Abstract

The noncontact optical measuring method is based on the focusing properties of simple optical elements. High sensitivity is achieved by the utilization of differential detection and electronic feedback systems. Preliminary measurements indicate that height sensitivities of the order of 0.1 μm with spatial resolution of better than 2 μm are easily obtained, while measurements in the millimeter range are also feasible. For optimization purposes, a mathematical analysis has been performed using operator algebra and the characteristics of the Wigner distribution function.

© 1982 Optical Society of America

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References

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    [CrossRef]
  5. S. Uchida, H. Sato, M. O-hori, Ann. CIRP 28, 419 (1979).
  6. M. Shiraishi, Bull. Japan Soc. Prec. Engg. 13, 133 (1979).
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    [CrossRef]
  9. Y. Fainman, E. Lenz, J. Shamir, Appl. Opt. 20, 158 (1981).
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  10. H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
    [CrossRef]
  11. A. Yariv, Introduction to Optical Electronics (Holt, Reinhart & Winston, New York, 1971).
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    [CrossRef]
  13. M. Nazarathy, J. Shamir, J. Opt. Soc. Am. 70, 150 (1980).
    [CrossRef]
  14. M. J. Bastiaans, “The Wigner Distribution Function and Its Applications to Optics,” ICO Mexico 1980 Conference on Optics in Four Dimensions (Pergamon, New York, 1980).
  15. H. O. Bartelt, H. H. Brenner, A. W. Lohnmann: Opt. Commun. 32, 32 (1980).
    [CrossRef]
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  17. J. C. Dainty, Prog. Opt. 14, 3 (1976).
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  20. S. Lowenthal, H. H. Arsenault, J. Opt. Soc. Am. 60, 1478 (1970).
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  21. J. Uozumi, H. Fujii, T. Asakura, J. Opt. Soc. Am. 67, 808 (1977).
    [CrossRef]

1981

1980

M. Nazarathy, J. Shamir, J. Opt. Soc. Am. 70, 150 (1980).
[CrossRef]

L. Parker, Laser Focus, 40–46, (1980).

J. Shamir, Opt. Eng. 19, 801 (1980).
[CrossRef]

J. Uozumi, T. Asakura, Opt. Acta 27, 1345 (1980).
[CrossRef]

H. O. Bartelt, H. H. Brenner, A. W. Lohnmann: Opt. Commun. 32, 32 (1980).
[CrossRef]

1979

F. T. Arecchi, D. Bertani, S. Ciliberto, Opt. Commun. 31, 263 (1979).
[CrossRef]

S. Uchida, H. Sato, M. O-hori, Ann. CIRP 28, 419 (1979).

M. Shiraishi, Bull. Japan Soc. Prec. Engg. 13, 133 (1979).

1978

T. Sawatari, R. B. Zipin, Proc. Soc. Photo-Opt. Eng. 153, 8 (1978).

1977

1976

1970

1966

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

Arecchi, F. T.

F. T. Arecchi, D. Bertani, S. Ciliberto, Opt. Commun. 31, 263 (1979).
[CrossRef]

Arsenault, H. H.

Asakura, T.

Bartelt, H. O.

H. O. Bartelt, H. H. Brenner, A. W. Lohnmann: Opt. Commun. 32, 32 (1980).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, “The Wigner Distribution Function and Its Applications to Optics,” ICO Mexico 1980 Conference on Optics in Four Dimensions (Pergamon, New York, 1980).

Bennett, J. M.

Bertani, D.

F. T. Arecchi, D. Bertani, S. Ciliberto, Opt. Commun. 31, 263 (1979).
[CrossRef]

Brenner, H. H.

H. O. Bartelt, H. H. Brenner, A. W. Lohnmann: Opt. Commun. 32, 32 (1980).
[CrossRef]

Ciliberto, S.

F. T. Arecchi, D. Bertani, S. Ciliberto, Opt. Commun. 31, 263 (1979).
[CrossRef]

Dainty, J. C.

J. C. Dainty, Prog. Opt. 14, 3 (1976).

Dancy, J. H.

Fainman, Y.

Fujii, H.

Goodman, J. W.

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, Heidelberg, 1975).
[CrossRef]

Kogelnik, H.

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

H. Kogelnik, “Propagation of Laser Beams,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York, 1979).
[CrossRef]

Lenz, E.

Li, T.

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

Lohnmann, A. W.

H. O. Bartelt, H. H. Brenner, A. W. Lohnmann: Opt. Commun. 32, 32 (1980).
[CrossRef]

Lowenthal, S.

