Abstract

A number of field experiments measuring the fluctuating intensity of a laser beam propagating along horizontal paths in the maritime environment is performed over sub-kilometer distances at the United States Naval Academy. Both above the ground and over the water links are explored. Two different detection schemes, one photographing the beam on a white board, and the other capturing the beam directly using a ccd sensor, gave consistent results. The probability density function (pdf) of the fluctuating intensity is reconstructed with the help of two theoretical models: the Gamma-Gamma and the Gamma-Laguerre, and compared with the intensity’s histograms. It is found that the on-ground experimental results are in good agreement with theoretical predictions. The results obtained above the water paths lead to appreciable discrepancies, especially in the case of the Gamma-Gamma model. These discrepancies are attributed to the presence of the various scatterers along the path of the beam, such as water droplets, aerosols and other airborne particles. Our paper’s main contribution is providing a methodology for computing the pdf function of the laser beam intensity in the maritime environment using field measurements.

© 2011 OSA

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References

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  1. A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice Hall, 1996).
  2. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961; reproduced by Dover, New York, 1967).
  3. S. F. Clifford, “The classical theory of wave propagation in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. Strohbehn, ed.(Springer, New York, 1978).
  4. R. L. Fante, “Wave propagation in random media: a system approach,” Progress in Optics Vol 22, E. Wolf, ed. (Elsevier, Amsterdam, 1985).
  5. L. C. Andrews and R. L. Phillips, Laser Beam Propagation in Random Media, 2nd ed. (SPIE Press, Bellingham, Washington, 2005).
  6. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Electromagnetic Beam Scintillation with Applications (SPIE Press, Bellingham, Washington, 2001).
  7. D. Wheelon, Electromagnetic Scintillation II Weak Scattering (Cambridge University Press, 2003)
  8. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  9. M. I. Mishchenko and L. D. Travis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University Press, 2002).
  10. M. P. J. L. Chang, C. O. Font, G. C. Gilbreath, and E. Oh, “Humidity’s influence on visible region refractive index structure parameter Cn2.,” Appl. Opt. 46(13), 2453–2459 (2007).
    [CrossRef] [PubMed]
  11. R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges,” Radio Sci. 13(6), 953–961 (1978).
    [CrossRef]
  12. C. A. Friehe, J. C. La Rue, F. H. Champagne, C. H. Gibson, and G. F. Dreyer, “Effects of temperature and humidity fluctuations on the optical refractive index in the marine boundary layer,” J. Opt. Soc. Am. 65(12), 1502–1511 (1975).
    [CrossRef]
  13. R. Weiss-Wrana, “Turbulence statistics in littoral area,” Proc. SPIE 6364, 63640F, 63640F-12 (2006).
    [CrossRef]
  14. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18(1), 173–184 (2008).
    [CrossRef]
  15. K. J. Mayer and C. Y. Young, “Effect of atmospheric spectrum models on scintillation in moderate turbulence,” J. Mod. Opt. 55(7), 1101–1117 (2008).
    [CrossRef]
  16. M. Reed and B. Simon, Fourier Analysis, Self-Adjointness, Vol. 2 of Methods of Modern Mathematical Physics (Academic, 1975), p. 341.
  17. J. A. Shehat and J. D. Tamarkin, The Problem of Moments (American Mathematical Society, New York, 1943).
  18. W. Feller, An Introduction to Probability Theory and its Applications (Wiley, 1971), Vol. II.
  19. M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
    [CrossRef]
  20. R. Barakat, “First-order intensity and log-intensity probability density functions of light scattered by the turbulent atmosphere in terms of lower-order moments,” J. Opt. Soc. Am. 16(9), 2269–2274 (1999).
    [CrossRef]

2008 (2)

J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18(1), 173–184 (2008).
[CrossRef]

K. J. Mayer and C. Y. Young, “Effect of atmospheric spectrum models on scintillation in moderate turbulence,” J. Mod. Opt. 55(7), 1101–1117 (2008).
[CrossRef]

2007 (1)

2006 (1)

R. Weiss-Wrana, “Turbulence statistics in littoral area,” Proc. SPIE 6364, 63640F, 63640F-12 (2006).
[CrossRef]

2001 (1)

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
[CrossRef]

1999 (1)

R. Barakat, “First-order intensity and log-intensity probability density functions of light scattered by the turbulent atmosphere in terms of lower-order moments,” J. Opt. Soc. Am. 16(9), 2269–2274 (1999).
[CrossRef]

1978 (1)

R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges,” Radio Sci. 13(6), 953–961 (1978).
[CrossRef]

1975 (1)

Al-Habash, M. A.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
[CrossRef]

Andrews, L. C.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
[CrossRef]

Barakat, R.

