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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 12, Iss. 9 — Sep. 1, 1995
  • pp: 1997–2008

Statistical properties of phase-shift algorithms

C. Rathjen  »View Author Affiliations

JOSA A, Vol. 12, Issue 9, pp. 1997-2008 (1995)

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Statistical properties of phase-shift algorithms are investigated for the case of additive Gaussian intensity noise. Based on a bivariate normal distribution, a generally valid probability-density function for the random phase error is derived. This new description of the random phase error shows properties that cannot be obtained through Gaussian error propagation. The assumption of a normally distributed phase error is compared with the derived probability-density function. For small signal-to-noise ratios the assumption of a normally distributed phase error is not valid. Additionally, it is shown that some advanced systematic-error-compensating algorithms have a disadvantageous effect on the random phase error.

© 1995 Optical Society of America

Original Manuscript: September 19, 1994
Revised Manuscript: March 15, 1995
Manuscript Accepted: March 29, 1995
Published: September 1, 1995

C. Rathjen, "Statistical properties of phase-shift algorithms," J. Opt. Soc. Am. A 12, 1997-2008 (1995)

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  32. This test is not the only possible one. The advantage of the test is the simple idea on which it is based.
  33. Averaging the single intensity pattern Ii(x, y) first will not yield this effect. This is the important difference between averaging before and averaging after phase calculation. The difference is caused by the nonlinearity of the arctangent function. Therefore the difference vanishes with an increasing SNR.
  34. In Ref. 5 an asymmetric distribution of the reference phase is published. Such an asymmetry may be caused by the calibration algorithm.
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  36. Regarding k1as a quadratic equation of α, one can show that there are no real roots for ρ2< 1. Therefore k1is always positive. For ρ2= 1, which corresponds to completely correlated noise terms of Dand N, no practical importance can be seen. However, the phase error Δφcan easily be derived from Fig. 1. Then we can obtain the probability-density function of Δφby transforming the probability-density function of the noise term in a manner similar to that in Eq. (A5).

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