Abstract

Some modes of two parallel, circular, lossless, dielectric rods are discussed for identical and nonidentical rods. The electromagnetic boundary-value problem is solved by use of an expansion of the mode field in circular harmonics. Examples of the distribution of flux density are given. The beat wavelength for the beating of energy between the rods computed for identical rods is compared with approximate results of coupled-mode theory.

© 1973 Optical Society of America

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References

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  1. M. F. Bracey, A. L. Cullen, E. F. F. Gillespie, and J. A. Staniforth, IRE Trans. Antennas Propag. 7, S219 (1959).
    [Crossref]
  2. E. Snitzer, in Advances in Quantum Electronics, edited by J. R. Singer (Columbia U. P., New York, 1961), p. 348.
  3. A. L. Jones, J. Opt. Soc. Am. 55, 261 (1965).
    [Crossref]
  4. N. S. Kapany, Fiber Optics (Academic, New York, London, 1967).
  5. J. J. Burke, J. Opt. Soc. Am. 57, 1056 (1967).
    [Crossref]
  6. R. Vanclooster and P. Phariseau, Physica (Utr.) 47, 485 (1970).
    [Crossref]
  7. R. Vanclooster and P. Phariseau, Physica (Utr.) 47, 501 (1970).
    [Crossref]
  8. A. W. Snyder, J. Opt. Soc. Am. 62, 1267 (1972).
    [Crossref]
  9. J. E. Goell, Bell Syst. Tech. J. 48, 2133 (1969).
    [Crossref]
  10. D. Marcuse, Bell Syst. Tech. J. 50, 1791 (1971).
    [Crossref]

1972 (1)

1971 (1)

D. Marcuse, Bell Syst. Tech. J. 50, 1791 (1971).
[Crossref]

1970 (2)

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

R. Vanclooster and P. Phariseau, Physica (Utr.) 47, 501 (1970).
[Crossref]

1969 (1)

J. E. Goell, Bell Syst. Tech. J. 48, 2133 (1969).
[Crossref]

1967 (1)

1965 (1)

1959 (1)

M. F. Bracey, A. L. Cullen, E. F. F. Gillespie, and J. A. Staniforth, IRE Trans. Antennas Propag. 7, S219 (1959).
[Crossref]

Bracey, M. F.

M. F. Bracey, A. L. Cullen, E. F. F. Gillespie, and J. A. Staniforth, IRE Trans. Antennas Propag. 7, S219 (1959).
[Crossref]

Burke, J. J.

Cullen, A. L.

M. F. Bracey, A. L. Cullen, E. F. F. Gillespie, and J. A. Staniforth, IRE Trans. Antennas Propag. 7, S219 (1959).
[Crossref]

Gillespie, E. F. F.

M. F. Bracey, A. L. Cullen, E. F. F. Gillespie, and J. A. Staniforth, IRE Trans. Antennas Propag. 7, S219 (1959).
[Crossref]

Goell, J. E.

J. E. Goell, Bell Syst. Tech. J. 48, 2133 (1969).
[Crossref]

Jones, A. L.

Kapany, N. S.

N. S. Kapany, Fiber Optics (Academic, New York, London, 1967).

Marcuse, D.

D. Marcuse, Bell Syst. Tech. J. 50, 1791 (1971).
[Crossref]

Phariseau, P.

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

R. Vanclooster and P. Phariseau, Physica (Utr.) 47, 501 (1970).
[Crossref]

Snitzer, E.

E. Snitzer, in Advances in Quantum Electronics, edited by J. R. Singer (Columbia U. P., New York, 1961), p. 348.

Snyder, A. W.

Staniforth, J. A.

M. F. Bracey, A. L. Cullen, E. F. F. Gillespie, and J. A. Staniforth, IRE Trans. Antennas Propag. 7, S219 (1959).
[Crossref]

Vanclooster, R.

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

R. Vanclooster and P. Phariseau, Physica (Utr.) 47, 501 (1970).
[Crossref]

Bell Syst. Tech. J. (2)

J. E. Goell, Bell Syst. Tech. J. 48, 2133 (1969).
[Crossref]

D. Marcuse, Bell Syst. Tech. J. 50, 1791 (1971).
[Crossref]

IRE Trans. Antennas Propag. (1)

M. F. Bracey, A. L. Cullen, E. F. F. Gillespie, and J. A. Staniforth, IRE Trans. Antennas Propag. 7, S219 (1959).
[Crossref]

J. Opt. Soc. Am. (3)

Physica (Utr.) (2)

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

R. Vanclooster and P. Phariseau, Physica (Utr.) 47, 501 (1970).
[Crossref]

Other (2)

N. S. Kapany, Fiber Optics (Academic, New York, London, 1967).

E. Snitzer, in Advances in Quantum Electronics, edited by J. R. Singer (Columbia U. P., New York, 1961), p. 348.

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Figures (8)

Fig. 1
Fig. 1

Geometry of the two-fiber problem.

