Abstract

Using a Rayleigh–Ritz formulation, modal propagation constants, group delays, and eigenfields are evaluated for practical multimode graded-index profiles by a solution of the scalar-wave equation. The fiber-index core may consist of a Taylor port, a central Gaussian dip, and damped ripples and is surrounded by a uniform cladding. Laguerre Gaussian expansion functions are applied, permitting analytical simplification of the general formulation. For a graded-index fiber with dip and V value of 20.8, modal group delays are determined with an accuracy of the order of 0.5 psec/km. For a larger fiber with V = 69, the method permits accurate and fast interpretation of differential-mode delay measurements.

© 1982 Optical Society of America

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Corrections

J. Hedegaard Povlsen, P. Danielsen, and G. Jacobsen, "Modal propagation constants, group delays, and eigenfield for practical multimode graded-index fibers: errata," J. Opt. Soc. Am. 73, 512-512 (1983)
https://www.osapublishing.org/josa/abstract.cfm?uri=josa-73-4-512

References

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  1. 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]
  2. J. M. Arnold, “Inhomogeneous dielectric waveguides: a uniform asymptotic theory,” J. Phys. A 13, 347–360 (1980).
    [CrossRef]
  3. J. M. Arnold, “Uniform theory of inhomogeneous waveguide modes near cut-off,” J. Phys. A 13, 361–372 (1980).
    [CrossRef]
  4. G. Jacobsen, “Evanescent-wave analysis of general clad graded, index fibers,” J. Opt. Soc. Am. 72, 699–710 (1982).
    [CrossRef]
  5. G. Jacobsen, “Multimode graded index optical fibers: comparison of two Wentzel–Kramers–Brillouin formulations,” J. Opt. Soc. Am. 71, 1492–1496 (1981).
    [CrossRef]
  6. G. Jacobsen and J. J. Ramskov Hansen, “Detailed error estimates for the first order WKB method,” Electron. Lett. 16, 540–541 (1980).
    [CrossRef]
  7. G. Jacobsen, “Evanescent wave theory describing propagation in guiding environments,” Ph.D. Thesis (Technical University of Denmark, Lyngby, Denmark, 1980), Chap. 5.
  8. S. Choudhary and L. B. Felsen, “Guided modes in graded index optical fibers,” J. Opt. Soc. Am. 67, 1192–1196 (1977).
    [CrossRef]
  9. 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]
  10. M. D. Feit and J. A. Fleck, “Mode properties and dispersion for two optical fiber-index profiles by the propagating beam method,” Appl. Opt. 19, 3140–3150 (1980).
    [CrossRef] [PubMed]
  11. M. D. Feit and J. A. Fleck, “Calculation of dispersion for two optical fiber profiles by the propagating beam method,” Radio Sci. 16, 501–509 (1981).
    [CrossRef]
  12. D. Gloge and E. A. J. Marcatili, “Multimode theory of graded core fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).
    [CrossRef]
  13. T. Okoshi and K. Okamoto, “Analysis of wave propagation in inhomogeneous optical fibers using a variational method,” IEEE Trans. Microwave Theory Tech. MTT-22, 938–945 (1974).
    [CrossRef]
  14. M. Geshiro, M. Matsuhara, and N. Kumagai, “Truncated parabolic-index fiber with minimum mode dispersion,” IEEE Trans. Microwave Theory Tech. MTT-26, 115–119 (1978).
    [CrossRef]
  15. O. Georg, “Use of orthogonal system of Laguerre–Gaussian functions in the theory of circular symmetric optical waveguides,” Appl. Opt. 21, 141–146 (1982).
    [CrossRef] [PubMed]
  16. J. A. Arnoud, Beam and Fiber Optics (Academic, New York, 1976), Chap. 3.
  17. S. H. Gould, Variational Methods for Eigenvalue Problems (U. Toronto Press, Toronto, Canada, 1957), Chap. 3.
  18. D. Gloge, “Dispersion in weakly guiding fibers,” Appl. Opt. 10, 2442–2445 (1971).
    [CrossRef] [PubMed]
  19. J. A. Fleck and M. D. Feit, Lawrence Livermore National Laboratory, Livermore, California 94550 (personal communication).
  20. J. Saijonmaa, A. B. Sharma, and S. J. Halme, “Selective excitation of parabolic-index optical fibers by Gaussian beams,” Appl. Opt. 19, 2442–2452 (1980).
    [CrossRef] [PubMed]
  21. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Chap. 22.

