Correction of static and non-common path aberrations in an adaptive optics system using inherent calibration data

Per Knutsson; Jörgen Thaung; Mette Owner-Petersen; Zoran Popović; Zoran Popović

doi:10.1364/OE.408954

1. Introduction

Adaptive optics (AO) has been applied in different areas of science, e.g., astronomy [1,2] and ophthalmology [3], to enhance the capabilities of imaging systems and also in beam control of lasers [4]. In environments where an imaging system is used to observe objects behind a continuously evolving phase curtain (atmosphere, ocular optics, heating effects, etc.), the AO system can effectively mitigate the effects of this medium to regain the loss of imaging performance within the control space of the deformable mirror. However, common to any sensory or active control system, phase errors outside the control space of the deformable mirror or the wavefront sensor will remain uncorrected.

The purpose of an AO system is to correct aberrations, or phase errors Φ(ξ), across the pupil thereby reducing the phase variance

(1)$$\sigma _{\mathit{\Phi }} ^2 = \mathop \int \limits_{{A_p}} {\left[ {{\mathit{\Phi }}({\boldsymbol{\mathrm{\xi }}} )- \mathop {\mathit{\Phi }} \limits^ - } \right]^2}d{\boldsymbol{\mathrm{\xi }}}/{A_p}$$

where ${\overline{\mathit{\Phi }}}$ is the average phase over the pupil area ${A_p}$. However, the wavefront sensor in many AO-systems is relative (e.g., a Shack-Hartmann sensor), meaning that static errors are unseen if not calibrated correctly. Even with an absolute wavefront sensor, non-common path (NCP) errors between the wavefront sensor and final image plane imply an aberrated optical path of the imaging optics. It is therefore crucial to reduce these effects to achieve optimal performance of the AO system.

The pupil phase error can be deduced from the final focal plane intensity by using phase retrieval or phase diversity, which has been applied to reduce static and NCP errors in telescopes [5–9]. Phase diversity needs duplicate measurements with a known pupil phase difference between the measurements, and thus a physical modification of the optical system or any other means to introduce controlled aberrations. Phase retrieval can also be used in this context but is likewise often dependent on a known alteration of the setup, e.g., amplitude masks or aberrations (single image phase diversity can be used in that case) and is generally computationally intense. Further, AO systems can include an interferometer to estimate these errors [10,11].

Other tools to improve imaging performance are image sharpening [12–16], where final image quality is enhanced by optimizing a focal plane quality metric or figure of merit, or sensorless adaptive optics, in which operation is based upon maximizing a detector signal or a related measure [17,18]. A recent paper describes a wavefront correction algorithm for AO systems operating on either a Zernike or generalized deformable mirror mode base that uses the square of the focal spot radius as a criterion to guide correction [19]. There are of course also options to use neural networks for wavefront sensing in the focal plane [20].

The patented [21–24] method presented in this paper is similar to the imaging sharpening method, but a novel figure of merit based on the optical transfer function (OTF) is used, and the inherent singular modes of the AO system are optimized. The purpose is to establish the optimal reference points for a Shack-Hartmann wavefront sensor. The scope of this paper is not to compare the proposed figure-of-merit to other image sharpening techniques and/or phase diversity methods, but that part is of course welcomed by the authors to do elsewhere. The scope is to present a method that exploits already existing data and functions in an AO system, and propose to handle this data in a computationally efficient way to correct for NCP errors. The only necessary addition to the system is a reference object, which is necessary also for other image sharpening and phase diversity techniques.

2. Method

The method presented below rests upon familiar simple theoretical relationships, and it is easily implemented in high-level programming environments such as LabVIEW [25] or MATLAB [26]. A figure of merit based on the optical transfer function (OTF) that is shown to reduce the focal plane aberrations is given in the next subsection. An iterative improvement of this figure using the singular value decomposition (SVD) modal content of the interaction matrix is described in the succeeding subsection.

