Abstract

We report experiments in which a fiber-coupled heterodyne laser system operating at a wavelength of 1.5 µm is used to measure the phase fluctuations induced on a laser beam by passage through a thin layer of turbulent air and subsequent propagation through free space. We investigate the statistical properties and power spectra of the phase and its rate of change, in addition to the intensity statistics. We find that the power spectrum of the rate of change of phase has a simple negative exponential form. We discuss our results in the context of the problem of detection of phase variations over an extended turbulent atmospheric path.

© 2002 Optical Society of America

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References

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  1. E. Jakeman, J. G. McWhirter, “Non-Gaussian scattering by a random phase screen,” Appl. Phys. B 26, 125–131 (1981).
    [CrossRef]
  2. G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “Focusing by a random phase screen,” Opt. Commun. 22, 195–201 (1977).
    [CrossRef]
  3. G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “The statistical and correlation properties of light scattered by a random phase screen,” in Coherence and Quantum Optics IV, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978), pp. 351–361.
  4. I. A. Joia, R. J. Perkins, B. J. Uscinski, G. Balmer, D. Jordan, E. Jakeman, “Optical properties of a planar turbulent jet,” Appl. Opt. 34, 7039–7053 (1995).
    [CrossRef] [PubMed]
  5. Y. Ohtsuka, I. Sasaki, “Light-beat method to process phase fluctuations of a laser light beam,” Appl. Phys. 7, 265–270 (1975).
    [CrossRef]
  6. K. Schätzel, “Interferometric analysis of deep random-phase screens,” Opt. Lett. 5, 389–391 (1980).
    [CrossRef] [PubMed]
  7. C. J. Karlsson, F. A. A. Olsson, D. Letalick, M. Harris, “All-fiber multifunction continuous-wave coherent laser radar at 1.55 µm for range, speed, vibration, and wind measurements,” Appl. Opt. 39, 3716–3726 (2000).
    [CrossRef]
  8. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 8.
  9. E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antenn as Propag. AP-24, 806–814 (1976).
    [CrossRef]
  10. S. O. Rice, “Statistical properties of a sine wave plus random noise,” Bell Syst. Tech. J. 27, 109–157 (1948).
    [CrossRef]
  11. E. Jakeman, K. D. Ridley, S. M. Watson, “Intensity-weighted phase-derivative statistics: application to phase screen scattering,” J. Opt. Soc. Am. A (2001).
    [CrossRef]
  12. M. V. Berry, “Diffractals,” J. Phys. A. 12, 781–797 (1979).
    [CrossRef]
  13. S. F. Clifford, “Temporal-frequency spectra for a spherical wave propagating through atmospheric turbulence,” J. Opt. Soc. Am. 61, 1285–1292 (1971).
    [CrossRef]

2001 (1)

E. Jakeman, K. D. Ridley, S. M. Watson, “Intensity-weighted phase-derivative statistics: application to phase screen scattering,” J. Opt. Soc. Am. A (2001).
[CrossRef]

2000 (1)

1995 (1)

1981 (1)

E. Jakeman, J. G. McWhirter, “Non-Gaussian scattering by a random phase screen,” Appl. Phys. B 26, 125–131 (1981).
[CrossRef]

1980 (1)

1979 (1)

M. V. Berry, “Diffractals,” J. Phys. A. 12, 781–797 (1979).
[CrossRef]

1977 (1)

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “Focusing by a random phase screen,” Opt. Commun. 22, 195–201 (1977).
[CrossRef]

1976 (1)

E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antenn as Propag. AP-24, 806–814 (1976).
[CrossRef]

1975 (1)

Y. Ohtsuka, I. Sasaki, “Light-beat method to process phase fluctuations of a laser light beam,” Appl. Phys. 7, 265–270 (1975).
[CrossRef]

1971 (1)

1948 (1)

S. O. Rice, “Statistical properties of a sine wave plus random noise,” Bell Syst. Tech. J. 27, 109–157 (1948).
[CrossRef]

Balmer, G.

Berry, M. V.

