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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 2 — Jan. 16, 2012
  • pp: 1649–1661
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Effect of fog on free-space optical links employing imaging receivers

Reza Nasiri Mahalati and Joseph M. Kahn  »View Author Affiliations


Optics Express, Vol. 20, Issue 2, pp. 1649-1661 (2012)
http://dx.doi.org/10.1364/OE.20.001649


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Abstract

We analyze free-space optical links employing imaging receivers in the presence of misalignment and atmospheric effects, such as haze, fog or rain. We present a detailed propagation model based on the radiative transfer equation. We also compare the relative importance of two mechanisms by which these effects degrade link performance: signal attenuation and image blooming. We show that image blooming dominates over attenuation, except under medium-to-heavy fog conditions.

© 2012 OSA

1. Introduction

An imaging receiver employs a lens, telescope or similar optical system to image a received signal onto an image sensor, which is subdivided into multiple pixels. Such an imaging receiver can separate a desired received signal from undesired ambient light and interfering transmissions, if present [16

16. P. Djahani and J. M. Kahn, “Analysis of infrared wireless links employing multi-beam transmitters and imaging diversity receivers,” IEEE Trans. Commun. 48(12), 2077–2088 (2000). [CrossRef]

]. Atmospheric effects, such as fog, can degrade the performance of imaging receivers by two mechanisms. First is attenuation of the signal, caused both by absorption and by scattering of light out of the field of view (FOV) of the receiver. The second mechanism is the blooming of the image spot at the receiver focal plane, which causes the spot to spread over a larger area, and thus a larger number of pixels, on the image sensor. When the signal power is spread over a larger number of pixels, each contributing noise, the receiver electrical signal-to-noise ratio (SNR) is reduced [16

16. P. Djahani and J. M. Kahn, “Analysis of infrared wireless links employing multi-beam transmitters and imaging diversity receivers,” IEEE Trans. Commun. 48(12), 2077–2088 (2000). [CrossRef]

].

2. Optical propagation through fog

As light propagates through the atmosphere, it can get scattered multiple times, which results in a glow around the light source in the image, as shown in Fig. 1
Fig. 1 As a result of the multiple scattering of light in the atmosphere, the image of a point source spreads out into a spot known as the atmospheric point spread function. When the distance between the transmitter and receiver is much larger than the focal length of the receiver lens, we can assume that multiple scattering only happens within a sphere around the source that fits into the FOV of the receiver (region of significant multiple scattering), and that the effect of propagation through the rest of the atmosphere is merely attenuation.
. Multiple scattering can be neglected in clear air or light rain, but becomes particularly important in haze or fog. There are different approaches for modeling propagation of light with multiple scattering. One class of methods involves numerical Monte-Carlo ray-tracing simulation [17

17. S. Antyufeev, “Monte Carlo method for solving inverse problems of radiative transfer,” in Inverse and Ill-Posed Problem Series (VSP Publishers, 2000).

]. A drawback of such methods is their high computational complexity, which can grow exponentially with the number of scattering events to which a ray is subjected.

An alternate approach, which we pursue here, is based on solving the radiative transfer equation (RTE) [18

18. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

]. When a ray of light enters a scattering particle, it can get scattered in multiple directions, as shown in Fig. 2
Fig. 2 The phase functionP(cosα)is the ratio of the intensity of light scattered at an angleα to the intensity of the incident light.
. The intensity of light scattered in different directions depends on the phase function of the particle, which is defined as [18

18. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

]
P(cosα)=P(θ,φ;θ,φ)=I(θ,φ)I(θ,φ),
(1)
where I(θ,φ) is the intensity of the light incident along the direction (θ,φ) and I(θ,φ) is the intensity of the light scattered along the direction (θ,φ). Under most atmospheric conditions, the phase function is only a function of α, which is the angle between the incident and scattered rays. It can be shown that the cosine of this angle can be expressed in terms of the directions of the incident and scattered light, as
cosα=μμ+(1μ2)(1μ2)cos(φφ),
(2)
where μ=cosθ and μ=cosθ.

