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NASA NTRS2007

Iterative-Transform Phase Retrieval Using Adaptive Diversity

A phase-diverse iterative-transform phase-retrieval algorithm enables high spatial-frequency, high-dynamic-range, image-based wavefront sensing. [The terms phase-diverse, phase retrieval, image-based, and wavefront sensing are defined in the first of the two immediately preceding articles, Broadband Phase Retrieval for Image-Based Wavefront Sensing (GSC-14899-1).] As described below, no prior phase-retrieval algorithm has offered both high dynamic range and the capability to recover high spatial-frequency components. Each of the previously developed image-based phase-retrieval techniques can be classified into one of two categories: iterative transform or parametric. Among the modifications of the original iterative-transform approach has been the introduction of a defocus diversity function (also defined in the cited companion article). Modifications of the original parametric approach have included minimizing alternative objective functions as well as implementing a variety of nonlinear optimization methods. The iterative-transform approach offers the advantage of ability to recover low, middle, and high spatial frequencies, but has disadvantage of having a limited dynamic range to one wavelength or less. In contrast, parametric phase retrieval offers the advantage of high dynamic range, but is poorly suited for recovering higher spatial frequency aberrations. The present phase-diverse iterative transform phase-retrieval algorithm offers both the high-spatial-frequency capability of the iterative-transform approach and the high dynamic range of parametric phase-recovery techniques. In implementation, this is a focus-diverse iterative-transform phaseretrieval algorithm that incorporates an adaptive diversity function, which makes it possible to avoid phase unwrapping while preserving high-spatial-frequency recovery. The algorithm includes an inner and an outer loop (see figure). An initial estimate of phase is used to start the algorithm on the inner loop, wherein multiple intensity images are processed, each using a different defocus value. The processing is done by an iterative-transform method, yielding individual phase estimates corresponding to each image of the defocus-diversity data set. These individual phase estimates are combined in a weighted average to form a new phase estimate, which serves as the initial phase estimate for either the next iteration of the iterative-transform method or, if the maximum number of iterations has been reached, for the next several steps, which constitute the outerloop portion of the algorithm. The details of the next several steps must be omitted here for the sake of brevity. The overall effect of these steps is to adaptively update the diversity defocus values according to recovery of global defocus in the phase estimate. Aberration recovery varies with differing amounts as the amount of diversity defocus is updated in each image; thus, feedback is incorporated into the recovery process. This process is iterated until the global defocus error is driven to zero during the recovery process. The amplitude of aberration may far exceed one wavelength after completion of the inner-loop portion of the algorithm, and the classical iterative transform method does not, by itself, enable recovery of multi-wavelength aberrations. Hence, in the absence of a means of off-loading the multi-wavelength portion of the aberration, the algorithm would produce a wrapped phase map. However, a special aberration-fitting procedure can be applied to the wrapped phase data to transfer at least some portion of the multi-wavelength aberration to the diversity function, wherein the data are treated as known phase values. In this way, a multiwavelength aberration can be recovered incrementally by successively applying the aberration-fitting procedure to intermediate wrapped phase maps. During recovery, as more of the aberration is transferred to the diversity function following successive iterations around the ter loop, the estimated phase ceases to wrap in places where the aberration values become incorporated as part of the diversity function. As a result, as the aberration content is transferred to the diversity function, the phase estimate resembles that of a reference flat.

Dean, Bruce H.↗
NASA NTRS2008

Adaptive MGS Phase Retrieval

Adaptive MGS Phase Retrieval software uses the Modified Gerchberg-Saxton (MGS) algorithm, an image-based sensing method that can turn any focal plane science instrument into a wavefront sensor, avoiding the need to use external metrology equipment. Knowledge of the wavefront enables intelligent control of active optical systems.

