Efficient wavefront sensorless adaptive optics based on large dynamic crosstalk-free holographic modal wavefront sensing

Ming Liu; Bing Dong

doi:10.1364/OE.453176

1. Introduction

Wavefront sensorless adaptive optics (WFSless AO) which performs wavefront sensing and correction with a single adaptive element like a deformable mirror (DM) or a spatial light modulator (SLM), has been developed rapidly in the last decades and applied on biomicroscopy [1], ophthalmoscopy [2], laser processing [3], laser communication [4] and many other applications [5–7]. Numerous WFSless AO algorithms have been proposed, either model-free or model-based. The model-free methods, such as stochastic parallel gradient descent [8], simulated annealing [9], genetic [10], etc., require a lot of iterations to achieve convergence. The model-based methods are generally more efficient by utilizing deterministic relationships between a set of particular modes and well-defined metric functions [11–13]. The aberration modes can be estimated from a sequence of images which are captured in turn after introducing predetermined bias modes using an adaptive element. The model-based methods can achieve convergence within 2∼3 correction cycles and avoid dropping into local optimum. However, since the bias modes are added one by one, the correction bandwidth of model-based WFSless AO is still seriously limited even using fast correction schemes [14,15].

To further accelerate the correction, we may recall an earlier model-based approach that uses a modal wavefront sensor (MWFS) to measure a set of orthogonal modes like Zernike polynomials [16]. The MWFS employs a pinhole photodetector to measure the intensity difference between adding a positive and a negative modal bias. The differential signal is proportional to the modal amplitude when the input aberration is small. The MWFS was initially used for fluorescence microscopes to detect specimen induced aberrations [17] and later in other scenarios [18–22]. In these applications, the aberration modes are still measured one by one. However, the MWFS can be implemented in a spatially multiplexed way that allows simultaneous measurement of multiple modes [16]. In this way, only a single-shot detection is needed for the whole wavefront estimation. To achieve this, a computer-generated hologram (CGH) into which all bias modes are encoded together is employed and usually generated by a liquid-crystal (LC) SLM [23,24]. An array of photodetectors or an image sensor can be used to collect the biased spots array. This kind of MWFS is referred to as the holographic modal wavefront sensor (HMWFS).

Despite the advantages in measurement efficiency and computational simplicity, the HMWFS has its drawbacks. Firstly, the linear measurement range of the HMWFS is only about ±1 rad RMS as given in [16] and [25], so it is only applicable to small aberrations. Secondly, the inter-modal crosstalk effect deteriorates the sensing accuracy and only some low-order modes can be precisely measured. Here the inter-modal crosstalk means the measurement of a given mode is affected by the presence of other modes in the incident beam. Many efforts have been made to improve the performance of HMWFS. The bias amplitude, detector size, output signal form and modal basis have been taken as variables to optimize the performance of HMWFS [25,26]. However, a trade-off still has to be made between the linear response range and the measurement sensitivity. Konwar et al. improved the linear response of the HMWFS by incorporating a variable magnitude of the sensor mode into the beam, but only three aberration modes were measured since at least five pairs of spots were needed for each mode [27]. Dong et al. extended the dynamic range by coding a low-resolution Shack-Hartmann sensor and a MWFS into one static binary-phase hologram [28]. The two sensors were used successively to measure the low-order and high-order aberration modes, which inevitably reduced the temporal bandwidth.

Although some improvements have been made, the crosstalk problem and the limitation on dynamic range of the HMWFS have not been fundamentally overcome. The measurement range of conventional HMWFS can be enlarged by increasing the bias and the pinhole size, but at the expense of reduced sensitivity. In this paper, we propose a novel HMWFS to achieve a large dynamic range and crosstalk-free wavefront estimation. In our approach, the second moment of each focal spot intensity, not just the intensity in a pinhole, is utilized to estimate aberrations and the dynamic range of each mode is only constrained by the imaging area. By using gradient-orthogonal Lukosz modes to represent the wavefront, the inter-modal crosstalk effect can be mostly eliminated. Different from the conventional HMWFS which commonly uses a binary CGH, a kinoform (phase holograms with a continuous phase profile) CGH with much higher diffraction efficiency is adopted in the improved HMWFS. We show how to optimize the kinoform to suppress speckle noises on the image plane. The kinoform is displayed by a phase-only LCSLM which also serves as a wavefront corrector in our WFSless AO system. Numerical simulation and experimental results are presented to prove the effectiveness and advantages of the improved HMWFS.

