Phase Retrieval

James R. Fienup used phase retrieval algorithms to determine the aberrations in the Hubble Space Telescope. The theory of phase retrieval is described in this paper. Other applications of phase retrieval, such as undersampled broadband images, lensless coherent imaging, and optical metrology, are also discussed. Introduction: James R. Fienup established the use of phase retrieval in the early 1980’s. Phase retrieval determines the phase error or aberrations of an optical system. Phase retrieval algorithms use computer modeling that often start with an educated guess. This model is run several times and a pattern is determined. It is from the comparison of this pattern to the known data of the actual optical system that the correct parameters can usually be determined and used to accurately model the system. From this model, the aberration coefficients can be determined. These aberration coefficients are used to calibrate an already working optical system. This is especially useful for optical systems in space, where direct testing cannot be used [1]. Phase retrieval was used to help correct the blurred images from the uncorrected Hubble Space Telescope [2]. There are other more recent applications for phase retrieval, such as undersampled broadband images [3], lensless coherent imaging [4], optical metrology [5], and 3-D locator sets of opaque objects [6]. Theory: In phase retrieval, the phase error can be found from some a priori information about the Fourier transform of the function [1]. For the case of mathematical modeling, a priori means to analyze data that is collected and look for patterns that are created from that model [7]. Phase retrieval creates a model of the point spread function of the optical system [1]. A point spread function shows the propagation of a light wave from a point source through an optical system [8]. In a perfect optical system, the point spread function from the object should be the same at the detector array. However, when aberrations are present in the optical system, the point-spread function at the entrance pupil of the system is disorganized at the detector array, and the image appears blurred. The phase retrieval algorithm calculates the point spread function of the system so the calibration aberration coefficient can be used to deblur the images. Many different kinds of aberrations can appear in optical systems. The most common are polynomial aberrations, such as astigmatism, chromatic, coma, and spherical (Fig.(1)). Spherical aberrations are the only on-axis aberrations and cause the focal point to actually be spread out along the optical axis. In other words, the rays closest to the optical axis cross the optical axis before the rays towards the edge of the lens do [9]. Astigmatism, coma, and chromatic aberrations are all off-axis aberrations. Astigmatism occurs when an off-axis bundle of light travels through an optical system. There will be a point where a vertical focus line and a horizontal focus line will appear, instead of circles of focused light [9]. Coma aberrations are similar, but occur when an off-axis bundle of light is not perfectly re-imaged, or not all the rays cross at the same off-axis point [9]. Finally, chromatic aberrations occur because the index of refraction is not the same for all wavelengths of light; therefore, the different wavelength rays focus at different points [9]. The phase-retrieval algorithm determines the Zernike coefficients, which come from the even Zernike polynomial (Eq.[1]) and the odd Zernike polynomial (Eq.[2]) that are typically used to describe aberrations [10]. ) cos( ) , ( φ φ ρ m R Z m n m n = (1) ) sin( ) , ( φ φ ρ m R Z m n m n = − , (2) where m and n are integers (n ≥ m), ρ is the normalized radial distance, ф is the azimuthal angle (radians), and Rn are the radial polynomials. Figure 1. Polynomial aberrations There are two approaches to determining the phase retrieval: gradient-search algorithms and iterative transform algorithms [1]. Both these algorithms are very complex, with many uses of multiple Fourier transforms. The models that are created from the algorithms are computer generated. The gradient-search algorithm starts with a weighted error metric, [ ] ∑ − = u u F u G u W E 2 ) ( ) ( ) ( (3) where F(u) is the magnitude of the optical field at the detector, W(u) is a weighting function, and G(u) is a computer generated model of the magnitude of an optical wave front, including the phase error [1]. A weighting function gives a chosen element more “weight” over other elements in the equation [11]. When these algorithms were applied to the Hubble Space Telescope, the spherical aberrations were far greater than the scientists originally thought [1]. The other polynomial aberrations were present but much smaller. The gradient-search algorithm for calculating the single-plane phase retrieval was used to find the Zernike coefficients. The normalized root mean square error (Eq.[4]) was computed to be within 0.001 waves of the correct Zernike coefficients for the optical telescope assembly, which is very accurate [2].