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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 17 — Aug. 15, 2011
  • pp: 15965–15975
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Atmospheric turbulence-induced fading channel model for space-to-ground laser communications links

Morio Toyoshima, Hideki Takenaka, and Yoshihisa Takayama  »View Author Affiliations


Optics Express, Vol. 19, Issue 17, pp. 15965-15975 (2011)
http://dx.doi.org/10.1364/OE.19.015965


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Abstract

The fading channel model for generating a random time-varying signal based on the atmospheric turbulence spectrum for space-to-ground laser links is discussed. The temporal frequency characteristics of the downlink are theoretically derived based on the von Karman spectrum. The rms wind speed based on the Bufton wind model is used as the transverse wind velocity, which makes the simulation simple. The time-varying signal is generated as functions of the receiver aperture diameter and the rms wind speed. The simulated result of the time-varying signal is presented and compared with the gamma-gamma distribution based on the scintillation theory in a moderate-to-strong-turbulence regime.

© 2011 OSA

1. Introduction

2. Temporal frequency characteristics of laser beam propagation through atmospheric turbulence

2.1. Derivation of the spatial spectrum

e(t)=e(Vt)=τra(r)s(r+Vt)dr.
(1)

Here, the receiver area for a(r) is πD 2/4, where the antenna diameter is represented by D, τr is the optics loss of the receiver, and V represents the transverse wind velocity between the air mass and the optical link. The function e(Vt) is the spatial signal form of the electrical signal, e(t), and is given by the following convolution:

e(r)=τra(r)s(r),
(2)

e(r)=1[We(κ)],=12πWe(κ)exp(j2πkr)dκ,=τr2πWa(κ)Ws*(κ)exp(j2πkr)dκ.
(4)

The temporal frequency spectrum is given by

We(f)=e(t)exp(jωt)dt,=e(Vt)exp(jωt)dt.
(5)

Here, substituting Eq. (4) into Eq. (5), we have

We(f)=[τr2πWa(κ)Ws*(κ)exp(j2πkVt)dκ]×exp(jωt)dt,=τr2πWa(κ)Ws*(κ)dκ×exp[j(2πkVω)t]dt,=τr2πWa(κ)Ws*(κ)dκδ(2πkVω).
(6)

Let κ and ϕ be the polar components of the two-dimensional vector κ = (κcosθ, κsinθ). We then obtain

Wa(κ)=Wa(κ,ϕ).
(7)

If the direction of the transverse wind velocity is represented by one axis, Eq. (6) can be written as
We(f)=τr2π0Wa(κ,ϕ)Ws*(κ,ϕ)κdκ×02π2πδ(Vκcosϕf)dϕ,=τr0Wa(κ,ϕ)Ws*(κ,ϕ)κdκ×02πδ(Vκcosϕf)dϕ,
(8)
where V is the transverse wind speed. Here, the integration of the impulse function is

f(t)δ(tt0)=f(t0).
(9)

Therefore,

We(f)=τrf/VWa(κ)Ws*(κ)κdκ1Vκ2(f/V)2,=τrV0Wa(κ2+f2V2)Ws*(κ2+f2V2)dκ.
(10)

2.2. Derivation of the temporal frequency spectrum

The power spectral component can be viewed as the probability of each frequency; we therefore consider the expected value of the frequency. When the spectrum Ws2(κ) can be described by the von Kármán spectrum,
Ws2(κ)=0.033Cn2exp(κ2/κm2)(κ2+κ02)11/6,
(11)
where κm=5.92/l0 and κ0=2π/L0. l 0 is an inner scale and L 0 is an outer scale. The window function a(r) is circular with a diameter D, as below:

{a(r)=1|r|D/2a(r)=0|r|>D/2.
(12)

The spatial spectrum of Wa(κ) is given by
Wa(κ)=πD242J1(πDκ)(πDκ),
(13)
where κ is the scalar spatial frequency. Using Eqs. (11) and (13), the temporal power spectrum is given by

We2(f)=|τrV0Wa(κ2+f2V2)Ws*(κ2+f2V2)dκ|2,=τr2V20(πD24)222J12(πDκ2+f2/V2)(πDκ2+f2/V2)2×0.033Cn2exp[(κ2+f2/V2)/κm2](κ2+f2/V2+κ02)11/6dκ,=0.033Cn2τr2D24V20J12(πDκ2+f2/V2)(κ2+f2/V2)×exp[(κ2+f2/V2)/κm2](κ2+f2/V2+κ02)11/6dκ.
(14)

