The dominant aberrations of the optical system that the DMs must correct are coma $Z3,1$ and spherical aberration $Z4,0$ [we use the American National Standards Institute (ANSI) $Zn,m$ indexing scheme for radial order $n$ and azimuthal frequency $m$], arising from the fact that the active focus control makes the objective lens operate at a focus location other than its optimum working distance. Other off-axis aberrations also come into play when the galvo scanners steer the beam to larger scanning angles, with appreciable contributions from both the relay optics and the objective lens. The simulation results show significant astigmatism $Z2,2$, secondary astigmatism $Z4,2$, secondary coma $Z5,1$, trefoil $Z3,3$, and secondary spherical aberration $Z6,0$. The range of simulated aberration coefficients are summarized in Table 1. Note that we are using the “unnormalized” polynomials for our Zernike basis, given explicitly here: Display Formula
$Defocus:\u2009\u2009Z2,0(\rho ,\theta )=2\rho 2\u22121,Astigmatism:\u2009\u2009Z2,2(\rho ,\theta )=\rho 2\u2009cos\u20092\theta ,Coma:\u2009\u2009Z3,1(\rho ,\theta )=(3\rho 2\u22122)\rho \u2009cos\u2009\theta ,Vertical trefoil:\u2009\u2009Z3,\u22123(\rho ,\theta )=\rho 3\u2009cos\u20093\theta ,Primary spherical:\u2009\u2009Z4,0(\rho ,\theta )=6\rho 4\u22126\rho 2+1,Secondary astigmatism:\u2009\u2009Z4,2(\rho ,\theta )=(4\rho 2\u22123)\rho 2\u2009cos\u20092\theta ,Secondary coma:\u2009\u2009Z5,1(\rho ,\theta )=(10\rho 4\u221212\rho 2+3)\rho \u2009cos\u2009\theta ,Secondary spherical:\u2009\u2009Z6,0(\rho ,\theta )=(20\rho 6\u221230\rho 4+12\rho 2\u22121).$