Basinger, Scott A.↗
NASA NTRS2007

Broadband Phase Retrieval for Image-Based Wavefront Sensing

A focus-diverse phase-retrieval algorithm has been shown to perform adequately for the purpose of image-based wavefront sensing when (1) broadband light (typically spanning the visible spectrum) is used in forming the images by use of an optical system under test and (2) the assumption of monochromaticity is applied to the broadband image data. Heretofore, it had been assumed that in order to obtain adequate performance, it is necessary to use narrowband or monochromatic light. Some background information, including definitions of terms and a brief description of pertinent aspects of image-based phase retrieval, is prerequisite to a meaningful summary of the present development. Phase retrieval is a general term used in optics to denote estimation of optical imperfections or aberrations of an optical system under test. The term image-based wavefront sensing refers to a general class of algorithms that recover optical phase information, and phase-retrieval algorithms constitute a subset of this class. In phase retrieval, one utilizes the measured response of the optical system under test to produce a phase estimate. The optical response of the system is defined as the image of a point-source object, which could be a star or a laboratory point source. The phase-retrieval problem is characterized as image-based in the sense that a charge-coupled-device camera, preferably of scientific imaging quality, is used to collect image data where the optical system would normally form an image. In a variant of phase retrieval, denoted phase-diverse phase retrieval [which can include focus-diverse phase retrieval (in which various defocus planes are used)], an additional known aberration (or an equivalent diversity function) is superimposed as an aid in estimating unknown aberrations by use of an image-based wavefront-sensing algorithm. Image-based phase-retrieval differs from such other wavefront-sensing methods, such as interferometry, shearing interferometry, curvature wavefront sensing, and Shack-Hartmann sensing, all of which entail disadvantages in comparison with image-based methods. The main disadvantages of these non-image based methods are complexity of test equipment and the need for a wavefront reference.

Dean, Bruce H.↗
NASA NTRS2011

Phase Retrieval for Radio Telescope and Antenna Control

Phase-retrieval is a general term used in optics to describe the estimation of optical imperfections or "aberrations." The purpose of this innovation is to develop the application of phase retrieval to radio telescope and antenna control in the millimeter wave band. Earlier techniques do not approximate the incoherent subtraction process as a coherent propagation. This approximation reduces the noise in the data and allows a straightforward application of conventional phase retrieval techniques for radio telescope and antenna control. The application of iterative-transform phase retrieval to radio telescope and antenna control is made by approximating the incoherent subtraction process as a coherent propagation. Thus, for systems utilizing both positive and negative polarity feeds, this approximation allows both surface and alignment errors to be assessed without the use of additional hardware or laser metrology. Knowledge of the antenna surface profile allows errors to be corrected at a given surface temperature and observing angle. In addition to imperfections of the antenna surface figure, the misalignment of multiple antennas operating in unison can reduce or degrade the signal-to-noise ratio of the received or broadcast signals. This technique also has application to the alignment of antenna array configurations.

Dean, Bruce↗
NASA NTRS2007

Status on Iterative Transform Phase Retrieval Applied to the GBT Data

This slide presentation reviews the use of iterative transform phase retrieval in the analysis of the Green Bank Radio Telescope (GBT) Data. It reviews the NASA projects that have used phase retrieval, and the testbed for the algorithm to be used for the James Webb Space Telescope. It shows the comparison of phase retrieval with an interferometer, and reviews the two approaches used for phase retrieval, iterative transform (ITA) or parametric (non-linear least squares model fitting). The concept of ITA Phase Retrieval is reviewed, and the application to Radio Antennas is reviewed. The presentation also examines the National Radio Astronomy Observatory (NRAO) data from the GBT, and the Fourier model that NRAO uses to analyze the data. The challenge for ITA phase retrieval is reviewed, and the coherent approximation for incoherent data is shown. The validity of the approximation is good for a large tilt. There is a review of the proof of concept of the Phase Review simulation using the input wavefront, and the initial sampling parameters estimate from the focused GBT data.

