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 Polynomials^{
a
}

Zernike

Formula

Config1

Config2

Config3

Config4

Config5

Config6

Config7

Config8

Z4

$\sqrt{3}(2{\rho}^{2}-1)$

$-6.261$

$-6.235$

$-6.245$

$-6.266$

$-6.266$

$-6.245$

$-6.235$

$-6.261$

Z5

$\sqrt{6}{\rho}^{2}\mathrm{sin}2\theta $

0.001

0.000

0.002

0.003

$-0.003$

$-0.002$

0.000

$-0.001$

Z6

$\sqrt{6}{\rho}^{2}\mathrm{cos}2\theta $

7.394

7.385

7.382

7.399

7.399

7.382

7.385

7.394

Z7

$\sqrt{8}(3{\rho}^{3}-2\rho )\mathrm{sin}\theta $

0.013

0.026

0.035

0.040

0.040

0.035

0.026

0.013

Z8

$\sqrt{8}(3{\rho}^{3}-2\rho )\mathrm{cos}\theta $

$-0.003$

$-0.002$

$-0.001$

0.000

0.000

0.001

0.002

0.003

Z9

$\sqrt{8}{\rho}^{3}\mathrm{sin}3\theta $

0.000

$-0.007$

$-0.012$

$-0.015$

$-0.015$

$-0.012$

$-0.007$

0.000

Z10

$\sqrt{8}{\rho}^{3}\mathrm{cos}3\theta $

0.001

0.001

0.001

0.000

0.000

$-0.001$

$-0.001$

$-0.001$

Each component is given as a root mean square value expressed in micrometers. The normalization radius is $2.6\text{\hspace{0.17em}}\mathrm{mm}$.

Table 2

Slow and Fast Axis Radii, Nominal Radii, and Tilt Angles for All Eight Component Types

Config Number

${R}_{x}$ (mm)

${R}_{y}$ (mm)

R (mm)

θ (Deg)

1

$-945$

$-84.9$

$-292.3$

72.78

2

$-963$

$-85.2$

$-294.8$

72.8

3

$-951$

$-85.1$

$-293.3$

72.52

4

$-944$

$-84.9$

$-292.1$

72.34

5

$-944$

$-84.9$

$-292.1$

72.34

6

$-951$

$-85.1$

$-293.3$

72.52

7

$-963$

$-85.2$

$-294.8$

72.8

8

$-945$

$-84.9$

$-292.3$

72.78

Table 3

Major Sources of Measurement Uncertainty by Origin and Broken Down into Zernike Polynomials^{
a
}

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 $2.6\text{\hspace{0.17em}}\mathrm{mm}$.

Table 4

Form Errors in Nanometers RMS for All Eight Components into the Nominal $\mathsf{5.2}\text{\hspace{0.17em}}\mathsf{mm}$ Diameter Pupil^{
a
}

Component Configuration

Zernike Term

1

2

3

4

5

6

7

8

Z4

$-18$

13

$-1$

17

37

10

56

$-2$

Z5

48

$-41$

$-8$

$-87$

2

$-59$

$-29$

$-16$

Z6

$-28$

$-58$

$-63$

$-95$

$-101$

$-67$

$-109$

$-38$

Z7

3

6

2

4

2

4

4

0

Z8

$-9$

$-6$

$-8$

$-8$

$-6$

4

$-18$

$-15$

Z9

6

5

14

12

8

7

4

$-3$

Z10

11

8

11

12

7

$-7$

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 Footprint^{
a
}

Component Configuration

Zernike Term

1

2

3

4

5

6

7

8

Z4

0

$-2$

$-4$

$-4$

$-1$

$-8$

$-2$

$-2$

Z5

$-3$

$-5$

$-8$

$-1$

1

5

1

$-4$

Z6

$-6$

$-6$

$-10$

$-8$

$-14$

$-7$

$-7$

$-6$

Z7

1

1

1

1

1

0

1

2

Z8

0

$-1$

$-1$

0

$-1$

$-1$

$-3$

$-1$

Z9

1

0

1

0

2

$-2$

0

1

Z10

1

0

$-1$

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 $2.45\text{\hspace{0.17em}}\mathrm{mm}\times 0.95\text{\hspace{0.17em}}\mathrm{mm}$. Errors are much smaller than those for the nominal $5.2\text{\hspace{0.17em}}\mathrm{mm}$ 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 Mirror^{
a
}

