Abstract

A fringe-projection system for microscopic applications, fringe-projecting microscopy, is developed and analyzed. Projection of the grating and imaging of the fringe system, modulated by the surface, are accomplished by the same high-aperture objective. The spectrum of the grating is spatially filtered and projected into the aperture with a lateral shift, which leads to a telecentric projection under oblique incidence and telecentric imaging. Topographies of specularly as well as diffusely reflecting surfaces can be obtained. The measurement of highly rough surfaces is described together with preprocessing steps. The resulting intensity distribution of the fringes is analyzed. Formulas for vertical and lateral resolution, measuring range, and dynamic range, based on noise considerations, are presented and verified by topographies of technical surfaces.

© 1994 Optical Society of America

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References

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  1. M. Takeda, K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977–3982 (1983).
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    [CrossRef] [PubMed]
  3. S. Toyooka, Y. Iwaasa, “Automated profilometry of 3-D diffuse objects by spatial phase detection,” Appl. Opt. 25, 1630–1633 (1936).
    [CrossRef]
  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. R. W. Wygant, S. P. Almeida, O. D. D. Soares, “Surface inspection via projection interferometry,” Appl. Opt. 27, 4626–4630 (1988).
    [CrossRef] [PubMed]
  6. G. Schmalz, Technische Oberflaechenkunde (Springer, Berlin, 1936).
  7. J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. XXVII, pp. 271–357.
    [CrossRef]
  8. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
  9. J. Li, X.-J. Su, L.-R. Guo, “Improved Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29, 1439–1444 (1990).
    [CrossRef]
  10. K. Creath, “Phase-measurement techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. XXVI, pp. 349–393.
    [CrossRef]
  11. K. Leonhardt, “Topometrie rauher technischer Oberflaechen durch Streifenprojektion in mikroskopischen Strahlengaengen,” presented at Jahrestagung der Deutsche Gesellscahft für augewandte Optik, Oldenburg, Germany, 21–25 May 1991; T. Ittner, “Strukturierte Beleuchtung für 3-D Topographien im Mikroskop,” Ph.D. dissertation (Institut fuer Technische Optik, Universitaet Stuttgart, Stuttgart, Germany, 1989).
  12. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

1990

J. Li, X.-J. Su, L.-R. Guo, “Improved Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29, 1439–1444 (1990).
[CrossRef]

1988

1984

1983

1982

1974

1936

Almeida, S. P.

Brangaccio, D. J.

Bruning, J. H.

Creath, K.

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

Gallagher, J. E.

Guo, L.-R.

J. Li, X.-J. Su, L.-R. Guo, “Improved Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29, 1439–1444 (1990).
[CrossRef]

Halioua, M.

Herriott, D. R.

Ina, H.

Iwaasa, Y.

Kobayashi, S.

Leonhardt, K.

K. Leonhardt, “Topometrie rauher technischer Oberflaechen durch Streifenprojektion in mikroskopischen Strahlengaengen,” presented at Jahrestagung der Deutsche Gesellscahft für augewandte Optik, Oldenburg, Germany, 21–25 May 1991; T. Ittner, “Strukturierte Beleuchtung für 3-D Topographien im Mikroskop,” Ph.D. dissertation (Institut fuer Technische Optik, Universitaet Stuttgart, Stuttgart, Germany, 1989).

Li, J.

J. Li, X.-J. Su, L.-R. Guo, “Improved Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29, 1439–1444 (1990).
[CrossRef]

Liu, H. C.

Mutoh, K.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

Rosenfeld, D. P.

Schmalz, G.

G. Schmalz, Technische Oberflaechenkunde (Springer, Berlin, 1936).

Schwider, J.

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. XXVII, pp. 271–357.
[CrossRef]

Soares, O. D. D.

Srinivasan, V.

Su, X.-J.

J. Li, X.-J. Su, L.-R. Guo, “Improved Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29, 1439–1444 (1990).
[CrossRef]

Takeda, M.

Toyooka, S.

