Abstract

A noncontact method for measuring the surface profile of a reflecting surface is presented. The scheme is based on the measurement of the phase difference between a surface reflected laser beam and a reference beam. Both beams are scanned past a photodetector by means of a rotating beam splitter. The slope of the surface at various points is electronically obtained, and then the surface profile in a plane is determined by direct integration. The accuracy of the method falls between the more elaborate interferometric techniques and mechanical contact schemes. A complete theoretical analysis of the method is presented as well as the results of various experimental tests.

© 1978 Optical Society of America

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References

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  1. T. P. Caudell, F. M. Smolka, Proc. Soc. Photo-Opt. Instrum. Eng. 115 (1977), no page number available.
  2. F. M. Smolka, H. A. Hill, Appl. Opt. 16, 292 (1977).
    [CrossRef] [PubMed]
  3. J. C. Wyant, Optical Testing (Optical Sciences Center, U. Ariz., 1976).
  4. D. Malacara, A. Cornejo, M. V. R. K. Murty, Appl. Opt. 14, 1065 (1975).
    [CrossRef] [PubMed]
  5. G. Randon, E. P. Wallerstein, Appl. Opt. 5, 737 (1966).
    [CrossRef]
  6. D. M. Meadows, W. O. Johnson, J. B. Allen, Appl. Opt. 9, 942 (1970).
    [CrossRef] [PubMed]
  7. J. D. Evans, Appl. Opt. 10, 995 (1971).
    [CrossRef]
  8. J. D. Evans, Appl. Opt. 11, 943 (1972).
    [CrossRef]
  9. N. Roth, Rev. Sci. Instrum. 40, 1509 (1969).
    [CrossRef] [PubMed]
  10. V. Bodlaj, E. Klement, Appl. Opt. 15, 1433 (1976).
    [CrossRef]
  11. A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971).
  12. H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).
  13. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  14. J. D. Zook, T. C. Lee, Appl. Opt. 11, 2140 (1972).
    [CrossRef] [PubMed]
  15. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (GPO, Washington, D.C., 1964).
  16. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), p. 43.

1977 (2)

T. P. Caudell, F. M. Smolka, Proc. Soc. Photo-Opt. Instrum. Eng. 115 (1977), no page number available.

F. M. Smolka, H. A. Hill, Appl. Opt. 16, 292 (1977).
[CrossRef] [PubMed]

1976 (1)

V. Bodlaj, E. Klement, Appl. Opt. 15, 1433 (1976).
[CrossRef]

1975 (1)

1972 (2)

1971 (1)

1970 (1)

1969 (1)

N. Roth, Rev. Sci. Instrum. 40, 1509 (1969).
[CrossRef] [PubMed]

1966 (2)

1965 (1)

H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).

Allen, J. B.

Bodlaj, V.

V. Bodlaj, E. Klement, Appl. Opt. 15, 1433 (1976).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), p. 43.

Caudell, T. P.

T. P. Caudell, F. M. Smolka, Proc. Soc. Photo-Opt. Instrum. Eng. 115 (1977), no page number available.

Cornejo, A.

Evans, J. D.

J. D. Evans, Appl. Opt. 11, 943 (1972).
[CrossRef]

J. D. Evans, Appl. Opt. 10, 995 (1971).
[CrossRef]

Hill, H. A.

Johnson, W. O.

Klement, E.

V. Bodlaj, E. Klement, Appl. Opt. 15, 1433 (1976).
[CrossRef]

Kogelnik, H.

H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
[CrossRef] [PubMed]

H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).

Lee, T. C.

Li, T.

Malacara, D.

Meadows, D. M.

Murty, M. V. R. K.

Randon, G.

Roth, N.

N. Roth, Rev. Sci. Instrum. 40, 1509 (1969).
[CrossRef] [PubMed]

Siegman, A. E.

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971).

Smolka, F. M.

T. P. Caudell, F. M. Smolka, Proc. Soc. Photo-Opt. Instrum. Eng. 115 (1977), no page number available.

F. M. Smolka, H. A. Hill, Appl. Opt. 16, 292 (1977).
[CrossRef] [PubMed]

Wallerstein, E. P.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), p. 43.

Wyant, J. C.

J. C. Wyant, Optical Testing (Optical Sciences Center, U. Ariz., 1976).

Zook, J. D.