1982 (2)

1981 (2)

G. Jacobsen, “Multimode graded index optical fibers: comparison of two Wentzel–Kramers–Brillouin formulations,” J. Opt. Soc. Am. 71, 1492–1496 (1981).
[CrossRef]

M. D. Feit and J. A. Fleck, “Calculation of dispersion for two optical fiber profiles by the propagating beam method,” Radio Sci. 16, 501–509 (1981).
[CrossRef]

1980 (5)

J. M. Arnold, “Inhomogeneous dielectric waveguides: a uniform asymptotic theory,” J. Phys. A 13, 347–360 (1980).
[CrossRef]

J. M. Arnold, “Uniform theory of inhomogeneous waveguide modes near cut-off,” J. Phys. A 13, 361–372 (1980).
[CrossRef]

G. Jacobsen and J. J. Ramskov Hansen, “Detailed error estimates for the first order WKB method,” Electron. Lett. 16, 540–541 (1980).
[CrossRef]

J. Saijonmaa, A. B. Sharma, and S. J. Halme, “Selective excitation of parabolic-index optical fibers by Gaussian beams,” Appl. Opt. 19, 2442–2452 (1980).
[CrossRef] [PubMed]

M. D. Feit and J. A. Fleck, “Mode properties and dispersion for two optical fiber-index profiles by the propagating beam method,” Appl. Opt. 19, 3140–3150 (1980).
[CrossRef] [PubMed]

1979 (1)

1978 (2)

M. Geshiro, M. Matsuhara, and N. Kumagai, “Truncated parabolic-index fiber with minimum mode dispersion,” IEEE Trans. Microwave Theory Tech. MTT-26, 115–119 (1978).
[CrossRef]

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]

1977 (1)

1974 (1)

T. Okoshi and K. Okamoto, “Analysis of wave propagation in inhomogeneous optical fibers using a variational method,” IEEE Trans. Microwave Theory Tech. MTT-22, 938–945 (1974).
[CrossRef]

1973 (1)

D. Gloge and E. A. J. Marcatili, “Multimode theory of graded core fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).
[CrossRef]

1971 (1)

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Chap. 22.

Arnold, J. M.

J. M. Arnold, “Inhomogeneous dielectric waveguides: a uniform asymptotic theory,” J. Phys. A 13, 347–360 (1980).
[CrossRef]

J. M. Arnold, “Uniform theory of inhomogeneous waveguide modes near cut-off,” J. Phys. A 13, 361–372 (1980).
[CrossRef]

Arnoud, J. A.

J. A. Arnoud, Beam and Fiber Optics (Academic, New York, 1976), Chap. 3.

Choudhary, S.

Feit, M. D.

M. D. Feit and J. A. Fleck, “Calculation of dispersion for two optical fiber profiles by the propagating beam method,” Radio Sci. 16, 501–509 (1981).
[CrossRef]

M. D. Feit and J. A. Fleck, “Mode properties and dispersion for two optical fiber-index profiles by the propagating beam method,” Appl. Opt. 19, 3140–3150 (1980).
[CrossRef] [PubMed]

J. A. Fleck and M. D. Feit, Lawrence Livermore National Laboratory, Livermore, California 94550 (personal communication).

Felsen, L. B.

Fleck, J. A.

M. D. Feit and J. A. Fleck, “Calculation of dispersion for two optical fiber profiles by the propagating beam method,” Radio Sci. 16, 501–509 (1981).
[CrossRef]

M. D. Feit and J. A. Fleck, “Mode properties and dispersion for two optical fiber-index profiles by the propagating beam method,” Appl. Opt. 19, 3140–3150 (1980).
[CrossRef] [PubMed]

J. A. Fleck and M. D. Feit, Lawrence Livermore National Laboratory, Livermore, California 94550 (personal communication).

Georg, O.

Geshiro, M.

M. Geshiro, M. Matsuhara, and N. Kumagai, “Truncated parabolic-index fiber with minimum mode dispersion,” IEEE Trans. Microwave Theory Tech. MTT-26, 115–119 (1978).
[CrossRef]

Gloge, D.

D. Gloge and E. A. J. Marcatili, “Multimode theory of graded core fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).
[CrossRef]

D. Gloge, “Dispersion in weakly guiding fibers,” Appl. Opt. 10, 2442–2445 (1971).
[CrossRef] [PubMed]

Gould, S. H.

