## Null test of an off-axis parabolic mirror. I. Configuration with spherical reference wave and flat return surface

Optics Express, Vol. 17, Issue 5, pp. 3196-3210 (2009)

http://dx.doi.org/10.1364/OE.17.003196

Acrobat PDF (3046 KB)

### Abstract

We demonstrate the precise figure measurement of a one-inch (25.4 mm) diamond-turned 90° off-axis commercial-quality parabolic mirror. The test is carried out with a phase-shifting Fizeau interferometer fitted with a spherical reference surface, auxiliary components and a flat return mirror. We present a detailed and systematic appraisal of the necessary steps for alignment and calibration of the instrument and the alignment of the parabolic mirror. Alignment errors and interferometric sensitivity variations are characterised and corrected, and the results give some insight into the diamond-turning process.

© 2009 Optical Society of America

## 1. Introduction

4. J. F. Cuttino, A. C. Miller Jr., and D. E. Schinstock, “Performance optimization of a fast tool servo for single-point diamond turning machines,” IEEE Trans. Mechatronics **4**, 169–179 (1999). [CrossRef]

_{2}coating, and the figure error specification is ¼ wave rms (

*λ*= 632.8 nm). Application examples for off-axis parabolic mirrors can be found in Refs. [5

5. K. J. Dana and J. Wang, “Device for convenient measurement of spatially varying bidirectional reflectance,” J. Opt. Soc. Am. A **21**,1–12 (2004). [CrossRef]

6. H. G. Jenniskens, A. Bot, P. W. F. Dorlandt, W. van Essenberg, E. de Haas, and A. W. Kleyn, “An ultrahigh vacuum (UHV) apparatus to study the interaction between adsorbates and photons,” Meas. Sci. Technol. **8**, 1313–1322 (1997). [CrossRef]

7. R. E. Parks, C. J. Evans, and L. Shao, “Test of a slow off-axis parabola at its center of curvature,” Appl. Opt. **34**, 7174–7178 (1995). [CrossRef] [PubMed]

8. Y. Pi and P. J. Reardon, “Determining parent radius and conic of an off-axis segment interferometrically with a spherical reference wave,” Opt. Lett. **32**, 1063–1065 (2007). [CrossRef] [PubMed]

10. U. Griesmann, Q. Wang, J. Soons, and R. Carakos, “A simple ball averager for reference sphere calibrations,” Proc. SPIE **5869**, 58690s (2005). [CrossRef]

## 2. Calibrations

### 2.1 Instrument alignment

16. J. Burke, B. Oreb, B. Piatt, and B. Nemati, “Precision metrology of dihedral angle error in prisms and corner cubes for the Space Interferometry Mission,” Proc. SPIE **5869**, 58690W (2005). [CrossRef]

### 2.2 Transmission sphere

10. U. Griesmann, Q. Wang, J. Soons, and R. Carakos, “A simple ball averager for reference sphere calibrations,” Proc. SPIE **5869**, 58690s (2005). [CrossRef]

17. K. Creath and J. C. Wyant, “Absolute measurement of surface roughness,” Appl. Opt. **29**, 3823–3827 (1990). [CrossRef] [PubMed]

*σ*, a sufficiently large

_{ball}*N*will ensure that

*σ*is a simple measure for the repeatability of a single measurement, which is determined by taking two measurements

_{single}*meas*

_{1}and

*meas*

_{2}in immediate succession, subtracting them and scaling the resulting rms to represent one measurement. Alternatively, a larger number of measurements can be taken and averaged to construct a standard “

*all*”, from which the repeatability can be determined by

10. U. Griesmann, Q. Wang, J. Soons, and R. Carakos, “A simple ball averager for reference sphere calibrations,” Proc. SPIE **5869**, 58690s (2005). [CrossRef]

17. K. Creath and J. C. Wyant, “Absolute measurement of surface roughness,” Appl. Opt. **29**, 3823–3827 (1990). [CrossRef] [PubMed]

*σ*that we are calculating does not refer to a spatially uniform random error, but instead to a figure error, where the departures from a best-fit sphere may be highly localised, depending on the actual figure of the calibration and reference spheres.