Nazarathy, M.

O-hori, M.

S. Uchida, H. Sato, M. O-hori, Ann. CIRP 28, 419 (1979).

Parker, L.

L. Parker, Laser Focus, 40–46, (1980).

Sato, H.

S. Uchida, H. Sato, M. O-hori, Ann. CIRP 28, 419 (1979).

Sawatari, T.

T. Sawatari, R. B. Zipin, Proc. Soc. Photo-Opt. Eng. 153, 8 (1978).

Shamir, J.

Shiraishi, M.

M. Shiraishi, Bull. Japan Soc. Prec. Engg. 13, 133 (1979).

Uchida, S.

S. Uchida, H. Sato, M. O-hori, Ann. CIRP 28, 419 (1979).

Uozumi, J.

Yariv, A.

A. Yariv, Introduction to Optical Electronics (Holt, Reinhart & Winston, New York, 1971).

Zipin, R. B.

T. Sawatari, R. B. Zipin, Proc. Soc. Photo-Opt. Eng. 153, 8 (1978).

Ann. CIRP

S. Uchida, H. Sato, M. O-hori, Ann. CIRP 28, 419 (1979).

Appl. Opt.

Bull. Japan Soc. Prec. Engg.

M. Shiraishi, Bull. Japan Soc. Prec. Engg. 13, 133 (1979).

J. Opt. Soc. Am.

Laser Focus

L. Parker, Laser Focus, 40–46, (1980).

Opt. Acta

J. Uozumi, T. Asakura, Opt. Acta 27, 1345 (1980).
[CrossRef]

Opt. Commun.

F. T. Arecchi, D. Bertani, S. Ciliberto, Opt. Commun. 31, 263 (1979).
[CrossRef]

H. O. Bartelt, H. H. Brenner, A. W. Lohnmann: Opt. Commun. 32, 32 (1980).
[CrossRef]

Opt. Eng.

J. Shamir, Opt. Eng. 19, 801 (1980).
[CrossRef]

Proc. IEEE

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

Proc. Soc. Photo-Opt. Eng.

T. Sawatari, R. B. Zipin, Proc. Soc. Photo-Opt. Eng. 153, 8 (1978).

Prog. Opt.

J. C. Dainty, Prog. Opt. 14, 3 (1976).

Other

A. Yariv, Introduction to Optical Electronics (Holt, Reinhart & Winston, New York, 1971).

H. Kogelnik, “Propagation of Laser Beams,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York, 1979).
[CrossRef]

M. J. Bastiaans, “The Wigner Distribution Function and Its Applications to Optics,” ICO Mexico 1980 Conference on Optics in Four Dimensions (Pergamon, New York, 1980).

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer, Heidelberg, 1975).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Schematic configuration of measuring system. The object, 0, mounted on an xy translation stage is illuminated by a collimated beam transmitted through beam splitter B1 and focused by lens L1. The filtered Fourier transform of the aperture, P, by filters, F, is analyzed by detectors, D, connected to differential amplifier, DA,

Fig. 2
Fig. 2

Sensitivity test of optical system: (a) Differential amplifier output with feedback disconnected as a function of mirror axial displacement; (b) Noise output of stationary system.

Fig. 3
Fig. 3

Measurements on triangular groove standard block: (a) Optical profilometer; (b) Stylus gauge with a 2.5 μm point; (c) Stylus with a 25-μm point.

Fig. 4
Fig. 4

Same as Fig. 3 but for a milled-steel surface

Fig. 5
Fig. 5

Optical measurement of rough emery-paper surface.

Fig. 6
Fig. 6

Optical measurement of a drill cutting edge. (Note the difference in the x and y scales.)

Fig. 7
Fig. 7

One arm of the optical system of Fig. 1: (a) The actual system without the beam splitter B2; (b) The unfolded equivalent form of (a).

Fig. 8
Fig. 8

The Wigner distribution function of a 1-D infinite aperture. The x-w plane has only meaning in an abstract mathematical space, while the actual field distribution is a delta function—a single point on the x axis. (a) Focused image, (b) misfocused image.

Fig. 9
Fig. 9

As in Fig. 8 but for a 1-D slit of infinite width. S represents the size of the central lobe of the sine function. (b) and (c) are misfocusing in opposite directions.

Fig. 10
Fig. 10

Profiling a triangular grid by a stylus point of radius r. The measure profile corresponds to the trajectory of the center of the curvature of the stylus tip. δa and δb are the measure for profile distortions.

