Abstract

An integral representation technique, which was developed for calculating the modal properties of a single homogeneous weakly guiding dielectric waveguide of arbitrary cross-sectional shape, is extended to the calculation of the modes of an ensemble of two or more such guides arrayed in arbitrary spatial configurations. Detailed numerical results are presented for two identical circular, two identical square, and two identical rectangular guides, and excellent agreement with almost all published data is found. Similar calculations are made for two dissimilar guides. The technique is also applied to a weakly coupled linear array of N identical, circular, uniformly spaced guides, where N = 3, 4, 5, and 6.

© 1981 Optical Society of America

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References

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  1. L. Eyges, P. Gianino, and P. Wintersteiner, “Modes of dielectric waveguides of arbitrary cross sectional shape,” J. Opt. Soc. Am. 69, 1226–1235 (1979).
    [Crossref]
  2. A. W. Snyder and W. R. Young, “Modes of optical waveguides,” J. Opt. Soc. Am. 68, 297–309 (1978).
    [Crossref]
  3. G. Meltz and E. Snitzer, “Thermal and dispersive characteristics of multicore fibers,” URSI Symposium on Electromagnetic Waves, Munich, Federal Republic of Germany, 1979 (to be published).
  4. W. Wijngaard, “Guided normal modes of two parallel circular dielectric rods,” J. Opt. Soc. Am. 63, 944–950, (1973).
    [Crossref]
  5. A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1267–1277 (1972).
    [Crossref]
  6. P. D. McIntyre and A. W. Snyder, “Power transfer between optical-fibers,” J. Opt. Soc. Am. 63, 1518–1527 (1973).
    [Crossref]
  7. E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
    [Crossref]

1979 (1)

1978 (1)

1973 (2)

1972 (1)

1969 (1)

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
[Crossref]

Eyges, L.

Gianino, P.

Marcatili, E. A. J.

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
[Crossref]

McIntyre, P. D.

Meltz, G.

G. Meltz and E. Snitzer, “Thermal and dispersive characteristics of multicore fibers,” URSI Symposium on Electromagnetic Waves, Munich, Federal Republic of Germany, 1979 (to be published).

Snitzer, E.

G. Meltz and E. Snitzer, “Thermal and dispersive characteristics of multicore fibers,” URSI Symposium on Electromagnetic Waves, Munich, Federal Republic of Germany, 1979 (to be published).

Snyder, A. W.

Wijngaard, W.

Wintersteiner, P.

Young, W. R.

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Figures (11)

Fig. 1
Fig. 1

Two identical guides a distance d apart.

Fig. 2
Fig. 2

Cosine modes of a composite system consisting of two identical circular guides. The three solid lines correspond to isolated guide modes with single-term field expansions [Φ = Js(γ1ρ)cos , with s = 0, 1, and 2]. The dashed lines correspond to modes that are symmetric (antisymmetric) with respect to the y axis when they fall above (below) the isolated guide curves. The separation distances d are the same for the antisymmetric modes as for the symmetric ones.

Fig. 3
Fig. 3

Normalized coupling constant from coupled-mode theory (solid lines) and the quantity ¼ V ( P S 2 - P A 2 ) (points) plotted versus d for the first symmetric and antisymmetric modes of a composite system consisting of two identical circular guides.

Fig. 4
Fig. 4

Same as. Fig. 3, except the second symmetric and antisymmetric modes are used. These correspond to dashed lines of the second family of Fig. 2.

Fig. 5
Fig. 5

Normalized beat length from coupled-mode theory (solid lines) and the quantity ( P S 2 - P A 2 ) - 1 (points) plotted versus V for the first symmetric and antisymmetric modes of a composite system consisting of two identical circular guides. Cutoff of the antisymmetric mode occurs near the left-most dot in each case.

Fig. 6
Fig. 6

Cosine modes of a composite system consisting of two identical rectangular guides with an aspect ratio of 2. The three solid lines correspond to isolated-guide modes. The dashed lines correspond to modes that are symmetric (antisymmetric) with respect to the y axis when they fall above (below) the isolated-guide curves. The separation distances are 4, 4.5, and 5 for each family.

Fig. 7
Fig. 7

Contours of constant squared amplitude Φ2 for that symmetric mode of a composite system, consisting of two square guides, that derives from the dominant mode of an isolated guide. The amplitude at the center of the pattern has magnitude unity, and on successively larger contours Φ2 changes by 0.1.

Fig. 8
Fig. 8

Same as Fig. 7, but for the antisymmetric mode.

Fig. 9
Fig. 9

Modes of a composite system consisting of two circular guides of radii 1.00 and 1.05. The solid lines are the modes of the larger and smaller guides in isolation with the mode of the larger guide lying above that of the smaller in each case. The dashed lines correspond to finite separations. d = 2.05 corresponds to touching guides.

Fig. 10
Fig. 10

As in Fig. 9, except the radii are 1.00 and 1.25.

