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NASA NTRSDate not supplied

Uncertainty Estimates for Fitting Zernike Polynomials to Discrete Data

Zernike polynomials are a widely used metric in modern optical analysis. They conveniently represent surfaces as a series of weighted terms corresponding to various optical aberrations. Ideally, each term is independent of others in the series, but Zernike polynomials lose this property when working with sets of discrete data. This gives rise to uncertainty in each polynomial’s actual contribution and affects metrology and simulation estimates of their relative weights. Several factors influencing these estimates are the number and arrangement of sample locations, the method for calculating the weights, and the total number of Zernike terms used in the calculation. Discussed is the uncertainty associated with linear regression using random sampling. Other topics reviewed are complex Zernike polynomials and vector spaces of functions.

Zernike Polynomials↗

DOE OSTI2025

ZERNIPAX: A fast and accurate Zernike polynomial calculator in Python

Zernike polynomials serve as an orthogonal basis on the unit disc, and have proven to be effective in optics simulations, astrophysics, and more recently in plasma simulations. Unlike Bessel functions, Zernike polynomials are inherently finite and smooth at the disc center (r=0), ensuring continuous differentiability along the axis. This property makes them particularly suitable for simulations, requiring no additional handling at the origin. We developed ZERNIPAX, an open-source Python package capable of utilizing CPU/GPUs, leveraging Google's JAX package and available on GitHub as well as the Python software repository PyPI. Furthermore, our implementation of the recursion relation between Jacobi polynomials significantly improves computation time compared to alternative methods by use of parallel computing while still performing more accurately for high-mode numbers.

Astrophysics↗

NASA NTRS1982

A recursive algorithm for Zernike polynomials

The analysis of a function defined on a rotationally symmetric system, with either a circular or annular pupil is discussed. In order to numerically analyze such systems it is typical to expand the given function in terms of a class of orthogonal polynomials. Because of their particular properties, the Zernike polynomials are especially suited for numerical calculations. Developed is a recursive algorithm that can be used to generate the Zernike polynomials up to a given order. The algorithm is recursively defined over J where R(J,N) is the Zernike polynomial of degree N obtained by orthogonalizing the sequence R(J), R(J+2), ..., R(J+2N) over (epsilon, 1). The terms in the preceding row - the (J-1) row - up to the N+1 term is needed for generating the (J,N)th term. Thus, the algorith generates an upper left-triangular table. This algorithm was placed in the computer with the necessary support program also included.

Davenport, J. W.↗

NASA NTRS2007

Zernike-like Orthogonal Basis Functions for Wavefront Characterization over Sampled, Irregular Apertures

For optical systems with circular apertures, wavefronts are often analyzed using Zernike polynomials, and individual Zernike functions are associated with familiar optical aberrations. For systems with noncircular apertures, or in practical situations in which wavefronts are measured at a limited number of points in the aperture, the Zernike polynomials are no longer an orthogonal basis for the measured data. Although there are an endless number of ways to create a basis for such measured data, a "Zernike-like" basis is useful to connect with our experience with the usual optical aberrations. In this paper, the steps required to identify a Zernike-like basis for wavefronts over sampled, irregular apertures are presented, based on the Gram-Schmidt orthogonalization technique. The benefits of analyzing optical wavefronts using an orthogonal basis specific to an optical system's aperture shape and wavefront sampling, instead of using the traditional Zernike polynomials, are detailed in two examples, from image-based wavefront sensing on a segmented-aperture telescope (the James Webb Space Telescope Testbed Telescope at Ball Aerospace) and from interferometer characterization for surface metrology of a hexagonal mirror segment.

Aronstein, David L.↗

NASA NTRS1982

Tolerance analysis of optical telescopes using coherent addition of wavefront errors

A near diffraction-limited telescope requires that tolerance analysis be done on the basis of system wavefront error. One method of analyzing the wavefront error is to represent the wavefront error function in terms of its Zernike polynomial expansion. A Ramsey-Korsch ray trace package, a computer program that simulates the tracing of rays through an optical telescope system, was expanded to include the Zernike polynomial expansion up through the fifth-order spherical term. An option to determine a 3 dimensional plot of the wavefront error function was also included in the Ramsey-Korsch package. Several assimulation runs were analyzed to determine the particular set of coefficients in the Zernike expansion that are effected by various errors such as tilt, decenter and despace. A 3 dimensional plot of each error up through the fifth-order spherical term was also included in the study. Tolerance analysis data are presented.

