Donald G. Burkhard, David L. Shealy, and George L. Strobel, "Imaging characteristics of a conical primary, aspheric secondary x-ray telescope," Appl. Opt. 21, 3713-3718 (1982)
The imaging effectiveness of an x-ray telescope consisting of a simple conical primary reflector and an aspheric secondary reflector is compared with a paraboloidal primary, hyperboloidal secondary of the same dimensions (Wolter type II). The rms spot size for the Wolter type II is considerably smaller being 1/28 as large as that of the cone–asphere for incident collimated light 1 min off the optical axis and ⅙ as large for incident light 12 min off the optical axis for a typical configuration. The cone–asphere spot size, on the other hand, is one half as large as that for a hyperboloidal primary, asphere secondary telescope of the same dimensions.
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Average Value of the rms Ray Positions Divided by the Average Ray Intercept Distance (Xav) from the Optical Axis for Rays Incident at Off-Axis Angles 2, 4, 6, 8, 10, 12 min of arc for Various Telescope Configurationsa
Config.
ZPI
XPI
XSI
XSF
ZSI
CL
AL
Average rms/Xav
av. Lost (%)
av. apt rms
1
300
10
9
10
290
10
13.5
0.040
1.4
1400
2
300
10
6
7
260
10
13
0.235
2.4
220
3
300
10
3
3.7
230
10
11.7
0.526
4.7
66
4
300
20
18
20.4
280
10
28
0.034
.9
3450
5
300
20
18
19.6
280
16
45
0.036
2.8
4880
6
300
20
12
14.7
160
16
36
0.212
2.8
8600
7
300
20
6
7.8
160
16
25
0.50
6.1
3000
8
600
10
9
9.7
590
10
12
0.046
0
640
9
600
10
6
6.6
560
10
11.5
0.242
2.8
97
10
600
10
3
3.5
530
10
10.8
0.532
4.7
28
11
600
20
18
19
580
10
15
0.046
0
1300
12
600
20
12
13
520
10
13
0.243
6.1
189
13
600
20
18
19.6
580
16
22
0.043
2.4
2130
14
600
20
12
13.2
520
16
20
0.238
4.2
240
15
600
30
27
28.3
570
10
18
0.044
2.8
1800
16
600
30
18
19
480
10
16
0.241
7.9
291
apt/rms is obtained by calculating apt/rms for α = 2,4,6,8,10,12 min of arc and then averaging the six values. at/rms is larger for smaller α. For example, for configuration 4 above, apt/rms has the values 10.7,5.3,3.4,2.5,2.0,1.6 each times 103 for α = 2,4,6,8,10, and 12, respectively. Detailed performances of configurations 4 and 1 are given in Tables II and III.
Table II
Characteristics of Cone–Asphere A Telescope Designa
CONE–ASPHERE A
ZPI = 300
XSI = 18
XSF = 20
CL = 10
XPI = 20
ZSI = 280
apt area = 67.8
AL = 28
α (min of arc)
Xav
rms (× 10−2)
Typical peak illuminance
ξ (sec of arc)
Lost (%)
0
0
0
—
∞
∞
0
0
1
0.088
0.29
32
2.1
106
1.92
2.6
2
0.176
0.58
32
1.05
5 × 105
3.84
2.6
4
0.352
1.2
33
0.52
5 × 104
7.86
2.6
6
0.529
1.7
32
0.37
2 × 104
11.5
3.9
8
0.705
2.3
33
0.27
104
15.4
4.2
10
0.882
2.8
32
0.22
104
19.2
5.5
12
1.059
3.4
32
0.18
104
23.0
6.0
WOLTER TYPE II FOR COMPARISON WITH CONE-ASPHERE Å
ZPI = 300
XSI = 18
XSF = 20
CL = 10
XPI = 20
ZSI = 280
apt area = 29.9
AL = 24.1
α (min of arc)
Xav
rms (× 10−2)
Typical peak illuminance
ξ (sec of arc)
Lost (%)
0
0
0
–
∞
∞
0
0
1
0.091
0.010
1.1
30
5 × 108
0.066
2.9
2
0.181
0.025
1.4
12
1 × 107
0.168
2.9
4
0.363
0.072
2.0
4.1
5 × 106
0.48
3.7
6
0.545
0.147
2.7
2.0
5 × 106
0.972
4.7
8
0.726
0.252
3.5
1.2
1 × 106
1.68
5.5
10
0.908
0.386
4.2
0.8
1 × 106
2.52
6.0
12
1.01
0.543
5.4
0.5
5 × 105
3.89
7.9
Tables II and III show characteristics of two cone–asphere telescope designs compared with a Wolter type II. The angular resolution ξ tabulated in the next to last column is obtained as follows: Xav ≅ αF, where F is the focal length. The rms corresponds to an angular increments ξ in α so that rms ≅ ξF. Thus ξ = α (rms)/Xav. ξ is tabulated in seconds of arc.
