Approximate formulas are derived for the axial coma resulting from tilt and decenter of a surface and for the spherical aberration resulting from a change in its axial position. These expressions include terms that represent aberrations induced by the subsystem preceding the surface in addition to other terms that are intrinsic contributions from the misaligned surface itself. This separation of the terms gives a simple method of designing a system that is insensitive to a misalignment at a given surface. The method is illustrated by applying it to a two-mirror astronomical telescope with corrector. Two examples are given—one for tilt and the other for despace. In both examples an appreciable reduction in the sensitivity is obtained. The limitations of these solutions and the problem of simultaneous correction for two types of misalignment are examined.
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The primary mirror had a diameter of 4 m; the EFL was 146.6 m.
Table 3
Misalignment Wave Aberrations (in units of ⊑ = 588 nm) for Tilt, Decenter, and Despace: Comparison among Different Calculation Methods
Misalignment Parameters
W
WI
WZ
WR
WSC
t = l mrad
2.170
−0.346
2.148
2.141
2.171
e = −1.751 mm
4.712
−0.346
4.664
4.646
4.713
δd = 10 mm
2.480
1.263
2.532
2.535
2.504
Table 4
Misalignment Wave Aberrations (in units of ⊑ = 588 nm) for Tilt, Decenter and Despace: Zonal Dependence of the Aberrations on the Aperture
Normalized Aperture
Tilt
Decenter
Despace
Wct
WZ
Wce
WZ
Wsd
WZ
1.0
2.170
2.148
4.712
4.664
2.480
2.532
0.8
1.111
1.104
2.413
2.397
1.018
1.043
0.5
0.271
0.271
0.589
0.588
0.156
0.160
Table 5
Dependence of the Aberrations (in units of ⊑) on the Misalignment Parameter: Tilt and Decenter
Aberration
W
WZ
t (mrad)
1
2.170
2.148
2
4.340
4.296
4
8.680
8.592
e (mm) (ec = t)
−1.751
4.712
4.664
−3.502
9.424
9.327
−7.004
18.848
18.652
Table 6
Dependence of the Aberrations (in units of ⊑) on the Misalignment Parameter: Despace
δd (mm)
W
WZ
F/N
δ(BFL)
0
0.000
0.000
36.7
0
5
1.240
1.250
39.7
997
10
2.480
2.532
43.4
2177
20
4.960
5.187
53.1
5331
40
9.920
10.878
96.4
19352
−10
−2.480
−2.411
31.7
−1593
Table 7
Design Data of a Catadioptric System Corrected for Spherical Aberration, OSC and Wcta
Surface
Component
Radius of Curvature (mm)
Separation (mm)
Conic Constant
1
Primary mirror
−21,621.6216
−10,000.0000
−1.08200
2
Secondary mirror
−1763.7463
6,359.4098
−2.89953
3
Lens (BK7)
−372.5302
15.0000
4
−515.1174
7.7638
5
Lens (BK7)
441.2292
33.8176
6
611.2117
3,996.7927
7
Image plane
—
EFL, 146.6 m; EPD, 4 m; λ = 588 nm.