Davenport, J. W.↗

NASA NTRS1980

Alignment and evaluation of the cryogenic corrected infrared astronomical satellite /IRAS/ telescope

Room temperature alignment and evaluation techniques for the Infrared Astronomical Satellite (IRAS) telescope, which has a primary mirror figured to correct for surface distortions and the 2 K operating temperature are discussed. Interferometric cryogenic testing of the 0.6 m, f/1.5 lightweighted beryllium primary mirror at its intended operating temperature reveals surface distortions that can be modeled with Zernike polynomials. With this model, it becomes possible to derive the 'inverse' of the cryowavefront error (ideal cryo mirror) and to figure the cryo correction into the primary mirror using Perkin-Elmer's Computer Controlled Polisher. It is recognized that during room temperature assembly of the system, misalignment of the secondary mirror can introduce additional unwanted aberrations that may cancel or distort the wavefront errors purposely introduced by the cryo figuring. To avoid this possible degradation and to ensure optimum telescope performance, the system Zernike polynomial coefficients and wavefront maps generated from the in-process alignment interferograms are monitored and compared to Zernike coefficients and wavefront maps for the cryo corrected primary mirror.

Harned, N.↗

NASA NTRS2000

Optical Performance Modeling of FUSE Telescope Mirror

We describe the Metrology Data Processor (METDAT), the Optical Surface Analysis Code (OSAC), and their application to the image evaluation of the Far Ultraviolet Spectroscopic Explorer (FUSE) mirrors. The FUSE instrument - designed and developed by the Johns Hopkins University and launched in June 1999 is an astrophysics satellite which provides high resolution spectra (lambda/Delta(lambda) = 20,000 - 25,000) in the wavelength region from 90.5 to 118.7 nm The FUSE instrument is comprised of four co-aligned, normal incidence, off-axis parabolic mirrors, four Rowland circle spectrograph channels with holographic gratings, and delay line microchannel plate detectors. The OSAC code provides a comprehensive analysis of optical system performance, including the effects of optical surface misalignments, low spatial frequency deformations described by discrete polynomial terms, mid- and high-spatial frequency deformations (surface roughness), and diffraction due to the finite size of the aperture. Both normal incidence (traditionally infrared, visible, and near ultraviolet mirror systems) and grazing incidence (x-ray mirror systems) systems can be analyzed. The code also properly accounts for reflectance losses on the mirror surfaces. Low frequency surface errors are described in OSAC by using Zernike polynomials for normal incidence mirrors and Legendre-Fourier polynomials for grazing incidence mirrors. The scatter analysis of the mirror is based on scalar scatter theory. The program accepts simple autocovariance (ACV) function models or power spectral density (PSD) models derived from mirror surface metrology data as input to the scatter calculation. The end product of the program is a user-defined pixel array containing the system Point Spread Function (PSF). The METDAT routine is used in conjunction with the OSAC program. This code reads in laboratory metrology data in a normalized format. The code then fits the data using Zernike polynomials for normal incidence systems or Legendre-Fourier polynomials for grazing incidence systems. It removes low order terms from the metrology data, calculates statistical ACV or PSD functions, and fits these data to OSAC models for the scatter analysis. In this paper we briefly describe the laboratory image testing of FUSE spare mirror performed in the near and vacuum ultraviolet at John Hopkins University and OSAC modeling of the test setup performed at NASA/GSFC. The test setup is a double-pass configuration consisting of a Hg discharge source, the FUSE off-axis parabolic mirror under test, an autocollimating flat mirror, and a tomographic imaging detector. Two additional, small fold flats are used in the optical train to accommodate the light source and the detector. The modeling is based on Zernike fitting and PSD analysis of surface metrology data measured by both the mirror vendor (Tinsley) and JHU. The results of our models agree well with the laboratory imaging data, thus validating our theoretical model. Finally, we predict the imaging performance of FUSE mirrors in their flight configuration at far-ultraviolet wavelengths.

