Abstract

An efficient method is introduced in this paper to compute the dispersion characteristics as well as the Poynting flux distribution of radially stratified fibers. Only 4 × 4 matrix operations were needed. Detailed results are given for several representative radially inhomogeneous fibers of practical interest.

© 1977 Optical Society of America

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References

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  1. J. G. Dil, H. Blok, Opto-electronics 5, 415 (1973); M. O. Vassell, Opto-electronics 6, 271 (1974).
    [CrossRef]
  2. G. L. Yip, Y. H. Ahmex, in Proc. URSI Symposium of Electromagnetic Wave Theory (1974), p. 272.
  3. P. J. B. Clarricoats, K. B. Chan, Electron. Lett. 6, 694 (1970).
    [CrossRef]

1973 (1)

J. G. Dil, H. Blok, Opto-electronics 5, 415 (1973); M. O. Vassell, Opto-electronics 6, 271 (1974).
[CrossRef]

1970 (1)

P. J. B. Clarricoats, K. B. Chan, Electron. Lett. 6, 694 (1970).
[CrossRef]

Ahmex, Y. H.

G. L. Yip, Y. H. Ahmex, in Proc. URSI Symposium of Electromagnetic Wave Theory (1974), p. 272.

Blok, H.

J. G. Dil, H. Blok, Opto-electronics 5, 415 (1973); M. O. Vassell, Opto-electronics 6, 271 (1974).
[CrossRef]

Chan, K. B.

P. J. B. Clarricoats, K. B. Chan, Electron. Lett. 6, 694 (1970).
[CrossRef]

Clarricoats, P. J. B.

P. J. B. Clarricoats, K. B. Chan, Electron. Lett. 6, 694 (1970).
[CrossRef]

Dil, J. G.

J. G. Dil, H. Blok, Opto-electronics 5, 415 (1973); M. O. Vassell, Opto-electronics 6, 271 (1974).
[CrossRef]

Yip, G. L.

G. L. Yip, Y. H. Ahmex, in Proc. URSI Symposium of Electromagnetic Wave Theory (1974), p. 272.

Electron. Lett. (1)

P. J. B. Clarricoats, K. B. Chan, Electron. Lett. 6, 694 (1970).
[CrossRef]

Opto-electronics (1)

J. G. Dil, H. Blok, Opto-electronics 5, 415 (1973); M. O. Vassell, Opto-electronics 6, 271 (1974).
[CrossRef]

Other (1)

G. L. Yip, Y. H. Ahmex, in Proc. URSI Symposium of Electromagnetic Wave Theory (1974), p. 272.

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Figures (20)

Fig. 1
Fig. 1

Geometry of the inhomogeneous fiber.

Fig. 2
Fig. 2

Refractive-index profiles of several fibers of interest.

Fig. 3
Fig. 3

Dispersion curves for homogeneous core fiber with n1 = 1.515 and n2 = 1.50. a is the radius of the core.

Fig. 4
Fig. 4

Dispersion curves for parabolic core (five-layer) fiber with nMAX = 1.515 and n2 = 1.50.

Fig. 5
Fig. 5

Staircase approximations of two doughnut refractive-index profiles.

Fig. 6
Fig. 6

Comparison of dispersion curves for two fibers with different doughnut refractive-index profiles.

Fig. 7
Fig. 7

Comparison of HE1,1 mode dispersion curves for homogeneous, doughnut, and parabolic refractive-index profile fibers.

Fig. 8
Fig. 8

Poynting flux characteristics of HE1,1 mode. Fiber profile is homogeneous, n1 = 1.515 and n2 = 1.500.

Fig. 9
Fig. 9

Poynting flux characteristics of EH1,2HE1,3 modes. Fiber profile is homogeneous, n1 = 1.515 and n2 = 1.500.

Fig. 10
Fig. 10

Poynting flux characteristics of HE2,1 mode. Fiber profile is homogeneous, n1 = 1.515 and n2 = 1.500.

Fig. 11
Fig. 11

Poynting flux characteristics of HE10,1 mode. Fiber profile is homogeneous, n1 = 1.515 and n2 = 1.500.

Fig. 12
Fig. 12

Poynting flux characteristics of HE1,1 mode. Fiber profile is homogeneous, n1 = 1.53 and n2 = 1.500.

Fig. 13
Fig. 13

Poynting flux characteristics of HE1,1 mode. Fiber profile is parabolic (five layer), nMAX = 1.515 and n2 = 1.500.

Fig. 14
Fig. 14

Poynting flux characteristics of HE2,1 mode. Fiber profile is parabolic (five layer), nMAX = 1.515 and n2 = 1.500.

Fig. 15
Fig. 15

Poynting flux characteristics of HE1,1 mode. Fiber profile is doughnut (five layer), nMAX = 1.515 and n2 = 1.500.

Fig. 16
Fig. 16

Poynting flux characteristics of HE2,1 mode. Fiber profile is doughnut (five layer), nMAX = 1.515 and n2 = 1.500.

Fig. 17
Fig. 17

Comparison of dispersion curves for homogeneous parabolic (five-layer) refractive-index profile fibers. Radius of parabolic core fiber is 1.66 times larger than homogeneous core radius.

Fig. 18
Fig. 18

Comparison of Poynting flux characteristics for homogeneous and parabolic (five-layer) core fibers. Radius of parabolic core fiber is 1.66 times larger than homogeneous core radius.

Fig. 19
Fig. 19

Staircase approximations of parabolic refractive-index profile.

Fig. 20
Fig. 20

Dispersion curves for HE1,1 mode of parabolic refracttive-index profile fiber approximated by five and ten layers.

