J. R. Fienup, J. C. Marron, T. J. Schulz, and J. H. Seldin, "Hubble Space Telescope characterized by using phase-retrieval algorithms," Appl. Opt. 32, 1747-1767 (1993)
We describe several results characterizing the Hubble Space Telescope from measured point spread functions by using phase-retrieval algorithms. The Cramer–Rao lower bounds show that point spread functions taken well out of focus result in smaller errors when aberrations are estimated and that, for those images, photon noise is not a limiting factor. Reconstruction experiments with both simulated and real data show that the calculation of wave-front propagation by the retrieval algorithms must be performed with a multiple-plane propagation rather than a simple fast Fourier transform to ensure the high accuracy ruired. Pupil reconstruction was performed and indicates a misalignment of the optical axis of a camera relay telescope relative to the main telescope. After we accounted for measured spherical aberration in the relay telescope, our estimate of the conic constant of the primary mirror of the HST was −1.0144.
You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Figure files are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Article tables are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.
Propagations involving x3′ include the imaginary pair of lenses before and after the PC obscurations. The pyramid is included in propagation x1 → x2.
Table 14
Example Evaluation of Spatial Scale Factors for λ = 889 nm and N = 256
Plane
In terms of Δxprev
In terms of Δx4
Δxk (mm)
CCD x4
0.01524
PC obscuration x3
0.2537
Pyramid x2
0.01226
OTA pupil x1
16.23
OTA pupil x1 (single-FFT case)
16.23
Table 15
Zernike Coefficients (Micrometer rms Wave-Front Error) for HARP1A images PC-6 F889N_P2 and PC-6 F889N_Q2
j
P2 Single
P2Multi(11)
P2 Multi (22)
New Parameter P2 Multi (11)
Q2 Single
4
−2.212
−2.227
−2.223
−2.306
0.73
5
−0.018
−0.003
0.006
−0.003
0.06
6
−0.025
0.025
0.026
0.031
−0.02
7
0.004
0.001
0.005
−0.001
0.01
8
0.017
0.010
0.009
0.013
0.06
9
−0.022
−0.020
−0.009
−0.020
−0.00
10
0.002
0.008
0.010
0.005
0.01
11
−0.280
−0.292
−0.295
−0.299
−0.281
12
0.008
(n/a)
(n/a)
0.01
16
−0.009
(n/a)
−0.004
(n/a)
0.01
20
0.006
(n/a)
(n/a)
0.00
22
0.005
0.006
0.007
0.008
0.04
Conic κ =
−1.0146
−1.0151
−1.0152
−1.0154
−1.0146
Root-mean-squared error =
0.1583
0.1352
0.1353
0.1428
0.2508
Table 16
Effect of the Plate Scale on Zernike Coefficients for Image PC-6 F889N_P2 (Micrometer rms Wave-Front Error)a
s (pixels/m)
With BQ = −0.000068
With BQ = −0.00005
With BQ = −0.000054
58.00
60.04
62.00
62.00
61.74
Plate scale (arcsec/pixel)
41.54
43.00
44.40
44.40
44.20
j = 4
−2.213
−2.227
−2.311
−2.320
−2.306
5
−0.028
−0.003
−0.001
−0.000
−0.003
6
−0.025
0.025
0.027
0.026
0.031
7
0.004
0.001
0.001
−0.001
−0.001
8
0.021
0.010
0.014
0.014
0.013
9
−0.017
−0.020
−0.023
−0.022
−0.020
10
0.005
0.008
0.002
0.003
0.005
11
−0.291
−0.292
−0.302
−0.299
−0.299
22
−0.001
0.006
0.007
0.008
0.008
Conic κ =
−1.0151
−1.0151
−1.0155
−1.0154
−1.0154
Root-mean-squared err =
0.1545
0.1442
0.1413
0.1415
0.1428
Using multiple-plane propagation, fit coefficients 1–11,22, N = 256, PSF weighted by diameter-220 circle, PC obscuration shifted by −4.25, −3.25 pixels.