The normalized frequency components can be viewed as the probability of occurrence at each frequency. The normalized power spectrum corresponding to the probability function is formulated as follows

PW(f)=fWe2(f)/0We2(x)dx.
(15)

The expected value of the frequency corresponding to the mean frequency can be obtained by multiplying the frequency by the weighting spectra:

f¯W=0PW(f)df.
(16)

If we modulate the amplitude and phase randomly in each complex frequency component of We(f), the power spectral density (PSD) for generating the random time-varying signals can be calculated from
We'(f)=We(f)×exp[jθrand(f)],
(17)
where θrand(f) is the random phase perturbation from 0 to 2π. Therefore, the time-varying electrical signals can be given by the inverse Fourier transform by

e(t)=|1[We'(f)]|.
(18)

3. Scintillation theory

3.1. Gamma-gamma distribution

σ2=1α+1β+1αβ.
(21)

3.2. Rms wind speed in the HV model

The structural parameter of the HV model is given by [24

24. L. C. Andrews and R. L. Phillips, eds., Laser Beam Propagation through Random Media, 2nd ed. (SPIE Press, 2005).

] and the Cn 2 model modified for the optical ground station at National Institute of Information and Communications Technology (NICT) is given by [25

25. M. Toyoshima, Y. Takayama, H. Kunimori, S. Yamakawa, and T. Jono, “Characteristics of laser beam propagation through the turbulent atmosphere in ground-to-low earth orbit satellite laser communications links,” IEICE Trans. Commun. J94-B(3), 409–418 (2011).

]
Cn2(h)=M×0.00594(v/27)2(105h)10exp(h/1000)+2.7×1016exp(h/1500)+Aexp(h/100),
(22)
where M is the new parameter added for the NICT optical ground station and h is the altitude of the optical ground station. When Cn 2 becomes larger than 10−14, the scintillation can be considered to be in the moderate-to-strong-turbulence regime. The rms wind speed in meters per second, v, is determined from
v=[115×1035×10320×103VB2(h)dh]1/2,
(23)
where VB(h) is described by the Bufton wind model, which is obtained as

VB(h)=ωgh+vg+30exp(h94004800).
(24)

Here, vg is the ground wind speed, and ωg is the slew rate associated with satellite motion relative to an observer on the ground. As shown in Fig. 2
Fig. 2 Rms wind speed as a function of the antenna slew rate at the optical ground station based on the Bufton wind model.
, the rms wind speed can exceed 200 m/s when the antenna slew rate is greater than 0.8°/s. The transverse wind velocity V in Section 2 is regarded as the rms wind speed v in this paper. Equation (23) is the conventional equation; therefore, it is useful to create linkages between the rms wind speed and the antenna slew rate for LEO satellites. As the outer scale is very sensitive for the peak frequency components in the power spectra, the outer scale can be properly fitted with the measured power spectra at the desired rms wind speed.

4. Numerical results

4.1. Temporal frequency spectrum

Figures 3(a)
Fig. 3 Normalized power spectra of fWe 2(f) as functions of the rms wind speed and the receiver aperture diameter. (a) Rms wind speed of v = 76 m/s and the receiver aperture diameters of 5 cm and 32 cm. (b) Rms wind speeds of 76 m/s and 117 m/s and the receiver aperture diameter of 32 cm.
and 3(b) show the temporal power spectra fWe 2(f) calculated for aperture diameters of 5 cm and 32 cm, and an rms wind speed of 76 m/s. The peak frequency for the 5-cm diameter is higher than that for the 32-cm diameter. The temporal power spectra are calculated based on Eq. (15). The power spectra at different rms wind speeds of 76 m/s and 117 m/s are plotted in Fig. 3(b). The peak frequency component for v = 117 m/s becomes higher than that for v = 76 m/s.

