Abstract

A propagation model that describes the temporal characteristics of singly scattered radiation in a homogeneous scattering and absorbing medium is presented. The model generalizes previous results in the area and is used to analyze the angular spectrum of singly scattered energy as well as the impulse-response durations and path losses of short-range non-line-of-sight optical communication systems. It is shown that the angular response starts to drop off significantly at an off-axis angle equal to the receiver half-field of view. It is also shown that lower path losses correspond to longer impulse responses so that a lower available bandwidth is indicated. These results are based on numerical examples motivated by the operation of non-line-of-sight communications links in the middle-ultraviolet wave band.

© 1991 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  2. R. S. Kennedy, “Communication through optical scattering channels: an introduction,” Proc. IEEE 58, 1651–1665 (1970).
    [Crossref]
  3. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.
  4. J. H. Shapiro, C. Warde, “Optical communication through low-visibility weather,” Opt. Eng. 20, 76–83 (1981).
    [Crossref]
  5. G. C. Mooradian, M. Geller, L. B. Stotts, D. H. Stephens, R. A. Krautwald, “Blue–green pulsed propagation through fog,” Appl. Opt. 18, 429–441 (1979).
    [Crossref] [PubMed]
  6. D. M. Reilly, C. M. Warde, “Temporal characteristics of single-scatter radiation,” J. Opt. Soc. Am. 69, 464–470 (1979).
    [Crossref]
  7. C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).
  8. M. Luettgen, “Trajectory estimation of an optically radiating source,” master’s thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1990).
  9. B. Carnahan, H. A. Luther, J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

1981 (1)

J. H. Shapiro, C. Warde, “Optical communication through low-visibility weather,” Opt. Eng. 20, 76–83 (1981).
[Crossref]

1979 (2)

1970 (1)

R. S. Kennedy, “Communication through optical scattering channels: an introduction,” Proc. IEEE 58, 1651–1665 (1970).
[Crossref]

Carnahan, B.

B. Carnahan, H. A. Luther, J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Flammer, C.

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).

Geller, M.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.

Kennedy, R. S.

R. S. Kennedy, “Communication through optical scattering channels: an introduction,” Proc. IEEE 58, 1651–1665 (1970).
[Crossref]

Krautwald, R. A.

Luettgen, M.

M. Luettgen, “Trajectory estimation of an optically radiating source,” master’s thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1990).

Luther, H. A.

B. Carnahan, H. A. Luther, J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Mooradian, G. C.

Reilly, D. M.

Shapiro, J. H.

J. H. Shapiro, C. Warde, “Optical communication through low-visibility weather,” Opt. Eng. 20, 76–83 (1981).
[Crossref]

Stephens, D. H.

Stotts, L. B.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Warde, C.

J. H. Shapiro, C. Warde, “Optical communication through low-visibility weather,” Opt. Eng. 20, 76–83 (1981).
[Crossref]

Warde, C. M.

Wilkes, J. O.

B. Carnahan, H. A. Luther, J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

J. H. Shapiro, C. Warde, “Optical communication through low-visibility weather,” Opt. Eng. 20, 76–83 (1981).
[Crossref]

Proc. IEEE (1)

R. S. Kennedy, “Communication through optical scattering channels: an introduction,” Proc. IEEE 58, 1651–1665 (1970).
[Crossref]

Other (5)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).

M. Luettgen, “Trajectory estimation of an optically radiating source,” master’s thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1990).

B. Carnahan, H. A. Luther, J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

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Figures (12)

Fig. 1
Fig. 1

Prolate-spheroidal coordinates.

Fig. 2
Fig. 2

Definition of ϕ = arctan (x, y).

Fig. 3
Fig. 3

Non-line-of-sight scatter communications link.

Fig. 4
Fig. 4

Instantaneous scattering area.

Fig. 5
Fig. 5

Reception geometries: isotropic transmitter case.

Fig. 6
Fig. 6

Prolate spheroid: receiver cone geometry (valid for βR <θR < π/4).

Fig. 7
Fig. 7

Reception geometries: isotropic receiver case.

Fig. 8
Fig. 8

Single-scatter impulse responses.

Fig. 9
Fig. 9

Single-scatter energy densities.

Fig. 10
Fig. 10

Non-line-of-sight system impulse and square pulse response.

