Self-calibrated general model-based wavefront sensorless adaptive optics for both point-like and extended objects

Hongxi Ren; Bing Dong

doi:10.1364/OE.454901

1. Introduction

A traditional adaptive optics (AO) system has a wavefront sensor (WFS) like Shack-Hartmann to detect aberrations and a wavefront corrector like a deformable mirror (DM) to perform conjugate correction. Traditional AO is mainly developed for ground-based astronomical telescopes to compensate fast changing aberrations induced by atmospheric turbulence [1]. However, using a dedicated WFS requires a separate detection path which increases system complexity and reduces throughput. Moreover, the direct wavefront sensing approach can be challenging due to strong scintillation, non-common path error, insufficient signal intensity and measurement uncertainty [2–6]. To avoid the difficulties of direct wavefront sensing, wavefront sensorless (WFSless) AO has been developed in the last decades and successfully demonstrated in many applications [7–16]. In WFSless AO, the WFS is discarded and the control signals of wavefront corrector are continuously optimized to find the extremum of an image-quality-related metric function. The residual aberration is minimized when the metric function converges to the extremum. The efficiency of optimization algorithms is a major concern when applying WFSless AO to correct dynamic aberrations [11] or taking a lot of image measurements is undesired such as in biomicroscopy [4]. Among the proposed optimization algorithms, the model-based approach is generally more efficient than the model-free stochastic approach by exploring the deterministic relationship between a set of specific modes and a well-chosen metric function [17–25]. The aberrations can be inferred from a sequence of image measurements that are implemented after introducing predetermined bias modes into the system through a wavefront corrector.

Many model-based algorithms have been proposed, either for point-like or extended objects. The term “model-based” means that the relationships between metric functions and aberration modes can be mathematically established by theoretical derivation or experiment. For point-like source, Strehl ratio measured by a pinhole photodetector was initially used as the metric function and the Zernike polynomials were chosen as the bias modes [17–18]. However, the quadratic relationship between Strehl ratio and Zernike coefficients is only valid for small aberrations. For large aberrations, root-mean-square (RMS) spot radius is a better choice for the metric function and the Lukosz polynomials can be the bias modes [19]. Using a metric function called masked detector signal (MDS) which is similar to that in [19], Huang & Rao proposed a general method insensitive to the selection of bias modes [20]. The above algorithms for point source require a minimum of N+1 image measurements to correct N aberration modes simultaneously. Alternatively, using gradient-orthogonal modes, a sequential (mode-by-mode) correction scheme can be implemented to reduce the time delay between corrections, requiring 2N images to correct N aberration modes [21]. For extended objects, Débarre et al. proposed to use the low spatial frequency contents of image as the metric function and the Lukosz polynomials as the bias modes [22]. Yang et al. further generalized Huang & Rao’s method to extended objects [23]. Recently, we improved Débarre’s method by reducing the measurement number from 2N+1 to N+2 in simultaneous correction [24] and proposed a fast sequential correction scheme that reduces the measurement number from 3N to 2N [25].

Although so many model-based algorithms have been proposed and demonstrated, the interrelationships between these algorithms are still unclear. Moreover, a common prerequisite of these algorithms is that the bias modes must be precisely generated on the pupil plane by an adaptive element with known amplitudes. Thus, the adaptive element, especially for a DM, should be calibrated in advance by a WFS or an interferometer. With calibrated influence functions (IFs) of a DM, we can fit them to analytic modes like the Zernike/Lukosz polynomials or derive a set of mirror modes that have appropriate orthogonality [26]. The WFS is unnecessary during correction for WFSless AO, but it will increase the whole system complexity and may be unavailable in some applications. To avoid using a WFS in calibration, Thayil & Booth proposed to obtain the IFs of DM by solving the membrane equation, however, requiring complete knowledge of the DM [27]. In [28–29], the relationships between the metric functions and the aberration modes are determined by measuring the metric function under a large number of test aberrations and then fitting a multidimensional ellipsoid, which is time-consuming and requires to eliminate initial system aberrations in advance. Antonello et al. proposed to calibrate the relationship using semidefinite programming method to ensure the metric function has a global extremum at zero aberration, also requiring a large input-output data set [30]. Moreover, the input basis modes in the experimental calibrations in [28–30] are low-order Zernike polynomials, which means the IFs of DM are still required to be measured before calibration to produce these modes.

In this paper, the relationships between metric functions and aberrations in typical model-based algorithms are generalized to a unified form and the calibration problem comes down to the measurement of a Gram matrix. We proposed a self-calibration procedure to determine the Gram matrix without using a WFS or prior knowledge of the DM. The self-calibration is implemented by applying control signals on DM actuators and recording related focal-plane images. The elements of the Gram matrix are inner-products of specific vectors that depend on what kind of metric function is used. The calibrated Gram matrix can be used directly for simultaneous correction if using the IFs of DM as the bias modes, requiring N+1 images to correct N aberration modes. Alternatively, orthogonal or gradient-orthogonal mirror modes obtained from the eigenvectors of the Gram matrix can be used as the modal basis to implement independent sequential correction that requires 2N images to correct N aberration modes. Simulations and experiments are performed to verify the feasibility of the proposed self-calibration procedure and correction methods for both point-like and extended objects.

2. Principle

2.1 General framework of model-based WFSless AO

In model-based WFSless AO, aberrations are estimated from the relationships between metric functions and aberration modes. The relationships in four typical model-based algorithms are briefly described as below, two for point-like objects and the other two for extended objects.