4.2. Mean frequency

The mean frequencies can be estimated using Eq. (26). The power spectra fWe 2(f) are integrated and the expected values of the frequencies are plotted in Fig. 4
Fig. 4 Mean frequency of the power spectra fWe 2(f) for receiver aperture diameters of 5 cm and 32 cm as a function of the antenna slew rate.
. The two curves were calculated with aperture diameters of 5 cm and 32 cm, indicated as “We(f) model” in Fig. 4. The measurement results obtained in space-to-ground laser communications experiments [19

19. M. Toyoshima, H. Takenaka, Y. Shoji, and Y. Takayama, “Frequency characteristics of atmospheric turbulence in space-to-ground laser links,” Proc. SPIE 7685, 76850G, 76850G-12 (2010). [CrossRef]

] are also plotted in the figure and these values were measured by a pin-photo diode (PD), an avalanche photo diode (APD), and a quadrant detector (QD) with aperture diameters of 5 cm, 31.8 cm and 31.8 cm, respectively.

4.3. Scintillation index

The normalized variance, which is called the scintillation index, can be calculated using the Cn 2 model in Ref [25

25. M. Toyoshima, Y. Takayama, H. Kunimori, S. Yamakawa, and T. Jono, “Characteristics of laser beam propagation through the turbulent atmosphere in ground-to-low earth orbit satellite laser communications links,” IEICE Trans. Commun. J94-B(3), 409–418 (2011).

]. The parameters used for the simulation are shown in Table 1

Table 1. Parameters for Calculating the Scintillation Index

table-icon
View This Table
. From Eq. (22), Cn 2 at an altitute of 122 m for the NICT optical ground station in Koganei can be estimated to be 2.68 x 10−14 m-2/3. The atmospheric coherence lengths, known as Fried’s parameter, are estimated to be 4.2 cm and 4.7 cm at elevation angles of 23.5° and 35.2°, which correspond to seeing angles of 5.3 arcsec and 4.7 arcsec, respectively. The scintillation indices can be estimated to be 0.093 and 0.083 for the aperutre diameter of 32 cm at elevation angles of 23.5° and 35.2° based on the parameters in Table 1. The scintillation indices are 0.61 and 0.71 for the aperture diameter of 5 cm.

4.4. Generation of random time-varying signals

Figures 5(a)
Fig. 5 Time-varying signals with different circular apertures and different rms wind speeds with the random phase pertubation in the PSDs. (a) Rms wind speed of 76 m/s and the receiver aperture diameters of 5 cm. (b) Rms wind speed of 76 m/s and the receiver aperture diameters of 32 cm. (c) Rms wind speed of 76 m/s and the receiver aperture diameter of 32 cm. (d) Rms wind speed of 117 m/s and the receiver aperture diameter of 32 cm.
and 5(b) show the random time-varying signals generated from the proposed model of Eq. (18) with aperture diameters of 5 cm and 32 cm, and rms wind speed of v = 76 m/s. At the smaller apertures, the time-varying signals have higher frequency fluctuations. At higher rms wind speeds, the random time-varying signals have higher frequency fluctuations, as shown in Figs. 5(c) and 5(d). The normalized intensities are rescaled by the scintillation indices estimated in Subsection 4.3.

4.5. Histogram of the time-varying signals

5. Conclusion

Acknowledgments

The authors would like to express their sincere gratitude to the Japan Aerospace Exploration Agency (JAXA) and NEC Toshiba Space Systems, Ltd., for conducting the NICT optical ground station to the Optical Inter-orbit Communications Engineering Test Satellite (OICETS) experiments.

References and links

1.

D. L. Fried, “Scintillation of a ground-to-space laser illuminator,” J. Opt. Soc. Am. 57(8), 980–983 (1967). [CrossRef]

2.

J. L. Bufton, “Scintillation statistics measured in an earth-space-earth retroreflector link,” Appl. Opt. 16(10), 2654–2660 (1977). [CrossRef] [PubMed]

3.

R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58(10), 1523–1545 (1970). [CrossRef]

4.

A. Ishimaru, “Temporal frequency spectra of multifrequency waves in turbulent atmosphere,” IEEE Trans. Antenn. Propag. 20(1), 10–19 (1972). [CrossRef]

5.

S. F. Clifford, “Temporal-frequency spectra for a spherical wave propagating through atmospheric turbulence,” J. Opt. Soc. Am. 61(10), 1285–1292 (1971). [CrossRef]

6.

D. P. Greenwood and D. L. Fried, “Power spectra requirements for wave-front-compensative systems,” J. Opt. Soc. Am. 66(3), 193–206 (1976). [CrossRef]

7.

D. P. Greenwood, “Bandwidth specification for adaptive optics systems,” J. Opt. Soc. Am. 67(3), 390–393 (1977). [CrossRef]

8.

