# 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 $$H(f_U, f_V) = P(-\lambda zf_U, -\lambda z f_V)$$

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.

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

## Results

### Coherent transfer function   