Abstract

Phase errors that arise in phase-stepping interferometry are discussed. Investigations were performed by use of a Twyman–Green interferometer equipped with a compensation plate with a variable and servo-controlled tilt angle. With this instrument, phase-stepping errors can be reduced to a negligible level. There are, however, phase errors that are caused by camera nonlinearities. Two methods for minimizing these errors are presented. The first method is based on the simple idea that the interference intensity at the output of a two-beam interferometer has an exact cosine shape. The camera signals were monitored as a function of the tilt angle of the compensation plate, and the deviation from the cosine form was used to produce a correction. The second method is based on the idea that, under specific conditions, errors of an average of two phase measurements may compensate for each other. Numerical calculations were performed and give evidence of this hypothesis. Each method, the signal-correction and the averaging method, drastically reduces errors in evaluation of phases. The combination of both methods is a powerful tool that allows precise phase data to be obtained with an uncertainty, in the range λ/2000 ≈ 0.3 nm, that is caused mainly by signal noise.

© 2002 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  7. K. Creath, “Phase-measuring interferometry: beware these errors,” in Laser Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE1553, 213–219 (1991).
    [CrossRef]
  8. J. van Wingerden, H. J. Frankena, C. Smorenburg, “Linear approximation for measurement errors in phase shifting interferometry,” Appl. Opt. 30, 2718–2729 (1991).
    [CrossRef] [PubMed]
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    [CrossRef]

1991

1986

G. T. Reid, “Automatic fringe pattern analysis: a review,” Opt. Lasers Eng. 7, 37–68 (1986).
[CrossRef]

1985

1983

1974

1966

P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Brangaccio, D. J.

Brohinsky, W. R.

Bruning, J. H.

Burow, R.

Carré, P.

P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Creath, K.

K. Creath, “Temporal phase measuring methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993).

K. Creath, “Phase-measuring interferometry: beware these errors,” in Laser Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE1553, 213–219 (1991).
[CrossRef]

K. Creath, “Comparison of phase-measurement algorithms,” in Surface Characterization and Testing, K. Kreath, ed., Proc. SPIE680, 19–27 (1986).
[CrossRef]

Elssner, K.-E.

Frankena, H. J.

Gallagher, J. E.

Grzanna, J.

Hariharan, P.

P. Hariharan, “Interferometry with lasers,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1987), Vol. 24, pp. 103–164.
[CrossRef]

Herriott, D. R.

Merkel, K.

Reid, G. T.

G. T. Reid, “Automatic fringe pattern analysis: a review,” Opt. Lasers Eng. 7, 37–68 (1986).
[CrossRef]

Rosenfeld, D. P.

Schwider, J.

Smorenburg, C.

Spolaczyk, R.

Stetson, K. A.

van Wingerden, J.

White, A. D.

Appl. Opt.

Metrologia

P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Opt. Lasers Eng.

G. T. Reid, “Automatic fringe pattern analysis: a review,” Opt. Lasers Eng. 7, 37–68 (1986).
[CrossRef]

Other

P. Hariharan, “Interferometry with lasers,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1987), Vol. 24, pp. 103–164.
[CrossRef]

K. Creath, “Temporal phase measuring methods,” in Interferogram Analysis: Digital Fringe Pattern Measurement Techniques, D. W. Robinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993).

K. Creath, “Comparison of phase-measurement algorithms,” in Surface Characterization and Testing, K. Kreath, ed., Proc. SPIE680, 19–27 (1986).
[CrossRef]

K. Creath, “Phase-measuring interferometry: beware these errors,” in Laser Interferometry IV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE1553, 213–219 (1991).
[CrossRef]

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Figures (19)

Fig. 1
Fig. 1

Schematic diagram of the interferometer.

Fig. 2
Fig. 2

Output signal detected with several camera pixels as a function of the phase shift (data points). There are two sets of each of the four steps (indicated by open and filled triangles, respectively). The internal step width is α, whereas ν is the separation between the two sets. The solid curve indicates a cosine function. The deviation of measured data from a cosine function is discussed below.

Fig. 3
Fig. 3

Phase difference ϕB* - ϕA* obtained from measurements at various separations ν (Fig. 2). Dotted lines, mean values ν* of ϕB* - ϕA*; numbers at the right, individual values of ν that were set.

Fig. 4
Fig. 4

rms of (ϕB* - ϕA* - ν*) for the data shown in Fig. 3.

