Noise propagation in wave-front sensing with phase diversity

Abstract

The phase diversity technique is studied as a wave-front sensor to be implemented with widely extended sources. The wave-front phase expanded on the Zernike polynomials is estimated from a pair of images (in focus and out of focus) by use of a maximum-likelihood approach. The propagation of the photon noise in the images on the estimated phase is derived from a theoretical analysis. The covariance matrix of the phase estimator is calculated, and the optimal distance between the observation planes that minimizes the noise propagation is determined. The phase error is inversely proportional to the number of photons in the images. The noise variance on the Zernike polynomials increases with the order of the polynomial. These results are confirmed with both numerical and experimental validations. The influence of the spectral bandwidth on the phase estimator is also studied with simulations.

Calibration and precompensation of noncommon path aberrations for extreme adaptive optics

“…According to Figure 5 and to the fact that precision of PD is inversely proportional to SNR in image, 6 a SNR of 10 4 is sufficient to have a precision of 0.4nm 2 on each of the 100 measured Zernike, i.e. a residual WFE of 6nm.…”

“…The WFS must therefore be based on the processing of the focal plane images recorded by the scientific camera itself. The phase diversity (PD) approach 4,5,6,7 is a simple and efficient candidate to perform such a measurement. In this section we are going to briefly describe the PD concept and its interest with respect to our particular problematic.…”

“…Then, the term P (φ) will easily solve the problem of the choice of the number of Zernike to be accounted for in the estimation of the phase. Let us mention that in the conventional PD approach, 5,6,11 the relatively arbitrary choice of a given number of Zernike (the N first) to be estimated, is in fact an implicit regularization of the estimation problem by the truncation of the phase expansion in order to avoid the noise propagation on the Zernike high orders. This corresponds to a reduction in the dimension of the solution space.…”

“…These aberrations have to be measured by a dedicated WFS tool, judiciously placed in the imaging camera focal plane, and then directly pre-compensated in the closed loop process. An efficient way to obtain such a calibration is to use a Phase Diversity (PD) algorithm 4,5,6 for the NCPA measurement.…”

Influence of phase-diversity image reconstruction techniques on circular polarization asymmetries

“…Additionally, the noise on the reconstructed images present a Gaussian distribution with a greater variance than that of the original focused and defocused images. The main problem is that this noise is spatially correlated (Meynadier et al 1999). This leads to an aesthetically unappealing web-like noise that avoids using many profiles in the quiet-Sun because their amplitudes are already very low.…”