Abstract

The phase-shifting technique is used in optical metrology to evaluate the local phase of a fringe pattern. Accurate calibration of the shifting device is often essential but sometimes hardly possible because of deviations of the fringe pattern from the ideal sinusoidal shape and because of a nonconstant phase shift between consecutive frames. We introduce a new technique for calculating the phase shift between frames even in the presence of high noise and nonsinusoidal fringe patterns. In addition, this technique permits the identification of different error sources such as low signal-to-noise ratio, higher harmonics contained in the fringe pattern, and nonconstant phase shift.

© 1998 Optical Society of America

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References

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  1. D. Malacara, ed., Optical Shop Testing, 2nd ed. (Wiley, New York, 1992).
  2. D. W. Robinson, G. T. Reid, eds., Interferogram Analysis (Institute of Physics, Bristol, UK, 1993).
  3. K. Hibino, B. F. Oreb, D. I. Farrant, K. G. Larkin, “Phase shifting of non-sinusodial wave-forms with phase-shift errors,” J. Opt. Soc. Am. A 12, 761–768 (1995).
    [CrossRef]
  4. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22, 3421–3432 (1983).
    [CrossRef] [PubMed]
  5. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1988), Vol. XXVI, pp. 349–393.
    [CrossRef]

1995

1983

Burow, R.

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1988), Vol. XXVI, pp. 349–393.
[CrossRef]

Elssner, K. E.

Farrant, D. I.

Grzanna, J.

Hibino, K.

Larkin, K. G.

Merkel, K.

Oreb, B. F.

Schwider, J.

Spolaczyk, R.

Appl. Opt.

J. Opt. Soc. Am. A

Other

D. Malacara, ed., Optical Shop Testing, 2nd ed. (Wiley, New York, 1992).

D. W. Robinson, G. T. Reid, eds., Interferogram Analysis (Institute of Physics, Bristol, UK, 1993).

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1988), Vol. XXVI, pp. 349–393.
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Coordinate system for the lattice-site representation. The rectangle denotes the range of numerator and denominator for 256 gray levels. The origin straight lines represent lattice sites with equal shift angles α.

Fig. 2
Fig. 2

Simulated fringe patterns with varying fringe visibility: (a) sinusoidal fringes and (b) a Ronchi grating.

Fig. 3
Fig. 3

Lattice-site representation of shift angles for simulated fringe patterns with additive intensity noise from Eq. (7) with α = 105°, Q = 0, γ = 0.75, I 0 = 127, and σ I = 10 (SNR, 16.6 dB). Two highly populated cores with width ω are shown. The line through the center of gravity includes an angle δ with the M axis.

Fig. 4
Fig. 4

Influence of noise on cores and tails in the lattice-site representation: (a), (b) Simulated fringe patterns from Eq. (7) with α = 105°, Q = 0, γ = 0.75, and σ I = 10. (a) I 0 = 127; SNR, 16.6 dB. (b) I 0 = 30; SNR, 4.0 dB. (c) Real measurement of a phase-shifted electronic speckle-pattern interferometry pattern with a low SNR, shifted by 50°.

Fig. 5
Fig. 5

Histograms for the fringe patterns used in Fig. 4.

Fig. 6
Fig. 6

Number of combinations of Z and M that yield the same shift angle α.

Fig. 7
Fig. 7

Illustration of the level of degeneracy for α = 60° and α = 60.13°. The line for α = 60° contains 254 possible combinations of integer values of Z and M (lattice sites), whereas there is only one combination (i.e., [510, 254]) that yields α = 60.13°.

Fig. 8
Fig. 8

Influence of higher harmonics on cores and tails in the lattice-site representation: (a), (b) Simulated fringe patterns from Eq. (7) with α = 72°, I 0 = 127, γ varying from 0.55 to 0.95, σ I = 3, and Q = 10. (a) σ k = 0.7 and (b) σ k = 2.0. (c) Real measurement in a projected-fringe application with a nondefocused Ronchi grating, shifted by 72°.