The aberration can be expanded by an arbitrary modal basis {K_i} (excluding piston) as

(1)$$\Phi ({\mathbf r} )= \sum\limits_{i = 1}^N {{a_i}{K_i}} ({\mathbf r} ),$$

where a_i is the coefficient of mode K_i.

Algorithm 1: For point-like objects and small aberrations, the relationship between Strehl ratio (M₁) and the modal coefficients can be expressed as [18]

(2)$${M_1} = \frac{{I({{x_0}} )}}{{{I_0}({{x_0}} )}} \approx 1 - \sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {{a_i}{a_j}\left( {\frac{1}{\pi }\int\!\!\!\int_P {{K_i} \cdot {K_j}\textrm{d}A} } \right)} } ,$$

where I is the aberrated image; I₀ is the diffraction-limited image; x₀ is the position of the central peak of I₀; P denotes circular pupil area.

Although M₁ is only valid for small aberrations, it can be measured by a photodetector with bandwidth up to megahertz. If the modes K are chosen as the Zernike polynomials that are orthogonal over the pupil, Eq. (2) can be simplified as

(3)$${M_1} \approx 1 - \sum\limits_{i = 1}^N {a_i^2} .$$

.

Algorithm 2: For point-like objects and large aberrations, the relationship between the metric function M₂, namely the MDS in [20], and the modal coefficients is given by

(4)$${M_2} = \frac{{\int\!\!\!\int_{|{\mathbf r} |\le R} {I({\mathbf r} )\cdot ({1 - {{{{|{\mathbf r} |}^2}} / {{R^2}}}} )\textrm{d}{\mathbf r}} }}{{\int\!\!\!\int_{|{\mathbf r} |\le R} {I({\mathbf r} )\textrm{d}{\mathbf r}} }} \approx 1 - {c_0}(R )\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {{a_i}{a_j}\left( {\int\!\!\!\int_P {\nabla {K_i} \cdot \nabla {K_j}\textrm{d}A} } \right)} } ,$$

where I is the aberrated image; R is detector radius; c₀(R) is a constant related to R; ∇ is the gradient operator. If the modes K are chosen as the Lukosz polynomials whose gradients are mutually orthogonal [19], Eq. (4) can be simplified as

(5)$${M_2} \approx 1 - {c_0}(R )\sum\limits_{i = 1}^N {a_i^2} .$$

.

Algorithm 3: For extended objects, the metric function can be the reciprocal of low spatial frequency content of images and its relationship with aberration is expressed as [22, 24, 25]

(6)$${M_3} = {\left( {\int_0^{2\pi } {\int_{{m_1}}^{{m_2}} {{S_J}({m,\xi } )} } \;m{\kern 1pt} {\kern 1pt} \textrm{d}m{\kern 1pt} {\kern 1pt} \textrm{d}\xi } \right)^{ - 1}} \approx {q_2} + {q_3}\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {{a_i}{a_j}\left( {\int\!\!\!\int_P {\nabla {K_i} \cdot \nabla {K_j}\textrm{d}A} } \right),} }$$

where S_J is the image spectral density, m is the spatial frequency; ξ is the polar coordinate angle; q₂ and q₃ are parameters related to the imaging system and objects

(7)$$\begin{array}{*{20}{c}} {{q_2} = {{\left( {\int_0^{2\pi } {\int_{{m_1}}^{{m_2}} {{S_{{J_0}}}({m,\xi } )\;m{\kern 1pt} {\kern 1pt} \textrm{d}m{\kern 1pt} {\kern 1pt} \textrm{d}\xi } } } \right)}^{\textrm{ - }1}},}&{\;\;\;{q_3}\textrm{ = }\frac{{\int_0^{2\pi } {\int_{{m_1}}^{{m_2}} {{S_{{J_0}}}({m,\xi } )\;\textrm{/}{H_0}({m,\xi } )\;{m^3}\textrm{d}m\textrm{d}\xi } } }}{{2\pi {{\left( {\int_0^{2\pi } {\int_{{m_1}}^{{m_2}} {{S_{{J_0}}}({m,\xi } )\;\textrm{d}m\textrm{d}\xi } } } \right)}^2}}}} \end{array},$$

where S_J0 is the image spectral density without aberration and H₀ is the normalized diffraction-limited optical transfer function (OTF). q₂ is actually the minimum of M₂ corresponding to zero aberration. Again, if the gradients of modes K are orthogonal, Eq. (6) can be simplified as

(8)$${M_3} \approx {q_2} + {q_3}\sum\limits_{i = 1}^N {a_i^2} .$$

.

Algorithm 4: The metric function for extended objects can also be defined in spatial domain. The relationship between the masked detector signal for extended objects (MDSE) and the modal coefficients is described by [23]

(9)$${M_4} = MDSE = \frac{{\int\!\!\!\int_{|{\mathbf r} |\le R} {I({\mathbf r} )\cdot {{|{\mathbf r} |}^2}\textrm{d}{\mathbf r}} }}{{\int\!\!\!\int_{|{\mathbf r} |\le R} {I({\mathbf r} )\textrm{d}{\mathbf r}} }} \approx MDS{E_0} + \frac{1}{{4{\pi ^3}}}\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {{a_i}{a_j}\left( {\int\!\!\!\int_P {\nabla {K_i} \cdot \nabla {K_j}\textrm{d}A} } \right)} } ,$$

where MDSE₀ denotes the metric function value without aberration.