Dean, Bruce↗
NASA NTRS2011

Phase Retrieval System for Assessing Diamond Turning and Optical Surface Defects

An optical design is presented for a measurement system used to assess the impact of surface errors originating from diamond turning artifacts. Diamond turning artifacts are common by-products of optical surface shaping using the diamond turning process (a diamond-tipped cutting tool used in a lathe configuration). Assessing and evaluating the errors imparted by diamond turning (including other surface errors attributed to optical manufacturing techniques) can be problematic and generally requires the use of an optical interferometer. Commercial interferometers can be expensive when compared to the simple optical setup developed here, which is used in combination with an image-based sensing technique (phase retrieval). Phase retrieval is a general term used in optics to describe the estimation of optical imperfections or aberrations. This turnkey system uses only image-based data and has minimal hardware requirements. The system is straightforward to set up, easy to align, and can provide nanometer accuracy on the measurement of optical surface defects.

Dean, Bruce↗
NASA NTRS2003

Genetic Algorithm Phase Retrieval for the Systematic Image-Based Optical Alignment Testbed

Phase retrieval requires calculation of the real-valued phase of the pupil fimction from the image intensity distribution and characteristics of an optical system. Genetic 'algorithms were used to solve two one-dimensional phase retrieval problem. A GA successfully estimated the coefficients of a polynomial expansion of the phase when the number of coefficients was correctly specified. A GA also successfully estimated the multiple p h e s of a segmented optical system analogous to the seven-mirror Systematic Image-Based Optical Alignment (SIBOA) testbed located at NASA s Marshall Space Flight Center. The SIBOA testbed was developed to investigate phase retrieval techniques. Tiphilt and piston motions of the mirrors accomplish phase corrections. A constant phase over each mirror can be achieved by an independent tip/tilt correction: the phase Conection term can then be factored out of the Discrete Fourier Tranform (DFT), greatly reducing computations.

Taylor, Jaime↗
NASA NTRS2016

Phase-Retrieval Uncertainty Estimation and Algorithm Comparison for the JWST-ISIM Test Campaign

Phase retrieval, the process of determining the exitpupil wavefront of an optical instrument from image-plane intensity measurements, is the baseline methodology for characterizing the wavefront for the suite of science instruments (SIs) in the Integrated Science Instrument Module (ISIM) for the James Webb Space Telescope (JWST). JWST is a large, infrared space telescope with a 6.5-meter diameter primary mirror. JWST is currently NASA's flagship mission and will be the premier space observatory of the next decade. ISIM contains four optical benches with nine unique instruments, including redundancies. ISIM was characterized at the Goddard Space Flight Center (GSFC) in Greenbelt, MD in a series of cryogenic vacuum tests using a telescope simulator. During these tests, phase-retrieval algorithms were used to characterize the instruments. The objective of this paper is to describe the Monte-Carlo simulations that were used to establish uncertainties (i.e., error bars) for the wavefronts of the various instruments in ISIM. Multiple retrieval algorithms were used in the analysis of ISIM phase-retrieval focus-sweep data, including an iterativetransform algorithm and a nonlinear optimization algorithm. These algorithms emphasize the recovery of numerous optical parameters, including low-order wavefront composition described by Zernike polynomial terms and high-order wavefront described by a point-by-point map, location of instrument best focus, focal ratio, exit-pupil amplitude, the morphology of any extended object, and optical jitter. The secondary objective of this paper is to report on the relative accuracies of these algorithms for the ISIM instrument tests, and a comparison of their computational complexity and their performance on central and graphical processing unit clusters. From a phase-retrieval perspective, the ISIM test campaign includes a variety of source illumination bandwidths, various image-plane sampling criteria above and below the Nyquist- Shannon critical sampling value, various extended object sizes, and several other impactful effects.

