Abstract

We demonstrate the breakdown of the usual WKB expressions for tunneling ray power transmission coefficients on multimode optical wavequides with power law refractive index dependence, when the profile approaches the step index limit. A qualitative analysis provides a solution valid in this transition region, which also quantifies the domain of applicability of the WKB analysis. The latter is much more restricted than the usual condition on the slope of the profile predicts, and is due to the abrupt change in slope at the core-cladding interface.

© 1978 Optical Society of America

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References

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  1. A. Ankiewicz and C. Pask, “On the geometric optics description of light acceptance and propagation in graded index fibers,” Opt. Quantum Electron. 9-2, 87–109 (1977).
    [CrossRef]
  2. A. W. Snyder and J. D Love, “Attenuation coefficient for tunnelling rays in graded fibres,” Electron. Lett. 12, 324–326 (1976).
    [CrossRef]
  3. D. Marcuse, “Light Transmission Optics,” (Van Nostrand, New York, 1972).
  4. J. D. Love and C. Winkler, “Attenuation and tunneling coefficients for leaky rays in multilayered optical waveguides,” J. Opt. Soc. Am. 67, 1627–1633 (1977).
    [CrossRef]
  5. L. Brekhovskikh, Waves in Layered Media (Academic, New York, 1961).

1977 (2)

A. Ankiewicz and C. Pask, “On the geometric optics description of light acceptance and propagation in graded index fibers,” Opt. Quantum Electron. 9-2, 87–109 (1977).
[CrossRef]

J. D. Love and C. Winkler, “Attenuation and tunneling coefficients for leaky rays in multilayered optical waveguides,” J. Opt. Soc. Am. 67, 1627–1633 (1977).
[CrossRef]

1976 (1)

A. W. Snyder and J. D Love, “Attenuation coefficient for tunnelling rays in graded fibres,” Electron. Lett. 12, 324–326 (1976).
[CrossRef]

Ankiewicz, A.

A. Ankiewicz and C. Pask, “On the geometric optics description of light acceptance and propagation in graded index fibers,” Opt. Quantum Electron. 9-2, 87–109 (1977).
[CrossRef]

Brekhovskikh, L.

L. Brekhovskikh, Waves in Layered Media (Academic, New York, 1961).

Love, J. D

A. W. Snyder and J. D Love, “Attenuation coefficient for tunnelling rays in graded fibres,” Electron. Lett. 12, 324–326 (1976).
[CrossRef]

Love, J. D.

Marcuse, D.

D. Marcuse, “Light Transmission Optics,” (Van Nostrand, New York, 1972).

Pask, C.

A. Ankiewicz and C. Pask, “On the geometric optics description of light acceptance and propagation in graded index fibers,” Opt. Quantum Electron. 9-2, 87–109 (1977).
[CrossRef]

Snyder, A. W.

A. W. Snyder and J. D Love, “Attenuation coefficient for tunnelling rays in graded fibres,” Electron. Lett. 12, 324–326 (1976).
[CrossRef]

Winkler, C.

Electron. Lett. (1)

A. W. Snyder and J. D Love, “Attenuation coefficient for tunnelling rays in graded fibres,” Electron. Lett. 12, 324–326 (1976).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Quantum Electron. (1)

A. Ankiewicz and C. Pask, “On the geometric optics description of light acceptance and propagation in graded index fibers,” Opt. Quantum Electron. 9-2, 87–109 (1977).
[CrossRef]

Other (2)

D. Marcuse, “Light Transmission Optics,” (Van Nostrand, New York, 1972).

L. Brekhovskikh, Waves in Layered Media (Academic, New York, 1961).

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Figures (5)

FIG. 1
FIG. 1

The power law refractive index profiles of Eq. (1) for various values of q. In the limit q → ∞ the continuous profile goes over to the step index profile with a discontinuity at y = d.

FIG. 2
FIG. 2

Plot of the position of ytp in microns for the power law profiles of Eq. (1) as q increases. ytp approaches d(=50 μ) as q → ∞.

FIG. 3
FIG. 3

The linearized profile used in the analysis corresponds to ABDE, where BD is tangent to the power law profile at y = d. At ytp the tunneling ray is reflected and partially transmitted to reappear in the cladding at y = yrad.

