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Invited Review Article: Measurement uncertainty of linear phase-stepping algorithms
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Image of FIG. 1.
FIG. 1.

Sketch of a simple phase-shifting interferometer. An expanded laser beam is divided by a beam splitter and, after reflection in each arm, recombined on a camera sensor (where the surface of the test flat should be in focus). The reference mirror is moved via a piezo-electric actuator on which it is mounted, thereby introducing the prescribed sequence of reference phase shifts.

Image of FIG. 2.
FIG. 2.

Top: set of four phase-shifted fringe patterns with 90° nominal phase shift per step. The pixel encircled in the centre sees a cosinusoidal brightness modulation as the reference phase is advanced. Bottom: the four brightness readings are graphed as white dots at their nominal phases; the best fit to these of a cosine is graphed in black, and the average brightness is denoted by a white line. This cosine fitting is implicitly done for every pixel in the image by application of the phase-shifting formula, and the resulting phase offset of the fitted cosine represents the desired phase φ for each pixel.

Image of FIG. 3.
FIG. 3.

Top left: “saw tooth” or “wrapped” phase map calculated from the four interferograms in Fig. 2; the phase between 0 and 2π is represented by gray values from 0 to 255, and the phase is usually made continuous (unwrapped) in a following data processing step. Top right: phase map from four different interferograms with 78° phase shift per step, but calculated with the same 90° formula as before. Bottom: plots of the central horizontal line for both saw tooth images, 90° nominal and actual (black) and 90° nominal and 78° actual (white). The phase shows a cyclic error when the actual and nominal phase steps do not match.

Image of FIG. 4.
FIG. 4.

Sensitivity of different phase stepping algorithms as expressed with the noise figure F. For definitions of the names used, see the Appendix.

Image of FIG. 5.
FIG. 5.

Average phase uncertainty [mrad] as a function of fluctuation r in illumination intensity or camera gain for N-bucket algorithms. Lines are plotted for different values of the visibility.

Image of FIG. 6.
FIG. 6.

Phase uncertainty in [mrad] due to shot noise for N-bucket algorithms, as a function of the number of counts. Lines are plotted for different values of the visibility.

Image of FIG. 7.
FIG. 7.

Average phase uncertainty due to 10% lowest contributing order (N – 1) detector nonlinearity, i.e., as a function of the number of frames of the N-bucket algorithm, Table I. Lines are plotted for different values of the visibility.

Image of FIG. 8.
FIG. 8.

Sum of squared phase uncertainties from 1% equivalent contributions for a visibility of v = 1 (a) and v = 0.25 (b). Values are for the average phase variance for the N-bucket algorithms, as given in Eq. (62), using an 8-bit camera. From Table I, the values used in the calculation are: S camnoise = 0.01 × 256GL, S LSB = 1GL, r(I) = 0.01, r(γ) = 0.01, u(β) = 0.01 × π/2, S B = 10000 cnts, ε = 0.01, e 2 = 0.01 × 2π/N/(N − 1), e 3 = 0.01 × 2π/N/(N − 1)2, r N − 1 = 0.01, , δdrift = 0.01 over 4 frames, and δillumdrift = 0.01 over 4 frames.


Generic image for table
Table I.

Influence quantities and references to the respective formula for quantitative evaluation. Explicit expressions for algorithms with sin (β k + δ k ) = 0 ∀k and for the N-bucket algorithms are included.

Generic image for table
Table II.

Common phase step algorithms with their coefficients, F-value and angle values.


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Scitation: Invited Review Article: Measurement uncertainty of linear phase-stepping algorithms