Abstract

The effects of scintillation on an optical data channel are analyzed and numerical results presented. Scintillation with a log normal distribution typical of atmospheric optical effects is assumed. The analysis is concerned with the target miss probability of a laser radar and the bit error probability of an optical binary communications link. Results are expressed in terms of a loss factor which is the extra number of dB signal-to-noise ratio necessary to keep the performance in the presence of scintillation up to the level achievable in the absence of scintillation. With even moderately weak scintillation large loss factors are encountered. A brief treatment of the effects of normally distributed scintillation is also presented.

© 1967 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. I. Davis, Appl. Opt. 5, 139 (1966).
    [CrossRef] [PubMed]
  2. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Co., Inc., New York, 1961), p. 209.
  3. Ref. 2, p. 210.
  4. D. L. Fried, G. E. Mevers, M. P. Keister, J. Opt. Soc. Am. 57, 787 (1967).
    [CrossRef]
  5. D. H. Höhn, Appl. Opt. 5, 1427 (1966).
    [CrossRef] [PubMed]
  6. Ref. 2, p. 229.
  7. D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966).
    [CrossRef]
  8. J. Heading, An Introduction to Phase-Integral Methods (John Wiley & Sons, Inc., New York, 1962).
  9. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), Chap. 7.
  10. V. I. Tatarskii, Soviet Phys.–JETP 22, 1083 (1966).
  11. Ref. 2, Eq. (7.94).
  12. D. L. Fried, J. B. Seidman, J. Opt. Soc. Am. 57, 181 (1967).
    [CrossRef]
  13. D. L. Fried, J. Opt. Soc. Am. 57, 169 (1967).
    [CrossRef]
  14. G. R. Heidbreder, R. L. Mitchell, IEEE Trans. AES-3, 5 (1967).

1967 (4)

1966 (4)

Abramowitz, M.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), Chap. 7.

Davis, J. I.

Fried, D. L.

Heading, J.

J. Heading, An Introduction to Phase-Integral Methods (John Wiley & Sons, Inc., New York, 1962).

Heidbreder, G. R.

G. R. Heidbreder, R. L. Mitchell, IEEE Trans. AES-3, 5 (1967).

Höhn, D. H.

Keister, M. P.

Mevers, G. E.

Mitchell, R. L.

G. R. Heidbreder, R. L. Mitchell, IEEE Trans. AES-3, 5 (1967).

Seidman, J. B.

Stegun, I.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), Chap. 7.

Tatarskii, V. I.

V. I. Tatarskii, Soviet Phys.–JETP 22, 1083 (1966).

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Co., Inc., New York, 1961), p. 209.

Appl. Opt. (2)

IEEE Trans. (1)

G. R. Heidbreder, R. L. Mitchell, IEEE Trans. AES-3, 5 (1967).

J. Opt. Soc. Am. (4)

Soviet Phys.–JETP (1)

V. I. Tatarskii, Soviet Phys.–JETP 22, 1083 (1966).

Other (6)

Ref. 2, Eq. (7.94).

Ref. 2, p. 229.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill Book Co., Inc., New York, 1961), p. 209.

Ref. 2, p. 210.

J. Heading, An Introduction to Phase-Integral Methods (John Wiley & Sons, Inc., New York, 1962).

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), Chap. 7.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

The loss factor L as a function of the target miss probability P M for an optical radar in the presence of log normal scintillation. (The loss factor is the extra number of dB signal-to-noise ratio required above that needed because of the regular noise of the system, to compensate for the adverse effect of the scintillation. The calculations are carried out for a range of values of the log amplitude variance σ l 2. In performing the calculations the threshold-to-noise ratio was assumed set at α = 3, correponding to a false alarm probability P FA = 1.35 × 10−3.)

Fig. 2
Fig. 2

Same as Fig. 1 except that α = 4, P FA = 3.17 × 10−5.

Fig. 3
Fig. 3

Same as Fig. 1 except that α = 5, P FA = 2.87 × 10−7.

Fig. 4
Fig. 4

Same as Fig. 1 except that α = 6, P FA = 1.0 × 10−9.

Fig. 5
Fig. 5

Same as Fig. 1 except that α = 7, P FA = 1.3 × 10−12.

Fig. 6
Fig. 6

The loss factor, L, as a function of the probability of error, P E for an optical binary communications link in the presence of log normal scintillation. (The loss factor is the extra number of dB signal-to-noise ratio required above that needed because of the regular noise of the system, to compensate for the adverse effect of scintillation. The calculations are carried out for a range of values of the log amplitude variance σ l 2.

