Abstract

The wavelength dependence of the extinction coefficient in fog and haze is investigated using Mie single scattering theory. It is shown that the effective radius of drop size distribution determines the slope of the log-log dependence of the extinction on wavelengths in the interval between 0.2 and 2 microns. The relation between the atmospheric visibility and the effective radius is derived from the empirical relationship of liquid water content and extinction. Based on these results, the model of the relationship between visibility and the extinction coefficient with different effective radii for fog and for haze conditions is proposed.

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References

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  1. H. Willebrand, and B. S. Ghuman, Free-Space Optics: Enabling Optical Connectivity in Today’s Networks (SAMS, Indianapolis, 2002), Chap. 3.
  2. O. Bouchet, H. Sizun, C. Boisrobert, F. de Fornel, and P. Favennec, Free-Space Optics, Propagation and Communication (ISTE, London, 2006), Chap. 4.
  3. A. K. Majumdar, and J. C. Ricklin, eds., Free-Space Laser Communications (Springer, New York, 2008).
  4. P. W. Kruse, L. D. McGlauchlin, and R. B. McQuistan, Elements of Infrared Technology: Generation, Transmission and Detection (Jonh Wiley & Sons, New York, 1962), Chap. 5.
  5. E. Ferdinandov, K. Dimitrov, A. Dandarov, and I. Bakalski, “A general model of the atmospheric scattering in the wavelength interval 300 – 1100 nm,” Radioengineering 18, 517–521 (2009).
  6. I. I. Kim, B. McArthur, and E. J. Korevaar, “Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications,” Proc. SPIE 4214, 26–37 (2001).
    [CrossRef]
  7. M. Al Naboulsi, H. Sizun, and F. de Fornel, “Fog attenuation prediction for optical and infrared waves,” Opt. Eng. 43(2), 319–329 (2004).
    [CrossRef]
  8. P. Corrigan, R. Martini, E. A. Whittaker, and C. Bethea, “Quantum cascade lasers and the Kruse model in free space optical communication,” Opt. Express 17(6), 4355–4359 (2009).
    [CrossRef] [PubMed]
  9. K. W. Fischer, M. R. Witiw, and E. Eisenberg, “Optical attenuation in fog at a wavelength of 1.55 micrometers,” Atmos. Res. 87(3-4), 252–258 (2008).
    [CrossRef]
  10. R. Nebuloni, “Empirical relationships between extinction coefficient and visibility in fog,” Appl. Opt. 44(18), 3795–3804 (2005).
    [CrossRef] [PubMed]
  11. H. C. van de Hulst, Light Scattering by Small Particles, (Dover Publications, New York, 1981).
  12. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions, (American Elsevier Pub. Co., New York, 1969), Chap. 4.
  13. D. Segelstein, The Complex Refractive Index of Water, (University of Missouri, Kansas City, 1981).
  14. R. G. Eldridge, “Haze and fog aerosol distributions,” J. Atmos. Sci. 23(5), 605–613 (1966).
    [CrossRef]
  15. P. Chýlek, “Extinction and liquid water content of fogs and clouds,” J. Atmos. Sci. 35, 296–300 (1978).
  16. M. Grabner, and V. Kvicera, “On the relation between atmospheric visibility and the drop size distribution of fog for FSO link planning,” in Proceedings of the 35th European Conference on Optical Communication (VDE VERLAG GMBH, Vienna, 2009), pp. 1–2.
  17. M. S. Awan, R. Nebuloni, C. Capsoni, L. Csurgai-Horváth, S. S. Muhammad, F. Nadeem, M. S. Khan, and E. Leitgeb, “Prediction of drop size distribution parameters for optical wireless communications through moderate continental fog,” Int. J. Satell. Commun. Network. 29(1), 97–116 (2011).
    [CrossRef]
  18. M. Grabner, and V. Kvicera, “Fog attenuation dependence on atmospheric visibility at two wavelengths for FSO link planning,” in Proceedings of Loughborough Antennas & Propagation Conference (Loughborough University, Loughborough, 2010), pp. 193–196.

2011 (1)

M. S. Awan, R. Nebuloni, C. Capsoni, L. Csurgai-Horváth, S. S. Muhammad, F. Nadeem, M. S. Khan, and E. Leitgeb, “Prediction of drop size distribution parameters for optical wireless communications through moderate continental fog,” Int. J. Satell. Commun. Network. 29(1), 97–116 (2011).
[CrossRef]

2009 (2)

P. Corrigan, R. Martini, E. A. Whittaker, and C. Bethea, “Quantum cascade lasers and the Kruse model in free space optical communication,” Opt. Express 17(6), 4355–4359 (2009).
[CrossRef] [PubMed]

E. Ferdinandov, K. Dimitrov, A. Dandarov, and I. Bakalski, “A general model of the atmospheric scattering in the wavelength interval 300 – 1100 nm,” Radioengineering 18, 517–521 (2009).