2. Principles

2.1 Conventional HMWFS

The schematic diagrams of MWFS and HMWFS are shown in Fig. 1(a) and 1(b) respectively. For the MWFS, a series of bias modes are introduced in turn by an adaptive element to the input wavefront which is then focused onto a pinhole detector located at the focal point of the lens. To measure a wavefront containing a single Zernike mode Z_n, i.e., $\Phi = {a_n}{Z_n}(x,y)$ where (x, y) are pupil coordinates, modal biases with equal and opposite amplitude ± b_n are added sequentially to the system and the output signals of the detector are measured as

(1)$$W_n^ \pm{=} \int\!\!\!\int_{r \le {r_0}} {{{|{{\cal F}\{{\exp \{ j[({a_n} \pm {b_n}){Z_n}]\} } \}} |}^2}dudv} ,$$

where ${\cal F}\{{} \}$ denotes the Fourier transform; (u, v) are coordinates in the image plane;$r = \sqrt {{u^2} + {v^2}}$ and r₀ is the pinhole radius.

Fig. 1. The schematic diagrams of (a) MWFS and (b)HMWFS.

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The differential signal ΔW_n is proportional to the modal amplitude when the aberration is small [25].

(2)$$\Delta {W_n} = \frac{{W_n^{\; + } - W_n^{\; - }}}{{W_n^{\; + } + W_n^{\; - }}} \approx {S_n}{a_n}\textrm{ },$$

where S_n is the sensitivity of MWFS to the mode Z_n and can be obtained numerically or experimentally in advance.

The modal coefficient a_n can be easily solved from Eq. (2). In reality, the input wavefront usually contains multiple aberration modes and we need to calibrate a sensitivity matrix S whose element S_ik represents the sensitivity of the sensor designed for the mode Z_i to the input mode Z_k. S is sparse and diagonal dominant. Once we obtain the differential signal vector ΔW, the modal coefficient vector a can be determined by

(3)$${\mathbf a} = {{\mathbf S}^{{\mathbf - 1}}}{\mathbf{\Delta W}}.$$

The measurement principle of HMWFS is similar to the MWFS but implemented in a multiplexed manner. Instead of introducing bias modes sequentially, all bias modes in HMWFS are encoded into a CGH whose transmittance is given by

(4)$${U_{CGH}} = \sum\limits_{n = 1}^N {\exp [j({b_n}{Z_n} + {\varphi _{pos,n}})]} \textrm{ + }\sum\limits_{n = 1}^N {\exp [ - j({b_n}{Z_n} + {\varphi _{pos,n}})]} ,$$

where b_n is the bias amplitude; φ_pos,n is a displacement phase pattern that situates the biased image of mode n at image plane position (u_n, v_n).

(5)$${\varphi _{pos,n}}(x,y) = \frac{{2\pi }}{{\lambda f}}(x{u_n} + y{v_n}).$$

where λ is the beam wavelength and f is the focal length of the lens in Fig. 1.

As the U_CGH in Eq. (4) is real, a binary CGH can be generated by taking the sign of it as

(6)$${U_{BCGH}} = sign({U_{CGH}})\textrm{ = }\left\{ {\begin{array}{ll} {1}&{\textrm{if}\;{U_{CGH}} > 0}\\ { - 1}&{\textrm{if}\;{U_{CGH}} \le 0} \end{array}} \right.$$

The binary CGH with value 1 or -1 can be produced by a fixed binary phase element or an LCSLM with phase value of 0 or π. After passing the CGH and the lens, N pairs of spatially separated spots will appear symmetrically on the focal plane and each pair is used to estimate a particular mode. The spots intensities related to the mode Z_n can be written as

(7)$$I_n^ \pm{=} {\left|{{\cal F}\left\{ {\exp \{ j[({a_n} \pm {b_n}){Z_n} \pm {\phi_{pos,n}} + \sum\limits_{m \ne n}^N {{a_m}{Z_m}} ]\} } \right\}} \right|^2}.$$

In the presence of additional aberration modes Z_m, the linear relationship between the differential signal ΔW_n and the targeted mode coefficient $a_n$ no longer holds accurately. In other words, ΔW_n responses to not only the targeted mode Z_n but also the other non-targeted modes Z_m_≠_n. This problem is called inter-modal crosstalk which undoubtedly degrades the measurement accuracy. The crosstalk problem could get worse if the input wavefront contains more aberration modes [29,30]. So the conventional HMWFS is typically constrained to detecting a few low order modes.

