Using a broadband long-wavelength channel to increase the capture range of segment piston phase retrieval for segmented-aperture systems.

Segmented-aperture systems, such as the James Webb Space Telescope (JWST), require fine piston alignment between primary mirror segments. Computer simulation experiments show that using a broadband long-wavelength channel, illustrated with the Mid Infrared Instrument (MIRI) onboard the JWST, can extend the capture range of segment piston phase retrieval significantly (in the case of JWST with MIRI, up to hundreds of microns), greatly reducing the requirements on coarse phasing.

Demonstration of multi-plane sharpness metric maximization on motion-compensated, multi-wavelength 3D digital holographic field data.

This Letter examines sharpness metric maximization methods on 3D images obtained at Table Mountain, Colorado. We employ multi-wavelength 3D imaging with digital holography and a pilot tone to obtain the aberrated images and use sharpness metric maximization to correct the aberrated images with both pupil-plane and multi-plane corrections. Image quality improves when sharpness metric maximization is used and particularly with multi-plane correction.

Phase retrieval with only a nonnegativity constraint.

We show that it may be possible to reconstruct a real-valued, nonnegative 2D object from the magnitude of its Fourier transform using only a nonnegativity constraint without the usual support constraint, even when significant noise is present in the Fourier intensity data.

3D multi-plane sharpness metric maximization with variable corrective phase screens.

Sharpness metric maximization is a method for reconstructing coherent images that have been aberrated due to distributed-volume turbulence. This method places one or more corrective phase screens in the digital-propagation path that serve to increase overall sharpness of the image. As such, this study uses sharpness metric maximization on 3D irradiances obtained via frequency-diverse digital holography. We vary the number of corrective phase screens in the propagation path and sharpen images of a realistic, extended object via multi-plane sharpness metric maximization. The results indicate that image reconstruction is possible when using fewer corrective screens than aberrating screens, but that image quality increases with a greater number of corrective screens.

Field stop diffraction and sampling effects in apodized pupil Lyot coronagraphs.

We study the predicted performance of two apodized pupil Lyot coronagraph designs in the presence of an occulter-plane field stop. We discuss techniques for capturing diffraction effects when the radius of the stop is larger than the field of view of an ordinary numerical diffraction model, including mask upsampling and analytical focal-plane envelope functions. We simulate a closed-loop coronagraphic wavefront control to assess the extent to which such diffraction effects can be compensated using deformable mirrors. We show that for the designs considered, field stop diffraction effects are significant at diameters considerably larger than the instrument field of view, suggesting the need to explicitly include a focal-plane stop in the design process.

Algorithm for exact area-weighted antialiasing of discrete circular apertures.

Accurately calculating diffraction from geometrical shapes such as circular apertures is important in computational Fourier optics. In this paper, we present an algorithm for exactly generating a discrete representation of a circular aperture whose pixel values are given by the integral of the true aperture within each pixel. We characterize the accuracy and runtime of the presented algorithm in comparison to approximate techniques such as binning high-resolution arrays, relative to the analytical Airy pattern.

Techniques for arbitrary sampling in two-dimensional Fourier transforms.

In this paper, we discuss two effective methods for computing optical propagations using two-dimensional (2D) discrete Fourier transforms: the matrix triple product (MTP) and the chirp z-transform (CZT) and analyze their performance both in theory and via benchmarks compared to the performance of a traditional padded fast Fourier transform (FFT). We show that, in many regimes of interest for phase-retrieval algorithms, the MTP or CZT is comparable to or better than the FFT in terms of run time while offering more flexible control over the sampling. We propose that for many applications, the CZT makes a robust general purpose alternative to the padded 2D FFT.

Verification of transverse translation diverse phase retrieval for concave optical metrology.

The surface figure error of a concave spherical mirror was measured using transverse translation diverse phase retrieval (TTDPR), an image-based wavefront sensing technique. Good reproducibility of the surface measurement is demonstrated. Additionally, the TTDPR measurement of the surface, with certain alignment terms removed, is shown to agree with interferometric measurements to 0.0060 waves root mean square.

Interferometric spatial mode analyzer with a bucket detector.

A spatial mode analyzer based on a Michelson interferometer with a bucket detector is experimentally implemented. The delay line in the interferometer is an optical implementation of the fractional Fourier transform (fFT) which enables the spatial mode analysis of a given input field in the Hermite-Gaussian (HG) mode basis. Modal weights for both 1D and 2D input fields are experimentally measured. Results for input fields comprising of multiple HG modes are also presented.

Machine learning for improved image-based wavefront sensing.