White, A. D.

Wygant, R. W.

Appl. Opt.

J. Opt. Soc. Am.

Opt. Eng.

J. Li, X.-J. Su, L.-R. Guo, “Improved Fourier transform profilometry for the automatic measurement of three-dimensional object shapes,” Opt. Eng. 29, 1439–1444 (1990).
[CrossRef]

Other

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

K. Leonhardt, “Topometrie rauher technischer Oberflaechen durch Streifenprojektion in mikroskopischen Strahlengaengen,” presented at Jahrestagung der Deutsche Gesellscahft für augewandte Optik, Oldenburg, Germany, 21–25 May 1991; T. Ittner, “Strukturierte Beleuchtung für 3-D Topographien im Mikroskop,” Ph.D. dissertation (Institut fuer Technische Optik, Universitaet Stuttgart, Stuttgart, Germany, 1989).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

G. Schmalz, Technische Oberflaechenkunde (Springer, Berlin, 1936).

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. XXVII, pp. 271–357.
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Schematic of fringe-projecting microscopy in a symmetric, single-aperture configuration: CCD, image plane with camera; OC, tube lens; O1, relay lens for the grating spectrum; F, filtering plane of the grating spectrum; G, grating plane; P, entrance pupil with lens-shaped aperture; L, lamp; BS, beam splitter; EP, aperture plane of objective MO with the spectrum of the projected grating; MO, microscope objective.

Fig. 2
Fig. 2

Grating spectrum in aperture EP. The −1st, 0th, and +1st orders of the grating spectrum are transmitted (hatched lens-shaped areas). The ±2nd orders are practically missing, and orders higher than ±2 are filtered out by the slit in plane F, The position of the borders of the slits are marked by S1–S1 and S2–S2.

Fig. 3
Fig. 3

Simplified arrangement of the grating projection and the spatial filtering showing the parameters of the theoretical analysis. Relay lens O1 is omitted for simplicity: G, grating plane; F, filtering plane conjugated to aperture plane EP; D, aperture diameter of MO; d, transversal shift of the grating spectrum.

Fig. 4
Fig. 4

Wave fronts of the underlying three-beam interference on a step CDAB on the surface. The path differences introduced by the step are 2h cos β i , i = −1, 0, +1.

Fig. 5
Fig. 5

Intensity distribution of the filtered and projected grating lines [Eq. (9)]: a, surface in focus; b, object height Δh = 28.5 μm, which corresponds to a focus deviation of W 20 = 12λ; c, very strong defocusing (practically irrelevant in applications). Only the second harmonic in Eq. (9) is effective. This corresponds to the Talbot effect for on-axis, free-space propagation of gratings.

Fig. 6
Fig. 6

Photograph of a monitor display with deformed and disrupted grating lines that appear when the deep-drawing sheet steel surface of Fig. 11 is measured.

Fig. 7
Fig. 7

Simulation of the topography of a staircase object under illumination with an inhomogeneous intensity distribution (low-frequency multiplicative noise) to show erroneous height modulation in the form of waves parallel to the grating lines if the direct process or the π-shift-and-subtraction process is used: (a) simulation of the object height field h(x, y), (b) calculated intensity distribution I(x, y) according Eqs. (16) with a linearly increasing multiplicative factor in the x direction, (c) topography h(x, y) calculated from (b) with the resulting erroneous waviness of the same period as the projected grating. If the π-shift-and-division process is used, no waviness in the topography occurs.

Fig. 8
Fig. 8

Intensity signal of the π-shift-and-division process, J div(x) [Eq. (25)], for a plane surface: c a = 0.05, r m = r a = 0, and 0.4 < γ < 1.0 (as a parameter).

Fig. 9
Fig. 9

Noise topography obtained, with a 50 × 0.85 microscope objective, by the π-shift-and-division process: (a) topography of a high-quality plane mirror after subtraction of a plane, fitted for a least-square deviation. (b) Autocorrelation function (ACF): vertical resolution δ h = ACF(0, 0)0,5 = 7.1 nm; the autocorrelation widths, which are defined as e −1 − width, are w x = 3.1 μm and w y = 1.8 μm.