2.2 Improved HMWFS

In this section, an improved HMWFS is proposed to overcome the drawbacks of the conventional HMWFS. Firstly, the second moment of the focal spot intensity is used as the metric function which enables measurement of arbitrarily large aberrations in principle if having unlimited imaging area [11]. Secondly, Lukosz polynomials whose gradients are orthogonal are used as the bias modes, which can effectively eliminate the inter-modal crosstalk.

It has been proved that the integral of modulus squared wavefront gradients is proportional to the second moment of the focal spot intensity (see Eq. 5 in [12]).

(8)$${c_0}\int\!\!\!\int_{r \le R} {I(u,v){r^2}dudv} + {c_1} \approx \int\!\!\!\int_P {{{|{\nabla \Phi (x,y)} |}^2}dxdy}$$

where P denotes the pupil; ∇ is the gradient operator; I is the focal intensity; R is the integration radius; c₀ and c₁ are constants depending only on R. Unlike the relationship in Eq. (2) that is only valid for small aberrations, the relationship in Eq. (8) is valid over all aberration amplitudes.

The input aberration can be expanded by Lukosz modes [11] as

(9)$$\Phi (x,y)\textrm{ = }\sum\limits_{n = 1}^N {{q_n}{L_n}(x,y)}$$

where L_n is the Lukosz mode and q_n is the corresponding modal coefficient.

The gradients of Lukosz modes are mutually orthogonal over a unit circle.

(10)$$\frac{1}{\pi }\int\!\!\!\int_P {\nabla {L_m} \cdot \nabla {L_n}dxdy} = \left\{ {\begin{array}{ll} {1}&m = n\\ {0}&m \ne n \end{array}} \right. .$$

Using the orthogonality of Lukosz modes, the second moment of the focal intensity with input aberration can be written as

(11)$${c_0}\int\!\!\!\int\limits_{r \le R} {{I_0}(u,v)} {r^2}dudv + {c_1} \approx \pi \sum\limits_{n = 1}^N {q_n^2}$$

where I₀ is the unbiased focal spot intensity with input aberration.

By introducing a positive bias mode $+ {b_k}{L_k}$ into the system, we can get

(12)$$\begin{aligned} {c_0}\int\!\!\!\int\limits_{r \le R} {{I_{k + }}} (u,v){r^2}dudv + {c_1} &\approx \int\!\!\!\int\limits_P {{{|{\nabla (\Phi + {b_k}{L_k})} |}^2}dxdy} \\ &= \int\!\!\!\int\limits_P {\{{{{|{\nabla \Phi } |}^2} + {b_k}^2{{|{\nabla {L_k}} |}^2} + 2{b_k}\nabla \Phi \cdot \nabla {L_k}} \}} dxdy\\ &\textrm{ = }\pi \left( {\sum\limits_{n = 1}^N {q_n^2} + {b_k}^2 + 2{q_k}{b_k}} \right) \end{aligned}$$

where I_k+ is the biased focal spot intensity and b_k is the bias amplitude.

The kth Lukosz mode’s coefficient can be estimated independently by

(13)$${q_k} \approx \frac{{{c_0}}}{{2\pi {b_k}}}\int\!\!\!\int\limits_{r \le R} {({{I_{k + }} - {I_0}} )} {r^2}dudv - \frac{{{b_k}}}{2}$$

The above modal estimation process can also be implemented in a multiplexed manner by adopting a CGH to obtain unbiased/biased images in a single shot. The desired complex-amplitude CGH is designed as

(14)$${U_{CGH}} = \exp (j{\varphi _{pos,0}}) + \sum\limits_{n = 1}^N {\exp [j({b_n}{L_n} + {\varphi _{pos,n}})]} .$$

where φ_pos,n is the wavefront tilt for the biased image and φ_pos,0 is the wavefront tilt for the unbiased image. After passing through the CGH and the lens, N+1 spots corresponding to N biased images (I₁₊ ∼ I_N+) and one unbiased image (I₀) will be arranged in order on the focal plane. Then, all aberration modes can be measured simultaneously using Eq. (13) from a single-shot.