By comparing the above four algorithms, we can find their similarities and generalize a unified expression to describe the relationships between the metric functions and the aberration modes as

(10)$$M \approx {M_e} - c\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {{a_i}{a_j}{\alpha _{ij}}} } ,$$

where M is the metric function related to image quality and M_e is the extremum of metric function for zero aberration. α_ij is defined by the inner products of a set of modes and is related to the metric function as

(11)$${\alpha _{ij}}\textrm{ = }\left\{ {\begin{array}{*{20}{c}} {\;\left\langle {{K_i},{K_j}} \right\rangle \textrm{ = }\int\!\!\!\int_P {{K_i}{K_j}\textrm{d}A} \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\;\;\;\;\;}&{\;\;\;\;\;\;\;\;For\;M = {M_1}} \end{array}}&{} \end{array}}&{}&{} \end{array}}&{}&{} \end{array}}\\ {\left\langle {\nabla {K_i},\nabla {K_j}} \right\rangle \begin{array}{*{20}{c}} { = \int\!\!\!\int_P {\nabla {K_i} \cdot \nabla {K_j}\textrm{d}A} }&{\;\;\;For\;M = {M_2}\;or\;{M_3}\;or\;{M_4}}&{} \end{array}} \end{array}} \right.$$

The parameter c in Eq. (10) is positive for maximization of M (i.e., M₁ and M₂) and is negative for minimization of M (i.e., M₃ and M₄). For M₁ and M₄, c is equal to 1/π and –1/4π³ respectively. For M₂, c depends on the detector radius R and can be obtained analytically [23] or numerically [20]. For M₃, c is equivalent to –q₃ which can be estimated from aberrated images as demonstrated in [25].

Equation (10) can be written in matrix form as

(12)$$M \approx {M_e} - c{{\mathbf A}^{\mathbf T}}{\mathbf G} {\mathbf A},$$

where

(13)$${\mathbf A} = \left[ {\begin{array}{*{20}{c}} {{a_1}}\\ \vdots \\ {{a_N}} \end{array}} \right]\begin{array}{*{20}{c}} {}&{}&{{\mathbf G} = \left[ {\begin{array}{*{20}{c}} {{\alpha_{11}}}&{{\alpha_{12}}}& \cdots &{{\alpha_{1N}}}\\ {{\alpha_{21}}}&{{\alpha_{22}}}& \cdots &{{\alpha_{2N}}}\\ \vdots & \vdots & \ddots & \vdots \\ {{\alpha_{N1}}}&{{\alpha_{N2}}}& \cdots &{{\alpha_{NN}}} \end{array}} \right]} \end{array}$$

A is the modal coefficient vector. G is an N×N Gram matrix which is symmetric and positive-semidefinite. For any possible A, we have ${{\mathbf A}^{\mathbf T}}{\mathbf G}{\mathbf A} \ge 0$, implying that a global extremum of metric function M can be achieved for zero aberration.

2.2 Self-calibration procedure

The traditional calibration approach is to measure the IFs of DM using a WFS and then use the IFs to fit particular bias modes such as Zernike [18], Lukosz [19] or mirror modes [26]. Although it is possible to use the IFs directly as bias modes, they are still needed to be measured to compute the matrix G. In this section, we provide a new way to calculate matrix G without using any wavefront sensing device to measure IFs.

Although the profiles of IFs are unknown, the control signals applied on actuators of DM are accessible. Using the IFs as basis functions to fit the aberration, Eq. (1) can be rewritten as

(14)$$\Phi ({\mathbf r} )= \sum\limits_{i = 1}^N {{v_i}{F_i}} ({\mathbf r} ),$$

where v_i and F_i are the control signal and the influence function of the ith actuator respectively. N is the actuator number of DM.

The metric function with initial system aberration is given by

(15)$${M_0} \approx {M_e} - c\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {{v_i}{v_j}{\alpha _{ij}}} } .$$

By applying control signal of + b and –b to the kth actuator, we can introduce a positive and a negative bias into the system in turn and the corresponding metric functions are written as

(16)$${M_{k + }} \approx {M_e} - c\left( {\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {{v_i}{v_j}{\alpha_{ij}}} } + {b^2}{\alpha_{kk}} + 2b\sum\limits_{i = 1}^N {{v_i}{\alpha_{ik}}} } \right),$$

(17)$${M_{k - }} \approx {M_e} - c\left( {\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^N {{v_i}{v_j}{\alpha_{ij}}} } + {b^2}{\alpha_{kk}} - 2b\sum\limits_{i = 1}^N {{v_i}{\alpha_{ik}}} } \right).$$

Subtracting Eq. (16) or Eq. (17) from Eq. (15), we have

(18)$${M_0} - {M_{k + }}\textrm{ = }c{b^2}{\alpha _{kk}} + 2cb\sum\limits_{i = 1}^N {{v_i}{\alpha _{ik}}} ,$$

(19)$${M_0} - {M_{k - }}\textrm{ = }c{b^2}{\alpha _{kk}} - 2cb\sum\limits_{i = 1}^N {{v_i}{\alpha _{ik}}} .$$

Combining Eq. (18) and Eq. (19), we can get

(20)$$2{M_0} - {M_{k + }} - {M_{k - }}\textrm{ = 2}c{b^2}{\alpha _{kk}}.$$

From Eq. (20), the diagonal elements of matrix G can be solved by

(21)$${\alpha _{kk}}\textrm{ = }\frac{{2{M_0} - {M_{k + }} - {M_{k - }}}}{{\textrm{2}c{b^2}}}.$$

Similarly, by applying control signals of + b or –b to the kth and the sth actuator simultaneously, we can get

(22)$$2{M_0} - {M_{k + s + }} - {M_{k - s - }}\textrm{ = 2}c{b^2}({{\alpha_{kk}} + 2{\alpha_{ks}} + {\alpha_{ss}}} ).$$

From Eq. (22), the off-diagonal elements of matrix G can be solved by

(23)$${\alpha _{ks}}\textrm{ = }\frac{{2{M_0} - {M_{k + s + }} - {M_{k - s - }}}}{{\textrm{4}c{b^2}}} - \frac{{{\alpha _{kk}} + {\alpha _{ss}}}}{2},$$

where ${\alpha _{kk}}$ and ${\alpha _{ss}}$ are obtained from Eq. (21).