2008 (1)

K. W. Fischer, M. R. Witiw, and E. Eisenberg, “Optical attenuation in fog at a wavelength of 1.55 micrometers,” Atmos. Res. 87(3-4), 252–258 (2008).
[CrossRef]

2005 (1)

2004 (1)

M. Al Naboulsi, H. Sizun, and F. de Fornel, “Fog attenuation prediction for optical and infrared waves,” Opt. Eng. 43(2), 319–329 (2004).
[CrossRef]

2001 (1)

I. I. Kim, B. McArthur, and E. J. Korevaar, “Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications,” Proc. SPIE 4214, 26–37 (2001).
[CrossRef]

1978 (1)

P. Chýlek, “Extinction and liquid water content of fogs and clouds,” J. Atmos. Sci. 35, 296–300 (1978).

1966 (1)

R. G. Eldridge, “Haze and fog aerosol distributions,” J. Atmos. Sci. 23(5), 605–613 (1966).
[CrossRef]

Al Naboulsi, M.

M. Al Naboulsi, H. Sizun, and F. de Fornel, “Fog attenuation prediction for optical and infrared waves,” Opt. Eng. 43(2), 319–329 (2004).
[CrossRef]

Awan, M. S.

M. S. Awan, R. Nebuloni, C. Capsoni, L. Csurgai-Horváth, S. S. Muhammad, F. Nadeem, M. S. Khan, and E. Leitgeb, “Prediction of drop size distribution parameters for optical wireless communications through moderate continental fog,” Int. J. Satell. Commun. Network. 29(1), 97–116 (2011).
[CrossRef]

Bakalski, I.

E. Ferdinandov, K. Dimitrov, A. Dandarov, and I. Bakalski, “A general model of the atmospheric scattering in the wavelength interval 300 – 1100 nm,” Radioengineering 18, 517–521 (2009).

Bethea, C.

Capsoni, C.

M. S. Awan, R. Nebuloni, C. Capsoni, L. Csurgai-Horváth, S. S. Muhammad, F. Nadeem, M. S. Khan, and E. Leitgeb, “Prediction of drop size distribution parameters for optical wireless communications through moderate continental fog,” Int. J. Satell. Commun. Network. 29(1), 97–116 (2011).
[CrossRef]

Chýlek, P.

P. Chýlek, “Extinction and liquid water content of fogs and clouds,” J. Atmos. Sci. 35, 296–300 (1978).

Corrigan, P.

Csurgai-Horváth, L.

M. S. Awan, R. Nebuloni, C. Capsoni, L. Csurgai-Horváth, S. S. Muhammad, F. Nadeem, M. S. Khan, and E. Leitgeb, “Prediction of drop size distribution parameters for optical wireless communications through moderate continental fog,” Int. J. Satell. Commun. Network. 29(1), 97–116 (2011).
[CrossRef]

Dandarov, A.

E. Ferdinandov, K. Dimitrov, A. Dandarov, and I. Bakalski, “A general model of the atmospheric scattering in the wavelength interval 300 – 1100 nm,” Radioengineering 18, 517–521 (2009).

de Fornel, F.

M. Al Naboulsi, H. Sizun, and F. de Fornel, “Fog attenuation prediction for optical and infrared waves,” Opt. Eng. 43(2), 319–329 (2004).
[CrossRef]

Dimitrov, K.

E. Ferdinandov, K. Dimitrov, A. Dandarov, and I. Bakalski, “A general model of the atmospheric scattering in the wavelength interval 300 – 1100 nm,” Radioengineering 18, 517–521 (2009).

Eisenberg, E.

K. W. Fischer, M. R. Witiw, and E. Eisenberg, “Optical attenuation in fog at a wavelength of 1.55 micrometers,” Atmos. Res. 87(3-4), 252–258 (2008).
[CrossRef]

Eldridge, R. G.

R. G. Eldridge, “Haze and fog aerosol distributions,” J. Atmos. Sci. 23(5), 605–613 (1966).
[CrossRef]

Ferdinandov, E.

E. Ferdinandov, K. Dimitrov, A. Dandarov, and I. Bakalski, “A general model of the atmospheric scattering in the wavelength interval 300 – 1100 nm,” Radioengineering 18, 517–521 (2009).

