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Chapter 10:
Zernike Polynomials and Wavefronts
A wavefront from a source at infinity arrives as a plane wave having no structure related to the nature of the source. However, as the wavefront is reflected from or passes through an optical system, it can become aberrated; i.e., the plane wave changes from being flat to taking on structure. The structure in the wavefront can be measured using wavefront sensors such as a Shack Hartmann sensor and modeled mathematically using Zernike polynomials. Starlight passing though the atmosphere will encounter regions of differing indices of refraction in the air before reaching a telescope’s aperture. These refractive index variations delay portions of the wavefront. When the starlight is focused by the telescope, the formed image is blurred such that the spot is no longer larger than the performance limits of the telescope would predict. Whether viewed at the focus of the optical system or at the pupil using a wavefront sensor, aberrations can be observed. Fritz Zernike developed the Zernike polynomials in 1934. These polynomials have a form that is similar to the aberrations encountered during optical testing. They are expressed in terms of the radial distance and a rotation angle, and can be combined to construct very complicated wavefronts. In this chapter we will explore how to simulate and combine Zernike polynomials to represent wavefronts. The context of a Shack–Hartman wavefront sensor will be used to illustrate how wavefronts can be measured.
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