If the bias amplitude is large and the input aberration is small, the intensity difference between I₀ and I_n+ may exceed the dynamic range of the image sensor in practice. In this case, we can use a negative bias mode -b_s L_s ($s \in [{1,N} ]$) to generate a focal spot I_s- to replace I₀ and the second moment of I₀ in Eq. (13) can be calculated indirectly by

(15)$${c_0}\int\!\!\!\int\limits_{r \le R} {{I_0}{r^2}} dudv = \frac{{{c_0}}}{2}\int\!\!\!\int\limits_{r \le R} {[{I_{s + }}} + {I_{s - }}]{r^2}dudv - \pi b_{_s}^2.$$

The corresponding CGH should be modified as

(16)$${U_{CGH}} = \exp [{j( - {b_s}{L_s} + {\varphi_{pos,0}})} ]+ \sum\limits_{n = 1}^N {\exp [{j({b_n}{L_n} + {\varphi_{pos,n}})} ]} .$$

3. Kinoform CGH optimization

The complex-amplitude CGH in Eq. (16) can reconstruct a perfect image. However, it is difficult to realize the complex-amplitude CGH with a single element in practice, since the currently available holographic devices such as SLMs can only provide either phase or amplitude modulation of light. Although some complex-amplitude CGH generating methods have been proposed [31–33], they are at the expense of precise alignment operation, low resolution or long-time delay. The commonly used approach is to convert the complex-amplitude CGH to a quantized amplitude-only or phase-only CGH. In the conventional HMWFS, a binary phase CGH is obtained by binarizing the real part of the complex-amplitude CGH (as Eq. (6)) and a pair of conjugated spots are produced for each aberration mode. However, the binary CGH is not the best choice for our HMWFS since only one biased image is required for each mode estimation and the conjugate diffraction order will be useless. Here we choose the kinoform CGH [34] that has a single diffraction order and much higher diffraction efficiency. Theoretically, the diffraction efficiency of an 8-bit quantized kinoform CGH is nearly 100%, in contrast to about 41% for a binary phase CGH.

The kinoform CGH is calculated by taking only the argument part of the complex-amplitude CGH as given by

(17)$${U_{KCGH}} = \arg ({{U_{CGH}}} ).$$

The reconstructed image quality of the kinoform CGH is degraded compared with the complex-amplitude CGH. The two CGHs are equivalent only if the amplitude information of the complex-amplitude CGH is negligible. The complex-amplitude CGH can be written as ${U_{CGH}}\textrm{ = }|{{U_{CGH}}} |\exp (j{\varphi _{CGH}})$. For a kinoform CGH, the phase distribution ${\varphi _{CGH}}$ is generally optimized to enhance the image quality by iterative methods with constraint $|{{U_{CGH}}} |$=1 [35–37]. However, these methods are not suitable for optimization of the ${U_{CGH}}$ in Eq. 16 where ${\varphi _{CGH}}$ is not an arbitrary variable but mostly determined by the phase distributions of the Lukosz modes to be estimated. In our case, the bias amplitude b and the sensing mode number N can be changed to optimize the reconstructed image. The peak signal-to-noise ratio (PSNR) is used to quantify the reconstructed image quality as given by

(18)$$PSNR = 10\log \left\{ {\frac{{max_{_I}^2}}{{MSE\{{{I_C} - {I_K}} \}}}} \right\},$$

where max_I is the maximum pixel value of the image; I_C and I_K are the intensities of the reconstructed images using the complex-amplitude CGH and the kinoform CGH respectively; MSE denotes the mean square error.

Assuming a constant bias amplitude for all modes, the PSNR of reconstructed image varying with the bias amplitude for a given sensing mode number is plotted in Fig. 2(a). It is shown that a larger bias amplitude and a smaller number of sensing modes are beneficial to the reconstructed image quality. However, it is always desired to detect more modes from a single-shot image to make an efficient measurement in practice. For a given sensing mode number, the PSNR is enhanced with the increasing of the bias amplitude and tends to converge for large bias amplitudes. Thus, it is reasonable that the sensing accuracy of HMWFS can be improved when using a large enough bias amplitude. To obtain the optimal bias amplitude for certain sensing mode number, we performed simulations to calculate the RMS of residual aberrations when using different bias amplitudes. For each sensing mode number, 100 random aberrations having the same mode number are produced with their RMS values all normalized to 2 rad. The mean RMS of residual errors after three iterations for different bias amplitudes and sensing mode numbers are shown in Fig. 2(b). Just as we expected, a larger bias amplitude should be used for a larger sensing mode number in order to achieve smaller enough residual error.

