Abstract

Temporal analysis of the irradiance at the detector plane is intended as the first step in the study of the mean fade time in a free optical communication system. In the present work this analysis has been performed for a Gaussian laser beam propagating in the atmospheric turbulence by means of computer simulation. To this end, we have adapted a previously known numerical method to the generation of long phase screens. The screens are displaced in a transverse direction as the wave is propagated, in order to simulate the wind effect. The amplitude of the temporal covariance and its power spectrum have been obtained at the optical axis, at the beam centroid and at a certain distance from these two points. Results have been worked out for weak, moderate and strong turbulence regimes and when possible they have been compared with theoretical models. These results show a significant contribution of beam wander to the temporal behaviour of the irradiance, even in the case of weak turbulence. We have also found that the spectral bandwidth of the covariance is hardly dependent on the Rytov variance.

© 2008 Optical Society of America

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References

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2006

2005

J. Recolons and F. Dios, "Accurate calculation of phase screens for the modelling of laser beam propagation through atmospheric turbulence," Proc. SPIE 5891, 51-62 (2005).

2004

2003

X. Zhu and J. M. Kahn, "Markov chain model in maximum-likelihood sequence detection for free-space optical communication through atmospheric turbulence channels," IEEE Trans. Commun. 51, 509-516 (2003)
[CrossRef]

X. Zhu, J. M. Kahn and J. Wang, "Mitigation of turbulence-induced scintillation noise in free-space optical links using temporal-domain detection techniques," IEEE Photon. Technol. Lett. 15, 623-625 (2003)
[CrossRef]

2001

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen and R. L. Phillips, "Theory of optical scintillation: Gaussian-beam wave model," Waves Random Media 11, 271-291 (2001)
[CrossRef]

2000

1999

1995

V. P. Lukin, "Investigation of the anisotropy of the atmospheric turbulence spectrum in the low-frequency range," Proc. SPIE 2471, 347-355 1995.
[CrossRef]

J. D. Shelton, "Turbulence-induced scintillation on Gaussian-beam waves: theoretical predictions and observations from a laser-illuminated satellite," J. Opt. Soc. Am. A 12, 2172-2181 (1995)
[CrossRef]

1992

R. G. Lane, A. Glindemann and J. C. Dainty, "Simulation of a Kolmogorov phase screen," Waves Random Media 2, 209-224 (1992)
[CrossRef]

1984

C. Macaskill and T. E. Ewart, "Computer simulation of two-dimensional random wave propagation", IMA J. Appl. Math. 33, 1-15 (1984).
[CrossRef]

1983

1971

1970

R. S. Lawrence and J. W. Strohbehn, "A survey of clear-air propagation effects relevant to optical communications," Proc. IEEE 58, 1523-1545 (1970)
[CrossRef]

1969

A. Ishimaru, "Fluctuations of a focused beam wave for atmospheric turbulence probing," Proc. IEEE 57, 407-414 (1969)
[CrossRef]

Appl. Opt.

IEEE Photon. Technol. Lett.

X. Zhu, J. M. Kahn and J. Wang, "Mitigation of turbulence-induced scintillation noise in free-space optical links using temporal-domain detection techniques," IEEE Photon. Technol. Lett. 15, 623-625 (2003)
[CrossRef]

IEEE Trans. Commun.

B. Hamzeh and M. Kavehrad, "OCDMA-coded free-space optical links for wireless optical-mesh networks," IEEE Trans. Commun. 52, 2165-2174 (2004)
[CrossRef]

X. Zhu and J. M. Kahn, "Markov chain model in maximum-likelihood sequence detection for free-space optical communication through atmospheric turbulence channels," IEEE Trans. Commun. 51, 509-516 (2003)
[CrossRef]

IMA J. Appl. Math.

C. Macaskill and T. E. Ewart, "Computer simulation of two-dimensional random wave propagation", IMA J. Appl. Math. 33, 1-15 (1984).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Express

Proc. IEEE

A. Ishimaru, "Fluctuations of a focused beam wave for atmospheric turbulence probing," Proc. IEEE 57, 407-414 (1969)
[CrossRef]

R. S. Lawrence and J. W. Strohbehn, "A survey of clear-air propagation effects relevant to optical communications," Proc. IEEE 58, 1523-1545 (1970)
[CrossRef]

Proc. SPIE

G. J. Baker and R. S. Benson, "Gaussian beam scintillation on ground to space paths: the importance of beam wander," Proc. SPIE 5550, 225-235 (2004)
[CrossRef]

V. P. Lukin, "Investigation of the anisotropy of the atmospheric turbulence spectrum in the low-frequency range," Proc. SPIE 2471, 347-355 1995.
[CrossRef]

J. Recolons and F. Dios, "Accurate calculation of phase screens for the modelling of laser beam propagation through atmospheric turbulence," Proc. SPIE 5891, 51-62 (2005).

