Abstract

The root mean square (rms) of the surface departure or wave-front deformation is an important value to extract from an optical test. The rms may be a tolerance that an optical fabricator is trying to meet, or it may be a parameter used by an optical designer to evaluate optical performance. Because the calculation of a rms involves a squaring operation, the rms of the measured data map is higher on average than the rms of the true surface or wave-front deformation, even if the noise is zero on average. The bias becomes significant as the scale of the noise becomes comparable to the true surface or wave-front deformation, as can be the case in the testing of ultraprecision optics. We describe and demonstrate a simple data analysis method to arrive at an unbiased estimate of the rms and a means to determine the measurement uncertainty.

© 2001 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 6th ed. (Pergamon, New York, 1980).
  2. C. J. Evans, “Absolute figure metrology of high precision optical surfaces,” Ph.D. dissertation (University of Birmingham, Birmingham, UK, 1996).
  3. R. E. Parks, L. Shao, C. J. Evans, “Pixel-based absolute test for three flats,” Appl. Opt. 37, 5951–5956 (1998).
    [CrossRef]
  4. See Ref. 2 and references therein, for example.
  5. D. Malacara, S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992).
  6. “U.S. guide to the expression of uncertainty in measurement,” (National Conference of Standards Laboratories, Boulder, Colorado, 1997), Sec. C.2.9.
  7. The starting signal-to-noise ratio level is somewhat arbitrary. We chose a starting noise level and values of N to cover a signal-to-noise ratio range slightly beyond what would be encountered in typical optical tests.
  8. Note that these bars should not be confused with estimates of the standard uncertainty for each mean variance; this uncertainty would be approximately a factor of 300 less than the height of the shaded bars.
  9. Ref. 6, Sec. 4.2.1.
  10. Ref. 6, Secs. 4.2 and 4.3.2.
  11. Ref. 6, Sec. 5.1.
  12. Ref. 6, Sec.3.3.
  13. Ref. 6, Sec. 4.2.3.
  14. R. E. Parks, C. J. Evans, L.-Z. Shao, “Calibration of interferometer transmission spheres,” poster presented at the Optical Fabrication and Testing Meeting, Kona, Hawaii, 8–11 June 1998.
  15. Commercially available products are identified to provide a complete description of the research. Such identification does not imply endorsement by the National Institute of Standards and Technology nor that they are necessarily best suited for the application.
  16. Ref. 6, Sec. E.4.3.

1998 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 6th ed. (Pergamon, New York, 1980).

DeVore, S. L.

D. Malacara, S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992).

Evans, C. J.

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

C. J. Evans, “Absolute figure metrology of high precision optical surfaces,” Ph.D. dissertation (University of Birmingham, Birmingham, UK, 1996).

R. E. Parks, C. J. Evans, L.-Z. Shao, “Calibration of interferometer transmission spheres,” poster presented at the Optical Fabrication and Testing Meeting, Kona, Hawaii, 8–11 June 1998.

Malacara, D.

D. Malacara, S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992).

Parks, R. E.

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

R. E. Parks, C. J. Evans, L.-Z. Shao, “Calibration of interferometer transmission spheres,” poster presented at the Optical Fabrication and Testing Meeting, Kona, Hawaii, 8–11 June 1998.

Shao, L.

Shao, L.-Z.

R. E. Parks, C. J. Evans, L.-Z. Shao, “Calibration of interferometer transmission spheres,” poster presented at the Optical Fabrication and Testing Meeting, Kona, Hawaii, 8–11 June 1998.

Wolf, E.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 6th ed. (Pergamon, New York, 1980).

Appl. Opt. (1)

Other (15)

See Ref. 2 and references therein, for example.

D. Malacara, S. L. DeVore, “Interferogram evaluation and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1992).

“U.S. guide to the expression of uncertainty in measurement,” (National Conference of Standards Laboratories, Boulder, Colorado, 1997), Sec. C.2.9.

The starting signal-to-noise ratio level is somewhat arbitrary. We chose a starting noise level and values of N to cover a signal-to-noise ratio range slightly beyond what would be encountered in typical optical tests.

Note that these bars should not be confused with estimates of the standard uncertainty for each mean variance; this uncertainty would be approximately a factor of 300 less than the height of the shaded bars.

Ref. 6, Sec. 4.2.1.

Ref. 6, Secs. 4.2 and 4.3.2.

Ref. 6, Sec. 5.1.

Ref. 6, Sec.3.3.

Ref. 6, Sec. 4.2.3.

R. E. Parks, C. J. Evans, L.-Z. Shao, “Calibration of interferometer transmission spheres,” poster presented at the Optical Fabrication and Testing Meeting, Kona, Hawaii, 8–11 June 1998.

Commercially available products are identified to provide a complete description of the research. Such identification does not imply endorsement by the National Institute of Standards and Technology nor that they are necessarily best suited for the application.

Ref. 6, Sec. E.4.3.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 6th ed. (Pergamon, New York, 1980).

C. J. Evans, “Absolute figure metrology of high precision optical surfaces,” Ph.D. dissertation (University of Birmingham, Birmingham, UK, 1996).

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Figures (2)

Fig. 1
Fig. 1

(a) One-dimensional simulated test results. The signal (wave-front profile) S is modeled as a sine wave. Noise (both pixel independent and correlated) is added to the signal to simulate a measurement W N=1. We modeled the improved signal-to-noise ratio by averaging N = 1 test results. An example for N = 60 is shown. (b) Plot of the mean var i (W N ) as a function of N determined from simulations of 300 statistically independent test results for each signal-to-noise ratio condition N. The inset is included for clarity on symbol definitions.

Fig. 2
Fig. 2

(a) One of the single measurements in the random ball test of an F/1.1 transmission sphere. (b) The final estimate of the reference surface of the F/1.1 transmission sphere, which is the average of 60 single measurements each with a different ball orientation. In the language of the optician, the quality of the reference surface is of the order of 1/20th of a wave, a high-quality optic. The rms of this data map is 5.73 nm.

Tables (2)

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Table 1 Summary of Method to Determine Short-Time-Scale Contributions to Eqs. (6) and (8)

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Table 2 Results of Random Ball Test of F/1.1 Transmission Spherea

Equations (12)

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variW=1P-1i=1Psi+li+i-w02,
EvariW=1P-1i=1Psi2+Ei2+Eli2.
EvariW=variS+Evari+Evari.
1P-1i=1P2si+2li+iA+iB2.
EvariWsum=4variS+4Evari+2Evari,
EvariWdif=2Evari.
variS=14EvariWsum-EvariWdif-Evari.
variSˆ=14variWsum¯s-variWdif¯d-variˆl,
μvˆ=14 μs2+14 μd2+μl21/2.
μsˆ=varkvariWk,sumN/21/2,μdˆ=varkvariWk,difN/21/2,
rmsˆ=variSˆ1/2.
μcˆ=μvˆ2variSˆ1/2.

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