Direct and inverse discrete Zernike transform

Spatiotemporal statistics of the turbulent piston-removed phase and Zernike coefficients for two distinct beams

In the context of adaptive optics for astronomy, one can rely on the statistics of the turbulent phase to assess a part of the system's performance. Temporal statistics with one source and spatial statistics with two sources are well known and widely used for classical adaptive optics systems. A more general framework, including both spatial and temporal statistics, can be useful for analysis of the existing systems and to support the design of future ones. In this paper, we propose an expression of the temporal cross power spectral densities of turbulent phases in two distinct beams, which is from two different sources to two different apertures. We consider the phase either as it is, without a piston, or as its decomposition on Zernike modes. The general formulas allow coverage of a wide variety of configurations, from single-aperture to interferometric telescopes equipped with adaptive optics, with the possibility to consider apertures of different sizes and/or sources at a finite distance. The presented approach should lead to similar results with respect to existing methods in the Fourier domain, but it is focused on temporal frequencies rather than spatial ones, which might be convenient for some aspects such as control optimization. To illustrate this framework with a simple application, we demonstrate that the wavefront residual due to the anisoplanatism error in a single-conjugated adaptive optics system is overestimated when it is computed from covariances without taking into account the temporal filtering of the adaptive optics loop. We also show this overestimation in the case of a small-baseline interferometer, for which the two beams are significantly correlated.

Golden angle based scanning for robust corneal topography with OCT

Corneal topography allows the assessment of the cornea's refractive power which is crucial for diagnostics and surgical planning. The use of optical coherence tomography (OCT) for corneal topography is still limited. One limitation is the susceptibility to disturbances like blinking of the eye. This can result in partially corrupted scans that cannot be evaluated using common methods. We present a new scanning method for reliable corneal topography from partial scans. Based on the golden angle, the method features a balanced scan point distribution which refines over measurement time and remains balanced when part of the scan is removed. The performance of the method is assessed numerically and by measurements of test surfaces. The results confirm that the method enables numerically well-conditioned and reliable corneal topography from partially corrupted scans and reduces the need for repeated measurements in case of abrupt disturbances.

Recursive formula to compute Zernike radial polynomials

In optics, Zernike polynomials are widely used in testing, wavefront sensing, and aberration theory. This unique set of radial polynomials is orthogonal over the unit circle and finite on its boundary. This Letter presents a recursive formula to compute Zernike radial polynomials using a relationship between radial polynomials and Chebyshev polynomials of the second kind. Unlike the previous algorithms, the derived recurrence relation depends neither on the degree nor on the azimuthal order of the radial polynomials. This leads to a reduction in the computational complexity.

Wavefront sensing with critical sampling

Different types of nonredundant sampling patterns are shown to guarantee completeness of the basis formed by the sampled partial derivatives of Zernike polynomials, commonly used to reconstruct the wavefront from its slopes (wavefront sensing). In the ideal noise-free case, this enables one to recover double the number of modes J than sampling points I (critical sampling J=2I). With real data, noise amplification makes the optimal number of modes lower I<J<2I. Our computer simulations show that optimized nonredundant sampling provides a significant improvement of wavefront reconstructions, with the number of modes recovered about 2.5 higher than with standard sampling patterns.