2.1 Figure of merit

Quality metrics for general objects have been considered elsewhere [12] but given the fact that the interaction matrix of most AO systems is calibrated with a point source, the discussion here will be limited to that. Image plane quality metrics can then be encircled energy radius or ${I^n}(\textrm{\textbf{x}} )$ where $I(\textrm{\textbf{x}} )$ is the focal plane intensity etc. [13,27]. A commonly used metric that describes the AO system performance is the Strehl ratio

(2)$$S = \frac{{I({0,0} )}}{{{I_\ast }({0,0} )}} = \frac{{\smallint \tilde{I}({\textrm{\textbf{f}}} )d{\textrm{\textbf{f}}}}}{{\smallint {{\tilde{I}}_\ast }({\textrm{\textbf{f}}})d{\textrm{\textbf{f}}}}} \approx \textrm{exp} ( - \sigma _{\mathit{\Phi }} ^2), $$

where $I({0,0} )$ is the on-axis image intensity, ${I_\ast }({0,0} )$ is the on-axis aberration-free image intensity in the focal plane and ~ denotes the Fourier transform. The second equality is a consequence of the definite integral theorem [28], and following that, the Marèchal approximation is given, valid for phase deviations conforming to Gaussian statistics [29]. The Marèchal approximation is seen to describe a Gaussian function of the RMS pupil phase error ${\sigma _{\mathit{\Phi }} }$, but is also commonly given in a quadratic form $S \approx 1 - \sigma _{\mathit{\Phi }} ^2$. Both approximations are valid for ${\sigma _{\mathit{\Phi }} } \ll 1$. Looking at simulated Strehl values for the Zernike modes from 2nd to 5th radial order it is seen that the observed Strehl value will approximate a Gaussian function $\textrm{exp}({ - \sigma_{\mathit{\Phi }} ^2} )$ over a larger aberration interval than the quadratic decay $1 - \sigma _{\mathit{\Phi }} ^2$ (see Fig. 1).

Fig. 1. Simulation of Strehl ratios for Zernike terms of 2nd to 5th radial order in grey, compared to Strehl approximations $S \approx \textrm{exp}({ - \sigma_{\mathit{\Phi }} ^2} )$ (dashed) and $S \approx 1 - \sigma _{\mathit{\Phi }} ^2$ (dotted).

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If the actual tip-tilt contribution is neglected, which does not affect the image quality, the peak intensity is found at ${\textbf{\textrm{x}}_{\textrm{max}}} = \textrm{arg}\; \textrm{ma}{\textrm{x}_\textrm{x}}I(\textbf{\textrm{x}} )$, and according to the shift theorem [28] it is found that

(3)$${S_ \bot } = \frac{{I({\textbf{x}_{\max }})}}{{{I_\ast }(0,0)}} = \frac{{\int {\tilde{I}(\textbf{f})\textrm{exp} (i2\pi \textbf{f}{\textbf{x}_{\max }})d\textbf{f}} }}{{\int {{{\tilde{I}}_\ast }(\textbf{f})d\textbf{f}} }} \approx \textrm{exp} ( - \sigma _{{{\mathit{\Phi }} _ \bot }}^2).$$

It is obvious from this expression that minimizing the phase error (purpose of the AO system) is identical to maximizing the numerators. A Nyquist-sampled or near thereto image will suffer from severe discretization implying that the spatial domain has been found less suitable for the task of estimating the Strehl value. In case it is used, it is common to change the sampling interval by zero-padding in the Fourier domain followed by inverse transformation [30]. Since all information is contained within the Fourier domain, the quality metric used here is based on the Fourier transformed image. For a discrete image ${I_{rs}}$ from the imaging sensor, where the pixel coordinates are given by the positive integers r and s, the subpixel shift in the image domain $({{x_{\textrm{max}}},{y_{\textrm{max}}}} )$ is estimated with a quadratic interpolation around the maximum pixel intensity value at ${I_{{r_{\textrm{max}}}{s_{\textrm{max}}}}}$ giving [31]

(4)$${x_{\max }} = ({r_{\max }} - \frac{m}{2} - 1) + \frac{{{I_{({r_{\max }} - 1){s_{\max }}}} - {I_{({r_{\max }} + 1){s_{\max }}}}}}{{2[{I_{({r_{\max }} - 1){s_{\max }}}} + {I_{({r_{\max }} + 1){s_{\max }}}} - 2{I_{{r_{\max }}{s_{\max }}}}]}},$$

and analogous for ${y_{\textrm{max}}}$. The continuous pixel coordinates x and y have an origin in the center if the discrete image, hence the subtraction of $m/2 + 1$ as the image format is $m \times m$. Centering of the PSF is needed if the singular modes in the AO system have significant tip-tilt content and the actual position of the PSF is irrelevant. The proposed figure of merit is then given by the discrete version of the Fourier domain numerator in Eq. (3)

