Centroid error due to non-uniform lenslet illumination in the Shack-Hartmann wavefront sensor

Statistical optimal parameters obtained by using clinical human ocular aberrations for high-precision aberration measurement

PURPOSE

Compared to Shack-Hartmann wavefront sensor (SHWS), the parameters of virtual SHWS (vSHWS) can be easily adjusted to obtain the optimal performance of aberration measurement. Its current optimal parameters are obtained with only a set of statistical aberrations and not statistically significant. Whether the above parameters are consistent with the statistical results of the optimal parameters corresponding to each set of aberrations, and which performance is better if not? The purpose of this study was to answer these questions.

METHODS

The optimal parameters to reconstruct 624 sets of clinical ocular aberrations in the highest accuracy, including the numbers of sub-apertures (NSAs) and the expansion ratios (ERs) of electric field zero-padding, were determined sequentially in this work. By using wavefront-reconstruction accuracy as an evaluation index, the statistical optimal parameter configuration was selected from some possible configurations determined by the optimal NSAs and ERs.

RESULTS

The statistical optimal parameters are consistent for normal and abnormal eyes. They are different from the optimal parameters obtained with a set of statistical aberrations from the same 624 sets of aberrations, and the performance using the former is better than that using the latter. The performance using a fixed set of statistical optimal parameters is even close to that using the respective optimal parameters corresponding to each set of aberrations.

CONCLUSION

The vSHWS configured with a fixed set of statistical optimal parameters can be used for high-precision aberration measurement of both normal and abnormal eyes. The statistical optimal parameters are more suitable for vSHWS than the parameters obtained with a set of statistical aberrations. These conclusions are significant for the designs of vSHWS and also SHWS.

Zernike-like Laguerre-Gaussian orthonormal polynomials for optical field reconstruction

Analytic closed form expressions for orthonormal polynomials exhibiting both rotational and Gaussian symmetries are derived for both circular and elliptical geometries. They exhibit a close correspondence to the Zernike polynomials but are of Gaussian shape and orthogonal over the (x,y) plane. Consequently, they may be expressed in terms of Laguerre polynomials. Formulas for calculating the centroid of a real function are also presented and, along with the analytic expressions for the polynomials, may prove to be of especial use in reconstruction of the intensity distribution incident on a Shack-Hartmann wavefront sensor.

Dynamic wavefront distortion in resonant scanners

Dynamic mirror deformation can substantially degrade the performance of optical instruments using resonant scanners. Here, we evaluate two scanners with resonant frequencies >12kHz with low dynamic distortion. First, we tested an existing galvanometric motor with a novel, to the best of our knowledge, mirror substrate material, silicon carbide, which resonates at 13.8 kHz. This material is stiffer than conventional optical glasses and has lower manufacturing toxicity than beryllium, the stiffest material currently used for this application. Then, we tested a biaxial microelectromechanical (MEMS) scanner with the resonant axis operating at 29.4 kHz. Dynamic deformation measurements show that wavefront aberrations in the galvanometric scanner are dominated by linear oblique astigmatism (90%), while wavefront aberrations in the MEMS scanner are dominated by horizontal coma (30%) and oblique trefoil (27%). In both scanners, distortion amplitude increases linearly with deflection angle, yielding diffraction-limited performance over half of the maximum possible deflection for wavelengths longer than 450 nm and over the full deflection range for wavelengths above 850 nm. Diffraction-limited performance for shorter wavelengths or over larger fractions of the deflection range can be achieved by reducing the beam diameter at the mirror surface. The small dynamic distortion of the MEMS scanner offers a promising alternative to galvanometric resonant scanners with desirable but currently unattainably high resonant frequencies.

Retinal adaptive optics imaging with a pyramid wavefront sensor

The pyramid wavefront sensor (P-WFS) has replaced the Shack-Hartmann (SH-) WFS as the sensor of choice for high-performance adaptive optics (AO) systems in astronomy. Many advantages of the P-WFS, such as its adjustable pupil sampling and superior sensitivity, are potentially of great benefit for AO-supported imaging in ophthalmology as well. However, so far no high quality ophthalmic AO imaging was achieved using this novel sensor. Usually, a P-WFS requires modulation and high precision optics that lead to high complexity and costs of the sensor. These factors limit the competitiveness of the P-WFS with respect to other WFS devices for AO correction in visual science. Here, we present a cost-effective realization of AO correction with a non-modulated P-WFS based on standard components and apply this technique to human retinal in vivo imaging using optical coherence tomography (OCT). P-WFS based high quality AO imaging was successfully performed in 5 healthy subjects and smallest retinal cells such as central foveal cone photoreceptors are visualized. The robustness and versatility of the sensor is demonstrated in the model eye under various conditions and in vivo by high-resolution imaging of other structures in the retina using standard and extended fields of view. As a quality benchmark, the performance of conventional SH-WFS based AO was used and successfully met. This work may trigger a paradigm shift with respect to the wavefront sensor of choice for AO in ophthalmic imaging.