Table 8
Evaluations of a Telescope with Reduced Sensitivity to Tilt (Wave Aberrations in Units of ⊑): Wave Aberrations for 1-mrad Tilt and Different Values of the Aperture
Normalized Pupil Coordinates
Corrected Telescope
Reference Telescope Wz
Wct
WI
Wz
WR
1.0
−0.0134
−2.562
−0.0459
−0.0527
2.148
0.8
−0.0063
−1.311
−0.0195
−0.0217
1.104
0.5
−0.0014
−0.320
−0.0036
−0.0038
0.271
Table 9
Evaluations of a Telescope with Reduced Sensitivity to Tilt (Wave Aberrations in Units of ⊑): Ray Trace Data of the Corrected System
Normalized Pupil Coordinates
h
ni
x − h
OSC
δWOSC
1.0
150
−0.99954
0.753
−6.01 10−5
−0.0307
0.8
120
−0.79963
0.386
−3.98 10−5
−0.0162
0.5
75
−0.49977
0.094
−1.59 10−5
−0.0041
Table 10
Evaluations of a Telescope with Reduced Sensitivity to Tilt (Wave Aberrations in Units of λ): rms Wave-Front Error (Wrms) for 1-mradTilt
Corrected System
Reference System
Untilted
Tilted
Untilted
Tilted
0.003
0.011
0.004
0.256
Table 11
Evaluations of a Telescope with Reduced Sensitivity to Tilt: Polychromatic Encircled Energy in Millimeters (Geometrical Point-Spread Function)
Energy Percentage
Corrected System
Reference System
Untilted
Tilted
Untilted
Tilted
50
0.0030
0.0050
0.0063
0.0928
80
0.0060
0.0113
0.0121
0.1745
90
0.0065
0.0125
0.0155
0.2116
Table 12
Telescope with Reduced Sensitivity to Despace: Corrected System with EFL = 49,035 mm
Surface
Component
Radius of Curvature
Separation
Conic Constant
1
Primary mirror
−26,666.6667
−10,000.0000
0.11480
2
Secondary mirror
−8748.9064
9622.9133
7.33700
3
Lens(BK7)
−379.6312
30.0000
—
4
−486.9341
186.8292
0.31958
5
Lens(BK7)
−793.4003
55.0000
—
6
−557.3587
4056.5717
—
7
Image plane
—
—
—
Table 13
Telescope with Reduced Sensitivity to Despace: Reference System with EFL = 49 035 mm
Surface
Component
Radius of Curvature
Separation
Conic Constant
1
Primary mirror
−26666.6667
−10000.0000
−1.05100
2
Secondary mirror
−9156.4359
12258.9257
−3.57400
3
Image plane
—
—
—
Table 14
Primary Spherical Aberration W40 Normalized Coma W31 and Wsd for a Two-Mirror System with Wct = Wce = 0a
BFL (m)
h2/h1
0.05
0.15
0.25
0.35
8.0
W40
−77.6
−188.1
−245.5
−257.1
W31/H
−4.6 × 10−3
−3.9 × 10−3
−3.5 × 10−3
−3.2 × 10−3
Wsd
3.23
2.91
2.50
2.06
14.0
W40
−76.5
−181.0
−231.6
−241.1
W31/H
−4.5 × 10−3
−3.5 × 10−3
−2.8 × 10−3
−2.2 × 10−3
Wsd
3.07
2.54
2.03
1.56
λ = 588 nm; H is the optical invariant.
Tables (14)
Table 1
Reference System Data: System Construction Parameters (mm)a
Component
Radius of Curvature
Separation
Conic Constant
Primary mirror
−21621.622
−10000.00
−1.000894
Secondary mirror
−1750.700
10997.067
−1.357460
Image plane
The primary mirror had a diameter of 4 m; the EFL was 146.6 m.
The primary mirror had a diameter of 4 m; the EFL was 146.6 m.