(5)$$Q = \sum\limits_{r = 1}^m {\sum\limits_{s = 1}^m {{{\tilde{I}}_{rs}}\textrm{exp} \left( {\frac{{i2\pi }}{m}\left[ {\left( {r - \frac{m}{2} - 1} \right){x_{\textrm{max}}} + \left( {s - \frac{m}{2} - 1} \right){y_{\max }}} \right]} \right)} .}$$

Maximizing this figure of merit, the integrated non-normalized tip-tilt-free optical transfer function, will minimize the wavefront error and maximize the Strehl ratio. ${\tilde{I}_{rs}}$ is the discrete Fourier transform (e.g., using FFT) of the point source image ${I_{rs}}$. Since the image intensity is a real function, its Fourier transform will be Hermitian, i.e., $\tilde{I}({ - {\textrm{\textbf{f}}}} )= {\tilde{I}^\ast }({\textrm{\textbf{f}}} )$, and the imaginary part will cancel out to give a real figure of merit. Hence the summation can be limited to the real part in half of the Fourier domain to speed up calculations.

2.2 Single value decomposition modal optimization

The phase in a pupil plane imposed by a deformable mirror with k actuators can be denoted by

(6)$${\mathit{\Phi}} ({\boldsymbol{\mathrm{\xi }}} )= \mathop \sum \nolimits_k {c_k}{\mathit{\Phi}} _k^\textrm{I}({\boldsymbol{\mathrm{\xi }}} )$$

where ${\mathit{\Phi}} _k^\textrm{I}({\boldsymbol{\mathrm{\xi }}} )$ is the point response function, or influence function, of a unit actuator command ${c_k} = 1$. The phase is measured by the wavefront sensor producing the measurement vector s. During calibration of an AO system, the interaction matrix s = Gc is obtained by measuring the unit response of each actuator and collecting these wavefront sensor measurements as columns in G. During closed loop the (truncated or Tikhonov regularized) pseudoinverse is used to update the shape of the deformable mirror c = G⁺s, and common in the control of AO systems is to obtain the singular value decomposition G = VΛU^T; for a reference on SVD see e.g., [32]. Λ is an $m \times n$ rectangular matrix with non-negative numbers on the diagonal referred to as the singular values of G, with their number equal to the rank of G. The columns of V and U, ${\textrm{\textbf{v}}_m}$ and ${\textrm{\textbf{u}}_n}$, are the left and right singular modes defining orthonormal vectors in sensor measurement space and actuator command space respectively. These are also ordered according to sensitivity, starting with the most sensitive modes. Hence each singular mode is an orthogonal phase distribution according to Eq. (7), i.e.

(7)$${{\mathit{\Phi}} _n}({\boldsymbol{\mathrm{\xi }}} )= \mathop \sum \nolimits_k {U_{kn}}{\mathit{\Phi}} _k^\textrm{I}({\boldsymbol{\mathrm{\xi }}} ), $$

and changing the magnitude of this phase mode to ${\alpha }_n {\mathit{\Phi }}_n$(ξ) corresponds to applying the actuator commands ${\alpha _n}{\textbf{\textrm{u}}_n}$. It is certainly common to use also other orthogonal expansions, e.g., Zernike polynomials, to describe the phase, but the SVD method offers to a natural decomposition of the AO system’s inputs and outputs without any further approximations, simultaneously grading the sensitivity of the singular modes. The method presented here exploits scanning of the orthogonal singular modes ${\mathit{\Phi }}_n$(ξ). According to Eqs. (1)–(4) the quality metric Q will follow a Gaussian function for small aberrations. Hence, for each scanned singular mode (changing ${\alpha _n}$ over j points, where j is odd to allow for even sampling around zero value), a least squares fit gives

