Abstract
For low-order adaptive optics systems, a method that is able to correct for system aberrations in the final focal plane is presented. The paper presents a novel figure of merit, corresponding to the integrated non-normalized tip-tilt-free optical transfer function. The inherent singular value decomposition modal content of the interaction matrix is used to optimize this figure of merit. The method has proven to be stable and robust, providing a simple mean to facilitate diffraction limited imaging in an experimental setup for ophthalmic applications.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
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
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
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
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
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, nmax, 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 rGS 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.
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 nmax 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.
The columns of U, e.g., un 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 nmax modes twice, since most of the energy is contained within these modes. The number of scan points was empirically chosen as jmax = 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.
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 nmax. 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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