Table 3
Misalignment Wave Aberrations (in units of ⊑ = 588 nm) for Tilt, Decenter, and Despace: Comparison among Different Calculation Methods
Misalignment Parameters
W
WI
WZ
WR
WSC
t = l mrad
2.170
−0.346
2.148
2.141
2.171
e = −1.751 mm
4.712
−0.346
4.664
4.646
4.713
δd = 10 mm
2.480
1.263
2.532
2.535
2.504
Table 4
Misalignment Wave Aberrations (in units of ⊑ = 588 nm) for Tilt, Decenter and Despace: Zonal Dependence of the Aberrations on the Aperture
Normalized Aperture
Tilt
Decenter
Despace
Wct
WZ
Wce
WZ
Wsd
WZ
1.0
2.170
2.148
4.712
4.664
2.480
2.532
0.8
1.111
1.104
2.413
2.397
1.018
1.043
0.5
0.271
0.271
0.589
0.588
0.156
0.160
Table 5
Dependence of the Aberrations (in units of ⊑) on the Misalignment Parameter: Tilt and Decenter
Aberration
W
WZ
t (mrad)
1
2.170
2.148
2
4.340
4.296
4
8.680
8.592
e (mm) (ec = t)
−1.751
4.712
4.664
−3.502
9.424
9.327
−7.004
18.848
18.652
Table 6
Dependence of the Aberrations (in units of ⊑) on the Misalignment Parameter: Despace
δd (mm)
W
WZ
F/N
δ(BFL)
0
0.000
0.000
36.7
0
5
1.240
1.250
39.7
997
10
2.480
2.532
43.4
2177
20
4.960
5.187
53.1
5331
40
9.920
10.878
96.4
19352
−10
−2.480
−2.411
31.7
−1593
Table 7
Design Data of a Catadioptric System Corrected for Spherical Aberration, OSC and Wcta
Surface
Component
Radius of Curvature (mm)
Separation (mm)
Conic Constant
1
Primary mirror
−21,621.6216
−10,000.0000
−1.08200
2
Secondary mirror
−1763.7463
6,359.4098
−2.89953
3
Lens (BK7)
−372.5302
15.0000
4
−515.1174
7.7638
5
Lens (BK7)
441.2292
33.8176
6
611.2117
3,996.7927
7
Image plane
—
EFL, 146.6 m; EPD, 4 m; λ = 588 nm.
Table 8
Evaluations of a Telescope with Reduced Sensitivity to Tilt (Wave Aberrations in Units of ⊑): Wave Aberrations for 1-mrad Tilt and Different Values of the Aperture
Normalized Pupil Coordinates
Corrected Telescope
Reference Telescope Wz
Wct
WI
Wz
WR
1.0
−0.0134
−2.562
−0.0459
−0.0527
2.148
0.8
−0.0063
−1.311
−0.0195
−0.0217
1.104
0.5
−0.0014
−0.320
−0.0036
−0.0038
0.271
Table 9
Evaluations of a Telescope with Reduced Sensitivity to Tilt (Wave Aberrations in Units of ⊑): Ray Trace Data of the Corrected System
Normalized Pupil Coordinates
h
ni
x − h
OSC
δWOSC
1.0
150
−0.99954
0.753
−6.01 10−5
−0.0307
0.8
120
−0.79963
0.386
−3.98 10−5
−0.0162
0.5
75
−0.49977
0.094
−1.59 10−5
−0.0041
Table 10
Evaluations of a Telescope with Reduced Sensitivity to Tilt (Wave Aberrations in Units of λ): rms Wave-Front Error (Wrms) for 1-mradTilt
Corrected System
Reference System
Untilted
Tilted
Untilted
Tilted
0.003
0.011
0.004
0.256
Table 11
Evaluations of a Telescope with Reduced Sensitivity to Tilt: Polychromatic Encircled Energy in Millimeters (Geometrical Point-Spread Function)
Energy Percentage
Corrected System
Reference System
Untilted
Tilted
Untilted
Tilted
50
0.0030
0.0050
0.0063
0.0928
80
0.0060
0.0113
0.0121
0.1745
90
0.0065
0.0125
0.0155
0.2116
Table 12
Telescope with Reduced Sensitivity to Despace: Corrected System with EFL = 49,035 mm
Surface
Component
Radius of Curvature
Separation
Conic Constant
1
Primary mirror
−26,666.6667
−10,000.0000
0.11480
2
Secondary mirror
−8748.9064
9622.9133
7.33700
3
Lens(BK7)
−379.6312
30.0000
—
4
−486.9341
186.8292
0.31958
5
Lens(BK7)
−793.4003
55.0000
—
6
−557.3587
4056.5717
—
7
Image plane
—
—
—
Table 13
Telescope with Reduced Sensitivity to Despace: Reference System with EFL = 49 035 mm
Surface
Component
Radius of Curvature
Separation
Conic Constant
1
Primary mirror
−26666.6667
−10000.0000
−1.05100
2
Secondary mirror
−9156.4359
12258.9257
−3.57400
3
Image plane
—
—
—
Table 14
Primary Spherical Aberration W40 Normalized Coma W31 and Wsd for a Two-Mirror System with Wct = Wce = 0a
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