S. H. Gould, Variational Methods for Eigenvalue Problems (U. Toronto Press, Toronto, Canada, 1957), Chap. 3.

Halme, S. J.

Jacobsen, G.

Kumagai, N.

M. Geshiro, M. Matsuhara, and N. Kumagai, “Truncated parabolic-index fiber with minimum mode dispersion,” IEEE Trans. Microwave Theory Tech. MTT-26, 115–119 (1978).
[CrossRef]

Marcatili, E. A. J.

D. Gloge and E. A. J. Marcatili, “Multimode theory of graded core fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).
[CrossRef]

Matsuhara, M.

M. Geshiro, M. Matsuhara, and N. Kumagai, “Truncated parabolic-index fiber with minimum mode dispersion,” IEEE Trans. Microwave Theory Tech. MTT-26, 115–119 (1978).
[CrossRef]

Nicolaisen, E.

Okamoto, K.

T. Okoshi and K. Okamoto, “Analysis of wave propagation in inhomogeneous optical fibers using a variational method,” IEEE Trans. Microwave Theory Tech. MTT-22, 938–945 (1974).
[CrossRef]

Okoshi, T.

T. Okoshi and K. Okamoto, “Analysis of wave propagation in inhomogeneous optical fibers using a variational method,” IEEE Trans. Microwave Theory Tech. MTT-22, 938–945 (1974).
[CrossRef]

Ramskov Hansen, J. J.

Saijonmaa, J.

Sharma, A. B.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Chap. 22.

Appl. Opt. (6)

Bell Syst. Tech. J. (1)

D. Gloge and E. A. J. Marcatili, “Multimode theory of graded core fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).
[CrossRef]

Electron. Lett. (1)

G. Jacobsen and J. J. Ramskov Hansen, “Detailed error estimates for the first order WKB method,” Electron. Lett. 16, 540–541 (1980).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

T. Okoshi and K. Okamoto, “Analysis of wave propagation in inhomogeneous optical fibers using a variational method,” IEEE Trans. Microwave Theory Tech. MTT-22, 938–945 (1974).
[CrossRef]

M. Geshiro, M. Matsuhara, and N. Kumagai, “Truncated parabolic-index fiber with minimum mode dispersion,” IEEE Trans. Microwave Theory Tech. MTT-26, 115–119 (1978).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Phys. A (2)

J. M. Arnold, “Inhomogeneous dielectric waveguides: a uniform asymptotic theory,” J. Phys. A 13, 347–360 (1980).
[CrossRef]

J. M. Arnold, “Uniform theory of inhomogeneous waveguide modes near cut-off,” J. Phys. A 13, 361–372 (1980).
[CrossRef]

Radio Sci. (1)

M. D. Feit and J. A. Fleck, “Calculation of dispersion for two optical fiber profiles by the propagating beam method,” Radio Sci. 16, 501–509 (1981).
[CrossRef]

Other (5)

G. Jacobsen, “Evanescent wave theory describing propagation in guiding environments,” Ph.D. Thesis (Technical University of Denmark, Lyngby, Denmark, 1980), Chap. 5.

J. A. Arnoud, Beam and Fiber Optics (Academic, New York, 1976), Chap. 3.

S. H. Gould, Variational Methods for Eigenvalue Problems (U. Toronto Press, Toronto, Canada, 1957), Chap. 3.

J. A. Fleck and M. D. Feit, Lawrence Livermore National Laboratory, Livermore, California 94550 (personal communication).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Chap. 22.

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Figures (2)

Fig. 1
Fig. 1

Modal group delay relative to a plane wave field of the cladding (τ) versus relative propagation constant [χ, Eq. (23)] for the profile of Eqs. (22), (24), and (25). (a) Profiles under consideration. Untruncated (- - -), with cladding and dip (—). (b) Results for the untruncated profile [Eq. (22)]. (c) Results for the profile with cladding [Eq. (24)]. (d) Results for the profile with cladding and dip [Eq. (25)].

Fig. 2
Fig. 2

Results for the profile of Eqs. (26)(29) with dip, ripples, and cladding. (a) Profile under consideration. (b) Modal group delay relative to a plane-wave field of the cladding (τ) versus relative propagation constant (χ) for all guided modes (+). Results for an untruncated α profile with α = 1.9 are included as a solid line. (c) τ versus χ for azimuthal mode number ν = 0 (Δ) and for azimuthal mode number ν = 8 (+). Results for the untruncated α-profile are included as a solid line. (d) Pulse spreading versus relative distance from the core center δ. The rms value is given for Dirac delta pulse excitation in time. The source is Gaussian with a spot size of 5 μm. Results are indicated for polynomial core with (1) cladding, (2) dip and cladding, (3) ripples and cladding, and (4) dip, ripples, and cladding.