Fig. 5
Fig. 5

Top, averaged output signal detected with ∼100 camera pixels, yielding similar signals as a function of the phase shift (crosses). These data obviously deviate from a cosine function (solid curve). Bottom, explicit illustration of this difference.

Fig. 6
Fig. 6

Deviation of the camera signal from the cosine function of Fig. 5 assigned to the camera signal (data points). A continuous function was generated (solid curve) that can be used as additive correction within the signal interval shown.

Fig. 7
Fig. 7

Averaged output signal detected with ∼100 camera pixels at reduced light intensity. Data points of the upper curve that exceed 450 were subjected to the additive correction of Fig. 6 and yielded values drawn as bow ties. The top solid curve is the fit of a cosine function to these data. From this curve a correction curve similar to that in Fig. 6 can be obtained (not shown), enhancing the signal-correction interval to values of ∼250. The same procedure was applied to the lower data points (open circles) and to the cosine fit (lower solid curve), enhancing the signal range even more.

Fig. 8
Fig. 8

Overall additive correction that resulted from the data shown in Figs. 5 7. The solid curve will be used as a continuous correction function within the signal range of approximately 140–3500. For illustration, the corresponding camera sensitivity is shown in the inset (solid upper curve); it clearly deviates from linearity (dashed lower curve).

Fig. 9
Fig. 9

rms of (ϕB* - ϕA* - ν*) if the camera signal correction from Fig. 8 is applied to each pixel. These data show a drastically reduced phase error compared with uncorrected data (Fig. 4).

Fig. 10
Fig. 10

Calculated phase errors ϕ′ - ϕ of evaluated phase ϕ′ distorted by camera nonlinearities as a function of input phase ϕ according to Eqs. (4 ). The data are shown for three different phase-step widths: α/2π = 0.24 (open circles), α/2π = 0.32 (filled circles), and α/2π = 0.39 (bow ties). Inset, the rms of ϕ′ - ϕ as a function of α.

Fig. 11
Fig. 11

ϕB - ϕA for several separations ν (whose values are indicated at the right) as a function of input phase data ϕ at α/2π = 0.32.

Fig. 12
Fig. 12

rms of ϕB - ϕA - ν as a function of ν and α, showing maxima at ν/2π = 1/6, 3/6, … , minima at ν/2π = 2/6, 4/6, … , and an unspecific dependence on α.

Fig. 13
Fig. 13

(ϕA + ϕB)/2 for various separations ν (whose values are indicated at the right) as a function of simulated phase data ϕ A at a step width α/2π = 0.32.

Fig. 14
Fig. 14

rms of (ϕA + ϕB - ν*)/2 - ϕ as a function of ν and α, showing minima at ν/2π = 1/6, 3/6, … , and maxima at ν/2π = 2/6, 4/6, … .

Fig. 15
Fig. 15

Data of Fig. 14 at ν/2π = 1/6 and ν/2π = 3/6 as a function of α.

Fig. 16
Fig. 16

Data of Fig. 14 at α/2π = 0.32 as a function of ν.

Fig. 17
Fig. 17

rms of ϕB12* - ϕA12* - ν* if the averaging method is applied to each pixel. These data have a drastically reduced phase error than the data of Fig. 4.

Fig. 18
Fig. 18

rms of ϕB12* - ϕA12* - ν* if both the averaging method and the camera signal correction from Fig. 8 are applied. These data have a slightly reduced phase error compared with those in Figs. 9 and 17.

Fig. 19
Fig. 19

Illustration of the effect of the methods illustrated above at νν 0: without correction, with camera additive correction, by the averaging method, by both methods together.

Equations (8)

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α=±2 arctan3I2-I3-I1-I4I2-I3+I1-I41/2,
ϕ=arctantanα2I1-I4+I2-I3I2+I3-I1+I41/2.
Ilow=cIhigh-d+d,
Iiϕ, α=I01+γ cosϕ+i-52α,i=1, 2, 3, 4,
Siϕ, α=IiIi-a0+a1Ii+a2Ii2+a3Ii3Ii<410410Ii<4096,
a0=-254.3, a1=0.754, a2=-3.624×10-4,a3=4.813×10-8,
ϕA12*=ϕA*+ϕB*ν0/2,
ϕB12*=ϕB1*ν+ϕB2*ν+ν0/2,

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