_{ball}*N*

^{-0.45}, in good approximation of the expected

*N*

^{-0.5}relation. The final calibration data are shown in Fig. 3. The centre of the transmission sphere does not coincide with the centre of the aperture, for two subtle reasons: (i) the optical axes of transmission sphere and OAP only need to intersect at the common focus, but need not be collinear – when they are not lined up, a shift between the respective centres will occur; (ii) even if the axes were perfectly aligned, the horizontal offset would remain, because the ray incidence angles in the aperture of the OAP are not symmetrical about 45°. Indeed, they span a range from 37° to 51°, as explained in more detail in Section 3.4, and therefore the centres cannot coincide; on comparing Fig. 3(b) with Fig. 14(b), the horizontal offsets match quite well.

### 2.3 Return Flat

18. B. F. Oreb, D. I. Farrant, C. J. Walsh, G. Forbes, and P. S. Fairman, “Calibration of a 300-mm-aperture phase-shifting Fizeau interferometer,” Appl. Opt. **39**, 5161–5171 (2000). [CrossRef]

19. M. Küchel, “A new approach to solve the three flat problem,” Optik **112**, 381–391 (2001). [CrossRef]

## 3. Testing of the parabolic mirror

*t*,

*r*, and

*z*, as sketched in Fig. 5. These coordinates are vaguely based on cylindrical coordinates, where

*t*is tangential to the shell that forms the surface of the rotation paraboloid, and has no

*z*component. The

*r*direction is “radial”, and

*z*is the direction of the optical axis. The coordinate system is attached to the laboratory (i.e. a 3-axis translation stage), not the OAP, so that the geometry shown in Fig. 5 is only applicable after the OAP has initially been rotated into this system by using the tilt controls and the rotary stage on which the OAP is mounted.

### 3.1 Preliminary alignment

*t*must be adjusted. We are not yet in a position to go by fringes, but it is possible to adjust ∆

*t*almost perfectly in the first iteration. While ∆

*r*and ∆

*z*influence each other, and ∆

*t*influences both of them, ∆

*t*is not altered by adjustments of either ∆

*r*or ∆

*z*. This is true of all off-axis paraboloids, and thus we can state the general rule that ∆

*t*should always be optimised first.

*r*and ∆

*z*, which is strongest for 90° OAPs, where the slant of the surface is 45° on the optical axis. The circular profile along the parabolic rotation surface leads to a weak coupling between ∆

*t*and the other two parameters, which disappears when ∆

*t*is adjusted properly, as then all tangential planes to the OAP surface in its central line of symmetry (cf. Fig. 5) are parallel to the

*t*direction. In Fig. 7-1B, the ∆

*t*adjustment has been made and the astigmatism is now vertical. We next adjust ∆

*z*until the spot pattern becomes more circular, as in 1C. We now also start to see the six-ray pattern coming from the cube corner. After adjusting ∆

*r*to remove the strong focus error, we obtain the image shown in Fig. 7-2A. Due to the coupling between ∆

*r*and ∆

*z*, removing the focus error has brought some of the astigmatism back, so we start the next iteration.

*z*, we can already go by a fringe pattern that we adjust to become circular, as in 2C. At this stage, if the transmission sphere is not aligned properly as described in 2.1, the fringe pattern will not be concentric with the vertex of the cube corner [14]. Removing ∆

*r*again starts the last iteration, in the third row of Fig. 7. Again, a small amount of error in

*t*has become visible in 3A, which we have removed in 3B. Finally we need to minimise the astigmatism (or ∆

*z*), which has been done in 3C. The entire process from 1A to 3C can be finished in less than a minute from almost any initial condition, as demonstrated in Fig. 8 and Media 1.

