Table 1
Expansion Coefficients
for a Radial Annular Polynomial
for
n
|
|
m
|
|
---|
0 | 0 | 0 | 1 |
1 | 1 | 1 |
|
2 | 0 | 0 |
|
2 | 2 | 0 |
|
2 | 2 | 2 |
|
3 | 1 | 1 |
|
3 | 3 | 1 |
|
3 | 3 | 3 |
|
4 | 0 | 0 |
|
4 | 2 | 0 |
|
4 | 4 | 0 |
|
4 | 4 | 0 |
|
4 | 4 | 2 |
|
4 | 4 | 4 |
|
5 | 1 | 1 |
|
5 | 3 | 1 |
|
5 | 5 | 1 |
|
5 | 3 | 3 |
|
5 | 5 | 3 |
|
5 | 5 | 5 |
|
6 | 0 | 0 |
|
6 | 2 | 0 |
|
6 | 4 | 0 |
|
6 | 6 | 0 |
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6 | 2 | 2 |
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6 | 4 | 2 |
|
6 | 6 | 2 |
|
6 | 4 | 4 |
|
6 | 6 | 4 |
|
6 | 6 | 6 |
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| | | |
Table 2
Expansion Coefficients
for a Polynomial
for
n
|
|
|
---|
0 | 0 | 1 |
1 | 1 |
ϵ
|
2 | 0 |
|
2 | 2 |
|
3 | 1 |
|
3 | 3 |
|
4 | 0 |
|
4 | 2 |
|
4 | 4 |
|
5 | 1 |
|
5 | 3 |
|
5 | 5 |
|
6 | 0 |
|
6 | 2 |
|
6 | 4 |
|
6 | 6 |
|
| | |
Table 3
Radial Part
of the Fourier Transform of a Zernike Annular Polynomial for
n
|
m
|
|
---|
0 | 0 |
|
1 | 1 |
|
2 | 0 |
|
2 | 2 |
|
3 | 1 |
|
3 | 3 |
|
4 | 0 |
|
4 | 2 |
|
4 | 4 |
|
5 | 1 |
|
5 | 3 |
|
5 | 5 |
|
6 | 0 |
|
6 | 2 |
|
6 | 4 |
|
6 | 6 |
|
| | |
Table 4
Variances of Zernike Annular Coefficients in Units of
for Various Values of Obscuration Ratio ϵ
ϵ
| 0 | 0.05 | 0.1 | 0.15 | 0.2 | 0.25 | 0.35 | 0.5 | 0.75 |
---|
| 0.44888 | 0.45000 | 0.45336 | 0.45893 | 0.46670 | 0.47660 | 0.50260 | 0.55610 | 0.67917 |
| 0.02322 | 0.02302 | 0.02245 | 0.02153 | 0.02033 | 0.01891 | 0.01560 | 0.01028 | 0.00292 |
| 0.02322 | 0.02328 | 0.02345 | 0.02375 | 0.02418 | 0.02475 | 0.02635 | 0.02997 | 0.03888 |
| 0.00619 | 0.00621 | 0.00624 | 0.00628 | 0.00630 | 0.00626 | 0.00593 | 0.00468 | 0.00162 |
| 0.00619 | 0.00621 | 0.00625 | 0.00633 | 0.00645 | 0.00660 | 0.00705 | 0.00815 | 0.01132 |
| 0.00245 | 0.00241 | 0.00227 | 0.00207 | 0.00183 | 0.00158 | 0.00109 | 0.00052 | 0.00007 |
| 0.00245 | 0.00246 | 0.00248 | 0.00251 | 0.00255 | 0.00260 | 0.00266 | 0.00247 | 0.00108 |
| 0.00245 | 0.00246 | 0.00248 | 0.00251 | 0.00256 | 0.00262 | 0.00280 | 0.00326 | 0.00480 |
Table 5
Covariances of Zernike Annular Coefficients in Units of
for Various Values of Obscuration Ratio ϵ
ϵ
| 0 | 0.05 | 0.1 | 0.15 | 0.2 | 0.25 | 0.35 | 0.5 | 0.75 |
---|
|
|
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| 0.00032 | 0.00030 | 0.00024 | 0.00017 | 0.00009 | 0.00002 |
|
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|
| 0.00365 | 0.01180 |
|
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| 0.00000 |
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Table 6
Mean Square Value
of the Residual Phase Error in Units of
When the First J Modes of Aberrations Are Corrected, for Several Values of ϵ
ϵ
| 0 | 0.25 | 0.5 | 0.75 |
---|
| 1.0324 | 1.0870 | 1.2461 | 1.5010 |
| 0.5835 | 0.6104 | 0.6900 | 0.8218 |
| 0.1347 | 0.1338 | 0.1339 | 0.1426 |
| 0.1115 | 0.1149 | 0.1236 | 0.1397 |
| 0.0882 | 0.0902 | 0.0937 | 0.1008 |
| 0.0650 | 0.0654 | 0.0637 | 0.0620 |
| 0.0588 | 0.0591 | 0.0590 | 0.0603 |
| 0.0526 | 0.0529 | 0.0543 | 0.0587 |
| 0.0464 | 0.0463 | 0.0462 | 0.0474 |
| 0.0402 | 0.0397 | 0.0380 | 0.0361 |
| 0.0378 | 0.0381 | 0.0375 | 0.0360 |
| 0.0353 | 0.0355 | 0.0350 | 0.0349 |
| 0.0329 | 0.0329 | 0.0325 | 0.0338 |
| 0.0304 | 0.0303 | 0.0293 | 0.0290 |
| 0.0280 | 0.0277 | 0.0260 | 0.0242 |
| 0.0268 | 0.0264 | 0.0244 | 0.0218 |
| 0.0256 | 0.0251 | 0.0229 | 0.0193 |
| 0.0244 | 0.0239 | 0.0215 | 0.0186 |
| 0.0232 | 0.0226 | 0.0201 | 0.0178 |
| 0.0220 | 0.0213 | 0.0185 | 0.0153 |
| 0.0208 | 0.0200 | 0.0169 | 0.0129 |
| 0.0202 | 0.0197 | 0.0168 | 0.0129 |
| 0.0195 | 0.0190 | 0.0159 | 0.0115 |
| 0.0189 | 0.0183 | 0.0151 | 0.0101 |
| 0.0182 | 0.0176 | 0.0142 | 0.0095 |
| 0.0176 | 0.0169 | 0.0134 | 0.0089 |
| 0.0169 | 0.0162 | 0.0125 | 0.0075 |
| 0.0162 | 0.0155 | 0.0117 | 0.0061 |
Table 7
Mode Shape Functions
and Cross-Mode Shape Functions
up to
for Zernike Circle Polynomials
Mode | Shape Function |
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