Abstract

Previous papers have treated power transfer between HE11, TE01, and TM01 modes propagating on identical cylindrical fibers. Here we extend the theory to include power transfer between modes of any order propagating on uniform circular fibers of different radii and dielectric constant. A simple analytical expression for the coupling coefficient is derived. The error in using the decoupled two-mode form of the coupled-mode equations is determined. Examples are given to illustrate the extension of the two-fiber results to arrays of fibers with different properties. All results are presented in a dimensionless form applicable to circularly cylindrical fibers of arbitrary physical parameters.

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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 (Lond.) 223, 526 (1970).
    [CrossRef]
  3. W. Wijngaard, J. Opt. Soc. Am. 61, 1187 (1971).
    [CrossRef] [PubMed]
  4. A. W. Snyder, Z. Vergl. Physiol. 76, 438 (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. L. Jones, J. Opt. Soc. Am. 55, 261 (1965).
    [CrossRef]
  8. R. Vanclooster and P. Phariseau, Physica (Utr.) 47, 485 (1970).
    [CrossRef]
  9. W. Wijngaard, J. Opt. Soc. Am. 63, 944 (1973).
    [CrossRef]
  10. A. W. Snyder, J. Opt. Soc. Am. 62, 1267 (1972).
    [CrossRef]
  11. A. W. Snyder, IEEE Trans. Microwave Theory Tech. 19, 402 (1971).
    [CrossRef]
  12. W. H. Louisell, Coupled Mode and Parametric Electronics (Wiley, New York, 1960), Ch. 1.
  13. A. W. Snyder and C. Pask, J. Opt. Soc. Am. 62, 998 (1972).
    [CrossRef]
  14. D. Gloge, Appl. Opt. 10, 2252 (1971).
    [CrossRef] [PubMed]
  15. C. C. Johnson, Field and Wave Electrodynamics (McGraw—Hill, New York, 1965), p. 314.
  16. A. W. Snyder, IEEE Trans. Microwave Theory Tech. 19, 720 (1971).
    [CrossRef]
  17. A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1130 (1969).
    [CrossRef]

1973 (1)

W. Wijngaard, J. Opt. Soc. Am. 63, 944 (1973).
[CrossRef]

1972 (4)

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

A. W. Snyder, Z. Vergl. Physiol. 76, 438 (1972).
[CrossRef]

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

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

1971 (4)

D. Gloge, Appl. Opt. 10, 2252 (1971).
[CrossRef] [PubMed]

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 19, 720 (1971).
[CrossRef]

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 19, 402 (1971).
[CrossRef]

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

1970 (3)

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

A. W. Snyder and P. A. V. Hall, Nature (Lond.) 223, 526 (1970).
[CrossRef]

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

1969 (1)

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1130 (1969).
[CrossRef]

1965 (1)

A. L. Jones, J. Opt. Soc. Am. 55, 261 (1965).
[CrossRef]

1961 (1)

E. Snitzer, J. Opt. Soc. Am. 51, 491 (1961)
[CrossRef]

Gloge, D.

D. Gloge, Appl. Opt. 10, 2252 (1971).
[CrossRef] [PubMed]

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

Hall, P. A. V.

A. W. Snyder and P. A. V. Hall, Nature (Lond.) 223, 526 (1970).
[CrossRef]

Johnson, C. C.

C. C. Johnson, Field and Wave Electrodynamics (McGraw—Hill, New York, 1965), p. 314.

Jones, A. L.

A. L. Jones, J. Opt. Soc. Am. 55, 261 (1965).
[CrossRef]

Louisell, W. H.

W. H. Louisell, Coupled Mode and Parametric Electronics (Wiley, New York, 1960), Ch. 1.

Pask, C.

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

Phariseau, P.

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

Snitzer, E.

E. Snitzer, J. Opt. Soc. Am. 51, 491 (1961)
[CrossRef]

Snyder, A. W.