Fig. 2. (a) The PSNR of reconstructed images and (b) the mean RMS of residual aberrations varying with the bias amplitudes for different sensing mode numbers.

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However, using a larger bias amplitude leads to a more spread-out biased image which may exceed the allocated imaging area for each mode and cause image overlapping. The bias amplitude may also be constrained by available imaging photons if the image detector noise is non-negligible. In practice, the bias amplitude and the sensing mode number should be carefully designed by considering the measurement accuracy, efficiency, dynamic range and available pixel number of LCSLM (see Section 6). In the following simulations, the bias amplitude is chosen to make the PSNR better than 54 dB for different sensing mode numbers. In our experiments, the sensing mode number is 15 and the bias amplitude is 20 rad.

4. Numerical simulation

We compared the performance of the conventional HMWFS and the improved HMWFS under various aberrations with different RMS values. The estimated aberration is fully corrected by a virtual wavefront corrector and the residual aberration is sent back to the HMWFS to form a closed-loop AO correction system.

One hundred random aberrations with initial RMS values normalized to 1 rad, 2 rad and 3 rad were simulated as the input. All of the aberrations contain 15 modes, Z₄∼ Z₁₈ for the conventional HMWFS and L₄∼ L₁₈ for the improved HMWFS. The RMS values of residual aberrations are evaluated after each time correction. Figure 3 shows the variation of the mean RMS values for five iterations and the horizontal dash line denotes 0.3rad RMS which corresponds to a Strehl ratio (SR) of about 0.9. From Fig. 3, the improved HMWFS obviously has a faster convergence speed than the conventional one. In all cases, the improved HMWFS can achieve convergence after two iterations and the resultant SR can be higher than 0.9 after the first-time correction. The bias amplitude of the conventional HMWFS is set as 1 rad and the pinhole radius is roughly equal to the diffraction limited spot’s radius [25]. We choose a relatively small bias amplitude and pinhole size to achieve high sensitivity at the expense of limited dynamic range. These values remain unchanged in following simulation and experiment.

Fig. 3. Residual aberrations’ RMS varying with five iterations of correction with the initial aberration RMS of 1 rad (a), 2 rad (b) and 3 rad (c).

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To investigate the performance of HMWFS for aberrations involving different mode numbers, 4×50 wavefronts containing 8, 15, 24 and 35 aberration modes are produced with an initial RMS of 1 rad. For aberrations with different mode numbers, the statistical distributions of the residual aberrations’ RMS after the first-time correction are shown in Fig. 4. With the increase of aberration mode number, the measurement accuracy of the conventional HMWFS is getting worse very quickly due to the inherent inter-modal crosstalk problem, whereas the performance of the improved HMWFS is almost unaffected.

Fig. 4. Residual aberrations’ RMS distributions of conventional HMWFS and improved HMWFS after one-time correction for input aberrations containing different number of modes.

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In Fig. 4, the input aberration mode number is equal to the sensing mode number of HMWFS. The crosstalk problem can be more significant if the number of aberration modes in the incident wavefront is larger than that is detected by an HMWFS. To demonstrate this, the sensing mode number of HMWFS is fixed at 15 (4th ∼18th) and the input aberration contains either 15 (4th ∼18th) or 35 (4th ∼38th) modes whose coefficients are all set to 0.5 rad. In the case of 15 aberration modes, all modes can be measured by the HMWFS. In the case of 35 aberration modes, only a part of the aberration modes can be measured, which is a more realistic scenario. The wavefront sensing results of the conventional HMWFS and the improved HMWFS under the two types of aberrations are shown in Fig. 5(a) and 5(b) respectively. The measurement result of the conventional HMWFS deviates far from the ground truth when the input mode number is larger than the sensing mode number. The measurement accuracy of the improved HMWFS is robust to the input mode number because each mode is estimated independently.

