Table I
Zernike Mode Definitions Used for Sensitivity Matrix Calculations (Unit Disk Normalization)
z1 = 2ρ cosθ | |
z2 = 2ρ sinθ | |
| Subaperture |
| ρ → ρ′ = x/r |
| θ → θ′ |
| Full Aperture |
| ρ → x/R |
| θ → θ |
| |
| |
Table II
Subaperture Zernike Mode Sensitivity Coefficients for Full Aperature Zernike Mode Aberrations
|
| 0 |
|
|
|
|
|
|
|
|
|
|
| 0 |
|
|
|
|
|
|
|
|
|
|
|
| 0 | 0 |
| 0 | 0 |
|
| 0 | 0 |
|
|
|
| 0 | 0 | 0 |
| 0 |
|
|
|
|
|
|
|
S (a,h) = | 0 | 0 | 0 | 0 |
|
|
|
|
|
|
|
|
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 |
|
|
|
| 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 |
|
|
|
| 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 |
|
|
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 |
|
|
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
|
|
|
| A1 | A2 | A3 | A4 | A5 | A6 | A7 | A8 | A9 | A10 |
| Full Aperture Mode |
Table III
Subaperture Placement Parameters for the Multiple Subaperture Flat Optical Test Configuration
Subaperture Number | Subaperture Offset | Offset Orientation Angle |
---|
k | hk | αk |
---|
|
---|
1 |
| 0° (0) |
2 |
| 60° (π/3) |
3 |
| 120° (2π/3) |
4 |
| 180° (π) |
5 |
| 240° (4π/3) |
6 |
| 300° (5π/3) |
7 | 0 | - |
Table IV
Full Aperture Zernike Mode inverse Covariance Matrix
|
| 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 |
|
|
| 0 |
| 0 | 0 | 0 | 0 |
| 0 | 0 | 0 |
|
|
| 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 |
|
|
|
| 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 |
|
|
HTWH = | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 |
|
|
|
| 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 |
|
|
| 0 |
| 0 | 0 | 0 | 0 |
| 0 | 0 | 0 |
|
|
| 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 |
|
|
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 |
|
|
| 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 |
|
|
|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
Table V
Symbolic Representation for the Nonzero Elements of the Matrix HTWH
Table VI
Symbolic Representation for the Full Aperture Zernike Coefficient Covariance Matrix
| | | | |
| | | | |
|
| | | | |
| | | |
| |
| | | | | | |
|
| | |
| | | | | | |
| | | |
| | | | | |
| | | | |
| | | | |
|
| | | | |
| | | |
| | | | | | |
| | |
| | | | | | | |
| |
| |
| | | | | | |
|
Table VII
Full Aperture Zernike Mode Estimation Accuracy as a Function of Subaperture Tilt Measurement Error (Ten-Full Aperture Modes Estimated Using Seven Subapertures with Ten Modes Each, σai = 0.006λ i > 3)
| FULL APERTURE (Standard Deviation wave) |
---|
|
|
---|
Subaperture Tilt | Total RSS | Focus | Astigmatism | Coma | Clover | Spherical | Tilt (single axis) |
---|
|
---|
0.0001λ | 0.0034λ | | | | | | |
0.0003 | 0.0035 | 0.0005 | 0.0003 | 0.0002 | 0.0003 | 0.0034 | .00036 |
0.001 | 0.0038 | | | | | | |
0.003 | 0.006 | 0.0017 | 0.0031 | 0.0021 | 0.0026 | 0.0034 | 0.0036 |
0.01 | 0.015 | | | | | | |
0.03 | 0.029 | 0.012 | 0.020 | 0.0087 | 0.012 | 0.0034 | 0.034 |
0.1 | 0.037 | | | | | | |
0.3 | 0.039 | | | | | | |
1.0 | 0.039 | 0.02 | 0.028 | 0.0094 | 0.013 | 0.0034 | 1.13 |
Table VIII
Full Aperture Zernike Mode Estimation Accuracy as a Function of Subaperture Tilt Measurement Error (Ten-Full Aperture Modes Estimated Using Seven Subapertures with Five Modes Each, σai = 0.006λ i > 3)
| FULL APERTURE (Standard Deviation wave) |
---|
|
|
---|
Subaperture Tilt | Total RSS | Focus | Astigmatism | Coma | Clover | Spherical | Tilt (single axis) |
---|
|
---|
0.0001λ | 0.0037λ | | | | | | |
0.0003 | 0.0038 | 0.0006 | 0.0003 | 0.0002 | 0.0021 | 0.0037 | 0.00035 |
0.001 | 0.0041 | | | | | | |
0.003 | 0.0062 | 0.0017 | 0.0032 | 0.0021 | 0.0027 | 0.0037 | 0.0036 |
0.01 | 0.0152 | | | | | | |
0.03 | 0.0292 | 0.0125 | 0.021 | 0.0087 | 0.012 | 0.0037 | 0.034 |
0.1 | 0.0377 | | | | | | |
0.3 | 0.0390 | | | | | | |
1.0 | 0.0392 | 0.02 | 0.028 | 0.0095 | 0.0135 | 0.0037 | 1.13 |
Table IX
Full Aperture Zernike Mode Estimation Accuracy as a Function of Subaperture Tilt Measurement Error (Ten-Full Aperture Modes Estimated Using Seven Subapertures with Ten Modes Each,
)
| FULL APERTURE (Standard Deviation wave) |
---|
|
|
---|
Subaperture Tilt | Total RSS | Focus | Astigmatism | Coma | Clover | Spherical | Tilt (single axis) |
---|
|
---|
0.0001λ | 0.0011 | | | | | | |
0.0003 | 0.0012 | 0.0002 | 0.0003 | 0.0002 | 0.0003 | 0.0011 | 0.00036 |
0.001 | 0.0019 | | | | | | |
0.003 | 0.0045 | 0.0016 | 0.003 | 0.0018 | 0.0023 | 0.0011 | 0.0035 |
0.01 | 0.0094 | | | | | | |
0.03 | 0.0118 | 0.0060 | 0.0088 | 0.0030 | 0.0042 | 0.0011 | 0.034 |
0.1 | 0.0123 | | | | | | |
0.3 | 0.0124 | | | | | | |
1.0 | 0.0124 | 0.0065 | 0.00091 | 0.0030 | 0.0042 | 0.0011 | 1.13 |
Table X
(HTWT)−1 for the Single Central Subaperture
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|
|
|
---|
1 |
| | | | |
| | | | |
|
|
2 | |
| | | | |
| | | |
|
|
3 | | |
| | | | | | | |
|
|
4 | | | |
| | | | | | |
|
|
5 | | | | |
| | | | | |
|
|
6 |
| | | | |
| | | | |
|
|
7 | |
| | | | |
| | | |
|
|
8 | | | | | | | |
| | |
|
|
9 | | | | | | | | |
| |
|
|
10 | | | | | | | | | |
|
RELATIVE TOTAL ESTIMATED rms ERROR