Abstract

Direct numerical integration of the wave equation is used to establish the validity of approximating the fundamental mode of graded-index fibers by a Gaussian function. We show that the fundamental modes of fibers, whose index profile can be expressed as a power law, are indeed very nearly Gaussian in shape (that is probably also true for graded-index fibers with convex profiles other than a power law). Graphs and empirical analytical expressions are presented for the optimum Gaussian beam width parameter and for the propagation constant of the fundamental mode.

© 1978 Optical Society of America

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References

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  1. W. A. Gambling, D. N. Payne, and H. Matsumura, “Cut-off Frequency in Radially Inhomogeneous Single-Mode Fiber,” Electron. Lett. 13, 139–140 (1977).
    [Crossref]
  2. J. D. Dil and H. Block, “Propagation of Electromagnetic Surface Waves in a Radially Inhomogeneous Optical Waveguide, Opto-Electron. 5, 415–428 (1973).
    [Crossref]
  3. K. Okamoto and T. Okoshi, “Analysis of Wave Propagation in Optical Fibers Having Core with α-Power Refractive-Index Distribution and Uniform Cladding,” IEEE Trans. Microwave Theory Techniques MTT-24, 416–421 (1976).
    [Crossref]
  4. C. N. Kurtz and W. S. Streifer, “Guided Waves in Inhomogeneous Focusing Media Part I: Formulation, Solution for Quadratic Inhomogeneity,” IEEE TRANS/Microwave Theory Techniques MTT-17, 11–15 (1969).
    [Crossref]
  5. K. Petermann, “Theory of Microbending Loss in Monomode Fibres with Arbitrary Refractive Index Profile,” Arch. Elctr. Uebertragung,  30, 337–342 (1976).
  6. D. Marcuse, “Mixrobending Losses of Single-Mode Step-Index and Multimode, Parabolic-Index Fibers,” Bell Syst. Tech. J. 55, 937–955 (1976).
    [Crossref]
  7. D. Marcuse, “Loss Analysis of Single-Mode Fiber Splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
    [Crossref]
  8. D. Marcuse, “Excitation of the Dominant Mode of a Round Fiber by a Gaussian Beam,” Bell Syst. Tech. J. 49, 1695–1703 (1970).
    [Crossref]
  9. D. Gloge, “Weakly Guiding Fibers,” Appl. Opt. 10, 2252–2258 (1971).
    [Crossref] [PubMed]
  10. D. M. Young and R. T. Gregory, A Survey of Numerical Analysis (Addison-Wesley, Reading, Mass., 1973), p. 610.

1977 (2)

W. A. Gambling, D. N. Payne, and H. Matsumura, “Cut-off Frequency in Radially Inhomogeneous Single-Mode Fiber,” Electron. Lett. 13, 139–140 (1977).
[Crossref]

D. Marcuse, “Loss Analysis of Single-Mode Fiber Splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[Crossref]

1976 (3)

K. Petermann, “Theory of Microbending Loss in Monomode Fibres with Arbitrary Refractive Index Profile,” Arch. Elctr. Uebertragung,  30, 337–342 (1976).

D. Marcuse, “Mixrobending Losses of Single-Mode Step-Index and Multimode, Parabolic-Index Fibers,” Bell Syst. Tech. J. 55, 937–955 (1976).
[Crossref]

K. Okamoto and T. Okoshi, “Analysis of Wave Propagation in Optical Fibers Having Core with α-Power Refractive-Index Distribution and Uniform Cladding,” IEEE Trans. Microwave Theory Techniques MTT-24, 416–421 (1976).
[Crossref]

1973 (1)

J. D. Dil and H. Block, “Propagation of Electromagnetic Surface Waves in a Radially Inhomogeneous Optical Waveguide, Opto-Electron. 5, 415–428 (1973).
[Crossref]

1971 (1)

1970 (1)

D. Marcuse, “Excitation of the Dominant Mode of a Round Fiber by a Gaussian Beam,” Bell Syst. Tech. J. 49, 1695–1703 (1970).
[Crossref]

1969 (1)

C. N. Kurtz and W. S. Streifer, “Guided Waves in Inhomogeneous Focusing Media Part I: Formulation, Solution for Quadratic Inhomogeneity,” IEEE TRANS/Microwave Theory Techniques MTT-17, 11–15 (1969).
[Crossref]

Block, H.

J. D. Dil and H. Block, “Propagation of Electromagnetic Surface Waves in a Radially Inhomogeneous Optical Waveguide, Opto-Electron. 5, 415–428 (1973).
[Crossref]

Dil, J. D.

J. D. Dil and H. Block, “Propagation of Electromagnetic Surface Waves in a Radially Inhomogeneous Optical Waveguide, Opto-Electron. 5, 415–428 (1973).
[Crossref]

Gambling, W. A.

W. A. Gambling, D. N. Payne, and H. Matsumura, “Cut-off Frequency in Radially Inhomogeneous Single-Mode Fiber,” Electron. Lett. 13, 139–140 (1977).
[Crossref]

Gloge, D.