Fig. 5. The modal coefficients measured by the conventional HMWFS (a) and the improved HMWFS (b). 15 or 35 aberration modes are involved in the input wavefront. The sensing mode number is set to 15 for both HMWFSs.

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5. Experiments

The WFSless AO system used to test the performance of HMWFSs is shown in Fig. 6(a). The fiber-coupled laser (670nm) is collimated by the lens L1, then reflected by the LCSLM (BNS 512×512 pixels), and finally focused on a CMOS camera (QHY 163M). The LCSLM plays multiple roles in our experiment. It is used to display the CGH for holographic modal wavefront sensing and also used for aberration generation and correction. The system’s initial aberration arising from optical elements manufacturing error and misalignment is corrected by the LCSLM using a sequential model-based WFSless AO algorithm developed in [38]. The first 30 Lukosz modes are corrected for three cycles to eliminate the initial aberration. The point spread function (PSF) before and after correction are shown in Fig. 6(b) and 6(c) respectively. The corrected PSF is taken as a reference to calculate the SR to evaluate the correction results of the WFSless AO system using HMWFSs.

Fig. 6. (a) Diagram of the experimental setup. L1: collimating lens. P: polarizer. BS: beam splitter. L2: focusing lens. The system’s PSFs before (b) and after (c) the initial aberration correction.

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The designed binary CGH and kinoform CGH, both encoding 15 (4th∼18th) aberration modes, are shown in Fig. 7(a) and 7(d) respectively. After applying the CGHs on the LCSLM, the resultant focal-plane images are shown in Fig. 7(b) and 7(e). Since the binary CGH produces multiple diffractive orders for each bias mode, the focal spots are arranged in an annular configuration to avoid mutual interference. The speckle noises of the kinoform CGH is suppressed by using a large bias amplitude of 20 rad. To make full use of the image detector, the spots of the kinoform CGH are arranged in a regular lattice configuration. The spot at the bottom-left corner in Fig. 7(e) corresponds to a negative bias mode of L₇. When a random phase aberration consisting of the 4th∼18th modes with 2 rad RMS is superposed on the two CGHs, the resultant focal images are illustrated in Fig. 7(c) and 7(f). The central bright spots on the focal plane are formed by the unmodulated light because of the limited fill factor of the LCSLM.

Fig. 7. Top row: binary phase CGH (a) and related focal spots with planar wavefront (b) and aberrated wavefront (c). Bottom row: kinoform CGH (d) and related focal spots with planar wavefront (e) and aberrated wavefront (f). The circles in (e) and (f) denote the allocated imaging area for each mode.

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Before correction, we calibrated the two HMWFSs experimentally. The sensitivity matrix S in Eq. (3) was obtained by introducing the 4th∼18th Zernike modes sequentially with amplitude of 0.4 rad. For each input mode, one focal image was captured and the differential signals of all sensing modes were used to calibrate one column of the sensitivity matrix. The calibrated sensitivity matrix S is shown in Fig. 8(a). The parameter c₀ in Eq. (13 ) was calibrated by introducing an aberration containing 4th∼18th Lukosz modes whose coefficients were all set to 1rad. Only a single focal image is required to calibrate c₀. Due to the noises on focal plane, the measured c₀ for different modes are slightly different as is shown in Fig. 8(b).

Fig. 8. (a) The calibrated sensitivity matrix S used for the conventional HMWFS. (b) The calibrated c₀ used for the improved HMWFS.

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To compare the performance of conventional and improved HMWFSs, we tested them under random aberrations with RMS values of 1 rad, 1.5 rad and 2 rad. The focal spots and corresponding SR before and after each time correction are shown in Fig. 9. For all three cases, the improved HMWF performs better than the conventional HMWFS. The aberrations can be almost fully corrected after two times correction when using the improved HMWFS, which is consistent with the simulation results. The convergence of conventional HMWFS becomes slow with the increase of aberration magnitude. For larger aberrations with RMS of 3 rad, 4 rad and 5 rad, the conventional HMWFS does not work but the improved HMWFS can still ensure convergent results after three times correction, as shown in Fig. 10.

Fig. 9. PSF images before and after each time correction using conventional or improved HMWFS for different initial aberrations with RMS of 1 rad (a), 1.5 rad (b) and 2 rad (c).

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Fig. 10. PSF images before and after each time correction using the improved HMWFS for large initial aberrations with RMS of 3 rad (a), 4 rad (b) and 5 rad (c).

