Abstract

Associated with a signal transmitted down a waveguide are multiple back and forth scatterings. These produce a delayed decaying interference trailing a sharp signal pulse. An analytical description of this phenomenon enables an estimate of its effect on an optical waveguide’s information-carrying capacity. For the example of a signal-to-interference ratio of 100 in a 1-km fiber of 20-dB attenuation, a bit rate limitation occurs when the fractional backscattered power captured per unit length exceeds 1.3 dB/km.

© 1972 Optical Society of America

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References

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  1. F. P. Kapron, D. B. Keck, Appl. Opt. 10, 1519 (1971).
    [CrossRef] [PubMed]
  2. R. B. Dyott, J. R. Stern, Electron. Lett. 7, 82 (1971).
    [CrossRef]
  3. D. Gloge, Appl. Opt. 10, 2442 (1971).
    [CrossRef] [PubMed]
  4. F. P. Kapron, N. F. Borrelli, D. B. Keck, IEEE J. Quantum Electron., QE-8, 222 (1972).
    [CrossRef]
  5. F. P. Kapron, D. B. Keck, R. D. Maurer, Appl. Phys. Lett. 17, 423 (1970).
    [CrossRef]
  6. R. B. Dyott, J. R. Stern, Electron. Lett. 7, 624 (1971).
    [CrossRef]
  7. A. R. Tynes, A. D. Pearson, D. L. Bisbee, J. Opt. Soc. Am. 61, 143 (1971).
    [CrossRef]
  8. A. W. Snyder, Electron. Lett. 5, 271 (1969).
    [CrossRef]
  9. D. Marcuse, Bell Syst. Tech. J. 49, 1665 (1970).
  10. Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, Eds. (Dover, New York, 1965), Chap. 29.
  11. Table of Integrals, Series and Products, translation ed. by A. Jeffrey (Academic, New York, 1965), Sec. 6.6.

1972 (1)

F. P. Kapron, N. F. Borrelli, D. B. Keck, IEEE J. Quantum Electron., QE-8, 222 (1972).
[CrossRef]

1971 (5)

1970 (2)

D. Marcuse, Bell Syst. Tech. J. 49, 1665 (1970).

F. P. Kapron, D. B. Keck, R. D. Maurer, Appl. Phys. Lett. 17, 423 (1970).
[CrossRef]

1969 (1)

A. W. Snyder, Electron. Lett. 5, 271 (1969).
[CrossRef]

Bisbee, D. L.

Borrelli, N. F.

F. P. Kapron, N. F. Borrelli, D. B. Keck, IEEE J. Quantum Electron., QE-8, 222 (1972).
[CrossRef]

Dyott, R. B.

R. B. Dyott, J. R. Stern, Electron. Lett. 7, 82 (1971).
[CrossRef]

R. B. Dyott, J. R. Stern, Electron. Lett. 7, 624 (1971).
[CrossRef]

Gloge, D.

Kapron, F. P.

F. P. Kapron, N. F. Borrelli, D. B. Keck, IEEE J. Quantum Electron., QE-8, 222 (1972).
[CrossRef]

F. P. Kapron, D. B. Keck, Appl. Opt. 10, 1519 (1971).
[CrossRef] [PubMed]

F. P. Kapron, D. B. Keck, R. D. Maurer, Appl. Phys. Lett. 17, 423 (1970).
[CrossRef]

Keck, D. B.

F. P. Kapron, N. F. Borrelli, D. B. Keck, IEEE J. Quantum Electron., QE-8, 222 (1972).
[CrossRef]

F. P. Kapron, D. B. Keck, Appl. Opt. 10, 1519 (1971).
[CrossRef] [PubMed]

F. P. Kapron, D. B. Keck, R. D. Maurer, Appl. Phys. Lett. 17, 423 (1970).
[CrossRef]

Marcuse, D.

D. Marcuse, Bell Syst. Tech. J. 49, 1665 (1970).

Maurer, R. D.

