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Applied Optics

Applied Optics


  • Vol. 40, Iss. 34 — Dec. 1, 2001
  • pp: 6203–6209

Estimating the root mean square of a wave front and its uncertainty

Angela Davies and Mark S. Levenson  »View Author Affiliations

Applied Optics, Vol. 40, Issue 34, pp. 6203-6209 (2001)

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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

OCIS Codes
(120.3940) Instrumentation, measurement, and metrology : Metrology
(120.4800) Instrumentation, measurement, and metrology : Optical standards and testing
(120.6650) Instrumentation, measurement, and metrology : Surface measurements, figure

Original Manuscript: April 26, 2001
Revised Manuscript: July 30, 2001
Published: December 1, 2001

Angela Davies and Mark S. Levenson, "Estimating the root mean square of a wave front and its uncertainty," Appl. Opt. 40, 6203-6209 (2001)

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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).
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  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.

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