Abstract

The ability to predict the light propagation characteristics in various practical multimode guiding structures is very important in optical fiber communications and in optical image transfer. The usual mode-by-mode analysis is impractical when the guiding structure is capable of supporting hundreds or thousands of modes. In this study a numerical technique is described which is capable of providing useful data on the propagation characteristics of optical multimode guiding structures whose index of refraction variation may be quite arbitrary. As a specific example, the problem of infinite or truncated Gaussian beam propagation in a radially inhomogeneous fiber with parabolic index profile is solved. The numerical results for the infinite Gaussian beam case are compared with exact analytical data and they are in complete agreement.

© 1978 Optical Society of America

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  1. S. E. Miller, E. A. J. Marcatili, and T. Li, Proc. IEEE 61, 1703 (1973).
    [CrossRef]
  2. E. A. J. Marcatili, Bell. Syst. Tech. J. 48, 2071 (1969); J. E. Goell, ibid., 48, 2133 (1969).
    [CrossRef]
  3. I. G. Dil and H. Blok, Opto-Electronics 5, 415 (1973); P. J. B. Clarricoats and K. B. Chan, Electronics Lett. 6, 694 (1970).
    [CrossRef]
  4. C. Yeh and G. Lindgren, Appl. Opt.16,(1977) (to be published).
    [CrossRef]
  5. C. Yeh, S. B. Dong, and W. Oliver, J. Appl. Phys. 46, 2125 (1975).
    [CrossRef]
  6. L. W. Casperson, Appl. Opt. 12, 2434 (1973).
    [CrossRef] [PubMed]
  7. H. Kita, I. Kotano, T. Uchida, and M. Furukawa, J. Am. Ceram. Soc. 54, 321 (1971).
    [CrossRef]
  8. W. A. Gambling and H. Matsumura, Opto-Electronics 5, 429 (1973).
    [CrossRef]
  9. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  11. M. K. Barnoski, M. D. Rourke, S. M. Jensen, and H. R. Friedrick, “Coupling Components for Single Optical Fibers,” presented at IEDM meeting, Washington, D. C., December, 1976 (unpublished).

1975 (1)

C. Yeh, S. B. Dong, and W. Oliver, J. Appl. Phys. 46, 2125 (1975).
[CrossRef]

1973 (4)

L. W. Casperson, Appl. Opt. 12, 2434 (1973).
[CrossRef] [PubMed]

S. E. Miller, E. A. J. Marcatili, and T. Li, Proc. IEEE 61, 1703 (1973).
[CrossRef]

I. G. Dil and H. Blok, Opto-Electronics 5, 415 (1973); P. J. B. Clarricoats and K. B. Chan, Electronics Lett. 6, 694 (1970).
[CrossRef]

W. A. Gambling and H. Matsumura, Opto-Electronics 5, 429 (1973).
[CrossRef]

1971 (1)

H. Kita, I. Kotano, T. Uchida, and M. Furukawa, J. Am. Ceram. Soc. 54, 321 (1971).
[CrossRef]

1969 (1)

E. A. J. Marcatili, Bell. Syst. Tech. J. 48, 2071 (1969); J. E. Goell, ibid., 48, 2133 (1969).
[CrossRef]

Barnoski, M. K.

M. K. Barnoski, M. D. Rourke, S. M. Jensen, and H. R. Friedrick, “Coupling Components for Single Optical Fibers,” presented at IEDM meeting, Washington, D. C., December, 1976 (unpublished).

Blok, H.

I. G. Dil and H. Blok, Opto-Electronics 5, 415 (1973); P. J. B. Clarricoats and K. B. Chan, Electronics Lett. 6, 694 (1970).
[CrossRef]

Casperson, L. W.

Dil, I. G.

I. G. Dil and H. Blok, Opto-Electronics 5, 415 (1973); P. J. B. Clarricoats and K. B. Chan, Electronics Lett. 6, 694 (1970).
[CrossRef]

Dong, S. B.

C. Yeh, S. B. Dong, and W. Oliver, J. Appl. Phys. 46, 2125 (1975).
[CrossRef]

Friedrick, H. R.

M. K. Barnoski, M. D. Rourke, S. M. Jensen, and H. R. Friedrick, “Coupling Components for Single Optical Fibers,” presented at IEDM meeting, Washington, D. C., December, 1976 (unpublished).

Furukawa, M.

H. Kita, I. Kotano, T. Uchida, and M. Furukawa, J. Am. Ceram. Soc. 54, 321 (1971).
[CrossRef]

Gambling, W. A.

W. A. Gambling and H. Matsumura, Opto-Electronics 5, 429 (1973).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Jensen, S. M.