Fig. 2
Fig. 2

Schematic representation of the transverse E field of the lowest-order Ez-sine modes. I0 and Iπ are the mode designations.

Fig. 3
Fig. 3

Flux-density patterns for the Ez-sine modes of two identical touching rods with υ1 = 3.5, δ1 = δ2 = 0.1. For the calculations M = 7.

Fig. 4
Fig. 4

Schematic representation of the transverse E field of some Ez-sine modes of two identical touching rods for δ ≪ 1.

Fig. 5
Fig. 5

The normalized beat wavelength γ for the beating of the HE1,1Ez-sine mode between two identical rods as a function of υ, with parameter a/ρ. γ was calculated by use of the approximate method, truncating the matrix of the system of equations to the minor M1,1. Curve C is a cut-off curve.

Fig. 6
Fig. 6

Effect of departure from a system of two identical rods. The amplitudes of modes I0 and Iπ, in nonidentical touching rods were chosen so that, for some z, Ey was 0 for r 1 = 0. α is then the ratio between the minimum and the maximum value of |Ey| for r 2 = 0. The departure from a system of two identical rods is measured by ρ1/ρ2 − 1. υ1 = 2, δ1 = δ2 = 0.1, M ≥ 5.

Fig. 8
Fig. 8

Flux-density patterns for the Ez-sine modes of two touching rods with υ1 = 3.5, δ1 = δ2 = 0.1, and ρ1/ρ2 = 2. The calculations were done with M = 7.

Fig. 7
Fig. 7

The mean flux density inside guide 2 in units of the mean flux density inside guide 1 for mode Iπ with υ1 = 3.5, δ1 = δ2 = 0.1. The parameter a = 2a/(ρ1 + ρ2) is the relative distance between the rods; for touching rods, a = 2. The calculations were done with M = 7.

Tables (2)

Tables Icon

Table I Convergence of solutions. u1 and g are given as functions of M. M indicates the truncation of the system of equations. g is a parameter indicating the exactness with which the calculated field of the mode satisfies the boundary conditions. The results for mode Iπ were calculated with υ1 = 3.5, δ1 = δ2 → 0, ρ1 = ρ2, and touching rods; the results for mode IIπ were calculated with υ1 = 3.5, δ1 = δ2 = 0.1, ρ1/ρ2 = 1.3, and touching rods.

Tables Icon

Table II Comparison of a precise and more-approximate method for the calculation of the normalized beat wavelength γ for the beats between modes I0 and Iπ for two identical touching rods. The precise results were calculated with M = 9. The approximate results were calculated using Eq. (A1). δ1 = δ2 → 0.

Equations (24)