Saha, Timo T.↗

NASA NTRS2010

Optical Imaging and Radiometric Modeling and Simulation

OPTOOL software is a general-purpose optical systems analysis tool that was developed to offer a solution to problems associated with computational programs written for the James Webb Space Telescope optical system. It integrates existing routines into coherent processes, and provides a structure with reusable capabilities that allow additional processes to be quickly developed and integrated. It has an extensive graphical user interface, which makes the tool more intuitive and friendly. OPTOOL is implemented using MATLAB with a Fourier optics-based approach for point spread function (PSF) calculations. It features parametric and Monte Carlo simulation capabilities, and uses a direct integration calculation to permit high spatial sampling of the PSF. Exit pupil optical path difference (OPD) maps can be generated using combinations of Zernike polynomials or shaped power spectral densities. The graphical user interface allows rapid creation of arbitrary pupil geometries, and entry of all other modeling parameters to support basic imaging and radiometric analyses. OPTOOL provides the capability to generate wavefront-error (WFE) maps for arbitrary grid sizes. These maps are 2D arrays containing digital sampled versions of functions ranging from Zernike polynomials to combination of sinusoidal wave functions in 2D, to functions generated from a spatial frequency power spectral distribution (PSD). It also can generate optical transfer functions (OTFs), which are incorporated into the PSF calculation. The user can specify radiometrics for the target and sky background, and key performance parameters for the instrument s focal plane array (FPA). This radiometric and detector model setup is fairly extensive, and includes parameters such as zodiacal background, thermal emission noise, read noise, and dark current. The setup also includes target spectral energy distribution as a function of wavelength for polychromatic sources, detector pixel size, and the FPA s charge diffusion modulation transfer function (MTF).

Ha, Kong Q.↗

NASA NTRS2012

Method for Pre-Conditioning a Measured Surface Height Map for Model Validation

This software allows one to up-sample or down-sample a measured surface map for model validation, not only without introducing any re-sampling errors, but also eliminating the existing measurement noise and measurement errors. Because the re-sampling of a surface map is accomplished based on the analytical expressions of Zernike-polynomials and a power spectral density model, such re-sampling does not introduce any aliasing and interpolation errors as is done by the conventional interpolation and FFT-based (fast-Fourier-transform-based) spatial-filtering method. Also, this new method automatically eliminates the measurement noise and other measurement errors such as artificial discontinuity. The developmental cycle of an optical system, such as a space telescope, includes, but is not limited to, the following two steps: (1) deriving requirements or specs on the optical quality of individual optics before they are fabricated through optical modeling and simulations, and (2) validating the optical model using the measured surface height maps after all optics are fabricated. There are a number of computational issues related to model validation, one of which is the "pre-conditioning" or pre-processing of the measured surface maps before using them in a model validation software tool. This software addresses the following issues: (1) up- or down-sampling a measured surface map to match it with the gridded data format of a model validation tool, and (2) eliminating the surface measurement noise or measurement errors such that the resulted surface height map is continuous or smoothly-varying. So far, the preferred method used for re-sampling a surface map is two-dimensional interpolation. The main problem of this method is that the same pixel can take different values when the method of interpolation is changed among the different methods such as the "nearest," "linear," "cubic," and "spline" fitting in Matlab. The conventional, FFT-based spatial filtering method used to eliminate the surface measurement noise or measurement errors can also suffer from aliasing effects. During re-sampling of a surface map, this software preserves the low spatial-frequency characteristic of a given surface map through the use of Zernike-polynomial fit coefficients, and maintains mid- and high-spatial-frequency characteristics of the given surface map by the use of a PSD model derived from the two-dimensional PSD data of the mid- and high-spatial-frequency components of the original surface map. Because this new method creates the new surface map in the desired sampling format from analytical expressions only, it does not encounter any aliasing effects and does not cause any discontinuity in the resultant surface map.

Sidick, Erkin↗

NASA NTRS2018

Wave-Front Error Measurements and Alignment of CLASP2 Telescope with a Dual-Band Pass Cold Mirror Coated Primary Mirror

