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Zernike Polynomials via Phase Recovery




The technology within this topic is restricted under the International Traffic in Arms Regulation (ITAR), 22 CFR Parts 120-130, which controls the export and import of defense-related material and services, including export of sensitive technical data, or the Export Administration Regulation (EAR), 15 CFR Parts 730-774, which controls dual use items. Offerors must disclose any proposed use of foreign nationals (FNs), their country(ies) of origin, the type of visa or work permit possessed, and the statement of work (SOW) tasks intended for accomplishment by the FN(s) in accordance with the Announcement. Offerors are advised foreign nationals proposed to perform on this topic may be restricted due to the technical data under US Export Control Laws.


OBJECTIVE: Technology capable of completely characterizing an optic under test by implementing phase recovery and a collimated, partially coherent light source.


DESCRIPTION: Developing this capability will enable improved resolution in the Army's ubiquitous direct view optical systems. This advance will improve soldier survivability and lethality due to increased situational awareness and greater ability to detect, classify, recognize and identify (DCRI) threats.


The United States Army requires the ability to completely characterize optical systems to validate their design and performance. Commercial systems can measure the Modulation Transfer Function (MTF), effective focal length, field curvature, and distortion. These systems monitor the Fourier transform plane of the lens under test (LUT). They are currently unable to extract coma and spherical aberrations. Hopkins or Seidel coefficients that pertain to ray traces do not form an orthonormal basis; however, the Zernike polynomials are a proper basis that describes wavefront error. Frame captures contain only intensity information, and recovering the Zernike coefficients requires phase recovery. The Hubble Space Telescope was characterized by Brady (2005) and Fienup (1993) using a variation of the Gerchberg-Saxton algorithm (Wittle, 2018). Phase recovery is an inverse problem and requires constraints. Normally one uses two planes: the image plane and the Fourier transform plane. Other methods may be employed, e.g., the use of a series of frame captures about the Fourier Transform plane (Dube, 2018; Zhou, 2021; Gureyev, 2004); Mehrabkhani, 2017; Volkov, 2001). Pinhole illumination is assumed in advance. This would be near the effective focal length of the LUT. Because this is near the waist of the caustic, there is a concern as to how much (Fisher) information is available. Thus, the auxiliary planes will need to be near the region of maximum curvature for the caustic. In the far field, one can use the Fraunhofer approximate for the more general Huygen-Fresnel (H-F) propagator. Whether a Fresnel approximation is valid may depend upon the f-number of the LUT.


The acquisition of off-axis terms will require that the lens be rotated about its second nodal point. This is also true for interferometric approaches (Gates, 1955; ZYGO). The rotation of the LUT also satisfies the conditions outlined in Zhou (2021). As implemented, it is a form of tomography. Neither compressive sensing (Candes, 2011; Li, 2020) nor any other solution that requires the addition of optical elements (Fuerschbach, 2014) such as phase screens or beam splitters into the optical path are of interest for the purpose of this topic. Assume that the aberrated Airy disk is commercially examined via a microscope objective and image sensing array.


PHASE I: Develop the algorithms needed for implementing phase recovery using a series of planes about the location of the Fourier transform plane of the LUT. Demonstrate that the algorithm can converge to solutions that are consistent with those derived from interferometric methods. Determine the criteria for setting the spacing between these planes for best performance.


PHASE II: Using the results from phase I, develop the hardware and software to realize the procedure on a commercial system. Plane locations can be manually entered and the images offloaded for processing.


PHASE III DUAL USE APPLICATIONS: Make the necessary hardware and software adjustments to a commercial platform, automate the acquisition procedure, and automate the extraction of the Zernike polynomials.



  1. Brady (2005), Gregory R. and James R. Fienup; “Phase retrieval as an optical metrology tool”; In Optifab 2005: Technical Digest (Vol. 10315, pp. 143-145). SPIE. Proceedings of SPIE - The International Society for Optical Engineering. 10.1117/12.605914.;
  2. Candes (2011), Emmanuel J., Thomas Strohmer and Vladislav Veroninski; “PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming”; arXiv:1109.4499v1 [cs.IT] 21 Sep 2011.;
  3. Dube (2018), Brandon D. “On the Use of Classical MTF Measurements to Perform Wavefront Sensing”; Thesis, The Institute of Optics Hajim School of Engineering and Applied Sciences;;
  4. Fienup (1993), James R.; “Phase-retrieval algorithms for a complicated optical system”. Applied optics, 32 10, 1737-46.;
  5. Fuerschbach (2014), Kyle, Kevin P. Thompson, and Jannick P. Rolland; Interferometric measurement of a concave, φ-polynomial, Zernike mirror”; Optics Letters / Vol. 39, No. 1 / January 1, 2014.;
  6. Gureyev (2004), T.E.; A Pogany, D.M Paganin, S.W Wilkins, “Linear algorithms for phase retrieval in the Fresnel region”; Optics Communications,Vol. 231; Issues 1–6, Pp. 53-70,; ISSN 0030-4018,;;
  7. Li (2020) Fanxing, Wei Yan, , Fupin Peng, Simo Wang and Jialin Du; “Enhanced Phase Retrieval Method Based on Random Phase Modulation”; Appl. Sci. 2020, 10, 1184; doi:10.3390/app10031184;
  8. Mehrabkhani (2017), Soheil & Kuester, Melvin. “Optimization of phase retrieval in the Fresnel domain by the modified Gerchberg-Saxton algorithm”;;
  9. Volkov (2001), V., & Zhu, Y.; “Phase Retrieval from Two Defocused Images by the Transport-Ofintensity Equation Formalism with Fast Fourier Transform”. Microscopy and Microanalysis, 7(S2), 430-431 Aug. 5-9 Long Beach CA; doi:10.1017/S1431927600028221;
  10. Wittle (2018), Lily. Investigating the Gerchberg-Saxton Phase Retrieval Algorithm. SIAM Undergraduate Research Online. 11. 10.1137/17S016610.;
  11. Zhou (2021) Guocheng, Shaohui Zhang, Yayu Zhai, Yao Hu, Qun Hao; “Single-Shot Through-Focus Image Acquisition and Phase Retrieval from Chromatic Aberration and Multi-Angle Illumination”; Frontiers in Physics, vol.9; DOI=10.3389/fphy.2021.648827;;;
  12. ZYGO “Typical Interferometer Setups”;]


KEYWORDS: Phase Recovery, Zernike Polynomials, Fresnel Propagator, Inverse problem, Constraints, Priors.

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