# 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.
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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