Abstract

Mode coupling theory is applied to the study of multimode optical fibers with graded-index cores. For coupling caused by random bends in the waveguide axis, the results predict the dependence of the induced losses on the fiber’s characteristics. The impulse response is determined for fibers with random bends having several different power spectra. The results are used to predict the transmitted power, the delay time, and the rms pulse width in fibers with graded-index cores.

© 1975 Optical Society of America

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  1. F. P. Kapron, D. B. Keck, R. D. Maurer, Appl. Phys. Lett. 17, 423 (1970).
    [CrossRef]
  2. D. B. Keck, R. D. Maurer, P. C. Schultz, Appl. Phys. Lett. 22, 307 (1973).
    [CrossRef]
  3. R. D. Maurer, Proc. IEEE 61, 452 (1973).
    [CrossRef]
  4. S. E. Miller, E. A. J. Marcatili, T. Li, Proc. IEEE 61, 1703 (1973).
    [CrossRef]
  5. S. E. Miller, Bell Syst. Tech. J. 44, 2017 (1965).
  6. S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech./ MIT-16, 814 (1968).
    [CrossRef]
  7. R. Bouille, J. R. Andrews, Electron. Lett. 8, 309 (1972).
    [CrossRef]
  8. D. Gloge, E. L. Chinnock, K. Koizumi, Electron. Lett. 8, 526 (1972).
    [CrossRef]
  9. K. Koizumi, Y. Ikeda, I. Kitano, M. Furukawa, T. Sumimoti, Appl. Opt. 13, 255 (1974).
    [CrossRef] [PubMed]
  10. S. D. Personick, Bell Syst. Tech. J. 50, 843 (1971).
  11. E. L. Chinnock, L. G. Cohen, W. S. Holden, R. D. Standley, D. B. Keck, Proc. IEEE (Lett.) 61, 1499 (1973).
    [CrossRef]
  12. D. Gloge, Bell Syst. Tech. J. 51, 1767 (1972).
  13. D. Gloge, Bell Syst. Tech. J. 52, 801 (1973).
  14. D. Marcuse, Bell Syst. Tech. J. 52, 1423 (1973).
  15. D. Marcuse, Bell Syst. Tech. J. 51, 229 (1972).
  16. D. Gloge, E. L. Chinnock, R. D. Standley, W. S. Holden, Electron. Lett. 8, 527 (1972).
    [CrossRef]
  17. D. B. Keck, Proc. IEEE 62, 649 (1974).
    [CrossRef]
  18. Bell Syst. Tech. J. 52, 1563 (1973).
  19. A. W. Snyder, J. Opt. Soc. Am. 62, 1267 (1972).
    [CrossRef]
  20. D. Marcuse, Bell Syst. Tech. J. 52, 817 (1973).
  21. D. B. Keck, Appl. Opt. 13, 1882 (1974).
    [CrossRef] [PubMed]
  22. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), p. 237.
  23. R. Olshansky, Appl. Opt. 14, 20 (1975).
    [CrossRef] [PubMed]

1975 (1)

1974 (3)

1973 (8)

E. L. Chinnock, L. G. Cohen, W. S. Holden, R. D. Standley, D. B. Keck, Proc. IEEE (Lett.) 61, 1499 (1973).
[CrossRef]

D. Gloge, Bell Syst. Tech. J. 52, 801 (1973).

D. Marcuse, Bell Syst. Tech. J. 52, 1423 (1973).

D. B. Keck, R. D. Maurer, P. C. Schultz, Appl. Phys. Lett. 22, 307 (1973).
[CrossRef]

R. D. Maurer, Proc. IEEE 61, 452 (1973).
[CrossRef]

S. E. Miller, E. A. J. Marcatili, T. Li, Proc. IEEE 61, 1703 (1973).
[CrossRef]

D. Marcuse, Bell Syst. Tech. J. 52, 817 (1973).

Bell Syst. Tech. J. 52, 1563 (1973).

