Polynomial expression

Meaning

Brief exp.

${\rho }^2\cos 2\theta$

Astigmatism 0˚ or 90˚

$Z_5$

${\rho }^2\sin 2\theta$

Astigmatism ±45˚

$Z_6$

$(3{\rho }^2-2)\rho \cos \theta$

X Coma and Tilt

$Z_7$

$(3{\rho }^2-2)\rho \sin \theta$

Y Coma and Tilt

$Z_8$

$6{\rho }^4-6{\rho }^2+1$

Spherical and Focus

$Z_9$