Tables (1)

Tables Icon

Table I Illustration of Error in Sz(r) due to Error in (βN,k0a) and P

Equations (13)

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[ E z ( 1 ) η H z ( 1 ) ρ E θ ( 1 ) η ρ H θ ( 1 ) ] = [ c 1 ( r ) 0 0 0 0 d 1 ( r ) 0 0 e 1 ( r ) f 1 ( r ) 0 0 q 1 ( r ) h 1 ( r ) 0 0 ] [ C 1 D 1 0 0 ] ;
[ E z ( m ) η H z ( m ) ρ E θ ( m ) η ρ H θ ( m ) ] = [ c m ( r ) 0 c m ( r ) 0 0 d m ( r ) 0 d m ( r ) e m ( r ) f m ( r ) e m ( r ) f m ( r ) g m ( r ) h m ( r ) g m ( r ) h m ( r ) ] [ C m D m C m D m ] ;
[ E z ( m + 1 ) η H z ( m + 1 ) ρ E θ ( m + 1 ) η ρ H θ ( m + 1 ) ] = [ 0 0 s ( r ) 0 0 0 0 τ ( r ) 0 0 u ( r ) v ( r ) 0 0 w ( r ) χ ( r ) ] [ 0 0 G F ] ;
c m ( r ) = J n ( p m r ) , c m ( r ) = N n ( p m r ) , d m ( r ) = i J n ( p m r ) , d m ( r ) = N n ( p m r ) i , e m ( r ) = - β n k 0 p m 2 J n ( p m r ) , e m ( r ) = - β n k 0 p m 2             N n ( p m r ) , f m ( r ) = - i k 0 p m k 0 r J n ( p m r ) i , f m ( r ) = - i k 0 p m k 0 r N n ( p m r ) i , g m ( r ) = i m 0 k 0 p m k 0 r J n ( p m r ) , g m ( r ) = m 0 k 0 p m k 0 r N n ( p m r ) , h m ( r ) = - n β k 0 p m 2 × J n ( p r ) i h m ( r ) = - n β k 0 p m N n ( p m r ) i , s ( r ) = K n ( q r ) , τ ( r ) = K n ( q r ) , u ( r ) = β n k 0 q 2 K n ( q r ) , v ( r ) = i k 0 q k 0 r K n ( q r ) i , w ( r ) = - i m + 1 0 k 0 q × k 0 r K n ( q r ) , χ ( r ) = β n k 0 q 2 K n ( q r ) i , p m 2 = ω 2 μ 0 m - β 2 , k m 2 = ω 2 μ 0 m , k 0 = ω ( μ 0 0 ) ½ , q 2 = β 2 - ω 2 μ 0 m + 1 , η = [ ( μ 0 ) / ( 0 ) ] ½ ,
[ c 1 0 0 0 0 0 d 1 0 e 1 0 f 1 0 g 1 0 h 1 0 ]             [ C 1 0 D 1 0 ] = [ c 2 c 2 0 0 0 0 d 2 d 2 e 2 e 2 f 2 f 2 g 2 g 2 h 2 h 2 ] [ C 2 C 2 D 2 D 2 ] [ c 2 c 2 0 0 0 0 d 2 d 2 e 2 e 2 f 2 f 2 g 2 g 2 h 2 h 2 ]             [ C 2 C 2 D 2 D 2 ] = [ c 3 c 3 0 0 0 0 d 3 d 3 e 3 e 3 f 3 f 3 g 3 g 3 h 3 h 3 ] [ C 3 C 3 D 3 D 3 ] [ c m - 1 c m - 1 0 0 0 0 d m - 1 d m - 1 e m - 1 e m - 1 f m - 1 f m - 1 g m - 1 g m - 1 h m - 1 h m - 1 ] [ C m - 1 C m - 1 D m - 1 D m - 1 ] = [ c m c m 0 0 0 0 d m d m e m e m f m f m g m g m h m h m ] [ C m C m D m D m ] [ c m c m 0 0 0 0 d m d m e m e m f m f m g m g m h m h m ]             [ C m C m D m D m ] = [ s 0 0 0 0 0 τ 0 u 0 v 0 w 0 χ 0 ]             [ E 0 F 0 ] ,
[ C 2 C 2 D 2 D 2 ]
[ C 3 C 3 D 3 D 3 ]
M 1 [ C 1 0 D 1 0 ] = M 2 M 2 - 1 M 3 M 3 - 1 M m M m - 1 M m + 1 [ E 0 F 0 ] ,
M 1 = [ c 1 0 0 0 0 0 d 1 0 e 1 0 f 1 0 g 1 - h 1 0 ] , M m = [ c m c m 0 0 0 0 d m d m e m e m f m f m g m g m h m h m ] , M m - 1 = [ c m c m 0 0 0 0 d m d m e m e m f m f m g m g m h m h m ] - 1 , M m + 1 = [ s 0 0 0 0 0 τ 0 u 0 v 0 w 0 χ 0 ] .
[ c 1 0 - M 1 1 - M 1 3 0 d 1 - M 2 1 - M 2 3 e 1 f 1 - M 3 1 - M 3 3 g 1 h 1 - M 4 1 - M 4 3 ] [ C 1 D 1 E F ] = 0 ,
M = M 2 M 2 - 1 M 3 M 3 - 1 M m M m - 1 M m + 1 .
S z ( m ) = 1 2 Re [ E r ( m ) H ϕ ( m ) * - E ϕ ( m ) H r ( m ) * ] .
R β = β N - n 2 n 1 - n 2 ; n 2 < β N < n 1 ,

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