Fig. 9
Fig. 9

Histograms for the fringe patterns used in Fig. 8.

Fig. 10
Fig. 10

Influence of a nonconstant shift angle on cores and tails in the lattice-site representation. Data for the simulation from Eq. (7): (a) Sinusoidal fringes, I 0 = 60, γ = 0.75, σ I = 3, Q = 0, and statistical shift errors of α = [75.0°, 71.3°, 71.8°, 56.0°]; (b) sinusoidal fringes, I 0 = 127, γ = 0.75, σ I = 3, Q = 0, and systematic shift errors of α = [96.0°, 109.0°, 121.5°, 134.0°]; and (c) Ronchi grating, Q = 10, and σ k = 0.7 [for other data see (b)].

Fig. 11
Fig. 11

Histograms for the fringe patterns used in Fig. 10.

Tables (2)

Tables Icon

Table 1 Shift-Angle Calculation by Several Methods for Simulated Fringe Patterns with Additive Intensity Noise [from Eq. (7) with Q = 0, γ = 0.75, and α = 105° for All Patterns] and for a Real Phase-Shift ESPI Measurement Shifted by α = 50°

Tables Icon

Table 2 Shift-Angle Calculation by Several Methods for Simulated Fringe Patterns with Higher Harmonicsa

Equations (24)

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I x ,   y = I 0 x ,   y 1 + γ x ,   y cos   ϕ x ,   y , 0 γ x ,   y 1 ,
I n x ,   y = I 0 x ,   y 1 + γ x ,   y cos ϕ x ,   y + n - 1 α ,
cos   α = 1 2 I 5 x ,   y - I 1 x ,   y I 4 x ,   y - I 2 x ,   y .
| cos   α | 1     | I 5 - I 1 | 2 | I 4 - I 2 |     | Z | | M | .
α = arccos tan   δ .
ϕ x ,   y = 2 π x / λ x + cos 2 π y / λ y .
f Δ I Δ I = 1 2 π σ I exp - Δ I 2 2 σ I 2 .
I n * x ,   y = I n x ,   y + Δ I σ I
SNR dB = 20   log 10 I 0 γ 2 σ I .
OTF k = exp - k 2 2 σ k 2 .
k 2 = 4 π 2 / λ x 2 + 4 π 2 / λ y 2 sin 2 2 π y / λ y .
R ϕ x ,   y = 1 Ω m = 1 Q + 1 exp - m 2 k 2 2 σ k 2 × sinc m π / 2 cos m ϕ x ,   y ,
Ω = m = 1 Q + 1 exp - m 2 k 2 2 σ k 2 sinc m π / 2 ;
I n x ,   y = I 0 x ,   y 1 + γ x ,   y R ϕ x ,   y + n - 1 α + Δ I σ I .
r 2 = Z 2 + M 2 = I 5 - I 1 2 + 4 I 4 - I 2 2 .
I 5 - I 1 = - 2 I 0 γ   sin ϕ + 2 α 2   sin   α   cos   α ,
I 4 - I 1 = - 2 I 0 γ   sin ϕ + 2 α sin   α .
r 2 = 4 I 0 γ   sin   α   sin ϕ x ,   y + 2 α 2 1 + cos 2   α .
f r r d r = g ϕ ϕ d ϕ ,
f r r = g ϕ ϕ d r d ϕ - 1 .
f r r = π 4 I 0 γ   sin   α 2 1 + cos 2   α - r 2 1 / 2 - 1 ,
r s = i   m i r i i   m i .
M COG = G = 1 255 Z = - 255 255   2 G × H 2 G ,   Z G = 1 255 Z = - 255 255   H 2 G ,   Z , Z COG = G = 1 255 Z = - 255 255   Z × H 2 G ,   Z G = 1 255 Z = - 255 255   H 2 G ,   Z ,
α = arccos Z COG M COG

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