Fig. 11
Fig. 11

Predicted impulse-response durations for non-line-of-sight systems.

Fig. 12
Fig. 12

Predicted path losses for non-line-of-sight systems.

Equations (63)

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ξ = ( r 1 + r 2 ) / r ( 1 ξ ) ,
η = ( r 1 r 2 ) / r ( 1 η 1 ) ,
ϕ = arctan ( x , y ) ( π ϕ π ) ,
r 1 = [ x 2 + y 2 + ( z + r / 2 ) 2 ] 1 / 2 ,
r 2 = [ x 2 + y 2 + ( z r / 2 ) 2 ] 1 / 2 ,
cos ψ 1 = ( 1 + ξ η ) / ( ξ + η ) ,
sin ψ 1 = [ ( ξ 2 1 ) ( 1 η 2 ) ] 1 / 2 / ( ξ + η ) ,
cos ψ 2 = ( 1 ξ η ) / ( ξ η ) ,
sin ψ 2 = [ ( ξ 2 1 ) ( 1 η 2 ) ] 1 / 2 / ( ξ η ) ,
cos θ s = ( 2 ξ 2 η 2 ) / ( ξ 2 η 2 ) .
η = ( ξ cos ψ 1 1 ) / ( ξ cos ψ 1 ) ,
η = ( 1 ξ cos ψ 2 ) / ( ξ cos ψ 2 ) .
H P = Q T exp ( k e r 2 ) Ω T ( r 2 ) 2 .
δ Q P = k s H p δ V = k s Q T exp ( k e r 2 ) Ω T ( r 2 ) 2 δ V .
δ R P = δ Q P P ( θ s ) / 4 π .
δ H R = δ R P cos ( ζ ) exp ( k e r 1 ) ( r 1 ) 2 = Q T k s cos ( ζ ) exp [ k e ( r 1 + r 2 ) ] 4 π Ω T ( r 1 r 2 ) 2 P ( θ s ) δ V ,
cos ( ζ ) = cos ( β R ) cos ( ψ 1 ) + sin ( β R ) sin ( ψ 1 ) cos ( ϕ ) ,
δ V = ( r 3 / 8 ) ( ξ 2 η 2 ) δ ξ δ η δ ϕ .
δ H R = Q T k s cos ( ζ ) exp ( k e r ξ ) 2 π Ω T r ( ξ 2 η 2 ) P ( θ s ) δ ξ δ η δ ϕ .
ξ = c t / r ,
δ ξ = c δ t / r .
δ E ( ξ ) = Q T c k s cos ( ζ ) exp ( k e r ξ ) 2 π Ω T r 2 ( ξ 2 η 2 ) P ( θ s ) δ ϕ δ η .
E ( ξ ) = { 0 ( ξ < ξ min ) Q T c k s exp ( k e r ξ ) 2 π Ω T r 2 η 1 ( ξ ) η 2 ( ξ ) ϕ 1 ( ξ , η ) ϕ 2 ( ξ , η ) cos ( ζ ) P ( θ s ) ξ 2 η 2 d ϕ d η ( ξ min ξ ξ max ) 0 ( ξ > ξ max ) ,
ξ min = { 1 ( β R θ R ) 0 or ( β T θ T ) 0 a + ( a 2 1 ) 1 / 2 0 < ( β R θ R ) + ( β T θ T ) < π π ( β R θ R ) + ( β T θ T ) ,
ξ max = { b + ( b 2 1 ) 1 / 2 0 < ( θ R + β R ) + ( θ T + β T ) < π , π ( θ R + β R ) + ( θ T + β T )
a = 1 + cos ( β R θ R ) cos ( β T θ T ) cos ( β R θ R ) + cos ( β T θ T ) ,
b = 1 + cos ( θ R + β R ) cos ( θ T + β T ) cos ( θ R + β R ) + cos ( θ T + β T ) .
ϕ 2 ( ξ , η ) = ϕ 1 ( ξ , η ) ,
E ( ξ ) = { 0 Q T c k s exp ( k e r ξ ) 2 π Ω T r 2 0 ( ξ < ξ min ) η 1 ( ξ ) η 2 ( ξ ) 2 g [ ϕ 2 ( ξ , η ) ] P ( θ s ) ξ 2 η 2 d η ( ξ min ξ ξ max ) , ( ξ > ξ min )