Using Eq. (21) and Eq. (23), we can calibrate all elements of matrix G by applying control signals to DM actuators and computing corresponding metric function values. The proposed calibration procedure is robust to the initial system aberration whose effect has already been involved in the measured metric functions. For a DM with N actuators, N²+ N+1 image measurements are required in the calibration procedure.

2.3 Correction methods based on calibrated matrix

2.3.1. Simultaneous correction scheme (N+1 algorithm)

After calibration, several schemes can be implemented to acquire the correction signals of DM. The most straightforward way is using the IFs of DM as the bias modes. By applying a control signal of + b to each actuator in turn and evaluating the metric functions, a series of equations similar to Eq. (18) can be obtained and written in matrix form as

(24)$${{\mathbf M}_{\mathbf b}} = 2cb{\mathbf G \mathbf V} + c{b^2}{{\mathbf G}_{\mathbf d}},$$

where G_d contains the diagonal elements of G and V is the control signal vector of DM.

(25)$${{\mathbf M}_\textrm{b}} = \left[ {\begin{array}{*{20}{c}} {{M_0} - {M_{1 + }}}\\ \vdots \\ {{M_0} - {M_{N + }}} \end{array}} \right]\begin{array}{*{20}{c}} {}&{\begin{array}{*{20}{c}} {{{\mathbf G}_\textrm{d}}\textrm{ = }\left[ {\begin{array}{*{20}{c}} {{\alpha_{11}}}\\ \vdots \\ {{\alpha_{NN}}} \end{array}} \right]}&{} \end{array}}&{{\mathbf V} = } \end{array}\left[ {\begin{array}{*{20}{c}} {{v_1}}\\ \vdots \\ {{v_N}} \end{array}} \right]$$

The control signal of DM to correct aberration Φ is calculated by

(26)$${\mathbf V} ={-} \;{{\mathbf G}^{ - 1}}\left( {\frac{{{{\mathbf M}_\textrm{b}} - c{b^2}{\kern 1pt} {{\mathbf G}_\textrm{d}}}}{{2cb}}} \right).$$

Because of the inevitable noises in measurement of metric functions, the calibrated matrix G may be indefinite and it is recommended to use the eigendecomposition to calculate the inverse matrix G^-1. The matrix G can be decomposed by

(27)$${\mathbf G} = {\mathbf Q}{\mathbf \Lambda }{{\mathbf Q}^{\mathbf T}}.$$

Q is a square matrix whose columns are the eigenvectors of G and Q^T= Q^-1. Λ is the diagonal matrix whose entries are the eigenvalues as Λ_ii = β_i. If G is positive-semidefinite, then all β_i≥ 0, otherwise, β_i can be negative. The condition number of G is given by the ratio of the maximum modulus eigenvalue to the minimum. The condition number is required to be small to achieve a stable correction. In practice, we can simply zero the negative and relatively small positive diagonal elements in Λ and then calculate G^-1 by

(28)$${{\mathbf G}^{ - 1}}= {\mathbf Q}{\tilde{{\mathbf \Lambda }}^{ - 1}}{{\mathbf Q}^{\mathbf T}},$$

where $\tilde{{\mathbf \Lambda }}$ is the modified diagonal matrix.

2.3.2. Sequential correction scheme (2N algorithm)

From Eq. (26), the control signal of DM is obtained from N+1 image measurements, including N biased images and one unbiased image. Then a simultaneous correction can be applied. Since N is the actuator number, the time delay between each correction will get too long if the DM has a large number of actuators. To reduce the time delay, it is better to use sequential correction scheme where the aberration modes are corrected independently [21, 25]. The sequential correction requires the bias modes having certain orthogonality. In Appendix 1, it is proved that the control signals of DM to generate mirror modes [26] that are either orthogonal or gradient-orthogonal are the columns of Q. The metric function in Eq. (12) can be rewritten as

(29)$$M \approx {M_e} - c{{\mathbf V}^{\mathbf T}}{\mathbf Q}{\mathbf \Lambda }{{\mathbf Q}^{\mathbf T}}{\mathbf V} = {M_e} - c{{\mathbf D}^{\mathbf T}}{\mathbf \Lambda}{\mathbf D} = {M_e} - c\sum\limits_{i = 1}^{} {{\beta _i}d_i^2} ,$$

where D = Q^TV is the coefficient vector of mirror modes; d_i is the coefficient of the ith mode.

By taking the initial metric function M₀ and then adding the ith mirror mode with amplitude + b to obtain metric function ${M_{i + }}$, we can get an equation similar to Eq. (18) as

(30)$${M_0} - {M_{i + }}\textrm{ = }c{b^2}{\beta _i} + 2cb{d_i}{\beta _i}.$$

The coefficient of the ith mirror mode is calculated by

(31)$${d_i} = \frac{{{M_0} - {M_{i + }}}}{{2cb{\beta _i}}} - \frac{b}{2}.$$

The control signal of DM to correct the ith mirror mode is given by

(32)$${{\mathbf V}_i} ={-} {d_i}{{\mathbf Q}_i}.$$

In sequential correction scheme, the correction for each mirror mode is performed immediately after getting the corrective signal. Since the unbiased metric function M₀ must be updated after each mode’s correction, 2N images in total are required to correct N mirror modes. Please note that the mirror modes with negative or relatively small eigenvalues should also be excluded from sequential correction.