A. W. Snyder, Z. Vergl. Physiol. 76, 438 (1972).
[CrossRef]

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

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

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

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 19, 720 (1971).
[CrossRef]

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 19, 402 (1971).
[CrossRef]

A. W. Snyder and P. A. V. Hall, Nature (Lond.) 223, 526 (1970).
[CrossRef]

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1130 (1969).
[CrossRef]

Vanclooster, R.

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

Wijngaard, W.

W. Wijngaard, J. Opt. Soc. Am. 63, 944 (1973).
[CrossRef]

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

Other (17)

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

A. W. Snyder and P. A. V. Hall, Nature (Lond.) 223, 526 (1970).
[CrossRef]

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

A. W. Snyder, Z. Vergl. Physiol. 76, 438 (1972).
[CrossRef]

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

E. Snitzer, J. Opt. Soc. Am. 51, 491 (1961)
[CrossRef]

A. L. Jones, J. Opt. Soc. Am. 55, 261 (1965).
[CrossRef]

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

W. Wijngaard, J. Opt. Soc. Am. 63, 944 (1973).
[CrossRef]

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

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 19, 402 (1971).
[CrossRef]

W. H. Louisell, Coupled Mode and Parametric Electronics (Wiley, New York, 1960), Ch. 1.

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

D. Gloge, Appl. Opt. 10, 2252 (1971).
[CrossRef] [PubMed]

C. C. Johnson, Field and Wave Electrodynamics (McGraw—Hill, New York, 1965), p. 314.

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 19, 720 (1971).
[CrossRef]

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1130 (1969).
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

Normalized eigenvalue U as a function of V = ( 2 π ρ / λ ) × n 1 ( δ ) 1 2. The propagation constant β is related to U by (ρβ)2 = (V2/δ)[1 − δ(U/V)2].

Fig. 2
Fig. 2

Maximum fiber length L ¯ as a function of V and dielectric difference δ = 1 − (n/n1)2 for modes P11 (solid line) and P21 (dashed line). The modes are degenerate along a fiber of length L and radius ρ if L / ρ L ¯.

Fig. 3
Fig. 3

Geometry of two-fiber coupling. Modes P l 1 m 1 , P l 2 m 2 propagate on fibers 1 and 2, respectively.

Fig. 4
Fig. 4

Values of V2, as a function of V1 and ρ1/ρ2, that satisfy β1 = β2 for the case of a P01 mode propagating on each fiber. Given 1δ1, 2δ2 can be found from the relation V 2 / V 1 = ρ 2 / ρ 1 ( 2 δ 2 / 1 δ 1 ) 1 2.

Fig. 5
Fig. 5

As for Fig. 4 but with a P11 mode propagating on each fiber.

Fig. 6
Fig. 6

As for Fig. 4 but with P01, P11 modes propagating on fibers 1 and 2, respectively.

Fig. 7
Fig. 7

The power transfer F (solid line) and the error T in percent (dashed line) as a function of Δβ = ρ1 (β1β2)/√δ1 for coupling between two P01 modes. D = d/ρ1 (see Fig. 3) and t indicates touching fibers. Unless stated otherwise, V(=V1) = 2. For coupling between like modes, the two ways of varying Δβ (see text) give almost identical curves. The changes in ρ1/ρ2 or δ2/δ1 corresponding to the changes in Δβ and expressed as a percentage of their values when β1 = β2, are Δ(ρ1/ρ2) = 80Δβ%, Δ(δ2/δ1) = −67Δβ%.

Fig. 8
Fig. 8

As for Fig. 7 but for coupling between P11 modes. V1 = 3 and α = 0° unless indicated otherwise. Here Δ(ρ1/ρ2) = 34Δβ%, Δ(δ2/δ1) = −50Δβ%.

Fig. 9
Fig. 9

As for Fig. 7 but for coupling between a P01 mode (fiber 1) and a P11 mode (fiber 2). For coupling between unlike modes, the two ways of varying Δβ give different curves. Here n1 = n2 (δ1 = δ2) and the ratio ρ1/ρ2 is varied. Δ(ρ1/ρ2) = 60Δβ%.

