We report the application of interferometry to the form measurement of a family of highly astigmatic optical surfaces. These measurements are based on a null test that employs a double-pass off-axis test arrangement with a tilted test surface and a reference sphere. This arrangement provides a perfect null test for an ellipsoid of revolution, or prolate spheroid. Its application is illustrated in detail in the presentation of results for the measurement of a specific family of eight differing surfaces that are incorporated into the K-Band Multi-Object Spectrometer Integral Field Unit. All surfaces measured here are sufficiently close to a prolate spheroid to justify its practical application. We discuss the application of the technique as a flexible low-cost approach for the generation of null interferograms in the measurement of a variety of complex surfaces.
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Form Definitions for the Eight Component Configurations, Expressed in Terms of Zernike Polynomialsa
Zernike
Formula
Config1
Config2
Config3
Config4
Config5
Config6
Config7
Config8
Z4
Z5
0.001
0.000
0.002
0.003
0.000
Z6
7.394
7.385
7.382
7.399
7.399
7.382
7.385
7.394
Z7
0.013
0.026
0.035
0.040
0.040
0.035
0.026
0.013
Z8
0.000
0.000
0.001
0.002
0.003
Z9
0.000
0.000
Z10
0.001
0.001
0.001
0.000
0.000
Each component is given as a root mean square value expressed in micrometers. The normalization radius is .
Table 2
Slow and Fast Axis Radii, Nominal Radii, and Tilt Angles for All Eight Component Types
Config Number
(mm)
(mm)
R (mm)
θ (Deg)
1
72.78
2
72.8
3
72.52
4
72.34
5
72.34
6
72.52
7
72.8
8
72.78
Table 3
Major Sources of Measurement Uncertainty by Origin and Broken Down into Zernike Polynomialsa
Zernike Polynomial
Error Source
Z4
Z5
Z6
Total
Fast axis error
0
5
0
5
Radius error
0.5
0
0.5
1
Turntable error
3
0
5
6
RSS sum
8
Numbering convention is as per Noll [25]. All figures are in nanometers RMS into a radius of .
Table 4
Form Errors in Nanometers RMS for All Eight Components into the Nominal Diameter Pupila
Component Configuration
Zernike Term
1
2
3
4
5
6
7
8
Z4
13
17
37
10
56
Z5
48
2
Z6
Z7
3
6
2
4
2
4
4
0
Z8
4
Z9
6
5
14
12
8
7
4
Z10
11
8
11
12
7
20
10
>Z10
11
19
9
10
13
11
19
12
Total
62
76
67
132
109
92
130
46
Form errors have been partitioned into Zernike polynomials. Nomenclature is as per Noll convention [25].
Table 5
Form Errors in Nanometers RMS for All Eight Components into Restricted Optical Footprinta
Component Configuration
Zernike Term
1
2
3
4
5
6
7
8
Z4
0
Z5
1
5
1
Z6
Z7
1
1
1
1
1
0
1
2
Z8
0
0
Z9
1
0
1
0
2
0
1
Z10
1
0
1
3
0
3
0
>Z10
9
8
6
9
6
7
8
13
Total
12
11
14
13
16
14
11
16
Footprint is elliptical . Errors are much smaller than those for the nominal diameter pupil. Form errors have been partitioned into Zernike polynomials. Nomenclature is as per Noll convention [25].
Table 6
Zernike Form Contributions for Outer Segment of the Proposed E-ELT Primary Telescope Mirrora
Zernike Term
Contribution
Z4
806144
Z5
Z6
Z7
Z8
Z9
27
Z10
7
Z11
1
Figures are in nanometers RMS.
Tables (6)
Table 1
Form Definitions for the Eight Component Configurations, Expressed in Terms of Zernike Polynomialsa
Zernike
Formula
Config1
Config2
Config3
Config4
Config5
Config6
Config7
Config8
Z4
Z5
0.001
0.000
0.002
0.003
0.000
Z6
7.394
7.385
7.382
7.399
7.399
7.382
7.385
7.394
Z7
0.013
0.026
0.035
0.040
0.040
0.035
0.026
0.013
Z8
0.000
0.000
0.001
0.002
0.003
Z9
0.000
0.000
Z10
0.001
0.001
0.001
0.000
0.000
Each component is given as a root mean square value expressed in micrometers. The normalization radius is .
Table 2
Slow and Fast Axis Radii, Nominal Radii, and Tilt Angles for All Eight Component Types
Config Number
(mm)
(mm)
R (mm)
θ (Deg)
1
72.78
2
72.8
3
72.52
4
72.34
5
72.34
6
72.52
7
72.8
8
72.78
Table 3
Major Sources of Measurement Uncertainty by Origin and Broken Down into Zernike Polynomialsa
Zernike Polynomial
Error Source
Z4
Z5
Z6
Total
Fast axis error
0
5
0
5
Radius error
0.5
0
0.5
1
Turntable error
3
0
5
6
RSS sum
8
Numbering convention is as per Noll [25]. All figures are in nanometers RMS into a radius of .
Table 4
Form Errors in Nanometers RMS for All Eight Components into the Nominal Diameter Pupila
Component Configuration
Zernike Term
1
2
3
4
5
6
7
8
Z4
13
17
37
10
56
Z5
48
2
Z6
Z7
3
6
2
4
2
4
4
0
Z8
4
Z9
6
5
14
12
8
7
4
Z10
11
8
11
12
7
20
10
>Z10
11
19
9
10
13
11
19
12
Total
62
76
67
132
109
92
130
46
Form errors have been partitioned into Zernike polynomials. Nomenclature is as per Noll convention [25].
Table 5
Form Errors in Nanometers RMS for All Eight Components into Restricted Optical Footprinta
Component Configuration
Zernike Term
1
2
3
4
5
6
7
8
Z4
0
Z5
1
5
1
Z6
Z7
1
1
1
1
1
0
1
2
Z8
0
0
Z9
1
0
1
0
2
0
1
Z10
1
0
1
3
0
3
0
>Z10
9
8
6
9
6
7
8
13
Total
12
11
14
13
16
14
11
16
Footprint is elliptical . Errors are much smaller than those for the nominal diameter pupil. Form errors have been partitioned into Zernike polynomials. Nomenclature is as per Noll convention [25].
Table 6
Zernike Form Contributions for Outer Segment of the Proposed E-ELT Primary Telescope Mirrora