3. Simulations

A model-based WFSless AO simulation system with a 37-channel DM is shown in Fig. 1 [24]. The pupil aberration is imaged on to the DM and then focused on to an image detector. The IFs of DM are modeled by a Gaussian function and the inter-actuator coupling factor is 0.15. We tested the performance of the proposed self-calibration and correction methods for both point-like and extended objects using all the metric functions (M₁∼M₄) in Section 2.1. The extended object used here is a USAF resolution test chart. Please note that the proposed calibration method is independent of the image content.

Fig. 1. (a) WFSless AO system; (b) Actuator layout of the 37-channel DM. The red inner-circle denotes the clear aperture of DM (77% in diameter).

Download Full Size | PDF

For comparison, we first calculate the matrix G based on the known IFs as the ground truth and the results are shown in Fig. 2 (a₁) and (a₂). The calibration accuracy can be evaluated by

(33)$$f = 1 - \frac{{var({{\mathbf G}_{cali}} - {{\mathbf G}_{truth}})}}{{var({{\mathbf G}_{truth}})}},$$

where G_truth is the ground truth of matrix G; G_cali is the calibrated matrix; var denotes the calculation of variance.

Fig. 2. Gram matrix obtained from known IFs (a₁) ∼ (a₂) and self-calibration procedure with zero system aberration (b₁) ∼ (b₄), with system aberrations (c₁) ∼ (c₄), with both system aberrations and image noises (d₁) ∼ (d₄). f with a maximum of 1 represents the calibration accuracy.

Download Full Size | PDF

Without the initial aberration, the self-calibrated matrices using the metric functions from M₁ to M₄ are shown in Fig. 2 (b₁)–2(b₄) respectively. In practice, there always exists system aberrations arising from non-ideal system design, misalignment and manufacturing. With random initial system aberrations of 0.2 rad RMS for M₁ and 2 rad RMS for M₂ to M₄, the self-calibrated matrices are depicted in Fig. 2 (c₁)–2(c₄). The initial aberration for M₁ (Strehl Ratio) is relatively small since it is only effective for small aberrations. It is shown that the self-calibrated matrix is insensitive to the initial aberration.

Another factor that must be considered in practice is the measurement error of metric functions due to inevitable image noises. With initial aberrations having the same RMS as in Fig. 2 (c₁)–2(c₄) and image peak signal-to-noise ratio (PSNR) of 45dB, the self-calibrated matrices are calculated and shown in Fig. 2 (d₁)–2(d₄). The PSNR in dB is defined by Eq. (34) where MSE is the mean squared error of the image. 45 dB is also the nominal PSNR of the camera used in our experiment. The self-calibration accuracy has a remarkable decline in consideration of image noises. Moreover, the calibrated matrices in this case are mostly indefinite with both positive and negative eigenvalues. Under lower PSNR, a larger bias amplitude should be applied [24]. Here the bias amplitude under PSNR of 45dB is 1 rad.

(34)$$PSNR = 20 \cdot {\log _{10}}\left( {\frac{{{2^{16}} - 1}}{{\sqrt {MSE} }}} \right).$$

To investigate the performance of simultaneous correction using Eq. (26), 100 random aberrations with RMS normalized to a fixed value (0.7 rad for M₁ and 3 rad for M₂∼M₄) are generated by the DM itself. For different metric functions and calibrated Gram matrices, the mean RMS values of residual aberrations varying with the correction cycles are plotted in Fig. 3. The aberrations can be effectively corrected after 2∼3 correction cycles, no matter what kind of metric function is used. When the matrix G is calibrated with measurement noises (Self-calibration 3), a relatively larger residual aberration can be observed but the convergence is still very stable. The negative eigenvalues together with the positive eigenvalues less than 1/30 of the maximum are set to zero in calculating G^-1 using Eq. (28).

Fig. 3. Simultaneous correction results using IFs of DM as bias modes. The RMS of residual aberrations varying with the correction cycles are plotted when using metric functions of (a) M₁, (b) M₂, (c) M₃, and (d) M₄. The calibrated Gram matrices used here are shown in Fig. 2.

Download Full Size | PDF

To correct aberration modes independently, we can use mirror modes as bias modes. Using the calibrated matrices in Fig. 2(d₁)–2(d₄) which are the most realistic results, the control signals of DM to generate mirror modes were obtained by calculating the eigenvectors of G as Eq. (27). The first 9 orthogonal mirror modes (OMM) obtained from M₁ and gradient-orthogonal mirror modes (GOMM) obtained from M₂, M₃ and M₄ are depicted in Fig. 4. The GOMMs obtained from M₂, M₃ and M₄ are theoretically identical, however, showing differences in ordering, sign and details of distribution in Fig. 4 (c)–4(d). The orthogonality matrices of the mirror modes are also given in Fig. 4. The orthogonality matrices are diagonally-dominant but not diagonal because the Gram matrices used to derive these modes are not ideal but calibrated with measurement noises. Please note that the phase distributions of mirror modes are unobtainable in practice if using the self-calibration method since the IFs of DM are unknown.

Fig. 4. Phase distributions of mirror modes (1∼9) from the results of “Self-calibration 3” in Fig. 2 and the orthogonality matrices of all modes.

Download Full Size | PDF

To investigate the performance of sequential independent correction using Eq. (32), aberrations are generated by applying random voltages on DM actuators. Using different metric functions and the first 30 mirror modes as bias modes, the RMS values of residual aberrations varying with the image sampling number are plotted in Fig. 5. The aberrations can be effectively corrected and reach convergence after 1∼2 correction cycles (60 image samplings per cycle), no matter using WFS-calibrated or self-calibrated matrix.