1972 (6)

A. W. Snyder, J. Opt. Soc. Am. 62, 1267 (1972).
[CrossRef]

R. Bouille, J. R. Andrews, Electron. Lett. 8, 309 (1972).
[CrossRef]

D. Gloge, E. L. Chinnock, K. Koizumi, Electron. Lett. 8, 526 (1972).
[CrossRef]

D. Marcuse, Bell Syst. Tech. J. 51, 229 (1972).

D. Gloge, E. L. Chinnock, R. D. Standley, W. S. Holden, Electron. Lett. 8, 527 (1972).
[CrossRef]

D. Gloge, Bell Syst. Tech. J. 51, 1767 (1972).

1971 (1)

S. D. Personick, Bell Syst. Tech. J. 50, 843 (1971).

1970 (1)

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

1968 (1)

S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech./ MIT-16, 814 (1968).
[CrossRef]

1965 (1)

S. E. Miller, Bell Syst. Tech. J. 44, 2017 (1965).

Andrews, J. R.

R. Bouille, J. R. Andrews, Electron. Lett. 8, 309 (1972).
[CrossRef]

Bouille, R.

R. Bouille, J. R. Andrews, Electron. Lett. 8, 309 (1972).
[CrossRef]

Chinnock, E. L.

E. L. Chinnock, L. G. Cohen, W. S. Holden, R. D. Standley, D. B. Keck, Proc. IEEE (Lett.) 61, 1499 (1973).
[CrossRef]

D. Gloge, E. L. Chinnock, R. D. Standley, W. S. Holden, Electron. Lett. 8, 527 (1972).
[CrossRef]

D. Gloge, E. L. Chinnock, K. Koizumi, Electron. Lett. 8, 526 (1972).
[CrossRef]

Cohen, L. G.

E. L. Chinnock, L. G. Cohen, W. S. Holden, R. D. Standley, D. B. Keck, Proc. IEEE (Lett.) 61, 1499 (1973).
[CrossRef]

Furukawa, M.

Gloge, D.

D. Gloge, Bell Syst. Tech. J. 52, 801 (1973).

D. Gloge, Bell Syst. Tech. J. 51, 1767 (1972).

D. Gloge, E. L. Chinnock, K. Koizumi, Electron. Lett. 8, 526 (1972).
[CrossRef]

D. Gloge, E. L. Chinnock, R. D. Standley, W. S. Holden, Electron. Lett. 8, 527 (1972).
[CrossRef]

Holden, W. S.

E. L. Chinnock, L. G. Cohen, W. S. Holden, R. D. Standley, D. B. Keck, Proc. IEEE (Lett.) 61, 1499 (1973).
[CrossRef]

D. Gloge, E. L. Chinnock, R. D. Standley, W. S. Holden, Electron. Lett. 8, 527 (1972).
[CrossRef]

Ikeda, Y.

Kapron, F. P.

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

Kawakami, S.

S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech./ MIT-16, 814 (1968).
[CrossRef]

Keck, D. B.

D. B. Keck, Proc. IEEE 62, 649 (1974).
[CrossRef]

D. B. Keck, Appl. Opt. 13, 1882 (1974).
[CrossRef] [PubMed]

D. B. Keck, R. D. Maurer, P. C. Schultz, Appl. Phys. Lett. 22, 307 (1973).
[CrossRef]

E. L. Chinnock, L. G. Cohen, W. S. Holden, R. D. Standley, D. B. Keck, Proc. IEEE (Lett.) 61, 1499 (1973).
[CrossRef]

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

Kitano, I.

Koizumi, K.

Li, T.

S. E. Miller, E. A. J. Marcatili, T. Li, Proc. IEEE 61, 1703 (1973).
[CrossRef]

Marcatili, E. A. J.

S. E. Miller, E. A. J. Marcatili, T. Li, Proc. IEEE 61, 1703 (1973).
[CrossRef]

Marcuse, D.

