Abstract

A set of coupled-mode equations is derived to describe mode propagation in uniform and slightly nonuniform cylindrical optical-fiber systems. The coupling between fibers of an array made up of n identical fibers each at the vertex of a polygon and one at the center, which is not necessarily the same as its n neighbors, is determined. Examples of this array are two fibers, three fibers in a row, and a hexagonal array with a fiber in the center. Very simple expressions for the coupling coefficients are presented. Mode coupling on a lossy fiber is investigated and a simple expression for the loss of a HE11 mode is given.

© 1972 Optical Society of America

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References

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  1. D. Gloge, Proc. IEEE 58, 1513 (1970).
    [CrossRef]
  2. A. W. Snyder and P. A. V. Hall, Nature 223, (1970).
  3. W. Wijngaard, J. Opt. Soc. Am. 61, 1187 (1971).
    [CrossRef] [PubMed]
  4. A. W. Snyder, J. Opt. Soc. Am. 62, 998 (1972).
    [CrossRef]
  5. A. W. Snyder, Vision Res. 12, 1389 (1972).
    [CrossRef]
  6. E. Snitzer, J. Opt. Soc. Am. 51, 491 (1961).
    [CrossRef]
  7. A. W. Snyder, IEEE Trans. MTT-19, 720 (1971).
  8. N. S. Kapany, Fiber Optics (Academic, New York, 1961).
  9. W. H. Louisell, Coupled Mode and Parametric Electronics (Wiley, New York, 1960).
  10. A. L. Jones, J. Opt. Soc. Am. 55, 261 (1965).
    [CrossRef]
  11. D. Marcuse, Bell System Tech. J. 49, 274 (1970).
  12. D. Marcuse and R. M. Derosier, Bell System Tech. J. 48, 3217 (1969).
    [CrossRef]
  13. R. Vanclooster and P. Phariseau, Physica 47, 485 (1970)
    [CrossRef]
  14. D. Marcuse, Bell System Tech. J. 51, 229 (1972).
    [CrossRef]
  15. A. W. Snyder, IEEE Trans. MTT-18, 383 (1970).
  16. A. W. Snyder, IEEE Trans. MTT-19, 402 (1971).
  17. A. W. Snyder, Ph.D. thesis, University of London (1966), Ch. 7.
  18. B. Friedman, Principles of Applied Mathematics (Wiley, New York, 1956).
  19. A. W. Snyder, IEEE Trans. MTT-18, 608 (1970).
  20. A. W. Snyder, Electron. Letters 6, 561 (1970).
    [CrossRef]
  21. A. W. Snyder, IEEE Trans. MTT-17, 1130 (1969).
  22. A. W. Snyder, Proc. IEEE 60, 757 (1972).
    [CrossRef]
  23. Processing of Optical Data by Organisms and by Machines, edited by W. Reichardt (Academic, New York, 1969).

1972 (4)

A. W. Snyder, Vision Res. 12, 1389 (1972).
[CrossRef]

D. Marcuse, Bell System Tech. J. 51, 229 (1972).
[CrossRef]

A. W. Snyder, Proc. IEEE 60, 757 (1972).
[CrossRef]

A. W. Snyder, J. Opt. Soc. Am. 62, 998 (1972).
[CrossRef]

1971 (3)

W. Wijngaard, J. Opt. Soc. Am. 61, 1187 (1971).
[CrossRef] [PubMed]

A. W. Snyder, IEEE Trans. MTT-19, 402 (1971).

A. W. Snyder, IEEE Trans. MTT-19, 720 (1971).

1970 (7)

D. Marcuse, Bell System Tech. J. 49, 274 (1970).

A. W. Snyder, IEEE Trans. MTT-18, 383 (1970).

D. Gloge, Proc. IEEE 58, 1513 (1970).
[CrossRef]

A. W. Snyder and P. A. V. Hall, Nature 223, (1970).

A. W. Snyder, IEEE Trans. MTT-18, 608 (1970).

A. W. Snyder, Electron. Letters 6, 561 (1970).
[CrossRef]

R. Vanclooster and P. Phariseau, Physica 47, 485 (1970)
[CrossRef]

1969 (2)

A. W. Snyder, IEEE Trans. MTT-17, 1130 (1969).

D. Marcuse and R. M. Derosier, Bell System Tech. J. 48, 3217 (1969).
[CrossRef]

1965 (1)

1961 (1)

Derosier, R. M.

