Abstract

The spatial resolution of the phase image derived from the interferogram by Fourier fringe analysis is limited by the necessity to isolate a first order in the Fourier plane. By use of the two complementary outputs of the interferometer, it is possible to eliminate the zero order and thus to improve the spatial resolution by a factor of approximately 2. The theory of this improvement is presented and confirmed experimentally.

© 2003 Optical Society of America

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References

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  1. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based tomography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  2. S. Kostianovski, S. G. Lipson, E. N. Ribak, “Interference microscopy and Fourier fringe analysis applied to measuring the spatial refractive-index distribution,” Appl. Opt. 32, 4744–4750 (1993).
    [CrossRef] [PubMed]
  3. K. A. Nugent, “Interferogram analysis using an accurate fully automatic algorithm,” Appl. Opt. 24, 3101–3105 (1985).
    [CrossRef] [PubMed]
  4. Z. Ge, F. Kobayashi, S. Matsuda, M. Takeda, “Coordinate transform technique for closed-fringe analysis by the Fourier-transform method,” Appl. Opt. 40, 1649–1657 (2001).
    [CrossRef]
  5. C. Roddier, F. Roddier, “Interferogram analysis using Fourier transform techniques,” Appl. Opt. 26, 1668–1673 (1987).
    [CrossRef] [PubMed]
  6. W. W. Macy, “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22, 3898–3901 (1983).
    [CrossRef] [PubMed]
  7. D. J. Bone, H.-A. Bachor, R. J. Sandeman, “Fringe pattern analysis using a 2-D Fourier transform,” Appl. Opt. 25, 1653–1660 (1986).
    [CrossRef]
  8. P. Hariharan, Optical Interferometry (Academic, Marrickville, Australia, 1985), pp. 157–159.
  9. W. Li, X. Su, “Real-time calibration algorithm for phase shifting in phase-measuring philometry,” Opt. Eng. 40, 761–766 (2001).
    [CrossRef]
  10. M. A. Herraez, D. R. Burton, M. J. Lalor, “Accelerating fast Fourier transform and filtering operations in Fourier fringe analysis for accurate measurement of three dimensional surfaces,” Opt. Laser Eng. 31, 131–145 (1999).
  11. S. G. Lipson, H. Lipson, D. S. Tanhauser, Optical Physics, 3rd ed. (Cambridge U. Press, Cambridge, UK, 1995), pp. 120–122.
  12. E. Hecht, Optics, 3rd ed. (Addison-Wesley Longman, New York, 1998).

2001 (2)

Z. Ge, F. Kobayashi, S. Matsuda, M. Takeda, “Coordinate transform technique for closed-fringe analysis by the Fourier-transform method,” Appl. Opt. 40, 1649–1657 (2001).
[CrossRef]

W. Li, X. Su, “Real-time calibration algorithm for phase shifting in phase-measuring philometry,” Opt. Eng. 40, 761–766 (2001).
[CrossRef]

1999 (1)

M. A. Herraez, D. R. Burton, M. J. Lalor, “Accelerating fast Fourier transform and filtering operations in Fourier fringe analysis for accurate measurement of three dimensional surfaces,” Opt. Laser Eng. 31, 131–145 (1999).

1993 (1)

1987 (1)

1986 (1)

1985 (1)

1983 (1)

1982 (1)

Bachor, H.-A.

Bone, D. J.

Burton, D. R.

M. A. Herraez, D. R. Burton, M. J. Lalor, “Accelerating fast Fourier transform and filtering operations in Fourier fringe analysis for accurate measurement of three dimensional surfaces,” Opt. Laser Eng. 31, 131–145 (1999).

Ge, Z.

Hariharan, P.

P. Hariharan, Optical Interferometry (Academic, Marrickville, Australia, 1985), pp. 157–159.

Hecht, E.

E. Hecht, Optics, 3rd ed. (Addison-Wesley Longman, New York, 1998).

Herraez, M. A.

M. A. Herraez, D. R. Burton, M. J. Lalor, “Accelerating fast Fourier transform and filtering operations in Fourier fringe analysis for accurate measurement of three dimensional surfaces,” Opt. Laser Eng. 31, 131–145 (1999).

Ina, H.

Kobayashi, F.

Kobayashi, S.

Kostianovski, S.

Lalor, M. J.