D. Marcuse, Bell Syst. Tech. J. 52, 1423 (1973).

D. Marcuse, Bell Syst. Tech. J. 52, 817 (1973).

D. Marcuse, Bell Syst. Tech. J. 51, 229 (1972).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), p. 237.

Maurer, R. D.

R. D. Maurer, Proc. IEEE 61, 452 (1973).
[CrossRef]

D. B. Keck, R. D. Maurer, P. C. Schultz, Appl. Phys. Lett. 22, 307 (1973).
[CrossRef]

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

Miller, S. E.

S. E. Miller, E. A. J. Marcatili, T. Li, Proc. IEEE 61, 1703 (1973).
[CrossRef]

S. E. Miller, Bell Syst. Tech. J. 44, 2017 (1965).

Nishizawa, J.

S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech./ MIT-16, 814 (1968).
[CrossRef]

Olshansky, R.

Personick, S. D.

S. D. Personick, Bell Syst. Tech. J. 50, 843 (1971).

Schultz, P. C.

D. B. Keck, R. D. Maurer, P. C. Schultz, Appl. Phys. Lett. 22, 307 (1973).
[CrossRef]

Snyder, A. W.

Standley, R. D.

E. L. Chinnock, L. G. Cohen, W. S. Holden, R. D. Standley, D. B. Keck, Proc. IEEE (Lett.) 61, 1499 (1973).
[CrossRef]

D. Gloge, E. L. Chinnock, R. D. Standley, W. S. Holden, Electron. Lett. 8, 527 (1972).
[CrossRef]

Sumimoti, T.

Appl. Opt. (3)

Appl. Phys. Lett. (2)

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

D. B. Keck, R. D. Maurer, P. C. Schultz, Appl. Phys. Lett. 22, 307 (1973).
[CrossRef]

Bell Syst. Tech. J. (8)

S. E. Miller, Bell Syst. Tech. J. 44, 2017 (1965).

S. D. Personick, Bell Syst. Tech. J. 50, 843 (1971).

Bell Syst. Tech. J. 52, 1563 (1973).

D. Gloge, Bell Syst. Tech. J. 51, 1767 (1972).

D. Gloge, Bell Syst. Tech. J. 52, 801 (1973).

D. Marcuse, Bell Syst. Tech. J. 52, 1423 (1973).

D. Marcuse, Bell Syst. Tech. J. 51, 229 (1972).

D. Marcuse, Bell Syst. Tech. J. 52, 817 (1973).

Electron. Lett. (3)

D. Gloge, E. L. Chinnock, R. D. Standley, W. S. Holden, Electron. Lett. 8, 527 (1972).
[CrossRef]

R. Bouille, J. R. Andrews, Electron. Lett. 8, 309 (1972).
[CrossRef]

D. Gloge, E. L. Chinnock, K. Koizumi, Electron. Lett. 8, 526 (1972).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech./ MIT-16, 814 (1968).
[CrossRef]

J. Opt. Soc. Am. (1)

Proc. IEEE (3)

D. B. Keck, Proc. IEEE 62, 649 (1974).
[CrossRef]

R. D. Maurer, Proc. IEEE 61, 452 (1973).
[CrossRef]

S. E. Miller, E. A. J. Marcatili, T. Li, Proc. IEEE 61, 1703 (1973).
[CrossRef]

Proc. IEEE (Lett.) (1)

E. L. Chinnock, L. G. Cohen, W. S. Holden, R. D. Standley, D. B. Keck, Proc. IEEE (Lett.) 61, 1499 (1973).
[CrossRef]

Other (1)

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), p. 237.

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Figures (6)

Fig. 1
Fig. 1

The coefficient C(α, p) of the steady state attenuation is plotted as a function of index gradient α for power spectra with p = 0, 1, 2.

Fig. 2
Fig. 2

The transmitted power P(z) is plotted as a function of length z for perturbations producing 2-dB/km, 10-dB/km, and 20-dB/km steady state losses.