D. Marcuse and R. M. Derosier, Bell System Tech. J. 48, 3217 (1969).
[CrossRef]

Friedman, B.

B. Friedman, Principles of Applied Mathematics (Wiley, New York, 1956).

Gloge, D.

D. Gloge, Proc. IEEE 58, 1513 (1970).
[CrossRef]

Hall, P. A. V.

A. W. Snyder and P. A. V. Hall, Nature 223, (1970).

Jones, A. L.

Kapany, N. S.

N. S. Kapany, Fiber Optics (Academic, New York, 1961).

Louisell, W. H.

W. H. Louisell, Coupled Mode and Parametric Electronics (Wiley, New York, 1960).

Marcuse, D.

D. Marcuse, Bell System Tech. J. 51, 229 (1972).
[CrossRef]

D. Marcuse, Bell System Tech. J. 49, 274 (1970).

D. Marcuse and R. M. Derosier, Bell System Tech. J. 48, 3217 (1969).
[CrossRef]

Phariseau, P.

R. Vanclooster and P. Phariseau, Physica 47, 485 (1970)
[CrossRef]

Snitzer, E.

Snyder, A. W.

A. W. Snyder, Proc. IEEE 60, 757 (1972).
[CrossRef]

A. W. Snyder, J. Opt. Soc. Am. 62, 998 (1972).
[CrossRef]

A. W. Snyder, Vision Res. 12, 1389 (1972).
[CrossRef]

A. W. Snyder, IEEE Trans. MTT-19, 402 (1971).

A. W. Snyder, IEEE Trans. MTT-19, 720 (1971).

A. W. Snyder and P. A. V. Hall, Nature 223, (1970).

A. W. Snyder, IEEE Trans. MTT-18, 383 (1970).

A. W. Snyder, IEEE Trans. MTT-18, 608 (1970).

A. W. Snyder, Electron. Letters 6, 561 (1970).
[CrossRef]

A. W. Snyder, IEEE Trans. MTT-17, 1130 (1969).

A. W. Snyder, Ph.D. thesis, University of London (1966), Ch. 7.

Vanclooster, R.

R. Vanclooster and P. Phariseau, Physica 47, 485 (1970)
[CrossRef]

Wijngaard, W.

Bell System Tech. J. (3)

D. Marcuse, Bell System Tech. J. 51, 229 (1972).
[CrossRef]

D. Marcuse, Bell System Tech. J. 49, 274 (1970).

D. Marcuse and R. M. Derosier, Bell System Tech. J. 48, 3217 (1969).
[CrossRef]

Electron. Letters (1)

A. W. Snyder, Electron. Letters 6, 561 (1970).
[CrossRef]

IEEE Trans. (5)

A. W. Snyder, IEEE Trans. MTT-17, 1130 (1969).

A. W. Snyder, IEEE Trans. MTT-18, 608 (1970).

A. W. Snyder, IEEE Trans. MTT-18, 383 (1970).

A. W. Snyder, IEEE Trans. MTT-19, 402 (1971).

A. W. Snyder, IEEE Trans. MTT-19, 720 (1971).

J. Opt. Soc. Am. (4)

Nature (1)

A. W. Snyder and P. A. V. Hall, Nature 223, (1970).

Physica (1)

R. Vanclooster and P. Phariseau, Physica 47, 485 (1970)
[CrossRef]

Proc. IEEE (2)

A. W. Snyder, Proc. IEEE 60, 757 (1972).
[CrossRef]

D. Gloge, Proc. IEEE 58, 1513 (1970).
[CrossRef]

Vision Res. (1)

A. W. Snyder, Vision Res. 12, 1389 (1972).
[CrossRef]

Other (5)

Processing of Optical Data by Organisms and by Machines, edited by W. Reichardt (Academic, New York, 1969).