Equations (40)

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w o f = f λ π w o i [ 1 + ( f λ / π w o i 2 ) ] - 1 / 2 ,
f / No . f / 2 w o i ,
w o f 2 λ f / No . π ,
2 w o f = 2 2 λ 2 π ( N . A . ) 500 nm .
w 2 ( z ) = w o f 2 [ 1 + ( z / b ) 2 ] ,
b = π λ w o f 2 ,
R ( z ) = z [ 1 + ( b / z ) 2 ] .
γ ( ρ ) exp [ ϕ ( ρ ) ] .
R [ Δ ] a ( ρ ) .
u r = γ ( ρ ) exp [ i ϕ ( ρ ) ] R [ Δ ] a ( ρ ) .
u 1 = V [ 1 λ f 1 ] F R [ Δ ] γ ( ρ ) exp [ i ϕ ( ρ ) ] R [ Δ ] a ( ρ ) ,
Q [ - 1 / f 2 ] exp [ - j k 2 f 2 ( x 2 + y 2 ) ] ,
u i = R [ d 2 + f 2 ] Q [ - 1 f 2 ] P ( ρ ) R [ d 1 + f 2 ] V [ 1 λ f 1 ] F R ( Δ ) γ ( ρ ) × exp [ i ϕ ( ρ ) R [ Δ ] a ( ρ ) .
u i R [ d 2 + f 2 ] P ( ρ ) Q [ - 1 f 2 ] R [ d 1 + f 2 ] V [ 1 λ f 1 ] F R [ 2 Δ ] δ ( ρ ) .
R [ d ] F - 1 Q [ - λ 2 d ] F
u i R [ d 2 + f 2 ] P ( ρ ) Q [ - 1 f 2 ] R [ d 1 + f 2 ] V [ 1 λ f 1 ] × F F - 1 Q [ - λ 2 2 Δ ] F δ ( ρ ) = R [ d 2 + f 2 ] P ( ρ ) Q [ - 1 f 2 ] R [ d 1 + f 2 ] Q [ - 2 Δ f 1 2 ] ,
R ( d ) Q ( 1 d ) V ( 1 λ d ) F Q ( 1 d ) ,
u 1 Q ( 1 d 2 + f 2 ) V [ 1 λ ( d 2 + f 2 ) ] F Q ( 1 d 2 + f 2 ) P ( ρ ) Q ( - 1 f 2 ) Q ( 1 f 2 ) × V ( 1 λ f 2 ) F Q ( 1 f 2 ) Q ( - 2 Δ f 1 2 ) .
u i Q ( 1 d 2 + f 2 ) V ( 1 λ ( d 2 + f 2 ) ) F P ( ρ ) × Q [ 1 d 2 + f 2 - 1 f 2 - 2 Δ f 2 2 / f 1 2 ] .
1 d 2 + f 2 - 1 f 2 - 2 Δ f 2 2 / f 1 2 = 0 ,
d 2 = - 2 Δ f 2 2 / f 1 2 .
u i Q ( 1 d 2 + f 2 ) V [ 1 λ ( f 2 + d 2 ) ] F P ( ρ ) ,
f 2 / f 1 1.
d 2 = - 2 Δ f 2 2 / f 1 2 .
d 2 f 2 ,
W ( x , w ) = - u ( x + ½ x ) u * ( x - ½ x ) exp ( - i w x ) d x
W ( x , w ) = - U ( w + ½ w ) U * ( w - ½ w ) exp ( - i x w ) d w ,
u ( x ) 2 = 1 2 π W ( x , w ) d w
U ( w ) 2 = W ( x , w ) d x .
W d ( x , w ) = W f ( x - d k w , w ) = W f ( x , w ) * δ ( x - d k w ) ,
I ( x ) = H ( x ) u d ( x ) 2 = 1 2 π H ( x ) - W d ( x , w ) d w = 1 2 π H ( x ) W d ( x , w ) d w .
E ( d ) = I ( x ) d x = 1 2 π H ( x ) W d ( x , w ) d x d w .
E ( d ) E ( 0 ) .
H ( x ) = { 1 ; u f ( x ) 2 0 0 ; u f ( x ) 2 = 0.
E ( d ) = - [ X W u ( x , w ) d x ] d w ,
W f ( x , w ) = δ ( x + ½ x ) δ * ( x - ½ x ) exp ( - i w x ) d x = δ ( x ) ,
W d ( x , w ) = δ ( x - d k w )
δ a = r / cos α - r ,
tan α = 2 a / b ,
δ b = 2 r sin α .

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