## The wavelength dependent model of extinction in fog and haze for free space optical communication |

Optics Express, Vol. 19, Issue 4, pp. 3379-3386 (2011)

http://dx.doi.org/10.1364/OE.19.003379

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### 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.

© 2011 OSA

## 1. Introduction

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]

### 1.1 Visibility and Kruse model

*γ*(1/km) and visibility

*V*(km). The specific optical attenuation is obtained from the extinction coefficient by

*A*(dB/km) = 10 log(e)

*γ.*

### 1.2 Other models

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]

*q*= 0) at the lowest visibilities (

*V*< 0.5 km).

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]

*V*(km) < 50 and in the range of wavelengths between 0.3 and 1.1 µm. It was derived by fitting the experimental data published in literature. It seems however that the used experimental data was not obtained during dense fog conditions.

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

*γ*=

*aV*fitting best the measured data at 1.2, 3.7 and 10.6 microns.

^{b}*V*< 0.5 km and it is similar to the Kruse model for higher visibilities

*V*> 1 km.

## 2. Scattering theory: extinction vs. effective radius

*γ*of the hydrometeor can be calculated (within the first order multiple scattering approximation) as:where

*r*is the particle radius,

*n*(

*r*) is the particle (drop) size distribution and

*Q*is the extinction efficiency factor [11]. A gamma type distribution is widely used for representing the microphysical properties of clouds and fogs [12]:where

_{ext}*a*,

*b*and

*α*are adjustable parameters of the distribution. It is appropriate to interpret these model parameters in terms of physical quantities. A particularly fruitful approach is to define an effective radius

*r*[11] of the distribution that well characterizes the whole distribution from the point of view of an integrated geometrical cross section of all particles. The effective radius is proportional to a ratio of the total volume and the total geometrical cross section of particles in the distribution:Substituting (4) into (5), one can find for the effective radius that

_{e}^{−3}) with the extinction efficiency factor

*Q*calculated using full Mie formulae [11]. The complex refractive index of water was taken from [13]. Figure 1 shows the wavelength dependence of the extinction coefficient for fixed

_{ext}*LWC*,

*α*and different values of

*r*. In the short wavelength limit, one can check in Fig. 1 that the extinction coefficient is given by

_{e}*γ*(km

^{−1}) = 1500/

*r*(µm) corresponding to (6) with

_{e}*LWC*=

*ρ*= 1. In the region where applicable (see Fig. 1), a simple relation between the extinction coefficient, the liquid water content and the effective radius is so obtained:where

*γ*(1/km),

*LWC*(g/m

^{3}) and

*r*(µm). The region of sufficient accuracy of (7) with a relative error less than 10% is determined by the condition: λ <

_{e}*r*/5. In order to analyze the extinction dependence on visibility, the

_{e}*relative*extinction coefficient measured relatively to the extinction coefficient at the 0.55 μm wavelength is of interest. The relative extinction is independent of

*LWC*provided

*r*and

_{e}*α*are fixed. It is also clear from Fig. 1 that there is no generally monotonic wavelength dependence except in some interval around 0.55 μm. It seems therefore that the exponent

*q*in (1) may serve as some approximation only in the interval 0.2 μm <

*λ*< 2 μm that is focused on in the following.

*λ*< 0.55 μm and

*λ*≥ 0.55 μm and a linear approximation of the log-log wavelength dependence was found in both of them, see the straight lines in Fig. 2.

*slope*of the log-log dependence

*s*(the exponent in (10) below) as a function of the effective radius was obtained. The

*s*(

*r*) function can be approximated by the following expression: where

_{e}*r*(μm) and the parameters

_{e}*p*for the two wavelength subintervals are summarized in the Table 1 .

_{i}*s*obtained for different

*r*(points) and the interpolating functions (8) (lines) with the parameter values stated in Table 1. The function (8) was chosen so that the limit of

_{e}*s*as

*r*→ 0 is

_{e}*s*= −4 which corresponds to the Rayleigh scattering. The limit of

*s*as

*r*becomes large is

_{e}*s*= 0 which corresponds to the geometrical optics approximation. Similarly as in (1), the wavelength dependence of the extinction coefficient can be expressed as:where

*γ*(1/km),

*λ*(μm). The extinction coefficient at the wavelength of 0.55 μm,

*γ*(0.55μm), is to be obtained from the visibility using its definition as in (2).

## 3. Relation of the effective radius and atmospheric visibility

- a) The relation between the extinction coefficient
*γ*and the liquid water content*LWC*is often (e.g [14].)14. R. G. Eldridge, “Haze and fog aerosol distributions,” J. Atmos. Sci.

**23**(5), 605–613 (1966). [CrossRef]*empirically*modeled by a power-law equation:

*γ*

_{0}stands for the extinction coefficient corresponding to the particular liquid water content

*LWC*

_{0}. The value of the exponent

*c*is usually between 0.5 and 1, but it is wavelength dependent generally.

