Abstract

The evanescent wave theory (ewt) is extended to treat modal propagation along a general graded-index optical-fiber structure surrounded by a uniform cladding. A step or a valley may be present at the core–cladding interface. The fiber core may be specified in a more general form than was the case for untruncated profiles. The reformulation of the original ewt method uses the uniform corrective method, developed by Arnold for WKB calculations, iteratively to specify nonintegral radial mode numbers for the clad index profile. The integral radial mode numbers of the unclad profile are used as first approximations. Our method gives corrected values for propagation constants and group delays of guided modes and complex propagation constants of leaky modes. Numerical results show good agreement with a numerical solution for a parabolic-core fiber. For other profiles they specify accurate asymptotic upper and lower limits for the cladding influence. For leaky modes of a truncated near-parabolic profile we show agreement with results of a simple WKB formulation. A valley, present at the core–cladding interface, causes a reduction in the maximum intermodal group-delay difference from 3.1 to 1.9 nsec/km because of a strong attenuation of high-order (leaky) modes.

© 1982 Optical Society of America

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References

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  1. S. Chaudhary 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. 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]
  4. D. Shanks, “Nonlinear transformations of divergent and slowly convergent sequences,” J. Math. Phys. 34, 1–42 (1955).
  5. C. M. Bender and A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), pp. 368–410.
  6. G. Jacobsen, “Evanescent-wave and nonlinear transformation analysis of graded index fibers,” J. Opt. Soc. Am. 70, 1338–1344 (1980); Errata 71, 788 (1981).
    [Crossref]
  7. G. Jacobsen and J. J. Ramskov Hansen, “Detailed error estimates for the first order WKB method,” Electron. Lett. 16, 540–541 (1980).
    [Crossref]
  8. G. Jacobsen and F. Ramskov Hansen, “Behavior of transients in propagation constant series of evanescent wave theory,” Radio Sci. 16, 519–524 (1981).
    [Crossref]
  9. G. Jacobsen, “Multimode graded index optical fibers: comparison of two Wentzel-Kramers-Brillouin formulations,” J. Opt. Soc. Am. 71, 1492–1496 (1981).
    [Crossref]
  10. D. Gloge, “Dispersion in weakly guiding fibers,” Appl. Opt. 10, 2442–2445 (1971).
    [Crossref] [PubMed]
  11. J. M. Arnold, “Inhomogeneous dielectric waveguides: A uniform asymptotic theory,” J. Phys. A 13, 347–360 (1980).
    [Crossref]
  12. J. M. Arnold, “Uniform theory of inhomogeneous waveguide modes near cutoff,” J. Phys. A 13, 361–372 (1980).
    [Crossref]
  13. G. A. E. Crone and J. M. Arnold, “Anamalous group delay in optical fibers,” Opt. Quantum Electron. 12, 511–517 (1980).
    [Crossref]
  14. D. Gloge and A. E. I. Marcatili, “Multimode theory of graded core fibers,” Bell. Syst. Tech. J. 52, 1563–1578 (1973).
    [Crossref]
  15. J. M. Arnold, “Unified asymptotic theory of asymmetrical planar waveguides,” J. Acoust. Soc. Am. 69, 17–24 (1981).
    [Crossref]
  16. G. Jacobsen, “Evanescent wave theory describing propagation in guiding environments,” Ph.D. Thesis (Technical University of Denmark, Copenhagen, 1980).
  17. G. Jacobsen, “Exact field expressions for guided modes of a general class of graded index fibers,” Electron. Lett. 14, 464–465 (1978).
    [Crossref]
  18. 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]
  19. R. Olshansky, “Leaky modes in graded index optical fibers,” Appl. Opt. 15, 2773–2777 (1976).
    [Crossref] [PubMed]
  20. 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]

1981 (3)

G. Jacobsen and F. Ramskov Hansen, “Behavior of transients in propagation constant series of evanescent wave theory,” Radio Sci. 16, 519–524 (1981).
[Crossref]

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

J. M. Arnold, “Unified asymptotic theory of asymmetrical planar waveguides,” J. Acoust. Soc. Am. 69, 17–24 (1981).
[Crossref]

1980 (6)

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]

G. Jacobsen, “Evanescent-wave and nonlinear transformation analysis of graded index fibers,” J. Opt. Soc. Am. 70, 1338–1344 (1980); Errata 71, 788 (1981).
[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. 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 cutoff,” J. Phys. A 13, 361–372 (1980).
[Crossref]