Gregory, R. T.

D. M. Young and R. T. Gregory, A Survey of Numerical Analysis (Addison-Wesley, Reading, Mass., 1973), p. 610.

Kurtz, C. N.

C. N. Kurtz and W. S. Streifer, “Guided Waves in Inhomogeneous Focusing Media Part I: Formulation, Solution for Quadratic Inhomogeneity,” IEEE TRANS/Microwave Theory Techniques MTT-17, 11–15 (1969).
[Crossref]

Marcuse, D.

D. Marcuse, “Loss Analysis of Single-Mode Fiber Splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[Crossref]

D. Marcuse, “Mixrobending Losses of Single-Mode Step-Index and Multimode, Parabolic-Index Fibers,” Bell Syst. Tech. J. 55, 937–955 (1976).
[Crossref]

D. Marcuse, “Excitation of the Dominant Mode of a Round Fiber by a Gaussian Beam,” Bell Syst. Tech. J. 49, 1695–1703 (1970).
[Crossref]

Matsumura, H.

W. A. Gambling, D. N. Payne, and H. Matsumura, “Cut-off Frequency in Radially Inhomogeneous Single-Mode Fiber,” Electron. Lett. 13, 139–140 (1977).
[Crossref]

Okamoto, K.

K. Okamoto and T. Okoshi, “Analysis of Wave Propagation in Optical Fibers Having Core with α-Power Refractive-Index Distribution and Uniform Cladding,” IEEE Trans. Microwave Theory Techniques MTT-24, 416–421 (1976).
[Crossref]

Okoshi, T.

K. Okamoto and T. Okoshi, “Analysis of Wave Propagation in Optical Fibers Having Core with α-Power Refractive-Index Distribution and Uniform Cladding,” IEEE Trans. Microwave Theory Techniques MTT-24, 416–421 (1976).
[Crossref]

Payne, D. N.

W. A. Gambling, D. N. Payne, and H. Matsumura, “Cut-off Frequency in Radially Inhomogeneous Single-Mode Fiber,” Electron. Lett. 13, 139–140 (1977).
[Crossref]

Petermann, K.

K. Petermann, “Theory of Microbending Loss in Monomode Fibres with Arbitrary Refractive Index Profile,” Arch. Elctr. Uebertragung,  30, 337–342 (1976).

Streifer, W. S.

C. N. Kurtz and W. S. Streifer, “Guided Waves in Inhomogeneous Focusing Media Part I: Formulation, Solution for Quadratic Inhomogeneity,” IEEE TRANS/Microwave Theory Techniques MTT-17, 11–15 (1969).
[Crossref]

Young, D. M.

D. M. Young and R. T. Gregory, A Survey of Numerical Analysis (Addison-Wesley, Reading, Mass., 1973), p. 610.

Appl. Opt. (1)

Arch. Elctr. Uebertragung (1)

K. Petermann, “Theory of Microbending Loss in Monomode Fibres with Arbitrary Refractive Index Profile,” Arch. Elctr. Uebertragung,  30, 337–342 (1976).

Bell Syst. Tech. J. (3)

D. Marcuse, “Mixrobending Losses of Single-Mode Step-Index and Multimode, Parabolic-Index Fibers,” Bell Syst. Tech. J. 55, 937–955 (1976).
[Crossref]

D. Marcuse, “Loss Analysis of Single-Mode Fiber Splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[Crossref]

D. Marcuse, “Excitation of the Dominant Mode of a Round Fiber by a Gaussian Beam,” Bell Syst. Tech. J. 49, 1695–1703 (1970).
[Crossref]

Electron. Lett. (1)

W. A. Gambling, D. N. Payne, and H. Matsumura, “Cut-off Frequency in Radially Inhomogeneous Single-Mode Fiber,” Electron. Lett. 13, 139–140 (1977).
[Crossref]

IEEE Trans. Microwave Theory Techniques (1)

K. Okamoto and T. Okoshi, “Analysis of Wave Propagation in Optical Fibers Having Core with α-Power Refractive-Index Distribution and Uniform Cladding,” IEEE Trans. Microwave Theory Techniques MTT-24, 416–421 (1976).
[Crossref]

IEEE TRANS/Microwave Theory Techniques (1)

C. N. Kurtz and W. S. Streifer, “Guided Waves in Inhomogeneous Focusing Media Part I: Formulation, Solution for Quadratic Inhomogeneity,” IEEE TRANS/Microwave Theory Techniques MTT-17, 11–15 (1969).
[Crossref]

Opto-Electron. (1)

J. D. Dil and H. Block, “Propagation of Electromagnetic Surface Waves in a Radially Inhomogeneous Optical Waveguide, Opto-Electron. 5, 415–428 (1973).
[Crossref]

Other (1)

D. M. Young and R. T. Gregory, A Survey of Numerical Analysis (Addison-Wesley, Reading, Mass., 1973), p. 610.

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Figures (14)

FIG. 1
FIG. 1

Normalized power law refractive index profiles for several values of the exponent g.

