Editing Optical aberration (section)

===Zernike model of aberrations===

[[File:ZernikeAiryImage.jpg|360px|thumb|Image plane of a flat-top beam under the effect of the first 21 Zernike polynomials.]]
[[File:ZernikeLogAiryImage.jpg|360px|thumb|Effect of Zernike aberrations in Log scale. The intensity minima are visible.]]

Circular wavefront profiles associated with aberrations may be mathematically modeled using [[Zernike polynomial]]s.  Developed by [[Frits Zernike]] in the 1930s, Zernike's polynomials are [[orthogonal]] over a circle of unit radius. A complex, aberrated wavefront profile may be [[curve-fitted]] with Zernike polynomials to yield a set of fitting [[coefficient]]s 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.

There are [[even and odd functions|even and odd]] Zernike polynomials. The even Zernike polynomials are defined as

<math display="block">Z^{m}_n(\rho,\phi) = R^m_n(\rho)\,\cos(m\,\phi)</math>

and the odd Zernike polynomials as

<math display="block">Z^{-m}_n(\rho,\phi) = R^m_n(\rho)\,\sin(m\,\phi)</math>

where {{mvar|m}} and {{mvar|n}} are nonnegative [[integer]]s with {{math|''n'' ≥ ''m''}}, {{mvar|ϕ}} is the [[azimuth]]al [[angle]] in [[radian]]s, and {{mvar|ρ}} is the normalized radial distance.  The radial polynomials <math>R^m_n</math> have no azimuthal dependence, and are defined as

<math display="block">R^m_n(\rho) = 
\begin{cases}
\sum_{k=0}^{(n-m)/2} \!\!\! \frac{(-1)^k\,(n-k)!}{k!\,\left({n+m \over 2}-k\right)!\,\left({n-m \over 2}-k\right)!} \;\rho^{n-2\,k}, & \text{if } n-m \text{ is even} \\
0, & \text{if } n-m \text{ is odd.}
\end{cases}</math>

The first few Zernike polynomials, multiplied by their respective fitting coefficients, are:<ref>{{Cite book|last=Schroeder, D. J.|url=https://www.worldcat.org/oclc/162132153|title=Astronomical optics|date=2000|publisher=Academic Press|isbn=978-0-08-049951-2|edition=2nd|location=San Diego|oclc=162132153}}</ref>
{|
|-
|<math>a_0 \times 1 </math>|| "Piston", equal to the [[mean value]] of the wavefront
|-
|<math>a_1\times \rho \cos(\phi)</math> || "X-Tilt", the deviation of the overall beam in the [[Sagittal ray#Optical systems|sagittal]] direction
|-
|<math>a_2\times \rho \sin(\phi)</math> || "Y-Tilt", the deviation of the overall beam in the [[Tangential ray#Optical systems|tangential]] direction
|-
|<math>a_3\times (2\rho^2-1)</math> || "Defocus", a [[Parabola|parabolic]] wavefront resulting from being out of focus
|-
|<math>a_4\times \rho^2 \cos(2\phi)</math> || "0° Astigmatism", a [[cylindrical]] shape along the X or Y axis
|-
|<math>a_5\times \rho^2 \sin(2\phi)</math> || "45° Astigmatism", a cylindrical shape oriented at ±45° from the X axis
|-
|<math>a_6\times (3\rho^2-2)\rho \cos(\phi)</math> || "X-Coma", comatic image flaring in the horizontal direction
|-
|<math>a_7\times (3\rho^2-2)\rho \sin(\phi)</math> || "Y-Coma", comatic image flaring in the vertical direction
|-
|<math>a_8\times (6\rho^4-6\rho^2+1)</math> || "Third order spherical aberration"
|}

where {{mvar|ρ}} is the normalized pupil radius with {{math|0 ≤ ''ρ'' ≤ 1}}, {{mvar|ϕ}} is the azimuthal angle around the pupil with {{math|0 ≤ ''ϕ'' ≤ 2''π''}}, and the fitting coefficients {{math|''a''{{sub|0}}, ..., ''a''{{sub|8}}}}  are the wavefront errors in wavelengths.

As in [[Fourier analysis|Fourier]] synthesis using [[sine]]s and [[cosine]]s, a wavefront may be perfectly represented by a sufficiently large number of higher-order Zernike polynomials.  However, wavefronts with very steep [[gradients]] or very high [[spatial frequency]] structure, such as produced by [[Wave propagation|propagation]] through [[atmospheric turbulence]] or [[turbulence|aerodynamic flowfields]], are not well modeled by Zernike polynomials, which tend to [[low-pass filter]] fine [[Three-dimensional space|spatial]] definition in the wavefront. In this case, other fitting methods such as [[fractals]] or [[singular value decomposition]] may yield improved fitting results.

The [[Zernike polynomials|circle polynomials]] were introduced by [[Frits Zernike]] to evaluate the point image of an aberrated optical system taking into account the effects of [[diffraction]]. The perfect point image in the presence of diffraction had already been described by [[George Biddell Airy|Airy]], as early as 1835. It took almost hundred years to arrive at a comprehensive theory and modeling of the point image of aberrated systems (Zernike and Nijboer). The  analysis by Nijboer and Zernike describes the intensity distribution close to the optimum focal plane. An extended theory that allows the calculation of the point image amplitude and  intensity over a much larger volume in the focal region was recently developed ([http://www.nijboerzernike.nl Extended Nijboer-Zernike theory]). This Extended Nijboer-Zernike theory of point image or 'point-spread function' formation has found applications in general research on image formation, especially for systems with a high [[numerical aperture]], and in characterizing optical systems with respect to their aberrations.<ref>{{cite book |title=[[Principles of Optics|Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light]]|first1=Max |last1=Born |first2=Emil |last2=Wolf |isbn=978-0521642224|date=1999-10-13 |publisher=Cambridge University Press }}</ref>