Abstract

The Fourier transform method is applied to analyze the initial phase of linear and equispaced Fizeau fringes. We develop an algorithm for high-precision phase measurement by using the Fourier coefficient that corresponds to the spatial frequency of the Fizeau fringes, and we describe methods for determining the fringe carrier frequency. Errors caused by carrier frequency fluctuation and data truncation are studied theoretically and by computer simulation. To demonstrate the method we apply it to the real-time calibration of a piezoelectric transducer mirror in a Twyman–Green interferometer.

© 1994 Optical Society of America

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References

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  1. T. Yatagai, M. Kobayashi, “Precise measurement of PZT mirror movement using FFT method,” in Optics and the Information Age (14th Congress of the International Commission for Optics), H. H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.813, 61–62 (1987).
  2. M. Kobayashi, “High precision fringe analysis,” M.S. thesis (University of Tsukuba, Tsubuka, Japan, 1986).
  3. G. Lai, “Studies on phase-shifting interferometry and its applications,” Ph.D. dissertation (University of Tsukuba, Tsukuba, Japan, 1990).
  4. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  5. M. Takeda, Q. Ru, “Computer-based highly sensitive electron-wave interferometry,” Appl. Opt. 24, 3068–3071 (1985).
    [CrossRef] [PubMed]
  6. R. Snyder, L. Hesselink, “High speed optical tomography for flow visualization,” Appl. Opt. 24, 4046–4051 (1985).
    [CrossRef] [PubMed]
  7. S. Nakadate, “Phase detection of equidistant fringes for highly sensitive optical sensing. I. Principle and error analyses,” J. Opt. Soc. Am. A 5, 1258–1264 (1988).
    [CrossRef]
  8. S. Nakadate, “Phase detection of equidistant fringes for highly sensitive optical sensing. II. Experiments,” J. Opt. Soc. Am. A 5, 1265–1269 (1988).
    [CrossRef]
  9. T. Yatagai, “Interpolation method of computer-generated filters for large object formats,” Opt. Commun 23, 347–351 (1977).
    [CrossRef]
  10. T. Yatagai, “Interpolation approach to computer-generated holograms,” in International Conference on Computer-Generated Holography, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.437, 19–23 (1983).
    [CrossRef]
  11. T. Yatagai, T. Kanou, “Aspherical surface testing with shearing interferometer using fringe scanning detection method,” Opt. Eng. 23, 357–360 (1984).
    [CrossRef]
  12. G. Lai, T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A 8, 822–827 (1991).
    [CrossRef]

1991 (1)

1988 (2)

1985 (2)

1984 (1)

T. Yatagai, T. Kanou, “Aspherical surface testing with shearing interferometer using fringe scanning detection method,” Opt. Eng. 23, 357–360 (1984).
[CrossRef]

1982 (1)

1977 (1)

T. Yatagai, “Interpolation method of computer-generated filters for large object formats,” Opt. Commun 23, 347–351 (1977).
[CrossRef]

Hesselink, L.

Ina, H.

Kanou, T.

T. Yatagai, T. Kanou, “Aspherical surface testing with shearing interferometer using fringe scanning detection method,” Opt. Eng. 23, 357–360 (1984).
[CrossRef]

Kobayashi, M.

T. Yatagai, M. Kobayashi, “Precise measurement of PZT mirror movement using FFT method,” in Optics and the Information Age (14th Congress of the International Commission for Optics), H. H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.813, 61–62 (1987).

M. Kobayashi, “High precision fringe analysis,” M.S. thesis (University of Tsukuba, Tsubuka, Japan, 1986).

Kobayashi, S.

Lai, G.

G. Lai, T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A 8, 822–827 (1991).
[CrossRef]

G. Lai, “Studies on phase-shifting interferometry and its applications,” Ph.D. dissertation (University of Tsukuba, Tsukuba, Japan, 1990).

Nakadate, S.

Ru, Q.

Snyder, R.

Takeda, M.

Yatagai, T.

G. Lai, T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A 8, 822–827 (1991).
[CrossRef]

T. Yatagai, T. Kanou, “Aspherical surface testing with shearing interferometer using fringe scanning detection method,” Opt. Eng. 23, 357–360 (1984).
[CrossRef]

T. Yatagai, “Interpolation method of computer-generated filters for large object formats,” Opt. Commun 23, 347–351 (1977).
[CrossRef]

T. Yatagai, “Interpolation approach to computer-generated holograms,” in International Conference on Computer-Generated Holography, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.437, 19–23 (1983).
[CrossRef]

T. Yatagai, M. Kobayashi, “Precise measurement of PZT mirror movement using FFT method,” in Optics and the Information Age (14th Congress of the International Commission for Optics), H. H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.813, 61–62 (1987).

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Commun (1)

T. Yatagai, “Interpolation method of computer-generated filters for large object formats,” Opt. Commun 23, 347–351 (1977).
[CrossRef]

Opt. Eng. (1)

T. Yatagai, T. Kanou, “Aspherical surface testing with shearing interferometer using fringe scanning detection method,” Opt. Eng. 23, 357–360 (1984).
[CrossRef]

Other (4)

T. Yatagai, M. Kobayashi, “Precise measurement of PZT mirror movement using FFT method,” in Optics and the Information Age (14th Congress of the International Commission for Optics), H. H. Arsenault, ed., Proc. Soc. Photo-Opt. Instrum. Eng.813, 61–62 (1987).

M. Kobayashi, “High precision fringe analysis,” M.S. thesis (University of Tsukuba, Tsubuka, Japan, 1986).

G. Lai, “Studies on phase-shifting interferometry and its applications,” Ph.D. dissertation (University of Tsukuba, Tsukuba, Japan, 1990).