Fig. 10
Fig. 10

Groove 6 of a PTB calibration standard. The certified depth is 9.00 ± 0.05 μm; the peak-to-valley value over the whole topography is evaluated as 9.08 μm.

Fig. 11
Fig. 11

Oil-bearing structure of deep-drawing sheet steel: (a) topography, (b) cross section, (c) histogram with bearing-area diagram.

Fig. 12
Fig. 12

Microchip topography measured with a 50 × 0.85 objective.

Equations (61)

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u ( x g ) = exp ( - i 2 π x g d λ f 2 ) [ n = - n = + δ ( n p g - x g ) * rect ( 2 x g p g ) ] × rect ( x g w g ) ,
u ˜ ( f x ) = δ ( f x - f x 0 ) * [ i = - + 1 p g δ ( f x - i p g ) p g 2 sinc ( f x p g 2 ) ] * w g sinc ( f x w g ) ,
u ˜ ( f x ) = w g 2 [ s δ ( f x - f x - 1 ) + s 0 δ ( f x - f x 0 ) + s δ ( f x - f x + 1 ) ] ,
s 0 = sinc ( 0 ) = 1 , s 1 = s - 1 = s = sinc ( 0.5 ) = 0.637.
u ( x ) = w g 2 j = - 1 1 s j exp { i k [ ( d f MO - λ p g j ) x - 2 cos β j h ( x ) ] } .
ϕ j = k W ( x f ) δ ( x f - x f j ) ,
W j = x r j 2 W 20 ,             j = - 1 , 0 , + 1 , x r j = x f j D / 2 ,
u ( x ) = w g 2 j = - 1 1 s j exp { i k [ ( d f MO - λ p g j ) x - 2 cos β j h ( x ) + W j ] } .
I ( x , y ) = I 0 ( 1 + 2 s cos α ) 2 = I 0 ( 1 + 2 s 2 + 4 s cos α + 2 s 2 cos 2 α ) ,
α ( x ) = 2 π p [ x - 2 h ( x ) sin β ] + k Δ W , Δ W ( W 3 - W 1 ) / 2.
Δ h = Δ 2 sin β ,             sin β = d f MO ,
J f ( x i , Y ) = a ( x i , Y ) + a 1 ( x i , Y ) cos ( 2 π i N ) + b 1 ( x i , Y ) sin ( 2 π i N ) ,
a 1 ( x i , Y ) = N 2 j = i i + N - 1 J ( x i , Y ) cos ( 2 π j / N ) ,
b 1 ( x i , Y ) = N 2 j = i i + N - 1 J ( x j , Y j ) sin ( 2 π j / N ) ,
a ( x i , Y ) = N j = i i + N - 1 J ( x j , Y ) ,
h ( x i , Y ) = p 4 π sin β arctan b 1 ( x i , Y ) a 1 ( x i , Y ) ,
I i = I 0 [ 1 + s 2 + 4 s cos ( α + Θ i ) + 2 s 2 cos ( 2 α + 2 Θ i ) ] ,             Θ i = 0 , π / 2 , π , 2 π / 3 , h ( x i , y i ) = p 4 π sin β arctan I 4 - I 2 I 1 - I 3 .
J n ( x ) = J ( x ) n m ( x ) + n a ( x ) ,
n a ( x ) = J [ c a + r a ( x ) ] ;             0 < c a 1 ,             r a max < c a , E { r a } = 0 ,             E { r a 2 } = σ a 2 ,
n m ( x ) = 1 + r m ( x ) ;             r m 1 ,             E { r m } = 0 , E { r m 2 } = σ m 2 .