M. A. Herraez, D. R. Burton, M. J. Lalor, “Accelerating fast Fourier transform and filtering operations in Fourier fringe analysis for accurate measurement of three dimensional surfaces,” Opt. Laser Eng. 31, 131–145 (1999).

Li, W.

W. Li, X. Su, “Real-time calibration algorithm for phase shifting in phase-measuring philometry,” Opt. Eng. 40, 761–766 (2001).
[CrossRef]

Lipson, H.

S. G. Lipson, H. Lipson, D. S. Tanhauser, Optical Physics, 3rd ed. (Cambridge U. Press, Cambridge, UK, 1995), pp. 120–122.

Lipson, S. G.

Macy, W. W.

Matsuda, S.

Nugent, K. A.

Ribak, E. N.

Roddier, C.

Roddier, F.

Sandeman, R. J.

Su, X.

W. Li, X. Su, “Real-time calibration algorithm for phase shifting in phase-measuring philometry,” Opt. Eng. 40, 761–766 (2001).
[CrossRef]

Takeda, M.

Tanhauser, D. S.

S. G. Lipson, H. Lipson, D. S. Tanhauser, Optical Physics, 3rd ed. (Cambridge U. Press, Cambridge, UK, 1995), pp. 120–122.

Appl. Opt. (6)

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

W. Li, X. Su, “Real-time calibration algorithm for phase shifting in phase-measuring philometry,” Opt. Eng. 40, 761–766 (2001).
[CrossRef]

Opt. Laser Eng. (1)

M. A. Herraez, D. R. Burton, M. J. Lalor, “Accelerating fast Fourier transform and filtering operations in Fourier fringe analysis for accurate measurement of three dimensional surfaces,” Opt. Laser Eng. 31, 131–145 (1999).

Other (3)

S. G. Lipson, H. Lipson, D. S. Tanhauser, Optical Physics, 3rd ed. (Cambridge U. Press, Cambridge, UK, 1995), pp. 120–122.

E. Hecht, Optics, 3rd ed. (Addison-Wesley Longman, New York, 1998).

P. Hariharan, Optical Interferometry (Academic, Marrickville, Australia, 1985), pp. 157–159.

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Figures (12)

Fig. 1
Fig. 1

Schematic diagram of the Fourier pattern (a) at one exit of the interferometer, (b) at the other exit, (c) the subtraction of (b) and (c).

Fig. 2
Fig. 2

Schematic diagram of the experimental setup: 1, 15-mW He-Ne laser; 2, beam expander consisting of 3 and 5 positive lenses, and 4, pinhole; 6, 7, 8, and 9, mirrors; 10, 50–50 beam splitter; 11 and 12, CCDs (737 horizontal × 575 vertical pixels); 13, frame grabber; 14, imaging lenses; 15, investigated sample; 16, polarizer.

Fig. 3
Fig. 3

Sample consists of groups of five rectangular strips with vertical edges and unit mark-space ratio. The scale decreases by a factor of 2 from group to group. The numbers on the target indicate the period of the strips in micrometers.

Fig. 4
Fig. 4

Sample fragment on CCD1.

Fig. 5
Fig. 5

Interferometric pattern (a) at CCD1 and (b) at CCD2.

Fig. 6
Fig. 6

FFT pattern (one interferogram). Black indicates signals with intensity greater than half of the maximum.

Fig. 7
Fig. 7

(a) FFT pattern (difference FFA), after subtraction. (b) Profile along a line going through the three orders before (the left plot) and after (the right plot) the subtraction.

Fig. 8
Fig. 8

Phase map: (a) FFA, Gaussian width is 6 × 4 pixels; (b) advanced FFA, Gaussian width is 12 × 8 pixels. The black dashed lines symbolize the profile shown in Fig. 10.

Fig. 9
Fig. 9

Amplitude map: (a) FFA, Gaussian width is 6 × 4 pixels; (b) advanced FFA, Gaussian width is 12 × 8 pixels.

Fig. 10
Fig. 10

Phase profile for ordinary FFA (dashed curve) and advanced FFA (solid curve). x = 400, y = 1:500.

Fig. 11
Fig. 11

Smallest detail size d as a function of n, where n = ω0/ω.

Fig. 12
Fig. 12

MTF as a function of spatial frequency of the reconstructed sample for FFA with one interferogram (dashed curve) and subtraction FFA (solid curve). The spatial frequencies measured correspond to the sample frequencies available on the target.