N. S. Kapany, Fiber Optics (Academic, New York, 1961).

W. H. Louisell, Coupled Mode and Parametric Electronics (Wiley, New York, 1960).

A. W. Snyder, Ph.D. thesis, University of London (1966), Ch. 7.

B. Friedman, Principles of Applied Mathematics (Wiley, New York, 1956).

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Figures (6)

F. 1
F. 1

(a) Medium characterized by a dielectric constant ( x , y , z ). (b) Uniform, lossless medium characterized by the dielectric constant (x,y), (x,y) = 1 within a cylinder of radius ρ and 2 outside. The fields of the system in (b) are assumed known.

F. 2
F. 2

Array made up of n identical fibers at vertices of a polygon (labeled 1,2,…,n) with a fiber at the center (labeled 0), which in general is different from its n neighbors. The angular spacing between fiber centers is ψ = 360°/n. Special cases of this arrangement are the n = 1, 2, 3, and 6 cases shown in the figure. d ¯ is the distance between the centers of fiber 0 and its nearest neighbors; d is the distance between each of the n fiber centers.

F. 3
F. 3

Possible excitation conditions for the case of zero power transfer and the case of A1(0) = 0 with A2(0) = 1, where A1and A2 are defined by Eqs. (23). The arrows represent the direction of the electric vector and the numbers the value of the amplitude coefficients ap(j) for mode p of fiber j.

F. 4
F. 4

Normal modes of a two-identical-fiber system in comparison with the local modes. The arrows represent the electric vectors.

F. 5
F. 5

HE11 mode-coupling coefficient for two identical circular cylinders as a function of the distance between their centers for several values of dimensionless frequency V, denned by Eq. (26a). The curves are nearly linear on the log-linear graph, indicating that the asymptotic expression for K0[W(d/ρ)] in Eq. (26a) can be used. As V decreases, the proportion of the HE11 modal power within the cylinders decreases, and coupling increases. δ is denned by Eq. (28).

F. 6
F. 6

HE11 modal power within a cylinder divided by the total modal power throughout all space. V is the dimensionless frequency defined by Eq. (27a).

Tables (1)

Tables Icon

Table I The percent maximum (%F) power transfer between the center and surrounding fibers of an array like Fig. 2 made up of identical elements, one of which is excited in the HE11 mode. n is the number of noncentral fibers, ψ° is the angle in degrees and d the distance between the centers of neighboring noncentral fibers, d ¯ is the distance between the centers of the central and surrounding fibers, %Δρ/ρ is the percent increase in radius of the central fiber required for total power transfer, and l is the length required for the transfer. For the calculations, V = 2.5, δ = 0.01, and ( d ¯ / ρ ) = 4.

Equations (71)