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E z = n = 0 { A n , i J n ( λ 1 , i r i ) cos ( n ϕ i ) + E n , i J n ( λ 1 , i r i ) sin ( n ϕ i ) } ,
H z = n = 0 { B n , i J n ( λ 1 , i r i ) sin ( n ϕ i ) + F n , i J n ( λ 1 , i r i ) cos ( n ϕ i ) } ,
E z = i = 1 2 n = 0 { C n , i K n ( λ 2 r i ) cos ( n ϕ i ) + G n , i K n ( λ 2 r i ) sin ( n ϕ i ) } ,
H z = i = 1 2 n = 0 { D n , i K n ( λ 2 r i ) sin ( n ϕ i ) + H n , i K n ( λ 2 r i ) cos ( n ϕ i ) } .
λ 1 , i = ( k 1 , i 2 h 2 ) 1 2 , λ 2 = ( h 2 k 2 2 ) 1 2 ,
K n ( λ 2 r 2 ) { cos ( n ψ ) sin ( n ψ ) } = m = + K n + m ( λ 2 a ) I m ( λ 2 r 1 ) { cos ( m θ ) sin ( m θ ) } .
E z = n = 0 { cos ( n ϕ 1 ) [ C n , 1 K n ( λ 2 r 1 ) + I n ( λ 2 r 1 ) m = 0 C m , 2 f ¯ 1 , n , m ] + sin ( n ϕ 1 ) [ G n , 1 K n ( λ 2 r 1 ) + I n ( λ 2 r 1 ) m = 0 G m , 2 f ¯ 4 , n , m ] } ,
H z = n = 0 { sin ( n ϕ 1 ) [ D n , 1 K n ( λ 2 r 1 ) + I n ( λ 2 r 1 ) m = 0 D m , 2 f ¯ 4 , n , m ] + cos ( n ϕ 1 ) [ H n , 1 K n ( λ 2 r 1 ) + I n ( λ 2 r 1 ) m = 0 H m , 2 f ¯ 1 , n , m ] } ,
f ¯ 1 , n , m = ( 1 ) m { K n + m ( λ 2 a ) + K n m ( λ 2 a ) } for n > 0 ,
f ¯ 1 , 0 , m = ( 1 ) m K m ( λ 2 a ) ,
f ¯ 4 , n , m = ( 1 ) m { K n + m ( λ 2 a ) + K n m ( λ 2 a ) } .
f ¯ ¯ 1 , n , m = ( 1 ) n m f ¯ 1 , n , m ;
f ¯ ¯ 4 , n , m = ( 1 ) n m f ¯ 4 , n , m .
E z = n = 0 E n , i J n ( λ 1 , i r i ) sin ( n ϕ i )
E z = n = 0 A n , i J n ( λ 1 , i r i ) cos ( n ϕ i )
δ i = 1 2 1 , i , u i = ρ i λ 1 , i w i = ρ i λ 2 , υ i = ( u i 2 + w i 2 ) 1 2 ( i = 1 , 2 ) .
g P = ( | E ϕ E ϕ | + | E r 2 E r / 1 , i | ) / E max .
M ( 1 , 1 ) = μ ω 2 h 2 K n ( w 1 ) δ n , m ( 1 , 1 η ¯ 1 , n + 2 η ¯ 3 , n ) × ( 1 δ 0 , n ) + δ 0 , n δ n , m K n ( w 1 ) , M ( 1 , 2 ) = n υ 1 2 u 1 2 w 1 2 K n ( w 1 ) δ n , m , M ( 1 , 3 ) = μ ω 2 h 2 I n ( w 1 ) f 4 , n , m ( 1 , 1 η ¯ 1 , n + 2 η ¯ 2 , n ) , M ( 1 , 4 ) = n υ 1 2 u 1 2 w 1 2 I n ( w 1 ) f 1 , n , m , M ( 2 , 1 ) = n υ 1 2 u 1 2 w 1 2 K n ( w 1 ) δ n , m , M ( 2 , 2 ) = K n ( w 1 ) δ n , m ( η ¯ 1 , n + η ¯ 3 , n ) , M ( 2 , 3 ) = n υ 1 2 u 1 2 w 1 2 I n ( w 1 ) f 4 , n , m , M ( 2 , 4 ) = I n ( w 1 ) f 1 , n , m ( η ¯ 1 , n + η ¯ 2 , n ) , u i = λ 1 , i ρ i , w i = λ 2 ρ i , υ i 2 = u i 2 + w i 2 , η ¯ 1 , n = J n ( u 1 ) u 1 J n ( u 1 ) , η ¯ ¯ 1 , n = J n ( u 2 ) u 2 J n ( u 2 ) , η ¯ 2 , n = I n ( w 1 ) w 1 I n ( w 1 ) , η ¯ ¯ 2 , n = I n ( w 2 ) w 2 I n ( w 2 ) , η ¯ 3 , n = K n ( w 1 ) w 1 K n ( w 1 ) , η ¯ ¯ 3 , n = K n ( w 2 ) w 2 K n ( w 2 ) , f 1 , n , m = K n + m ( λ 2 a ) + K n m ( λ 2 a ) ( n > 0 ) , f 1 , 0 , m = K m ( λ 2 a ) , f 4 , n , m = K n + m ( λ 2 a ) + K n m ( λ 2 a ) , δ n , m = 0 for n m , δ n , m = 1 for n = m .
N ( 1 , 1 ) = μ ω 2 h 2 K n ( w ) δ n , m ( 1 η 1 , n + 2 η 3 , n ) ( 1 δ 0 , n ) + δ 0 , n δ n , m K n ( w ) + μ ω 2 h 2 I n ( w ) c f 4 , n , m × ( 1 η 1 , n + 2 η 2 , n ) , N ( 1 , 2 ) = n υ 2 u 2 w 2 ( K n ( w ) δ n , m + I n ( w ) c f 1 , n , m ) , N ( 2 , 1 ) = n υ 2 u 2 w 2 ( K n ( w ) δ n , m + I n ( w ) c f 4 , n , m ) , N ( 2 , 2 ) = K n ( w ) δ n , m ( η 1 , n + η 3 , n ) + I n ( w ) c f 1 , n , m ( η 1 , n + η 2 , n ) .
Φ 1 = c w 2 K 1 2 ( w ) { k 2 2 f 4 , 1 , 1 ( η 1 , 1 + η 3 , 1 ) } + f 1 , 1 , 1 ( k 1 2 η 1 , 1 + k 2 2 η 3 , 1 ) } ,
Φ n = ( k 1 2 η 1 , n + k 2 2 η 3 , n ) · ( η 1 , n + η 3 , n ) n 2 h 2 { υ 2 u 2 w 2 } 2 .
h h 0 = c w 2 K 1 2 ( w ) Φ 1 ( h 0 ) { k 2 2 f 4 , 1 , 1 ( η 1 , 1 + η 3 , 1 ) } + f 1 , 1 , 1 ( k 1 2 η 1 , 1 + k 2 2 η 3 , 1 ) } .
λ B = π ( 1 δ 1 , 1 ) | c 1 , 1 , 1 , 1 , 1 , 2 + | ,
| c 1 , 1 , 1 , 1 , 1 , 2 + | = Δ h 0