Tables (4)

Tables Icon

Table 1 Recursation Constants and Zero–Zero Elements of Taylor and Dip Matrices

Tables Icon

Table 2 Convergence of β50 and τ50 for the Profiles of Eqs. (22), (24), and (25) for Increasing Matrix Dimension n

Tables Icon

Table 3 Comparison of β00, β50, τ00, and τ50 for the Profile of Eqs. (22) and (24) Evaluated by Using the Evanescent Wave Theorya and the Rayleigh–Ritz Method

Tables Icon

Table 4 Comparison of Modal Propagation Constants and Group Delays for Guided Modes with ν = 0 and ν = 1 of the Profile of Eqs. (22), (24), and (25) Evaluated by Using the Propagating Beam Methoda and the Rayleigh–Ritz Method

Equations (46)

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[ 2 + k 2 n 2 ( r , λ ) ] ψ μ ν ( r , θ , z ) = 0 ,
[ T 2 + k 2 n 2 ( r , λ ) ] ψ μ ν ( r , θ ) = k 2 β μ ν 2 ψ μ ν ( r , θ ) ,
τ μ ν = 1 c d ( k β μ ν ) d k ,
τ μ ν = 1 c β μ ν | ψ μ ν ( r , θ ) | 2 n 2 ( r , λ ) [ 1 + p ( r , λ ) ] r d θ d r .
p ( r , λ ) = λ log [ n 2 ( r , λ ) ] .
ψ μ ν ( r , θ ) = i = 1 b i μ ν ϕ i ( r , θ ) .
[ L + k 2 N 2 ] b μ ν = k 2 β μ ν 2 b μ ν ,
L i j = ϕ i * T 2 ϕ j r d θ d r , i 0 , j 0 .
N i j 2 = ϕ i * n 2 ( r , λ ) ϕ j r d θ d r , i 0 , j 0 ,
τ μ ν = 1 c β μ ν ( b t μ ν ) * N 2 ( E + P ) b μ ν .
P i j = ϕ i * p ( r , λ ) ϕ j r d θ d r ,
β μ ν 0 β μ ν m 1 + m 1 β μ ν m 1 m β μ ν m 1 m + 1 β μ ν ( ) = β μ ν .
n p 2 ( r ) = n 0 2 λ 1 2 k 2 r 2 ,
( T 2 + λ 1 2 r 2 ) ϕ μ ν ( r , θ ) = E μ ν ϕ μ ν ( r , θ ) ,
ϕ μ ν ( r , θ ) = [ λ 1 μ ! π ( μ + | ν | ) ! ] 1 / 2 × ( r λ 1 ) | ν | exp ( r 2 λ 1 2 ) L μ | ν | ( r 2 λ 1 ) e i ν θ
E μ ν = 2 λ 1 ( 2 μ + | ν | + 1 ) .
λ 1 = 2 k a 2 { 0 a [ n 2 ( r ) n 1 2 ] r d r } 1 / 2 ,
L i j ν = λ 1 { [ ( j + 1 ) ( j + ν + 1 ) ] 1 / 2 δ i , j + 1 + ( 2 j + ν + 1 ) δ i j + j ( j + ν ) 1 / 2 δ i , j 1 } ,
n 2 ( r ) = { l = 0 m a 2 l r 2 l n d 2 exp ( r 2 / σ d 2 ) n 1 2 , r > a n r 2 cos [ 2 π ( r d ) 2 + ϕ 1 ] exp ( r 2 / ϕ r 2 ) , r a
N 2 = l = 0 m a 2 l T ( l , a ) n d 2 D ( σ d , a ) n r 2 R ( σ r , d , ϕ 1 , a ) + n 1 2 N clad 2 ( a ) .
