Sparse Reconstruction of Wavefronts using an Over-Complete Phase Dictionary

Abstract

Wavefront reconstruction is a critical component in various optical systems, including adaptive optics, interferometry, and phase contrast imaging. Traditional reconstruction methods often employ either the Cartesian (pixel) basis or the Zernike polynomial basis. While the Cartesian basis is adept at capturing high-frequency features, it is susceptible to overfitting and inefficiencies due to the high number of degrees of freedom. The Zernike basis efficiently represents common optical aberrations but struggles with complex or non-standard wavefronts such as optical vortices, Bessel beams, or wavefronts with sharp discontinuities. This paper introduces a novel approach to wavefront reconstruction using an over-complete phase dictionary combined with sparse representation techniques. By constructing a dictionary that includes a diverse set of basis functions - ranging from Zernike polynomials to specialized functions representing optical vortices and other complex modes - we enable a more flexible and efficient representation of complex wavefronts. Furthermore, a trainable affine transform is implemented to account for misalignment. Utilizing principles from compressed sensing and sparse coding, we enforce sparsity in the coefficient space to avoid overfitting and enhance robustness to noise.

Figures (4)

• Figure 1: The expansion of a number of wavefronts in terms of the Pixel and Zernike bases. In order to describe wavefront function, $\Psi$, the coefficient for mode $n$ in basis $\Phi$ was calculated by $c_n = \braket{\Psi|\Phi_{n}}$. a) To express a single pixel requires many Zernike modes. b) A single Zernike mode requires many modes in the Pixel basis. c) There exist wavefronts, such as the optical vortex, which are not efficiently represented in either the Zernike or Pixel basis. The Zernike modes are ordered according to their Noll index noll1976zernike, and the colorbar applies for all wavefronts.
• Figure 2: The wavefront was synthesized by the random initialization of 28 Zernike modes, and the Axicon mode. The top row displays the modal coefficient values, with each subplot showing a zoom of the first 29 coefficients - the Zernikes and Axicon (marked with a red star). The bottom row shows the predicted wavefront, with all plotted on the same color scale. One sees that even in the presence of significant noise (parameterized by a percentage, $\eta$, of the maximum value of the derivative), PROD is able to accurately extract the true coefficient values.
• Figure 3: Demonstrating the utility of PROD's affine transform module to efficiently represent de-centered modes, in this case the $Z_{3}^{3}$ mode. Here PROD learns the de-center parameter, and identifies the single Zernike mode. Also displayed for comparison is the LASSO technique, which is given the same over-complete dictionary, but possesses no ability to transform the modes. All wavefronts are plotted on the same color scale. If its L1 penalization parameter, $\alpha$, is large, it cannot represent the wavefront accurately, whereas if $\alpha$ is small, it must utilize lower order Zernike modes to help represent the wavefront.
• Figure 4: The use of PROD to characterize an experimental optical vortex. Once the gradients are extracted from the Shack-Hartmann pattern, the technique is used to find the modal coefficients for the vortex, and lower order Zernike modes. The coefficient for the vortex (the topological charge) was 2.09, in good agreement with the theoretical value for this setup of 2.