Optimization of Virtual Shack-Hartmann Wavefront Sensing

Virtual Shack–Hartmann wavefront sensing (vSHWS) can flexibly adjust parameters to meet different requirements without changing the system, and it is a promising means for aberration measurement. However, how to optimize its parameters to achieve the best performance is rarely discussed. In this work, the data processing procedure and methods of vSHWS were demonstrated by using a set of normal human ocular aberrations as an example. The shapes (round and square) of a virtual lenslet, the zero-padding of the sub-aperture electric field, sub-aperture number, as well as the sequences (before and after diffraction calculation), algorithms, and interval of data interpolation, were analyzed to find the optimal configuration. The effect of the above optimizations on its anti-noise performance was also studied. The Zernike coefficient errors and the root mean square of the wavefront error between the reconstructed and preset wavefronts were used for performance evaluation. The performance of the optimized vSHWS could be significantly improved compared to that of a non-optimized one, which was also verified with 20 sets of clinical human ocular aberrations. This work makes the vSHWS’s implementation clearer, and the optimization methods and the obtained results are of great significance for its applications.

Shack-Hartmann wavefront sensor optical dynamic range

The widely used lenslet-bound definition of the Shack-Hartmann wavefront sensor (SHWS) dynamic range is based on the permanent association between groups of pixels and individual lenslets. Here, we formalize an alternative definition that we term optical dynamic range, based on avoiding the overlap of lenslet images. The comparison of both definitions for Zernike polynomials up to the third order plus spherical aberration shows that the optical dynamic range is larger by a factor proportional to the number of lenslets across the SHWS pupil. Finally, a pre-centroiding algorithm to facilitate lenslet image location in the presence of defocus and astigmatism is proposed. This approach, based on the SHWS image periodicity, is demonstrated using optometric lenses that translate lenslet images outside the projected lenslet boundaries.

Multi-layer Shack-Hartmann wavefront sensing in the point source regime

The Shack-Hartmann wavefront sensor (SHWS) is often operated under the assumption that the sensed light can be described by a single wavefront. In biological tissues and other multi-layered samples, secondary wavefronts from axially and/or transversely displaced regions can lead to artifactual aberrations. Here, we evaluate these artifactual aberrations in a simulated ophthalmic SHWS by modeling the beacons that would be generated by a two-layer retina in human and mouse eyes. Then, we propose formulae for calculating a minimum SHWS centroid integration area to mitigate these aberrations by an order of magnitude, potentially benefiting SHWS-based metrology and adaptive optics systems such as those used for retinal imaging and microscopy.

Dynamic distortion in resonant galvanometric optical scanners

High-speed optical systems are revolutionizing biomedical imaging in microscopy, DNA sequencing, and flow cytometry, as well as numerous other applications, including data storage, display technologies, printing, and autonomous vehicles. These systems often achieve the necessary imaging or sensing speed through the use of resonant galvanometric optical scanners. Here, we show that the optical performance of these devices suffers due to the dynamic mirror distortion that arises from the variation in torque with angular displacement. In one of two scanners tested, these distortions result in a variation of signal-to-noise (Strehl) ratio by an order of magnitude across the field of view, degrading transverse resolution by more than a factor of 2. This mirror distortion could be mitigated through the use of stiffer materials, such as beryllium or silicon carbide, at the expense of surface roughness, as these cannot be polished to the same degree of smoothness as common optical glasses. The repeatability of the dynamic distortion indicates that computational and optical corrective methods are also possible.

Average gradient of Zernike polynomials over polygons

Wavefront estimation from slope sensor data is often achieved by fitting measured slopes with Zernike polynomial derivatives averaged over the sampling subapertures. Here we discuss how the calculation of these average derivatives can be reduced to one-dimensional integrals of the Zernike polynomials, rather than their derivatives, along the perimeter of each subaperture. We then use this result to derive closed-form expressions for the average Zernike polynomial derivatives over polygonal areas, only requiring evaluation of polynomials at the polygon vertices. Finally, these expressions are applied to simulated Shack-Hartmann wavefront sensors with 7 and 23 fully illuminated lenslets across a circular pupil, with their accuracy and calculation time compared against commonly used integration methods.

Wavefront reconstruction of a Shack-Hartmann sensor with insufficient lenslets based on an extreme learning machine

In a standard Shack-Hartmann wavefront sensor, the number of effective lenslets is the vital parameter that limits the wavefront restoration accuracy. This paper proposes a wavefront reconstruction algorithm for a Shack-Hartmann wavefront sensor with an insufficient microlens based on an extreme learning machine. The neural network model is used to fit the nonlinear corresponding relationship between the centroid displacement and the Zernike model coefficients under a sparse microlens. Experiments with a 6×6 lenslet array show that the root mean square (RMS) relative error of the proposed method is only 4.36% of the initial value, which is 80.72% lower than the standard modal algorithm.

Intensity-enhanced deep network wavefront reconstruction in Shack-Hartmann sensors

The Shack-Hartmann wavefront sensor (SH-WFS) is known to produce incorrect measurements of the wavefront gradient in the presence of non-uniform illumination. Moreover, the most common least-squares phase reconstructors cannot accurately reconstruct the wavefront in the presence of branch points. We therefore developed the intensity/slopes network (ISNet), a deep convolutional-neural-network-based reconstructor that uses both the wavefront gradient information and the intensity of the SH-WFS's subapertures to provide better wavefront reconstruction. We trained the network on simulated data with multiple levels of turbulence and compared the performance of our reconstructor to several other reconstruction techniques. ISNet produced the lowest wavefront error of the reconstructors we evaluated and operated at a speed suitable for real-time applications, enabling the use of the SH-WFS in stronger turbulence than was previously possible.