J s ( X , y ) = 1 m u = - k + k J ( X , y - u ) ;             k = ( m - 1 ) / 2 ,             m = 3 , 5 , 7 , .
J sub = [ 1 + 2 s 2 + 4 s cos ( α ) + 2 s 2 cos ( 2 α ) ] n m + n a - [ 1 + 2 s 2 - 4 s cos ( α ) + 2 s 2 cos ( 2 α ) ] n m - n a , J sub ( x ) = 8 s cos ( α ) n m ( x ) .
S 2 = 1 2 σ m 2 .
J div ( x ) = { 1 + 2 s γ cos [ α ( x ) ] } 2 n m ( x ) + c a + r a ( x ) { 1 - 2 s γ cos [ α ( x ) ] } 2 n m ( x ) + c a + r a ( x ) .
n a ( x ) I 0 n m ( x ) = v ( x ) c a + r a ( x ) - c a r m ( x ) ,
J div ( x ) = 1 + v ( x ) + 4 s γ cos α ( x ) + 4 s 2 γ 2 cos 2 α ( x ) 1 + v ( x ) - 4 s γ cos α ( x ) + 4 s 2 γ 2 cos 2 α ( x ) .
J div ( α ) = 1 + 8 s γ cos ( α ) [ 1 - c a - r a ( α ) + c a r m ( α ) ] + .
J ˜ ( f x ) = δ ( f x ) + 1 p 8 ( 1 - c a ) s γ II ( f x p ) + 1 p 8 s γ c a r ˜ m ( f x ) * II ( f x p ) - 1 p 8 s γ r ˜ a ( f x ) * II ( f x p ) .
S div 2 = ( 1 - c a ) 2 2 ( c a 2 σ m 2 + σ a 2 ) ,
h ( x i , y i ) = h ( x i , y i ) - h ref ( x i , y i ) .
h ( x i , y i ) = [ a I ( x i , y i ) + a I π ( x i , y i ) ] h I ( x i , y i ) + [ a II ( x i , y i ) + a II π ( x i , y i ) ] h II ( x i , y i ) a I ( x i , y i ) + a I π ( x i , y i ) + a II ( x i , y i ) + a II π ( x i , y i ) .
h i = p 4 π sin β ( b i a i ) .
R h h ( t ) = 1 N t i = 1 N t h i h i + t = p 2 16 π 2 sin 2 β N t i = 1 N t b i b i + t a i a i + t ,
R h h ( t ) p 2 16 8 s 2 sin 2 β π 2 N { ( 1 + 2 s 2 ) 2 [ R m ( t ) + R a ( t ) ] + ( 8 s 2 + 2 s 4 ) R m ( t ) } .
S 2 = 8 s 2 ( 1 + 2 s 2 ) ( σ m 2 + σ a 2 ) + ( 8 s 2 + 4 s 4 ) σ m 2 ,
δ h dir = σ h = p 4 π sin β S N ,
δ h dir = Δ x c N M - 1 4 π sin β S = N L 4 π sin β N tot S .
J j = 0.5 [ 1 + K cos ( 2 π N j ) ] ( 1 + r m ) + c a + r a ,
δ h coh = p 4 π sin β N S .
S 2 = K 2 / [ 2 ( σ m 2 + σ a 2 ) + K 2 σ m 2 / 2 ] - 1 .
R h h ( t ) = p 2 σ m 2 16 π 2 sin 2 β N cos ( 2 t ) .
δ h sub = p 4 2 π sin β N S = N L 4 2 π sin β N t S ,
δ h div = p 4 2 π sin β S N = N L 4 2 π sin β N t S ,
S div 2 = S dir 2 = ( 1 - c a ) 2 2 ( c a 2 σ m 2 + σ a 2 ) 1 2 σ a 2 .
H t = m H p ,             H p = p 2 sin β .
B dir coh = 2 m N π S .
B div sub = 2 2 m π S N .
δ h min = λ f MO 2 π N sin β S D c = λ 4 π N sin β ( N . A . - sin β ) S ,