(8)$$\left[ {\begin{array}{c} {{{\hat{a}}_n}}\\ {{{\hat{b}}_n}}\\ {{{\hat{k}}_n}} \end{array}} \right] = \mathop {\arg \min }\limits_{{a_n},{b_n},{k_n}} {\sum\limits_{j = 1}^{{j_{\max }}} {\left|\left|{{Q_j} - {k_n}\textrm{exp} ( - \frac{{{{({a_{n,j}} - {a_n})}^2}}}{{b_n^2}})} \right|\right|} ^2},$$

where the last term in the squared norm defines a Gaussian function, given by the parameters ${\alpha _n}$, ${b_n}$ and ${k_n}$. The peak of the estimated function is found at ${\hat{a}_n}$, and hence for a specific singular mode the Strehl is maximized and the wavefront error is minimized for ${\hat{a}_n}{{\mathit{\Phi}} _n}({\boldsymbol{\mathrm{\xi }}} )$. As all singular modes have been optimized the mirror shape that optimizes the AO performance, without externally introduced aberrations that are corrected during close loop operation, will be ${\textbf{\textrm{c}}_\ast } = \mathop \sum \limits_n {\hat{a}_n}{\textbf{\textrm{u}}_n}$. A flow chart describing the method is given in Fig. 2. The optimal vector is translated to an additive to the reference Hartmann points through $\Delta {\textbf{\textrm{s}}_\ast } = \textbf{G}{\textbf{\textrm{c}}_\ast }$, defining the wavefront sensor signal that reduces externally induced phase errors and system internal NCP errors.

Fig. 2. Flow chart of optimization procedure. An initial shape of the DM different from zero can be set, e.g., from a previous calibration, hence the motivation for the first step “Apply original command c_*”. The n_max singular modes are sampled over j points.

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This method is proposed, since all parameters that are needed to achieve the optimization are already available in most AO systems and it can be executed immediately after the interaction matrix has been calibrated without any alteration of the setup. No further assumptions need to be made, nor is any additional equipment or alteration of the optical system needed. However, it does require that the pupil is defined by an aperture along the imaging path, and that this aperture is matched to the closed-loop pupil. It is of course also important to consider the limited intervals of ${\alpha _n}$ due to the limited actuator stroke. For ill-sensed modes ($n$ high) this method will also elucidate which modes that are sensible to correct in the reconstruction matrix that is obtained by truncating SVD modes, and the optimized ${\textbf{\textrm{c}}_\ast }$ will contain only the controlled modes.

2.3 Level of correction

There are of course limitations on the possible level of correction for the method. Due to the orthogonality of the singular modes, the method will converge to a local maximum. However, several iterations of the method are needed for the case of severe initial aberrations (no core in the PSF). For the practical implementation of the correction procedure, when a new calibration is needed, we have found it useful to start the new calibration from the preceding calibration of the DM, i.e., from the old command vector ${\textbf{\textrm{c}}_\ast }$, since the quasi-static aberrations in the imaging path are likely similar. Likewise, the first 10 modes are scanned twice, since loss of alignment and thermally induced errors will plausibly introduce low-order aberrations such as astigmatism and coma, and these are commonly present among the well-sensed modes of the AO system.

Given that the method will find the global maximum, the fitting error, or ability of the deformable mirror to reproduce the spatial phase deviation, will limit the level of correction. Since this is an error that is dependent on the individual characteristics of the system, it will not be considered here. Likewise, a truncation of the number of SVD modes to optimize, n_max, is a form of fitting error that will limit the achievable correction. When lower-order misalignment aberrations are present, these can certainly be corrected by the deformable mirror and it is assumed that the fitting error is small.

3. Results

The method presented above has been implemented in two ophthalmic AO instruments [33,34]. The current instrument [34] features a Hi-Speed DM52-1.5 as the pupil DM and a Hi-Speed DM97-1.5 as the field-correcting DM (both from ALPAO S.A.S., Grenoble, France), as well as a multi-object Shack-Hartmann wavefront sensor with 5 × 89 lenslets, yielding a reconstructor matrix of 149 × 890 elements.

Five single mode optical fibers were used as point sources ($\lambda $ = 632 nm) in the retinal conjugate plane R1 of the AO-instrument described in [34] (Fig. 3(A), top left). Their positions roughly correspond to the positions of the five guide objects on the retina (Fig. 3(B)), where r_GS corresponds to a visual angle of 3.1 deg (880 µm). The sampling of the CCD detector corresponds to Nyquist sampling and the PSFs were optimized according to the method above.

Fig. 3. A) Cutout of optical layout of an ophthalmic AO instrument showing the NCP calibration path. The single mode optical fiber point sources are located at the retinal conjugate plane R1 to the upper left. The optical path used for NCP calibration includes two deformable mirrors (DM1 and DM2), relay optics, a cold mirror (CM), imaging path optics, and a retinal camera. B) The optical fibers are arranged in a manner roughly corresponding to the positions of the guide objects on the retina.