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× = i ω μ H ,
× H = i ω ,
· F = i ω ( * ) E · * ,
F = E × H * + * × H .
A · F d A = z A F · d A + L A F · n ̂ d L ,
E p ( x , y , z ) = e p ( x , y ) e i β p z ,
H p ( x , y , z ) = h p ( x , y ) e i β p z ,
β p = β p ,
h p = h p ,
e p = e p .
A · e p × h q * d A = ± δ p q ,
F p = e p × H * + * × h p ,
( z + i β p ) A F p · d A = i ω A ( * ) e p · * d A L A F p · n ̂ d L .
t * = q a q ( z ) e q t * ( x , y ) ,
H t * = q a q ( z ) h q t * ( x , y ) ,
d a p ( z ) d z + i β p a p ( z ) = γ p i ω 2 A ( * ) e p · * d A ,
z * = * ( x , y , z ) ( x , y ) q a q ( z ) e q z * ,
H z * = q a q ( z ) h q z * ,
d a p ( z ) d z + i β p a p ( z ) = i q a q C p q ,
C p q = γ p ω 2 A ( * ) ( e p t · e q t * + * e p z · e q z * ) d A ;
| a p ( z ) | 2 = 1 | a q ( z ) | 2 ,
| a q ( z ) | 2 = F p q sin 2 β b z
F p q = { 1 + ( β p β q 2 | C p q | ) 2 } 1
β b = 1 ( F p q ) 1 2 | C p q | .
d a p ( j ) d z + i β p ( j ) a p ( j ) = i s j a p ( s ) C p p ( j ) ( s ) ,
C p p ( j ) ( s ) = ω 2 A ( s ) ( ( s ) ) e p ( j ) · e p ( s ) * d A .
C p p ( j ) ( s ) = { 1 1 + Δ p p ( j ) ( s ) } × L s ( e p ( j ) × h p * ( s ) + e p * ( s ) × h p ( j ) ) · n ̂ d L ,
Δ p q = A s ( e p ( j ) × h p * ( s ) + e p * ( s ) × h p ( j ) ) · d A .
j = 1 n ( d a p ( j ) d z + i β p ( 1 ) a p ( j ) ) = i 2 γ n C p p ( 1 ) ( 2 ) j = 1 n a p ( j ) + n C p p ( 0 ) ( 1 ) a p ( 0 ) ,
d a p ( 0 ) d z + i β p ( 0 ) a p ( 0 ) = C p p ( 0 ) ( 1 ) j = 1 n a p ( j ) ,
A 1 ( z ) = a 0 ( z ) ,
A 2 ( z ) = 1 n j = 1 n a p ( j ) ( z )
d A 2 d z + i β 2 A 2 = i C 12 A 1 ,
d A 1 d z + i β 1 A 1 = i C 12 A 2 ,
β 2 = β p ( 1 ) 2 γ n C p p ( 1 ) ( 2 ) ,
β 1 = β p ( 0 )
C 12 = n C p p ( 0 ) ( 1 ) .
ρ C p p ( i ) ( j ) = δ U 2 V 3 K 0 [ W ( d i j / ρ ) ] K 1 2 ( W )
V > 4 5.8 ( 2 δ π ( d i j / ρ ) V 5 ) 1 2 ( 1 1 V ) 2 e ( V d i j / ρ 2 ) ,
V 2 = ( 2 π ρ n 1 λ ) 2 δ
= U 2 + W 2 .
δ = 1 ( n 2 n 1 ) 2 .
U K 0 ( W ) J 1 ( U ) = W K 1 ( W ) J 0 ( U )
U 2.405 e ( 1 δ / 2 ) / V .
F = 1 , 1 n < 3
= [ 1 + { C p p ( 1 ) ( 2 ) n C p p ( 0 ) ( 1 ) } 2 ] 1 , n 3 ,
l = π F 2 n C p p ( 0 ) ( 1 ) ,
Δ ρ ρ 2 u 2 ( V 1 1 / V ) [ ρ C p p ( 1 ) ( 2 ) δ ] ,
* = i ,
| a p ( z ) | 2 = | a p ( 0 ) | 2 e 2 | C p p | Z ,
exp { ( | C p p | + | C q q | ) z } .
| β p β q 2 C p q | δ V 2 ρ α
α = ( 2 ω / c ) n ;
| a p ( z ) | 2 = | a p ( 0 ) | 2 e α η p z ,
F = 1 , 1 n < 3
= [ 1 + { C p p ( 1 ) ( 2 ) n C p p ( 0 ) ( 1 ) } 2 ] 1 , n 3 ,
l = π F 2 n C p p ( 0 ) ( 1 ) .
V = 2 π ρ λ n 1 δ ,
δ = 1 ( n 2 n 1 ) 2 ,
P ( z ) = P ( 0 ) e α η z ,
d d ¯
% Δ ρ ρ
l ρ × 10 3
H z = z · × i ω μ = t · ( z × t ) i ω μ ,
z = z · × i ω = t · ( z × H t ) i ω .
γ n C p p ( 1 ) ( 2 ) = C p p ( 1 ) ( 2 ) + C p p ( 1 ) ( 3 ) + C p p ( 1 ) ( q ) + C p p ( 1 ) ( q + 1 ) / 2 ,
γ n C p p ( 1 ) ( 2 ) = C p p ( 1 ) ( 2 ) + C p p ( 1 ) ( 3 ) + C p p ( 1 ) ( q )
γ n C p p ( 1 ) ( 2 ) = C p p ( 1 ) ( 2 ) .
K 0 ( W r ρ ) = l = ( 1 ) l cos l ϕ 2 I l ( W r 2 ρ ) K l ( W d ρ )
z I 0 ( W z ) J 0 ( U z ) z dz = z U 2 + W 2 { W J 0 ( U z ) I 1 ( W z ) + U I 0 ( W z ) J 1 ( U z ) } ,
I 1 ( W ) K 0 ( W ) + I 0 ( W ) K 1 ( W ) = 1 W .