M i , j + 1 ν = C 0 ν [ i ! ( i + ν ) ! ] 1 / 2 [ ( j + 1 ) ! ( j + ν + 1 ) ! ] 1 / 2 × ( λ 1 a 2 ) ν + 1 L i ν ( λ 1 a 2 ) L i ν ( λ 1 a 2 ) exp ( λ 1 a 2 ) + C 1 ν { ( j + 1 j + ν + 1 ) 1 / 2 C 2 ν M i j ν + [ j ( j + 1 ) ( j + ν + 1 ) ( j + ν ) ] 1 / 2 × C 3 ν M i j 1 ν + [ i ( j + 1 ) ( i + ν ) ( j + ν + 1 ) ] 1 / 2 C 4 ν M i 1 , j ν } .
a 0 = 2.25 , a 2 = 1.54107396 × 10 2 × ( 20 μ m ) 2 , a 4 = 6.42936056112 × 10 3 × ( 20 μ m ) 4 , a 6 = 6.705823065248154 × 10 4 × ( 20 μ m ) 6 , a 2 i = 0 , i > 3 .
χ μ ν ( a 0 β μ ν ) / [ a 0 n ( 20 μ m ) ] ,
n 1 n ( 20 μ m ) = 1.492477576894325 , r 20 μ m
n d 2 = ( a 0 n 1 2 ) , σ d = 1.2 μ m ,
a 0 = 2.25 , a 2 = 3.935842 × 10 2 × ( 50 μ m ) 2 , a 4 = 4.64559 × 10 3 × ( 50 μ m ) 4 , a 6 = 1.25662 × 10 3 × ( 50 μ m ) 6 , a 8 = 1.1345 × 10 4 × ( 50 μ m ) 8 , a 2 i = 0 , i > 4 ,
n 1 = n ( 50 μ m ) = 1.4880 , r 50 μ m ,
n d 2 = a 0 n 1 2 , σ d = 3 μ m ,
n r 2 = 1 10 ( a 0 n 1 2 ) , d = 1 30 × 50 μ m , σ r = 1 3 × 50 μ m , ϕ 1 = 0 .
a 2 l j + 1
1 j + 1
1 λ 1 l ( 1 e λ 1 a 2 n = 0 ν + l ( λ 1 a 2 ) n n ! )
exp ( a 2 / θ d 2 ( j + 1 ) ( 1 + 1 λ 1 σ d 2 )
1 ( j + 1 ) ( 1 + 1 λ 1 σ d 2 )
2 j + ν + 1 λ 1 σ d 2 + j i
( j + ν ) λ 1 σ d 2
1 ( 1 + 1 λ 1 σ d 2 ) ν + 1 { 1 exp [ a 2 ( λ 1 + 1 σ d 2 ) ] n = 0 ν [ a 2 ( λ 1 + 1 σ d 2 ) ] n n ! }
M i j ν = [ ( j + ν ) ! j ! ] 1 / 2 [ ( i + ν ) ! i ! ] 1 / 2 M i , j ν , i 0 , j 0 ,
T i j l ν = 1 λ i l 0 λ 1 a 2 L i ν ( y ) L j ν ( y ) e y y ν y l d y ,
D i j ν = 0 λ 1 a 2 L i ν ( y ) L j ν ( y ) e y y ν exp [ y ( λ 1 σ d 2 ) ] d y .
n ! L n ν ( x ) x ν e x = d n d x n ( e x x n x ν ) d n 1 d x n 1 ( e x x n x ν ) = ( n 1 ) ! x i = 0 n 1 e x x ν L i ν ( x ) ,
x d d x L n ν ( x ) = n L n ν ( x ) ( n + ν ) L n 1 ν ( x ) ,
x L n ν ( x ) = ( n + 1 ) L n + 1 ν ( x ) + ( 2 n + ν + 1 ) L n ν ( x ) ( n + ν ) L n 1 ν ( x ) .
j T i j l ν = j = 0 j 1 1 λ 1 l L i ν ( λ 1 a 2 ) e λ 1 a 2 ( λ 1 a 2 ) l + ν + 1 j = 0 j 1 0 λ 1 a 2 y d d y [ L i ν ( y ) y l ] L j ν ( y ) e y y ν d y ,
j D i j ν = j = 0 j 1 L i ν ( λ 1 a 2 ) L j ν ( λ 1 a 2 ) e λ 1 a 2 e ( a / σ d ) 2 ( λ 1 a 2 ) ν + 1 j = 0 j 1 0 λ 1 a 2 y d d y { L i ν ( y ) exp y / ( λ 1 σ d 2 ] } × L j ν ( y ) e y y ν d y .
( j + 1 ) T i , j + 1 l ν j T i j l ν and ( j + 1 ) D i , j + 1 ν j D i j ν ,