# PSF Estimation of Space-Variant Ultra-Wide Field of View Imaging Systems
## 2 Wavefront Aberration Functions  
An ideal diffraction limited imaging system transforms a spherical input wave with an inclination equal to the plane wave direction. As described by [PERSON] (1994), real imaging systems convert the input plane wave into a deformed wavefront. The difference between an ideal wavefront and an aberrated wavefront in the exit aperture can be expressed as  
\[W(x,\,y)=W_{ab}(x,\,y)-W_{sp}(x,\,y) \tag{1}\]  
where \(W(x,\,y)\) are wavefronts--the surface of points characterized by the same phase. Figure 1 illustrates the shape of the ideal spherical wavefront \(W_{sp}(x,\,y)\) and aberrated wavefront \(W_{ab}(x,\,y)\). It can be useful to introduce the coordinate system illustrated in Figure 2. The object plane is described by the \(\xi\), \(\eta\) coordinates; the exit pupil uses \(x\), \(y\) notation, and the image plane uses \(u\), \(\
u\) notation. Then we introduce the \(\hat{x},\hat{y}\) normalized coordinates (see Figure 2b,c) at the exit pupil and \(\varrho\), \(\theta\) polar coordinates. The normalized image plane coordinates are \(\hat{u},\hat{\varrho}\) and \(H\), \(\varphi\), respectively. The aberrated wavefront can be represented by a set of orthonormal base functions, known as Zernike polynomials. Zernike polynomials, which are described in [PERSON] (1957) and in [PERSON] (1958); [PERSON] (1959); [PERSON] (1960), are a set of functions orthogonal over the unit circle, usually described in polar coordinates as \(Z_{n}^{m}(\rho,\theta)\), where \(\rho\) is the radius and \(\theta\) is the angle with respect to the \(\hat{x}\)-axis in the exit aperture (see Figure 2b). They represent functions of optical distortions that classify each aberration using a set of polynomials. The set of Zernike polynomials is defined in [PERSON] (1957); other adoptions are in [PERSON] (1958); [PERSON] (1960); [PERSON] (1960), and it can be written as  
\[Z_{n}^{m}(\rho,\theta)=N_{n}^{m}R_{n}^{|m|}(\rho)\cos(m\theta) \quad m\geq 0 \tag{2}\] \[Z_{n}^{m}(\rho,\theta)=-N_{n}^{m}R_{n}^{|m|}(\rho)\sin(m\theta) \quad m<0\]  
with \(n\) describing the power of the radial polynomial and \(m\) describing the angular frequency. \[N_{n}^{m}=\sqrt{\frac{2(n+1)}{1+\delta_{m0}}}, \tag{3}\]  
Figure 1: The difference between an ideal spherical wavefront and an aberrated wavefront. is the normalization factor with the Kronecker delta function \(\delta_{m0}=1\) for \(m=0\), and \(\delta_{m0}=0\) for \(m\
eq 0\), and  
\[R_{n}^{|m|}(\rho)=\sum_{s=0}^{(n-|m|)/2}\frac{(-1)^{s}(n-s)!}{s!\left[\frac{(n +|m|)}{2}-s\right]!\left[\frac{(n-|m|)}{2}-s\right]!}\rho^{(n-2s)}, \tag{4}\]  
is the radial part of the Zernike polynomial. Any wavefront phase distortion over a circular aperture of unit radius can be expanded as a sum of the Zernike polynomials as  
\[W(\rho,\,\theta)=\sum_{n}^{k}\sum_{m=-n}^{n}A_{n}^{m}Z_{n}^{m}(\rho,\,\theta), \tag{5}\]  
which can be rewritten using Equation (2) as  
\[W(\rho,\,\theta)=\sum_{n}^{k}\sum_{m=0}^{n}A_{n}^{m}N_{n}^{m}R_{n}^{|m|}(\rho )\cos(m\theta)-\sum_{n}^{k}\sum_{m=-n}^{-1}A_{n}^{m}N_{n}^{m}R_{n}^{|m|}(\rho )\sin(m\theta), \tag{6}\]  
where \(m\) has values of \(-n,-n+2,\ldots n\), \(k\) is the order of the polynomial expansion, and \(A_{n}^{m}=A_{n}^{m}(H,\varphi)\) is the coefficient of the \(Z_{n}^{m}\) mode in the expansion, i.e., it is equal to the root mean square (RMS) phase difference for that mode. The wavefront aberration function across the field of view of the optical system can then be described as  
Figure 2: Graphical representation of the adopted coordinate systems. Subfigure (**a**) illustrates all planes, subfigure (**b**) illustrates normalized exit pupil and (**c**) illustrates notation of normalized Image plane. Figure 2.0  
where the wavefront is described with normalized polar coordinates. A better understanding can be provided by Table 1 in Appendix A and by Figure 3, which describes the indexing of Zernike polynomials and the kind of aberration associated with an index of Zernike polynomials.