σ=3.912V(m1).
(4)

Also,F(θ,φ) is the source function defined by

F(θ,φ)=14π02π0πP(cosα)I(θ,φ)dθdφ.
(5)

When a light source is isotropic and the medium in which the light propagates has spherical symmetry, the radiance at each point in space is only a function of the distance from the source, R, and the angle with respect to the radial direction, θ. In this case,ds describes an element of length in the direction θ at a distance R from the source, and we have

dR=cosθds      and         Rdθ=sinθds.
(6)

Therefore, the RTE becomes

cosθIRsinθRIθ=σ[I(R,θ)F(R,θ)].
(7)

Writing the RTE in terms of μ, as defined above, and defining an optical thickness T=σR, we obtain

μIT+1μ2TIμ=I(T,μ)+14π02π11P(cosα)I(T,μ)dμdθ.
(8)

Different phase functions are used to describe scattering in different media. It has been shown that the Henyey-Greenstein phase function can model very well a wide range of atmospheres, from clear air to dense fog [19

19. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

]. This phase function is given by

P(cosα)=1q2(1+q22qcosα)3/2.
(9)

Narasimhan and Nayar [22

22. S. G. Narasimhan and S. K. Nayar, “Shedding light on the weather,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 2003), pp. 665–672.

] have shown that for an isotropic point source, by expanding the Henyey-Greenstein phase function in a series of Legendre polynomials, the following solution to the RTE is obtained:
I(T,μ)=I0n=0(gn(T)+gn+1(T))Ln(μ),
(10)
where I0 is the radiant intensity of the isotropic source (W/sr), Ln() is the Legendre polynomial of order n, and

g0(T)=0gn(T)=exp(βnTαnlnT),n0αn=n+1βn=2n+1n(1qn1).
(11)

Closed-form approximations have been used to speed up computation of (10) in some applications, such as computer vision [23

23. S. Metari and F. Deschnes, “A new convolutional kernel for atmospheric point spread function applied to computer vision,” in Proceedings of IEEE International Conference on Computer Vision (IEEE, 2003), pp. 1–8.

].

The image of a point source with unit radiant intensity obtained in an imaging system under a given atmospheric condition is called the atmospheric point spread function (APSF). The importance of the APSF is that if it is known for a given atmospheric condition, then the image of any light source of arbitrary shape and size can be computed by a two-dimensional convolution. Given an imaging system, to find the APSF as a function of image-plane coordinates (x,y), on should project I(T,μ) onto the image plane, i.e., find a mapping between the variables (T,μ) and (x,y). It has been shown [22

22. S. G. Narasimhan and S. K. Nayar, “Shedding light on the weather,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 2003), pp. 665–672.

] that when the distance between the light source and the receiver, d, is much greater than the focal length of the lens in the receiver, f, one can assume that multiple scattering happens only inside a sphere that surrounds the source and lies within the receiver FOV. In Fig. 1, this sphere is indicated as the region of significant multiple scattering. The effect of propagation through the rest of the atmosphere can be modeled simply as a loss of exp(T), where T=σd is the optical thickness between the point source and imaging system. As seen in Fig. 1, assuming the point source is isotropic, because of azimuthal symmetry in the system, the APSF is only a function of the radial distance from the center of the image plane, ρ=(x2+y2)1/2. Working out the relationship between ρ and θ and considering multiple scattering only in the region of significant multiple scattering, we obtain
APSF(x,y)=APSF(ρ)=I(T,μ(ρ))I0eT(m2),
(12)
where

μ(ρ)=cos(θ(ρ))θ(ρ)=φ+ψ=tan1(ρf)+sin1(hR)h=bb2acaa=ρ2+f2b=dfρc=(d2R2)ρ2R=dsin(FOV2).
(13)