Fischer, K. W.

K. W. Fischer, M. R. Witiw, and E. Eisenberg, “Optical attenuation in fog at a wavelength of 1.55 micrometers,” Atmos. Res. 87(3-4), 252–258 (2008).
[CrossRef]

Khan, M. S.

M. S. Awan, R. Nebuloni, C. Capsoni, L. Csurgai-Horváth, S. S. Muhammad, F. Nadeem, M. S. Khan, and E. Leitgeb, “Prediction of drop size distribution parameters for optical wireless communications through moderate continental fog,” Int. J. Satell. Commun. Network. 29(1), 97–116 (2011).
[CrossRef]

Kim, I. I.

I. I. Kim, B. McArthur, and E. J. Korevaar, “Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications,” Proc. SPIE 4214, 26–37 (2001).
[CrossRef]

Korevaar, E. J.

I. I. Kim, B. McArthur, and E. J. Korevaar, “Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications,” Proc. SPIE 4214, 26–37 (2001).
[CrossRef]

Leitgeb, E.

M. S. Awan, R. Nebuloni, C. Capsoni, L. Csurgai-Horváth, S. S. Muhammad, F. Nadeem, M. S. Khan, and E. Leitgeb, “Prediction of drop size distribution parameters for optical wireless communications through moderate continental fog,” Int. J. Satell. Commun. Network. 29(1), 97–116 (2011).
[CrossRef]

Martini, R.

McArthur, B.

I. I. Kim, B. McArthur, and E. J. Korevaar, “Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications,” Proc. SPIE 4214, 26–37 (2001).
[CrossRef]

Muhammad, S. S.

M. S. Awan, R. Nebuloni, C. Capsoni, L. Csurgai-Horváth, S. S. Muhammad, F. Nadeem, M. S. Khan, and E. Leitgeb, “Prediction of drop size distribution parameters for optical wireless communications through moderate continental fog,” Int. J. Satell. Commun. Network. 29(1), 97–116 (2011).
[CrossRef]

Nadeem, F.

M. S. Awan, R. Nebuloni, C. Capsoni, L. Csurgai-Horváth, S. S. Muhammad, F. Nadeem, M. S. Khan, and E. Leitgeb, “Prediction of drop size distribution parameters for optical wireless communications through moderate continental fog,” Int. J. Satell. Commun. Network. 29(1), 97–116 (2011).
[CrossRef]

Nebuloni, R.

M. S. Awan, R. Nebuloni, C. Capsoni, L. Csurgai-Horváth, S. S. Muhammad, F. Nadeem, M. S. Khan, and E. Leitgeb, “Prediction of drop size distribution parameters for optical wireless communications through moderate continental fog,” Int. J. Satell. Commun. Network. 29(1), 97–116 (2011).
[CrossRef]

R. Nebuloni, “Empirical relationships between extinction coefficient and visibility in fog,” Appl. Opt. 44(18), 3795–3804 (2005).
[CrossRef] [PubMed]

Sizun, H.

M. Al Naboulsi, H. Sizun, and F. de Fornel, “Fog attenuation prediction for optical and infrared waves,” Opt. Eng. 43(2), 319–329 (2004).
[CrossRef]

Whittaker, E. A.

Witiw, M. R.

K. W. Fischer, M. R. Witiw, and E. Eisenberg, “Optical attenuation in fog at a wavelength of 1.55 micrometers,” Atmos. Res. 87(3-4), 252–258 (2008).
[CrossRef]

Appl. Opt. (1)

Atmos. Res. (1)

K. W. Fischer, M. R. Witiw, and E. Eisenberg, “Optical attenuation in fog at a wavelength of 1.55 micrometers,” Atmos. Res. 87(3-4), 252–258 (2008).
[CrossRef]

Int. J. Satell. Commun. Network. (1)

M. S. Awan, R. Nebuloni, C. Capsoni, L. Csurgai-Horváth, S. S. Muhammad, F. Nadeem, M. S. Khan, and E. Leitgeb, “Prediction of drop size distribution parameters for optical wireless communications through moderate continental fog,” Int. J. Satell. Commun. Network. 29(1), 97–116 (2011).
[CrossRef]

J. Atmos. Sci. (2)

R. G. Eldridge, “Haze and fog aerosol distributions,” J. Atmos. Sci. 23(5), 605–613 (1966).
[CrossRef]

P. Chýlek, “Extinction and liquid water content of fogs and clouds,” J. Atmos. Sci. 35, 296–300 (1978).

Opt. Eng. (1)

