Abstract

We discuss the limits of optical range sensing by shearing interferometry for diffusely reflecting objects. The basic principle is the following: the radius of a wave that is scattered at the object under test is measured by shearing interferometry. This radius is the desired distance. We show that the limits of the method are mainly determined by speckle. With coherent light depth resolution cannot be increased considerably beyond the Rayleigh depth of focus. With partially coherent light a rms depth resolution of 68 μm at a distance of 380 mm (1:5500) was achieved. This resolution is 25 times better than the Rayleigh limit. The working aperture is very small (0.013); shading problems are minimized.

© 1988 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Häusler, J. Hutfless, M. Maul, H. Weissmann, “Range Sensing Based on Shearing Interferometry,” Appl. Opt. 27, 4638 (15Nov.1988).
    [CrossRef] [PubMed]
  2. O. Bryngdahl, “Applications of Shearing Interferometry,” Prog. Opt. 4, 37 (1965).
    [CrossRef]
  3. J. Dyson, “Image Position Measurement by Polarimetry,” Opt. Acta 10, 171 (1963).
    [CrossRef]
  4. J. N. Butters, J. A. Leendertz, “A Double Exposure Technique for Speckle Pattern Interferometry,” J. Phys. E 4, 277 (1971).
    [CrossRef]
  5. G. E. Sommargren, “Optical Heterodyne Profilometry,” Appl. Opt. 20, 610 (1981).
    [CrossRef] [PubMed]
  6. O. Kwon, J. C. Wyant, C. R. Hayslett, “Rough Surface Interferometry at 10.6 μm,” Appl. Opt. 19, 1862 (1980).
    [CrossRef] [PubMed]
  7. Y.-Y. Cheng, J. C. Wyant, “Two-Wavelength Phase Shifting Interferometry,” Appl. Opt. 23, 4539 (1984).
    [CrossRef] [PubMed]
  8. A. F. Fercher, H. Z. Hu, U. Vry, “Rough Surface Interferometry with a Two-Wavelength Heterodyne Speckle Interferometer,” Appl. Opt. 24, 2181 (1985).
    [CrossRef] [PubMed]
  9. U. Vry, “Absolute Statistical Error in Two-Wavelength Rough-Surface Interferometry (ROSI),” Opt. Acta 33, 1221 (1986).
    [CrossRef]
  10. J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer-Verlag, Berlin, 1984), p. 9.
  11. M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1970), p. 514.

1988 (1)

1986 (1)

U. Vry, “Absolute Statistical Error in Two-Wavelength Rough-Surface Interferometry (ROSI),” Opt. Acta 33, 1221 (1986).
[CrossRef]

1985 (1)

1984 (1)

1981 (1)

1980 (1)

1971 (1)

J. N. Butters, J. A. Leendertz, “A Double Exposure Technique for Speckle Pattern Interferometry,” J. Phys. E 4, 277 (1971).
[CrossRef]

1965 (1)

O. Bryngdahl, “Applications of Shearing Interferometry,” Prog. Opt. 4, 37 (1965).
[CrossRef]

1963 (1)

J. Dyson, “Image Position Measurement by Polarimetry,” Opt. Acta 10, 171 (1963).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1970), p. 514.

Bryngdahl, O.

O. Bryngdahl, “Applications of Shearing Interferometry,” Prog. Opt. 4, 37 (1965).
[CrossRef]

Butters, J. N.

J. N. Butters, J. A. Leendertz, “A Double Exposure Technique for Speckle Pattern Interferometry,” J. Phys. E 4, 277 (1971).
[CrossRef]

Cheng, Y.-Y.

Dyson, J.

J. Dyson, “Image Position Measurement by Polarimetry,” Opt. Acta 10, 171 (1963).
[CrossRef]

Fercher, A. F.

Goodman, J. W.

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer-Verlag, Berlin, 1984), p. 9.

Häusler, G.

Hayslett, C. R.

Hu, H. Z.

Hutfless, J.

Kwon, O.

Leendertz, J. A.

J. N. Butters, J. A. Leendertz, “A Double Exposure Technique for Speckle Pattern Interferometry,” J. Phys. E 4, 277 (1971).
[CrossRef]

Maul, M.

Sommargren, G. E.

Vry, U.

U. Vry, “Absolute Statistical Error in Two-Wavelength Rough-Surface Interferometry (ROSI),” Opt. Acta 33, 1221 (1986).
[CrossRef]

A. F. Fercher, H. Z. Hu, U. Vry, “Rough Surface Interferometry with a Two-Wavelength Heterodyne Speckle Interferometer,” Appl. Opt. 24, 2181 (1985).
[CrossRef] [PubMed]

Weissmann, H.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1970), p. 514.

Wyant, J. C.

