Abstract

The principal purpose of this paper is to compare calculated and measured impulse responses of multimode optical fibers to substantiate the validity of the theoretical approach now being used by investigators. To perform the comparison efficiently, a new, practical method of analysis of the impulse response of inhomogeneous optical fibers is presented first. The necessary input data for calculating the impulse response are refractive indices at N points, refractive index in the cladding, core radius, and the wavelength of the light source. The calculated impulse responses are compared with the measured ones for several multimode optical fibers, showing good agreement.

© 1979 Optical Society of America

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References

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  1. D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).
  2. P. J. B. Clarricoats, K. B. Chan, Electron. Lett. 6, 694 (1970).
    [Crossref]
  3. T. Tanaka, Y. Suematsu, Trans. Inst. Electron. Commun. Eng. Japan 59-E (1976).
  4. E. Bianciardi, V. Rizzoli, Opt. Quantum Electron. 121 (1977).
  5. C. Yeh, G. Lindgren, Appl. Opt. 6, 483 (1977).
    [Crossref]
  6. T. Okoshi, K. Okamoto, IEEE Trans. Microwave Theory Tech. MTT-22, 938 (1974).
    [Crossref]
  7. K. Okamoto, T. Okoshi, IEEE Trans. Microwave Theory Tech. MTT-24, 416 (1976).
    [Crossref]
  8. K. Okamoto, T. Okoshi, IEEE Trans. Microwave Theory Tech. MTT-25, 213 (1977).
    [Crossref]
  9. K. Okamoto, T. Okoshi, IEEE Trans. Microwave Theory Tech., MTT-26, 109 (1978).
    [Crossref]
  10. R. Olshansky, D. B. Keck, Appl. Opt. 15, 483 (1976).
    [Crossref] [PubMed]
  11. T. Okoshi, K. Okamoto, K. Hotate, Fundamentals of Optical Fibers (in Japanese) (Ohm Publishing Company, Tokyo, 1977), p. 64.
  12. R. Olshansky, S. M. Oaks, Appl. Opt. 17, 1830 (1978).
    [Crossref] [PubMed]

1978 (2)

K. Okamoto, T. Okoshi, IEEE Trans. Microwave Theory Tech., MTT-26, 109 (1978).
[Crossref]

R. Olshansky, S. M. Oaks, Appl. Opt. 17, 1830 (1978).
[Crossref] [PubMed]

1977 (3)

K. Okamoto, T. Okoshi, IEEE Trans. Microwave Theory Tech. MTT-25, 213 (1977).
[Crossref]

E. Bianciardi, V. Rizzoli, Opt. Quantum Electron. 121 (1977).

C. Yeh, G. Lindgren, Appl. Opt. 6, 483 (1977).
[Crossref]

1976 (3)

R. Olshansky, D. B. Keck, Appl. Opt. 15, 483 (1976).
[Crossref] [PubMed]

T. Tanaka, Y. Suematsu, Trans. Inst. Electron. Commun. Eng. Japan 59-E (1976).

K. Okamoto, T. Okoshi, IEEE Trans. Microwave Theory Tech. MTT-24, 416 (1976).
[Crossref]

1974 (1)

T. Okoshi, K. Okamoto, IEEE Trans. Microwave Theory Tech. MTT-22, 938 (1974).
[Crossref]

1973 (1)

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).

1970 (1)

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

Bianciardi, E.

E. Bianciardi, V. Rizzoli, Opt. Quantum Electron. 121 (1977).

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]

Gloge, D.

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).

Hotate, K.

T. Okoshi, K. Okamoto, K. Hotate, Fundamentals of Optical Fibers (in Japanese) (Ohm Publishing Company, Tokyo, 1977), p. 64.

Keck, D. B.

Lindgren, G.

C. Yeh, G. Lindgren, Appl. Opt. 6, 483 (1977).
[Crossref]

Marcatili, E. A. J.

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).

Oaks, S. M.

Okamoto, K.

K. Okamoto, T. Okoshi, IEEE Trans. Microwave Theory Tech., MTT-26, 109 (1978).
[Crossref]

K. Okamoto, T. Okoshi, IEEE Trans. Microwave Theory Tech. MTT-25, 213 (1977).
[Crossref]

K. Okamoto, T. Okoshi, IEEE Trans. Microwave Theory Tech. MTT-24, 416 (1976).
[Crossref]

T. Okoshi, K. Okamoto, IEEE Trans. Microwave Theory Tech. MTT-22, 938 (1974).
[Crossref]

T. Okoshi, K. Okamoto, K. Hotate, Fundamentals of Optical Fibers (in Japanese) (Ohm Publishing Company, Tokyo, 1977), p. 64.

Okoshi, T.

K. Okamoto, T. Okoshi, IEEE Trans. Microwave Theory Tech., MTT-26, 109 (1978).
[Crossref]

K. Okamoto, T. Okoshi, IEEE Trans. Microwave Theory Tech. MTT-25, 213 (1977).
[Crossref]

K. Okamoto, T. Okoshi, IEEE Trans. Microwave Theory Tech. MTT-24, 416 (1976).
[Crossref]

T. Okoshi, K. Okamoto, IEEE Trans. Microwave Theory Tech. MTT-22, 938 (1974).
[Crossref]

T. Okoshi, K. Okamoto, K. Hotate, Fundamentals of Optical Fibers (in Japanese) (Ohm Publishing Company, Tokyo, 1977), p. 64.

