Abstract

A characterizing technique for radially asymmetric multimode optical waveguides is proposed. By using a precisely measured refractive-index distribution of an asymmetric optical waveguide, ray equations are solved numerically, and practical ray tracing in the asymmetric multimode optical waveguide is realized. The behavior of the asymmetric optical waveguide is evaluated by tracing a number of rays in the asymmetric waveguide. As an example of the technique, some characterizing results are demonstrated. This technique is accurate, fast, and applicable to analyses of arbitrary multimode optical waveguides.

© 1989 Optical Society of America

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References

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  1. E. Okuda, I. Tanaka, T. Yamasaki, “Planar gradient-index glass waveguide and its applications to a 4-port branched circuit and star coupler,” Appl. Opt. 23, 1745–1748 (1984).
    [CrossRef] [PubMed]
  2. C. Yeh, S. B. Dong, W. Oliver, “Arbitrarily shaped inhomogeneous optical fiber or integrated optical waveguides,” J. Appl. Phys. 46, 2125–2129 (1975).
    [CrossRef]
  3. C. Yeh, K. Ha, S. B. Dong, W. P. Brown, “Single-mode optical waveguides,” Appl. Opt. 18, 1490–1504 (1979).
    [CrossRef] [PubMed]
  4. K. Oyamada, T. Okoshi, “Two-dimensional finite-element method calculation of propagation characteristics of axially non-symmetrical optical fiber,” Radio Sci. 17, 109–116 (1982).
    [CrossRef]
  5. W. Streifer, K. B. Paxton, “Analytic solution of ray equations in cylindrically inhomogeneous guiding media. 1: Meridional rays,” Appl. Opt. 10, 769–775 (1971).
    [CrossRef] [PubMed]
  6. K. B. Paxten, W. Streifer, “Analytic solutions of ray equations in cylindrically inhomogeneous guiding media. 2: Skew rays,” Appl. Opt. 10, 1164–1171 (1971).
    [CrossRef]
  7. D. T. Moore, “Ray tracing in gradient-index media,”J. Opt. Soc. Am. 65, 451–455 (1975).
    [CrossRef]
  8. J. Rogers, M. E. Harrigan, R. P. Loce, “The Y–Y¯ diagram for radial gradient systems,” Appl. Opt. 27, 452–458 (1988).
    [CrossRef] [PubMed]
  9. E. W. Marchand, “Rapid ray tracing in radial gradients,” Appl. Opt. 27, 465–467 (1988).
    [CrossRef] [PubMed]
  10. X. Tian, G. Lai, T. Yatagai, M. Seki, “Characterization of asymmetric optical waveguides using high precision refractive index measurement and accurate ray tracing,” in Technical Digest of Microoptics Conference ’87 (Japan Society of Applied Physics, Tokyo, 1987), pp. 26–30.
  11. X. Tian, G. Lai, T. Yatagai, M. Seki, “High-precision refractive index distribution measurement of optical waveguides,” Opt. Commun. 69, 11–14 (1988).
    [CrossRef]
  12. E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), p. 6.
  13. H. T. Davis, Introduction to Nonlinear Differential and Integral Equations (Dover, New York, 1962), p. 251.
  14. T. Okoshi, Optical Fibers (Academic, New York, 1982), p. 39.
  15. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 61.

1988 (3)

1984 (1)

1982 (1)

K. Oyamada, T. Okoshi, “Two-dimensional finite-element method calculation of propagation characteristics of axially non-symmetrical optical fiber,” Radio Sci. 17, 109–116 (1982).
[CrossRef]

1979 (1)

1975 (2)

C. Yeh, S. B. Dong, W. Oliver, “Arbitrarily shaped inhomogeneous optical fiber or integrated optical waveguides,” J. Appl. Phys. 46, 2125–2129 (1975).
[CrossRef]

D. T. Moore, “Ray tracing in gradient-index media,”J. Opt. Soc. Am. 65, 451–455 (1975).
[CrossRef]

1971 (2)

Brown, W. P.

Davis, H. T.

H. T. Davis, Introduction to Nonlinear Differential and Integral Equations (Dover, New York, 1962), p. 251.

Dong, S. B.