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The scan interval of each mode was adjusted individually, with an increasing interval of ${\alpha _n}$ for each mode, roughly corresponding to ${\sigma _{\mathit{\Phi }}} \approx [{ - 0.15,0.15} ]$ waves for which the Gaussian approximation has a small deviation from observed values (c.f. Figure 1). Likewise, the number of modes n_max to optimize was limited since it was obvious that that ill-sensed modes did not follow the Marèchal approximation, and the threshold was set according to this criterion.

Table 1. Estimated Strehl values of PSFs in Fig. 4 for the cases of no correction, correction without NCP optimization, and correction with NCP optimization.

View Table

The columns of U, e.g., u_n are the actuator command space singular modes from the SVD decomposition. They are ordered according to sensitivity, starting with the most sensitive modes. We consequently chose to optimize the first 10 of the n_max modes twice, since most of the energy is contained within these modes. The number of scan points was empirically chosen as j_max = 11, with five samples below and above the nominal value to allow for a robust gaussian fit to the data. Typical PSF and Strehl results of the method are given in Fig. 4 and Table 1, respectively.

Fig. 4. Log-intensity-plot of PSFs, with mean background subtracted to reveal fine details, from five optical fiber point sources located in the retinal conjugate plane R1 in Fig. 3(A): top row – uncorrected, middle row – AO corrected with a regular Hartmann spot pattern according to lenslet pitch used as wavefront sensor reference and no NCP optimization, bottom row – AO corrected with NCP optimization. The dominating residual static aberrations in the middle row are coma and astigmatism (bow-tie pattern). This is no longer visible after NCP calibration. Image size 32 × 32 pixels (retinal scale 0.982 µm/pixel)

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For each guide object, the optimal vector is translated to an additive to the reference points of the Shack-Hartmann wavefront sensor through $\Delta {\textbf{\textrm{s}}_\ast } = \textbf{G}{\textbf{\textrm{c}}_\ast }$. Alternatively, the reference points can be exposed once the mirror shape is optimized.

4. Discussion

This method being computationally simple and exploiting only already available data and functions, has proven to be very useful and reliable when used in an ophthalmic AO system [33,34], allowing diffraction limited imaging. Here, the method has been used on five guide objects. The method is being run immediately after the calibration of the interaction matrix and it has been found to be autonomous, once tuned to proper scan intervals of ${\alpha _n}$ and number of modes to optimize n_max. Misalignment aberrations such as astigmatism and coma are most well-sensed modes by the AO system, and thus corrected early on in the iterative process. Arguably for a high-order AO system, e.g., extreme AO, the method would be time consuming, and why phase diversity is a better option.

One option to speed up the proposed method would be to temporally multiplex the modal optimization procedure, as in [35]. However, due to its simplicity it has been very useful in a low-order AO system, which is often the case for ophthalmic applications. Potential error sources are an uneven illumination of the pupil, and a mismatch between the aperture and pupil in case an aperture is needed during the calibration, but these are error sources for phase diversity as well [7].

The quality metric Q is similar to the integrated optical transfer function OTF, it is actually the integrated non-normalized tip-tilt-free OTF. During the development of the method the MTF or modulus of the OTF was used, but this often resulted in residual symmetrical tails in the optimized PSF. The centering of the PSF according to Eq. (4) was needed since many of the singular modes in the AO system had significant tip-tilt content and the actual position of the PSF is irrelevant in our application.

Funding

De Blindas Vänner (10/07).

Disclosures

PK, JT, MOP, ZP: Profundus AB, Gothenburg, Sweden (I, P).

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Strehl	PSF_1	PSF_2	PSF_3	PSF_4	PSF_5
Uncorrected	0.19	0.14	0.18	0.19	0.19
AO corrected	0.22	0.24	0.33	0.30	0.30
NCP corrected	0.73	0.68	0.71	0.79	0.75

Correction of static and non-common path aberrations in an adaptive optics system using inherent calibration data

Abstract

1. Introduction

2. Method

2.1 Figure of merit

2.2 Single value decomposition modal optimization

2.3 Level of correction

3. Results

4. Discussion

Funding

Disclosures

References

Cited By

Figures (4)

Tables (1)

Equations (8)

Optics Express