Design Analysis↗
NASA NTRS2009

Phase Retrieval with Signal Bias

The effect of a uniform measurement bias, due to background light, stray light, detector dark current, or detector offset, on phase retrieval wavefront sensing algorithms is analyzed. Simulation results indicate that the root-mean-square error of the retrieved phase can be more sensitive to an unaccounted-for signal bias than to random noise in practical scenarios. Three methods for reducing the impact of signal bias are presented

Thurman, Samuel T.↗
NASA NTRS1993

Phase-retrieval algorithms for a complicated optical system

Phase-retrieval algorithms have been developed that handle a complicated optical system that requires multiple Fresnellike transforms to propagate from one end of the system to the other including the absorption by apertures in more than one plane and allowance for bad detector pixels. Gradient-search algorithms and generalizations of the iterative-transform phase-retrieval algorithms are derived. Analytic expressions for the gradient of an error metric, with respect to polynomial coefficients and with respect to point-by-point phase descriptions, are given. The entire gradient can be computed with the number of transforms required to propagate a wave front from one end of the optical system to the other and back again, independent of the number of coefficients or phase points. This greatly speeds the computation. The reconstruction of pupil amplitude is also given. A convergence proof of the generalized iterative transform algorithm is given. These improved algorithms permit a more accurate characterization of complicated optical systems from their point spread functions.

Fienup, J. R.↗
NASA NTRS2016

Phase Retrieval Implementation for the WFIRST Coronagraph Development Testbed

The Wide-Field Infrared Survey Telescope (WFIRST) is a 2.4m diameter space telescope NASA program. The payload will include two instruments, including a coronagraph instrument (CGI). The CGI will be the first active, high-contrast stellar coronagraph in space intended for imaging, discovery, and spectral characterization of exoplanets and debris discs. The CGI designs under development use deformable mirrors (DM) to create a point spread function (PSF) with a dark region around the obscured star object. Each DM consists of an array of actuators, and each actuator causes a local displacement to a thin, mirror-coated, glass face sheet. The actuator displacements are designed to create the PSF. Typically, phase retrieval is used to set the DM actuator displacements to their designed pattern, and then the actuators are further modified by an iterative control loop to correct for system non-idealities. To observe an exoplanet near the star object, the contrast in the annular region must be better than 10^8 (10^9 after post processing) relative to the unobstructed image of the star [1,2]. The coronagraph testbed at Jet Propulsion Laboratory is used to test coronagraph design and control algorithms. The optical system (Fig. 1) is maintained in a vacuum tank so that the PSF is stable for long integration times. Phase retrieval is an important tool for the testbed and serves several purposes, in addition to setting initial DM actuator displacements. Other uses include evaluation of systematic wavefront error introduced by system components, and alignment of the DMs and pupil masks. For example, the location of specific DM actuators relative to the Lyot mask can be determined by displacing the actuator, then measuring the location of the phase disturbance in the pupil plane with phase retrieval. In the next section we further describe the optical train of the testbed and the collection of source and pupil images available as inputs for phase retrieval, as well as one phase retrieval algorithm employed. Section 3 shows example phase retrieval results.

Marx, David↗
NASA NTRS2011

Fast Approximate Broadband Phase Retrieval for Segmented Systems

Broadband phase retrieval needed when: a) Narrow spectral filters are unavailable. b) Dim sources. c) Low throughput due to misalignment. d) Short exposures times. i.e., Pointing instability (space); and Atmospheric instability (ground based AO). Traditional approach is computationally burdensome for extreme bandwidths. Approximate approach: a) Substitute monochromatic model. b) Blur model and data. Test case performance: a) approx.270x reduction in computational cost for FGS-like test case. b) Good accuracy for monolithic system. c) Acceptable accuracy for segmented systems. i.e., Reduced by diffraction and Reduced by higher order segment model.