E. Ryznar, “Dependency of optical scintillation frequency on wind speed,” Appl. Opt. 4(11), 1416–1418 (1965). [CrossRef]

9.

T. J. Gilmartin and R. R. Horning, “Spectral characteristics of intensity fluctuations on a laser beam propagating in a desert atmosphere,” IEEE J. Quantum Electron. 3(6), 254–256 (1967). [CrossRef]

10.

D. H. Höhn, “Effects of atmospheric turbulence on the transmission of a laser beam at 6328 A. II-frequency spectra,” Appl. Opt. 5(9), 1433–1436 (1966). [CrossRef] [PubMed]

11.

T. S. Chu, “On the wavelength dependence of the spectrum of laser beams traversing the atmosphere,” Appl. Opt. 6(1), 163–163 (1967). [CrossRef] [PubMed]

12.

G. A. Tyler, “Bandwidth considerations for tracking through turbulence,” J. Opt. Soc. Am. A 11(1), 358–367 (1994). [CrossRef]

13.

D. L. Fried, “Aperture averaging of scintillation,” J. Opt. Soc. Am. 57(2), 169–175 (1967). [CrossRef]

14.

P. J. Titterton, “Scintillation and transmitter-aperture averaging over vertical paths,” J. Opt. Soc. Am. 63(4), 439–444 (1973). [CrossRef]

15.

J. H. Churnside, “Aperture averaging of optical scintillations in the turbulent atmosphere,” Appl. Opt. 30(15), 1982–1994 (1991). [CrossRef] [PubMed]

16.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Aperture averaging of optical scintillations: power fluctuations and the temporal spectrum,” Waves Random Media 10(1), 53–70 (2000). [CrossRef]

17.

M. Toyoshima, Y. Takayama, H. Kunimori, T. Jono, and K. Arai, “Data analysis results from the KODEN experiments,” Proc. SPIE 6709, 1–8 (2007).

18.

M. Toyoshima, K. Takizawa, T. Kuri, W. Klaus, M. Toyoda, H. Kunimori, T. Jono, Y. Takayama, N. Kura, K. Ohinata, K. Arai, and K. Shiratama, “Ground-to-OICETS laser communication experiments,” Proc. SPIE 6304, 63040B (2006).

19.

M. Toyoshima, H. Takenaka, Y. Shoji, and Y. Takayama, “Frequency characteristics of atmospheric turbulence in space-to-ground laser links,” Proc. SPIE 7685, 76850G, 76850G-12 (2010). [CrossRef]

20.

R. J. Hill and R. G. Frehlich, “Probability distribution of irradiance for the onset of strong scintillation,” J. Opt. Soc. Am. A 14(7), 1530–1540 (1997). [CrossRef]

21.

L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16(6), 1417–1429 (1999). [CrossRef]

22.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Scintillation model for a satellite communication link at large zenith angles,” Opt. Eng. 39(12), 3272–3280 (2000). [CrossRef]

23.

A. Belmonte, “Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance,” Appl. Opt. 39(30), 5426–5445 (2000). [CrossRef] [PubMed]

24.

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

25.

M. Toyoshima, Y. Takayama, H. Kunimori, S. Yamakawa, and T. Jono, “Characteristics of laser beam propagation through the turbulent atmosphere in ground-to-low earth orbit satellite laser communications links,” IEICE Trans. Commun. J94-B(3), 409–418 (2011).

OCIS Codes
(010.1300) Atmospheric and oceanic optics : Atmospheric propagation
(010.3310) Atmospheric and oceanic optics : Laser beam transmission
(060.2605) Fiber optics and optical communications : Free-space optical communication

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: June 20, 2011
Revised Manuscript: July 27, 2011
Manuscript Accepted: July 27, 2011
Published: August 4, 2011

Citation
Morio Toyoshima, Hideki Takenaka, and Yoshihisa Takayama, "Atmospheric turbulence-induced fading channel model for space-to-ground laser communications links," Opt. Express 19, 15965-15975 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-17-15965