M. Al Naboulsi, H. Sizun, and F. de Fornel, “Fog attenuation prediction for optical and infrared waves,” Opt. Eng. 43(2), 319–329 (2004).
[CrossRef]

Opt. Express (1)

Proc. SPIE (1)

I. I. Kim, B. McArthur, and E. J. Korevaar, “Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications,” Proc. SPIE 4214, 26–37 (2001).
[CrossRef]

Radioengineering (1)

E. Ferdinandov, K. Dimitrov, A. Dandarov, and I. Bakalski, “A general model of the atmospheric scattering in the wavelength interval 300 – 1100 nm,” Radioengineering 18, 517–521 (2009).

Other (9)

H. Willebrand, and B. S. Ghuman, Free-Space Optics: Enabling Optical Connectivity in Today’s Networks (SAMS, Indianapolis, 2002), Chap. 3.

O. Bouchet, H. Sizun, C. Boisrobert, F. de Fornel, and P. Favennec, Free-Space Optics, Propagation and Communication (ISTE, London, 2006), Chap. 4.

A. K. Majumdar, and J. C. Ricklin, eds., Free-Space Laser Communications (Springer, New York, 2008).

P. W. Kruse, L. D. McGlauchlin, and R. B. McQuistan, Elements of Infrared Technology: Generation, Transmission and Detection (Jonh Wiley & Sons, New York, 1962), Chap. 5.

M. Grabner, and V. Kvicera, “On the relation between atmospheric visibility and the drop size distribution of fog for FSO link planning,” in Proceedings of the 35th European Conference on Optical Communication (VDE VERLAG GMBH, Vienna, 2009), pp. 1–2.

M. Grabner, and V. Kvicera, “Fog attenuation dependence on atmospheric visibility at two wavelengths for FSO link planning,” in Proceedings of Loughborough Antennas & Propagation Conference (Loughborough University, Loughborough, 2010), pp. 193–196.

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

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions, (American Elsevier Pub. Co., New York, 1969), Chap. 4.

D. Segelstein, The Complex Refractive Index of Water, (University of Missouri, Kansas City, 1981).

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Figures (6)

Fig. 1
Fig. 1

Extinction coefficient for different effective radii, LWC = 1 g/m3, α = 5

Fig. 2
Fig. 2

Relative extinction and the linear approximation of its log-log wavelength dependence, α = 5

Fig. 3
Fig. 3

The exponent of wavelength dependence γ ~λs , α = 5

Fig. 4
Fig. 4

The scheme of the relation between the effective radius and the liquid water content

Fig. 5
Fig. 5

Liquid water content, LWC, particulate surface area, PSA, and effective radius of fog measured during a fog event observed on 20-21 January 2009 in Prague, the Czech Republic.

Fig. 6
Fig. 6

Specific optical attenuation vs visibility predicted by different models and measured data [18], the wavelength 1.55 μm shifted downwards (multiplied by 1/10) for better clarity.

Tables (1)

Tables Icon

Table 1 Parameters of the model (8) for two wavelength subintervals

Equations (14)

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γ ( λ ) = 3.91 V ( λ 0.55 ) q   where:  q = 0.585 V 1 / 3   for   V < 6   km .
I I 0 = 0.02 = e γ ( 0.55 μm ) V γ ( 0.55 μm ) = ln 0.02 V
γ = 0 π r 2 n ( r ) Q e x t   d r
n ( r ) = a r α e b r
r e = 0 r 3 n ( r )  d r 0 r 2 n ( r )  d r = 3 4 total volume total geom . cross section .
γ = 2 0 π r 2 n ( r )   d r = 3 L W C 2 ρ 1 r e
γ = 1500 L W C r e
s = 2 ( tanh ( p 1 ( w + p 4 ) ) 1 ) + p 2 exp ( p 3 ( w + p 5 ) 2 )
w = log 10 r e
γ ( λ ) = γ ( 0.55 μm ) ( λ 0.55 ) s
γ = γ 0 ( L W C L W C 0 ) c
r e r e 0 = γ 1 γ = ( L W C L W C 0 ) 1 c   where   γ 1 = γ 0 ( L W C L W C 0 ) 1
r e = r e 0 ( L W C L W C 0 ) 1 c = r e 0 ( ( γ γ 0 ) 1 / c ) 1 c = r e 0 ( γ γ 0 ) ( 1 / c ) 1 = r e 0 ( V 0 V ) ( 1 / c ) 1 .
r e = 10 ( 0.05 V ) 1 / 2

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