# Zernike Aberrations

## Introduction

This page computes and plots variuos characteristics of the Zernike polynominals. Optical system is assumed to be circular in shape. $x_p$ and $y_p$ are the normalized exit pupil coordinates, where the $x_p$ axis defines the sagittal plane and the $y_p$ axis defines the meridional plane.

Each aberration is specified using two subscripts $n$ and $m$. The wavefront is the Zernike polynomial series $$\begin{eqnarray} W(\rho, \theta) = \overline{\Delta W} + \sum_{n=1}^{\infty}\left [ A_n \sum_{s=0}^{n}(-1)^s \frac{(2n -s)! \rho^{2(n-s)}}{s!(n-s)!(n-s)!} + \\ \sum_{m=1}^{n} \sum_{s=0}^{n-m}(-1)^s \frac{(2n -m -s)! \rho^{2(n-m-s)}}{s!(n-s)!(n-m-s)!} \rho^m (B_{nm} \cos m\theta' + C_{nm} \sin m\theta')\right ] \end{eqnarray}$$ where $\rho = \sqrt{x_p^2+y_p^2}$, $\rho \cos \theta = x_p$, $\overline{\Delta W}$ is the mean wavefront ODP and $A_n$, $B_{nm}$, and $C_{nm}$ are the Zernike coefficients.

## Coherent transfer function

The first step in computing the point spread functon and modulation transfer function is to find the coherent transfer function. First, the aberrated pupil function is $$P(x_p, y_p) = circ\left ( \frac{\sqrt{x_p^2 + y_p^2}}{r_{xp}} \right )e^{-ikW(x_p,y_p)}$$ where $r_{xp}$ is the exit pupil radius and $k$ is the wave number. Then, the coherent transfer function is computed as $$H(f_U, f_V) = P(-\lambda zf_U, -\lambda z f_V)$$## Point spread function

The point spread function (PSF) is calculated by $$\left | h(u,v) \right |^2 = \left | \Im ^{-1}(H(f_U,f_V)) \right |^2$$ where $\Im$ is the Fourier Transform function; this page uses a 512-point FFT.## Modulation transfer function

The modulation transfer function (MTF) is calculated as $$MTF(f_U, f_V) = \left | \Im \left ( \left | h(u,v) \right | ^2 \right ) \right |$$ normalized to 1.
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*References:*

*1. Computational Fourier Optics by David Voelz, SPIE Press, 2011*

2. Numerical Simulation of Optical Wave Propagation with Examples in MATLAB by Jason D. Schmidt, SPIE Press, 2010

2. Numerical Simulation of Optical Wave Propagation with Examples in MATLAB by Jason D. Schmidt, SPIE Press, 2010