Analytical representation of the phase and its mode components reconstructed according to the wave-front slopes

Vector polynomials for direct analysis of circular wavefront slope data

In the aberration analysis of a circular wavefront, Zernike circle polynomials are used to obtain its wave aberration coefficients. To obtain these coefficients from the wavefront slope data, we need vector functions that are orthogonal to the gradients of the Zernike polynomials, and are irrotational so as to propagate minimum uncorrelated random noise from the data to the coefficients. In this paper, we derive such vector functions, which happen to be polynomials.

What is a Hartmann test?

In this paper we will review some of the many different practical arrangements that have been obtained to measure the transversal aberrations of optical systems based on the odd and vulnerable Hartmann test. There are many optical testing configurations that apparently are not related to the original Hartmann test. However, they are really the same thing and can be considered just a variation of the same basic arrangement, as will be described here.

Modal processing of Hartmann and Shack-Hartmann patterns by means of a least squares fitting of the transverse aberrations

Instead of measuring the wavefront deformations, Hartmann and Shack-Hartmann tests measure wavefront slopes, which are equivalent to ray transverse aberrations. Numerous integration methods have been described in the literature to obtain the wavefront deformations from these measurements. Basically, they can be classified in two different categories, i.e., modal and zonal. Frequently, a least squares fit of the transverse aberrations in the x direction and a least squares fit of the transverse aberrations in the y direction is performed to obtain the wavefront. In this work, we briefly describe a modal method to integrate Hartmann and Shack-Hartmann patterns by means of a single least squares fit of the transverse aberrations simultaneously instead of the traditional x-y separate method. The proposed method uses monomial calculation instead of using Zernike polynomials, to simplify numerical calculations. Later, a method is proposed to convert from monomials to Zernike polynomials. An important obtained result is that if polar coordinates are used, angular transverse aberrations are not actually needed to obtain all wavefront coefficients.

Modal integration of Hartmann and Shack-Hartmann patterns

Instead of measuring the wavefront deformations directly, Hartmann and Shack-Hartmann tests measure the wavefront slopes, which are equivalent to the ray transverse aberrations. Numerous different integration methods have been described in the literature to obtain the wavefront deformations from these measurements. Basically, they can be classified in two different categories, i.e., modal and zonal. In this work we describe a modal method to integrate Hartmann and Shack-Hartmann patterns using orthogonal wavefront slope aberration polynomials, instead of the commonly used Zernike polynomials for the wavefront deformations.

Theory of singular-phase reconstruction for an optical speckle field in the turbulent atmosphere

Analytical expressions are derived and computational algorithms are constructed for retrieving optical-field phase distribution under strong scintillation. The input data for the phase reconstruction are the wave-front slopes registered by a Hartmann sensor or shearing interferometer. The theory is based on representing the slope-vector field as the sum of its potential and solenoid components; it introduces the concept of phase-source and phase-vortex density and uses strict integral expressions relating these quantities to the wave-front slopes. To overcome the difficulties arising from the singular character of phase distribution, use is made of regularization of the wave-front slopes. The slopes can be measured with an ideal point wave-front sensor. It is shown that the slopes measured at the output of a nonideal sensor can be treated as regularized values of these slopes. Numerical simulation of phase unwrapping from the reference values of the wave-front slopes has shown that the algorithm designed for visualization of local phase singularities and those for phase reconstruction are very helpful in eliminating the measurement noise.

Potential and vortex features of optical speckle fields and visualization of wave-front singularities

The feasibility of noninterferometric methods to measure phase distribution in a laser beam cross section for visualization of the vortex dislocations of an optical speckle-field wave front is analyzed. Peculiarities of the phase retrieved from the measured intensity distribution (the phase problem in optics) and from the wave-front slopes measured by a Hartmann sensor are discussed. A concept of the vortex and the potential parts of the phase is introduced. An analytic formula to retrieve the potential phase from the measured intensity has been obtained. We show that the considered means of measurements allow the positions of the dislocation centers to be sensed and the spatial configuration of the intensity zero lines to be reconstructed.

Determination of phase mode components in terms of local wave-front slopes: an analytical approach

An analytical formulation that relates the modal expansion coefficients of a given wave front to its local transverse phase derivatives is proposed. The modal coefficients are calculated as a weighted integral over the wave-front slopes. The weighting functions for each mode are the components of a two-dimensional vector whose divergence equals the corresponding mode function. This approach is useful for analytical phase reconstruction from the input data provided by shearing interferometers or Hartmann-Shack wave-front sensors. Numerical results for a simulated experiment in terms of a set of Zernike polynomials are given.