Abstract

Lasers used in free-space optics propagate a beam within a truncated cone. Because of this shape, the intensity cannot follow the Beer–Lambert law. In the case of a homogeneous atmosphere, we calculate the gap from the cylinder case. We will see that the gap exists but is generally very weak and, therefore, that the use of the Beer–Lambert law is a justified approximation.

© 2009 Optical Society of America

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References

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    [CrossRef]
  2. N. Araki and H. Yashima, “A channel model of optical wireless communications during rainfall,” in 2nd International Symposium on Wireless Communication Systems (IEEE, 2005), pp. 205-209.
    [CrossRef]
  3. M. Gebhart, E. Leitgeb, and J. Bregenzer, “Atmospheric effects on optical wireless links,” in 7th International Conference on Telecommunications--ConTEL (IEEE, 2003), pp. 395-401.
  4. V. Kvicera, M. Grabner, and O. Fiser, “Visibility and attenuation due to hydrometeors at 850 nm on a 850 m path,” in 6th International Symposium on Communication Systems, Networks and Digital Signal Processing (IEEE, 2008), pp. 270-272.
    [CrossRef]
  5. S. S. Muhammad, P. Köhldorfer, E. Leitgeb, “Channel modeling for terrestrial free space optical links,” in Proceedings of the 2005 7th International Conference on Transparent Optical Networks (IEEE, 2005), pp. 407-410.
    [CrossRef]
  6. S. Bloom, E. Korevaar, J. Schuster, and H. Willebrand, “Understanding the performance of free-space optics,” J. Opt. Netw. 2, 178-200 (2003).
  7. A. J. Cox, A. J. DeWeerd, and J. Linden, “An experiment to measure Mie and Rayleigh total scattering cross sections,” Am. J. Phys. 70, 620-624 (2002).
    [CrossRef]
  8. H. Abitan, H. Bohr, and P. Buchhave, “Correction to the Beer-Lambert-Bouguer law for the optical absorption,” Appl. Opt. 47, 5354-5357 (2008).
    [CrossRef] [PubMed]
  9. M. Musculus and L. Pickett, “Diagnostic considerations for optical laser-extinction measurements of soot in high-pressure transient combustion environments,” Combust. Flame 141, 371-391 (2005).
    [CrossRef]
  10. B. Lacaze and M. Chabert, “Power spectra for laser-extinction measurements,” Opt. Express 14, 6011-6019 (2006).
    [CrossRef] [PubMed]
  11. H. Jiang, M. Sano, and M. Sekine, “Weibull raindrop-size distribution and its application to rain attenuation,” IEE Proc. Microwaves Antennas Propag. 144 (3), 197-200 (1997).
    [CrossRef]

2008 (2)

2006 (1)

2005 (1)

M. Musculus and L. Pickett, “Diagnostic considerations for optical laser-extinction measurements of soot in high-pressure transient combustion environments,” Combust. Flame 141, 371-391 (2005).
[CrossRef]

2003 (1)

2002 (1)

A. J. Cox, A. J. DeWeerd, and J. Linden, “An experiment to measure Mie and Rayleigh total scattering cross sections,” Am. J. Phys. 70, 620-624 (2002).
[CrossRef]

1997 (1)

H. Jiang, M. Sano, and M. Sekine, “Weibull raindrop-size distribution and its application to rain attenuation,” IEE Proc. Microwaves Antennas Propag. 144 (3), 197-200 (1997).
[CrossRef]

Abitan, H.

Akiba, M.

Araki, N.

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

Bloom, S.

Bohr, H.

Bregenzer, J.

M. Gebhart, E. Leitgeb, and J. Bregenzer, “Atmospheric effects on optical wireless links,” in 7th International Conference on Telecommunications--ConTEL (IEEE, 2003), pp. 395-401.

Buchhave, P.

Chabert, M.

Cox, A. J.

A. J. Cox, A. J. DeWeerd, and J. Linden, “An experiment to measure Mie and Rayleigh total scattering cross sections,” Am. J. Phys. 70, 620-624 (2002).
[CrossRef]

DeWeerd, A. J.

A. J. Cox, A. J. DeWeerd, and J. Linden, “An experiment to measure Mie and Rayleigh total scattering cross sections,” Am. J. Phys. 70, 620-624 (2002).
[CrossRef]

Fiser, O.