Appl. Opt. (9)

Bell Syst. Tech. J. (1)

H. Kogelnik, Bell Syst. Tech. J. 44, 455 (1965).

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

T. P. Caudell, F. M. Smolka, Proc. Soc. Photo-Opt. Instrum. Eng. 115 (1977), no page number available.

Rev. Sci. Instrum. (1)

N. Roth, Rev. Sci. Instrum. 40, 1509 (1969).
[CrossRef] [PubMed]

Other (4)

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971).

J. C. Wyant, Optical Testing (Optical Sciences Center, U. Ariz., 1976).

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (GPO, Washington, D.C., 1964).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), p. 43.

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Figures (9)

Fig. 1
Fig. 1

Geometrical configuration of the surface profile measurement scheme. The rotating beam splitter sweeps the laser beam past the detector. (a) The directly reflected beam hits the detector; (b) the surface reflected beam is directed to the detector. The phase relationship between these two pulses gives information about the tilt angle.

Fig. 2
Fig. 2

The change in the phase between the two pulses as a function of tilt angle α. The various curves represent different values of the geometrical parameter h/t.

Fig. 3
Fig. 3

Calculated pulse shape as a function of various detector size parameters (Ω) and for a fixed beam size (Ө). τ0 is the pulse location for an infinitely small detector and beam size.

Fig. 4
Fig. 4

Calculated pulse shape as a function of various beam size parameters (Ө) and for a fixed detector size (Ω). τ0 is the pulse location for an infinitely small detector and beam size.

Fig. 5
Fig. 5

Experimental phase change angle δ for different tilt angles α and for two values of t.

Fig. 6
Fig. 6

Experimental train of pulses for the calibration type experiments. The small pulse is for the surface reflected signal and the large one for the directly reflected beam.

Fig. 7
Fig. 7

Experimental surface reflected pulse shape.

Fig. 8
Fig. 8

Ratio of surface reflected to directly reflected pulse height as a function of tilt angle. The laser beam was polarized at 30° with respect to the incidence plane.

Fig. 9
Fig. 9

Measured surface profile of a planoconvex glass lens. The statistical error bars are smaller than the dot size.

Equations (21)

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d y / d x = tan α .
h t = sin 2 α sin ( γ 0 2 β r 2 α ) .
β d = ( π + γ 0 ) / 2 .
h t = sin 2 α sin [ 2 α 2 ( β d β r ) ] .
β d β r = ( π / 2 ) + δ ,
h t = sin 2 α sin ( 2 δ 2 α ) .
u = t sin 2 α sin ( 2 α + β r ) .
α m = ( d / t ) sin ( γ 0 / 2 ) 2 + ( d / t ) ( S 2 ) cos ( γ 0 / 2 ) ,
S 1 + 1 / ( d / t ) .
d V d t + V R C + i d C = 0 ,
V ( t ) = ( χ C ) exp ( t / R C ) 0 t L ( ξ ) exp ( ξ / R C ) d ξ .
L ( t ) = I ( t ν ) D ( ν ) d ν / D ( ν ) d ν ,
I ( t ) = 1 π σ exp ( t 2 σ 2 ) ,
W 2 = W 1 2 { [ 1 + Δ z ( 1 R 2 R S ) ] 2 + ( λ Δ z π w 1 2 ) 2 } ,
V ( t ) = [ χ / 4 C Ω ] ( υ 1 + υ 2 + υ 3 υ 4 ) ,
υ 1 = erf ( τ τ 0 + Ω Ө ) erf ( τ τ 0 Ω Ө ) ,
υ 2 = exp ( τ ) [ erf ( τ 0 Ω Ө ) erf ( τ 0 + Ω Ө ) ] ,
υ 3 = exp ( τ 0 τ Ω + Ө 2 4 ) [ erf ( τ 0 τ Ω Ө Ө 2 ) erf ( τ 0 Ω Ө + Ө 2 ) ] ,
υ 4 = exp ( τ 0 τ Ω + Ө 2 4 ) [ erf ( τ 0 τ + Ω Ө + Ө 2 ) erf ( τ 0 + Ω Ө + Ө 2 ) ] ,
δ ( d y d x ) = | R C ω S m | ,
f ( x j ) = i = 1 j tan α i · Δ x i ,

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