Abstract

We show analytically that, at a critical point where the effective refractive index of a mode coincides with the bulk index of a substrate region, for certain graded-index planar waveguides there exists a family of bound modes with algebraic (power-law) tails in the evanescent field.

© 1995 Optical Society of America

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References

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  1. H. Kogelnik, in Integrated Optics, T. Tamir, ed. (Springer-Verlag, Berlin, 1982), pp. 13–81.
  2. M. Suzuki, R. Kubo, J. Phys. Soc. Jpn. 24, 51 (1968).
    [CrossRef]
  3. P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987); Phys. Rev. A 38, 364 (1988).
    [CrossRef] [PubMed]

1987 (1)

P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987); Phys. Rev. A 38, 364 (1988).
[CrossRef] [PubMed]

1968 (1)

M. Suzuki, R. Kubo, J. Phys. Soc. Jpn. 24, 51 (1968).
[CrossRef]

Bak, P.

P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987); Phys. Rev. A 38, 364 (1988).
[CrossRef] [PubMed]

Kogelnik, H.

H. Kogelnik, in Integrated Optics, T. Tamir, ed. (Springer-Verlag, Berlin, 1982), pp. 13–81.

Kubo, R.

M. Suzuki, R. Kubo, J. Phys. Soc. Jpn. 24, 51 (1968).
[CrossRef]

Suzuki, M.

M. Suzuki, R. Kubo, J. Phys. Soc. Jpn. 24, 51 (1968).
[CrossRef]

Tang, C.

P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987); Phys. Rev. A 38, 364 (1988).
[CrossRef] [PubMed]

Wiesenfeld, K.

P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987); Phys. Rev. A 38, 364 (1988).
[CrossRef] [PubMed]

J. Phys. Soc. Jpn. (1)

M. Suzuki, R. Kubo, J. Phys. Soc. Jpn. 24, 51 (1968).
[CrossRef]

Phys. Rev. Lett. (1)

P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987); Phys. Rev. A 38, 364 (1988).
[CrossRef] [PubMed]

Other (1)

H. Kogelnik, in Integrated Optics, T. Tamir, ed. (Springer-Verlag, Berlin, 1982), pp. 13–81.

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Figures (1)

Fig. 1
Fig. 1

Plots of (a) the refractive-index [Eq. (15a)] and (b) the modal-field [Eq. (16) with f0 = 1] distributions along the normalized confinement axis X ≡ (Δn)1/2x (m = 1).

Equations (19)

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f x x + [ n 2 ( x ) - n eff 2 ] f = 0 ,
n 2 ( x ) = n s 2 + 2 Δ n g ( x ) ,
f x x + 2 Δ n g ( x ) f = 0.
f ( x ) = f 0 ( α x 2 + 1 ) - 1 / m
g ( x ) = ( Δ n ) - 1 ( α / m ) [ - ( 2 / m + 1 ) ( α x 2 + 1 ) - 1 + 2 ( 1 / m + 1 ) ( α x 2 + 1 ) - 2 ] .
g ( 0 ) = α / ( m Δ n ) .
x c = [ m ( m + 2 ) - 1 α - 1 ] 1 / 2 .
g x ( x ) = ( Δ n ) - 1 ( 2 α 2 x / m ) ( α x 2 + 1 ) - 3 × [ ( 2 / m + 1 ) α x 2 - ( 2 / m + 3 ) ] .
x d = [ ( 3 m + 2 ) ( m + 2 ) - 1 α - 1 ] 1 / 2 .
α = m Δ n .
x c = [ ( m + 2 ) Δ n ] - 1 / 2 ,
x d = [ ( 3 m + 2 ) / m ] - 1 / 2 x c .
g ( ± x d ) = - ( m + 2 ) 2 / [ 8 m ( m + 1 ) ] .
f ( x ) ~ C x - 2 / m             as x ,
g ( x ) = - 3 ( Δ n x 2 + 1 ) - 1 + 4 ( Δ n x 2 + 1 ) - 2 ,
x c = ( 3 Δ n ) - 1 / 2 ,
x d = 5 1 / 2 x c ,
g ( ± x d ) = - 9 / 16.
f ( x ) = f 0 ( Δ n x 2 + 1 ) - 1 .

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