V. Kvicera, M. Grabner, and O. Fiser, “Visibility and attenuation due to hydrometeors at 850 nm on a 850 m path,” in 6th International Symposium on Communication Systems, Networks and Digital Signal Processing (IEEE, 2008), pp. 270-272.
[CrossRef]

Gebhart, M.

M. Gebhart, E. Leitgeb, and J. Bregenzer, “Atmospheric effects on optical wireless links,” in 7th International Conference on Telecommunications--ConTEL (IEEE, 2003), pp. 395-401.

Grabner, M.

V. Kvicera, M. Grabner, and O. Fiser, “Visibility and attenuation due to hydrometeors at 850 nm on a 850 m path,” in 6th International Symposium on Communication Systems, Networks and Digital Signal Processing (IEEE, 2008), pp. 270-272.
[CrossRef]

Ito, S.

Jiang, H.

H. Jiang, M. Sano, and M. Sekine, “Weibull raindrop-size distribution and its application to rain attenuation,” IEE Proc. Microwaves Antennas Propag. 144 (3), 197-200 (1997).
[CrossRef]

Kodate, K.

Köhldorfer, P.

S. S. Muhammad, P. Köhldorfer, E. Leitgeb, “Channel modeling for terrestrial free space optical links,” in Proceedings of the 2005 7th International Conference on Transparent Optical Networks (IEEE, 2005), pp. 407-410.
[CrossRef]

Korevaar, E.

Kvicera, V.

V. Kvicera, M. Grabner, and O. Fiser, “Visibility and attenuation due to hydrometeors at 850 nm on a 850 m path,” in 6th International Symposium on Communication Systems, Networks and Digital Signal Processing (IEEE, 2008), pp. 270-272.
[CrossRef]

Lacaze, B.

Leitgeb, E.

M. Gebhart, E. Leitgeb, and J. Bregenzer, “Atmospheric effects on optical wireless links,” in 7th International Conference on Telecommunications--ConTEL (IEEE, 2003), pp. 395-401.

S. S. Muhammad, P. Köhldorfer, E. Leitgeb, “Channel modeling for terrestrial free space optical links,” in Proceedings of the 2005 7th International Conference on Transparent Optical Networks (IEEE, 2005), pp. 407-410.
[CrossRef]

Linden, J.

A. J. Cox, A. J. DeWeerd, and J. Linden, “An experiment to measure Mie and Rayleigh total scattering cross sections,” Am. J. Phys. 70, 620-624 (2002).
[CrossRef]

Muhammad, S. S.

S. S. Muhammad, P. Köhldorfer, E. Leitgeb, “Channel modeling for terrestrial free space optical links,” in Proceedings of the 2005 7th International Conference on Transparent Optical Networks (IEEE, 2005), pp. 407-410.
[CrossRef]

Musculus, M.

M. Musculus and L. Pickett, “Diagnostic considerations for optical laser-extinction measurements of soot in high-pressure transient combustion environments,” Combust. Flame 141, 371-391 (2005).
[CrossRef]

Ogawa, K.

Pickett, L.

M. Musculus and L. Pickett, “Diagnostic considerations for optical laser-extinction measurements of soot in high-pressure transient combustion environments,” Combust. Flame 141, 371-391 (2005).
[CrossRef]

Sano, M.

H. Jiang, M. Sano, and M. Sekine, “Weibull raindrop-size distribution and its application to rain attenuation,” IEE Proc. Microwaves Antennas Propag. 144 (3), 197-200 (1997).
[CrossRef]

Schuster, J.

Sekine, M.

H. Jiang, M. Sano, and M. Sekine, “Weibull raindrop-size distribution and its application to rain attenuation,” IEE Proc. Microwaves Antennas Propag. 144 (3), 197-200 (1997).
[CrossRef]

Wakamori, K.

Willebrand, H.

Yashima, H.