Equations (76)

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

( 2 π ) - 1 2 5 exp ( - t 2 / 2 ) d t = 2.87 × 10 - 5 .
1 2 ( ( 2 π ) - 1 2 4 ( exp - t 2 / 2 ) d t ) + 1 2 ( ( 2 π ) - 1 2 6 ( - t 2 / 2 ) d t ) = 1 2 ( 3.17 × 10 - 3 ) + 1 2 ( 2 × 10 - 7 ) = 1.59 × 10 - 3 .
P F A = ( 2 π σ N 2 ) - 1 2 T exp ( - τ / 2 2 σ N 2 ) .
P M = ( 2 π σ N 2 ) - 1 2 - T exp [ - ( τ - S ) 2 / 2 σ N 2 ] d τ .
P E = ( 2 π σ N 2 ) - 1 2 - 0 exp [ - ( τ - S ) 2 / 2 σ N 2 ] d τ .
P E = lim T 0 P M .
P M = - + d s P f ( S ) ( 2 π σ N 2 ) - 1 2 - T exp [ - ( τ - S ) 2 / 2 σ N 2 ] d τ .
l = 1 2 ln ( S / S 0 ) ,
l ¯ = - σ l 2 .
Prob ( l , l + d l ) = ( 2 π σ l 2 ) - ½ exp [ - ( l + σ l 2 ) 2 / 2 σ l 2 ] d l .
d l = 1 2 d S / S .
P f ( S ) d S = { ( 2 π σ l 2 ) - 1 2 ( 2 S ) - 1 exp [ - ( l + σ l 2 ) 2 / 2 σ l 2 ] d S ,             if S 0 0 ,             if S < 0.
t = 1 - ( τ - S ) / T ,
d t = - d τ / T ,
α = T / σ N ,
P M = - + d s P f ( S ) ( α 2 / 2 π ) 1 2 S / T d t exp ( - 1 2 α 2 ( t - 1 ) 2 ] .
S / T = exp ( u T - u ) ,
d S / S = - d u ,
u T = ln ( S 0 / T ) - 2 σ l 2
u = ln ( S 0 / S ) - 2 σ l 2 = - 2 ( l + σ l 2 ) ,
P M = ( α 4 π σ l ) - + d u exp ( - u 2 / 8 σ l 2 ) exp ( u T - u ) + d t exp [ - 1 2 α 2 ( t - 1 ) 2 ] .
P M = ( α / 4 π σ l ) 0 d t u T - ln t d u exp [ - 1 2 α 2 ( t - 1 ) 2 - u 2 / 8 σ l 2 ] .
w = u + ln t ,
d w = d u ,
P M = ( α / 4 π σ l ) 0 d t u T d w exp [ - 1 2 α 2 ( t - 1 ) 2 - ( w - ln t ) 2 / 8 σ l 2 ] .
P M = ( α / 4 π σ l ) u T d w 0 d t exp [ - f ( w , t ) ] ,
f ( w , t ) = 1 2 α 2 ( t - 1 ) 2 + ( w - ln t ) 2 / 8 σ l 2 .
f ( w , t ) = f ( w , z ) + f t ( w , z ) ( t - z ) / 1 ! + f t t ( w , z ) ( t - z ) 2 / 2 ! + ,
f t ( w , t ) = α 2 ( t - 1 ) - ( w - ln t ) / 4 σ l 2 t ,
f t t ( w , t ) = α 2 + 1 / 4 σ l 2 t 2 + ( w - ln t ) / 4 σ l 2 t 2 .
f t ( w , z ) = 0 ,
0 d t exp [ - f ( w , t ) ] exp [ - f ( w , z ) ] - + d t exp [ - 1 2 f t t ( w , z ) ( t - z ) 2 ] = ( 2 π ) 1 2 [ f t t ( w , z ) ] - 1 2 exp - [ f ( w , z ) ] .
4 σ l 2 α 2 ( z 2 - z ) - w + ln z = 0.
f t t ( w , z ) = [ 1 + 4 σ l 2 α 2 ( 2 z 2 - z ) ] / 4 σ l 2 z 2 .
P M = ( α 2 2 π ) 1 2 u T d w z exp [ - 1 2 α 2 ( z - 1 ) 2 - ( w - ln z ) 2 / 8 σ l 2 ] [ 1 + 4 σ l 2 α 2 ( 2 z 2 - 2 ) ] ½ .
d w = [ 4 σ l 2 α 2 ( 2 z - 1 ) + z - 1 ] d z ,
( w - ln z ) 2 / 8 σ l 2 = 2 σ l 2 α 4 ( z - 1 ) 2 z 2 ,
P M ( α 2 / 2 π ) 1 2 z T d z [ 1 + 4 σ l 2 α 2 ( 2 z - z ) ] 1 2 exp [ - 1 2 α 2 ( z - 1 ) 2 ( 1 + 4 σ l 2 α 2 z 2 ) ] .