C. Yeh, K. Ha, S. B. Dong, W. P. Brown, “Single-mode optical waveguides,” Appl. Opt. 18, 1490–1504 (1979).
[CrossRef] [PubMed]

C. Yeh, S. B. Dong, W. Oliver, “Arbitrarily shaped inhomogeneous optical fiber or integrated optical waveguides,” J. Appl. Phys. 46, 2125–2129 (1975).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 61.

Ha, K.

Harrigan, M. E.

Lai, G.

X. Tian, G. Lai, T. Yatagai, M. Seki, “High-precision refractive index distribution measurement of optical waveguides,” Opt. Commun. 69, 11–14 (1988).
[CrossRef]

X. Tian, G. Lai, T. Yatagai, M. Seki, “Characterization of asymmetric optical waveguides using high precision refractive index measurement and accurate ray tracing,” in Technical Digest of Microoptics Conference ’87 (Japan Society of Applied Physics, Tokyo, 1987), pp. 26–30.

Loce, R. P.

Marchand, E. W.

Moore, D. T.

Okoshi, T.

K. Oyamada, T. Okoshi, “Two-dimensional finite-element method calculation of propagation characteristics of axially non-symmetrical optical fiber,” Radio Sci. 17, 109–116 (1982).
[CrossRef]

T. Okoshi, Optical Fibers (Academic, New York, 1982), p. 39.

Okuda, E.

Oliver, W.

C. Yeh, S. B. Dong, W. Oliver, “Arbitrarily shaped inhomogeneous optical fiber or integrated optical waveguides,” J. Appl. Phys. 46, 2125–2129 (1975).
[CrossRef]

Oyamada, K.

K. Oyamada, T. Okoshi, “Two-dimensional finite-element method calculation of propagation characteristics of axially non-symmetrical optical fiber,” Radio Sci. 17, 109–116 (1982).
[CrossRef]

Paxten, K. B.

Paxton, K. B.

Rogers, J.

Seki, M.

X. Tian, G. Lai, T. Yatagai, M. Seki, “High-precision refractive index distribution measurement of optical waveguides,” Opt. Commun. 69, 11–14 (1988).
[CrossRef]

X. Tian, G. Lai, T. Yatagai, M. Seki, “Characterization of asymmetric optical waveguides using high precision refractive index measurement and accurate ray tracing,” in Technical Digest of Microoptics Conference ’87 (Japan Society of Applied Physics, Tokyo, 1987), pp. 26–30.

Streifer, W.

Tanaka, I.

Tian, X.

X. Tian, G. Lai, T. Yatagai, M. Seki, “High-precision refractive index distribution measurement of optical waveguides,” Opt. Commun. 69, 11–14 (1988).
[CrossRef]

X. Tian, G. Lai, T. Yatagai, M. Seki, “Characterization of asymmetric optical waveguides using high precision refractive index measurement and accurate ray tracing,” in Technical Digest of Microoptics Conference ’87 (Japan Society of Applied Physics, Tokyo, 1987), pp. 26–30.

Yamasaki, T.

Yatagai, T.

X. Tian, G. Lai, T. Yatagai, M. Seki, “High-precision refractive index distribution measurement of optical waveguides,” Opt. Commun. 69, 11–14 (1988).
[CrossRef]

X. Tian, G. Lai, T. Yatagai, M. Seki, “Characterization of asymmetric optical waveguides using high precision refractive index measurement and accurate ray tracing,” in Technical Digest of Microoptics Conference ’87 (Japan Society of Applied Physics, Tokyo, 1987), pp. 26–30.

Yeh, C.

