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The phase function is retrieved by the unknown-coefficient weighted product with (known values) of Zernike polynomial across the unit grid. Hence, coefficients can also be found by solving a linear system, for instance by matrix inversion.
A complex, aberrated wavefront profile may be curve-fitted with Zernike polynomials to yield a set of fitting coefficients that individually represent different types of aberrations. These Zernike coefficients are linearly independent, thus individual aberration contributions to an overall wavefront may be isolated and quantified separately.
Among these the most important Zernike coefficients affecting visual quality are coma, spherical aberration, and trefoil. [6] Zernike polynomials are usually expressed in terms of polar coordinates (ρ,θ), where ρ is radial coordinate and θ is the angle.
To take full advantage of a higher resolution medium, defocus and other aberrations must be minimized. Defocus is modeled in Zernike polynomial format as a ( 2 ρ 2 − 1 ) {\displaystyle a(2\rho ^{2}-1)} , where a {\displaystyle a} is the defocus coefficient in wavelengths of light.
The piston coefficient is typically expressed in wavelengths of light at a particular wavelength. Its main use is in curve-fitting wavefronts with Cartesian polynomials or Zernike polynomials . However, similar to a real engine piston moving up and down in its cylinder, optical piston values can be changed to bias the wavefront phase mean value ...
The and coefficients are typically expressed as a fraction of a chosen wavelength of light. Piston and tilt are not actually true optical aberrations, as they do not represent or model curvature in the wavefront. Defocus is the lowest order true optical aberration. If piston and tilt are subtracted from an otherwise perfect wavefront, a perfect ...
The following figures shows the two-dimensional equivalent of the ideal and the imperfect system discussed earlier, for an optical system with trefoil, a non-rotational-symmetric aberration. Optical transfer functions are not always real-valued. Period patterns can be shifted by any amount, depending on the aberration in the system.
In mathematics, pseudo-Zernike polynomials are well known and widely used in the analysis of optical systems. They are also widely used in image analysis as shape descriptors . Definition