Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-12-01T09:02:47.816Z Has data issue: false hasContentIssue false
  • English
  • Français

Matched asymptotics for a generalisation of a model equation for optical tunnelling

Published online by Cambridge University Press:  16 July 2009

Jinsong Liu  and
Alastair Wood
Jinsong Liu
Affiliation:
School of Mathematical Sciences, Dublin city University, Dublin 9, Ireland
Alastair Wood
Affiliation:
School of Mathematical Sciences, Dublin city University, Dublin 9, Ireland
Get access Rights & Permissions [Opens in a new window]

Abstract

We apply the method of matched asymptotic expansions to link the outgoing wave solution at infinity of the differential equation y″(x)+(lgr;+єxn)y(x) = 0, x∈(0, ∞),λ∈Cє>0, n∈N across the turning point x = (—λ/є)1/n nearest to the positive real axis to a linear homogeneous boundary condition at the origin. The equation with n = 1 models the leakage of energy from the core of a bent fibre optic waveguide, the rate of leakage corresponding to Im λ, which was shown to be exponentially small like O(exp[— 1/є]) by Paris & Wood (1989). The extension n = 2 by Brazel et al. (1990) obtained Im λ = O(exp[— 1/є1/2]). Both these papers involve delicate analysis of the asymptotics of special functions near to Stokes' lines. When n > 2 no special functions are available, and completely different methods must be employed to obtain the result Im λ = O(exp[— 1/є1/n]).

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 2 , Issue 3 , September 1991 , pp. 223 - 231
Copyright
Copyright © Cambridge University Press 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. & Stegun, I. (eds.) 1965 Handbook of mathematical functions. Dover.Google Scholar
Bender, C. M. & Orszag, S. A. 1978 Advanced mathematical methods for Scientist and Engineers. McGraw-Hill.Google Scholar
Berry, M. V. 1989 Uniform asymptotic smoothing of Stokes discontinuities. Proc. Roy. Soc. London A422, 7–21.Google Scholar
Brazel, N., Lawless, F. R. & Wood, A. D. 1990 Exponential asymptotics for the eigenvalues of a problem involving parabolic cylinder functions. (To appear in Proc. Am. Math. Soc.)Google Scholar
Kath, W. L. & Kriegsmann, G. A. 1989 Optical tunnelling: radiation losses in bent fibre optic waveguides. IMA J. Applied Math. 41, 85–103.CrossRefGoogle Scholar
Myer, R. E. 1980 Exponential asymptotics. SIAM Rev. 22, 213–244.CrossRefGoogle Scholar
Olver, F. W. J. 1990 On Stokes phenomenon and converging factors. In Asymptotic and Computational Analysis (ed. Wong, R.). Marcel Dekker.Google Scholar
Paris, R. B. & Wood, A. D. 1989 A model equation for optical tunnelling. IMA J. Applied Math. 43, 273–284.CrossRefGoogle Scholar
Segur, H. & Kruskal, M. D. 1987 Asymptotics beyond all orders in a model of dendrites. Phys. Rev. Lett. 58, 747–750.CrossRefGoogle Scholar