Appl. Opt. (5)

J. Phys. E (1)

J. N. Butters, J. A. Leendertz, “A Double Exposure Technique for Speckle Pattern Interferometry,” J. Phys. E 4, 277 (1971).
[CrossRef]

Opt. Acta (2)

U. Vry, “Absolute Statistical Error in Two-Wavelength Rough-Surface Interferometry (ROSI),” Opt. Acta 33, 1221 (1986).
[CrossRef]

J. Dyson, “Image Position Measurement by Polarimetry,” Opt. Acta 10, 171 (1963).
[CrossRef]

Prog. Opt. (1)

O. Bryngdahl, “Applications of Shearing Interferometry,” Prog. Opt. 4, 37 (1965).
[CrossRef]

Other (2)

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, Ed. (Springer-Verlag, Berlin, 1984), p. 9.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1970), p. 514.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Range sensing by shearing interferometry: basic setup.

Fig. 2
Fig. 2

Fringe patterns differently disturbed by speckle: (a) without any speckle reduction; (b) with speckle reduction by a boiling pupil; averaging time 4 ms; (c) with partially coherent illumination.

Fig. 3
Fig. 3

Statistical phase error Δφ between sheared speckle fields vs the ratio s/ds (shear/speckle diameter). The phase errors are smaller than Δφ with a probability W: (a) for large ratios s/ds; (b) for small ratios s/ds, which is the interesting range for speckle shearing interferometry.

Fig. 4
Fig. 4

Statistical error δz vs shear s for different diameters of the projected light spot (theoretical result).

Fig. 5
Fig. 5

Experimental result: standard deviation of measured distance z vs diameter do of the projected laser spot. All the other parameters are kept constant. Triangles, distance derived from the phase difference between two points in the interferogram by heterodyning. Circles, distance found by averaging over the whole observation aperture.

Fig. 6
Fig. 6

Intensity scan across the fringes: (a) with averaging in the fringe direction; (b) without averaging.

Fig. 7
Fig. 7

Demonstration of noise reduction by a boiling pupil. Several measurements are performed on different locations of the object surface at a constant distance z. The rms error is ~80 μm (1:2700).

Fig. 8
Fig. 8

Modified setup. A cylindrical lens integrates intensity along the fringe direction (partial coherent illumination is used).

Fig. 9
Fig. 9

Experimental results of range sensing with rough objects by shearing interferometry with partial coherent illumination. The setup is that of Fig. 8. A rms resolution of 68 μm is achieved (1:5500); the aperture is only 0.013.

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

p = λ z / s .
δ z R = ± ½ · λ / sin 2 u ,
μ ( x 1 , x 2 , y 1 , y 2 ) = A 1 ( x 1 , y 1 ) A 2 * ( x 2 , y 2 ) A 1 ( x 1 , y 1 ) 2 1 / 2 A 2 ( x 2 , y 2 ) 2 1 / 2 .
μ ( s x , s y ) = - P ( ζ , η ) 2 · exp [ i 2 π / ( λ z ) · ( ζ s x + η s y ) ] d ζ d η - P ( ζ , η ) 2 d ζ d η ,
P ( ζ , η ) 2 ~ exp [ - ( 2 ζ 2 + η 2 / d 0 ) 2 ] .
μ ( s ) = exp { - ( π / 2 ) 2 · [ s d o / ( λ z ) ] 2 } .
p ( θ ) = 1 - μ 2 2 π ( 1 - β 2 ) - 3 / 2 [ β arcsin ( β ) + π β 2 + ( 1 - β 2 ) 1 / 2 ] , β = μ · cos ( θ + ψ ) ;             - π θ π ;             ψ : phase of μ .
W = - Δ φ Δ φ p ( θ ) d θ .
σ = [ - g p π θ 2 p ( θ ) d θ ] 1 / 2             ( θ = 0 )
W σ = - σ + σ p ( θ ) d θ
Δ φ 2.1 · s / d s .
I ( x ) ~ 1 + cos [ 2 π s x / ( λ z ) ] .
I ( x ) ~ 1 + cos [ 2 π s x / ( λ z ) + Δ φ ] .
z = 2 π · s · b λ · φ ,
δ z = z φ · δ φ ,
δ φ = 2 Δ φ .
δ z = 2 · Δ φ · λ · z 2 2 π · s · b .
δ z z 2.1 · 2 · s · λ · z 2 π · s · b · d s = 2.1 · 2 · d o 2 π · 2 · b d o 2 b .
d o = 2 λ · z i b i ,
δ z = ¼ λ · 1 ½ b i / z i · 1 ½ b / z = ¼ λ · 1 sin u i · 1 sin u ,
δ z = ¼ λ · 1 sin 2 u .
a s = ( λ · z d o ) 2 · π 4 .
n s = a a s = 4 · b 2 · d o 2 π · ( λ z ) 2 .
δ z = z · d o b · ( n s ) - 1 = π 2 · λ · z 2 b 2 λ · z 2 b 2 ,
δ z ¼ λ 1 sin 2 u .

Metrics