R. Barakat, “First-order intensity and log-intensity probability density functions of light scattered by the turbulent atmosphere in terms of lower-order moments,” J. Opt. Soc. Am. 16(9), 2269–2274 (1999).
[CrossRef]

Champagne, F. H.

Chang, M. P. J. L.

Dreyer, G. F.

Font, C. O.

Friehe, C. A.

Gibson, C. H.

Gilbreath, G. C.

Grayshan, J.

J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18(1), 173–184 (2008).
[CrossRef]

Hill, R. J.

R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges,” Radio Sci. 13(6), 953–961 (1978).
[CrossRef]

La Rue, J. C.

Mayer, K. J.

K. J. Mayer and C. Y. Young, “Effect of atmospheric spectrum models on scintillation in moderate turbulence,” J. Mod. Opt. 55(7), 1101–1117 (2008).
[CrossRef]

Oh, E.

Phillips, R. L.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
[CrossRef]

Vetelino, F. S.

J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18(1), 173–184 (2008).
[CrossRef]

Weiss-Wrana, R.

R. Weiss-Wrana, “Turbulence statistics in littoral area,” Proc. SPIE 6364, 63640F, 63640F-12 (2006).
[CrossRef]

Young, C. Y.

J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18(1), 173–184 (2008).
[CrossRef]

K. J. Mayer and C. Y. Young, “Effect of atmospheric spectrum models on scintillation in moderate turbulence,” J. Mod. Opt. 55(7), 1101–1117 (2008).
[CrossRef]

Appl. Opt. (1)

J. Mod. Opt. (1)

K. J. Mayer and C. Y. Young, “Effect of atmospheric spectrum models on scintillation in moderate turbulence,” J. Mod. Opt. 55(7), 1101–1117 (2008).
[CrossRef]

J. Opt. Soc. Am. (2)

R. Barakat, “First-order intensity and log-intensity probability density functions of light scattered by the turbulent atmosphere in terms of lower-order moments,” J. Opt. Soc. Am. 16(9), 2269–2274 (1999).
[CrossRef]

C. A. Friehe, J. C. La Rue, F. H. Champagne, C. H. Gibson, and G. F. Dreyer, “Effects of temperature and humidity fluctuations on the optical refractive index in the marine boundary layer,” J. Opt. Soc. Am. 65(12), 1502–1511 (1975).
[CrossRef]

Opt. Eng. (1)

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
[CrossRef]

Proc. SPIE (1)

R. Weiss-Wrana, “Turbulence statistics in littoral area,” Proc. SPIE 6364, 63640F, 63640F-12 (2006).
[CrossRef]

Radio Sci. (1)

R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges,” Radio Sci. 13(6), 953–961 (1978).
[CrossRef]

Waves Random Complex Media (1)

J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18(1), 173–184 (2008).
[CrossRef]

Other (12)

M. Reed and B. Simon, Fourier Analysis, Self-Adjointness, Vol. 2 of Methods of Modern Mathematical Physics (Academic, 1975), p. 341.

J. A. Shehat and J. D. Tamarkin, The Problem of Moments (American Mathematical Society, New York, 1943).

W. Feller, An Introduction to Probability Theory and its Applications (Wiley, 1971), Vol. II.

A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice Hall, 1996).

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961; reproduced by Dover, New York, 1967).

S. F. Clifford, “The classical theory of wave propagation in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. Strohbehn, ed.(Springer, New York, 1978).

R. L. Fante, “Wave propagation in random media: a system approach,” Progress in Optics Vol 22, E. Wolf, ed. (Elsevier, Amsterdam, 1985).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation in Random Media, 2nd ed. (SPIE Press, Bellingham, Washington, 2005).

