Examples of performance metrics that deliver more than a single number, and thus contain more information, are the point spread function (PSF), the modulation transfer function (MTF), wavefront maps, and spot diagrams. Examples of performance metrics that deliver a single numerical value are the Strehl ratio ( S), the RMS wavefront error ( σ), and image-sharpness metrics. This immediately allows ranking of different optical systems, optimization of an optical system during its design, or finding the optimum state of an active or adaptive optics system. It is often desirable to state the performance of an optical system by a single number. Once the optical system is near the Maréchal limit, the Zernike modes become orthogonal to each other with respect to the merit function. This is represented by the second transition in the graphic representation in Fig. We aim at the optimization of severely aberrated systems with several λ of aberration and low Strehl ratio. They are also orthogonal to each other with respect to our merit function for small aberrations, but we demonstrate that this is not valid for aberrations of more than λ/8 RMS. Zernike modes are orthogonal to each other over a unit circle with respect to the wavefront. In this paper we discuss the landscape of an image-sharpness metric used as merit function when we control the surface of the active element with Zernike modes. Using an image-based method, active optics would allow image optimization for different objectives, e.g., for maximization of contrast in high or in low spatial frequencies, depending on the science target. It can handle large aberrations, in contrast to wavefront sensing that has limited dynamic range. Our image-based method evaluates the image of the science camera and adapts the surface of the active element to increase a merit function. In this sense, phase diversity is not an image-based method. Although phase diversity is technically image-based, here we use the term “image-based” to refer to correction methods that do not require the wavefront information. Indirect wavefront sensing uses the science camera and an iterative method, e.g., phase diversity, to retrieve the wavefront. In ground-based telescopes, anisoplanatism is reduced with sophisticated concepts such as multi-conjugate adaptive optics, by using several guide stars, wavefront sensors, and corrective elements. In addition, the angular separation between the science target and the guide star leads to anisoplanatism. Direct wavefront sensing using a dedicated wavefront sensor requires a bright guide star and results in non-common path errors. Therefore, in the case of a segmented mirror, the segments should be already co-phased by using another technique.Ĭonventional adaptive optics measures the wavefront and applies its inverse to the corrective active element. The method we discuss here requires a continuous surface of the active element, because we use the Zernike modes to describe the active element. Active optics at the primary mirror and/or in a plane conjugate to the primary mirror will be required to co-phase the segments, align the optical telescope, and correct for manufacturing errors and slow drifts caused by thermo-elastic effects and gravitational release. Such telescopes will need to have segmented, lightweight primaries in order to reduce mass and stowed volume. Space telescopes with 10-m-class primary mirrors are currently being studied for astronomy, in particular for characterization of exoplanets, and for Earth observation from the geostationary orbit. The non-orthogonality of the Zernike modes with respect to the merit function should be taken into account when designing the algorithm for image-based wavefront correction, because it may slow down the process or lead to premature convergence.
Coma x zernike full#
We show that for combinations of Zernike modes with the same azimuthal order, a flatter wavefront in the central region of the aperture is more important than the RMS wavefront error across the full aperture for achieving a better merit function.
Using wavefront maps, the PSF, and the MTF, we discuss the physical causes for the non-orthogonality of the Zernike modes with respect to the merit function. In severely aberrated systems, the Zernike modes are not orthogonal to each other with respect to this merit function. We use an image-sharpness metric as merit function to evaluate the image quality, and the Zernike modes as control variables. With a view to future large space telescopes, we investigate image-based wavefront correction with active optics.
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