Table 1
Fringe Zernike coefficients {α_{n}, n = 1,…, 37}, peaktovalley, RMS, and variance, provided by ZEMAX for the highly aberrated test lens. Unlisted coefficients are zero.
Index  Aberration Type [λ]  Design Value [λ] 

1
 Piston
 50.73020573

4
 Defocus
 75.17884184

9
 Spherical Aberration, Primary
 23.81690475

16
 Spherical Aberration, Secondary
 −0.60555615

25
 Spherical Aberration, Tertiary
 0.04113341

36
 Spherical Aberration, Quaternary
 −0.00287742

37
 Spherical Aberration,
12thorder Term
 −0.01310320

Peaktovalley [λ]
 N/A
 149.14956942

RMS [λ]
 N/A
 44.18240916

Variance [λ^{2}]  N/A  1952.08527937 
Table 2
Fringe Zernike Polynomials {Z_{n}, n = 1,…, 9, 16, 25, 36, 37}.
n  Fringe Zernike Polynomial  Aberration Type 

1
 1
 Piston

2
 ρcosθ
 Distortion  Tilt (xaxis)

3
 ρsinθ
 Distortion  Tilt (yaxis)

4
 2ρ ^{2} − 1
 Defocus  Field Curvature

5
 ρ ^{2}cos2θ
 Astigmatism, Primary (0° or 90°)

6
 ρ ^{2}sin2θ
 Astigmatism, Primary ( ± 45°)

7
 (3ρ ^{3} − 2ρ)cos2θ
 Coma, Primary (xaxis)

8
 (3ρ ^{3} − 2ρ)sin2θ
 Coma, Primary (yaxis)

9
 6ρ ^{4} − 6ρ ^{2} + 1
 Spherical Aberration, Primary

16
 20ρ ^{6} − 30ρ ^{4} + 12ρ ^{2} − 1
 Spherical Aberration, Secondary

25
 70ρ ^{8} − 140ρ ^{6} + 90ρ ^{4} − 20ρ ^{2} + 1
 Spherical Aberration, Tertiary

36
 252ρ ^{10} − 630ρ ^{8} + 560ρ ^{6} − 210ρ ^{4} + 30ρ ^{2} − 1
 Spherical Aberration, Quaternary

37  924ρ ^{12} − 2772ρ ^{10} + 3150ρ ^{8} − 1680ρ ^{6} + 420ρ ^{4} − 42ρ ^{2} + 1  Spherical Aberration, 12th order 
Table 3
Squareroot of the CRB for Fringe Zernike coefficients {α_{n}, n = 2,…, 9, 16} in the exit pupil of the highly aberrated test lens at λ = 0.6328 μm. Units are in waves λ.
Index  (CRB)^{1/2} [λ]  Index  (CRB)^{1/2} [λ] 

2
 6.6 × 10^{−10}
 7
 3.6 × 10^{−10}

3
 6.6 × 10^{−10}
 8
 3.6 × 10^{−10}

4
 1.2 × 10^{−10}
 9
 2.2 × 10^{−9}

5
 6.8 × 10^{−10}
 16
 1.5 × 10^{−10}

6  1.5 × 10^{−9}   
Table 4
Range in likelihood surface plots for Fringe Zernike coefficients {α_{n}, n = 2,…, 9, 16} in the exit pupil of the highly aberrated test lens. Units are in waves.λ.
Index  True value [λ]  Range 

2
 0
 ± 4λ

3
 0
 ± 4λ

4
 75.17884184
 ± 10λ

5
 0
 ± 4λ

7
 0
 ± 2λ

8
 0
 ± 2λ

9
 23.81690475
 ± 4λ

16  −0.60555615  ± λ/2 
Table 5
ML estimates of wavefront parameters for the highly aberrated test lens at λ = 0.6328 μm, including their standard deviations and the starting point in the search. Units are in waves λ.
Index  True Value [λ]  Start Point [λ]  Estimate [λ] 

2
 0
 2.33765863
 0.11 ± 0.34

3
 0
 3.67593941
 −0.09 ± 0.30

4
 75.17884184
 81.29481398
 75.5 ± 1.0

5
 0
 −3.71430657
 −0.09 ± 0.28

6
 0
 2.79303444
 0.17 ± 0.23

7
 0
 1.73597299
 0.08 ± 0.33

8
 0
 −1.81531443
 −0.10 ± 0.36

9
 23.81690475
 20.60294575
 23.58 ± 0.59

16  −0.60555615  −0.29345783  −0.604 ± 0.047 
Table 6
Fringe Zernike coefficients {α_{n}, n = 1,…, 37}, peaktovalley, RMS, and variance, provided by ZEMAX for the highly aberrated test lens. Unlisted coefficients are zero.
Index  Aberration Type [λ]  Design Value [λ] 

1
 Piston
 10.15431626

4
 Defocus
 15.26099035

9
 Spherical Aberration, Primary
 5.12576917

16
 Spherical Aberration, Secondary
 0.01863364

25
 Spherical Aberration, Tertiary
 −0.00048193

36
 Spherical Aberration, Quaternary
 −0.00002109

37
 Spherical Aberration,
12thorder Term
 −0.00000061

Peaktovalley [λ]
 N/A
 30.55920575

RMS [λ]
 N/A
 8.99563403

Variance [λ^{2}]  N/A  80.92143161 
Table 7
Squareroot of the CRB for Fringe Zernike coefficients {α_{n}, n = 2,…, 9, 16} in the exit pupil of the highly aberrated test lens at λ = 0.6328 μm. Units are in waves λ.
Index  (CRB)^{1/2} [λ]  Index  (CRB)^{1/2} [λ] 

2
 8.3 × 10^{−9}
 7
 5.7 × 10^{−9}

3
 8.3 × 10^{−9}
 8
 5.7 × 10^{−9}

4
 4.2 × 10^{−9}
 9
 3.5 × 10^{−9}

5
 3.9 × 10^{−9}
 16
 2.4 × 10^{−9}

6  3.9 × 10^{−9}   
Table 8
ML estimates of wavefront parameters for the spherical test lens at λ = 0.6328 μm, including their standard deviations. Design values were used as a starting point in the search. Ranges in the search space were relative to the design values. Units are in waves λ.
Index  Design Value [λ]  Range  ML Estimate [λ] 

2
 0
 ± 10λ
 0.10 ± 0.30

3
 0
 ± 10λ
 −0.13 ± 0.28

4
 15.26099035
 ± 20λ
 15.57 ± 0.64

5
 0
 ± 5λ
 −0.09 ± 0.19

6
 0
 ± 5λ
 0.13 ± 0.26

7
 0
 ± 5λ
 0.08 ± 0.18

8
 0
 ± 5λ
 −0.11 ± 0.23

9
 5.12576917
 ± 10λ
 4.92 ± 0.32

16  0.01863364  ± λ  0.0196 ± 0.0014 