# Seidel Aberrations

## Introduction

This page computes and plots various characteristics of the Seidel aberrations. Optical system is assumed to be circular in shape. $H$ is the radial position of the image plane, i.e. the field height or field vector normalized to 1. $x_p$ 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 three subscripts, where each subscript is the power of each of the wavefront power series expansion terms: $$W_{ijk} = H^{i}\rho ^{j}\cos ^{k}\theta$$

## Wave Fans

Wavefans are the combination of the Seidel aberration power series expansion terms evaluated from $-1 \leq (x_p, y_p) \leq 1$:$$W\left ( x_p,y_p \right ) = \sum W_{ijk}(x_p,y_p) \; \forall \;(i,j,k)$$

## 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.

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