Equations (26)

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gA1+cosω0x+φ,
gB1-cosω0x+φ,
GAδu, ν+δu-ω0δν * Fexpiφ+δu+ω0δν * Fexp-iφ,GBδu, ν-δu-ω0δν * Fexpiφ-δu+ω0δν * Fexp-iφ.
GA-GB/2=δu-ω0δν * Fexpiφ+δu+ω0δν * Fexp-iφ.
g01x, y=a1x, y+2b1x, ycosω0x+ϕ01x, y, g02x, y=a2x, y-2b2x, ycosω0x+ϕ02x, y;
g1x, y=d1x, ya1x, y+2b1x, ycosω0x+ϕ1x, y+ϕ01x, y, g2x, y=d2x, ya2x, y-2b2x, ycosω0x+ϕ2x, y+ϕ02x, y.
rx, y=|rx, y|expiφx, y,
amplitude=|rx, y| =d cosϕ01x, y+ϕ1x, y-ϕ02x, y+ϕ2x, ycosϕ01x, y-ϕ02x, y2, phase=anglerx, y=ϕ1x, y+ϕ2x, y2;
amplitude=|rx, y|=d1b1+d2b2b1+b2, phase=anglerx, y=ϕx, y.
ω0σ=3.
1+cosω0x+ωx.
δu+δu-ω0 * δu-ω+c.c.=δu+δu-ω0-ω+c.c.
ω0-ωσ=3.
ω0max=π2,
δϕ=exp-ω022σ2=0.01 rad.
ω0-ωσ=3,  d=σ-1=3ω0-ω=3ω0-ω0n=3nn-1ω0.
g01x, y=a1x, y+2b1x, ycosω0x+ϕ01x, y, g02x, y=a2x, y-2b2x, ycosω0x+ϕ02x, y.
g1x, y=d1x, ya1x, y+2b1x, ycosω0x+ϕ1x, y+ϕ01x, y, g2x, y=d2x, ya2x, y-2b2x, ycosω0x+ϕ2x, y+ϕ02x, y.
G01u, ν=A1u, ν+B1u, ν * δu-ω0δν * P01u, ν+δu+ω0δν * P01*-u, -ν,  G02u, ν=A2u, ν-B2u, ν * δu-ω0δν * P02u, ν+δu+ω0δν * P02*-u, -ν, G1u, ν=D1u, ν * A1u, ν+B1u, ν * δu-ω0δν * P01u, ν * P1u, ν+δu+ω0δν * P01*-u, -ν  * P1*-u, -ν,  G2u, ν=D2u, ν * A2u, ν-B2u, ν * δu-ω0δν * P02u, ν * P2u, ν+δu+ω0δν * P02-u, -ν * P2*-u, -ν.
G01u, ν-G02u, ν2=G012u, ν =A1u, ν-A2u, ν2+12 δu-ω0δν* B1u, ν* P01u, ν+B2u, ν* P01u, ν+c.c.,
G1u, ν-G2u, ν2   =G12u, ν   =D1u, ν * A1u, ν-D2u, ν * A2u, ν2+12 δu-ω0δν * D1u, ν * B1u, ν * P01u, ν * P1u, ν+D2u, ν * B2u, ν * P02u, ν * P2u, ν+c.c.
G012u, ν=12B1u, ν * P01u, ν+B2u, ν * P02u, ν, G12u, ν=12D1u, ν * B1u, ν * P01u, ν * P1u, ν+D2u, ν * B2u, ν * P02u, ν * P2u, ν.
g0x, y=12b1 expiϕ01x, y+b2 expiϕ02x, y, g12x, y=12d1b1 expiϕ01x, yexpiϕ1x, y+d2b2 expiϕ02x, yexpiϕ2x, y.
rx, y=g12x, yg0x, y=d1b1 expiϕ01x, yexpiϕ1x, y+d2b2 expiϕ02x, yexpiϕ2x, yb1 expiϕ01x, y+b2 expiϕ02x, y.
amplitude=|rx, y|=d cosϕ01x, y+ϕ1x, y-ϕ02x, y+ϕ2x, ycosϕ01x, y-ϕ02x, y2, phase=anglerx, y=ϕ1x, y+ϕ2x, y2.
amplitude=|rx, y|=d1b1+d2b2b1+b2, phase=anglerx, y=ϕx, y.

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