Abstract

For a general class of graded-index fiber profiles, modal propagation constants and group delays were previously evaluated asymptotically to O(k−9) by using the systematic calculation scheme of evanescent wave theory. Here, the numerical results obtained directly are improved by using a nonlinear transformation technique developed by Shanks. This technique extracts information from the diverging as well as converging part of an asymptotic series, whereas the direct method utilizes the converging part alone. We obtain accurate results for a profile far from parabolic shape as well as for one of near-parabolic shape. For the profile far from parabolic shape the results quantify the error in group delay for the WKB technique for the lowest-order mode to approximately 30 ps/km. For high-order modes the WKB results agree with the transformed results.

© 1980 Optical Society of America

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Corrections

Gunnar Jacobsen, "Evanescent-wave and nonlinear transformation analysis of graded-index fibers: errata," J. Opt. Soc. Am. 71, 788-788 (1981)
https://www.osapublishing.org/josa/abstract.cfm?uri=josa-71-6-788

References

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  1. S. Choudhary and L. B. Felsen, “Guided modes in graded index optical fibers,” J. Opt. Soc. Am. 67, 1192–1196 (1977).
    [Crossref]
  2. G. Jacobsen and J. J. Ramskov Hansen, “Propagation constants and group delays of guided modes in graded index fibers: a comparison of three theories,” Appl. Opt. 18, 2837–2842 (1979).
    [Crossref] [PubMed]
  3. R. Robey, PL/1—Formac Interputer User’s Reference Manual, IBM 1967/69, 360 D—03.3.004 (unpublished).
  4. G. Jacobsen and J. J. Ramskov Hansen, “Modified evanescent wave theory for evaluation of propagation constants and group delays of graded index fibers,” Appl. Opt. 18, 3719–3720 (1979).
    [Crossref] [PubMed]
  5. D. Shanks, “Non-linear transforms of divergent and slowly convergent sequences,” J. Math.Phys.1–42 (1955).
  6. J. J. Ramskov Hansen and E. Nicolaisen, “Propagation in graded index fibers: comparison between experiment and three theories,” Appl. Opt. 17, 2831–2835 (1978).
    [Crossref]
  7. H. Ikuno, “Propagation constants of guided modes in graded index fiber with polynomial-profile core,” Electron. Lett. 15(23), 762–763 (1979).
    [Crossref]

1979 (3)

1978 (1)

1977 (1)

1955 (1)

D. Shanks, “Non-linear transforms of divergent and slowly convergent sequences,” J. Math.Phys.1–42 (1955).

Choudhary, S.

Felsen, L. B.

Ikuno, H.

H. Ikuno, “Propagation constants of guided modes in graded index fiber with polynomial-profile core,” Electron. Lett. 15(23), 762–763 (1979).
[Crossref]

Jacobsen, G.

Nicolaisen, E.

Ramskov Hansen, J. J.

Robey, R.

R. Robey, PL/1—Formac Interputer User’s Reference Manual, IBM 1967/69, 360 D—03.3.004 (unpublished).

Shanks, D.

D. Shanks, “Non-linear transforms of divergent and slowly convergent sequences,” J. Math.Phys.1–42 (1955).

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Figures (3)

FIG. 1
FIG. 1

Sketch of Be1Be8 [Eq. (5)] for a mode of profile 2.

FIG. 2
FIG. 2

Error intervals for β2 of mode LP00 (profile 2) specified directly (1), via one (2), two (3) or three (4) successive first-order transforms, via one second-order transform (5), or via one third-order transform (6).

FIG. 3
FIG. 3

Sketch of Be1Be8 [Eq. (5)] for a mode of profile 1.

Tables (8)

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TABLE I Three successive first-order transforms for β2 of mode LP00 (profile 2)

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TABLE II β of mode LP00 (profile 2)

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TABLE III Three successive first-order transforms for β2 of mode LP50 (profile 2)

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TABLE IV β of mode LP50 (profiie 2)

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TABLE V β2 of mode LP00 specified directly and using second- and third-order transforms (profile 2)

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TABLE VI β2 of mode LP00 specified directly and using second- and third-order transforms (profile 1)

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TABLE VII β of mode LP00 and LP50 (profile 1)

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TABLE VIII Comparison between WKB results and EWT results of profile 2 using three successive first-order nonlinear transformations

Equations (18)

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n 2 ( r ) = n 0 2 a 0 2 r 2 ( 1 a 1 r 2 ) 2 0 r ,
[ 2 n 2 ( r ) k 2 ] u ( r ) = 0 ,
r = ( r , θ , z ) .
β μ ν 2 = i = 0 B i ( n 0 , a 0 , a 1 , μ , ν ) k i ,
B e j = i = 0 j B i k i j = , 0 , 1 , , 8 .
β = i = 0 p i ( n 0 , a 0 , a 1 , μ , ν ) k i ,
B i = j = 0 i p j p i j .
τ μ ν 1 c ( k 2 β 2 ) k / 2 k β = 1 c ( n 0 2 + j = 1 2 j 2 B j k j ) / ( i = 0 B i k i ) 1 / 2
τ μ ν = 1 c i = 0 p i ( 1 i ) k i ,
n 0 = 1.5 , a 0 = 6.207 × 10 3 μ m 1 , a 1 = 5.215 × 10 4 μ 2
n 0 = 1.5 , a 0 = 2.5 × 10 3 μ m 1 , a 1 = 5.0 × 10 3 μ m 2 .
Δ B e j = B e j + 1 B e j ,
e k ( B e n ) | B e n k B e n 1 B e n Δ B e n k Δ B e n 1 Δ B e n Δ B e n 1 Δ B e n + k 2 Δ B e n + k 1 , 1 1 1 Δ B e n k Δ B e n 1 Δ B e n Δ B e n 1 Δ B e n + k 2 Δ B e n + k 1 |
e 1 ( B e n ) = B e n + Δ B e n Δ B e n 1 Δ B e n 1 Δ B e n .
B e n = A + α q n ,
e 1 ( B e n ) = A + α q n + α ( q n + 1 q n ) α ( q n q n 1 ) α ( q n q n 1 ) α ( q n + 1 q n ) = A α q n α q n ( q 1 ) ( q n q n 1 ) q n 1 q n 1 2 q n = A .
B e n = A j = 1 i α j q j n ( A , α j , q j real quantities ) ,
e k ( B e n ) = A .