Figure 3
Fig. 3 Cross section of the APSF for (a) a thin atmosphere (T = 1.2) and (b) a thick atmosphere (T = 4.1). In a thick atmosphere, where the density of scatterers is high, the FWHM of the ASPF is almost independent of the value of the forward scattering parameterq. By contrast, in a thin atmosphere, where the density of the scatterers is low, the FWHM of the APSF is larger for smaller values of q.
shows the cross section of the APSF for an imaging system using a 28-mm f/2.8 lens and an image sensor of size 19×19mm2 in a thin (T<2) and a thick (T>3) atmosphere. It is seen that in a thin atmosphere, the full-width at half-maximum (FWHM) of the APSF depends strongly on q. As it is seen in Fig. 3(a), for an atmosphere with T=1.2, the FWHM of the APSF is 3.9 mm, 3.4 mm and 2.4 mm for q equal to 0.2, 0.75 and 0.9, respectively. The FWHM of APSF is a measure of how much the image blooms due to the atmospheric condition. As expected, the FWHM is smaller for bigger values of the forward scattering parameter q, because particles with larger q tend to scatter light more in the forward direction and less in transverse directions.

In contrast to the thin-atmosphere regime, in the thick atmosphere regime, the FWHM of the APSF is nearly independent of the forward scattering parameter. As seen in Fig. 3(b), for a thick atmosphere with T=4.1, the FWHM of the APSF is 7.3 mm for q equal to 0.2, 0.75 and 0.9. Comparison of the peak values of the APSF in Figs. 3(a) and 3(b) shows that the amount of attenuation strongly depends on the optical thickness, but depends only weakly on the forward scattering parameter. Note that the peak values of the APSF for T=4.1 are about three orders of magnitude smaller than those for T=1.2.

3. Effect of fog on free-space optical links

3.1 Link analysis

Figure 4
Fig. 4 Geometry of an FSO link.
shows the general geometry of an FSO link. Assuming the line joining the transmitter to the receiver makes angles φ and ψ with respect to the transmitter and receiver surface normals, respectively, the total average power detected by the receiver (either imaging or non-imaging) is given by [16

16. P. Djahani and J. M. Kahn, “Analysis of infrared wireless links employing multi-beam transmitters and imaging diversity receivers,” IEEE Trans. Commun. 48(12), 2077–2088 (2000). [CrossRef]

]
Prec=I(φ,d)TF(ψ)TL(ψ)Acosψ,
(14)
where I(φ,d) is the irradiance incident on the receiver (W/m2), TF(ψ) is the optical filter transmission factor (W/W), TL(ψ)is the lens transmission factor (W/W) and A is the receiver light collection area at normal incidence (m2). The transmission factors TF(ψ) and TL(ψ) lie between 0 and 1. The lens transmission factor rapidly approaches zero as the incidence angle ψ approaches the receiver FOV.

If light enters the receiver through a lens having f-number Nf and focal length f, the light collection area is given by.

A=π(f2Nf)2.
(15)

The image formed on the photodetector array in the absence of atmospheric effects is described by the irradiance distribution Irec(x,y). This image is a function of the geometry of the system, the lens focal length f, average received power Prec and the atmospheric conditions. In the absence of fog, one can find an approximation to this image using well-known equations of geometrical optics. Throughout this work, we neglect lens aberrations, whose effect is negligible in comparison to that of the atmospheric effects of interest. For example, assuming the transmitter emits light in a circular disk of uniform irradiance, the image is an ellipse of uniform irradiance, whose size and location can be found from the geometry of conic sections and the lens magnification. More generally, the image on the photodetector array can be expressed as
Irec(x,y)=L(S,Prec,f,d,ψ,θ,φ)(W/m2),
(17)
where L is a linear operator that models the propagation of light from transmitter to the receiver based on geometrical optics, and S specifies the transmitter irradiance distribution (e.g., circular disk with uniform irradiance). Atmospheric effects, such as fog, can strongly affect the image formed on the photodetector array. Assuming atmospheric effects make the imaging process fully incoherent [18

18. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

], by the principle of linear supersposition, the image formed in the presence of atmospheric effects is
Irec,fog(x,y)=Irec(x,y)APSF(x,y),
(18)
where denotes a two-dimensional convolution and APSF(x,y) is given by (12). The APSF (12), which was derived for an isotropic source, is assumed approximately valid for a first-order Lambertian source when the angleφ, as shown in Fig. 4, is small.