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The correction speed of our WFSless AO system is mainly dependent on the refresh rate of the LCSLM and the image sensor. Although a photodetector can be used for the traditional HMWFS to achieve a measurement rate up to MHz, the correction speed of AO system is constrained by the LCSLM with refresh rate only up to KHz which is also the typical frame rate of the image sensor. It is possible to build a WFSless AO system using our improved HMWFS for real-time correction purpose like the compensation of atmospheric turbulence.

In our simulation and experiment, the wavefront tip and tilt that will cause spots shifting are not included in the input aberrations. Although the tip/tilt errors are usually not sensed by the HMWFS in previous publications, they will deteriorate the sensing accuracy of other modes and should be effectively corrected. The improved HMWFS can tolerate larger residual tip/tilt errors than the conventional one because the measurement can be performed as long as the spots are still in their subregions. Please note that centers of gravity of the diffraction spots are not used in the measurement process of the improved HMWFS. The origin of coordinates of the integration domain for each diffraction spot in Eq. (13) are (u_n, v_n) determined by the displacement phase pattern given in Eq. (5) and are not changed with the shifting of spot.

6. Discussion

The measurement dynamic range and sensing mode number of the improved HMWFS cannot be chosen arbitrarily but constrained by the characteristics of LCSLM. Because of the pixelated structure of LCSLM, the usable region on the focal plane is confined by (λ f / p)², as illustrated in Fig. 11. The wavefront sensing mode number N should satisfy

(19)$$(N + 1){d^2} \le {\left( {\frac{{\lambda f}}{p}} \right)^2},$$

where p is the pixel pitch of LCSLM.

Fig. 11. Arrangement of focal spots of improved HMWFS.

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Each focal-plane spot occupies a square subregion with width d which is weighted by the imaging system’s diffraction limitation.

(20)$$d = \gamma \frac{{\lambda f}}{D} = \gamma \frac{{\lambda f}}{{{N_{SLM}}p}},$$

where γ is a factor that determines the measurement dynamic range of each mode; D is the pupil diameter; N_SLM is the pixel number of LCSLM along the pupil diameter.

Substituting Eq. (20) in Eq. (19), we can get

(21)$$({N + 1} ){\gamma ^2} \le N_{SLM}^2 = SW$$

where SW is the spatial-bandwidth product [39].

From Eq. (21), there is a trade-off between the dynamic range and sensing mode number for a given LCSLM. The most effective way to improve both of them is to use an LCSLM with a larger number of pixels.

In our experiment, the resolution of each subregion on the detector is about 650×650 and the sampling rate of the images is roughly three times higher than the Nyquist rate. A lower sampling rate may cause errors when calculating the second moment of the spots and decrease the sensing accuracy. The influence of image sampling rate on the performance of the HMWFS will be further investigated in our future work.

7. Conclusion

WFSless AO offers a simple and compact structure for aberration correction but suffers from a large number of image measurements or a time-consuming optimization process. In this paper, an improved HMWFS that has a large dynamic range and no crosstalk is proposed to form a highly-efficient WFSless AO system. By leveraging the multiplexing feature of the CGH, spots array corresponding to multiple biased aberration modes can be obtained in a single-shot image and the modal coefficients can be estimated simultaneously. The large dynamic range is achieved by using the linear relationship between the second moment of the focal spot intensities and the integral of modulus squared wavefront gradients. The crosstalk problem is overcome by using Lukosz modes whose gradients are orthogonal to represent aberrations. The kinoform CGH is adopted in the proposed HMWFS to replace the traditional binary CGH to achieve much higher diffraction efficiency. Simulation and experiments have been done to demonstrate the superior performance of the proposed HMWFS over the conventional one. To enhance the dynamic range and the sensing mode number simultaneously, an LCSLM with a large number of pixels is preferred for the proposed HMWFS.

Funding

National Natural Science Foundation of China (11874087).

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 maybe obtained from the authors upon reasonable request.

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Efficient wavefront sensorless adaptive optics based on large dynamic crosstalk-free holographic modal wavefront sensing

Abstract

1. Introduction

2. Principles

2.1 Conventional HMWFS

2.2 Improved HMWFS

3. Kinoform CGH optimization

4. Numerical simulation

5. Experiments

6. Discussion

7. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (11)

Equations (21)

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