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

Am. J. Phys. (1)

A. J. Cox, A. J. DeWeerd, and J. Linden, “An experiment to measure Mie and Rayleigh total scattering cross sections,” Am. J. Phys. 70, 620-624 (2002).
[CrossRef]

Appl. Opt. (2)

Combust. Flame (1)

M. Musculus and L. Pickett, “Diagnostic considerations for optical laser-extinction measurements of soot in high-pressure transient combustion environments,” Combust. Flame 141, 371-391 (2005).
[CrossRef]

IEE Proc. Microwaves Antennas Propag. (1)

H. Jiang, M. Sano, and M. Sekine, “Weibull raindrop-size distribution and its application to rain attenuation,” IEE Proc. Microwaves Antennas Propag. 144 (3), 197-200 (1997).
[CrossRef]

J. Opt. Netw. (1)

Opt. Express (1)

Other (4)

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

M. Gebhart, E. Leitgeb, and J. Bregenzer, “Atmospheric effects on optical wireless links,” in 7th International Conference on Telecommunications--ConTEL (IEEE, 2003), pp. 395-401.

V. Kvicera, M. Grabner, and O. Fiser, “Visibility and attenuation due to hydrometeors at 850 nm on a 850 m path,” in 6th International Symposium on Communication Systems, Networks and Digital Signal Processing (IEEE, 2008), pp. 270-272.
[CrossRef]

S. S. Muhammad, P. Köhldorfer, E. Leitgeb, “Channel modeling for terrestrial free space optical links,” in Proceedings of the 2005 7th International Conference on Transparent Optical Networks (IEEE, 2005), pp. 407-410.
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Beam geometry.

Fig. 2
Fig. 2

μ 75.10 4 function of l. The stars correspond to Eq. (10).

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

b = a + l tan θ .
A ( l ) = A ( 0 ) exp [ - μ l ] ,
μ = - 1 l E [ ln A ( l ) A ( 0 ) ] ,
A ( l ) = A ( 0 ) j J ( 1 - B j S j ) .
{ f ( s ) = 3 2 π 3 / 2 s ( a + l tan θ ) 3 - a 3 , π a 2 < s < π b 2 Pr [ s < S j < s + d s ] f ( s ) d s ,
E [ ln A ( l ) A ( 0 ) ] = n | V | E [ ln ( 1 - B j S j ) ] = n | V | π a 2 π b 2 E [ ln ( 1 - B s ) ] f ( s ) d s .
E [ ln A ( l ) A ( 0 ) ] = n 2 π tan θ π a 2 π b 2 E [ ln ( 1 - B s ) ] s d s .
μ = - n 2 π l tan θ π a 2 π b 2 E [ ln ( 1 - B s ) ] s d s .
μ = n E [ B ] .
μ n E [ B ] + n E [ B 2 ] 2 π a b ,
Δ μ μ = E [ B 2 ] 2 π a b E [ B ] ,
Var [ ln A ( l ) A ( 0 ) ] = n V [ E [ ln 2 ( 1 - B S ) ] - E 2 [ ln ( 1 - B S ) ] ] .
Var [ ln A ( l ) A ( 0 ) ] n l π [ E [ B 2 ] a b 3 E 2 [ B ] a 2 + a b + b 2 ] .
Var [ ln A ( l ) A ( 0 ) ] n l B 2 π ( a b ) 2 a b ( a 2 + a b + b 2 ) .
f ( l ) = e α l ,
f ( x + y ) = f ( x ) f ( y ) .
f ( 0 , x + y ) = f ( 0 , x ) f ( x , x + y ) ,
f ( x , y ) = A ( y ) / A ( x ) ,
P r [ s < S < s + d s ] = s d x | V | ,
{ s = π ( a + x tan θ ) 2 , d x = 1 2 tan θ d s π s .
f ( s ) = 3 s 2 π 3 / 2 ( b 3 a 3 ) .
E [ ln A ( l ) A ( 0 ) ] = n | V | π a 2 π b 2 E [ ln ( 1 B s ) ] f ( s ) d s ,
E [ ln A ( l ) A ( 0 ) ] = n 2 π tan θ π a 2 π b 2 E [ ln ( 1 B s ) ] s d s .
E [ ln A ( l ) A ( 0 ) ] = E { n π 3 l tan θ [ b 3 ln ( 1 B π b 2 ) a 3 ln ( 1 B π a 2 ) ] 2 n B 3 n B 3 / 2 3 π l tan θ ln ( 1 B / b π 1 + B / b π 1 + B / a π 1 B / a π ) } ,
B j S j S j ( S j S j ) B j S j . 1 S j S j = B j S j .

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