Table A.1

Zernike integer indices, mode orders, and polynomials, with wavefront aberration and symmetry.

Integer indices n m Mode order (Noll) Mode order (OSA/ANSI) Normalized Zernike polynomials, Z n m $ {Z}_n^m$ Wavefront aberration Symmetry (xi, yi)
0 0 J = 1 0 1 / π    1 $ \sqrt{1/\pi }\hspace{1em}\hspace{1em}1$ Piston (Even, Even)
1 1 J = 2 2 4 / π ρ cos ( θ ) $ \sqrt{4/\pi }\hspace{1em}\rho \mathrm{cos}(\theta )$ Tilt (along xi) (Odd, Even)
1 –1 J = 3 1 4 / π ρ sin ( θ ) $ \sqrt{4/\pi }\hspace{1em}\rho \mathrm{sin}(\theta )$ Tip (along yi) (Even, Odd)
2 0 J = 4 4 3 / π ( 2 ρ 2 - 1 ) $ \sqrt{3/\pi }\hspace{1em}(2{\rho }^2-1)$ Defocus (Even, Even)
2 –2 J = 5 3 6 / π ρ 2 sin ( 2 θ ) $ \sqrt{6/\pi }\hspace{1em}{\rho }^2\mathrm{sin}(2\theta )$ Primary astigmatism (at 45°) (Odd, Odd)
2 2 J = 6 5 6 / π ρ 2 cos ( 2 θ ) $ \sqrt{6/\pi }\hspace{1em}{\rho }^2\mathrm{cos}(2\theta )$ Primary astigmatism (at 0°) (Even, Even)
3 –1 J = 7 7 8 / π ( 3 ρ 3 - 2 ρ ) sin ( θ ) $ \sqrt{8/\pi }\hspace{1em}(3{\rho }^3-2\rho )\mathrm{sin}(\theta )$ Primary coma (along yi) (Even, Odd)
3 1 J = 8 8 8 / π ( 3 ρ 3 - 2 ρ ) cos ( θ ) $ \sqrt{8/\pi }\hspace{1em}(3{\rho }^3-2\rho )\mathrm{cos}(\theta )$ Primary coma (along xi) (Odd, Even)
3 –3 J = 9 6 8 / π ρ 3 sin ( 3 θ ) $ \sqrt{8/\pi }\hspace{1em}{\rho }^3\mathrm{sin}(3\theta )$ Trefoil (at 30°) (Even, Odd)
3 3 J = 10 9 8 / π ρ 3 cos ( 3 θ ) $ \sqrt{8/\pi }\hspace{1em}{\rho }^3\mathrm{cos}(3\theta )$ Trefoil (at 0°) (Odd, Even)
4 0 J = 11 12 5 / π ( 6 ρ 4 - 6 ρ 2 + 1 ) $ \sqrt{5/\pi }\hspace{1em}(6{\rho }^4-6{\rho }^2+1)$ Primary spherical aberration (Even, Even)
4 2 J = 12 13 10 / π   ( 4 ρ 4 - 3 ρ 2 ) cos ( 2 θ ) $ \sqrt{10/\pi }\enspace (4{\rho }^4-3{\rho }^2)\mathrm{cos}(2\theta )$ Secondary astigmatism (at 0°) (Even, Even)
4 –2 J = 13 11 10 / π   ( 4 ρ 4 - 3 ρ 2 ) sin ( 2 θ ) $ \sqrt{10/\pi }\enspace (4{\rho }^4-3{\rho }^2)\mathrm{sin}(2\theta )$ Secondary astigmatism (at 45°) (Odd, Odd)
4 4 J = 14 14 10 / π   ρ 4 cos ( 4 θ ) $ \sqrt{10/\pi }\enspace {\rho }^4\mathrm{cos}(4\theta )$ Tetrafoil (at 0°) (Even, Even)
4 –4 J = 15 10 10 / π   ρ 4 sin ( 4 θ ) $ \sqrt{10/\pi }\enspace {\rho }^4\mathrm{sin}(4\theta )$ Tetrafoil (at 22.5°) (Odd, Odd)
5 1 J = 16 18 12 / π   ( 10 ρ 5 - 12 ρ 3 + 3 ρ ) cos ( θ ) $ \sqrt{12/\pi }\enspace (10{\rho }^5-12{\rho }^3+3\rho )\mathrm{cos}(\theta )$ Secondary coma (along xi) (Odd, Even)

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