Zernike Term

Contribution

Z4

806144

Z5

$-25889$

Z6

$-21964$

Z7

$-258$

Z8

$-558$

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 Polynomials^{
a
}

Zernike

Formula

Config1

Config2

Config3

Config4

Config5

Config6

Config7

Config8

Z4

$\sqrt{3}(2{\rho}^{2}-1)$

$-6.261$

$-6.235$

$-6.245$

$-6.266$

$-6.266$

$-6.245$

$-6.235$

$-6.261$

Z5

$\sqrt{6}{\rho}^{2}\mathrm{sin}2\theta $

0.001

0.000

0.002

0.003

$-0.003$

$-0.002$

0.000

$-0.001$

Z6

$\sqrt{6}{\rho}^{2}\mathrm{cos}2\theta $

7.394

7.385

7.382

7.399

7.399

7.382

7.385

7.394

Z7

$\sqrt{8}(3{\rho}^{3}-2\rho )\mathrm{sin}\theta $

0.013

0.026

0.035

0.040

0.040

0.035

0.026

0.013

Z8

$\sqrt{8}(3{\rho}^{3}-2\rho )\mathrm{cos}\theta $

$-0.003$

$-0.002$

$-0.001$

0.000

0.000

0.001

0.002

0.003

Z9

$\sqrt{8}{\rho}^{3}\mathrm{sin}3\theta $

0.000

$-0.007$

$-0.012$

$-0.015$

$-0.015$

$-0.012$

$-0.007$

0.000

Z10

$\sqrt{8}{\rho}^{3}\mathrm{cos}3\theta $

0.001

0.001

0.001

0.000

0.000

$-0.001$

$-0.001$

$-0.001$

Each component is given as a root mean square value expressed in micrometers. The normalization radius is $2.6\text{\hspace{0.17em}}\mathrm{mm}$.

Table 2

Slow and Fast Axis Radii, Nominal Radii, and Tilt Angles for All Eight Component Types

Config Number

${R}_{x}$ (mm)

${R}_{y}$ (mm)

R (mm)

θ (Deg)

1

$-945$

$-84.9$

$-292.3$

72.78

2

$-963$

$-85.2$

$-294.8$

72.8

3

$-951$

$-85.1$

$-293.3$

72.52

4

$-944$

$-84.9$

$-292.1$

72.34

5

$-944$

$-84.9$

$-292.1$

72.34

6

$-951$

$-85.1$

$-293.3$

72.52

7

$-963$

$-85.2$

$-294.8$

72.8

8

$-945$

$-84.9$

$-292.3$

72.78

Table 3

Major Sources of Measurement Uncertainty by Origin and Broken Down into Zernike Polynomials^{
a
}

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 $2.6\text{\hspace{0.17em}}\mathrm{mm}$.

Table 4

Form Errors in Nanometers RMS for All Eight Components into the Nominal $\mathsf{5.2}\text{\hspace{0.17em}}\mathsf{mm}$ Diameter Pupil^{
a
}

Component Configuration

Zernike Term

1

2

3

4

5

6

7

8

Z4

$-18$

13

$-1$

17

37

10

56

$-2$

Z5

48

$-41$

$-8$

$-87$

2

$-59$

$-29$

$-16$

Z6

$-28$

$-58$

$-63$

$-95$

$-101$

$-67$

$-109$

$-38$

Z7

3

6

2

4

2

4

4

0

Z8

$-9$

$-6$

$-8$

$-8$

$-6$

4

$-18$

$-15$

Z9

6

5

14

12

8

7

4

$-3$

Z10

11

8

11

12

7

$-7$

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 Footprint^{
a
}

Component Configuration

Zernike Term

1

2

3

4

5

6

7

8

Z4

0

$-2$

$-4$

$-4$

$-1$

$-8$

$-2$

$-2$

Z5

$-3$

$-5$

$-8$

$-1$

1

5

1

$-4$

Z6

$-6$

$-6$

$-10$

$-8$

$-14$

$-7$

$-7$

$-6$

Z7

1

1

1

1

1

0

1

2

Z8

0

$-1$

$-1$

0

$-1$

$-1$

$-3$

$-1$

Z9

1

0

1

0

2

$-2$

0

1

Z10

1

0

$-1$

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 $2.45\text{\hspace{0.17em}}\mathrm{mm}\times 0.95\text{\hspace{0.17em}}\mathrm{mm}$. Errors are much smaller than those for the nominal $5.2\text{\hspace{0.17em}}\mathrm{mm}$ 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 Mirror^{
a
}