Fig. 5. Sequential correction results using mirror modes as biases. The residual aberrations’ RMS values varying with the image sampling number are plotted when using metric function of (a) M₁, (b) M₂, (c) M₃, and (d) M₄. The calibrated Gram matrices used here are shown in Fig. 2.

Download Full Size | PDF

4. Experiments

The experimental system is shown in Fig. 6. The point-like target is mimicked by a single-mode fiber laser (λ = 635nm) and the extended target is a resolution plate illuminated by a LED (λ₀ = 625nm). Both targets are placed on a motorized translation stage and can be switched back and forth. DM₁ (Thorlabs) is a 40-actuator piezoelectric DM used to generate aberrations. DM₂ which is conjugated to DM₁ is a 37-channel membrane DM (OKO Tech) used for correction. The focal plane camera is a 12-bit CMOS sensor (Point Grey, GS3-U3-23S6M-C).

Fig. 6. Experimental system layout.

Download Full Size | PDF

The Gram matrices calibrated using a commercial Shack-Hartmann WFS (Imagine Optic, HASO3-76GE) using 59×59 subapertures or obtained by self-calibration methods using different metric functions are depicted in Fig. 7. The WFS-based calibration results are taken as the ground truth to evaluate the self-calibration accuracy using Eq. (33). Because of the measurement noises, the self-calibrated matrices are all indefinite with 3∼5 negative eigenvalues. The self-calibration results are consistent with the simulation results in Fig. 2 (d₁)–2(d₄).

Fig. 7. Calibrated matrix G using different metric functions and calibration methods.

Download Full Size | PDF

Using the IFs of DM₂ as bias modes, the simultaneous correction results using different metric functions and calibrated Gram matrices are shown in Fig. 8. The first row in Fig. 8 shows the aberrated images and the initial metric function values. To compare the correction results of different algorithms with a general metric, we also use the image sharpness J (sum of squared intensity) to evaluate the images. After three correction cycles, the corrected images using WFS-calibrated or self-calibrated Gram matrices are shown in the second row and the third row of Fig. 8 respectively. The correction results using the self-calibrated Gram matrices are quite close to the WFS-calibrated ones, which is as expected because of the high similarity of calibrated matrices as shown in Fig. 7.

Fig. 8. Simultaneous correction results using IFs of DM₂ as bias modes. (a₁∼d₁) Aberrated images. Corrected images after three correction cycles using WFS-calibrated Gram matrices (a₂∼d₂) or self-calibrated Gram matrices (a₃∼d₃). J is the image sharpness.

Download Full Size | PDF

To perform sequential correction, the control signals of DM to generate mirror modes are obtained by eigendecomposition of the calibrated Gram matrices. Using the first 15 mirror modes as biases, the sequential correction results with different metric functions are shown in Fig. 9. It is shown that using the mirror modes derived from the self-calibrated Gram matrices also has similar correction results with those from the WFS-calibrated Gram matrices.

Fig. 9. Sequential correction results using the first 15 mirror modes as biases. The metric functions of (a) M₁, (b) M₂, (c) M₃, and (d) M₄ varying with image sampling number are plotted.

Download Full Size | PDF

5. Discussion

We noticed that the calibration method proposed by Débarre et al. in [31] (also given in Section 3A of [30]) can also be implemented without using a WFS if the basis modes are chosen as the IFs instead of Zernike polynomials. Taking the IFs as basis modes in Débarre’s method, the Gram matrices calibrated without initial aberration and with 1rad RMS aberration are illustrated in Fig. 10(a) and 10(b) respectively. With the same aberration, the Gram matrix obtained from our calibration method is shown in Fig. 10(c). The calibration accuracy is evaluated by f defined in Eq. (33). The calibration accuracy of Débarre’s method declines significantly if calibrated with aberration. The minimum number of image measurements for Débarre’s method is (3N²+ N)/2 [30] and much more images are required in practice to improve the calibration accuracy [31]. In contrast, only N²+ N+1 image measurements are needed in our method.

Fig. 10. The Gram matrices calibrated (a) without aberration and (b) with 1rad RMS aberration using Débarre’s method. (c) The Gram matrix calibrated with the same aberration using our calibration method. f is defined in Eq. (33) to evaluate the calibration accuracy. (d) The RMS of residual aberrations varying with the correction cycles. The metric function is M₃.

Download Full Size | PDF

For different calibrated matrices, the RMS of residual aberration varying with the correction cycles are plotted in Fig. 10(d). The residual aberration of Débarre’s method is relatively large when using the matrix calibrated with aberration. So the initial aberration must be eliminated from the system before implementing Débarre’s calibration method. In contrast, our calibration method is insensitive to the initial aberration. In Eq. (12) of [30], a linear term is introduced to relax the zero-aberration assumption, but the tolerance range of initial aberration is unclear.

In our correction schemes, either N+1 or 2N, the calibrated matrix G is involved in the wavefront estimation. For an uncalibrated system, we can simply use the IFs as basis modes and apply the quadratic optimization that takes three measurements for each mode (3N algorithm [32]) and doesn’t need to calibrate the matrix G. However, this kind of optimization is inefficient because of the crosstalk between IF modes [32].

A limitation of our calibration method is that it may not work with DM having a large number of actuators. The IFs of this kind of DM can be very localized in the system pupil and would hardly influence the metric function. In addition, in consideration of the measurement noises in calibration, it is not guaranteed that the derived mirror modes don’t contain tip/tilt which leads to image shift. To obtain a set of displacement-free modes, we may refer to the method in [27] that removes the tip/tilt from the basis modes (i.e., IFs in our method) by experiment.