Waves Random Media

R. G. Lane, A. Glindemann and J. C. Dainty, "Simulation of a Kolmogorov phase screen," Waves Random Media 2, 209-224 (1992)
[CrossRef]

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen and R. L. Phillips, "Theory of optical scintillation: Gaussian-beam wave model," Waves Random Media 11, 271-291 (2001)
[CrossRef]

Other

G. J. Baker and R. S. Benson, "Gaussian-beam weak scintillation on ground-to-space paths: compact descriptions and Rytov-method applicability," Opt. Eng. 44, 106002 1-10 (2005)
[CrossRef]

L. C. Andrews, R. L. Phillips, R. J. Sasiela and R.R. Parenti, "Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects," Opt. Eng. 45, 076001 1-12 (2006)
[CrossRef]

A. Comeron, F. Dios, A. Rodriguez and J. A. Rubio, Artemis laser link for atmospheric turbulence statistics, WP 1300 report, ESA AO/1-3930/01/NL/CK (European Space Agency, 2002)

V. I. Tatarskii, Wave Propagation in a Turbulent Medium, (McGraw-Hill, New York, 1961)

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE Press, Washington, 2005)
[CrossRef]

H. R. Anderson, Fixed Broadband Wireless. System Design, (Wiley & Sons, West Sussex, England, 2003)
[CrossRef]

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Figures (12)

Fig. 1.
Fig. 1.

Progressive loss of the Gaussian shape in the long term analysis of a laser beam propagating in the atmosphere. This effect is negligible under weak turbulence conditions (C2 n=10-16 m-2/3). The deviation of the field profile from the Gaussian shape increases with the accumulated turbulence along the beam path. Dashed black lines have been obtained by using Eq. (6), dot-dashed blue lines have been obtained by using Eq. (7), and red solid lines have been obtained by using Eq. (8). The grid parameters correspond to an effective inner scale value l0 =2.4 mm.

Fig. 2.
Fig. 2.

Simulated average field profile (blue lines) obtained at a distance of 5000 m from the transmitter, for the same values of the turbulence strength as in the previous figure. Red and magenta lines represent the theoretical Gaussian shape for each case. The numerical results again match well with the theoretical Gaussian profile for a structure constant Cn2=10-16 m-2/3, whereas for higher values the difference becomes more and more apparent.

Fig. 3.
Fig. 3.

From the exact low-resolution phase-screen (above in the figure) a certain number of columns is taken to create a smaller high-resolution screen (below). L y is the length of the initial screen in the moving direction.

Fig. 4.
Fig. 4.

Generation of the high-resolution screen by interpolating the low-resolution positions (black stars). The red points on the right cannot be properly calculated. The blue ones are correct and they will have to be taken into account when interpolating the next block to ensure continuity (below).

Fig. 5.
Fig. 5.

Example of two different phase screens interpolated from the same low-resolution grid. The leftmost section is identical in both screens. The figure shows two possible continuations in the middle and right sections, which have been generated independently.

Fig. 6.
Fig. 6.

Structure function D(y) in the horizontal direction Y for screens calculated with the method presented in this work (blue line). The black line represents the theoretical Kolmogorov structure function D=6.884 (y/r 0)5/3, where y is the coordinate in the wind direction, and r 0 is the Fried parameter

Fig. 7
Fig. 7

(a) Temporal irradiance covariance and (b) spectral power density of the irradiance at the reception plane obtained at a propagation distance L=500 m from the transmitter, in the weak turbulence regime. The curves in black represent the theoretical covariance and spectrum at the optical axis. In both figures, the horizontal axis has been normalized with respect to the theoretical cut-off frequency ω t for plane waves.

Fig. 8.
Fig. 8.

(a) Temporal covariance and (b) spectral power density of the irradiance in the same case as Fig. 7 but for points situated at a distance W LT/2 from the two centers considered in the beam.

Fig. 9.
Fig. 9.

(a) Temporal covariance and (b) power spectrum of the irradiance at the reception plane, after a propagation distance L=1500 m, obtained both at the optical axis and at the beam centroid, in the moderate turbulence regime. The normalizing frequency is ft =10 Hz for a 1m/s wind velocity.

Fig. 10.
Fig. 10.

Irradiance spectral power density curves obtained for the case of moderate turbulence, at points located at a WLT/2 distance, both from the optical axis and from the beam centroid.

Fig. 11.
Fig. 11.

(a) Irradiance spectral power density obtained at the theoretical center of the beam and at the centroid in the case of strong turbulence. Gray and green curves correspond to the moderate turbulence case. (b) Id. at a distance W LT/2 from the two previous points.

Fig. 12.
Fig. 12.

(a) Temporal covariance and (b) power spectrum of the irradiance for a plane wave, simulated by a super-Gaussian beam, in regime of strong turbulence. The difference between the behaviour at the center (r=0) and at the centroid becomes negligible.

Tables (1)

Tables Icon

Table 1: Several radius definitions for a Gaussian beam as a function of the power fraction encompassed by circle centered at the peak of irradiance

Equations (12)

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ρ = V τ
ξ ρ = V τ
γ ρ = V τ
S I ( ω ) = 2 0 B I ( τ ) exp ( j ω τ ) d τ
T ( F 0 ) P { F < F 0 } n ( F 0 )
W = 2 r 39
W = 2 r 63
W = r 86
B ( r 1 , r 2 ) B ( r , p )
I = I 0 exp ( 2 r 2 W LT 2 )
B I ( τ ) = 2.172 σ R 2 Re { ( 2 j + Λ ) 5 6 1 F 1 ( 5 6 ; 1 ; ω t 2 τ 2 2 j + Λ ) Λ 5 6 1 F 1 ( 5 6 ; 1 ; ω t 2 τ 2 Λ ) }
u ( x , y ; 0 ) exp ( ( x 2 + y 2 W 2 ) 8 )

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