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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 20 — Oct. 2, 2006
  • pp: 9169–9177
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Azimuthal position error correction algorithm for absolute test of large optical surfaces

Hyug-Gyo Rhee, Yun-Woo Lee, and Seung-Woo Kim  »View Author Affiliations


Optics Express, Vol. 14, Issue 20, pp. 9169-9177 (2006)
http://dx.doi.org/10.1364/OE.14.009169


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Abstract

Absolute test needs test part rotation to separate errors of the interferometer itself from errors due to the test surfaces. At this time, previous absolute test algorithms assume no azimuthal position error during part rotation. For large optics whose diameters are 0.6 m and over, however, exact rotations are physically difficult. Motivated by this, we propose a new algorithm that adopts least squares technique to determine the true azimuthal positions of part rotation and consequently eliminates testing errors caused by rotation inaccuracy.

© 2006 Optical Society of America

1. Introduction

2. Basic theory

The resulting wavefront W from any interferometric optical testing is composed of two partial wavefront components of

W=T+P
(1)

where T is the systematic error of the instrument including the reference surface, while P is the surface of the part to be measured. The above simple linear superposition of the two wavefronts is not strictly true but generally valid if they are of small orders as usual in most cases of optical shop testing. To separate the two wavefronts from each other, P is rotated about the optical axis by some physical means while T remains stationary. Then, let Wj be the wavefront of W sampled when P is stationed at an azimuthal position of αj . The subscript j indicates the rotation index ranging from 0 to N-1, where α0 =0 and N is the total number of part rotations. T undergoing no changes vanishes in the difference wavefront Dj , which is intermediately defined as the subtraction of

Dj=WjW0=(T+Pj)(T+P0)=PjP0.
(2)

Now, with the intention of reconstruct P0 from Dj , the expression of Zernike polynomials is adopted to decompose P0 such as

P0=P(r,θ)=l,kRlk(r)[clkcos(kθ)+dlksin(kθ)].
(3)

where r and θ are the normalized radial and angular coordinates; Rlk(r) the radial polynomials; clk and dlk the coefficients of the angular terms. In line with P0 expressed in Eq. (3), the rotated wavefront Pj is also described as

Pj=P(r,θ+αj)=l,kRlk(r)[clkcos(kθ+αj)+dlksin(kθ+αj)]
=l,kRlk(r)[clkcos(kθ)+dlksin(kθ)]
whereclk=clkcos(kαj)+dlksin(kαj)and
dlk=dlkcos(kαj)clksin(kαj).
(4)

Equations (3) and (4) indicate that the Zernike coefficients clk and dlk comply with the well-known transformation rule of vector rotation when they are regarded as the two orthogonal magnitude components of a two-dimensional vector. Then, substituting both the Zernike expressions of P0 and Pj into Eq. (2) allows the coefficients of the difference wavefront Dj to be obtained such as Δclk≡clk-c’lk and Δdlk≡dlk-d’ lk. Therefore, once Dj have actually been sampled and fitted to solve for Δclk and Δdlk , the coefficients of the original part wavefront P0 are readily determined as

clk=12[Δclk+Δdlksin(kαj)(1cos(kαj))],anddlk=12[Δdlk+Δdlksin(kαj)(1cos(kαj))].
(5)

Now we proceed to the averaging algorithm[8

8. C. Evans and R. Kestner, “Test optics error removal,” Appl. Opt. 35, 1015–1021 (1996). [CrossRef] [PubMed]

, 11

11. R. E. Parks, L. Shao, and C. J. Evans, “Pixel-based absolute topography test for three flats,” Appl. Opt. 37, 5951–5956 (1998). [CrossRef]

, 16

16. Ulf Griesmann, “Three-flat test solutions based on simple mirror symmetry” Appl. Opt. 45, 5856–5865 (2006). [CrossRef] [PubMed]