In an imaging receiver, the photodetector array has multiple pixels. The image of the transmitter, depending on its size and location, may overlap more than one pixel. The ith pixel receives a fraction of the total power that can be expressed as
Prec,i=si(x,y)Irec,fog(x,y)dxdyi=1,2,,N,
(19)
where the integral is carried over the photodetector, Nis the number of pixels and si(x,y) is an indicator function given by

si(x,y)={1,if(x,y)intheinterioroftheithpixel0,otherwise
(20)

The total noise variance in each pixel is given by,
σtot,i2=σshot,i2+σth,i2i=1,2,,N,
(21)
where σshot,i2 and σth,i2 describe, respectively, the shot noise and thermal noise in the ith pixel.

FOV=2tan1(w2f),
(25)

If the ambient light is uniform over the receiver FOV, to first order, the shot noise variance scales linearly with the pixel area and decreases as the number of pixels increases.

We assume that each pixel incorporates a transimpedance preamplifier. In general, thermal noise arises both from the feedback resistor and the transistors in the preamplifier, and has both white and non-white components. At sufficiently low bit rates (typically below 10 Mb/s), the white component is dominant [25

25. J. M. Kahn and J. R. Barry, “Wireless infrared communications,” Proc. IEEE 85(2), 265–298 (1997). [CrossRef]

], and contributes a variance
σth,i2=4kTRFFnΔfn,
(26)
where kis Boltzmann’s constant, Tis the absolute temperature, RFis the feedback resistance and Fn is the noise figure. As before, Δfn is the equivalent noise bandwidth. The cutoff frequency (at −3 dB) is given by B=G/2πRFCd, where G is the open-loop voltage gain and Cdis the capacitance of a single pixel [16

16. P. Djahani and J. M. Kahn, “Analysis of infrared wireless links employing multi-beam transmitters and imaging diversity receivers,” IEEE Trans. Commun. 48(12), 2077–2088 (2000). [CrossRef]

]. Assuming the detector has a fixed capacitance per unit area η, we have Cd=ηAd, where Adis the area of a single pixel. We assume that the total power consumption of all N pixel preamplifiers is constrained. It can be shown that this is equivalent to constraining G. If the receiver is required to achieve a fixed cutoff frequency B, the feedback resistance must scale asRF=G/2πBCd=G/2πBηAd, which is inversely proportional to the pixel capacitance or pixel area. The thermal noise variance becomes:
σth,i2=8πkTGηAdFnBΔfn,
(27)
which is proportional to the pixel area, and thus inversely proportional to the number of pixels. Note that typically, both B and Δfn scale in proportion to the bit rate.

Because the signal spot can overlap more than one pixel in an imaging receiver, the receiver can use different algorithms for detecting the signal. The simplest one is the select-best (SB) algorithm, where the receiver simply selects the pixel with maximum SNR and uses its output to detect the signal. In this algorithm, the outputs of all the other pixels are ignored. The SNR using SB is [16

16. P. Djahani and J. M. Kahn, “Analysis of infrared wireless links employing multi-beam transmitters and imaging diversity receivers,” IEEE Trans. Commun. 48(12), 2077–2088 (2000). [CrossRef]

]

SNRSB=maxi(r2Prec,i2σtot,i2)=maxiSNRi1iN.
(28)

