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Steven Johnson's Abstract
Extending perturbative methods to high-contrast nanophotonics
Steven G. Johnson, MIT Applied Mathematics
Perturbative methods, which take solutions for an idealized system and
exploit them to treat slightly modified structures, are of great utility
for the study of many practical problems in electromagnetism that consist
of small perturbations -- e.g., waveguide tapers, cavity tuning, nonlinear
effects, and scattering from fabrication disorder. However, we show that
several standard perturbative methods, from coupled-wave theories [1-3] to
eigenvalue perturbations [4] to Born approximations (a.k.a. volume-current
methods) [5] must be modified in order to accurately describe structures
with high index contrast and/or strong periodicity, such as photonic
crystals. We derive the corrected forms and also conclude a variety of
general theorems: from the conditions for an adiabatic theorem in periodic
structures [1], to the effect of a photonic band gap on radiative and
reflective scattering in disordered waveguides [3], to scaling laws for
group-velocity dependence of losses [1,5].
[1] S. G. Johnson et al., Phys. Rev. E 66, 066608 (2002).
[2] M. Skorobogatiy et al., Opt. Express 21, 1227 (2002).
[3] M. L. Povinelli et al., Appl. Phys. Lett. 84, 3639 (2004).
[4] S. G. Johnson et al., Phys. Rev. E. 65, 066611 (2002).
[5] S. G. Johnson et al., Appl. Phys. B, in press (summer 2005).
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