C. Yeh, K. Ha, S. B. Dong, W. P. Brown, “Single-mode optical waveguides,” Appl. Opt. 18, 1490–1504 (1979).
[CrossRef] [PubMed]

C. Yeh, S. B. Dong, W. Oliver, “Arbitrarily shaped inhomogeneous optical fiber or integrated optical waveguides,” J. Appl. Phys. 46, 2125–2129 (1975).
[CrossRef]

Appl. Opt. (6)

J. Appl. Phys. (1)

C. Yeh, S. B. Dong, W. Oliver, “Arbitrarily shaped inhomogeneous optical fiber or integrated optical waveguides,” J. Appl. Phys. 46, 2125–2129 (1975).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

X. Tian, G. Lai, T. Yatagai, M. Seki, “High-precision refractive index distribution measurement of optical waveguides,” Opt. Commun. 69, 11–14 (1988).
[CrossRef]

Radio Sci. (1)

K. Oyamada, T. Okoshi, “Two-dimensional finite-element method calculation of propagation characteristics of axially non-symmetrical optical fiber,” Radio Sci. 17, 109–116 (1982).
[CrossRef]

Other (5)

X. Tian, G. Lai, T. Yatagai, M. Seki, “Characterization of asymmetric optical waveguides using high precision refractive index measurement and accurate ray tracing,” in Technical Digest of Microoptics Conference ’87 (Japan Society of Applied Physics, Tokyo, 1987), pp. 26–30.

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), p. 6.

H. T. Davis, Introduction to Nonlinear Differential and Integral Equations (Dover, New York, 1962), p. 251.

T. Okoshi, Optical Fibers (Academic, New York, 1982), p. 39.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 61.

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Figures (11)

Fig. 1
Fig. 1

Interference pattern of an asymmetric optical waveguide.

Fig. 2
Fig. 2

Refractive-index distribution of an asymmetric optical waveguide, measured by the phase-shift interferometric method.

Fig. 3
Fig. 3

Coordinates used in the analyses.

Fig. 4
Fig. 4

Errors of the numerical ray-tracing algorithm, for evalutions with (curve A) three terms, (curve B) four terms, and (curve C) five terms.

Fig. 5
Fig. 5

Relation between the propagation distance and the errors of the algorithm. Evaluations were made with (curve A) three terms, (curve B) four terms, and (curve C) five terms. The number of dividing points in the 10-mm waveguide is 1000.

Fig. 6
Fig. 6

Ray trace in the asymmetric optical waveguide.

Fig. 7
Fig. 7

Projection of the ray trace to the YX plane.

Fig. 8
Fig. 8

Beam of rays in an asymmetric optical waveguide.

Fig. 9
Fig. 9

Ray-tracing results of eight rays from a point.

Fig. 10
Fig. 10

Plot of intersections of rays with the exit face of an asymmetric optical waveguide.

Fig. 11
Fig. 11

Ray intersection plot at a location 100 mm from the exit face.

Equations (39)