The SB algorithm is not optimal when the signal spot overlaps more than one pixel. In such cases, the optimal algorithm is maximal-ratio combining (MRC), where the receiver assigns different weights to different pixels and then sums the weighted outputs of all the pixels to form one signal, which is employed for signal decoding. The optimal weights for MRC are ωi=rPrec,i/σtot,i2 [16

16. P. Djahani and J. M. Kahn, “Analysis of infrared wireless links employing multi-beam transmitters and imaging diversity receivers,” IEEE Trans. Commun. 48(12), 2077–2088 (2000). [CrossRef]

] and, therefore, the SNR using MRC is

SNRMRC=(i=1NωirPrec,i)2i=1Nωi2σtot,i2=i=1Nr2Prec,i2σtot,i2=i=1NSNRi.
(29)

3.2. Image blooming vs. attenuation

δbloom=10log10(SNRMRC,fogSNRMRC,air)=10log10(Neff,airNeff,fog)(dB).
(31)

In the first equality, SNRMRC,fog and SNRMRC,air are the SNR values in fog (or other atmospheric conditions) and clear air, respectively, computed using (29), assuming the noise is uniform across all pixels and is independent of weather conditions. The second equality defines Neff,fog and Neff,air as the effective number of pixels over which the image spot spreads in fog (or other atmospheric conditions) and clear air, respectively. Note that Neff,fog and Neff,air need not be integer-valued. Note also that, as defined, δbloom does not depend on the operating SNR or the relative contributions of thermal noise and ambient-light shot noise.

3.3 Overall link performance

4. Discussion

5. Conclusion

Acknowledgments

We thank the Bosch Research and Technology Center in Palo Alto for their financial support.

References and Links

1.

A. K. Majumdar, “Optical communication between aircraft in low-visibility atmosphere using diode lasers,” Appl. Opt. 24(21), 3659–3665 (1985). [CrossRef] [PubMed]

2.

B. R. Strickland, M. J. Lavan, E. Woodbridge, and V. Chan, “Effects of fog on the bit-error rate of a free-space laser communication system,” Appl. Opt. 38(3), 424–431 (1999). [CrossRef] [PubMed]

3.

D. Kedar and S. Arnon, “Optical wireless communication through fog in the presence of pointing errors,” Appl. Opt. 42(24), 4946–4954 (2003). [CrossRef] [PubMed]

4.

X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002). [CrossRef]

5.

X. Zhu and J. M. Kahn, “Performance bounds for coded free-space optical communications through atmospheric turbulence channels,” IEEE Trans. Commun. 51(8), 1233–1239 (2003). [CrossRef]

6.

A. A. Farid and S. Hranilovic, “Outage probability for free-space optical systems over slow fading channels with Pointing Errors,” in Proceedings of 19th Annual Meeting of the IEEE Lasers and Electro-Optics Society (IEEE LEOS, 2006), pp. 82–83.

7.

N. Perlot, “Turbulence-induced fading probability in coherent optical communication through the atmosphere,” Appl. Opt. 46(29), 7218–7226 (2007). [CrossRef] [PubMed]

8.

T. Komine and M. Nakagawa, “Fundamental analysis for visible-light communication system using LED lights,” IEEE Trans. on Consum. Electron. 50, 100–107 (2004).

9.

D. C. O'Brien, L. Zeng, H. Le-Minh, G. Faulkner, J. W. Walewski, and S. Randel, “Visible light communication: challenges and possibilities,” in Proceedings of IEEE 19th International Symposium on Personal, Indoor and Mobile Radio Communications (IEEE, 2008), pp. 1–5.

10.

J. Grubor, S. Randel, K. Langer, and J. W. Walewski, “Broadband information broadcasts using LED-based interior lighting,” J. Lightwave Technol. 26(24), 3883–3892 (2008). [CrossRef]

11.

N. Araki and H. Yashima, “A channel model of optical wireless communications during rainfall,” in Proceedings of 2nd International Symposium on Wireless Communication Systems (2005), pp. 205–209.

12.