"Chromospheric LAyer Spectro-Polarimeter (CLASP2)" is the next sounding rocket experiment of the "Chromospheric Lyman-Alpha Spectro-Polarimeter (CLASP)" that succeeded in observing for the first time the linear polarization spectra in the hydrogen Lyman- line (121.6 nm) and is scheduled to be launched in 2019. In CLASP2, we will carry out full Stokes-vector spectropolarimetric observations in the Mg II h & k lines near 280 nm with the spectro-polarimeter (SP), while imaging observations in the Lyman- line will be conducted with the slitjaw optics (SJ). For the wavelength selection of CLASP2, the primary mirror of the telescope uses a new dual-band pass cold mirror coating targeting both at 121.6 nm and 280 nm. Therefore, we have to perform again the alignment of the telescope after the installation of the recoated primary mirror. Before unmounting the primary mirror from the telescope structure, we measured the wave-front error (WFE) of the telescope. The measured WFE map was consistent with what we had before the CLASP flight, clearly indicating that the telescope alignment has been maintained even after the flight. After the re-coated primary mirror was installed the WFE was measured, and coma aberration was found to be larger. Finally, the secondary mirror shim adjustments were carried out based on the WFE measurements. In CLASP2 telescope, we improved a fitting method of WFE map (applying 8th terms circular Zernike polynomial tting instead of 37th terms circular Zernike fitting) and the improved method enables to achieve better performance than CLASP telescope. Indeed, WFE map obtained after the final shim adjustment indicated that the required specification (< 5:5 m RMS spot radius) that is more stringent than CLASP telescope was met.

chromosphere; polarimetry; sounding rocket↗

NASA NTRSDate not supplied

Tailoring the Stiffness of Large Mirrors to Reduce Specific Optical Aberrations

Presented is a method that tailors the geometry of a mirror’s core structure as well as the mounting interface such that a desired set of aberrations is reduced. It will be shown that targeting specific aberrations as opposed to reducing overall surface figure error (SFE) is a valid method to reduce weight or improve performance. Analyzed first is the self-weight deflection of a thin circular plate of uniform thickness continuously supported along its outer edge. The Kirchhoff-Love plate bending equations are related to the radially symmetric Zernike polynomials of piston, defocus, and primary spherical aberration. Next, the mirror’s radial thickness is optimized to eliminate primary and secondary spherical aberrations when the outer edge is simply supported. As an example, the method is demonstrated using a 2-meter diameter aluminum mirror where surface deflections can have an appreciable effect at mid-to-near infrared wavelengths

mirror stiffness optimization↗

NASA NTRS1992

Faster Computation Of Far Fields Of Dish Antennas

Modified Zernike-polynomial approach reduces number of terms in numerical integration. Yields greater or equal accuracy with fewer terms in series expansion and fewer sampling points across aperture and, less computation. Method based on Jacobi-Bessel-expansion concept.

Jamnejad, Vahraz↗

NASA NTRS1991

Image inversion analysis of the HST OTA (Hubble Space Telescope Optical Telescope Assembly), phase A

Technical work during September-December 1990 consisted of: (1) analyzing HST point source images obtained from JPL; (2) retrieving phase information from the images by a direct (noniterative) technique; and (3) characterizing the wavefront aberration due to the errors in the Hubble Space Telescope (HST) mirrors, in a preliminary manner. This work was in support of JPL design of compensating optics for the next generation wide-field planetary camera on HST. This digital technique for phase retrieval from pairs of defocused images, is based on the energy transport equation between these image planes. In addition, an end-to-end wave optics routine, based on the JPL Code 5 prescription of the unaberrated HST and WFPC, was derived for output of the reference phase front when mirror error is absent. Also, the Roddier routine unwrapped the retrieved phase by inserting the required jumps of +/- 2(pi) radians for the sake of smoothness. A least-squares fitting routine, insensitive to phase unwrapping, but nonlinear, was used to obtain estimates of the Zernike polynomial coefficients that describe the aberration. The phase results were close to, but higher than, the expected error in conic constant of the primary mirror suggested by the fossil evidence. The analysis of aberration contributed by the camera itself could be responsible for the small discrepancy, but was not verified by analysis.

Litvak, M. M.↗

NASA NTRS1993

Artificial neural network for the determination of Hubble Space Telescope aberration from stellar images

An artificial-neural-network method, first developed for the measurement and control of atmospheric phase distortion, using stellar images, was used to estimate the optical aberration of the Hubble Space Telescope. A total of 26 estimates of distortion was obtained from 23 stellar images acquired at several secondary-mirror axial positions. The results were expressed as coefficients of eight orthogonal Zernike polynomials: focus through third-order spherical. For all modes other than spherical the measured aberration was small. The average spherical aberration of the estimates was -0.299 micron rms, which is in good agreement with predictions obtained when iterative phase-retrieval algorithms were used.