g [ ϕ 2 ( ξ , η ) ] = ϕ 2 ( ξ , η ) cos ( β R ) cos ( ψ 1 ) + sin ( β R ) sin ( ψ 1 ) sin [ ϕ 2 ( ξ , η ) ] .
H R = t min t max E ( c t / r ) d t .
θ T = π ,
β T = 0 .
ξ min = 1 ,
ξ max = .
η 1 = η 1 , R = { 1 ( β R + θ R 0 ) ξ cos ( β R + θ R ) 1 ξ cos ( β R + θ R ) ( β R + θ R 0 ) ,
η 2 = η 2 , R = { 1 ( β R θ R 0 ) ξ cos ( β R θ R ) 1 ξ cos ( β R θ R ) ( β R θ R 0 ) ,
cos ϕ 2 , R ( ξ , η ) = ( s 2 2 + s 3 2 s 4 2 ) / ( 2 s 2 s 3 ) .
s 2 = s 1 sin β R ,
s 3 = s 1 cos β R tan ψ 1 ,
s 4 2 = s 1 2 + s 5 2 2 s 1 s 5 cos θ R ,
s 5 = s 1 cos β R sec ψ 1 .
ϕ 2 , R ( ξ , η ) = arccos ( cos θ R cos β R cos ψ 1 sin β R sin ψ 1 ) = arccos { cos θ R cos β R 1 + ξ η ξ + η sin β R [ ( ξ 2 1 ) ( 1 η 2 ) ] 1 / 2 ξ + η } = f R ( η , ξ , θ R , β R ) .
ϕ 2 , R ( ξ , η ) = { f R ( η , ξ , θ R , β R ) ( η 1 , R η η R ) π ( η R η 1 ) ,
η R = ξ cos ( β R θ R ) 1 ξ cos ( β R θ R ) .
ϕ 2 , R ( ξ , η ) = { π ( η 1 , R η η R and η 1 , R η R ) f R ( η , ξ , θ R , β R ) ( η R η η R and η R η R ) π ( η R η η 2 , R and η R η 2 , R ) ,
η R = ξ cos ( θ R + β R ) 1 ξ cos ( β R θ R ) ,
η R = ξ cos ( θ R β R ) 1 ξ cos ( β R θ R ) .
θ R = π ,
β R = 0.
ξ min = 1 ,
ξ max = ,
η 1 = η 1 , T = { 1 ( β T θ T 0 ) 1 ξ cos ( β T θ T ) ξ cos ( β T θ T ) ( β T θ T 0 ) ,
η 2 = η 2 , T = { 1 ( θ T + β T π ) 1 ξ cos ( θ T + β T ) ξ cos ( θ T β T ) ( θ T + β T π ) ,
ϕ 2 , T ( ξ , η ) = { π ( η 1 , T η η T and η 1 , T η T ) f T ( η , ξ , θ T , β T ) ( η T η η T and η T η T ) π ( η T η η 2 , T and η T η 2 , T ) ,
f T ( η , ξ , θ T , β T ) = arccos { cos θ T = cos β T 1 ξ η ξ η sin β T [ ( ξ 2 1 ) ( 1 η 2 ) ] 1 / 2 ξ η } = f R ( η , ξ , θ T , β T ) ,
η T = 1 ξ cos ( β T θ T ) ξ cos ( β T θ T ) ,
η T = 1 ξ cos ( β T + θ T ) ξ cos ( β T + θ T ) .
η 1 ( ξ ) = max { η 1 , R , η 1 , T } ( ξ min ξ ξ max ) ,
η 2 ( ξ ) = min { η 2 , R , η 2 , T } ( ξ min ξ ξ max ) ,
ϕ 2 ( ξ , η ) = min { ϕ 2 , R ( ξ , η ) ϕ 2 , T ( ξ , η ) } [ η 1 ( ξ ) η η 2 ( ξ ) ] .
P Rayleigh ( θ s ) = 3 4 + 3 4 cos 2 ( θ s ) = 3 4 + 3 4 ( 2 ξ 2 η 2 ) 2 ( ξ 2 η 2 ) .
t = t ξ min r / c .

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