] that takes multiple part rotations to give the arithmetic mean of the measured wavefronts as the solution. To explain its underlying principles, by substituting Eqs. (3) and (4) into Eq. (2), the difference wavefront is rearranged in the form of

Dj=P0[cos(kαj)1]+P˜0sin(kαj),
whereP˜0=l,kRlk(r)[dlkcos(kθ)clksin(kθ)].
(6)

Then, summing up all the difference wavefronts leads to the expression of

j=0N1Dj=P0[j=0N1cos(kαj)N]+P˜0j=0N1sin(kαj).
(7)

The key idea of the averaging algorithm is to take the azimuthal angles to be equally spaced such as αj=2πj/N, with the intention of making the most of the invariant properties of the sine and cosine harmonic functions of

j=0N1sin(kαj)=0forallk,and
j=0N1cos(kαj)=0forknotbeingintegermultipliesofN,otherwise
j=0N1cos(kαj)=N.
(8)

Thus, if both the sums of sine and cosine terms are zero, Eq. (7) leads to the final form of the averaging algorithm of

P0=1Nj=0N1Dj.
(9)

This result was first proposed by Evans and Kestner [8

8. C. Evans and R. Kestner, “Test optics error removal,” Appl. Opt. 35, 1015–1021 (1996). [CrossRef] [PubMed]

], and named the multi-step averaging algorithm. In comparison with the algorithm using only 180° rotation, this arithmetic algorithm improves calibration accuracy and achieves a low uncertainty because any sampling error in a single wavefront is averaged out. However the error due to the rotation inaccuracy is still remaining.

3. Least square algorithm

The key idea of the newly proposed algorithm is that the rotated angles αj are treated as additional unknowns together with the coefficients clk and dlk . Then their actual values are determined from the measured wavefronts Dj using least squares technique. Our new algorithm beings with decomposing the part wavefront in terms of the angular order in k. Letting L(k) be the maximum radial order to be considered for each k, the partial sum is made up in detail such as

pk(r,θ)=lL(k)Rlk(r)[clkcos(kθ)+dlksin(kθ)]=lL(k)ξlkZlk(r,θ)
whereZlk(r,θ)Rlk(r){cossin}(kθ)
(10)

In Eq. (10), ξlk denotes another form of Zernike coefficients. Similarly, the sampled wavefront difference Dj is also expressed as Dj (r, θ)=k=1K Djk (r, θ) in which

Djk(r,θ)=P0k(r,θ)[cos(kαj)1]+P˜0k(r,θ)sin(kαj)
=lL(k){ξ0lkZlk(r,θ)[cos(kαj)1]+ξ˜0lkZlk(r,θ)sin(kαj)}
=lL(k)XljkZlk(r,θ).
(11)

For convenience, the subscript i is newly introduced to replace the notation (r,θ) such as Dijk =lL(k) XljkZlik. Now, let D^ijk be the actually measured value of D^ijk . Then computation for Zernike fitting of D^ijk and subsequent partial summing of the coefficients with same order of k allows Xljk of Eq. (11) to be computed as X^ljk . In doing that, if the values of αj are not correctly estimated, the computed values of X^ljk never equal the true values of Xljk . The computational errors in X^ljk are arranged in the form of two cost functions, which are defined for each k such as

Elk=j=0N1{XljkX̂ljk}2=j=0N1{ξ0lk[cos(kαj)1]+ξ˜0lksin(kαj)X̂ljk}2,
(12)

and

Ejk=lL(k){XljkX̂ljk}2=lL(k){ξ0lk[cos(kαj)1]+ξ˜0lksin(kαj)X̂ljk}2.
(13)

The former Elk represents the partial sum of errors induced in the radial coefficients of Zernike fitting by inaccurate estimation of rotation angles αj . On the other hand, the latter Ejk is the partial sum of errors resulting in the j-th wavefront. The cost functions should be minimized to determine the true values of the unknowns of ξ0lk, ξ˜k0l and αj . The necessary conditions are derived as