- b) The extinction coefficient
*γ*is linearly proportional to*LWC*when both the effective radius*r*and the shape parameter_{e}*α*are fixed. It means*c*= 1 in such a case. - c) For a
*fixed LWC*(and fixed*α*) and*λ*= 0.55 μm, the effective radius*r*is_{e}*inversely proportional*to the extinction coefficient provided*r*>_{e}*r*_{e}_{min}where*r*_{e}_{min}is about 0.5 μm. - d) It is from the definition of visibility that
*γ*/*γ*_{0}=*V*_{0}/*V*for the wavelength*λ*= 0.55 μm where*V*_{0}stands for the visibility corresponding to*γ*_{0}and*LWC*_{0}.

*α*constant, since it seems to be the least sensitive parameter in our context. Then it is evident from the points a) and b) above that if the empirical exponent

*c*≠ 1 then

*r*has to vary with

_{e}*LWC*. Consider the effective radius

*r*

_{e}_{0}corresponding to the

*LWC*

_{0}. The further step is to find the values of effective radius

*r*corresponding to the values of

_{e}*LWC*different from

*LWC*

_{0}. Figure 4 shows that (for example) for

*c*= 2/3 and

*LWC*= 0.01 g/m

^{3}, the effective radius

*r*<

_{e}*r*

_{e}_{0}(see the point c) above) and it depends on the ratio of extinction coefficients

*γ*and

*γ*

_{1}.

*c*was obtained for the wavelength

*λ*= 0.55 μm. The relation (13) gives the possibility to deduce the effective radius from the visibility if the particular

*r*

_{e}_{0},

*V*

_{0}and

*c*are known from experiments. For example, consider typical values

*c*= 2/3 (from [14

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

*r*

_{e}_{0}= 10 μm ([16]),

*V*

_{0}= 0.05 km. The Eq. (13) then reads:and it can be substituted to (9) to calculate the exponent

*s*as a function of visibility

*V*.

## 4. Model summary

*r*is estimated from visibility using (13). Second, the exponent of wavelength dependence

_{e}*s*is calculated using (8) and (9). Finally the extinction coefficient

*γ*is obtained using (10).

## 5. Comparison with measured data

*LWC*(g/m

^{3}) and particulate surface area

*PSA*(cm

^{2}/m

^{3}). The effective radius,

*r*(μm) is then calculated using its definition (5) as:

_{e}*r*= 30000∙

_{e}*LWC*/

*PSA*. It can be seen in Fig. 5 that the

*r*fluctuates around about 10 microns during the developed fog phase. It is expected, however, that the typical effective radius depends on local climatic conditions [17

_{e}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]

## 6. Conclusion

*λ*< 2 μm and for visibilities lower than 10 km. The presented modeling approach reveals more explicitly the connection between the extinction coefficient and the microphysical parameters of fog and haze. It also provides some degree of flexibility to adapt its properties according to locally specific atmospheric conditions.

## Acknowledgement

## References and links

1. | H. Willebrand, and B. S. Ghuman, |

2. | O. Bouchet, H. Sizun, C. Boisrobert, F. de Fornel, and P. Favennec, |

3. | A. K. Majumdar, and J. C. Ricklin, eds., |

4. | P. W. Kruse, L. D. McGlauchlin, and R. B. McQuistan, |

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 |

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 |

7. | M. Al Naboulsi, H. Sizun, and F. de Fornel, “Fog attenuation prediction for optical and infrared waves,” Opt. Eng. |

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 |

9. | K. W. Fischer, M. R. Witiw, and E. Eisenberg, “Optical attenuation in fog at a wavelength of 1.55 micrometers,” Atmos. Res. |

10. | R. Nebuloni, “Empirical relationships between extinction coefficient and visibility in fog,” Appl. Opt. |

11. | H. C. van de Hulst, |

12. | D. Deirmendjian, |

13. | D. Segelstein, |

14. | R. G. Eldridge, “Haze and fog aerosol distributions,” J. Atmos. Sci. |

15. | P. Chýlek, “Extinction and liquid water content of fogs and clouds,” J. Atmos. Sci. |

16. | M. Grabner, and V. Kvicera, “On the relation between atmospheric visibility and the drop size distribution of fog for FSO link planning,” in |

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. |

18. | M. Grabner, and V. Kvicera, “Fog attenuation dependence on atmospheric visibility at two wavelengths for FSO link planning,” in |

**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: December 13, 2010

Revised Manuscript: February 1, 2011

Manuscript Accepted: February 2, 2011

Published: February 4, 2011

**Citation**

Martin Grabner and Vaclav Kvicera, "The wavelength dependent model of extinction in fog and haze for free space optical communication," Opt. Express **19**, 3379-3386 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-3379

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### References

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

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