G. A. E. Crone and J. M. Arnold, “Anamalous group delay in optical fibers,” Opt. Quantum Electron. 12, 511–517 (1980).
[Crossref]

1979 (2)

1978 (2)

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]

G. Jacobsen, “Exact field expressions for guided modes of a general class of graded index fibers,” Electron. Lett. 14, 464–465 (1978).
[Crossref]

1977 (1)

1976 (1)

1973 (1)

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

1971 (1)

1955 (1)

D. Shanks, “Nonlinear transformations of divergent and slowly convergent sequences,” J. Math. Phys. 34, 1–42 (1955).

Arnold, J. M.

J. M. Arnold, “Unified asymptotic theory of asymmetrical planar waveguides,” J. Acoust. Soc. Am. 69, 17–24 (1981).
[Crossref]

G. A. E. Crone and J. M. Arnold, “Anamalous group delay in optical fibers,” Opt. Quantum Electron. 12, 511–517 (1980).
[Crossref]

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 cutoff,” J. Phys. A 13, 361–372 (1980).
[Crossref]

Bender, C. M.

C. M. Bender and A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), pp. 368–410.

Chaudhary, S.

Crone, G. A. E.

G. A. E. Crone and J. M. Arnold, “Anamalous group delay in optical fibers,” Opt. Quantum Electron. 12, 511–517 (1980).
[Crossref]

Feit, M. D.

Felsen, L. B.

Fleck, J. A.

Gloge, D.

D. Gloge and A. E. I. 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]

Jacobsen, G.

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

G. Jacobsen and F. Ramskov Hansen, “Behavior of transients in propagation constant series of evanescent wave theory,” Radio Sci. 16, 519–524 (1981).
[Crossref]

G. Jacobsen, “Evanescent-wave and nonlinear transformation analysis of graded index fibers,” J. Opt. Soc. Am. 70, 1338–1344 (1980); Errata 71, 788 (1981).
[Crossref]

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

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]

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]

G. Jacobsen, “Exact field expressions for guided modes of a general class of graded index fibers,” Electron. Lett. 14, 464–465 (1978).
[Crossref]

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

Marcatili, A. E. I.

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

Nicolaisen, E.

Olshansky, R.

Orszag, A.

C. M. Bender and A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), pp. 368–410.

Ramskov Hansen, F.

G. Jacobsen and F. Ramskov Hansen, “Behavior of transients in propagation constant series of evanescent wave theory,” Radio Sci. 16, 519–524 (1981).
[Crossref]

Ramskov Hansen, J. J.

Shanks, D.

D. Shanks, “Nonlinear transformations of divergent and slowly convergent sequences,” J. Math. Phys. 34, 1–42 (1955).

Appl. Opt. (6)

Bell. Syst. Tech. J. (1)

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

Electron. Lett. (2)

G. Jacobsen, “Exact field expressions for guided modes of a general class of graded index fibers,” Electron. Lett. 14, 464–465 (1978).
[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. Acoust. Soc. Am. (1)

J. M. Arnold, “Unified asymptotic theory of asymmetrical planar waveguides,” J. Acoust. Soc. Am. 69, 17–24 (1981).
[Crossref]

J. Math. Phys. (1)

D. Shanks, “Nonlinear transformations of divergent and slowly convergent sequences,” J. Math. Phys. 34, 1–42 (1955).

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 cutoff,” J. Phys. A 13, 361–372 (1980).
[Crossref]

Opt. Quantum Electron. (1)

G. A. E. Crone and J. M. Arnold, “Anamalous group delay in optical fibers,” Opt. Quantum Electron. 12, 511–517 (1980).
[Crossref]

Radio Sci. (1)

G. Jacobsen and F. Ramskov Hansen, “Behavior of transients in propagation constant series of evanescent wave theory,” Radio Sci. 16, 519–524 (1981).
[Crossref]

Other (2)

C. M. Bender and A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978), pp. 368–410.

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

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Figures (2)

Fig. 1
Fig. 1

Profile 1 with (1A) the profile having a step [Eqs. (55) and (56)] and (B) the profile having a valley [Eq. (57)], at the core–cladding interface.

Fig. 2
Fig. 2

Profiles 1, 2, and 3 of Eqs. (48)(54) shown truncated (solid curves) and untruncated (dashed curves).