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References

  1. D. L. Fried, “Scintillation of a ground-to-space laser illuminator,” J. Opt. Soc. Am. 57(8), 980–983 (1967). [CrossRef]
  2. J. L. Bufton, “Scintillation statistics measured in an earth-space-earth retroreflector link,” Appl. Opt. 16(10), 2654–2660 (1977). [CrossRef] [PubMed]
  3. R. S. Lawrence and J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58(10), 1523–1545 (1970). [CrossRef]
  4. A. Ishimaru, “Temporal frequency spectra of multifrequency waves in turbulent atmosphere,” IEEE Trans. Antenn. Propag. 20(1), 10–19 (1972). [CrossRef]
  5. S. F. Clifford, “Temporal-frequency spectra for a spherical wave propagating through atmospheric turbulence,” J. Opt. Soc. Am. 61(10), 1285–1292 (1971). [CrossRef]
  6. D. P. Greenwood and D. L. Fried, “Power spectra requirements for wave-front-compensative systems,” J. Opt. Soc. Am. 66(3), 193–206 (1976). [CrossRef]
  7. D. P. Greenwood, “Bandwidth specification for adaptive optics systems,” J. Opt. Soc. Am. 67(3), 390–393 (1977). [CrossRef]
  8. E. Ryznar, “Dependency of optical scintillation frequency on wind speed,” Appl. Opt. 4(11), 1416–1418 (1965). [CrossRef]
  9. T. J. Gilmartin and R. R. Horning, “Spectral characteristics of intensity fluctuations on a laser beam propagating in a desert atmosphere,” IEEE J. Quantum Electron. 3(6), 254–256 (1967). [CrossRef]
  10. D. H. Höhn, “Effects of atmospheric turbulence on the transmission of a laser beam at 6328 A. II-frequency spectra,” Appl. Opt. 5(9), 1433–1436 (1966). [CrossRef] [PubMed]
  11. T. S. Chu, “On the wavelength dependence of the spectrum of laser beams traversing the atmosphere,” Appl. Opt. 6(1), 163–163 (1967). [CrossRef] [PubMed]
  12. G. A. Tyler, “Bandwidth considerations for tracking through turbulence,” J. Opt. Soc. Am. A 11(1), 358–367 (1994). [CrossRef]
  13. D. L. Fried, “Aperture averaging of scintillation,” J. Opt. Soc. Am. 57(2), 169–175 (1967). [CrossRef]
  14. P. J. Titterton, “Scintillation and transmitter-aperture averaging over vertical paths,” J. Opt. Soc. Am. 63(4), 439–444 (1973). [CrossRef]
  15. J. H. Churnside, “Aperture averaging of optical scintillations in the turbulent atmosphere,” Appl. Opt. 30(15), 1982–1994 (1991). [CrossRef] [PubMed]
  16. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Aperture averaging of optical scintillations: power fluctuations and the temporal spectrum,” Waves Random Media 10(1), 53–70 (2000). [CrossRef]
  17. M. Toyoshima, Y. Takayama, H. Kunimori, T. Jono, and K. Arai, “Data analysis results from the KODEN experiments,” Proc. SPIE 6709, 1–8 (2007).
  18. M. Toyoshima, K. Takizawa, T. Kuri, W. Klaus, M. Toyoda, H. Kunimori, T. Jono, Y. Takayama, N. Kura, K. Ohinata, K. Arai, and K. Shiratama, “Ground-to-OICETS laser communication experiments,” Proc. SPIE 6304, 63040B (2006).
  19. M. Toyoshima, H. Takenaka, Y. Shoji, and Y. Takayama, “Frequency characteristics of atmospheric turbulence in space-to-ground laser links,” Proc. SPIE 7685, 76850G, 76850G-12 (2010). [CrossRef]
  20. R. J. Hill and R. G. Frehlich, “Probability distribution of irradiance for the onset of strong scintillation,” J. Opt. Soc. Am. A 14(7), 1530–1540 (1997). [CrossRef]
  21. L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16(6), 1417–1429 (1999). [CrossRef]
  22. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Scintillation model for a satellite communication link at large zenith angles,” Opt. Eng. 39(12), 3272–3280 (2000). [CrossRef]
  23. A. Belmonte, “Feasibility study for the simulation of beam propagation: consideration of coherent lidar performance,” Appl. Opt. 39(30), 5426–5445 (2000). [CrossRef] [PubMed]
  24. L. C. Andrews and R. L. Phillips, eds., Laser Beam Propagation through Random Media, 2nd ed. (SPIE Press, 2005).
  25. M. Toyoshima, Y. Takayama, H. Kunimori, S. Yamakawa, and T. Jono, “Characteristics of laser beam propagation through the turbulent atmosphere in ground-to-low earth orbit satellite laser communications links,” IEICE Trans. Commun. J94-B(3), 409–418 (2011).

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