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

Subspace wavefront estimation using image sharpening and predictive dynamic digital holography

Image sharpening algorithms used for phase retrieval to reconstruct images in digital holography are computationally intensive, requiring iterative virtual wavefront propagation and hill-climbing algorithms to optimize sharpness criteria. Recently, it was shown that minimum-variance wavefront prediction can be integrated with digital holography and image sharpening to significantly reduce the large number of costly sharpening iterations normally required to achieve near-optimal wavefront estimation [J. Opt. Soc. Am. A35, 923 (2018)JOAOD60740-323210.1364/JOSAA.35.000923]. This paper demonstrates further gains in computational efficiency with a new subspace sharpening method in conjunction with predictive dynamic digital holography for real-time applications. The method sharpens local regions of interest in an image plane by parallel independent wavefront estimation on reduced-dimension subspaces of the complex field in a pupil plane. Through wave-optics simulations, this paper shows that the new subspace method produces results comparable to those of conventional global and local sharpening, and that subspace wavefront estimation and sharpening coupled with wavefront prediction achieve orders-of-magnitude increases in processing speed.

Jacobi circle and annular polynomials: modal wavefront reconstruction from wavefront gradient

Jacobi circle polynomials, which are orthogonal on the unit circle with orthogonal radial derivatives, have been developed previously. As the classical Zernike mode can be represented as a linear combination of Jacobi modes, Zernike wavefront modes can be reconstructed using Jacobi modes. Comparison of the Jacobi and Zernike modes for the modal approach indicates that a modal approach incorporating the Gram matrix with the Jacobi modes has potential application in high-sampling wavefront gradient sensors. The Gram matrix method using the Jacobi modes can be extended to annular pupils.

Predictive dynamic digital holography and image sharpening

Digital holography is often combined with image sharpening for wavefront estimation and correction, and this combination has received recent attention for many imaging and sensing applications. A significant obstacle for digital holography and image sharpening in high-speed real-time applications is the fact that the process is computationally intensive, requiring iterative virtual wavefront propagation and hill-climbing algorithms to optimize sharpness criteria. This paper introduces a method for accelerating dynamic digital holography and image sharpening by wavefront prediction. The approach here integrates optimal state-space prediction filters with digital holography and image sharpening to short-circuit the computationally intensive process of virtual wavefront propagation and sharpness optimization.

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.

Method of obtaining wavefront slope data from through-focus point spread function measurements

A method is described for analyzing point source imagery collected with various amounts of defocus to obtain wavefront slope data at the exit pupil of an imaging system. Integration of this slope data yields the system wavefront aberration function. The method is based on a geometric optics interpretation of intensity point spread function measurements in the caustic region near focus. Algorithm performance is examined through Monte Carlo simulations. Application of the method to segmented-aperture systems is also explored. The proposed method is used to generate initial wavefront estimates for an iterative phase retrieval algorithm, significantly improving the capture range over a blind phase retrieval approach when segment tilt errors are large.

Variational solution for modal wave-front projection functions of minimum-error norm

Common wave-front sensors such as the Hartmann or curvature sensor provide measurements of the local gradient or Laplacian of the wave front. The expression of wave fronts in terms of a set of orthogonal basis functions thus generally leads to a linear wave-front-estimation problem in which modal cross coupling occurs. Auxiliary vector functions may be derived that effectively restore the orthogonality of the problem and enable the modes of a wave front to be independently and directly projected from slope measurements. By using variational methods, we derive the necessary and sufficient condition for these auxiliary vector functions to have minimum-error norm. For the specific case of a slope-based sensor and a basis set comprising the Zernike circular polynomials, these functions are precisely the Gavrielides functions.