M. S. Awan, L. C. Horwath, S. S. Muhammad, E. Leitgeb, F. Nadeem, and M. S. Khan, “Characterization of fog and snow attenuations for free-space optical propagation,” J. Commun. 4, 533–545 (2009).

13.

B. Wu, Z. Hajjarian, and M. Kavehrad, “Free space optical communications through clouds: analysis of signal characteristics,” Appl. Opt. 47(17), 3168–3176 (2008). [CrossRef] [PubMed]

14.

Z. Hajjarian and M. Kavehrad, “Using MIMO transmissions in free space optical communications in presence of clouds and turbulence,” Proc, SPIE 7199, 1–12 (2009).

15.

W. Popoola, Z. Ghassemlooy, M. S. Awan, and E. Leitgeb, “Atmospheric channel effects on terrestrial free-space optical communication links,” in Proceedings of 3rd International Conference on Electronics, Computers and Artificial Intelligence (2009), pp. 17–23.

16.

P. Djahani and J. M. Kahn, “Analysis of infrared wireless links employing multi-beam transmitters and imaging diversity receivers,” IEEE Trans. Commun. 48(12), 2077–2088 (2000). [CrossRef]

17.

S. Antyufeev, “Monte Carlo method for solving inverse problems of radiative transfer,” in Inverse and Ill-Posed Problem Series (VSP Publishers, 2000).

18.

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

19.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

20.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

21.

W. E. K. Middleton, Vision through the Atmosphere (University of Toronto Press, Toronto, 1968).

22.

S. G. Narasimhan and S. K. Nayar, “Shedding light on the weather,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 2003), pp. 665–672.

23.

S. Metari and F. Deschnes, “A new convolutional kernel for atmospheric point spread function applied to computer vision,” in Proceedings of IEEE International Conference on Computer Vision (IEEE, 2003), pp. 1–8.

24.

A. P. Tang, J. M. Kahn, and K. P. Ho, “Wireless infrared communication links using multi-beam transmitters and imaging receivers,” in Proceedings of IEEE International Conference on Communications (IEEE, 1996), pp. 180–186.

25.

J. M. Kahn and J. R. Barry, “Wireless infrared communications,” Proc. IEEE 85(2), 265–298 (1997). [CrossRef]

26.

A. A. Kokhanovsky, Cloud Optics (Springer, 2006).

27.

R. E. Bird, R. L. Hulstrom, and L. J. Lewis, “Terrestrial solar spectral data sets,” Sol. Energy 30(6), 563–573 (1983). [CrossRef]

OCIS Codes
(010.1310) Atmospheric and oceanic optics : Atmospheric scattering
(060.2605) Fiber optics and optical communications : Free-space optical communication

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: September 19, 2011
Revised Manuscript: October 18, 2011
Manuscript Accepted: October 18, 2011
Published: January 11, 2012

Citation
Reza Nasiri Mahalati and Joseph M. Kahn, "Effect of fog on free-space optical links employing imaging receivers," Opt. Express 20, 1649-1661 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-2-1649