Barrett, Todd K.↗

NASA NTRS2004

[Development of Nanotechnology for X-Ray Astronomy Instrumentation]

This Research Grant provides support for development of nanotechnology for x-ray astronomy instrumentation. MIT has made significant progress in several development areas. In the last year we have made considerable progress in demonstrating the high-fidelity patterning and replication of x-ray reflection gratings. We developed a process for fabricating blazed gratings in silicon with extremely smooth and sharp sawtooth profiles, and developed a nanoimprint process for replication. We also developed sophisticated new fixturing for holding thin optics during metrology without causing distortion. We developed a new image processing algorithm for our Shack-Hartmann tool that uses Zernike polynomials. This has resulted in much more accurate and repeatable measurements on thin optics.

Schattenburg, Mark L.↗

NASA NTRS2007

Wavefront Aberrations Due to Alignment and Figure Compensation of the NASA James Webb Space Telescope

This viewgraph presentation describes wavefront aberrations due to the alignment and improper compensation of the NASA James Webb Space Telescope. The contents include: 1) James Webb Space Telescope (JWST); 2) Optical design of JWST; 3) Alignment Observables for JWST; 4) Low order Zernike Polynomials; 5) PM SM Ability to Target Low Order Aberrations; 6) Compensator definitions and Modes; 7) Field impact from compensation; 8) PM align error compensated by PM figure; 9) PM align error compensated by SM alignment; 10) SM align error compensated by PM figure; 11) SM figure error compensated by SM alignment; 12) Worst Case Pupil Maps; 13) Worst Case Pupil Maps at BEST FOCUS; 14) Field impact from compensation (+/- 1 arcmin FOV); and 15) Concluding Remarks.

Howard, Joseph↗

NASA NTRS1995

Method for reducing sidelobe impact of low order aberration in a coronagraph

The invention relates to a method for reducing a sidelobe impact of low order aberrations using a coronagraph (2) having an apodized occulting mask (10), comprising the steps of: (a) providing in the coronagraph (2) the apodized occulting disk (10) having a transmission profile which graduates from opaque to transparent along its radius and the negative of whose amplitude transmission is a Gaussian profile; (b) determining a predicted sidelobe impact of the aberrations from a particular mix of low order aberration measured in a system as described by the Zernike polynomials; (c) applying the coronagraph to a system point spread function using a given rms width for the Gaussian profile describing the apodized occulting mask (10) and determining an attenuation level of the aberration sidelobes; (d) scaling the Gaussian occulting mask (10) profile to a wider rms width if the sidelobe attenuation level is too low; and (e) repeating the steps (b) through (d) until the attenuation level is acceptable.

Ftaclas, Christ↗

NASA NTRS2010

Reflector Surface Error Compensation in Dual-Reflector Antennas

By probing the field on a small subreflector at a minimal number of points, the main reflector surface errors can be obtained and subsequently used to design a phase-correction subreflector that can compensate for main reflector errors. The compensating phase-error profile across the subreflector can be achieved either by a surface deformation or by the use of an array of elements such as patch antennas that can cause a phase shift between the incoming and outgoing fields. The second option is of primary interest here, but the methodology can be applied to either case. The patch array is most easily implemented on a planar surface. Therefore, the example of a flat subreflector and a parabolic main reflector (a Newtonian dual reflector system) is considered in this work. The subreflector is assumed to be a reflector array covered with patch elements. The phase variation on a subreflector can be detected by a small number of receiving patch elements (probes). By probing the phase change at these few selected positions on the subreflector, the phase error over the entire surface can be recovered and used to change the phase of all the patch elements covering the subreflector plane to compensate for main reflector errors. This is accomplished by using a version of sampling theorem on the circular aperture. The sampling is performed on the phase-error function on the circular aperture of the main reflector by a method developed using Zernike polynomials. This method is based upon and extended from a theory previously proposed and applied to reflector aperture integration. This sampling method provides for an exact retrieval of the coefficients of up to certain orders in the expansion of the phase function, from values on a specifically calculated set of points in radial and azimuthal directions in the polar coordinate system, on the circular reflector aperture. The corresponding points on the subreflector are then obtained and, by probing the fields at these points, a set of phase values is determined that is then transferred back to the main reflector aperture for recovering the phase function. Once this function is recovered, the corresponding phase function on the subreflector is calculated and used to compensate for main reflector surface errors. In going back and forth between sub and main reflectors, geometrical (ray) optics is employed, which even though it ignores edge diffraction and other effects, is shown to be accurate for phase recovery.

Jamnejad, Vahraz↗