Table 1.

Zernike coefficients of the first five orders.^a

Zernike order	Radial/azimuthal degree $(n,m)$	Expression	Aberration
First	$(1,-1)$	$2\rho \sin (\theta )$	Tilt
	(1, 1)	$2\rho \cos (\theta )$	Tip
Second	(2, 0)	$\sqrt{3}(2{\rho }^2-1)$	Defocus
	$(2,-2)$	$\sqrt{6}{\rho }^2\sin (2\theta )$	Oblique astigmatism
	(2, 2)	$\sqrt{6}{\rho }^2\cos (2\theta )$	Vertical astigmatism
Third	$(3,-1)$	$\sqrt{8}(3{\rho }^3-2\rho )\sin (\theta )$	Vertical coma
	(3, 1)	$\sqrt{8}(3{\rho }^3-2\rho )\cos (\theta )$	Horizontal coma
	$(3,-3)$	$\sqrt{8}{\rho }^3\sin (3\theta )$	Vertical trefoil
	(3, 3)	$\sqrt{8}{\rho }^3\cos (3\theta )$	Oblique trefoil
Fourth	(4, 0)	$\sqrt{5}(6{\rho }^4-6{\rho }^2+1)$	Primary spherical
	$(4,-2)$	$\sqrt{10}(4{\rho }^4-3{\rho }^2)\sin (2\theta )$	Oblique secondary astigmatism
	(4, 2)	$\sqrt{10}(4{\rho }^4-3{\rho }^2)\cos (2\theta )$	Vertical secondary astigmatism
	$(4,-4)$	$\sqrt{10}{\rho }^4\sin (4\theta )$	Oblique quadrafoil
	(4, 4)	$\sqrt{10}{\rho }^4\cos (4\theta )$	Vertical quadrafoil
Fifth	$(5,-1)$	$\sqrt{12}(10{\rho }^5-12{\rho }^3+3\rho )\sin (\theta )$	Vertical secondary coma
	(5, 1)	$\sqrt{12}(10{\rho }^5-12{\rho }^3+3\rho )\cos (\theta )$	Horizontal secondary coma
	$(5,-3)$	$\sqrt{12}(5{\rho }^5-4{\rho }^3)\sin (3\theta )$	Vertical secondary trefoil
	(5, 3)	$\sqrt{12}(5{\rho }^5-4{\rho }^3)\cos (3\theta )$	Oblique secondary trefoil
	$(5,-5)$	$\sqrt{12}{\rho }^5\sin (5\theta )$	Vertical pentafoil
	(5, 5)	$\sqrt{12}{\rho }^5\cos (5\theta )$	Oblique pentafoil

^aFrom Ref. 57.