Fig. 3
Fig. 3

The rms pulse width is shown as a function of index gradient α in the case of no mode coupling.

Fig. 4
Fig. 4

The rms pulse width is shown as a function of the index gradient α in the case of mode coupling caused by random bends with power spectra characterized by p = 0, 1, 2.

Fig. 5
Fig. 5

The dispersion of the rms pulse width as a function of z is shown for the case α = 4 and p = 1.

Fig. 6
Fig. 6

The parameter R2β, which characterizes the tradeoff between the reduction R of pulse width and the increased loss β, is shown as a function of α for the cases p = 0, 1, 2.

Equations (107)

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σ u ( τ max τ min ) L .
σ c ( L / d ) 1 / 2 .
P n z + τ n P n t = γ n P n + m = 1 N d n m ( P m P n ) n = 1 N ,
P n = P n ( z , t )
d n m = d n + 1 δ m , n + 1 + d m + 1 δ n , m + 1 .
P n z + τ n P n t = γ n P n + d n + 1 ( P n + 1 P n ) d n ( P n P n 1 ) .
P n + 1 P n [ ( P n + 1 ) / n ] Δ n .
P n z + τ n P n t = γ n P n + d n + 1 P n + 1 n d n P n n .
P ( n ) z + τ ( n ) P ( n ) t = γ ( n ) P ( n ) + n [ d ( n ) P ( n ) n ] ,
n 2 ( r ) = n 1 2 [ 1 2 Δ ( r / a ) α ] 0 r a , n 2 ( r ) = n 1 2 ( 1 2 Δ ) = n 2 2 a r ,
β n = n 1 k { 1 2 Δ [ n / N ( α ) ] α / α + 2 } 1 / 2 ,
N ( α ) = [ α / ( α + 2 ) ] a 2 k 2 n 1 2 Δ ,
m = 2 μ + | ν | .
ν = m , m 2 , ( m 2 ) , m
m = 1 M 2 ( m + 1 ) M 2 = N .
β m = n 1 k { 1 2 Δ [ m / M ( α ) ] 2 α / α + 2 } 1 / 2 ,
M 2 ( α ) = N ( α ) .
Δ m = ± 1 Δ ν = ± 1 , 1
2 ( m + 1 ) P ¯ m = Σ P n ,
2 ( m + 1 ) [ P ¯ m z + τ ( m ) P ¯ m t ] = Σ [ P n z + τ n P n t ] .
P ¯ ( m ) z + τ ( m ) P ¯ ( m ) t = γ ( m ) P ¯ ( m ) + 1 m m m d ( m ) P ¯ ( m ) m ,
τ ( m ) = 1 c β k n 1 c [ 1 + α 2 α + 2 Δ ( m M ) 2 α / α + 2 + 1 2 3 α 2 α + 2 Δ 2 ( m M ) 4 α / α + 2 ] + 0 ( Δ 3 ) .
γ 0 ( m ) = γ 0 m M c γ 0 ( m ) = M c < m M ( α ) ,
d ( m ) = d 0 ( m / M c ) 2 q
P ( m ) = 0 for m M c .
P = 0 M c 2 m P ¯ ( m ) d m ,
P / z = γ 0 P + { 2 m d ( m ) [ P ¯ ( m ) / m ] } 0 M c
m d ( m ) P ¯ ( m ) m | m = 0 = 0.
[ P ¯ ( m , z , t ) ] / t = 0.