M. V. Berry, “Diffractals,” J. Phys. A. 12, 781–797 (1979).
[CrossRef]

Clifford, S. F.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 8.

Harris, M.

Jakeman, E.

E. Jakeman, K. D. Ridley, S. M. Watson, “Intensity-weighted phase-derivative statistics: application to phase screen scattering,” J. Opt. Soc. Am. A (2001).
[CrossRef]

I. A. Joia, R. J. Perkins, B. J. Uscinski, G. Balmer, D. Jordan, E. Jakeman, “Optical properties of a planar turbulent jet,” Appl. Opt. 34, 7039–7053 (1995).
[CrossRef] [PubMed]

E. Jakeman, J. G. McWhirter, “Non-Gaussian scattering by a random phase screen,” Appl. Phys. B 26, 125–131 (1981).
[CrossRef]

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “Focusing by a random phase screen,” Opt. Commun. 22, 195–201 (1977).
[CrossRef]

E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antenn as Propag. AP-24, 806–814 (1976).
[CrossRef]

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “The statistical and correlation properties of light scattered by a random phase screen,” in Coherence and Quantum Optics IV, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978), pp. 351–361.

Joia, I. A.

Jordan, D.

Karlsson, C. J.

Letalick, D.

McWhirter, J. G.

E. Jakeman, J. G. McWhirter, “Non-Gaussian scattering by a random phase screen,” Appl. Phys. B 26, 125–131 (1981).
[CrossRef]

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “Focusing by a random phase screen,” Opt. Commun. 22, 195–201 (1977).
[CrossRef]

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “The statistical and correlation properties of light scattered by a random phase screen,” in Coherence and Quantum Optics IV, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978), pp. 351–361.

Ohtsuka, Y.

Y. Ohtsuka, I. Sasaki, “Light-beat method to process phase fluctuations of a laser light beam,” Appl. Phys. 7, 265–270 (1975).
[CrossRef]

Olsson, F. A. A.

Parry, G.

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “Focusing by a random phase screen,” Opt. Commun. 22, 195–201 (1977).
[CrossRef]

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “The statistical and correlation properties of light scattered by a random phase screen,” in Coherence and Quantum Optics IV, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978), pp. 351–361.

Perkins, R. J.

Pusey, P. N.

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “Focusing by a random phase screen,” Opt. Commun. 22, 195–201 (1977).
[CrossRef]

E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antenn as Propag. AP-24, 806–814 (1976).
[CrossRef]

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “The statistical and correlation properties of light scattered by a random phase screen,” in Coherence and Quantum Optics IV, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978), pp. 351–361.

Rice, S. O.

S. O. Rice, “Statistical properties of a sine wave plus random noise,” Bell Syst. Tech. J. 27, 109–157 (1948).
[CrossRef]

Ridley, K. D.

E. Jakeman, K. D. Ridley, S. M. Watson, “Intensity-weighted phase-derivative statistics: application to phase screen scattering,” J. Opt. Soc. Am. A (2001).
[CrossRef]

Sasaki, I.

Y. Ohtsuka, I. Sasaki, “Light-beat method to process phase fluctuations of a laser light beam,” Appl. Phys. 7, 265–270 (1975).
[CrossRef]

Schätzel, K.

Uscinski, B. J.

Watson, S. M.

E. Jakeman, K. D. Ridley, S. M. Watson, “Intensity-weighted phase-derivative statistics: application to phase screen scattering,” J. Opt. Soc. Am. A (2001).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. (1)

Y. Ohtsuka, I. Sasaki, “Light-beat method to process phase fluctuations of a laser light beam,” Appl. Phys. 7, 265–270 (1975).
[CrossRef]

Appl. Phys. B (1)

E. Jakeman, J. G. McWhirter, “Non-Gaussian scattering by a random phase screen,” Appl. Phys. B 26, 125–131 (1981).
[CrossRef]

Bell Syst. Tech. J. (1)

S. O. Rice, “Statistical properties of a sine wave plus random noise,” Bell Syst. Tech. J. 27, 109–157 (1948).
[CrossRef]