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( n r ) = n ,
n ( r ¨ - r θ ˙ 2 ) = ( 1 + r ˙ 2 + r 2 θ ˙ 2 ) ( n r - r ˙ n Z ) ,
n ( r 2 θ ¨ + 2 r r ˙ θ ˙ ) = ( 1 + r ˙ 2 + r 2 θ ˙ 2 ) ( n θ - r 2 θ ˙ n Z ) ,
n x ¨ = ( 1 + x ˙ 2 + y ˙ 2 ) ( n x - x ˙ n z ) ,
n y ¨ = ( 1 + x ˙ 2 + y ˙ 2 ) ( n y - y ˙ n z ) ,
n ( r ¨ - r θ ˙ 2 ) = ( 1 + r ˙ 2 + r 2 θ ˙ 2 ) n r ,
n ( r 2 θ ¨ + 2 r r ˙ θ ˙ ) = ( 1 + r ˙ 2 + r 2 θ ˙ 2 ) n θ .
1 + r ˙ 2 + r 2 θ ˙ 2 = C 1 n 2 ( r , θ ) ,
C 1 = 1 + r ˙ 0 2 + r 0 2 θ ˙ 0 2 n 2 ( r 0 , θ 0 )
r ¨ - r θ ˙ 2 = C 1 2 ( n 2 r ) ,
r 2 θ ¨ + 2 r r ˙ θ ˙ = C 1 2 ( n 2 θ ) .
r n + 1 = r n + r ˙ n Δ z + ½ r ¨ n ( Δ z ) 2 + r n ( Δ z ) 3 + ,
θ n + 1 = θ n + θ ˙ n Δ z + ½ θ ¨ n ( Δ z ) 2 + θ n ( Δ z ) 3 + ,
r ˙ n + 1 = r ˙ n + r ¨ n Δ z + ½ r n ( Δ z ) 2 + ,
θ ˙ n + 1 = θ ˙ n + θ ¨ n Δ z + ½ θ n ( Δ z ) 2 + ,
r ¨ n = r n θ ˙ n 2 + C 1 2 [ n 2 ( r , θ ) ] r | r = r n θ = θ n ,
r n = r ˙ n θ ˙ n 2 + 2 r n θ ˙ n θ ¨ n + C 1 2 { 2 [ n 2 ( r , θ ) ] r 2 | r = r n θ = θ n r ˙ n + 2 [ n 2 ( r , θ ) ] r θ | r = r n θ = θ n θ ˙ n } , ;
θ ¨ n = C 1 2 r n 2 [ n 2 ( r , θ ) ] θ | r = r n θ = θ n - 2 r ˙ n θ ˙ n r n ,
θ n = - r ˙ n C 1 r n 3 [ n 2 ( r , θ ) ] θ | r = r n θ = θ n + C 1 2 r n 2 × { 2 [ n 2 ( r , θ ) ] θ 2 | r = r n θ = θ n θ ˙ n + 2 [ n 2 ( r , θ ) ] θ r | r = r n θ = θ n r ˙ n } - 2 r n ( r ¨ n θ ˙ n + r ˙ n θ ¨ n ) - r ˙ n 2 θ ˙ n r n 2 , .
1 + r ˙ 2 + r 2 θ ˙ 2 = C 1 n 2 ( r , θ ) ,
L = z 0 z n ( 1 + r ˙ 2 + r 2 θ ˙ 2 ) 1 / 2 d z = C 1 z 0 z n 2 d z ,
n X ¨ = ( 1 + X ˙ 2 + y ˙ 2 ) n X ,
n y ¨ = ( 1 + X ˙ 2 + y ˙ 2 ) n y .
1 + X ˙ 2 + y ˙ 2 = C 2 n 2 ( x , y ) ,
C 2 = 1 + X ˙ 0 2 + y ˙ 0 2 n 2 ( x 0 , y 0 )
X ¨ = C 2 2 ( n 2 X ) ,
y ¨ = C 2 2 ( n 2 y ) .
X ¨ n = C 2 2 [ n 2 ( x , y ) X ] | x = x n y = y n ,
X n = C 2 2 { 2 [ n 2 ( x , y ) ] X 2 | x = x n y = y n X ˙ n + 2 [ n 2 ( x , y ) ] X y | x = x n y = y n y ˙ n } , ;
y ¨ n = C 2 2 [ n 2 ( x , y ) y ] | x = x n y = y n ,
y n = C 2 2 { 2 [ n 2 ( x , y ) ] y 2 | x = x n y = y n y ˙ n + 2 [ n 2 ( x , y ) ] y X | x = x n y = y n X ˙ n } , .
X n + 1 = X n + X ˙ n Δ Z + ½ X ¨ n ( Δ Z ) 2 + X n ( Δ Z ) 3 + ,
y n + 1 = y n + y ˙ n Δ Z + ½ y ¨ n ( Δ Z ) 2 + y n ( Δ Z ) 3 + ,
X ˙ n + 1 = X ˙ n + X ¨ n Δ Z + ½ X n ( Δ Z ) 2 + ,
y ˙ n + 1 = y ˙ n + y ¨ n Δ Z + ½ y n ( Δ Z ) 2 + .
1 + X ˙ 2 + y ˙ 2 = C 2 n 2 ( x , y ) ,
L = z 0 z n ( 1 + X ˙ 2 + y ˙ 2 ) 1 / 2 d z = C 2 z 0 z n 2 d z .
n 2 ( x , y ) = N 0 2 - b 2 ( x 2 + y 2 ) ,
X 0 = 0.02 mm , X ˙ 0 = 0.0 , Y 0 = 0.0 mm , Y ˙ 0 = 0.0 ,

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