F. P. Kapron, D. B. Keck, R. D. Maurer, Appl. Phys. Lett. 17, 423 (1970).
[CrossRef]

Pearson, A. D.

Snyder, A. W.

A. W. Snyder, Electron. Lett. 5, 271 (1969).
[CrossRef]

Stern, J. R.

R. B. Dyott, J. R. Stern, Electron. Lett. 7, 624 (1971).
[CrossRef]

R. B. Dyott, J. R. Stern, Electron. Lett. 7, 82 (1971).
[CrossRef]

Tynes, A. R.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

F. P. Kapron, D. B. Keck, R. D. Maurer, Appl. Phys. Lett. 17, 423 (1970).
[CrossRef]

Bell Syst. Tech. J. (1)

D. Marcuse, Bell Syst. Tech. J. 49, 1665 (1970).

Electron. Lett. (3)

R. B. Dyott, J. R. Stern, Electron. Lett. 7, 624 (1971).
[CrossRef]

R. B. Dyott, J. R. Stern, Electron. Lett. 7, 82 (1971).
[CrossRef]

A. W. Snyder, Electron. Lett. 5, 271 (1969).
[CrossRef]

IEEE J. Quantum Electron. (1)

F. P. Kapron, N. F. Borrelli, D. B. Keck, IEEE J. Quantum Electron., QE-8, 222 (1972).
[CrossRef]

J. Opt. Soc. Am. (1)

Other (2)

Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, Eds. (Dover, New York, 1965), Chap. 29.

Table of Integrals, Series and Products, translation ed. by A. Jeffrey (Academic, New York, 1965), Sec. 6.6.

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Figures (4)

Fig. 1
Fig. 1

Absorption and scattering by a waveguide section of semimicroscopic length. α, absorption coefficient; f, forward-scattering coefficient; b, backward-scattering coefficient; c, total-scattering coefficient ≥ b + f.

Fig. 2
Fig. 2

Approximate normalized interference for a rectangular pulse of duration T for the cases of BT = 1 (high scattering or long pulse), BT = 10−2 (low scattering or short pulse).

Fig. 3
Fig. 3

Two signal-to-interference criteria for pulse-packing density. For clarity, high scattering levels are assumed.

Fig. 4
Fig. 4

Graphs of σN in (a) and (σA)−1 in (b) vs b/β for two values of attenuation β(dB/km) and length L(km).

Equations (39)

