Monte Carlo Analysis of Encircled and Ensquared Energy Variability with RMS Wavefront Error

Abstract

Analytical calculations for encircled and ensquared energy have limitations. When aberrations are present, analytical methods are often intractable. Modelling and simulation can be an alternative approach. A Monte Carlo analysis is performed to generate unique combinations of Zernike wavefront errors that yield the same overall rms error of the optical system. Zernike errors from orders 2nd through 5th are considered individually, as well as errors from 4th and 5th orders combined. Finally, all orders from 2nd to 5th are considered in combination. The optical systems simulated are f/1.4, f/4, f/6, and f/8 at operating wavelengths of 0.633 um, 1 um and 3.39 um. Relationships between the encircled and ensquared energy radius are determined for rms wavefront error < 0.18 waves. The most realistic relationship derived is for the case of 2nd to 5th Order Zernike considered in combination. Given a known rms wavefront error (wfe) in unit of waves, the approximated 85% encircled energy radius is 1.50 λf/# e^((5.21 wfe)/λ) , and ensquared energy radius is 1.31 λf/# e^((5.53 wfe)/λ). The Montel Carlo derived trendlines match the simulated aberrated systems to within a residual sum of squares, RSS, error of 5%. An additional equation is derived for simplification purposes with slightly higher RSS error. The approximated 83.8% Encircled energy radius is 1.22 λ f/# e^((2π wfe)/λ) with RSS fitting error of 10%.