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References

  1. A. K. Majumdar, “Optical communication between aircraft in low-visibility atmosphere using diode lasers,” Appl. Opt.24(21), 3659–3665 (1985). [CrossRef] [PubMed]
  2. B. R. Strickland, M. J. Lavan, E. Woodbridge, and V. Chan, “Effects of fog on the bit-error rate of a free-space laser communication system,” Appl. Opt.38(3), 424–431 (1999). [CrossRef] [PubMed]
  3. D. Kedar and S. Arnon, “Optical wireless communication through fog in the presence of pointing errors,” Appl. Opt.42(24), 4946–4954 (2003). [CrossRef] [PubMed]
  4. X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun.50(8), 1293–1300 (2002). [CrossRef]
  5. X. Zhu and J. M. Kahn, “Performance bounds for coded free-space optical communications through atmospheric turbulence channels,” IEEE Trans. Commun.51(8), 1233–1239 (2003). [CrossRef]
  6. A. A. Farid and S. Hranilovic, “Outage probability for free-space optical systems over slow fading channels with Pointing Errors,” in Proceedings of 19th Annual Meeting of the IEEE Lasers and Electro-Optics Society (IEEE LEOS, 2006), pp. 82–83.
  7. N. Perlot, “Turbulence-induced fading probability in coherent optical communication through the atmosphere,” Appl. Opt.46(29), 7218–7226 (2007). [CrossRef] [PubMed]
  8. T. Komine and M. Nakagawa, “Fundamental analysis for visible-light communication system using LED lights,” IEEE Trans. on Consum. Electron.50, 100–107 (2004).
  9. D. C. O'Brien, L. Zeng, H. Le-Minh, G. Faulkner, J. W. Walewski, and S. Randel, “Visible light communication: challenges and possibilities,” in Proceedings of IEEE 19th International Symposium on Personal, Indoor and Mobile Radio Communications (IEEE, 2008), pp. 1–5.
  10. J. Grubor, S. Randel, K. Langer, and J. W. Walewski, “Broadband information broadcasts using LED-based interior lighting,” J. Lightwave Technol.26(24), 3883–3892 (2008). [CrossRef]
  11. N. Araki and H. Yashima, “A channel model of optical wireless communications during rainfall,” in Proceedings of 2nd International Symposium on Wireless Communication Systems (2005), pp. 205–209.
  12. M. S. Awan, L. C. Horwath, S. S. Muhammad, E. Leitgeb, F. Nadeem, and M. S. Khan, “Characterization of fog and snow attenuations for free-space optical propagation,” J. Commun.4, 533–545 (2009).
  13. B. Wu, Z. Hajjarian, and M. Kavehrad, “Free space optical communications through clouds: analysis of signal characteristics,” Appl. Opt.47(17), 3168–3176 (2008). [CrossRef] [PubMed]
  14. Z. Hajjarian and M. Kavehrad, “Using MIMO transmissions in free space optical communications in presence of clouds and turbulence,” Proc, SPIE7199, 1–12 (2009).
  15. W. Popoola, Z. Ghassemlooy, M. S. Awan, and E. Leitgeb, “Atmospheric channel effects on terrestrial free-space optical communication links,” in Proceedings of 3rd International Conference on Electronics, Computers and Artificial Intelligence (2009), pp. 17–23.
  16. P. Djahani and J. M. Kahn, “Analysis of infrared wireless links employing multi-beam transmitters and imaging diversity receivers,” IEEE Trans. Commun.48(12), 2077–2088 (2000). [CrossRef]
  17. S. Antyufeev, “Monte Carlo method for solving inverse problems of radiative transfer,” in Inverse and Ill-Posed Problem Series (VSP Publishers, 2000).
  18. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  19. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  20. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  21. W. E. K. Middleton, Vision through the Atmosphere (University of Toronto Press, Toronto, 1968).
  22. S. G. Narasimhan and S. K. Nayar, “Shedding light on the weather,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 2003), pp. 665–672.
  23. S. Metari and F. Deschnes, “A new convolutional kernel for atmospheric point spread function applied to computer vision,” in Proceedings of IEEE International Conference on Computer Vision (IEEE, 2003), pp. 1–8.
  24. A. P. Tang, J. M. Kahn, and K. P. Ho, “Wireless infrared communication links using multi-beam transmitters and imaging receivers,” in Proceedings of IEEE International Conference on Communications (IEEE, 1996), pp. 180–186.
  25. J. M. Kahn and J. R. Barry, “Wireless infrared communications,” Proc. IEEE85(2), 265–298 (1997). [CrossRef]
  26. A. A. Kokhanovsky, Cloud Optics (Springer, 2006).
  27. R. E. Bird, R. L. Hulstrom, and L. J. Lewis, “Terrestrial solar spectral data sets,” Sol. Energy30(6), 563–573 (1983). [CrossRef]

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