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P 1 ( z + Δ z , t + Δ t ) = ( 1 β Δ z ) P 1 + b Δ z P 2 where Δ t = Δ z / υ .
β = α + s f ,
P 1 z + 1 υ P 1 t = β P 1 + b P 2 , P 2 z + 1 υ P 2 t = β P 2 + b P 1 .
p 1,2 ( z , s ) = 0 exp ( s t ) P 1,2 ( z , t ) d t L P 1,2
2 p 1 z 2 + p 1 [ b 2 ( β + s / υ ) 2 ] = P 1 ( z , t ) υ 2 t | t = 0 + P 1 ( z , 0 + ) υ ( 2 β + s / υ ) .
P 1 ( 0 , t ) = δ ( t ) .
p 1 ( L , s ) = p 1 ( 0 , s ) exp { ± [ ( β + s / υ ) 2 b 2 ] 1 2 L } .
L 1 exp [ A ( s 2 a 2 ) 1 2 ] = δ ( t A ) + a 2 A I 1 [ a ( t 2 A 2 ) 1 2 ] a ( t 2 A 2 ) 1 2 u ( t A ) ,
P 1 ( L , t ) = exp ( β L ) δ ( t L / V ) + b 2 L υ exp ( β υ t ) I 1 [ b ( υ 2 t 2 L 2 ) 1 2 ] b ( υ 2 t 2 L 2 ) 1 2 u ( t L / υ ) .
P 1 ( L , t ) = exp ( β L ) [ δ ( t L / υ ) + N ( t L / υ ) ] .
N ( t ) = b 2 L υ exp ( β υ t ) I 1 { b [ υ ( υ t + 2 L ) ] 1 2 } b [ υ t ( υ t + 2 L ) ] 1 2 u ( t ) ,
η ( s ) = exp ( L υ { ( s + β υ ) [ ( s + β υ ) 2 b 2 υ 2 ] 1 2 } 1 2 ) .
F ( t ) = F ( x ) δ ( t x ) d x ,
N F ( t ) = t F ( x ) N ( t x ) d x = 0 F ( t x ) N ( x ) d x .
N 0 N ( t ) d t = exp { [ β ( β 2 b 2 ) 1 2 ] L } 1
1 d x exp ( A x ) ( x 2 1 ) 1 2 J ν [ γ ( x 2 1 ) 1 2 ] = I ν / 2 [ 1 2 ( A 2 + γ 2 ) 1 2 A ] × K ν / 2 [ 1 2 ( A 2 + γ 2 ) 1 2 + A ]
N 1 ¯ 2 β L / ( b L ) 2 for b 2 4 β 2 , 2 β / L .
N ( t ) = A exp ( B t ) u ( t ) where A , B > 0 .
N ( 0 ) = A = 1 2 b 2 L υ .
N = A / B so B = 1 2 b 2 L υ exp { [ β ( β 2 b 2 ) 1 2 ] L } 1 .
B ¯ β υ .
N F ( t ) = N ( t ) t F ( x ) exp ( B x ) d x ,
R ( t ) = E / T 0 t T = 0 otherwise .
N R ( t ) = ( E N / T ) [ 1 exp ( B t ) ] t T = N R ( T ) exp [ B ( t T ) ] t T .
lim T 0 N R ( t ) = E N ( t ) , lim T T N R ( ) = E N
G ( t ) = ( a π 1 2 ) 1 exp ( t 2 / a 2 ) ,
N G ( t ) = 1 2 exp ( B 2 a 2 / 4 ) N ( t ) { 1 + erf [ ( t / a ) 1 2 B a ] } ,
erf z = 2 π 1 2 0 z exp ( x 2 ) d x .
σ = ( E / T ) × [ n = 1 N R ( n M 1 ) ] 1
n = 1 exp ( n x ) = ( e x 1 ) 1 ( x > 0 ) ,
M 1 = B ( log { 1 + σ N [ exp ( B T ) 1 ] } ) 1 .
σ = [ ( E / T ) + N R ( T ) ] × [ n = 0 N R ( T + n / M 2 ) ] 1 .
M 2 = B ( log { 1 + N [ 1 exp ( B T ) ] 1 ( σ 1 ) N [ 1 exp ( B T ) ] } ) 1 .
M 3 ¯ ( σ T N ) 1 ( σ B T N σ A T 1 , σ N > 1 ) ¯ 2 β L [ σ T ( b L ) 2 ] 1 ( b 2 2 β , 2 β / L ) .
L P 2 = b 1 { ( β + s / υ ) [ ( β + s / υ ) 2 b 2 ] 1 2 } p 2 ( z , s ) .
P 2 ( 0 , t ) = b υ exp ( β υ t ) ( b υ t ) 1 I 1 ( b υ t ) .
0 exp ( A x ) J ν ( γ x ) x 1 d x = [ ( A 2 + γ 2 ) 1 2 A ] ν × ( ν γ ν ) 1 ( Re ν > 0 , Re A > | Im γ | ) ,
N ˆ = ( β / b ) [ ( β / b ) 2 1 ] 1 2 ¯ b / 2 β if b 2 β 2 .
b = ( 2 / L ) log ( 1 + N ) [ N 2 + 1 ( N ˆ 2 + 2 N ˆ 2 ) 1 2 ] 1 + β = 1 2 b ( N ˆ 2 + 1 ) .

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