Abstract

Fourier-transform profilometry (FTP) and data-dependent systems profilometry (DDSP) are two methods that are available for recovering one-dimensional fine surface profiles from the phase of a single interferogram. FTP has already been extended to two-dimensional surfaces; a similar extension of DDSP is introduced here. Inasmuch as this extension involves autoregressive modeling of the rows or columns of an interferogram, the feasibility of using a common model order is explored. The common order reduces not only the amount of computation but also the errors caused by the heterodyned phase-removal procedure. As autoregression requires masking the first few data values, the length of the mask is determined by means of a Green’s function. A comparison shows that DDSP outperforms FTP in roughness measurements in terms of rms and center-line average. The comparison also shows that DDSP is able to recover a detailed surface, whereas FTP outlines only the global features. An interferogram regeneration procedure provides a reference surface for the verification of results.

© 1999 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 topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  2. S. M. Pandit, N. Jordache, “Interferogram analysis based on the data-dependent systems method for nanometrology applications,” Appl. Opt. 34, 6695–6703 (1995).
    [CrossRef] [PubMed]
  3. 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] [PubMed]
  4. S. M. Pandit, D. P. Chan, “Comparison of Fourier-transform and data-dependent systems profilometry using interferometric regeneration,” Appl. Opt. 38, 4095–4102 (1999).
    [CrossRef]
  5. S. M. Pandit, S. M. Wu, Time Series and System Analysis with Applications (Wiley, New York, 1983; reprinted with corrections by Krieger, Malabar, Fla., 1993).
  6. M. Kujawinska, “Spatial phase measurement,” in Interferogram Analysis, D. Robinson, G. T. Reid, eds. (Institute of Physics, University of Reading, Berkshire, UK, 1993), pp. 141–193.
  7. S. M. Pandit, N. Jordache, “Data-dependent-systems and Fourier-transform methods for single-interferogram analysis,” Appl. Opt. 34, 5945–5951 (1995).
    [CrossRef] [PubMed]
  8. S. M. Pandit, N. Jordache, G. A. Joshi, “Data-dependent systems methodology for noise-insensitive phase unwrapping in laser interferometric surface characterization,” J. Opt. Soc. Am. A 11, 2584–2592 (1994).
    [CrossRef]
  9. S. M. Pandit, Modal and Spectrum Analysis: Data Dependent Systems in State Space (Wiley, New York, 1991).

1999 (1)

1995 (2)

1994 (1)

1986 (1)

1982 (1)

Bachor, H. A.

Bone, D. J.

Chan, D. P.

Ina, H.

Jordache, N.

Joshi, G. A.

Kobayashi, S.

Kujawinska, M.

M. Kujawinska, “Spatial phase measurement,” in Interferogram Analysis, D. Robinson, G. T. Reid, eds. (Institute of Physics, University of Reading, Berkshire, UK, 1993), pp. 141–193.

Pandit, S. M.

Sandeman, R. J.

Takeda, M.

Wu, S. M.

S. M. Pandit, S. M. Wu, Time Series and System Analysis with Applications (Wiley, New York, 1983; reprinted with corrections by Krieger, Malabar, Fla., 1993).

Appl. Opt. (4)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Other (3)

S. M. Pandit, S. M. Wu, Time Series and System Analysis with Applications (Wiley, New York, 1983; reprinted with corrections by Krieger, Malabar, Fla., 1993).

M. Kujawinska, “Spatial phase measurement,” in Interferogram Analysis, D. Robinson, G. T. Reid, eds. (Institute of Physics, University of Reading, Berkshire, UK, 1993), pp. 141–193.

S. M. Pandit, Modal and Spectrum Analysis: Data Dependent Systems in State Space (Wiley, New York, 1991).

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Figures (19)

Fig. 1
Fig. 1

2-D Fourier spectrum of an interferogram.

Fig. 2
Fig. 2

Y-axis view of a Fourier spectrum.

Fig. 3
Fig. 3

X-axis view of a Fourier spectrum.

Fig. 4
Fig. 4

Flow chart of the interferogram regeneration procedure.

Fig. 5
Fig. 5

Interferogram of a ground surface.

Fig. 6
Fig. 6

Intensity plot of column 220.

Fig. 7
Fig. 7

Regenerated intensity plot of Column 220.

Fig. 8
Fig. 8

Comparison of profiles recovered with different model orders.

Fig. 9
Fig. 9

2-D reference surface.

Fig. 10
Fig. 10

Regenerated 2-D interferogram.

Fig. 11
Fig. 11

Reference and recovered profiles of column 50.

Fig. 12
Fig. 12

Reference and recovered profiles of column 80.

Fig. 13
Fig. 13

Unwrapped phase obtained with different model orders.

Fig. 14
Fig. 14

Masking zones in DDSP.

Fig. 15
Fig. 15

Comparison of FTP and DDSP recovered profiles of column 50.

Fig. 16
Fig. 16

Comparison of FTP and DDSP recovered profiles of row 200.

Fig. 17
Fig. 17

2-D surface recovered by DDSP.

Fig. 18
Fig. 18

2-D surface recovered by FTP.

Fig. 19
Fig. 19

Flow chart of DDSP procedure for 2-D surface recovery from an interferogram image.

Tables (4)

Tables Icon

Table 1 RSS for Column 220 Recovered with Different Orders

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Table 2 Orders of Lowest-Order Adequate Models for Selected Columns

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Table 3 RMS and RA Values of Columns 50 and 80 Obtained with a Common Model Order

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Table 4 RMS and RA Comparisons for 2-D Surfaces Recovered by Means of FTP and DDSP

Equations (20)

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Ix, y=k22I0x, y+Γx, y+Γ*x, y,
Ix, y=2k2I0x, y1+γˆx, ycosϕx, y+ϕ0+υx, y,
Ir=br+mrcosϕsr+2πf0·r,
If=Bf+0.5MQf-f0+0.5MQ*-f-f0,
Ix=ϕ1Ix-1+ϕ2Ix-2 +ϕnIx-n+ax,
Γx=j=0x giλijax-j,
Γx=λiΓx-1+giax,
gi=λin-11jinλi-λj.
Υ0=d1+d2++dn,
di=gig11-λiλ1+gig21-λiλ2++gign1-λiλnσa2
argΓr=atanImΓr/ReΓr.
ϕ0x, y=ϕ0yx, y-ϕ0yx, L+ϕ0xx, L,
ω˜=A¯θ+a
θˆ=A¯A¯-1A¯ω
υr=υxy=θˆ1y+θˆ2x+θˆ3.
h=λϕr/4π,
γˆr=|Γr|max|Γr|.
Gx=λsurfx
τ=-1lnλsurf.
xmask=n+4.5τ,

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