T. Yatagai, “Interpolation approach to computer-generated holograms,” in International Conference on Computer-Generated Holography, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.437, 19–23 (1983).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Phase measurement error caused by the discrete Fourier transform to the fringe profile with Rect and Hanning windows. In the case of the Hanning window, the phase error is multiplied by 103. (a) Error changes periodically with the initial phase of the fringes when the number of fringes and the decimal part of the fringes are 20 and 0.1, respectively. (b) Maximum error changes with the decimal part of the fringe carrier frequency.

Fig. 2
Fig. 2

Principle of the band-limited FFT interpolation method showing (a) the original fringe profile (b) the Fourier spectrum of (a), (c) the extended fringe profile with the Hanning window, (d) its densely sampled Fourier transform of (c).

Fig. 3
Fig. 3

1-D phase distribution calculated from the first-order diffraction of sinusoidal fringes showing (a) a fringe profile simulated with 20.5 periods and (b) a phase distribution calculated with a carrier frequency of 21 fringes. The total phase change over the sampling area was calculated by (Δϕ/Δn)N, where N represents the total number of sampled points.

Fig. 4
Fig. 4

Setup for demonstrating the proposed phase analysis of Fizeau fringes using the FFT method by measuring the movement of a PZT mirror: M1 and M2, mirrors; BS, beam splitter; L1–L3, lenses; D/A, digital to analog; A/D, analog to digital.

Fig. 5
Fig. 5

Fringe intensity profiles that correspond to different voltages supplied on the PZT.

Fig. 6
Fig. 6

Mirror displacement as a function of voltage supplied on the PZT, obtained by phase analysis of Fizeau fringes.

Equations (29)

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I ( x ) = a ( x ) [ 1 + γ ( x ) cos ( 2 π f x + ϕ ) ] ,
Î ( υ ) = I ( x ) exp ( 2 π i υ x ) d x = â ( υ ) + ĉ ( υ f ) exp ( i ϕ ) + ĉ ( υ + f ) exp ( i ϕ ) ,
â ( υ ) = a ( x ) exp ( 2 π i υ x ) d x ,
ĉ ( υ ) = ½ a ( x ) γ ( x ) exp ( 2 π i υ x ) d x ,
log [ Î ( f ) ] = log [ ĉ ( 0 ) ] + i ϕ ,
φ = ϕ + Im { log [ ĉ ( 0 ) ] } ,
I ( x ) = a ( x ) { 1 + γ ( x ) cos [ 2 π f x + ω ( x ) + ϕ ] } ,
ĉ ( υ ) = ½ a ( x ) γ ( x ) exp [ i ω ( x ) ] exp ( 2 π i υ x ) d x .
H ( n / N T ) = k = 0 N 1 h ( k t ) exp ( 2 π i k n / N ) ,
h ( k ) = 1 N n = 0 N 1 H ( n / N T ) exp ( 2 π i k n / N ) ,
I ( k ) = a ( k ) [ 1 + γ ( k ) cos ( 2 π f k + ϕ ) ] , ( k = 0 , 1 , , N 1 ) .
Î ( n / N ) = â ( n / N ) + ĉ ( n / N f ) exp ( i ϕ ) + ĉ ( n / N + f ) exp ( i ϕ ) , ( n = 0 , 1 , , N 1 ) ,
â ( n / N ) = k = 0 N 1 a ( k ) exp ( 2 π i k n / N ) ,
ĉ ( n / N ) = ½ k = 0 N 1 a ( k ) γ ( k ) exp ( 2 π i k n / N ) .
log [ Î ( f ) ] = log [ ĉ ( n / N f ) ] + i ϕ ,
f = ( n f + Δ n ) / N ,
log [ Î ( f = n f / N ) ] = log [ ĉ ( 0 ) ] + i ϕ ,
ϕ = Im [ log Î ( n f / N ) ] .
log [ Î ( f ) ] = log [ ĉ ( Δ n / N ) ] + i ϕ ,
ĉ ( Δ n / N ) = k = 0 N 1 a ( k ) γ ( k ) exp ( 2 π i k Δ n / N ) .
ĉ ( Δ n / N ) = a γ 1 exp ( 2 π i Δ n ) 1 exp ( 2 π i Δ n / N ) .
1 exp ( i 2 π Δ n ) = 1 cos ( 2 π Δ n ) + i sin ( 2 π Δ n ) = 2 | sin ( π Δ n ) | exp [ i ( π / 2 π Δ n ) ] .
log [ 1 exp ( 2 π i Δ n ) ] = log 2 | sin π Δ n | + i ( π / 2 π Δ n ) .
log [ Î ( f ) ] = log a γ | sin π Δ n | | sin π Δ n / N | + i [ ϕ + ( 1 / N 1 ) π Δ n ] .
Im { log [ Î ( f ) ] } = ϕ ( 1 1 / N ) π Δ n ,
I ( k ) = 1 + cos [ 2 π ( n + Δ n ) N k + ϕ ] , ( k = 0 , 1 , 2 , , N 1 ) ,
Î ( f ) = Î ( n / N ) sin π N ( f n / N ) π N ( f n / N ) .
Î ( n / N + Δ n / N ) = k = 0 N 1 I ( k ) exp [ 2 π i k ( n + Δ n ) / N ] = k = 0 N 1 [ I ( k ) exp ( 2 π i k Δ n / N ) ] exp ( 2 π i k n / N ) .
ϕ = 2 π λ 2 L .

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