Fig. 10
Fig. 10

As for Fig. 9 but ρ1 = ρ2 and δ2 is varied. To a good approximation, the curves are independent of the value of δ1 used. Δ(δ2/δ1) = −34Δβ%.

Fig. 11
Fig. 11

The coupling constant κ as a function of D1 = d/ρ1 (see Fig. 3), V1 = (2πρ/λ)n1δ1, and R = ρ1/ρ2, for P01P01 coupling. In this case, κ is independent of the angle of orientation α. The fibers are touching when D1 = 1 + ρ2/ρ1.

Fig. 12
Fig. 12

As for Fig. 11 but for P11P11 coupling and α = 0°.

Fig. 13
Fig. 13

As for Fig. 11 but for P01P11 coupling and α = 0°. For P01P11 coupling at an angle α, κ(α) = κ(0) cosα.

Fig. 14
Fig. 14

Array consisting of a central fiber surrounded by six fibers at the vertices of a regular hexagon (n = 6).

Tables (2)

Tables Icon

Table I Effect of fiber parameters on power transfer. Δβ/β1 gives the percent difference of the propagation constants necessary to reduce the power transfer F to 0.1. Δ(ρ1/ρ2) or Δ(δ2/δ1) are the corresponding variations of the fiber parameters, l is the distance for total power transfer. T indicates touching fibers. The two examples are (a) P01P11 coupling with V1 = 2, δ1 = δ2. The equal β condition (Fig. 6) gives V2 = 3.89, ρ1/ρ2 = 0.515. We take δ1 = δ2 = 0.1, α = 0°. (b) P01P11 coupling with V1 = 4, ρ1 = ρ2, δ1 = 0.01. Then, for equal β’s, V2 = 4.7, δ2/δ1 = 1.374. Take α = 0°.

Tables Icon

Table II The number of equations to be solved for an array of n fibers surrounding a central fiber, assuming a regular array and that the appropriate symmetry groups are present (see text). If this is not the case, more equations are needed to describe coupling on the array.

Equations (53)