Olshansky, R.

Rizzoli, V.

E. Bianciardi, V. Rizzoli, Opt. Quantum Electron. 121 (1977).

Suematsu, Y.

T. Tanaka, Y. Suematsu, Trans. Inst. Electron. Commun. Eng. Japan 59-E (1976).

Tanaka, T.

T. Tanaka, Y. Suematsu, Trans. Inst. Electron. Commun. Eng. Japan 59-E (1976).

Yeh, C.

C. Yeh, G. Lindgren, Appl. Opt. 6, 483 (1977).
[Crossref]

Appl. Opt. (3)

Bell Syst. Tech. J. (1)

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).

Electron. Lett. (1)

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

IEEE Trans. Microwave Theory Tech. (4)

T. Okoshi, K. Okamoto, IEEE Trans. Microwave Theory Tech. MTT-22, 938 (1974).
[Crossref]

K. Okamoto, T. Okoshi, IEEE Trans. Microwave Theory Tech. MTT-24, 416 (1976).
[Crossref]

K. Okamoto, T. Okoshi, IEEE Trans. Microwave Theory Tech. MTT-25, 213 (1977).
[Crossref]

K. Okamoto, T. Okoshi, IEEE Trans. Microwave Theory Tech., MTT-26, 109 (1978).
[Crossref]

Opt. Quantum Electron. (1)

E. Bianciardi, V. Rizzoli, Opt. Quantum Electron. 121 (1977).

Trans. Inst. Electron. Commun. Eng. Japan (1)

T. Tanaka, Y. Suematsu, Trans. Inst. Electron. Commun. Eng. Japan 59-E (1976).

Other (1)

T. Okoshi, K. Okamoto, K. Hotate, Fundamentals of Optical Fibers (in Japanese) (Ohm Publishing Company, Tokyo, 1977), p. 64.

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Figures (10)

Fig. 1
Fig. 1

Finite element representation of R(r).

Fig. 2
Fig. 2

Sampling function Ψi(r).

Fig. 3
Fig. 3

Flow chart for the calculation of impulse response.

Fig. 4
Fig. 4

The accuracy of the finite element method (FEM) versus the mode number. Parameter V1l is the normalized cutoff frequency of a uniform core fiber for the LP1l mode obtained by the FEM, whereas J0l is the exact solution.

Fig. 5
Fig. 5

Impulse responses of α-power refractive index profiles: (a) α = 1, (b) α = 2, and (c) α = 4.

Fig. 6
Fig. 6

Differential mode attenuation of the multimode optical fiber.

Fig. 7
Fig. 7

(a) Refractive index profile of B-10. (b) Impulse response of B-10 (calculated). L = 1000 m. (c) Impulse response of B-10 (experiment). L = 1705 m.

Fig. 8
Fig. 8

(a) Refractive index profile of V-279. (b) Impulse response of V-279 (calculated). L = 1000 m. (c) Impulse response of V-279 (experiment). L = 540 m.

Fig. 9
Fig. 9

(a) Refractive index profile of V-270. (b) Impulse response of V-270 (calculated). L = 1000 m. (c) Impulse response of V-270 (experiment). L = 940 m.

Fig. 10
Fig. 10

(a) Refractive index profile of V-278. (b) Impulse response of V-278 (calculated). L = 1000m. (c) Impulse response of V-278 (experiment). L = 998 m.

Tables (1)

Tables Icon

Table I Relative Error Between Calculated and Measured Pulse Widths

Equations (33)