P ¯ ( x , z ) z = γ 0 P ¯ ( x , z ) + d 0 M c 2 1 x x x 1 2 q P ¯ ( x , z ) x ,
x = m / M c
P ¯ j ( x , z ) = P ¯ j ( x ) exp ( γ j z ) j = 1,2 ,
P ¯ j ( x ) = N j x q J ν ( λ j x i + q ) ,
γ j = ( d 0 / M c 2 ) ( 1 + q ) 2 λ j 2 .
υ = ± q / ( 1 + q ) .
J υ ( λ j ) = 0.
λ j = z j ( υ ) ,
0 1 2 x P ¯ j ( x ) P ¯ k ( x ) d x = δ j k .
N j = ( 1 + q ) 1 / 2 | J υ + 1 [ Z j ( υ ) ] | 1 .
P ¯ ( x , z ) = j = 1 I j P ¯ j ( x ) exp ( γ j z ) ,
I j = 0 1 2 x P ¯ j ( x ) I ( x ) d x .
P ( x , z ) = 2 x P ¯ ( x , z ) .
P ¯ ( x , z ) I 1 P ¯ 1 ( x ) exp ( γ 1 z ) .
R z + τ ( x ) R t = d 0 M c 2 1 x x x 1 2 q R x .
σ i ( x , z ) = 0 d t t i R ( x , z , t ) .
r ( x , z , s ) = 0 d t exp ( s t ) R ( x , z , t ) .
r z + s τ ( x ) = d 0 M c 2 1 x x x 1 2 q r x .
σ i ( x , z ) = ( 1 ) i d i r ( x , z , s ) d s i | s = 0 .
r ( x , z , s ) = j = 1 d j ( z , s ) P ¯ ( x ) exp [ ( γ j + s n 1 / c ) z ] .
d j ( z , s ) z + s k = 1 M j k d k ( z , s ) exp [ ( γ j γ k ) z ] = 0 ,
M j k = 0 1 2 x [ τ ( x ) n 1 / c ] P ¯ j ( x ) P ¯ k ( x ) d x .
d j ( z , s ) = k = 0 d j k ( z ) s i .
R ( x , 0 , t ) = I ( x ) δ ( t ) ,
r ( x , 0 , s ) = I ( x ) .
d j 0 ( 0 ) = I j d j 0 ( 0 ) = 0 i = 1,2 , .
d j 1 ( z ) = [ j k I k M k j E k j ( z ) + I j M j j z ] ,
d j 2 ( z ) = ( k l l j I l M l k M k j E j l ( z ) γ k γ l k j l k I l M l k M k j E j k ( z ) γ k γ l + z l j I j | M l j | 2 γ 1 γ j z M j j j l I l M l j γ j γ l + 1 2 z 2 I j M j j 2 + j l I l M l l M l j { z exp [ ( γ j γ l ) z ] E j l ( z ) γ j γ t } )
E j k ( z ) = { exp [ ( γ j γ k ) z ] 1 } / ( γ j γ k ) .
σ i ( z ) = 0 1 2 x σ i ( x , z ) d x
P j = 0 1 2 x P ¯ j ( x ) d x ,
P j = [ 4 / ( 1 + q ) ] 1 / 2 { [ ( 1 ) j + 1 ] / [ Z j ( υ ) ] } .
r ( z , s ) = k = 0 j = 1 s k d j k ( z ) P j exp [ ( γ j + s n 1 / c ) z ]
P ( z ) = j = 1 I j P j exp ( γ j z ) .
P ( z ) I 1 P 1 exp ( γ 1 z ) ,
τ ( z ) = σ 1 ( z ) / P ( z ) .
τ ( z ) = n 1 z / c j = 1 d j 1 ( z ) P j exp ( γ j z ) / P ( z ) .
τ ( z ) [ ( n 1 / c ) + M 11 ] z + c τ ,
c τ = j = 2 I 1 M 1 j P j + I j M j P l γ 1 γ j I 1 P 1
σ ( z ) = [ σ 2 ( z ) / P ( z ) τ 2 ( z ) ] 1 / 2 .