M. K. Barnoski, M. D. Rourke, S. M. Jensen, and H. R. Friedrick, “Coupling Components for Single Optical Fibers,” presented at IEDM meeting, Washington, D. C., December, 1976 (unpublished).

Kita, H.

H. Kita, I. Kotano, T. Uchida, and M. Furukawa, J. Am. Ceram. Soc. 54, 321 (1971).
[CrossRef]

Kotano, I.

H. Kita, I. Kotano, T. Uchida, and M. Furukawa, J. Am. Ceram. Soc. 54, 321 (1971).
[CrossRef]

Li, T.

S. E. Miller, E. A. J. Marcatili, and T. Li, Proc. IEEE 61, 1703 (1973).
[CrossRef]

Lindgren, G.

C. Yeh and G. Lindgren, Appl. Opt.16,(1977) (to be published).
[CrossRef]

Marcatili, E. A. J.

S. E. Miller, E. A. J. Marcatili, and T. Li, Proc. IEEE 61, 1703 (1973).
[CrossRef]

E. A. J. Marcatili, Bell. Syst. Tech. J. 48, 2071 (1969); J. E. Goell, ibid., 48, 2133 (1969).
[CrossRef]

Matsumura, H.

W. A. Gambling and H. Matsumura, Opto-Electronics 5, 429 (1973).
[CrossRef]

Miller, S. E.

S. E. Miller, E. A. J. Marcatili, and T. Li, Proc. IEEE 61, 1703 (1973).
[CrossRef]

Oliver, W.

C. Yeh, S. B. Dong, and W. Oliver, J. Appl. Phys. 46, 2125 (1975).
[CrossRef]

Rourke, M. D.

M. K. Barnoski, M. D. Rourke, S. M. Jensen, and H. R. Friedrick, “Coupling Components for Single Optical Fibers,” presented at IEDM meeting, Washington, D. C., December, 1976 (unpublished).

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

Uchida, T.

H. Kita, I. Kotano, T. Uchida, and M. Furukawa, J. Am. Ceram. Soc. 54, 321 (1971).
[CrossRef]

Yeh, C.

C. Yeh, S. B. Dong, and W. Oliver, J. Appl. Phys. 46, 2125 (1975).
[CrossRef]

C. Yeh and G. Lindgren, Appl. Opt.16,(1977) (to be published).
[CrossRef]

Appl. Opt. (1)

Bell. Syst. Tech. J. (1)

E. A. J. Marcatili, Bell. Syst. Tech. J. 48, 2071 (1969); J. E. Goell, ibid., 48, 2133 (1969).
[CrossRef]

J. Am. Ceram. Soc. (1)

H. Kita, I. Kotano, T. Uchida, and M. Furukawa, J. Am. Ceram. Soc. 54, 321 (1971).
[CrossRef]

J. Appl. Phys. (1)

C. Yeh, S. B. Dong, and W. Oliver, J. Appl. Phys. 46, 2125 (1975).
[CrossRef]

Opto-Electronics (2)

I. G. Dil and H. Blok, Opto-Electronics 5, 415 (1973); P. J. B. Clarricoats and K. B. Chan, Electronics Lett. 6, 694 (1970).
[CrossRef]

W. A. Gambling and H. Matsumura, Opto-Electronics 5, 429 (1973).
[CrossRef]

Proc. IEEE (1)

S. E. Miller, E. A. J. Marcatili, and T. Li, Proc. IEEE 61, 1703 (1973).
[CrossRef]

Other (4)

C. Yeh and G. Lindgren, Appl. Opt.16,(1977) (to be published).
[CrossRef]

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

M. K. Barnoski, M. D. Rourke, S. M. Jensen, and H. R. Friedrick, “Coupling Components for Single Optical Fibers,” presented at IEDM meeting, Washington, D. C., December, 1976 (unpublished).

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Figures (9)

FIG. 1
FIG. 1

Normalized beam waist w2/w1 as a function of normalized axial distance ξ along the fiber axis for various α. w1 and w2 are, respectively, the beam waist at the input and that along the fiber axis.

FIG. 2
FIG. 2

Selected grey-scale intensity patterns along the fiber axis for an infinite Gaussian beam propagating in a radially inhomogeneous fiber with parabolic index profile. The values in the brackets represent the beam waists in μm of the beam while the other unbracketed values represent the highest intensity values for these patterns. Note that the Gaussian profile of the beam is retained throughout the propagation path. For the chosen parameters given in the text, it takes a 35124-μm-long axial distance to complete one cycle as shown.

