Abstract

In this paper we analyze the probability density function of the superheterodyne signal obtained in a two-wavelength interferometer from the beat of a local oscillator laser beam with a speckled return beam from a rough target. Theoretical investigation shows that, by using an increased number of spatially separated detectors, one can improve noticeably the detection probability of the superheterodyne signal. Experimental results obtained with a four-quadrant detector are in good agreement with theory.

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. C. Wyant, “Holographic and moire techniques,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), pp. 397–402.
  2. C. R. Tilford, “Analytical procedure for determining lengths from fractional fringes,” Appl. Opt. 16, 1857–1860 (1977).
    [CrossRef] [PubMed]
  3. J. C. Wyant, “Testing aspherics using two-wavelength holography,” Appl. Opt. 10, 2113–2118 (1971).
    [CrossRef] [PubMed]
  4. C. Polhemus, “Two-wavelength interferometry,” Appl. Opt. 12, 2071–2077 (1973).
    [CrossRef] [PubMed]
  5. A. F. Fercher, H. Z. Hu, U. Vry, “Rough surface interferometry with a two-wavelength heterodyne speckle interferometer,” Appl. Opt. 24, 2181–2188 (1985).
    [CrossRef] [PubMed]
  6. R. Dändliker, R. Thalmann, D. Prongué, “Two-wavelength laser interferometry using superheterodyne detection,” Opt. Lett. 13, 339–343 (1988).
    [CrossRef] [PubMed]
  7. J. H. Shapiro, “Correlation scales of laser speckle in heterodyne detection,” Appl. Opt. 24, 12, 1883–1888 (1985).
    [CrossRef]
  8. R. Thalmann, “Study and evaluation of a metrology concept for large structure in space,” ESA Report 220 EC 08/87, ESA contract SPM 6848/86/NL/JG (European Space Agency, Neuchatel, 1987).
  9. J. W. Goodman, “Statistical properties of laser speckles and related phenomena,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), pp. 9–75.
    [CrossRef]
  10. A. F. Fercher, U. Vry, W. Werner, “Two wavelength speckle interferometry on rough surfaces using a mode hopping diode laser,” Opt. Lasers Eng. 11, 271–279 (1989).
    [CrossRef]
  11. P. Lutzmann, R. Ebert, “Speckle reduction in CO2 laser radar systems,” in CO2 Lasers and Applications II, H. Opower, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1276, 288–293 (1990).
  12. U. Vry, A. F. Fercher, “Higher-order statistical properties of speckle fields and their application to rough-surface interferometry,” J. Opt. Soc. Am. A 3, 988–1000 (1986).
    [CrossRef]

1989 (1)

A. F. Fercher, U. Vry, W. Werner, “Two wavelength speckle interferometry on rough surfaces using a mode hopping diode laser,” Opt. Lasers Eng. 11, 271–279 (1989).
[CrossRef]

1988 (1)

1986 (1)

1985 (2)

1977 (1)

1973 (1)

1971 (1)

Dändliker, R.

Ebert, R.

P. Lutzmann, R. Ebert, “Speckle reduction in CO2 laser radar systems,” in CO2 Lasers and Applications II, H. Opower, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1276, 288–293 (1990).

Fercher, A. F.

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckles and related phenomena,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), pp. 9–75.
[CrossRef]

Hu, H. Z.

Lutzmann, P.

P. Lutzmann, R. Ebert, “Speckle reduction in CO2 laser radar systems,” in CO2 Lasers and Applications II, H. Opower, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1276, 288–293 (1990).

Polhemus, C.

Prongué, D.

Shapiro, J. H.

J. H. Shapiro, “Correlation scales of laser speckle in heterodyne detection,” Appl. Opt. 24, 12, 1883–1888 (1985).
[CrossRef]

Thalmann, R.

R. Dändliker, R. Thalmann, D. Prongué, “Two-wavelength laser interferometry using superheterodyne detection,” Opt. Lett. 13, 339–343 (1988).
[CrossRef] [PubMed]

R. Thalmann, “Study and evaluation of a metrology concept for large structure in space,” ESA Report 220 EC 08/87, ESA contract SPM 6848/86/NL/JG (European Space Agency, Neuchatel, 1987).

Tilford, C. R.

Vry, U.

Werner, W.