Table III
Characteristics of Cone–asphere B Telescope Designa
CONE–ASPHERE B
ZPI = 300
XSI = 9
XSF = 10
CL = 10
XPI = 10
ZSI = 290
apt area = 31.4
AL = 13.5
α (min of arc)
Xav
rms (× 10−3)
Typical peak illuminance
ξ (sec of arc)
Lost (%)
0
0
0
—
∞
∞
0
0
2
0.180
7.2
40
0.43
5 × 104
4.8
2.6
4
0.359
14.5
40
0.22
2 × 104
9.6
2.6
6
0.539
21.7
40
0.14
104
14.4
2.6
8
0.719
29.0
40
0.11
4 × 103
19.2
2.6
12
1.08
43.2
40
0.07
3 × 103
28.8
3.9
40
3.60
143.6
40
0.02
4 × 102
96
12.8
WOLTER TYPE II FOR COMPARISON WITH CONE–ASPHERE B
ZPI = 300
XSI = 9
XSF = 10
CL = 10
XPI = 10
ZSI = 290
apt area = 15
AL = 13.5
α (min of arc)
Xav
rms (× 10−2)
Typical peak illuminance
ξ (sec of arc)
Lost (%)
0
0
0
—
∞
—
0
0
2
0.188
0.20
1
7.5
2 × 107
0.12
2.9
4
0.375
0.80
2
1.88
1 × 107
0.48
2.9
6
0.563
1.79
3
0.84
5 × 106
1.08
4.9
8
0.751
3.17
4
0.47
1 × 106
1.92
4.6
12
1.13
7.14
6
0.21
3 × 106
4.32
5.6
40
3.76
8.38
20
0.18
3 × 103
48.00
16.7
Tables II and III show characteristics of two cone–asphere telescope designs compared with a Wolter type II. The angular resolution ξ tabulated in the next to last column is obtained as follows: Xav ≅ αF, where F is the focal length. The rms corresponds to an angular increment ξ in a so that rms ≅ ξF. Thus ξ = α (rms)/Xav. ξ is tabulated in seconds of arc.
Table IV
a (min of arc)
1
28
52
0.0332
0.069
2
24.7
39
0.0332
0.075
4
16.3
22.6
0.0332
0.069
6
11.7
15.4
0.0326
0.066
8
9.1
11.9
0.0326
0.067
10
9.8
0.0326
0.065
12
6.2
Column 2: rms spot size for cone–asphere telescope A divided by rms spot size for Wolter type II having similar dimensions.
Column 3: rms spot size hyperboloid–asphere divided by rms spot size for Wolter type II having similar dimensions.
Column 4: rms spot size for cone–asphere A divided by average position of the image for various α.
Column 5: ratio of rms spot size for hyperboloid asphere to corresponding image position. Telescope has the same dimensions as the cone–asphere telescope of column 4.
Tables (4)
Table I
Average Value of the rms Ray Positions Divided by the Average Ray Intercept Distance (Xav) from the Optical Axis for Rays Incident at Off-Axis Angles 2, 4, 6, 8, 10, 12 min of arc for Various Telescope Configurationsa
Config.
ZPI
XPI
XSI
XSF
ZSI
CL
AL
Average rms/Xav
av. Lost (%)
av. apt rms
1
300
10
9
10
290
10
13.5
0.040
1.4
1400
2
300
10
6
7
260
10
13
0.235
2.4
220
3
300
10
3
3.7
230
10
11.7
0.526
4.7
66
4
300
20
18
20.4
280
10
28
0.034
.9
3450
5
300
20
18
19.6
280
16
45
0.036
2.8
4880
6
300
20
12
14.7
160
16
36
0.212
2.8
8600
7
300
20
6
7.8
160
16
25
0.50
6.1
3000
8
600
10
9
9.7
590
10
12
0.046
0
640
9
600
10
6
6.6
560
10
11.5
0.242
2.8
97
10
600
10
3
3.5
530
10
10.8
0.532
4.7
28
11
600
20
18
19
580
10
15
0.046
0
1300
12
600
20
12
13
520
10
13
0.243
6.1
189
13
600
20
18
19.6
580
16
22
0.043
2.4
2130
14
600
20
12
13.2
520
16
20
0.238
4.2
240
15
600
30
27
28.3
570
10
18
0.044
2.8
1800
16
600
30
18
19
480
10
16
0.241
7.9
291
apt/rms is obtained by calculating apt/rms for α = 2,4,6,8,10,12 min of arc and then averaging the six values. at/rms is larger for smaller α. For example, for configuration 4 above, apt/rms has the values 10.7,5.3,3.4,2.5,2.0,1.6 each times 103 for α = 2,4,6,8,10, and 12, respectively. Detailed performances of configurations 4 and 1 are given in Tables II and III.