6. Conclusion

A general model-based WFSless AO framework for both point-like and extended objects is derived by investigating the relationships between metric functions and aberration modes in typical model-based algorithms. Under the general framework, the calibration of model-based WFSless AO comes down to the measurement of a Gram matrix. A novel self-calibration method is proposed to calculate the Gram matrix without using a WFS. The self-calibrated matrix is insensitive to initial system aberrations but can be indefinite due to the measurement noise. The negative and small positive eigenvalues of self-calibrated matrix should be discarded to ensure a stable correction.

The calibrated Gram matrix can be used directly for simultaneous correction if using the influence functions of DM as the bias modes. Alternatively, orthogonal or gradient-orthogonal mirror modes obtained from the eigenvectors of the Gram matrix can be used as the modal basis to implement independent sequential correction. The effectiveness of the proposed self-calibration procedure and correction methods has been demonstrated both by simulation and experiments. It is shown that the correction results using the self-calibrated matrix are quite close to those using the matrix calibrated by a WFS. By completely getting rid of the use of WFS, the proposed self-calibration and correction methods can ease the application of model-based WFSless AO in many fields.

Appendix 1

Here we prove that the control signals of DM to generate mirror modes are actually the eigenvectors of Gram matrix G whose elements are inner products of IFs of DM. The mirror modes are linear superposition of IFs and have certain orthogonality that can simplify the metric function as shown in Eq. (3), Eq. (5) and Eq. (8).

The IFs of DM written in matrix is ${\mathbf F} = \left[ {\begin{array}{cccc} {{{\mathbf F}_1}}&{{{\mathbf F}_2}}& \cdots &{{{\mathbf F}_N}} \end{array}} \right]$. If G is calibrated using M₁ as the metric function, then ${\mathbf G = }{{\mathbf G}_{\mathbf 1}}{\mathbf = }{{\mathbf F}^{\mathbf T}}{\mathbf F}$. For other three metric functions, ${\mathbf G = }{{\mathbf G}_{\mathbf 2}}{\mathbf = }\nabla {{\mathbf F}^{\mathbf T}} \cdot \nabla {\mathbf F}$. The mirror modes Y which is a linear transform of F can be expressed by

(35)$${\mathbf Y} = {\mathbf F}{\mathbf E}$$

The columns of E are the control signals of DM to generate mirror modes.

For orthogonal mirror modes, we have

(36)$${{\mathbf \Lambda }_1} = {{\mathbf Y}^{\mathbf T}}{\mathbf Y} = {{\mathbf E}^{\mathbf T}}{{\mathbf F}^{\mathbf T}}{\mathbf F}{\mathbf E}\textrm{ = }{{\mathbf E}^{\mathbf T}}{{\mathbf G}_1}{\mathbf E}$$

For gradient-orthogonal mirror modes, we have

(37)$${{\mathbf \Lambda }_2} = \nabla {{\mathbf Y}^{\mathbf T}} \cdot \nabla {\mathbf Y} = {{\mathbf E}^{\mathbf T}}\nabla {{\mathbf F}^{\mathbf T}} \cdot \nabla {\mathbf F}{\mathbf E} = {{\mathbf E}^{\mathbf T}}{{\mathbf G}_2}{\mathbf E}$$

${{\mathbf \Lambda }_1}$ and ${{\mathbf \Lambda }_2}$are diagonal matrices. Comparing Eq. (36) and Eq. (37) with Eq. (27), we can conclude that E is equivalent to Q. So the control signals of DM to generate mirror modes are the columns of Q, i.e., the eigenvectors of G. The orthogonality of mirror modes depends on which metric function is used in calibration of the matrix G.

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.

References

1. R. Davies and M. Kasper, “Adaptive Optics for Astronomy,” Annu. Rev. Astron. Astrophys. 50(1), 305–351 (2012). [CrossRef]

2. T. Weyrauch and M. A. Verontsov, “Atmospheric compensation with a speckle beacon in strong scintillation conditions: directed energy and laser communication applications,” Appl. Opt. 44(30), 6388–6401 (2005). [CrossRef]

3. H. Sun, N. J. Kasdin, and R. Vanderbei, “Efficient wavefront sensing for space-based adaptive optics,” J. Astron. Telesc. Instrum. Syst. 6(1), 019001 (2020). [CrossRef]

4. M. J. Booth, “Adaptive optical microscopy: The ongoing quest for a perfect image,” Light: Sci. Appl. 3(4), e165 (2014). [CrossRef]

5. T. I. M. van Werkhoven, J. Antonello, H. H. Truong, M. Verhaegen, H. C. Gerritsen, and C. U. Keller, “Snapshot coherence-gated direct wavefront sensing for multi-photon microscopy,” Opt. Express 22(8), 9715–9733 (2014). [CrossRef]

6. D. Ren, B. Dong, Y. Zhu, and D. Christian, “Correction of non-common path error for extreme adaptive optics,” Publ. Astron. Soc. Pac. 124(913), 247–253 (2012). [CrossRef]

7. M. A. Vorontsov and V. P. Sivokon, “Stochastic parallel-gradient-descent technique for high-resolution wave-front phase-distortion correction,” J. Opt. Soc. Am. A 15(10), 2745–2758 (1998). [CrossRef]

8. M. J. Booth, M. A. A. Neil, R. Juškaitis, and T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. 99(9), 5788–5792 (2002). [CrossRef]

9. H. Hofer, N. Sredar, H. Queener, C. Li, and J. Porter, “Wavefront sensorless adaptive optics ophthalmoscopy in the human eye,” Opt. Express 19(15), 14160–14171 (2011). [CrossRef]

10. L. Huang, “Coherent beam combination using a general model-based method,” Chin. Phys. Lett. 31(9), 094205 (2014). [CrossRef]

11. B. Dong, Y. Li, X. Han, and B. Hu, “Dynamic aberration correction for conformal window of high-speed aircraft using optimized model-based wavefront sensorless adaptive optics,” Sensors 16(9), 1414 (2016). [CrossRef]

12. B. Dong and R. Wang, “Laser beam cleanup using improved model-based wavefront sensorless adaptive optics,” Chin. Opt. Lett. 14(3), 031406 (2016).