A. F. Fercher, U. Vry, W. Werner, “Two wavelength speckle interferometry on rough surfaces using a mode hopping diode laser,” Opt. Lasers Eng. 11, 271–279 (1989).
[CrossRef]

Wyant, J. C.

J. C. Wyant, “Testing aspherics using two-wavelength holography,” Appl. Opt. 10, 2113–2118 (1971).
[CrossRef] [PubMed]

J. C. Wyant, “Holographic and moire techniques,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), pp. 397–402.

Appl. Opt. (5)

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

Opt. Lasers Eng. (1)

A. F. Fercher, U. Vry, W. Werner, “Two wavelength speckle interferometry on rough surfaces using a mode hopping diode laser,” Opt. Lasers Eng. 11, 271–279 (1989).
[CrossRef]

Opt. Lett. (1)

Other (4)

J. C. Wyant, “Holographic and moire techniques,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978), pp. 397–402.

P. Lutzmann, R. Ebert, “Speckle reduction in CO2 laser radar systems,” in CO2 Lasers and Applications II, H. Opower, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1276, 288–293 (1990).

R. Thalmann, “Study and evaluation of a metrology concept for large structure in space,” ESA Report 220 EC 08/87, ESA contract SPM 6848/86/NL/JG (European Space Agency, Neuchatel, 1987).

J. W. Goodman, “Statistical properties of laser speckles and related phenomena,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, New York, 1975), pp. 9–75.
[CrossRef]

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 (7)

Fig. 1
Fig. 1

Experimental setup for investigation of the PDF of the SHS from a diffuse target: PBS, polarizing beam splitter; AOM, acousto-optic modulator; HWP, half-wave plate; QWP, quarter-wave plate; OPT ISOL, optical isolator; P, polarizer; BS, beam splitter.

Fig. 2
Fig. 2

Block diagram of the analog electronic chain for demodulation of the 100-kHz SHS.

Fig. 3
Fig. 3

Probability density function for the sum of SHS’s from Q independent detectors calculated from Eq. (14).

Fig. 4
Fig. 4

Integrated probability versus X min calculated from Eq. (15). Q is the number of independent detectors.

Fig. 5
Fig. 5

Block diagram of the four-channel detection chain system.

Fig. 6
Fig. 6

Experimental results for the PDF using Q = 1, 2, 3, or 4 quadrants of the detector.

Fig. 7
Fig. 7

Comparison of theoretical and experimental results for the integrated probability when 1, 2, 3, and 4 detectors are used.

Equations (15)

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

p ( α ) = 1 π α 2 exp ( - α x 2 + α y 2 α 2 ) ,
p ( I ) = 1 I exp ( - I I ) ,
i h 1 , 2 = 2 R A LO Re { α 1 , 2 exp [ j ( 2 π f 1 , 2 t + φ 1 , 2 + β 1 , 2 ) ] } ,
p ( W h ) = 1 W h exp ( - W h W h ) ,
V = 2 G R A LO 2 [ a 1 cos ( 2 π f 1 t + φ 1 + β 1 ) + a 2 cos ( 2 π f 2 t + φ 2 + β 2 ) ] ,
S = 2 K G R A LO 2 a 1 cos ( 2 π f 1 t + φ 1 + β 1 ) a 2 × cos ( 2 π f 2 t + φ 2 + β 2 ) ,
μ = exp ( - 4 π 2 σ h 2 Λ 2 ) ,
S = K G R A LO 2 a 2 { cos [ ( ω 1 + ω 2 ) t + φ 1 + φ 2 + 2 β s ] + cos [ ( ω 1 - ω 2 ) t + φ 1 - φ 2 ] } ,
V sh = A sh cos ( 2 π Δ f t + Δ φ ) ,
A sh = K G R A LO 2 a 2 = K G R I s I LO ,
p ( A sh ) = 1 A sh exp ( - A sh A sh ) ,
i N = 2 R A LO Re { n = 0 N α n exp [ j ( 2 π f 1 , 2 t + φ 1 , 2 ) ] } ,
V Q = K G R I LO n = 1 Q I n cos ( 2 π Δ f t + Δ φ ) ,
p Q ( X sh ) = Q Q X sh Q - 1 Γ ( Q - 1 ) exp ( - Q X sh ) ,
P Q ( X min ) = 0 X min p Q ( X sh ) d X sh .

Metrics