δ h min = λ 4 π N sin β N . A . S .
J j = ( 1 + 2 s cos α j ) 2 ( 1 + r m j ) + J ( C a + r a j ) ; J = 1 + 2 s 2 .
b i = j = i i + N - 1 [ ( 1 + 2 s 2 ) ( 1 + c a ) sin α j ] + ( 4 s cos α j + 2 s 2 cos 2 α j ) sin α j + ( 1 + 2 s 2 ) ( r m j + r a j ) sin α j + ( 4 s cos     α j + 2 s 2 cos 2 α j ) r m j sin α j ] .
j = 1 N sin ( 2 π k j N ) = j = 1 N cos ( 2 π l j N ) = j = 1 N sin ( 2 π k j N ) cos ( 2 π l j N ) = 0             k , l = ± 1 , ± 2 , , j = 1 N cos ( 2 π k j N ) cos ( 2 π l j N ) = j = 1 N sin ( 2 π k j N ) sin ( 2 π l j N ) = N 2 δ k l ,
a i j = 1 i + N - 1 4 s ( cos α j ) 2 = 4 s N 1 2 ,
b i b i + t a i a i + t 1 4 s 2 N 2 j 1 = i i + N - 1 j 2 = i + t i + t + N - 1 { [ ( 1 + 2 s 2 ) 2 × ( r m j 1 r m j 2 + r a j 1 r a j 2 ) ] sin α j 1 sin α j 2 + 16 s 2 r m j 1 r m j 2 cos α j 1 sin α j 1 cos α j 2 sin α j 2 + 4 s 4 r m j 1 r m j 2 cos 2 α j 1 sin α j 1 sin α j 2 cos 2 α j 2 } .
R h h ( t ) p 2 64 s 2 sin 2 β π 2 N 2 N t j 1 = 1 N j 2 = t t + N N t { ( 1 + 2 s 2 ) 2 × [ R m ( t ) + R a ( t ) ] sin α j 1 sin α j 2 δ j 1 j 2 + 16 s 2 R m ( t ) sin α j 1 sin α j 2 cos α j 1 cos α j 2 δ j 1 j 2 + 4 s 2 R m ( t ) sin α j 1 sin α j 2 cos 2 α j 1 cos 2 α j 2 δ j 1 j 2 } ,
R h h p 2 64 s 2 sin 2 β π 2 N 2 { ( 1 + 2 s 2 ) 2 × [ R m ( 2 π t N ) + 4 R a ( 2 π t N ) ] j = 1 N ( sin α j ) 2 + 16 s 2 R m ( 2 π t N ) j = 1 N ( sin α j ) 2 ( cos α j ) 2 + 4 s 4 R m ( 2 π t N ) j = 1 N ( sin α j ) 2 ( cos 2 α j ) 2 } ,
R h h ( t ) p 2 64 s 2 sin 2 β π 2 N × { ( 1 + 2 s 2 ) 2 [ R m ( 2 π t N ) + R a ( 2 π t N ) ] 1 2 + 4 s 2 R m ( 2 π t N ) + s 4 R m ( 2 π t N ) } .
b j = 0.5 j = 1 i + N - 1 [ ( r m j + r a j ) sin α j + r m j K cos α j sin α j ] ,
a i 0.5 Σ K cos 2 α j = ¼ K N .
R h h ( t ) p 2 4 K 2 N 2 N t i = 1 N t j 1 = i i + N - 1 j 2 = i + t i + t + N - 1 ( r m j 1 + 2 r a j 1 ) × ( r m j 2 + r a j 2 ) ] sin α j 1 sin α j 2 + K ( r m j 1 + r a j 1 ) r m j 2 sin α j 1 cos α j 2 sin α j 2 + K ( r m j 2 + r a j 2 ) r m j 1 sin α j 2 cos α j 1 sin α j 1 + K 2 r m j 1 r m j 2 cos α j 1 sin α j 1 cos α j 2 sin α j 2 ,
R h h ( t ) p 2 8 π 2 sin 2 β N K 2 ( R m + R a + 1 2 K 2 R m ) .

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