13. Z. Li and X. Zhao, “BP artificial neural network based wave front correction for sensor-less free space optics communication,” Opt. Commun. 385, 219–228 (2017). [CrossRef]

14. D. Saha, U. Schmidt, Q. Zhang, A. Barbotin, Q. Hu, N. Ji, M. J. Booth, M. Weigert, and E. W. Myers, “Practical sensorless aberration estimation for 3D microscopy with deep learning,” Opt. Express 28(20), 29044–29053 (2020). [CrossRef]

15. I. Vishniakou and J. D. Seelig, “Differentiable model-based adaptive optics for two-photon microscopy,” Opt. Express 29(14), 21418–21427 (2021). [CrossRef]

16. H. Ren, B. Dong, and Y. Li, “Alignment of the active secondary mirror of a space telescope using model-based wavefront sensorless adaptive optics,” Appl. Opt. 60(8), 2228–2234 (2021). [CrossRef]

17. M. J. Booth, “Wave front sensorless adaptive optics, modal wave front sensing, and sphere packings,” Proc. SPIE 5553, 150–158 (2004). [CrossRef]

18. M. J. Booth, “Wave front sensor-less adaptive optics: a model-based approach using sphere packings,” Opt. Express 14(4), 1339–1352 (2006). [CrossRef]

19. M. J. Booth, “Wavefront sensorless adaptive optics for large aberrations,” Opt. Lett. 32(1), 5–7 (2007). [CrossRef]

20. L. Huang and C. Rao, “Wavefront sensorless adaptive optics: a general model-based approach,” Opt. Express 19(1), 371–379 (2011). [CrossRef]

21. W. Lianghua, P. Yang, Y. Kangjian, C. Shanqiu, W. Shuai, L. Wenjing, and B. Xu, “Synchronous model-based approach for wavefront sensorless adaptive optics system,” Opt. Express 25(17), 20584–20597 (2017). [CrossRef]

22. D. Débarre, M. J. Booth, and T. Wilson, “Image based adaptive optics through optimisation of low spatial frequencies,” Opt. Express 15(13), 8176–8190 (2007). [CrossRef]

23. H. Yang, O. Soloviev, and M. Verhaegen, “Model-based wavefront sensorless adaptive optics system for large aberrations and extended objects,” Opt. Express 23(19), 24587–24601 (2015). [CrossRef]

24. H. Ren and B. Dong, “Improved model-based wavefront sensorless adaptive optics for extended objects using N + 2 images,” Opt. Express 28(10), 14414–14427 (2020). [CrossRef]

25. H. Ren and B. Dong, “Fast dynamic correction algorithm for model-based wavefront sensorless adaptive optics in extended objects imaging,” Opt. Express 29(17), 27951–27960 (2021). [CrossRef]

26. B. Wang and M. J. Booth, “Optimum deformable mirror modes for sensorless adaptive optics,” Opt. Commun. 282(23), 4467–4474 (2009). [CrossRef]

27. A. Thayil and M. J. Booth, “Self calibration of sensorless adaptive optical microscopes,” J. Eur. Opt. Soc. Rapid Publ. 6, 11045 (2011). [CrossRef]

28. D. Débarre, E. J. Botcherby, M. J. Booth, and T. Wilson, “Adaptive optics for structured illumination microscopy,” Opt. Express 16(13), 9290–9305 (2008). [CrossRef]

29. D. Débarre, E. J. Botcherby, T. Watanabe, S. Srinivas, M. J. Booth, and T. Wilson, “Image-based adaptive optics for two-photon microscopy,” Opt. Lett. 34(16), 2495–2497 (2009). [CrossRef]

30. J. Antonello, M. Verhaegen, R. Fraanje, T. Werkhoven, H. C. Gerritsen, and C. U. Keller, “Semidefinite programming for model-based sensorless adaptive optics,” J. Opt. Soc. Am. A 29(11), 2428–2438 (2012). [CrossRef]

31. D. Débarre, A. Facomprez, and E. Beaurepaire, “Assessing correction accuracy in image-based adaptive optics,” Proc. SPIE 8253, 82530F (2012). [CrossRef]

32. A. Facomprez, E. Beaurepaire, and D. Débarre, “Accuracy of correction in modal sensorless adaptive optics,” Opt. Express 20(3), 2598–2612 (2012). [CrossRef]

Self-calibrated general model-based wavefront sensorless adaptive optics for both point-like and extended objects

Abstract

1. Introduction

2. Principle

2.1 General framework of model-based WFSless AO

2.2 Self-calibration procedure

2.3 Correction methods based on calibrated matrix

2.3.1. Simultaneous correction scheme (N+1 algorithm)

2.3.2. Sequential correction scheme (2N algorithm)

3. Simulations

4. Experiments

5. Discussion

6. Conclusion

Appendix 1

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (10)

Equations (37)

Optics Express