Jurling, Alden S.↗
NASA NTRS2012

Qualification of a Null Lens Using Image-Based Phase Retrieval

In measuring the figure error of an aspheric optic using a null lens, the wavefront contribution from the null lens must be independently and accurately characterized in order to isolate the optical performance of the aspheric optic alone. Various techniques can be used to characterize such a null lens, including interferometry, profilometry and image-based methods. Only image-based methods, such as phase retrieval, can measure the null-lens wavefront in situ - in single-pass, and at the same conjugates and in the same alignment state in which the null lens will ultimately be used - with no additional optical components. Due to the intended purpose of a Dull lens (e.g., to null a large aspheric wavefront with a near-equal-but-opposite spherical wavefront), characterizing a null-lens wavefront presents several challenges to image-based phase retrieval: Large wavefront slopes and high-dynamic-range data decrease the capture range of phase-retrieval algorithms, increase the requirements on the fidelity of the forward model of the optical system, and make it difficult to extract diagnostic information (e.g., the system F/#) from the image data. In this paper, we present a study of these effects on phase-retrieval algorithms in the context of a null lens used in component development for the Climate Absolute Radiance and Refractivity Observatory (CLARREO) mission. Approaches for mitigation are also discussed.

Bolcar, Matthew R.↗
NASA NTRS2003

Phase Retrieval Using a Genetic Algorithm on the Systematic Image-Based Optical Alignment Testbed

NASA s Marshall Space Flight Center s Systematic Image-Based Optical Alignment (SIBOA) Testbed was developed to test phase retrieval algorithms and hardware techniques. Individuals working with the facility developed the idea of implementing phase retrieval by breaking the determination of the tip/tilt of each mirror apart from the piston motion (or translation) of each mirror. Presented in this report is an algorithm that determines the optimal phase correction associated only with the piston motion of the mirrors. A description of the Phase Retrieval problem is first presented. The Systematic Image-Based Optical Alignment (SIBOA) Testbeb is then described. A Discrete Fourier Transform (DFT) is necessary to transfer the incoming wavefront (or estimate of phase error) into the spatial frequency domain to compare it with the image. A method for reducing the DFT to seven scalar/matrix multiplications is presented. A genetic algorithm is then used to search for the phase error. The results of this new algorithm on a test problem are presented.

Taylor, Jaime R.↗
NASA NTRS2002

Phase retrieval camera for testing NGST optics

The NGST Phase Retrieval Camera (PRC) is a portable wavefront sensor useful for optical testing in high-vibration environments. Primary applications of the PRC are testing and experimenting with NGST technology demonstrator mirrors, along with exploring other wavefront sensing and control problems not easily studied using WCT. An overview of the hardware and testing results will be presented.

NGST↗
NASA NTRS1991

Phase retrieval for the Hubble Space Telescope using iterative propagation algorithms

Phase retrieval algorithms, including the iterative transform algorithm and gradient search algorithms, were generalized to include the effects of propagation through a complicated optical system and to discount the effects of bad CCD pixels. For the gradient search algorithms, analytic gradients were derived that greatly speed up the computation over finite difference methods. For the Hubble Space Telescope (HST), the aperture function was reconstructed and the phase errors were retrieved. This information is useful to design correction optics for the telescope and for the deconvolution of blurred images from the HST.

Fienup, J. R.↗
NASA NTRS2013

System and method for phase retrieval for radio telescope and antenna control

Disclosed herein are systems, methods, and non-transitory computer-readable storage media for radio phase retrieval. A system practicing the method gathers first data from radio waves associated with an object observed via a first aperture, gathers second data from radio waves associated with the object observed via an introduced second aperture associated with the first aperture, generates reduced noise data by incoherently subtracting the second data from the first data, and performs phase retrieval for the radio waves by modeling the reduced noise data using a single Fourier transform. The first and second apertures are at different positions, such as side by side. This approach can include determining a value Q which represents a ratio of wavelength times a focal ratio divided by pixel spacing. This information can be used to accurately measure and correct alignment errors or other optical system flaws in the apertures.

Dean, Bruce H.↗