Tables (23)

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Table 1 U2/V for LPμ,ν Modes of a Truncated Parabolic Profile with V = 12

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Table 2 Cladding Influence on τμ,ν of Modes of Table 1

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Table 3 β0,02, Δμ of Mode LP0,0 of Profile 1

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Table 4 τ0,0 of Mode LP0,0 of Profile 1

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Table 5 β5,02, Δμ of Mode LP5,0 of Profile 1

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Table 6 τ5,0 of Mode LP5,0 of Profile 1

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Table 7 β0,02, Δμ of Mode LP0,0 of Profile 3

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Table 8 τ0,0 of Mode LP0,0 of Profile 3

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Table 9 β4,12, Δμ of Mode LP4,1 of Profile 3

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Table 10 τ4,1 of Mode LP4,1 of Profile 3

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Table 11 β0,02, Δμ, τ0,0 of Mode LP0,0 of Profile 2

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Table 12 β5,02, Δμ, τ5,0 of Mode LP5,0 of Profile 2

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Table 13 β0,112, Δμ of Mode LP0,11 of Profile 1

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Table 14 β0,172, Δμ of Mode LP0,17 of Profile 1

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Table 15 Comparison of WKB and Evanescent Wave Theory Results for the Imaginary Part of the Propagation Constants of Leaky Modes of Profile 1

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Table 16 β0,02, Δμ of Mode LP0,0 of Profile 1 with a Step at the Core–Cladding Interface

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Table 17 β5,02, Δμ of Mode LP5,0 of Profile 1 with a Step at the Core–Cladding Interface

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Table 18 τ5,0 of Mode LP5,0 of Profile 1 with a Step at the Core–Cladding Interface

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Table 19 β0,112, Δμ of Mode LP0,11 of Profile 1 with a Step at the Core–Cladding Interface

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Table 20 τ0,11 of Mode LP0,11 of Profile 1 with a Step at the Core–Cladding Interface

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Table 21 β0,02, Δμ of Mode LP0,0 of Profile 1 with a Valley at the Core–Cladding Interface

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Table 22 β5,02, Δμ of Mode LP5,0 of Profile 1 with a Valley at the Core–Cladding Interface

Tables Icon

Table 23 β0,112, Δμ of Mode LP0,11 of Profile 1 with a Valley at the Core–Cladding Interface

Equations (58)