Fig. 11
Fig. 11

Results for modes of composite systems consisting of N identical equally spaced circular guides in a row. The solid lines [corresponding to the right-hand side of Eq. (51)] represent our numerical results; the dashed lines [the right-hand side of Eq. (47)] represent the predictions of coupled-mode theory.3 These modes all derive from the dominant mode of an isolated guide. (a) V = 5, d = 2; (b) V = 5, d = 2.25; (c) V = 2.5, d = 2; (d) V = 2.5, d = 2.25.

Tables (3)

Tables Icon

Table 1 Coefficients C s ( 1 ) and C s ( 2 ) of Eq. (44) for the First Mode of the Composite System of Fig. 9 for Different Values of d and V = 4a

Tables Icon

Table 2 Coefficients C s ( 1 ) and C s ( 2 ) of Eq. (44) for the Second Mode of the Composite System of Fig. 9 for Different Values of d and V = 4a

Tables Icon

Table 3 Field Amplitudes at the Centers of Each Guide (m) in a Linear Array of Four Identical Equally Spaced Circular Guides, Normalized to a Maximum of 1.00, for Each Mode (r), Various Guide Separations, and V = 5a

Equations (54)

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ψ ( ρ , z , t ) = Φ ( ρ ) exp [ i ( k g z - ω t ) ] ,
( 2 + γ 1 2 ) Φ = 0             inside the guide ,
( 2 - γ 2 2 ) Φ = 0             outside the guide .
0 = L [ g γ 2 ( ρ , ρ ) Φ ( ρ ) n | - - Φ ( ρ ) - g γ 2 ( ρ , ρ ) n ] d L .
( 2 - γ 2 2 ) g γ 2 ( ρ , ρ ) = δ ( ρ - ρ ) .
g γ 2 ( ρ , ρ ) = - i 4 l = 0 l J l ( γ 2 ρ ) H l ( γ 2 ρ ) × ( cos l φ cos l φ + sin l φ sin l φ ) ,
0 = i = 1 N L i ( g γ 2 Φ n | - - Φ - g γ 2 n ) d L = 0 ,
Φ ( i ) ( ρ i ) = s = 0 J s ( γ 1 ( i ) ρ i ) × [ C s ( i ) cos s φ i + D s ( i ) sin s φ i ] .
δ = 1 - n 2 2 n 1 2 ,
V = b k 0 ( n 1 2 - n 2 2 ) 1 / 2 = b k 0 n 1 δ .
P 2 = k g 2 - k 2 2 k 1 2 - k 2 2 = γ 2 2 γ 1 2 + γ 2 2 .
[ - 2 + U ( ρ ) ] Φ = E Φ .
U ( ρ ) = { - U 0 ρ in A 0 ρ outside A ,
[ 2 + ( U 0 + E ) ] Φ = 0 ρ in A , ( 2 + E ) Φ = 0 ρ outside A .
U 0 = γ 1 2 + γ 2 2 = k 0 2 ( n 1 2 - n 2 2 ) .
C s ( 2 ) = ± ( - 1 ) s C s ( 1 ) ,
D s ( 2 ) = ( - 1 ) s D s ( 1 ) .
cosine { S A } { Φ ( 1 ) = s C s J s ( γ 1 ρ 1 ) cos s φ 1 Φ ( 2 ) = ± s ( - 1 ) s C s J s ( γ 1 ρ 2 ) cos s φ 2 ,
sine { S A } { Φ ( 1 ) = s D s J s ( γ 1 ρ 1 ) sin s φ 1 Φ ( 2 ) = s D s ( - 1 ) s J s ( γ 1 ρ 2 ) sin s φ 2 ,
H l ( γ 2 ρ 2 ) cos l φ 2 = 1 2 k = 0 k J k ( γ 2 ρ 1 ) cos k φ 1 [ H k - l ( γ 2 d ) + H k + l ( γ 2 d ) ( - 1 ) l ] ,
H l ( γ 2 ρ 2 ) sin l φ 2 = k = 1 J k ( γ 2 ρ 1 ) sin k φ 1 [ H k - l ( γ 2 d ) - H k + l ( γ 2 d ) ( - 1 ) l ] .
cosine { S A } s = 0 C s { l = 0 [ δ k l R l s ( 1 ) + α k l ( - 1 ) l - s R ¯ l s ( 2 ) ] } = 0 ,             k = 0 , 1 , 2 , , ,