αElkξ0lk=0,Elkξ˜0lk=0,Ejkcos(kαj)=0,andEjksin(kαj)=0.
(14)

The above conditions are arranged in the form of matrix equations such as

[j=0N1[cos(kαj)1]2j=0N1sin(kαj)[cos(kαj)1]j=0N1sin(kαj)[cos(kαj)1]j=0N1sin2(kαj)][ξ0lkξ̃0lk]=[j=0N1X̂ljk[cos(kαj)1]j=0N1X̂ljksin(kαj)]
(15)
[lL(k)[ξ0lk]2lL(k)ξ0lkξ˜0lklL(k)ξ0lkξ˜0lklL(k)[ξ˜0lk]2][cos(kαj)sin(kαj)]=[lL(k){X̂ljkξ0lk+[ξ0lk]2}lL(k){X̂ljkξ0lk+ξ0lkξ˜0lk}]
(16)

4. Simulation and experiment

Fig. 1. Comparison of simulation results when true values of αj are not equally spaced mistakenly as 0°, 61°, 121°, 179°, 241°, 299°. (a) Original wavefront generated for simulation with all the Zernike coefficients being 0.1λ for k=1-5. (b) Fringe map of the original wavefront (P-V: 1.118µm, rms: 0.110λ). (c) The wavefront error extracted by the 6-step averaging (P-V: 0.017µm, rms: 0.001λ). (d) The wavefront error computed by the least-squares algorithm (P-V: 0.001µm, rms: 3×10-7λ).
Fig. 2. Suppression capabilities of high frequency components when αj are intentionally taken as 0°, 61°, 121°, 179°, 241°, 299°. (a) Original wavefront generated for simulation with all the Zernike coefficients being 0.1λ for k=1-8 (P-V: 1.763µm, rms: 0.11λ). (b) The wavefront error extracted by the 6-step averaging (P-V: 0.409µm, rms: 0.001λ). (c) The wavefront error computed by the least squares algorithm (P-V: 0.012µm, rms: 2×10-5λ).

Figure 2 describes another case in which higher order instrument errors are dominant in the range of k=6 to 8 as shown in (a). If the number of part rotation is taken as 6, the averaging algorithm fails to remove the 6th angular harmonic error components as illustrated in (b). On the other hand, the least squares algorithm suppresses the higher order instrument errors even with the same number of part rotations, demonstrating that higher order surface irregularities of the part are examined accurately. Detailed numerical data for comparison are listed in the Fig. 2.

Fig. 3. Simulated result after random noise inserting.

Figure 3 shows the simulation result after random noise inserting in the original wavefront. The simulation procedure is here. 1) The original wavefront was generated with all the Zernike coefficients being 0.1λ for k=1-8 It has 1.763 µm in P-V value and 0.11 λ in RMS as show in Fig. 2(a). The diameter was 480 pixels. 2) Using the original wavefront, six rotated wavefronts were generated. The rotated angles were 0°, 60°, 120°, 180°, 240°, and 300°. 3) Using C++ code, six different random noise sets were automatically generated within the wavefront area. The P-V value of the noise was 0.01736 µm (1 % magnitude of the original wavefront) 4) Each random noise set (Step 3) was added on each rotated wavefront (Step 2) pixel by pixel. 5) We applied both algorithms, and calculated the residual errors due to the random noise. 6) We repeated the simulation procedure with different noise magnitude (2%, 3%, 4%, 5%, and 10% of the original wavefront.) The result is shown in Fig. 3. Our algorithm can reduce the random noise effect by 94.4 %.

To check our algorithm by experiment we tested 0.6 m spherical concave mirror with a Fizeau interferometer. Actual testing diameter was about 0.5 m, and the rotated angles αj were taken as 0°, 90°, 180°, 270° (4-step) instead of 6-step for experimental convenience. The least square algorithm gave us the estimated values of αj as 0°, 92.17°, 180.98°, 271.20°. Table 1 shows typical aberration coefficients.