σ ( z ) = { 2 j = 1 d j ( z ) P j exp ( γ j z ) / P ( z ) [ j = 1 d j ( z ) P j exp ( γ j z ) / P ( z ) ] 2 } 1 / 2
σ ( z ) ( 2 z j = 2 | M j 1 | 2 γ j γ 1 ) 1 / 2 + 0 ( z 1 / 2 ) .
M j k n 1 Δ c α 2 α + 2 0 1 2 x 2 α / α + 2 P ¯ j ( x ) P ¯ k ( x ) x d x + 0 ( Δ 2 ) .
d n n = | 1 L + K n n ( z ) exp [ i ( β n β n ) z ] d z | 2 ,
K n n ( z ) = ω 0 4 ( β n β n ) 0 r d r 0 2 π d ϕ n 2 z E n * E n .
β n 2 k ( 0 / μ 0 ) 1 / 2 0 r d r 0 2 π d ϕ E n * E n = δ n n .
r = { [ x f ( z ) ] 2 + [ y g ( z ) ] 2 } 1 / 2 ,
n 2 z 1 2 n 2 r [ exp ( i ϕ ) d f + d z + exp ( i ϕ ) d f d z ] ,
tan ϕ = x / y ,
f ± = f i g .
ν = ν ± 1.
d ( 2 , m ) = 1 2 ( n k / a ) 2 Δ 2 [ m / M ( 2 ) ] F ( Δ β )
d ( , m ) = 2 ( n k / a ) 2 Δ 2 [ m / M ( ) ] 4 F ( Δ β )
F ± ( Δ β ) = | 1 L 0 L d z f ± ( z ) exp ( i Δ β z ) | 2 ,
F ( Δ β ) = F + ( Δ β ) = F ( Δ β ) .
c ( z ) = ( d 2 f ) / d z 2
F ( Δ β ) = [ 1 / ( Δ β ) 4 ] C ( Δ β ) ,
Δ β = [ 2 Δ / a ] ( α / α + 2 ) 1 / 2 [ m / M ( α ) ] ( α 2 ) / ( α + 2 ) .
d ( 2 , m ) = 1 8 ( n k a ) 2 [ m / M ( 2 ) ] C ( Δ β )
d ( , m ) = 1 8 ( n k a ) 2 C ( Δ β ) .
d ( α , m ) = 1 8 ( n k a ) 2 [ m / M ( α ) ] 4 / ( α + 2 ) C ( Δ β ) .
C ( Δ β ) = d ( Δ β D ) 2 p
d ( α , m ) = 1 8 ( n 1 k a ) 2 [ a 2 / Δ D 2 ] p [ ( α + 2 ) / 4 α ] p [ m / M ( α ) 2 q ] ,
q = [ p ( α 2 ) 2 ] / ( α + 2 ) .
γ 1 ( α ) = [ C ( α , p ) / Δ ] ( a 2 / Δ ) p ,
C ( α , p ) = d 2 1 D 2 p ( α + 2 4 α ) 1 + p ( 1 + q ) 2 Z 1 2 ( υ ) [ M ( α ) M c ( g α ) ] 2 ( 1 + q ) .
I ( x ) = 1.
I j = P j .
P ( z ) P 1 exp ( γ 1 z )
σ ( z ) = σ u ( α ) n 1 Δ z / c ,
α u ( α ) = α [ ( α + 2 ) ( 3 α + 2 ) ] 1 / 2 [ ( α 2 2 α + 2 ) 2 + Δ ( α 2 ) ( 3 α 2 ) ( 4 α + 2 ) ( α + 1 ) + Δ 2 ( 3 α 2 ) 2 ( 5 α + 2 ) ( 3 α + 2 ) ] 1 / 2 ,
σ ( z ) = σ c ( α , p ) ( z / γ 1 ) 1 / 2 ,
σ c ( α , p ) = ( 2 Z 1 2 l = 2 | M 1 l | 2 z l 2 z 1 2 ) 1 / 2 .
L c = ( 1 / γ 1 ) [ σ c ( α , p ) / σ u ( α ) ] 2 .
R [ σ c ( α , p ) / σ u ( α ) ] [ 1 / ( z γ 1 ) 1 / 2 ]
β 4.3 γ 1 z .
R 2 β 4.3 [ σ c ( α , p ) / σ u ( α ) ] 2 .
R 2 β = 4.3 γ 1 L c ,

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