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d a p ( j ) d z + i β p ( j ) a p ( j ) = i s = l s j n all modes q C p q ( j ) ( s ) a q ( s ) .
C p q ( j ) ( s ) = γ p ω 2 A ( s ) ( ( s ) ) e p ( j ) · e q ( s ) d A ,
E p ( j ) ( x , y , z ) = e p ( j ) ( x , y ) e i β p z .
E t ( j ) * ( x , y , z ) = p a p ( j ) ( z ) e p ( j ) * ( x , y ) ,
| C p q ( j ) ( s ) | | β p ( j ) | , | β q ( s ) |
( ρ β l m ) 2 = ( ρ k ) 2 U l m 2
= V 2 δ ( 1 δ U l m 2 V 2 ) ,
V = ρ ω n 1 ( μ δ ) 1 2 = 2 π ρ λ ( n 1 2 n 2 )
U l m J l + 1 ( U l m ) J l ( U l m ) = W l m K l + 1 ( W l m ) K l ( W l m )
V 2 = U l m 2 + W l m 2 .
L / ρ L ¯ = π ρ β diff
ρ β diff δ 3 2 ( U V ) 2 δ 3 2 ,
κ p q ( j ) ( s ) = ( ρ j ρ s ) 1 2 C p q ( j ) ( s ) ( δ j δ s ) 1 4 .
κ p q ( j ) ( s ) = ( 1 ) l 2 γ p ( γ 1 γ 2 2 ) U 1 U 2 V 1 ( V 2 V 1 ) 1 2 [ K l 1 + l 2 ( W 1 D 1 ) cos ( l 1 + l 2 ) α + K l 1 l 2 ( W 1 D 1 ) cos ( l 1 l 2 ) α ] [ K l 1 1 ( W 1 ) K l 1 + 1 ( W 1 ) K l 2 1 ( W 2 ) K l 2 + 1 ( W 2 ) ] 1 2 × [ W ¯ 2 I l 2 + 1 ( W ¯ 2 ) K l 2 ( W 2 ) + W 2 I l 2 ( W ¯ 2 ) K l 2 + 1 ( W 2 ) ] [ U 2 2 + W ¯ 2 2 ] ,
d a 1 d z + i β 1 a 1 = i C 12 a 2 ,
d a 2 d z + i β 2 a 2 = i C 21 a 1 ,
T = | F 12 F 21 F 12 | = | C 12 C 21 C 12 |
| C i j | β = δ ρ β | κ i j | δ | κ i j | V
| a 3 | 2 = | C 13 β 1 β 3 | 2 ,
Δ β β 1 = δ 1 Δ β ρ 1 β 1 δ 1 Δ β V 1 = 10 Δ β V 1 % .
W 1 ρ 1 = W 2 ρ 2
κ = κ 12 = κ 21 = ( 1 ) l 2 ( γ 1 γ 2 2 ) U 1 U 2 ( V 1 V 2 ) 3 2 [ K l 1 + l 2 ( W 1 D 1 ) cos ( l 1 + l 2 ) α + K l 1 l 2 ( W 1 D 1 ) cos ( l 1 l 2 ) α ] [ K l 1 1 ( W 1 ) K l 1 + 1 ( W 1 ) K l 2 + 1 ( W 2 ) ] 1 2 ,
κ ( α ) = κ ( 0 ) cos α .
| a 1 ( z ) | 2 = 1 F 12 sin 2 β b z ,
| a 2 ( z ) | 2 = F 21 sin 2 β b z ,
F 12 = [ γ 12 + ( Δ β ) 2 4 | C 12 C 21 | ] 1 ,
F 21 = | C 21 C 12 | F 12 ,
β b = ( | C 12 C 21 | F 12 ) 1 2 ;
l = π 2 β b .
A ± = a 1 + ( γ ± η ) a 2 ,
γ = β 1 β 2 2 C 12 , η = ( C 21 F C 12 ) .
A 1 ( z ) = a 0 ( z ) ,
A 2 ( z ) = 1 2 j = 1 4 a j ( z ) ,
A 3 ( z ) = 1 2 ( a 5 ( z ) + a 6 ( z ) ) ,
d A 1 d z + i β 1 A 1 = i C 12 A 2 + i C 13 A 3 ,
d A 2 d z + i β 2 A 2 = i C 21 A 1 + i C 23 A 3 ,
d A 3 d z + i β 3 A 3 = i C 31 A 1 + i C 32 A 2 ,
β 1 = β p ( 0 ) ,
β 2 = β p ( 1 ) C p p ( 1 ) ( 2 ) ,
β 3 = β p ( 1 ) ,
C 12 = 2 C p p ( 0 ) ( 1 ) ,
C 13 = 2 C p p ( 0 ) ( 5 ) ,
C 23 = 2 C p p ( 1 ) ( 5 ) ;
A 1 ( z ) = a 0 ( z ) ,
A 2 ( z ) = 1 6 j = 1 6 a j ( z ) ,
β 1 = β p ( 0 ) ,
β 2 = β q ( 1 ) 2 C p q ( 1 ) ( 2 ) ,
C 12 = 6 C p q ( 0 ) ( 1 ) C 21 .
F = { 1 + ( Δ β 2 C 12 ) 2 } 1 ,
Δ β = β 1 β 2
( ρ i β i ) 2 = V i 2 δ i ( 1 δ i ( U i V i ) 2 ) .
l = π F 2 | C 12 |
κ = ( ρ 1 ρ 2 ) 1 2 ( δ 1 δ 2 ) 1 4 C 12 = ( γ 1 γ 2 2 ) U 1 U 2 ( V 1 V 2 ) 3 2 ( 1 ) l 2 [ K l 1 + l 2 ( W 1 D 1 ) cos ( l 1 + l 2 ) α + K l 1 l 2 ( W 1 D 1 ) cos ( l 1 l 2 ) α ] [ K l 1 1 ( W 1 ) K l 1 + 1 ( W 1 ) K l 2 1 ( W 2 ) K l 2 + 1 ( W 2 ) ] 1 2 .