Abstract

A method based on ray-optics techniques is used to evaluate the dispersion characteristics of a symmetrical-slab waveguide with an index distribution that varies from a quasi-parabolic to the quasi-step profile. The validity and accuracy of the method are checked by comparison with results that can be obtained by different methods, either for the cases of a particular graded index or for a step-index profile.

© 1975 Optical Society of America

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References

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  1. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 286.
  2. N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972).
  3. Reference 2, p. 74.
  4. H. Kogelnik, H. P. Weber, and J. Opt. Soc. Am. 64, 174 (1974).
    [Crossref]
  5. M. Ikeda, IEEE J. QE-10, 362 (1974).
    [Crossref]
  6. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice–Hall, Englewood Cliffs, N. J., 1973), Secs. 1.7-b and 5.8.
  7. IBM 1130 Scientific Subroutine Package (1130-CM-02X) Fortran Program RKGS, pp. 92–95.
  8. Reference 6, Sec. 5.9-b.
  9. D. Gloge, Appl. Opt. 10, 2252 (1971).
    [Crossref] [PubMed]
  10. These notations have been introduced for a step-index distribution, hence are quite adequate for the quasi-step profiles (large values of C). For continuity, they were used also for the quasi-parabolic case.

1974 (2)

H. Kogelnik, H. P. Weber, and J. Opt. Soc. Am. 64, 174 (1974).
[Crossref]

M. Ikeda, IEEE J. QE-10, 362 (1974).
[Crossref]

1971 (1)

Burke, J. J.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972).

Felsen, L. B.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice–Hall, Englewood Cliffs, N. J., 1973), Secs. 1.7-b and 5.8.

Gloge, D.

Ikeda, M.

M. Ikeda, IEEE J. QE-10, 362 (1974).
[Crossref]

Kapany, N. S.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972).

Kogelnik, H.

H. Kogelnik, H. P. Weber, and J. Opt. Soc. Am. 64, 174 (1974).
[Crossref]

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 286.

Marcuvitz, N.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice–Hall, Englewood Cliffs, N. J., 1973), Secs. 1.7-b and 5.8.

Opt, J.

H. Kogelnik, H. P. Weber, and J. Opt. Soc. Am. 64, 174 (1974).
[Crossref]

Weber, H. P.

H. Kogelnik, H. P. Weber, and J. Opt. Soc. Am. 64, 174 (1974).
[Crossref]

Appl. Opt. (1)

IEEE J. (1)

M. Ikeda, IEEE J. QE-10, 362 (1974).
[Crossref]

Soc. Am. (1)

H. Kogelnik, H. P. Weber, and J. Opt. Soc. Am. 64, 174 (1974).
[Crossref]

Other (7)

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice–Hall, Englewood Cliffs, N. J., 1973), Secs. 1.7-b and 5.8.

IBM 1130 Scientific Subroutine Package (1130-CM-02X) Fortran Program RKGS, pp. 92–95.

Reference 6, Sec. 5.9-b.

These notations have been introduced for a step-index distribution, hence are quite adequate for the quasi-step profiles (large values of C). For continuity, they were used also for the quasi-parabolic case.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), p. 286.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972).

Reference 2, p. 74.

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Figures (10)

FIG. 1
FIG. 1

Paths of rays trapped by continuous refraction in a slab waveguide.

FIG. 2
FIG. 2

(a) The inverse-square index profile, and (b) the corresponding ray picture.

FIG. 3
FIG. 3

Inverse-square-profile case, the normalized propagation constant β/k0 plotted versus k0 for a number of modes. The crosses correspond to values obtained from the numerically computed ray paths.

FIG. 4
FIG. 4

Inverse-square-profile case, normalized group delay plotted versus k0 in the same cases as in Fig. 3.

FIG. 5
FIG. 5

Index distributions ranging from the quasi-parabolic (C = 4) ····· to the quasi-step profile (C = 200) —; C = 40, - - -; and C = 10, -·-.

FIG. 6
FIG. 6

Normalized propagation constant b plotted versus the normalized frequency v for different modes and for four values of the parameter C that characterizes the index profile. C = 4, ·····; C = 10, -·-; C = 40, - - -; and C = 200, –.

FIG. 7
FIG. 7

Normalized group delay d(vb)/dv versus normalized frequency v in the same cases as in Fig. 6. C = 4, ····; C = 10, -·-; C = 40, - - -; and C = 200, –.

FIG. 8
FIG. 8

Three ray configurations for light propagation in a slab waveguide. (a) zig–zag ray; (b) zig–zag ray plus lateral shift; (c) curved zig–zag ray.

FIG 9
FIG 9

Normalized group delay d(vb)/dv plotted versus normalized frequency v for a number of modes. The continuous lines correspond to the approximate method (curved zig–zag model) in the quasi-step case (C = 200). The dotted lines correspond to the step case with a zig–zag model and the dashed lines again to the step case with a zig–zag plus lateral ray shift.

FIG. 10
FIG. 10

Ray-penetration depth (ys) plotted versus the initial angle α between the ray and the waveguide axis. n1 = 1.5, n2 = 1.4, 2a = 50 μm. C = 200, –; ray shift, ---.

Equations (17)

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( S ) 2 = n 2 ( r ) ,
n / n 2 k 0 1 ,             k 0 = 2 π / λ 0 .
d d s ( n d z d s ) = 0 ,             d d s ( n d y d s ) = d n d y ,
k 0 - y m y m [ n 2 ( y ) - τ m 2 ] 1 / 2 d y - π / 2 = m π ,
τ g = L c d β d k 0 = S L c ,
n 2 ( y ) = 1 - a 0 2 / ( a - y ) 2             for y > 0
n 2 ( y ) = 1 - a 0 2 / ( a + y ) 2             for y < 0.
u = y J s ( k 0 y y ) exp [ - i k 0 z z ] ,
k 0 y 2 + k 0 z 2 = k 0 2 .
u = A 2 k 0 y cos ( k 0 y y - s π / 2 - π / 4 ) exp [ - i k 0 z z ] .
2 [ k 0 y a - s π / 2 - π / 4 ] = m π ,
n ( y ) = n 2 + n 1 - n 2 1 + exp [ C ( y / a - 1 ) ] ,
v = a k 0 ( n 1 2 - n 2 2 ) 1 / 2 ,
b = β / k 0 - n 2 n 1 - n 2 .
Δ = ( n 1 - n 2 ) / n 2 1 ,
τ g = L c { d ( n k 0 ) d k 0 + n Δ d ( v b ) d v } ,
τ g = L c { n 2 + ( n 1 - n 2 ) d ( v b ) d v } .