Table 1. Typical aberration coefficients before and after absolute tests. The unit is λ (632.8 nm).

table-icon
View This Table

5. Conclusions

References and links

1.

D. Malacara, “Phase shifting interferometry,” in Optical Shop Testing, 2nd ed., (Wiley, New York, 1992), Chap.14.

2.

A. E. Jensen, “Absolute calibration method for laser Twyman-Green wave-front testing interferometers,” J. Opt. Soc. Am. 63, 1313A (1973).

3.

K. L. Shu, “Ray-trace analysis and data reduction methods for the Ritchey-Common test,” Appl. Opt. 22, 1879–1886 (1983). [CrossRef] [PubMed]

4.

B. S. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23, 379–383 (1984).

5.

K.-E. Elssner, R. Burow, J. Grzanna, and R. Spolaczyk, “Absolute sphericity measurement,” Appl. Opt. 28, 4649–4661 (1989). [CrossRef] [PubMed]

6.

G. Shulz and J. Grzanna, “Absolute flatness testing by the rotation method with optimal measuring error compensation,” Appl. Opt. 31, 3767–3780 (1992). [CrossRef]

7.

C. Ai and J. C. Wyant, “Absolute testing of flats using even and odd functions,” Appl. Opt. 32, 4698–4703 (1993). [CrossRef] [PubMed]

8.

C. Evans and R. Kestner, “Test optics error removal,” Appl. Opt. 35, 1015–1021 (1996). [CrossRef] [PubMed]

9.

P. Hariharan, “Interferometric testing of optical surfaces: absolute measurement of flatness,” Opt. Eng. 36, 2478–2481 (1997). [CrossRef]

10.

W. T. Estler, C. J. Evans, and L. Z. Shao, “Uncertainty estimation for multi-position form error metrology,” Prec. Eng. 21, 72–82 (1997). [CrossRef]

11.

R. E. Parks, L. Shao, and C. J. Evans, “Pixel-based absolute topography test for three flats,” Appl. Opt. 37, 5951–5956 (1998). [CrossRef]

12.

V. Greco, R. Tronconi, C. D. Vecchio, M. Trivi, and G. Molesini, “Absolute measurement of planarity with Fritz’s method: uncertainty evaluation,” Appl. Opt. 38, 2018–2027 (1999). [CrossRef]

13.

P. E. Murphy, T. G. Brown, and D. T. Moore, “Interference imaging for aspheric surface testing,” Appl. Opt. 39, 2122–2129 (2000). [CrossRef]

14.

K. R. Freischlad, “Absolute interferomtric testing based on reconstruction of rotational shear,” Appl. Opt. 401637–1648 (2001). [CrossRef]

15.

S. Reichelt, C. Pruss, and H. J. Tiziani, “Absolute interferometric test of aspheres by use of twin computer-generated holograms,” Appl. Opt. 42, 4468–4479 (2003). [CrossRef] [PubMed]

16.

Ulf Griesmann, “Three-flat test solutions based on simple mirror symmetry” Appl. Opt. 45, 5856–5865 (2006). [CrossRef] [PubMed]

17.

C. Evans, R. Hocken, and W. Estler, “Self-calibration: reversal, redundancy, error separation, and absolute testing,” Annals of the CIRP 45, 617–634 (1996). [CrossRef]

18.

S. Kim, H. S. Yang, Y. W. Lee, and S. W. Kim, “Merit Function regression method for efficient alignment control of two-mirror optical system,” Opt. Express (to be published). [PubMed]

19.