20. E. W. Weisstein, “Curvature,” in *MathWorld* A. Wolfram Web Resource, http://mathworld.wolfram.com/Curvature.html.

*p*= 1/4

*f*= 1/(101.6 mm), and its radial rate of change, given by

*z*/∂

*r*=0.5, and simpler tests with best-fit reference wavefronts [7

7. R. E. Parks, C. J. Evans, and L. Shao, “Test of a slow off-axis parabola at its center of curvature,” Appl. Opt. **34**, 7174–7178 (1995). [CrossRef] [PubMed]

8. Y. Pi and P. J. Reardon, “Determining parent radius and conic of an off-axis segment interferometrically with a spherical reference wave,” Opt. Lett. **32**, 1063–1065 (2007). [CrossRef] [PubMed]

21. J. Burke, K. Wang, and A. Bramble, “Testing of a diamond-turned off-axis parabolic mirror,” Proc. SPIE **7063**, 706312 (2008). [CrossRef]

### 3.2 Final alignment

### 3.3 Corrections of alignment errors

*r*,

*z*, and

*t*from their perfect settings. Since the surface is not perfect, there will always be remaining fringes and it may be hard to judge what the best setting is for taking the measurements. In tests of plane surfaces, we usually subtract only tilt; in tests of spherical surfaces, we subtract tilt and power; but can we simply subtract astigmatism and coma in a test of an OAP and consider them to be alignment errors? In general, this is not the case. It has been found that for each degree of freedom of (mis)alignment, some coupling between different Zernike or Seidel terms exists, which can be modelled by error functionals [22, 23

23. B. Dörband and H. J. Tiziani, “Testing aspheric surfaces with computer-generated holograms: analysis of adjustment and shape errors,” Appl. Opt. **24**, 2604–2611 (1985). [CrossRef] [PubMed]

23. B. Dörband and H. J. Tiziani, “Testing aspheric surfaces with computer-generated holograms: analysis of adjustment and shape errors,” Appl. Opt. **24**, 2604–2611 (1985). [CrossRef] [PubMed]

24. T. Dresel, N. Lindlein, and J. Schwider, “Empirical strategy for detection and removal of misalignment aberrations in interferometry,” Optik **112**, 304–308 (2001). [CrossRef]

21. J. Burke, K. Wang, and A. Bramble, “Testing of a diamond-turned off-axis parabolic mirror,” Proc. SPIE **7063**, 706312 (2008). [CrossRef]

24. T. Dresel, N. Lindlein, and J. Schwider, “Empirical strategy for detection and removal of misalignment aberrations in interferometry,” Optik **112**, 304–308 (2001). [CrossRef]

*n*Zernike error terms created by misalignment in the

*k*

^{th}degree of freedom. We let

*n*run from 3 to

*N*=24, since we want to capture trefoil (Z

_{9}and Z

_{10}) and quatrefoil (Z

_{16}and Z

_{17}). We start with the power term and avoid including Z

_{1}and Z

_{2}(tilt), because tilt can also be associated with the return mirror and therefore eludes unambiguous compensation. We can then visualise the experimentally-determined measurement responses to deliberate displacements as error maps, as in Fig. 12. In the experiments, the actual misalignments for each degree of freedom (

*r*,

*z*and

*t*) were about half a wave; however, for the display in Fig. 12, we have normalised the principal Zernike errors to one wave, but removed them from the phase map so that the coupled second-order effects can be seen.

*m*is for “measured”. This gives us a solution for (∆

*r*, ∆

*z*, ∆

*t*) and a corrected wavefront map for the OAP. A well-aligned interferogram and the final optimised wavefront map are shown in Fig. 13.

*r*, ∆

*z*, ∆

*t*) = (0.0685, −0.0805, −0.0903) waves, so the final alignment has been within 1/10 fringe, which is quite good, but still not perfect. No Zernike terms other than tilt have been subtracted from Fig. 13(b); and after the above considerations it should be clear that removing isolated terms such as focus or astigmatism is now neither necessary nor allowed. The fitted displacement vector changes only in the third digit when

*N*= 15 Zernike coefficients are used, and in the second digit when only 8 coefficients are used; however, as we have seen, it is important to capture at least trefoil.

### 3.4 Correction for sensitivity errors

*h*(

*x*,

*y*) is the true surface height,

*w*(

*x*,

*y*) is the error map in waves (cf. Fig. 11–Fig. 13), and

*λ*is the wavelength. Since we are dealing with a paraboloid, the slope ∂

*z*/∂

*x*is linear in

*x*; the correction, however, is not, because it is determined by the cosine of the slope. For conceptual clarity, let us note that we are using the angle between the

*x*axis and the OAP’s surface tangent as a proxy for the angle between the

*z*axis (collimated beam direction) and the normal to the OAP’s surface here. It is this latter angle that determines the interferometric sensitivity.

*z*/∂

*x*, the difficulty arises that we are viewing the OAP surface along the

*x*axis (which we can take to be the

*r*direction in Fig. 5(a) in a polar co-ordinate system. This means that we do not have a linear correspondence of an image co-ordinate (here: horizontal) with the

*x*axis of the OAP. However, the correction to apply is still linear because for lower numerical apertures, the image co-ordinates in a spherical Fizeau interferogram are proportional to the ray angles, and the ray angles are equal to arctan(∂

*z*/∂

*x*), so that all we need to do is replace arctan(∂

*z*/∂

*x*) in Eq. (9) by the horizontal image co-ordinate, with proper scaling and offset. The fact that we are observing the OAP from its centre of symmetry also accounts for the fact that the surface slope is independent of the vertical image co-ordinate, which expresses only a rotation of the parabola. The correction map is shown in Fig. 14(a).

*r*<50.8 mm, ∂

*z*/∂

*x*<1, angle of incidence <45° and hence more sensitive measurement, corrected by values <1) on the left, and the “outer” part (r >50.8 mm, ∂

*z*/∂

*x*>1, angle of incidence >45° and hence less sensitive measurement, corrected by values >1) on the right side.

*r*will create focus error and a smaller amount of coma; and errors ∆

*z*and ∆

*t*will create astigmatism (0° and 45° respectively), and a smaller amount of trefoil error (of odd-even and even-odd symmetry, respectively). This effect provides a good explanation for the aberrations summarised in Fig. 12, as can be seen in simulated results in Fig. 15. Here, one wave of focus (Z

_{3}), 0° astigmatism (Z

_{4}), or 45° astigmatism (Z

_{5}), respectively, have been added, then the errors due to the non-constant sensitivity were simulated (taking the map of Fig. 14 (a) for division, not multiplication) and the main misalignment subtracted again to inspect the secondary effects.

## 4. Conclusion

*λ*/12, reaching the specification of

*λ*/4 rms very easily.

## References and links

1. | R. N. Wilson, |

2. | D. Malacara, |

3. | R. N. Shagam, R. E. Sladky, and J. C. Wyant, “Optical figure inspection of diamond-turned metal mirrors,” Opt. Eng. |

4. | J. F. Cuttino, A. C. Miller Jr., and D. E. Schinstock, “Performance optimization of a fast tool servo for single-point diamond turning machines,” IEEE Trans. Mechatronics |

5. | K. J. Dana and J. Wang, “Device for convenient measurement of spatially varying bidirectional reflectance,” J. Opt. Soc. Am. A |

6. | H. G. Jenniskens, A. Bot, P. W. F. Dorlandt, W. van Essenberg, E. de Haas, and A. W. Kleyn, “An ultrahigh vacuum (UHV) apparatus to study the interaction between adsorbates and photons,” Meas. Sci. Technol. |

7. | R. E. Parks, C. J. Evans, and L. Shao, “Test of a slow off-axis parabola at its center of curvature,” Appl. Opt. |

8. | Y. Pi and P. J. Reardon, “Determining parent radius and conic of an off-axis segment interferometrically with a spherical reference wave,” Opt. Lett. |

9. | R. E. Parks, C. J. Evans, and L. Shao, “Calibration of interferometer transmission spheres,” in |

10. | U. Griesmann, Q. Wang, J. Soons, and R. Carakos, “A simple ball averager for reference sphere calibrations,” Proc. SPIE |

11. | R. E. Parks, “Alignment of off-axis conic mirrors,” in |

12. | C. Bond and C. A. Pipan, “How to align an off-axis parabolic mirror,” Proc. SPIE |

13. | L. C. Maxey, W. B. Dress, J. Rogers, and K. W. Tobin, “Automated alignment system for aspheric mirrors,” Proc. SPIE |

14. | L. C. Maxey, “Automated interferometric alignment system for paraboloidal mirrors,” US Patent 5249033,1–14 (1993). |

15. | WYKO 6000 is a 150 mm aperture Fizeau interferometer made by WYKO Corp. now owned by Veeco Tucson, Inc., 2650 E. Elvira Rd., Tucson, AZ 85706, USA. |

16. | J. Burke, B. Oreb, B. Piatt, and B. Nemati, “Precision metrology of dihedral angle error in prisms and corner cubes for the Space Interferometry Mission,” Proc. SPIE |

17. | K. Creath and J. C. Wyant, “Absolute measurement of surface roughness,” Appl. Opt. |

18. | B. F. Oreb, D. I. Farrant, C. J. Walsh, G. Forbes, and P. S. Fairman, “Calibration of a 300-mm-aperture phase-shifting Fizeau interferometer,” Appl. Opt. |

19. | M. Küchel, “A new approach to solve the three flat problem,” Optik |

20. | E. W. Weisstein, “Curvature,” in |

21. | J. Burke, K. Wang, and A. Bramble, “Testing of a diamond-turned off-axis parabolic mirror,” Proc. SPIE |

22. | J. Schwider and R. Burow, “Wave aberrations caused by misalignments of aspherics and their elimination,” Opt. Appl. |

23. | B. Dörband and H. J. Tiziani, “Testing aspheric surfaces with computer-generated holograms: analysis of adjustment and shape errors,” Appl. Opt. |

24. | T. Dresel, N. Lindlein, and J. Schwider, “Empirical strategy for detection and removal of misalignment aberrations in interferometry,” Optik |

**OCIS Codes**

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.3930) Instrumentation, measurement, and metrology : Metrological instrumentation

(120.4800) Instrumentation, measurement, and metrology : Optical standards and testing

(220.1250) Optical design and fabrication : Aspherics

(110.2650) Imaging systems : Fringe analysis

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: December 12, 2008

Revised Manuscript: January 28, 2009

Manuscript Accepted: February 6, 2009

Published: February 17, 2009

**Citation**

Jan Burke, Kai Wang, and Adam Bramble, "Null test of an off-axis parabolic mirror.
I. Configuration with spherical reference wave
and flat return surface," Opt. Express **17**, 3196-3210 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-5-3196

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### References

- R. N. Wilson, Reflecting Telescope Optics (Springer, Heidelberg, 2004), Chap. 1.
- D. Malacara, Optical Shop Testing (John Wiley & Sons, New York, 1992), Appendix 2.
- R. N. Shagam, R. E. Sladky, and J. C. Wyant, "Optical figure inspection of diamond-turned metal mirrors," Opt. Eng. 16, 375-380 (1977).
- J. F. Cuttino, A. C. Miller Jr., and D. E. Schinstock, "Performance optimization of a fast tool servo for single-point diamond turning machines," IEEE Trans. Mechatronics 4, 169-179 (1999). [CrossRef]
- K. J. Dana and J. Wang, "Device for convenient measurement of spatially varying bidirectional reflectance," J. Opt. Soc. Am. A 21, 1-12 (2004). [CrossRef]
- H. G. Jenniskens, A. Bot, P. W. F. Dorlandt, W. van Essenberg, E. de Haas, and A. W. Kleyn, "An ultrahigh vacuum (UHV) apparatus to study the interaction between adsorbates and photons," Meas. Sci. Technol. 8, 1313-1322 (1997). [CrossRef]
- R. E. Parks, C. J. Evans, and L. Shao, "Test of a slow off-axis parabola at its center of curvature," Appl. Opt. 34, 7174-7178 (1995). [CrossRef] [PubMed]
- Y. Pi and P. J. Reardon, "Determining parent radius and conic of an off-axis segment interferometrically with a spherical reference wave," Opt. Lett. 32, 1063-1065 (2007). [CrossRef] [PubMed]
- R. E. Parks, C. J. Evans, and L. Shao, "Calibration of interferometer transmission spheres," in Optical Fabrication and Testing Workshop, Vol. 12 of 1998 OSA Technical Digest Series (Optical Society of America, 1998), pp. 80-83.
- U. Griesmann, Q. Wang, J. Soons, and R. Carakos, "A simple ball averager for reference sphere calibrations," Proc. SPIE 5869, 58690S (2005). [CrossRef]
- R. E. Parks, "Alignment of off-axis conic mirrors," in Optical Fabrication and Testing, OSA Technical Digest Series (Optical Society of America, 1980), paper TuB4.
- C. Bond and C. A. Pipan, "How to align an off-axis parabolic mirror," Proc. SPIE 1113, 236-248 (1989).
- L. C. Maxey, W. B. Dress, J. Rogers, and K. W. Tobin, "Automated alignment system for aspheric mirrors," Proc. SPIE 1776, 130-139 (1992).
- L. C. Maxey, "Automated interferometric alignment system for paraboloidal mirrors," US Patent 5249033, 1-14 (1993).
- WYKO 6000 is a 150 mm aperture Fizeau interferometer made by WYKO Corp. now owned by Veeco Tucson, Inc., 2650 E. Elvira Rd., Tucson, AZ 85706, USA.
- J. Burke, B. Oreb, B. Platt, and B. Nemati, "Precision metrology of dihedral angle error in prisms and corner cubes for the Space Interferometry Mission," Proc. SPIE 5869, 58690W (2005). [CrossRef]
- K. Creath and J. C. Wyant, "Absolute measurement of surface roughness," Appl. Opt. 29, 3823-3827 (1990). [CrossRef] [PubMed]
- B. F. Oreb, D. I. Farrant, C. J. Walsh, G. Forbes, and P. S. Fairman, "Calibration of a 300-mm-aperture phase-shifting Fizeau interferometer," Appl. Opt. 39, 5161-5171 (2000). [CrossRef]
- M. Küchel, "A new approach to solve the three flat problem," Optik 112, 381-391 (2001). [CrossRef]
- E. W. Weisstein, "Curvature," in MathWorld A. Wolfram Web Resource, http://mathworld.wolfram.com/Curvature.html.
- J. Burke, K. Wang, and A. Bramble, "Testing of a diamond-turned off-axis parabolic mirror," Proc. SPIE 7063, 706312 (2008). [CrossRef]
- J. Schwider and R. Burow, "Wave aberrations caused by misalignments of aspherics and their elimination," Opt. Appl. 9, 33-38 (1979).
- B. Dörband and H. J. Tiziani, "Testing aspheric surfaces with computer-generated holograms: analysis of adjustment and shape errors," Appl. Opt. 24, 2604-2611 (1985). [CrossRef] [PubMed]
- T. Dresel, N. Lindlein, and J. Schwider, "Empirical strategy for detection and removal of misalignment aberrations in interferometry," Optik 112, 304-308 (2001). [CrossRef]

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