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Electromagnetic Beam Scintillation with Applications (SPIE Press, Bellingham, Washington, 2001).

D. Wheelon, Electromagnetic Scintillation II Weak Scattering (Cambridge University Press, 2003)

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

M. I. Mishchenko and L. D. Travis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University Press, 2002).

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Figures (5)

Fig. 1
Fig. 1

(a) Sherman Field site with a 400 m long laser link. The experiment was performed on 15 June 2010. (b) College Creek site with a 400 m long laser link. The experiment was performed on 18 June 2010. For both experiments data was collected by sequencing 30 frames per second over three minutes. Since the beam projection on a white board was photographed the pixel size was effectively measuring 0.3 mm2. Sensing area in this case was a photograph of a white board 1 m by 1 m. Spatial coherence radius for the Gaussian beam used in the experiment is on the order of 1 cm (estimated using following assumptions [5]).

Fig. 2
Fig. 2

(a) Sherman Field site (b) College Creek site. Link setup for experiments carried on March 15, 2011, with 300 m long laser link. The beam is sensed directly by the ccd sensor with red notch filter. The sensing area is 7.6 mm (horizontal) × 6.2 mm (vertical) with pixel size of (4.65)2 μm2. Each frame was recorded at the rate of 10 Hz.

Fig. 3
Fig. 3

Normalized intensity Qc/M of each realization 1 through 1800 collected on March 15, 2011 at the College Creek site. See Fig. 2b.

Fig. 4
Fig. 4

PDF of the fluctuating intensity for the experiments done on June 2010. Labels: * histogram, red curve Gamma-Laguerre model, blue curve Gamma-Gamma model. Video camera taking photographs beam projected onto the white board. (a) Experiment above the land (given on Fig. 1(a) and Table 1); (b) Experiment above water (given on Fig. 1(b) and Table 2). Histogram was constructed using 30 bins and as such compared to the models. The intensity was normalized to the first moment or the mean intensity as given in Eq. (13) section 4 in the text.

Fig. 5
Fig. 5

Pdf of the light intensity fluctuations for the experiments done in March 15. Labels: * histogram, red curve Gamma-Laguerre model, blue curve Gamma-Gamma model. Video camera is detects the beam directly. (A) Experiment above the land (given on Fig. 2(a) and Table 3); (B) Experiment above water (given on Fig. 2(b) and Table 3). Histogram was constructed using 30 bins and as such compared to the models. The intensity was normalized to the first moment or the mean intensity as given in Eq. (13) section 4 in the text.

Tables (3)

Tables Icon

Table 1 Weather conditions at Hospital Point on June 15, 2010

Tables Icon

Table 2 Weather conditions at Hospital Point on June 18, 2010

Tables Icon

Table 3 Weather conditions at Hospital Point on March 15, 2011

Equations (15)

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Probability(a<h<b)= a b W(h)dh .
h n = 0 W(h) h n dh.
W GL (h)= W g (h) n=0 W n L n (β1) ( βh μ ),
W g (h)= 1 Γ( β ) ( β μ ) β h β1 exp( βh μ ),
μ= h ,β= h 2 / ( h 2 h 2 ) .
W n =n!Γ(β) k=0 n ( β/μ ) k h k k!(nk)!Γ( β+k ) .
L n (β1) (x)= k=0 n ( n+β1 n1 ) (x) k k! .
W GG (h)= 2 (αβ) α+β 2 Γ(α)Γ(β) h α+β 2 1 K αβ (2 αβh ),
α= 1 exp( σ lnx 2 )1 ,β= 1 exp( σ lny 2 )1 ,
σ lnx 2 = 0.49 σ B 2 [1+0.56(1+θ) σ B 12/5 ] 7/6 and σ lny 2 = 0.51 σ B 2 [1+0.69 σ B 12/5 ] 5/6 ,
I s (x,y)= 1 N j=1 N I j (x,y) .
I max ( x beamcenter , y beamcenter )= max x,y { I s (x,y)}.
I k = 1 N j=1 N Q cj k M k .
T M = D C .
N A = T M / j=1 256 T M j .

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