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In mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. ... Defocus (longitudinal position) ...
Defocus is modeled in Zernike polynomial format as (), where is the defocus coefficient in wavelengths of light. This corresponds to the parabola -shaped optical path difference between two spherical wavefronts that are tangent at their vertices and have different radii of curvature .
The theory of aberrated point spread functions close to the optimum focal plane was studied by Zernike and Nijboer in the 1930–40s. A central role in their analysis is played by Zernike's circle polynomials that allow an efficient representation of the aberrations of any optical system with rotational symmetry. Recent analytic results have ...
Transfer function and example image of an f/4 optical imaging system at 500 nm with spherical aberration with standard Zernike coefficient of 0.25. As the ideal lens system, the contrast reaches zero at the spatial frequency of 500 cycles per millimeter.
Zernike polynomials are usually expressed in terms of polar coordinates (ρ,θ), where ρ is radial coordinate and θ is the angle. The advantage of expressing the aberrations in terms of these polynomials includes the fact that the polynomials are independent of one another.
Defocus is the lowest order true optical aberration. If piston and tilt are subtracted from an otherwise perfect wavefront, a perfect, aberration-free image is formed. If piston and tilt are subtracted from an otherwise perfect wavefront, a perfect, aberration-free image is formed.
1: Imaging by a lens with chromatic aberration. 2: A lens with less chromatic aberration. In optics, aberration is a property of optical systems, such as lenses, that causes light to be spread out over some region of space rather than focused to a point. [1]
Tilt quantifies the average slope in both the X and Y directions of a wavefront or phase profile across the pupil of an optical system. In conjunction with piston (the first Zernike polynomial term), X and Y tilt can be modeled using the second and third Zernike polynomials: