Abstract

A method for detecting a phase of equidistant and straight fringe patterns is presented that is based on an arctangent calculation of Fourier cosine and sine integrals of the fringe profile. In the phase calculation, phase errors caused by a spatial truncation, nonlinearity, and sampling of the light intensity and by a random noise are analyzed theoretically. From the analyses, it is concluded that (1) the phase error due to the truncation of data decreases as the frequency of the fringes increases, (2) the Hanning windowing makes the error small, and (3) the phase fluctuation due to the random noise depends mainly on the signal-to-noise ratio (SNR) of the fringe and the number of sample points. The total accuracy of the phase measurement is determined by the sum of three phase errors due to the calculation of a phase, a random noise, and a frequency deviation of a laser light. The total error is reduced to one thousandth of the wavelength of the laser light when the number of sample points is 256, the SNR is 100, and the frequency stability of the laser light is 1 × 10−7.

© 1988 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), pp. 260–323, 505–518.
  2. E. Archbold, A. E. Ennos, “Displacement measurement from double-exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
    [CrossRef]
  3. G. B. Hocker, “Fiber-optic sensing of pressure and temperature, Appl. Opt. 18, 1445–1448 (1979).
    [CrossRef] [PubMed]
  4. L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, London, 1975), pp. 91–93.
  5. W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958.
  6. W. R. Bennett, “Methods of solving noise problems,” Proc. IRE 44, 609–638 (1956).
    [CrossRef]
  7. J. H. Bruning, “Fringe scanning interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978).

1979 (1)

1972 (1)

E. Archbold, A. E. Ennos, “Displacement measurement from double-exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
[CrossRef]

1956 (1)

W. R. Bennett, “Methods of solving noise problems,” Proc. IRE 44, 609–638 (1956).
[CrossRef]

Archbold, E.

E. Archbold, A. E. Ennos, “Displacement measurement from double-exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
[CrossRef]

Bennett, W. R.

W. R. Bennett, “Methods of solving noise problems,” Proc. IRE 44, 609–638 (1956).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), pp. 260–323, 505–518.

Bruning, J. H.

J. H. Bruning, “Fringe scanning interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978).

Davenport, W. B.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958.

Ennos, A. E.

E. Archbold, A. E. Ennos, “Displacement measurement from double-exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
[CrossRef]

Gold, B.

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, London, 1975), pp. 91–93.

Hocker, G. B.

Rabiner, L. R.

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, London, 1975), pp. 91–93.

Root, W. L.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), pp. 260–323, 505–518.

Appl. Opt. (1)

Opt. Acta (1)

E. Archbold, A. E. Ennos, “Displacement measurement from double-exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
[CrossRef]

Proc. IRE (1)

W. R. Bennett, “Methods of solving noise problems,” Proc. IRE 44, 609–638 (1956).
[CrossRef]

Other (4)

J. H. Bruning, “Fringe scanning interferometers,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), pp. 260–323, 505–518.

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, London, 1975), pp. 91–93.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958.

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Figures (12)

Fig. 1
Fig. 1

Schematic illustration of the Fourier spectrum of Young’s fringes. All components of the spectrum are plotted together to facilitate a comparison of the sidelobes.

Fig. 2
Fig. 2

Phase error due to a truncation of fringes. (a) Perspective and (b) contour representations, where the fringe contrast and the frequency deviation are 0.6 and 0.1, respectively. The interval between contours is 50.

Fig. 3
Fig. 3

Maximum phase error as the phase of the fringe is changed from 0 to 2π, where the window function is rect(x) and the frequency error is 0.1. Fringe contrasts are a, 1.0; b, 0.6; and c, 0.4.

Fig. 4
Fig. 4

(a) Profile of the Hanning window, (b) Fourier spectra of a, Hanning window and b, rect window.

Fig. 5
Fig. 5

Maximum phase error as the phase of the fringe is changed from 0 to 2π, where the window function is the Hanning and the frequency error is 0.1. Fringe contrasts are a, 1.0; b, 0.6; and c, 0.4.

Fig. 6
Fig. 6

Fourier spectrum of a nonsinusoidal function, where spectra at harmonic frequencies are plotted together to facilitate a comparison of the sidelobes of all the spectra.

Fig. 7
Fig. 7

Schematic illustration of the sampling of Young’s fringes.

Fig. 8
Fig. 8

Fourier spectrum in Eq. (14), S ˜ 0 ( f ), of a sampled rect window (line a). Fourier spectrum, sinc(f), of a continuous rect window (line b). The number of sample points N is 50 for line a.

Fig. 9
Fig. 9

Reduction factor of fringe contrast, sinc(p), where p is the ratio of a sampling width to a fringe spacing 1/f0.

Fig. 10
Fig. 10

Maximum phase error for the sampling of the fringes. Fringe contrasts are a, 1.0; b, 0.6; and c, 0.4. The number of sample points N is 256, the ratio p of the sampling width to the period of the fringe is 0.1, and the frequency error Δf is 0.1.

Fig. 11
Fig. 11

Maximum phase error for the sampling of the fringe with the Hanning windowing. Fringe contrasts are a, 1.0; b, 0.6; and c, 0.4. The number of sample points is 256, the ratio p is 0.1, and the frequency error Δf is 0.1.

Fig. 12
Fig. 12

Vector plane spanned by cosine and sine coordinates. nc and ns are noise components due to random noise, and Δϕ is the phase error due to (nc, ns).

Equations (27)

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I ( x ) = W ( x ) [ a ( x ) + b ( x ) cos ( 2 π f 0 x ϕ ) ] ,
C = I ( x ) cos ( 2 π f 0 x ) d x ,
S = I ( x ) sin ( 2 π f 0 x ) d x .
ϕ = tan 1 ( S / C ) .
I ˜ ( f 0 ) = A ˜ ( f 0 ) + 1 2 B ˜ ( f 0 + f 0 ) exp ( i ϕ ) + 1 2 B ˜ ( Δ f ) exp ( i ϕ ) ,
W ( x ) = [ β + ( 1 β ) cos 2 π x ] rect ( x ) ,
W ˜ ( f ) = sinc ( f ) + 1 2 sinc ( f 1 ) + 1 2 sinc ( f + 1 ) ,
I ( x ) = W ( x ) g ( x ) .
g ( x ) = n = C n exp ( i 2 π f 0 n x ) ,
C n = 1 / 2 1 / 2 g ( ξ ) exp ( i 2 π n ξ ) d ξ ,
I ˜ ( f ) = n = C n W ˜ ( f f 0 n ) exp ( i n ϕ ) .
S A ( x ) = n = 1 N [ X ( n ) / 2 X ( n ) + / 2 I ( y ) d y ] δ [ x X ( n ) ] ,
S ˜ A ( x ) = N [ S ˜ 0 ( f ) + α 2 sinc ( f 0 ) S ˜ 0 ( f + f 0 ) exp ( i ϕ ) + α 2 sinc ( f 0 ) S ˜ 0 ( f f 0 ) exp ( i ϕ ) ] ,
S ˜ 0 ( f ) = sin ( N N 1 π f ) N sin ( π f N 1 ) .
S ˜ ( f 0 ) = N 2 [ S ˜ h ( f 0 ) + α 2 sinc ( f 0 ) S ˜ h ( f 0 + f 0 ) exp ( i ϕ ) + α 2 sinc ( f 0 ) S ˜ h ( Δ f ) exp ( i ϕ ) ] ,
S ˜ h ( f ) = [ S ˜ 0 ( f ) + 1 2 S ˜ 0 ( f 1 ) + 1 2 S ˜ 0 ( f + 1 ) ] .
n c = W ( x ) n ( x ) cos ( 2 π f 0 x ) d x ,
n s = W ( x ) n ( x ) sin ( 2 π f 0 x ) d x .
ϕ + Δ ϕ = tan 1 ( S + n s C + n c ) .
C = α 2 1 / 2 1 / 2 W ( x ) d x ,
σ 2 = n c 2 = n s 2 = σ 2 2 1 / 2 1 / 2 W 2 ( x ) d x ,
n ( x 1 , t 1 ) n ( x 2 , t 2 ) = σ 2 δ ( x 1 x 2 ) .
p ( Δ ϕ ) = C σ 2 π exp ( C 2 2 σ 2 2 π Δ ϕ 2 ) ,
σ ϕ 2 = ( σ C ) = 1 S N 2 1 / 2 1 / 2 W ( x ) d x [ 1 / 2 1 / 2 W ( x ) d x ] 2 .
σ ϕ 2 = 1 S N 2 l = 1 N W l 2 ( l = 1 N W l ) 2 ,
σ ϕ = 1 S N N 1 3 / 2 .
Δ ϕ = 2 π ( Δ λ λ ) ( Δ l λ ) ,

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