For large amounts of wavefront error, gradient-based optimization methods for image-based wavefront sensing are unlikely to converge when the starting guess for the wavefront differs greatly from the true wavefront. We use machine learning operating on a point-spread function to determine a good initial estimate of the wavefront. We show that our trained convolutional neural network provides good initial estimates in the presence of simulated detector noise and is more effective than using many random starting guesses for large amounts of wavefront error.

Transverse translation diverse phase retrieval using soft-edged illumination.

Transverse translation diverse phase retrieval (TTDPR), a ptychographic image-based wavefront-sensing technique, is a viable method for optical shop testing due to its high accuracy and relatively simple experimental arrangement. However, when measuring a reflective optic, a normally hard-edged translating illumination will become soft-edged due to diffraction, which may reduce the accuracy of TTDPR by suppressing fine structures in measured data. In this Letter, we quantitatively explore the wavefront-sensing accuracy of TTDPR in the presence of soft-edged translating illumination.

Phase retrieval in generalized optical interferometry systems.

Modal analysis of an optical field via generalized interferometry (GI) is a novel technique that treats said field as a linear superposition of transverse modes and recovers the amplitudes of modal weighting coefficients. We use phase retrieval by nonlinear optimization to recover the phase of these modal weighting coefficients. Information diversity increases the robustness of the algorithm by better constraining the solution. Additionally, multiple sets of random starting phase values assist the algorithm in overcoming local minima. The algorithm was able to recover nearly all coefficient phases for simulated fields consisting of up to 21 superpositioned Hermite Gaussian modes from simulated data and proved to be resilient to shot noise.

Extending capture range for piston retrieval in segmented systems.

Image-based wavefront sensing is a powerful technique for measuring the aberrations of optical systems and surfaces. It often fails for segmented systems with large piston errors per segment. We propose a method for finding these errors using broadband light and a specialized grid search as part of a more global search. We show that this method has a high rate of success for a case where nonlinear optimization gets stuck in local minima. We also explore points of failure.

Wide-field spatiospectral interferometry: theory and imaging properties.

The emerging astronomical technique known as wide-field spatiospectral interferometry can provide hyperspectral images with spatial resolutions that are unattainable with a single monolithic-aperture observatory. The theoretical groundwork for operation and data measurement is presented in full detail, including relevant coherence theory. We also discuss a data processing technique for recovering a hyperspectral image from an interferometric data set as well as the unusual effective transfer function of the system.

Ptychography for optical metrology with limited translation knowledge.

We introduce unknown-transverse translation diversity phase retrieval: a ptychographic algorithm for optical metrology when a subaperture is translating through a plane conjugate to the exit pupil in a very poorly known fashion. The algorithm estimates the direction of translation and the distance traveled by the subaperture from one point spread function (PSF) to the next. It also estimates unknown point target motion and rotations of the subaperture between PSF acquisitions from the PSF data.

Subaperture translation estimation accuracy in transverse translation diversity phase retrieval.

For optical metrology by transverse translation diversity phase retrieval (or ptychography), information theoretic limits on the ability to estimate subaperture translation, essential for accurate metrology, are assessed as a function of the optical aberrations of the system being measured. Special attention is given to the case that an unknown linear phase aberration, or equivalent detector or target motion, is present that varies with each point spread function in the measured data.

Phase retrieval with unknown sampling factors via the two-dimensional chirp z-transform.

We derive the analytic gradient of a phase retrieval error metric with respect to the sampling factor or the f-number that produced the measured point-spread function. This allows us to efficiently optimize over the sampling factor, thereby improving the accuracy of the phase estimate. Computer simulation results show its effectiveness.

Applications of algorithmic differentiation to phase retrieval algorithms.

In this paper, we generalize the techniques of reverse-mode algorithmic differentiation to include elementary operations on multidimensional arrays of complex numbers. We explore the application of the algorithmic differentiation to phase retrieval error metrics and show that reverse-mode algorithmic differentiation provides a framework for straightforward calculation of gradients of complicated error metrics without resorting to finite differences or laborious symbolic differentiation.

Extended capture range for focus-diverse phase retrieval in segmented aperture systems using geometrical optics.

Extending previous work by Thurman on wavefront sensing for segmented-aperture systems, we developed an algorithm for estimating segment tips and tilts from multiple point spread functions in different defocused planes. We also developed methods for overcoming two common modes for stagnation in nonlinear optimization-based phase retrieval algorithms for segmented systems. We showed that when used together, these methods largely solve the capture range problem in focus-diverse phase retrieval for segmented systems with large tips and tilts. Monte Carlo simulations produced a rate of success better than 98% for the combined approach.

Phase retrieval algorithms: a personal tour [Invited].

This paper gives the reader a personal tour through the field of phase retrieval and related works that lead up to or cited the paper "Phase Retrieval Algorithms: a Comparison," [Appl. Opt.21, 2758 (1982)].