H. S. Yang, Y. W. Lee, J. B. Song, and I. W. Lee, “Null Hartmann test for fabrication of large aspheric surfaces,” Opt. Express 6, 1839–1847 (2005). [CrossRef]

OCIS Codes
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.3940) Instrumentation, measurement, and metrology : Metrology
(120.6650) Instrumentation, measurement, and metrology : Surface measurements, figure

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: August 22, 2006
Revised Manuscript: September 14, 2006
Manuscript Accepted: September 21, 2006
Published: October 2, 2006

Citation
Hyug-Gyo Rhee, Yun-Woo Lee, and Seung-Woo Kim, "Azimuthal position error correction algorithm for absolute test of large optical surfaces," Opt. Express 14, 9169-9177 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-20-9169


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References

  1. D. Malacara, "Phase shifting interferometry," in Optical Shop Testing, 2nd ed., (Wiley, New York, 1992), Chap.14.
  2. A. E. Jensen, "Absolute calibration method for laser Twyman-Green wave-front testing interferometers," J. Opt. Soc. Am. 63, 1313A (1973).
  3. K. L. Shu, "Ray-trace analysis and data reduction methods for the Ritchey-Common test," Appl. Opt. 22, 1879-1886 (1983). [CrossRef] [PubMed]
  4. B. S. Fritz, "Absolute calibration of an optical flat," Opt. Eng. 23, 379-383 (1984).
  5. K.-E. Elssner, R. Burow, J. Grzanna, and R. Spolaczyk, "Absolute sphericity measurement," Appl. Opt. 28, 4649-4661 (1989). [CrossRef] [PubMed]
  6. G. Shulz and J. Grzanna, "Absolute flatness testing by the rotation method with optimal measuring error compensation," Appl. Opt. 31, 3767-3780 (1992). [CrossRef]
  7. C. Ai and J. C. Wyant, "Absolute testing of flats using even and odd functions," Appl. Opt. 32, 4698-4703 (1993). [CrossRef] [PubMed]
  8. C. Evans, R. Kestner, "Test optics error removal," Appl. Opt. 35, 1015-1021 (1996). [CrossRef] [PubMed]
  9. P. Hariharan, "Interferometric testing of optical surfaces: absolute measurement of flatness," Opt. Eng. 36, 2478-2481 (1997). [CrossRef]
  10. W. T. Estler, C. J. Evans, and L. Z. Shao, "Uncertainty estimation for multi-position form error metrology," Prec. Eng. 21, 72-82 (1997). [CrossRef]
  11. R. E. Parks, L. Shao, and C. J. Evans, "Pixel-based absolute topography test for three flats," Appl. Opt. 37, 5951-5956 (1998). [CrossRef]
  12. V. Greco, R. Tronconi, C. D. Vecchio, M. Trivi, and G. Molesini, "Absolute measurement of planarity with Fritz’s method: uncertainty evaluation," Appl. Opt. 38, 2018-2027 (1999). [CrossRef]
  13. P. E. Murphy, T. G. Brown, and D. T. Moore, "Interference imaging for aspheric surface testing," Appl. Opt. 39, 2122-2129 (2000). [CrossRef]
  14. K. R. Freischlad, "Absolute interferomtric testing based on reconstruction of rotational shear," Appl. Opt. 401637-1648 (2001). [CrossRef]
  15. S. Reichelt, C. Pruss, and H. J. Tiziani, "Absolute interferometric test of aspheres by use of twin computer-generated holograms," Appl. Opt. 42, 4468-4479 (2003). [CrossRef] [PubMed]
  16. Ulf Griesmann, "Three-flat test solutions based on simple mirror symmetry" Appl. Opt. 45, 5856-5865 (2006). [CrossRef] [PubMed]
  17. C. Evans, R. Hocken, and W. Estler, "Self-calibration: reversal, redundancy, error separation, and absolute testing," Annals of the CIRP 45, 617-634 (1996). [CrossRef]
  18. S. Kim, H. S. Yang, Y. W. Lee, and S. W. Kim, "Merit Function regression method for efficient alignment control of two-mirror optical system," Opt. Express (to be published). [PubMed]
  19. H. S. Yang, Y. W. Lee, J. B. Song, and I. W. Lee, "Null Hartmann test for fabrication of large aspheric surfaces," Opt. Express 6, 1839-1847 (2005). [CrossRef]

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