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I [ R ] = 0 a [ ( d R d r ) 2 + m 2 r 2 R 2 ] r d r - 0 a [ k 2 n 2 ( r ) - β 2 ] R 2 r d r - Ω β R 2 ( a ) ,
w = ( β 2 - k 2 n clad 2 ) 1 / 2 a ,
Ω β = w K m ( w ) K m ( w ) .
R i = R ( r i )             i = 0 , 1 , 2 , , N .
R ( r ) = i = 0 N R i Ψ i ( r ) ,
Ψ i ( r ) = { 0 , 0 r r i - 1 ( r - r i - 1 ) / ( r i - r i - 1 ) , r i - 1 r r i ( r i + 1 - r ) / ( r i + 1 - r i ) , r i r r i + 1 0 r i + 1 r r N .
1 I 2 R 0 = 0 r 1 ( d R d r d Ψ 0 d r + m 2 r 2 R Ψ 0 ) r d r - 0 r 1 [ k 2 n 2 ( r ) - β 2 ] R Ψ 0 r d r = 0 ,
1 I 2 R i = r i - 1 r i + 1 ( d R d r d Ψ i d r + m 2 r 2 R Ψ i ) r d r - r i - 1 r i + 1 [ k 2 n 2 ( r ) - β 2 ] R ψ i r d r = 0             ( i = 1 , 2 , , N - 1 ) ,
1 I 2 R N = r N - 1 r N [ d R d r d Ψ N d r + m 2 r 2 R Ψ N ] r d r - r N - 1 r N [ k 2 n 2 ( r ) - β 2 ] R Ψ N r d r - Ω β R N = 0.
n ( r ) = [ i = 0 N n i 2 Ψ i ( r ) ] 1 / 2 ,
1 I 2 R 0 = S 00 R 0 + S 01 R 1 = 0 ,
1 I 2 R i = S i , i - 1 R i - 1 + S i , i R i + S i , i + 1 R i + 1 = 0             ( i = 1 , 2 , , N - 1 ) ,
1 I 2 R N = S N , N - 1 R N - 1 + S N , N R N = 0.
S 00 = ( 1 2 - u 2 12 N 2 ) + ( 5 - 3 q 0 - 2 q 1 ) v 2 120 N 2 Δ ,
S 01 = - ( 1 2 + u 2 12 N 2 ) + ( 5 - 2 q 0 - 3 q 1 ) v 2 120 N 2 Δ ,
S 11 = [ 4 m 2 l n 2 - 2 ( m 2 - 1 ) - 2 u 2 3 N 2 ] + ( 40 - 30 q 0 - 30 q 1 - 7 q 2 ) × v 2 120 N 2 Δ
S i , i = [ ( i - 1 ) 2 m 2 l n i ( i - 1 ) + ( i + 1 ) 2 m 2 l n ( i + 1 ) i - 2 i ( m 2 - 1 ) - 2 i u 2 3 N 2 ] + [ 40 i - ( 5 i - 2 ) q i - 1 - 30 i q i - ( 5 i + 2 ) q i + 1 ] v 2 120 N 2 Δ             ( i = 2 , 3 , , N - 1 ) ,
S i , i + 1 = S i + 1 , i = [ ( i + 1 2 ) ( m 2 - 1 ) - i ( i + 1 ) m 2 l n ( i + 1 ) i - ( 2 i + 1 ) u 2 12 N 2 ] + [ 5 ( 2 i + 1 ) - ( 5 i + 2 ) q i - ( 5 i + 3 ) q i + 1 ] v 2 120 N 2 Δ             ( i = 1 , 2 , , N - 1 ) ,
S N , N = [ ( N - 1 ) 2 m 2 l n N ( N - 1 ) - ( N - 1 2 ) ( m 2 - 1 ) + m ( m + 1 ) + w K m - 1 ( w ) K m ( w ) - ( 4 N - 1 ) u 2 12 N 2 ] + [ 5 ( 4 N - 1 ) - ( 5 N - 2 ) q N - 1 - 3 ( 5 N - 1 ) q N ] v 2 120 N 2 Δ ,
q i = n i 2 n core 2             ( i = 0 , 1 , 2 , , N ) ,
n core = Max . [ n ( r ) ] ,
Δ = ( n core 2 - n clad 2 ) 2 n core 2 ,
u 2 = ( k 2 n core 2 - β 2 ) a 2 ,
v 2 = k 2 n core 2 a 2 2 Δ = u 2 + w 2 .
det ( S i j ) = 0
β 2 = 0 k 2 n 2 ( r ) Φ 2 ( r ) r d r - 0 [ ( d Φ d r ) 2 + m 2 r 2 Φ 2 ] r d r 0 Φ 2 ( r ) r d r ,
Φ ( r ) = { R ( r ) ( 0 r a ) R ( a ) K m ( w ) K m ( w r / a ) ( a r ) } .
d β d k = 1 2 β 0 d ( k 2 n 2 ) d k Φ 2 ( r ) r d r 0 Φ 2 ( r ) r d r .
τ = 1 c d β d k = k R T UR c β R T TR ,
{ T 00 = 5 T 01 = 5 T i i = 40 i             ( i = 1 , 2 , , N - 1 ) T i , i + 1 = T i + 1 , i = 5 ( 2 i + 1 )             ( i = 1 , 2 , , N - 1 ) T N N = 5 ( 4 N - 1 ) + 30 N 2 [ 1 / ξ m ( w ) - 1 ] } ,
{ u 00 = 3 n 0 N 0 + 2 n 1 N 1 U 01 = 2 n 0 N 0 + 3 n 1 N 1 U i i = 30 i n i N i + ( 5 i - 2 ) n i - 1 N i - 1 + ( 5 i + 2 ) n i + 1 N i + 1             ( i = 1 , 2 , , N - 1 ) U i , i + 1 = U i + 1 , i = ( 5 i + 2 ) n i N i + ( 5 i + 3 ) n i + 1 N i + 1             ( i = 1 , 2 , , N - 1 ) U N N = 3 n N N N ( 5 N - 1 ) + ( 5 N - 2 ) n N - 1 N N - 1 + 30 n clad N clad N 2 [ 1 / ξ m ( w ) - 1 ] } ,
N i = [ d ( k n i ) ] / ( d k ) ,
ξ m = K m 2 ( w ) K m - 1 ( w ) K m + 1 ( w ) .

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