FIG. 3
FIG. 3

Selected grey-scale intensity patterns along the fiber axis for a truncated Gaussian beam propagating in a radially inhomogeneous fiber with parabolic index profile. The values in the brackets represent the beam waists in μm of the beam while the other unbracketed values represent the highest-intensity values for these patterns. Note that the truncated Gaussian profile of the beam is not preserved along the propagation path. At some points along the path, the intensity at the center of the beam takes a dip. Owing to the presence of the diffraction effects, the beam is not completely symmetrical about a certain “focal point” where the beam achieves the smallest beam waist. The chosen parameters for this figure are the same as those for the infinite Gaussian beam case except the truncation radius is 75 μm.

FIG. 4
FIG. 4

Normalized beam waist ratios (w2/w1) as a function of the normalized axial distance ξ for various values of beam truncation radius (b) in μm. The coefficient for the parabolic index profile is chosen to be n2/n0 = 2 × 10−5 (μm)−2.

FIG. 5
FIG. 5

Normalized beam waist ratios (w2/w1) as a function of the normalized axial distance ξ for various values of beam truncation radius (b) in μm. The coefficient for the parabolic index profile is chosen to be n2/n0 = 1 × 10−5 (μm)−2.

FIG. 6
FIG. 6

Normalized beam waist ratios (w2/w1) as a function of the normalized axial distance ξ for various values of beam truncation radius (b) in μm. The coefficient for the parabolic index profile is chosen to be n2/n0 = 2 × 10−6 (μm)−2.

FIG. 7
FIG. 7

Relative peak irradiance as a function of normalized axial distance 2ξ/π for various values of beam truncation radius (b) in μm. The coefficient for the parabolic index profile is chosen to be n2/n0 = 8 × 10−9 (μm)−2.

FIG. 8
FIG. 8

Relative peak irradiance as a function of normalized axial distance 2ξ/π for various values of beam truncation radius (b) in μm. The coefficient for the parabolic index profile is chosen to be n2/n1 = 4 × 10−9 (μm)−2.

FIG. 9
FIG. 9

Relative peak irradiance as a function of normalized axial distance 2ξ/π for various values of beam truncation radius (b) in μm. The coefficient for the parabolic index profile is chosen to be n2/n1 = 8 × 10−10 (μm)−2.

Equations (21)

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[ 2 + k 2 n 2 ( x , z ) ] u ( x , z ) = 0 ,
( i 2 k n 0 z + T 2 + k 2 [ n 2 ( x , z ) - n 0 2 ] ) A ( x , z ) = - 2 A ( x , z ) z 2 ,
( i 2 k n 0 z + 2 x 2 + 2 y 2 + k 2 [ n 2 ( x , z ) - n 0 2 ] ) A ( x , z ) = 0.
A ( x , 0 ) = u ( x , 0 )
A ( ± , z ) = 0.
u ( x , y , 0 ) = u 0 exp ( - r 2 / w 2 ) for 0 r b , = 0 for r > b ,
A ( x , z ) = exp [ Γ ( x , z ) ] v ( x , z ) ,
Γ ( x , z ) = i k 2 n 0 z 0 z [ n 2 ( x , y , z ) - n 0 2 ] d z .
i 2 k n 0 z v ( x , z ) + e - Γ T 2 [ e Γ v ( x , z ) ] = 0.
( i 2 k n 0 z + T 2 ) v ( x , z ) = 0 ,
v ( x , y , 0 ) = u ( x , y , 0 ) .
v ( m , n , z ) = m , n = 0 N - 1 V ( m , n , z ) exp ( i 2 π N ( m m + n n ) ) ,
( i 2 k n 0 z + f ( m , n ) ( Δ x ) 2 ) V ( m , n , z ) = 0 ,
V ( m , n , z i ) = 1 N 2 m , n = 0 N - 1 v ( m , n , z i ) exp [ Γ ( m , n , z i ) ] × exp ( - i 2 π N ( m m + n n ) ) .
T 2 v = [ 1 / ( Δ x ) 2 ] [ v ( m + 1 , n , z ) - 2 v ( m , n , z ) + v ( m - 1 , n , z ) + v ( m , n + 1 , z ) - 2 v ( m , n , z ) + v ( m , n - 1 , z ) ] ,
f ( m , n ) = - 4 [ sin 2 ( π m / N ) + sin 2 ( π n / N ) ] .
V ( m , n , z i ) = V ( m , n z i ) exp ( - i f ( m , n ) 2 k ( Δ x ) 2 n 0 ( z - z i ) ) ,
n ( x , y ) = n 0 - 1 2 n 2 r 2 ,
u ( x , y , 0 ) = exp ( - r 2 / w 2 ) ,
ξ = z ( n 2 n 0 ) 1 / 2 ,             n 0 α = λ π w 1 2 ( n 0 n 2 ) 1 / 2 ,
n 0 = 2.0 , n 2 / n 0 = 8.0 × 10 - 9 ( μ m ) - 2 , = 70.7 μ m , λ = 0.8 μ m , Δ z = 878.1 μ m , z min = 17562 μ m ,