IEEE Trans. Antenn as Propag. (1)

E. Jakeman, P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antenn as Propag. AP-24, 806–814 (1976).
[CrossRef]

J. Opt. Soc. Am. (1)

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

E. Jakeman, K. D. Ridley, S. M. Watson, “Intensity-weighted phase-derivative statistics: application to phase screen scattering,” J. Opt. Soc. Am. A (2001).
[CrossRef]

J. Phys. A. (1)

M. V. Berry, “Diffractals,” J. Phys. A. 12, 781–797 (1979).
[CrossRef]

Opt. Commun. (1)

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “Focusing by a random phase screen,” Opt. Commun. 22, 195–201 (1977).
[CrossRef]

Opt. Lett. (1)

Other (2)

G. Parry, P. N. Pusey, E. Jakeman, J. G. McWhirter, “The statistical and correlation properties of light scattered by a random phase screen,” in Coherence and Quantum Optics IV, L. Mandel, E. Wolf, eds. (Plenum, New York, 1978), pp. 351–361.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 8.

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Figures (15)

Fig. 1
Fig. 1

Experimental layout. The fibers are represented by dotted lines and the electronic connections by solid lines. I, in-phase output; Q, quadrature output; A/D, analog to digital.

Fig. 2
Fig. 2

Phase structure function. The solid curve shows a fitted Kolmogorov 5/3 power law. For reference, the dotted curves show r and r 2 power laws.

Fig. 3
Fig. 3

Examples of time series at four different propagation distances with a phasor representation of the field. Each time series is 0.5 s in duration.

Fig. 4
Fig. 4

Histograms of the intensity variations at the four different propagation distances of Fig. 3. The solid curve in the final plot gives the result for a K distribution with the same mean and variance.

Fig. 5
Fig. 5

Histograms from Fig. 4 plotted to higher normalized intensity values with a logarithmic vertical axis to show the high-intensity tails. As in Fig. 4, the final plot shows the solid curve of the K distribution.

Fig. 6
Fig. 6

Example of the time variation of the phasor in the near field (upper plot, 0.5 in duration) and the corresponding unwrapped phase time series (lower plot).

Fig. 7
Fig. 7

Power spectrum of the unwrapped phase in the absence of the phase screen, showing the background noise level.

Fig. 8
Fig. 8

Power spectrum of the unwrapped phase in the presence of the phase screen, plotted on both logarithmic and linear vertical scales, with different ranges on the x axes.

Fig. 9
Fig. 9

Autocorrelation of the phase obtained by Fourier transform of the spectrum in Fig. 8.

Fig. 10
Fig. 10

Histogram of the phase shifts imposed by the phase screen with a fitted Gaussian. The lower plot has a logarithmic scale to show the behavior of the wings.

Fig. 11
Fig. 11

Power spectrum of the near-field phase derivative on logarithmic and linear axes. The solid curves are a fitted negative exponential.

Fig. 12
Fig. 12

Histogram of the near-field phase derivative, again with logarithmic and linear axes. The solid curves are Gaussian.

Fig. 13
Fig. 13

Histograms of the phase derivative at four different propagation distances. The far-field result has a fitted Student’s t probability density function.

Fig. 14
Fig. 14

Two far-field power spectra with the same set of data. (a) The phase derivative, (b) the intensity-weighted phase derivative. A single-tone frequency modulation was applied to the laser beam at 173 Hz.

Fig. 15
Fig. 15

Results for the second moment of the intensity (squares) and the variance of the IWPD statistic normalized by its value at the phase screen (circles) plotted against distance from the screen. The values plotted on the x axis are effective free-space distances given by Eq. (3).

Equations (8)

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Dr=|ϕ1-ϕ2|2.
Dr=2.91k2Cn2Lr5/3,
z=11/z-1/f,
pI=b/IΓvbI2v Kv-1bI,
n2=I2I2=21+1v.
Pϕ˙=12w1+ϕ˙w2-3/2.
ϕ˙=VTA.
PVT,ϕ˙>ϕ˙1=PVT, A<A1.

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