Fig. 2
Fig. 2

The matching parameter (transmission coefficient) T indicating the quality of the match between the fundamental mode field and a Gaussian function is shown as a function of the frequency parameter V for several values of the exponent g of the refractive index power law.

FIG. 3
FIG. 3

Cutoff value Vc of the normalized frequency parameter as a function of the exponent g of the refractive index power law. Vc is the cutoff value of the second mode.

FIG. 4
FIG. 4

Comparison of the shape of the fundamental mode field (open circles) with the optimally matched Gaussian field distribution (solid dots) for V = 3.5 and g = 2.

FIG. 5
FIG. 5

Same as Fig. 4 with V = 10 and g = 2.

FIG. 6
FIG. 6

Same as Fig. 4 with V = 3 and g = 4.

FIG. 7
FIG. 7

Same as Fig. 4 with V = 10 and g = 4.

FIG. 8
FIG. 8

Same as Fig. 4 with V = 2.4 and g = ∞.

FIG. 9
FIG. 9

Same as Fig. 4 with V = 10 and g = ∞.

FIG. 10
FIG. 10

Normalized Gaussian beam width parameter w/a as a function of V for several values of the exponent of the refractive index power law g.

FIG. 11
FIG. 11

Comparison of w/a, obtained from numerical integration of the wave equation and subsequent optimization of its width (solid lines), with the approximation obtained from (15) and (16) (dotted lines) for g = 1 and 4.

FIG. 12
FIG. 12

Same as Fig. 11 with g = 2 and 8.

FIG. 13
FIG. 13

Same as Fig. 11 with g = 6 and ∞.

FIG. 14
FIG. 14

The propagation parameter κa is shown as a function of V. The solid lines are obtained from the numerical solution of the wave equation, the dotted lines are the result of solving (15) and (16). The diagonal dotted line indicates the location of the cutoff of the second mode.

Equations (30)

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n 1 - n 2 1.
d 2 F d r 2 + 1 r d F d r + ( [ n ( r ) k ] 2 - β 2 - ν 2 r 2 ) F = 0.
E = F ( r ) cos ν ϕ e - i β z .
n ( r ) = { n 1 [ 1 - 2 ( r a ) g Δ ] 1 / 2 n 2 = n 1 ( 1 - 2 Δ ) 1 / 2 n 1 ( 1 - Δ ) .
Δ = n 1 2 - n 2 2 2 n 1 2 n 1 - n 2 n 1 .
E G = 2 w 1 ( 2 π ) 1 / 2 e - ( r / w ) 2 e - i β z
c = 1 P 0 2 π d ϕ 0 r Ê Ê G d r
P = 0 2 π d ϕ 0 r Ê 2 d r .
T = c 2 .
V = n 1 k a ( 2 Δ ) 1 / 2 ,
w a = A V 2 / ( g + 2 ) + B V 3 / 2 + C V 6 .
A = { 2 5 [ 1 + 4 ( 2 g ) 5 / 6 ] } 1 / 2 ,
B = e 0.298 / g - 1 + 1.478 ( 1 - e - 0.077 g ) ,
C = 3.76 + exp ( 4.19 / g 0.418 ) .
β 2 = { 4 k 2 w 2 0 n 2 ( r ) exp ( - 2 r 2 / w 2 ) r d r } - 2 w 2 ,
0 [ 4 w 2 ( r 2 w 2 - 1 ) + n 2 ( r ) k 2 - β 2 ] × ( 2 r 2 w 2 - 1 ) exp ( - 2 r 2 / w 2 ) r d r = 0.
κ = ( n 1 2 k 2 - β 2 ) 1 / 2 .
κ a = exp { [ A + ( 1 - A ) exp ( - B V C ) ] ln V } ,
A = 0.1771 + 0.266 / g + 0.3834 exp ( - 0.3 g ) ,
B = 0.304 arctan [ 1.4 ln ( 1 + 0.3675 g 1.347 ) ] ,
C = 0.5425 + 0.6417 / g 0.6214 .
- d 2 F d x 2 - 1 x d F d x + ( V 2 x g + ν 2 x 2 ) F = ( κ a ) 2 F ,
1 1 - x 2 ( - d 2 F d x 2 - 1 x d F d x + ν 2 x 2 F ) = V c 2 F .
d F d x - F ( x + 2 h ) + 8 F ( x + h ) - 8 F ( x - h ) + F ( x - 2 h ) 12 h ,
d 2 F d x 2 - F ( x + 2 h ) + 16 F ( x + h ) - 30 F ( x ) + 16 F ( x - h ) - F ( x - 2 h ) 12 h 2 .
F ( 1 ) = ( 1 - h ) ν F ( 1 - h )
F ( 1 + h ) = ( 1 - h 1 + h ) ν F ( 1 - h ) .
F ( 0 ) = [ 4 F ( h ) - F ( 2 h ) ] / 3.
A F = ( κ a ) 2 F .
F / [ ( κ a ) 2 - K ] = ( A - K I ) - 1 F