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n 2 ( r ) = { n 0 2 - a 0 2 r 2 ( 1 + a 1 r 2 ) 2 , 0 r < a , n 2 2 , a r .
n c 2 = n 0 2 - a 0 2 a 2 ( 1 + a 1 a 2 ) 2 ,
n 2 < > n c .
[ 2 + k 2 n 2 ( r ) ] u μ ν ( r ˜ ) = 0.
{ r d d r ( r d d r ) + r 2 [ k 2 n 2 ( r ) - k 2 β μ ν 2 - ν 2 r 2 ] } u μ ν ( r ) = 0 ,
β μ ν 2 = j = 0 B j ( n 0 , a 0 , a 1 , μ , ν ) k j
β μ ν = 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 2 c i = 0 ( 2 - i ) B i / k i ( j = 0 B j k j ) 1 / 2
τ μ ν = 1 c i = 0 p i ( 1 - i ) k i .
V 2 = k 2 a 2 [ n 2 ( 0 ) - n 2 ( a ) ] = k 2 a 2 ( n 0 2 - n 2 2 ) ,
U 2 = k 2 a 2 ( n 0 2 - β μ ν 2 ) ,
W 2 = k 2 a 2 ( β μ ν 2 - n 2 2 )
U μ ν ( r ) = ρ - 1 / 2 Φ μ ν ( ρ ) ,
ρ = r / a .
{ d 2 d ρ 2 + [ U 2 - V 2 ρ 2 ( 1 + a 1 a 2 ρ 2 ) 2 ( 1 + a 1 a 2 ) 2 + 1 / 4 - ν 2 ρ 2 ] } × Φ μ ν ( ρ ) = 0 ,             0 ρ < 1
[ d 2 d ρ 2 + ( - W 2 + 1 / 4 - ν 2 ρ 2 ) ] Φ μ ν ( ρ ) = 0 ,             1 ρ .
Φ μ ν ( ρ ) = ρ 1 / 2 K ν ( W ρ ) ,
d Φ μ ν ( ρ ) d ρ | ρ = 1 = K 0 Φ μ ν ( 1 ) ,
K 0 = 1 2 + W K ν ( W ) K ν ( W ) .
Φ μ ν ( ρ ) = ρ 1 / 2 H ν ( 1 ) ( i - W 2 ρ ) ,
K 0 = 1 2 + i - W 2 H ν ( 1 ) ( i - W 2 ) H ν ( 1 ) ( i - W 2 ) ,
- τ ( d τ d ρ ) 2 = U 2 V 2 - ρ 2 ( 1 + a 1 a 2 ρ 2 ) 2 ( 1 + a 1 a 2 ) 2 - ν 2 - 1 / 4 ρ 2 + h ( τ , ρ ) ,
h ( τ , ρ ) = - 1 V 2 ( d τ d ρ ) 1 / 2 d 2 d ρ 2 ( d τ d ρ ) - 1 / 2 .
τ = { 3 2 ρ 2 ρ [ - U V 2 + ρ 2 ( 1 + a 1 a 2 ρ 2 ) 2 ( 1 + a 1 a 2 ) 2 + ( ν 2 - 1 / 4 ) ρ 2 ] 1 / 2 d ρ } 2 / 3 ,
- τ ( d τ d ρ ) 2 = - τ [ d ( - τ ) d ρ ] 2
τ ( ρ ) = τ 0 ( ρ ) + τ 1 ( ρ ) V 2 .
Φ μ ν 0 ( ρ ) = ( d τ d ρ ) - 1 / 2 Φ μ ν 0 ( τ )
[ d 2 d τ 2 - V 2 τ ] Φ μ ν 0 ( τ ) = 0.
Φ μ ν 0 ( τ ) = A 1 A i ( V 2 / 3 τ ) + A 2 B i ( V 2 / 3 τ ) ,
Φ μ ν 0 ( τ ) = cos [ ( μ + Δ μ ) π ] A i ( V 2 / 3 τ ) - sin [ ( μ + Δ μ ) π ] B i ( V 2 / 3 τ ) ,
- 1 < Δ μ < 1.
d Φ μ ν 0 ( τ ) d τ | τ = τ 0 = K 1 Φ μ ν 0 ( τ 0 ) ,
K 1 = ( d τ d ρ | ρ = 1 ) - 1 [ K 0 + 1 2 ( d τ d ρ | ρ = 1 ) - 1 d 2 τ d ρ 2 | ρ = 1 ]
τ 0 = lim τ ρ 1 .
tan ( Δ μ π ) = A i ( V 2 / 3 τ 0 ) - V - 2 / 3 K 1 A i ( V 2 / 3 τ 0 ) B i ( V 2 / 3 τ 0 ) - V - 2 / 3 K 1 B i ( V 2 / 3 τ 0 ) .
τ μ ν = 1 c [ ( k β ) k | μ = const + ( k β ) μ | k = const μ k ] .
( k β ) μ = k j = 0 B j μ / k j 2 j = 0 B j / k j ,
Δ μ = Δ μ r + i Δ μ i ,
e 2 π i Δ μ = 1 + i A i ( V 2 / 3 τ 0 ) - V - 2 / 3 K 1 A i ( V 2 / 3 τ 0 ) B i ( V 2 / 3 τ 0 ) - V - 2 / 3 K 1 B i ( V 2 / 3 τ 0 ) 1 - i A i ( V 2 / 3 τ 0 ) - V - 2 / 3 K 1 A i ( V 2 / 3 τ 0 ) B i ( V 2 / 3 τ 0 ) - V - 2 / 3 K 1 B i ( V 2 / 3 τ 0 ) .
λ = 0.86 μ m ,
n 0 = 1.5 ,
a 0 = 1.507481343168133 μ m - 1 ,
a 1 = 0 ,
a = 20 μ m ,
n 2 = n c = n 0 / 1.02.
U 2 / V = 2 ( 2 μ + ν + 1 ) .
c = 2.9979 m / sec ,
a 0 = 6.207 × 10 - 3 μ m - 1 ,
a 1 = 5.215 × 10 - 4 μ m - 2 ,
a 0 = 2.5 × 10 - 3 μ m - 1 ,
a 1 = 5.0 × 10 - 3 μ m - 2 ,
a 0 = 8.987 × 10 - 3 μ m - 1 ,
a 1 = - 4.135 × 10 - 3 μ m - 2 .
n 2 = n c = 1.49248.
n c = 1.492477576894325 ,
n 2 = 1.491725334583757.
n 2 = 1.493229819204892.