure 5(a) illustrates the plots of the three gamma curves corresponding to the proposed method, the polarimetric method, and the self-interference method, respectively. It can be observed from Fig. 5(a) that the curves of the proposed method and the polarimetric method align closely, indicating good agreement between them. However, the result of the self-interference method deviates from the other two curves. This discrepancy can be attributed to the fact that the self-interference method employs a Fourier transform-based approach to calculate the phase response, while a more accurate algorithm based on phase-shifting is applied in the proposed method and polarimetric method.
Fig. 5. Recalibration results of phase response. (a) The gamma curves used to correct the nonlinear phase response. (b) The comparison of idea phase response and remeasurement results. (c) The absolute phase error of three calibration methods.
Later, to verify the accuracy of the calibration method, the phase response of SLM (after gamma correction) is re-measured using the same calibration method. This recalibration process is conducted using the three calibration methods mentioned earlier, and their measurement results are compared with the ideal phase response, as shown in Fig. 5(b). Additionally, the absolute phase difference between the ideal case and the curves obtained from the three calibration methods is plotted in Fig. 5(c). From Fig. 5(b) and (c), it is evident that the proposed method achieves higher accuracy compared to the other two methods, with the mean absolute errors of our method being lower than 1/250 $\lambda $ (0.0247 rad). This result demonstrates that the nonlinear phase response of SLM can be calibrated with high precision using the proposed method.
3.2 Compensation of static aberration
After correcting the nonlinear phase response, the static aberration of SLM can be compensated. By loading the plane kinoform shown in Fig. 1(c1), four interferograms are captured synchronously. These four interferograms, one of which is shown in Fig. 6(a1), is demodulated by Eq. (9) and then unwrapped by DZPT (tilt phase removed by setting the first three coefficients of the Zernike polynomial to 0), obtaining the true phase map of SLM’s static aberration. The corresponding wrapped and unwrapped phase maps are shown in Fig. 6(a2) and (a3) respectively. It is worth noting that this operation was already performed in the prior process of calibrating the nonlinear phase response. Next, the compensation kinoform is calculated using Eq. (10), as depicted in Fig. 6(a4). To verify the accuracy of the compensation, the calculated kinoform is loaded onto SLM, and four additional interferograms are captured simultaneously. One of these interferograms is shown in Fig. 6(a5). It can be observed that the static aberration of the SLM is effectively compensated since the interferogram after compensation contains only straight fringes. To evaluate the compensation accuracy quantitatively, the four interferograms after compensation are demodulated according to Eq. (9), unwrapped by DZPT, and tilt phase removed using least-squares fitting. The resulting tilt-free residual phase map is shown in Fig. 6(a6) with a root mean square error (RMSE) lower than 1/100 $\lambda $ (0.0620 rad), demonstrating the high precision compensation capability of the proposed method.
Fig. 6. Compensation results of the static aberration. (a1-a7), (b1-b7) and (c1-c7) The results of the proposed method, the four-frames Twyman-Green method and the twenty-frames Twyman-Green method respectively. (a1-c1) The interferograms of the static aberration. (a2-c2) The wrapped phase maps of the static aberration. (a3-c3) The unwrapped phase maps of the static aberration. (a4-c4) The compensation kinoforms. (a5-c5) The interferograms after compensation. (a6-c6) The tilt-free residual phase maps after compensation.
As a comparison, the static aberration of the SLM is also calibrated using the traditional Twyman-Green interferometric method. Two different phase-shifting technologies are employed to evaluate the impact of environmental disturbances and vibrations: the four-steps phase-shifting technology and the twenty-steps phase-shifting technology. The demodulation algorithm used is AIA (described in Ref. [40]), and the phase maps are unwrapped using the DZPT method. The experimental results obtained using the four-steps approach are shown in Fig. 6(b1-b6), while the results obtained using the twenty-steps approach are depicted in Fig. 6(c1-c6). In Fig. 6(b5), the compensated interferogram of the four-steps approach exhibits bended fringes in the upper right corner, as marked by the red circle. These bended fringes originate from the demodulation error of AIA, which can handle random space-uniform phase shifts but is susceptible to unknown tilt phase shifts caused by environmental vibrations. In contrast, Fig. 6(c5) does not show these bended fringes because the error caused by the unknown tilt phase shift is averaged by using 20 frames of interferograms. This conclusion is further supported by Fig. 6(b6) and Fig. 6(c6), as the RMSE of the twenty-steps approach is lower than that of the four-steps approach. Unlike the traditional Twyman-Green method, the proposed method can obtain four interferograms synchronously, thereby avoiding the impact of tilt phase errors. Figure 6(a6-c6) demonstrates that the proposed method achieves the lowest RMSE value compared to the other methods, indicating its superior accuracy in compensating for static aberrations.
4. Conclusion
In this paper, a comprehensive calibration of SLM is presented based on polarization multiplexing. Both the static aberration and the nonlinear phase response of SLM can be simultaneously calibrated with only two exposures of polarization camera. On the one hand, benefitted from the synchronous capture of four interferograms, our method is more robust against environmental disturbances than traditional Twyman-Green method in static aberration compensation, which is demonstrated by experiment that the RMSE of the proposed method is lower than 1/100 $\lambda $ (0.0620 rad). On the other hand, a novel kinoform is designed to eliminate the effect of tilt phase shift between two shots of camera for a better stability. Compared with other algorithms, our approach shows the same stability (smaller than 1/100 $\lambda $) and higher recalibration precision (lower than 1/250 $\lambda $). In conclusion, the proposed method provides a fast, robust and accurate means for calibrating SLM, addressing both the static aberration and the nonlinear phase response.
Besides, our method also can be applied to the calibration of transmissive SLMs by replacing the optical setup with Mach–Zehnder. The other advantage of our method is that the edge effect can be avoided by utilizing a 4-f system. However, the multi-wavelength calibration is still a time-consuming task for our method since it is trouble to changing the coherent light sources.
Funding
National Natural Science Foundation of China (No. 61975161, No.52175516).
Acknowledgments
This work is financially supported by the National Natural Science Foundation of China (No.52175516, No. 61975161). We are very grateful to the anonymous reviewers for their valuable comments and suggestions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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