sine { S A } s = 1 D s { l = 1 [ δ k l T l s ( 1 ) β k l ( - 1 ) l - s T ¯ l s ( 2 ) ] } = 0 ,             k = 1 , 2 , 3 , , ,
R l s ( 1 ) = 2 0 π d φ 1 [ F l s ( ρ 1 ) cos l φ 1 cos s φ 1 + E l s ( ρ 1 , φ 1 ) × ( s cos l φ 1 sin s φ 1 - l sin l φ 1 cos s φ 1 ) ] ,
R ¯ l s ( 2 ) = 2 0 π d φ 2 [ H l s ( ρ 2 ) cos l φ 2 cos s φ 2 + G l s ( ρ 2 , φ 2 ) × ( s cos l φ 2 sin s φ 2 - l sin l φ 2 cos s φ 2 ) ] ,
T l s ( 1 ) = 2 0 π d φ 1 [ F l s ( ρ 1 ) sin l φ 1 sin s φ 1 - E l s ( ρ 1 , φ 1 ) × ( s sin l φ 1 cos s φ 1 - l cos l φ 1 sin s φ 1 ) ] ,
T ¯ l s ( 2 ) = 2 0 π d φ 2 [ H l s ( ρ 2 ) sin l φ 2 sin s φ 2 - G l s ( ρ 2 , φ 2 ) × ( s sin l φ 2 cos s φ 2 - l cos l φ 2 sin s φ 2 ) ] ,
F l s ( ρ ) = l { 2 V ρ π b [ ( 1 - P 2 ) 1 / 2 K l ( γ 2 ρ ) J s - 1 ( γ 1 ρ ) + P K l - 1 ( γ 2 ρ ) J s ( γ 1 ρ ) ] + ( l - s ) K l ( γ 2 ρ ) J s ( γ 1 ρ ) } ,
H l s ( ρ ) = l { V ρ b [ ( 1 - P 2 ) 1 / 2 I l ( γ 2 ρ ) J s - 1 ( γ 1 ρ ) - P I l - 1 ( γ 2 ρ ) J s ( γ 1 ρ ) ] + ( l - s ) I l ( γ 2 ρ ) J s ( γ 1 ρ ) } ,
E l s ( ρ , φ ) = l 2 π K l ( γ 2 ρ ) J s ( γ 1 ρ ) ρ φ / ρ ,
G l s ( ρ , φ ) = l I l ( γ 2 ρ ) J s ( γ 1 ρ ) ρ φ / ρ ,
δ k l = Kronecker delta , α k l = k π [ K k - l ( γ 2 d ) + K k + l ( γ 2 d ) ] ,
β k l = 2 π [ K k - l ( γ 2 d ) - K k + l ( γ 2 d ) ] ,
γ 1 ρ = V ( 1 - P 2 ) 1 / 2 ρ b ,
γ 2 ρ = V P ρ b .
K l ( x ) = π 2 i l + 1 H l ( i x ) , I l ( x ) = i - l J l ( i x ) .
Φ S = Φ ( 1 ) + Φ ( 2 ) ,
Φ A = Φ ( 1 ) - Φ ( 2 ) .
ψ ( z ) = 1 2 { Φ S exp [ i k g ( S ) z ] + Φ A exp [ i k g ( A ) z ] } = 1 2 exp [ i k g ( S ) z ] ( Φ S + Φ A exp { i [ k g ( A ) - k g ( S ) ] } z ) .
[ k g ( S ) - k g ( A ) ] L = π ,
ψ ( L ) = Φ ( 2 ) exp [ i k g ( S ) L ] ,
γ = 1 2 δ n 1 ( λ B λ ) ,
γ = ( P S 2 - P A 2 ) - 1 { 1 - 1 2 δ [ 1 - 1 2 ( P S 2 + P A 2 ) ] } .
K = π b λ B ( δ ) - 1 / 2 .
K ¯ = 1 4 V ( P S 2 - P A 2 ) { 1 - 1 2 δ [ 1 - 1 2 ( P S 2 + P A 2 ) ] } - 1
Φ ( 1 ) = s = 0 C s ( 1 ) J s ( γ 1 ρ 1 ) cos s φ 1 Φ ( 2 ) = s = 0 C s ( 2 ) J s ( γ 1 ρ 2 ) cos s φ 2 ,
[ R k s ( 1 ) l = 0 ( - 1 ) l α k l R ¯ l s ( 2 ) ( - 1 ) k l = 0 α k l R ¯ l s ( 1 ) R k s ( 2 ) ] [ C s ( 1 ) C s ( 2 ) ] = ( 0 ) .
P 2 P 1 2 ,     P 2 2 , P N 2 , k g k g ( 1 ) , k g ( 2 ) , , k g ( N ) .
k g ( r ) - k g 1 2 k 1 δ ( P r 2 - P 2 ) × { 1 + 1 2 δ [ 1 - 1 2 ( P r 2 + P 2 ) ] } .
k g ( r ) - k g 2 Δ β = cos ( r π N + 1 ) ,             r = 1 , 2 , , N ,
2 Δ β = k g ( S ) - k g ( A ) .
Λ m ( r ) = [ 2 / ( N + 1 ) ] 1 / 2 sin ( m r π N + 1 ) .
k g ( S ) - k g ( A ) 1 2 k 1 δ ( P S 2 - P A 2 ) × { 1 + 1 2 δ [ 1 - 1 2 ( P S 2 + P A 2 ) ] } .
k g ( r ) - k g ( ) k g ( S ) - k g ( A ) = P r 2 - P 2 P S 2 - P A 2 ,             r = 1 , 2 , , N .