FIG. 4
FIG. 4

Plots of the tunneling coefficients against q. The stop solution corresponds to q = ∞ for the curved profile ADE of Fig. 2, the “exact” solution to BDE of Fig. 2, and the WKB solution to the curved profile ADE. Parameter values are: λ = 1 μm, d = 50 μm, n(0) = 1.50842, n(d) = 1.5, Δ = 5.57 × 10−3, k β ˜ = 9.43 × 106.

FIG. 5
FIG. 5

Variation of the ratio of transmission coefficients of Eq. (13) with q. For q ≲ 20 the Tgrad solution is too small and for q > 20 becomes too large.

Equations (19)

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n 2 ( y ) = { n 2 ( 0 ) [ 1 - 2 Δ ( y / d ) q ] for 0 y d , n 2 ( 0 ) [ 1 - 2 Δ ] for y d ,
n 2 ( y ) = { n 2 ( 0 ) for 0 y < d , n 2 ( 0 ) [ 1 - 2 Δ ] for y > d .
β ˜ n ( y ) cos θ ( y ) ,             l ˜ ( y / d ) n ( y ) sin θ ( y ) cos ϕ ( y ) ,
T grad = exp ( - 2 y tp y rad k y ( ξ ) d ξ ) ,
k y ( y ) 2 = k 2 [ n 2 ( y ) - β ˜ 2 - ( l ˜ d / y ) 2 ] ,
( λ / n 2 ) ( d n / d y ) 1.
q max = ( d n / λ ) ( 1 - 2 Δ ) / Δ .
T step = T F exp ( - 2 d y rad k y ( ξ ) d ξ ) ,
n 2 ( y ) = { n 2 ( 0 ) for 0 y y B , n 2 ( 0 ) [ 1 + 2 Δ ( q - 1 ) - 2 Δ q y / d ] for y B y < d ,
n 2 ( y ) = n 2 ( 0 ) [ ( 1 - 2 Δ ) + 2 Δ ( y - d ) / d ]             for y > d .
k y ( y ) 2 = k 2 [ n 2 ( y ) - β ˜ 2 ] ,
d 2 ψ d y 2 + k 2 [ n 2 ( y ) - β ˜ 2 ] ψ = 0 ,
T exact = - 4 K 1 K 2 / [ π 3 ( X 2 + Y 2 ) ] ,
X = K 1 A i ( s ) [ B i ( v ) A i ( u ) - B i ( u ) A i ( v ) ] + K 2 A i ( s ) [ A i ( v ) B i ( u ) - A i ( u ) B i ( v ) ] + B i ( s ) [ A i ( u ) B i ( v ) - A i ( v ) B i ( u ) ] + K 2 B i ( s ) [ B i ( u ) A i ( v ) - A i ( u ) B i ( v ) ] ,
Y = K 1 B i ( s ) [ A i ( v ) B i ( u ) - B i ( v ) A i ( u ) ] + K 1 K 2 B i ( s ) [ A i ( u ) B i ( v ) - B i ( u ) A i ( v ) ] + A i ( s ) [ A i ( u ) B i ( v ) - A i ( v ) B i ( u ) ] + K 2 A i ( s ) [ B i ( u ) A i ( v ) - A i ( u ) B i ( v ) ] ,
K 1 = ( α k 2 ) 1 / 3 / k y ( d ) ,             K 2 = - ( α / α ¯ ) 1 / 3 , u = ( y B - y tp ) ( α k 2 ) 1 / 3 ,             v = ( d - y tp ) ( α k 2 ) 1 / 3 , s = ( y rad - d ) ( α ¯ k 2 ) 1 / 3 ,             α = 2 Δ n ( 0 ) 2 q / d , α ¯ = 2 Δ n ( 0 ) 2 / d ,
lim q T exact = T step .
R = π Z 4 v 1 / 2 exp ( - y tp d k y ( ξ ) d ξ ) ,
Z = [ B i ( v ) + v 1 / 2 B i ( v ) ] 2 + [ A i ( v ) + v 1 / 2 A i ( v ) ] 2 ,