Table II
Characteristics of Cone–Asphere A Telescope Designa
CONE–ASPHERE A
ZPI = 300
XSI = 18
XSF = 20
CL = 10
XPI = 20
ZSI = 280
apt area = 67.8
AL = 28
α (min of arc)
Xav
rms (× 10−2)
Typical peak illuminance
ξ (sec of arc)
Lost (%)
0
0
0
—
∞
∞
0
0
1
0.088
0.29
32
2.1
106
1.92
2.6
2
0.176
0.58
32
1.05
5 × 105
3.84
2.6
4
0.352
1.2
33
0.52
5 × 104
7.86
2.6
6
0.529
1.7
32
0.37
2 × 104
11.5
3.9
8
0.705
2.3
33
0.27
104
15.4
4.2
10
0.882
2.8
32
0.22
104
19.2
5.5
12
1.059
3.4
32
0.18
104
23.0
6.0
WOLTER TYPE II FOR COMPARISON WITH CONE-ASPHERE Å
ZPI = 300
XSI = 18
XSF = 20
CL = 10
XPI = 20
ZSI = 280
apt area = 29.9
AL = 24.1
α (min of arc)
Xav
rms (× 10−2)
Typical peak illuminance
ξ (sec of arc)
Lost (%)
0
0
0
–
∞
∞
0
0
1
0.091
0.010
1.1
30
5 × 108
0.066
2.9
2
0.181
0.025
1.4
12
1 × 107
0.168
2.9
4
0.363
0.072
2.0
4.1
5 × 106
0.48
3.7
6
0.545
0.147
2.7
2.0
5 × 106
0.972
4.7
8
0.726
0.252
3.5
1.2
1 × 106
1.68
5.5
10
0.908
0.386
4.2
0.8
1 × 106
2.52
6.0
12
1.01
0.543
5.4
0.5
5 × 105
3.89
7.9
Tables II and III show characteristics of two cone–asphere telescope designs compared with a Wolter type II. The angular resolution ξ tabulated in the next to last column is obtained as follows: Xav ≅ αF, where F is the focal length. The rms corresponds to an angular increments ξ in α so that rms ≅ ξF. Thus ξ = α (rms)/Xav. ξ is tabulated in seconds of arc.
Table III
Characteristics of Cone–asphere B Telescope Designa
CONE–ASPHERE B
ZPI = 300
XSI = 9
XSF = 10
CL = 10
XPI = 10
ZSI = 290
apt area = 31.4
AL = 13.5
α (min of arc)
Xav
rms (× 10−3)
Typical peak illuminance
ξ (sec of arc)
Lost (%)
0
0
0
—
∞
∞
0
0
2
0.180
7.2
40
0.43
5 × 104
4.8
2.6
4
0.359
14.5
40
0.22
2 × 104
9.6
2.6
6
0.539
21.7
40
0.14
104
14.4
2.6
8
0.719
29.0
40
0.11
4 × 103
19.2
2.6
12
1.08
43.2
40
0.07
3 × 103
28.8
3.9
40
3.60
143.6
40
0.02
4 × 102
96
12.8
WOLTER TYPE II FOR COMPARISON WITH CONE–ASPHERE B
ZPI = 300
XSI = 9
XSF = 10
CL = 10
XPI = 10
ZSI = 290
apt area = 15
AL = 13.5
α (min of arc)
Xav
rms (× 10−2)
Typical peak illuminance
ξ (sec of arc)
Lost (%)
0
0
0
—
∞
—
0
0
2
0.188
0.20
1
7.5
2 × 107
0.12
2.9
4
0.375
0.80
2
1.88
1 × 107
0.48
2.9
6
0.563
1.79
3
0.84
5 × 106
1.08
4.9
8
0.751
3.17
4
0.47
1 × 106
1.92
4.6
12
1.13
7.14
6
0.21
3 × 106
4.32
5.6
40
3.76
8.38
20
0.18
3 × 103
48.00
16.7
Tables II and III show characteristics of two cone–asphere telescope designs compared with a Wolter type II. The angular resolution ξ tabulated in the next to last column is obtained as follows: Xav ≅ αF, where F is the focal length. The rms corresponds to an angular increment ξ in a so that rms ≅ ξF. Thus ξ = α (rms)/Xav. ξ is tabulated in seconds of arc.
Table IV
a (min of arc)
1
28
52
0.0332
0.069
2
24.7
39
0.0332
0.075
4
16.3
22.6
0.0332
0.069
6
11.7
15.4
0.0326
0.066
8
9.1
11.9
0.0326
0.067
10
9.8
0.0326
0.065
12
6.2
Column 2: rms spot size for cone–asphere telescope A divided by rms spot size for Wolter type II having similar dimensions.
Column 3: rms spot size hyperboloid–asphere divided by rms spot size for Wolter type II having similar dimensions.
Column 4: rms spot size for cone–asphere A divided by average position of the image for various α.
Column 5: ratio of rms spot size for hyperboloid asphere to corresponding image position. Telescope has the same dimensions as the cone–asphere telescope of column 4.
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