Table 17
Modified Zernike Polynomials
j
Normalization Factor
Polynomial
2
1.8992573
x
3
1.8992573
y
4
3.8874443
(r2 − 0.554450)
5
2.3137662
(x2−y2)
6
2.3137662
2xy
7
8.3345629
x(r2 − 0.673796)
8
8.3345629
y(r2 − 0.673796)
9
2.6701691
x(x2 − 3y2)
10
2.6701691
y(3x2 − y2)
11
16.895979
(r4 − 1.108900r2 + 0.241243)
12
12.033645
(x2 − y2)(r2 − 0.750864)
13
12.033645
2xy(r2 − 0.750864)
14
2.9851527
(r4 − 8x2y2)
15
2.9851527
4xy(x2 − y2)
16
36.321412
x(r4 − 1.230566r2 + 0.323221)
17
36.321412
y(r4 − 1.230566r2 + 0.323221)
18
16.372202
x(x2 − 3y2)(r2 − 0.800100)
19
16.372202
y(3x2- y2)(r2 − 0.800100)
20
3.2700486
x(x4 − 10x2y2 + 5y4)
21
3.2700486
y(5x4 − 10x2y2 + y4)
22
74.782446
(r6 − 1.663350r4 + 0.803136r2 − 0.104406)
Tables (17)
Table 1
Normalized Lower Bounds on E[(âj − aj)2] for Aperture 1
j
a4 = 0.0
a4 = 1.0
a4 = 3.0
4
96.460854
40.463634
1.017365
5
0.032564
0.021035
0.000830
6
0.047582
0.034603
0.000560
7
0.026248
0.011118
0.000346
8
0.029077
0.009690
0.000346
9
0.013022
0.018398
0.000415
10
0.013116
0.016309
0.000423
11
4.385449
1.764731
0.045521
22
0.019322
0.007839
0.000197
Table 2
Normalized Lower Bounds on E[(âj − aj)2] for Aperture 2
j
a4 = 0.0
a4 = 1.0
a4 = 3.0
4
23.568291
43.709743
0.952042
5
0.024185
0.053788
0.000745
6
0.024064
0.048569
0.000090
7
0.013642
0.007164
0.000171
8
0.013959
0.006924
0.000174
9
0.006207
0.029878
0.000153
10
0.006186
0.030008
0.000157
11
0.979072
1.784631
0.039075
22
0.004710
0.008210
0.000181
Table 3
Normalized Lower Bounds on E[(âj − aj)2] for Aperture 1 when a4 is Known
j
a4 = 0.0
a4 = 1.0
a4 = 3.0
4
Known
Known
Known
5
0.032412
0.020977
0.000829
6
0.047395
0.034598
0.000560
7
0.025872
0.010668
0.000345
8
0.028869
0.009541
0.000345
9
0.013019
0.018265
0.000414
10
0.013089
0.016304
0.000423
11
0.025298
0.010237
0.000368
22
0.000003
0.000004
0.000000
Table 4
Normalized Lower Bounds on E[(âj − aj)2] for Aperture 2 when a4 is Known
Propagations involving x3′ include the imaginary pair of lenses before and after the PC obscurations. The pyramid is included in propagation x1 → x2.
Table 14
Example Evaluation of Spatial Scale Factors for λ = 889 nm and N = 256
Plane
In terms of Δxprev
In terms of Δx4
Δxk (mm)
CCD x4
0.01524
PC obscuration x3
0.2537
Pyramid x2
0.01226
OTA pupil x1
16.23
OTA pupil x1 (single-FFT case)
16.23
Table 15
Zernike Coefficients (Micrometer rms Wave-Front Error) for HARP1A images PC-6 F889N_P2 and PC-6 F889N_Q2
j
P2 Single
P2Multi(11)
P2 Multi (22)
New Parameter P2 Multi (11)
Q2 Single
4
−2.212
−2.227
−2.223
−2.306
0.73
5
−0.018
−0.003
0.006
−0.003
0.06
6
−0.025
0.025
0.026
0.031
−0.02
7
0.004
0.001
0.005
−0.001
0.01
8
0.017
0.010
0.009
0.013
0.06
9
−0.022
−0.020
−0.009
−0.020
−0.00
10
0.002
0.008
0.010
0.005
0.01
11
−0.280
−0.292
−0.295
−0.299
−0.281
12
0.008
(n/a)
(n/a)
0.01
16
−0.009
(n/a)
−0.004
(n/a)
0.01
20
0.006
(n/a)
(n/a)
0.00
22
0.005
0.006
0.007
0.008
0.04
Conic κ =
−1.0146
−1.0151
−1.0152
−1.0154
−1.0146
Root-mean-squared error =
0.1583
0.1352
0.1353
0.1428
0.2508
Table 16
Effect of the Plate Scale on Zernike Coefficients for Image PC-6 F889N_P2 (Micrometer rms Wave-Front Error)a
s (pixels/m)
With BQ = −0.000068
With BQ = −0.00005
With BQ = −0.000054
58.00
60.04
62.00
62.00
61.74
Plate scale (arcsec/pixel)
41.54
43.00
44.40
44.40
44.20
j = 4
−2.213
−2.227
−2.311
−2.320
−2.306
5
−0.028
−0.003
−0.001
−0.000
−0.003
6
−0.025
0.025
0.027
0.026
0.031
7
0.004
0.001
0.001
−0.001
−0.001
8
0.021
0.010
0.014
0.014
0.013
9
−0.017
−0.020
−0.023
−0.022
−0.020
10
0.005
0.008
0.002
0.003
0.005
11
−0.291
−0.292
−0.302
−0.299
−0.299
22
−0.001
0.006
0.007
0.008
0.008
Conic κ =
−1.0151
−1.0151
−1.0155
−1.0154
−1.0154
Root-mean-squared err =
0.1545
0.1442
0.1413
0.1415
0.1428
Using multiple-plane propagation, fit coefficients 1–11,22, N = 256, PSF weighted by diameter-220 circle, PC obscuration shifted by −4.25, −3.25 pixels.