4 σ l 2 α 2 ( z T 2 - z T ) - u T + ln z T = 0.
z = 1 + v / 2 σ l α ,
d z = d v / 2 σ l α ,
P M = ( 8 π σ l 2 ) - 1 2 2 σ l α ( z T - 1 ) d v [ 1 + 2 ( v + σ l α ) ( v + 2 σ l α ) ] 1 2 exp { - ( v 2 / 8 σ l 2 ) [ 1 + ( v + 2 σ l α ) 2 ] } .
P M ( 8 π σ l 2 ) - 1 2 2 σ l α ( z T - 1 ) d v [ 1 + 8 σ l 2 α 2 z T ( z T - 1 2 ) ] exp [ - ( v 2 / 8 σ l 2 ) ( 1 + 4 σ l 2 α 2 z T 2 ) ] .
erfc ( x ) = 2 ( π ) - 1 2 x e - t 2 d t ,
P M = 1 2 [ 1 + u R / ( 1 + u S ) ] 1 2 erfc [ u R ( 1 + 1 / u S ) 1 2 ( 8 σ l 2 ) - 1 2 ] ,
u R = 4 σ l 2 α 2 ( z T 2 - z T ) ,
u S = 4 σ l 2 α 2 z T 2 .
β = S 0 / σ N .
β = α z T exp ( u R + 2 σ l 2 ) .
P M = 1 2 erfc [ ( β - α ) / ( 2 ) 1 2 ] .
L = 20 log 10 ( β / β )             ( dB ) .
P M = ( α / 4 π σ l ) u T d w 0 d t exp [ - 1 2 α 2 ( t - 1 ) 2 - ( w - ln t ) 2 / 8 σ l 2 ] .
t = ( α / β ) t ,
w = w - ln ( β / α ) ,
P M = ( β / 4 π σ l ) - 2 σ l 2 d w 0 d t exp [ - 1 2 β 2 ( t - α / β ) 2 - ( w - ln t ) 2 / 8 σ l 2 ] ,
P E = ( β / 4 π σ l ) - 2 σ l 2 d w 0 d t exp [ - g ( w , t ) ] ,
g ( w , t ) = 1 2 β 2 t 2 + ( w - ln t ) 2 / 8 σ l 2 .
g ( w , t ) g ( w , z ) + g t ( w , z ) ( t - z ) / 1 ! + g t t ( w , z ) ( t - z ) 2 / 2 ! ,
g t ( w , t ) = β 2 t - ( w - ln t ) / 4 σ l 2 t ,
g t t ( w , t ) = β 2 + 1 / 4 σ l 2 t 2 + ( w - l n t ) / 4 σ l 2 t 2 ,
g t ( w , z ) = 0.
w = 4 β 2 σ l 2 z 2 + ln z ,
g ( w , z ) = 1 2 β 2 z 2 + 2 σ l 2 β 4 z 4 ,
g t t ( w , z ) = 2 β 2 + 1 / 4 σ 1 2 z .
P E ( β 2 / 2 π σ l 2 ) 1 2 - 2 σ l 2 d w [ ( 8 β 2 σ l 2 ) + 1 / z 2 ] - 1 2 exp ( - 1 2 β 2 z 2 - 2 σ l 2 β 4 z 4 ) .
P E = ( β 2 / 8 π ) 1 2 x T d x ( 8 β 2 σ l 2 + 1 x ) 1 2 exp ( - 1 2 β 2 x - 2 σ l 2 β 4 x 2 ) ,
σ l 2 = - ( ln x T ) / ( 4 + 8 β 2 x T ) .
[ ( / x ) ( - 1 2 β 2 x - 2 σ l 2 β 4 x 2 ) ] x = x T = - 1 2 β 2 - 4 σ l 2 β 4 x T ,
P E ( β 2 / 8 π ) 1 2 x T d x ( 8 β 2 σ l 2 + 1 / x T ) 1 2 exp [ - ( 1 2 β 2 x T + 2 σ l 2 β 4 x T 2 ) - ( x - x T ) ( 1 2 β 2 + 4 σ l 2 β 4 x T ) ] .
P E = [ 2 π ( 8 β 4 σ l 2 x T 2 + β 2 x T ) ] - 1 2 exp [ - 1 2 ( 4 β 4 σ l 2 x T 2 + β 2 x T ) ] .
P f ( S ) = ( 2 π σ S 2 ) - 1 2 exp [ - 1 2 ( S - S 0 ) 2 / σ S 2 ] ,
P M = ( 2 π σ S σ N ) - 1 - + d S - T d τ exp { - 1 2 [ ( S - S 0 ) 2 / σ S 2 + ( τ - S ) 2 / σ N 2 ] } .
P M = ( 2 Σ 2 ) - 1 2 - T d τ exp [ - 1 2 ( τ - S 0 ) 2 / Σ 2 ] .
Σ 2 = σ N 2 + σ S 2 .
L = 20 log 10 [ α + ( β - α ) Σ / σ N α + β ] .
L = 20 log 10 ( Σ / σ N ) , ( binary communications ) .

Metrics