Scattering Resonances
The increasing use of micro- and nano-scale components in optical, electrical and mechanical systems makes the understanding of loss mechanisms and their quantification issues of fundamental importance. In many situations performance-limiting loss is due to scattering and radiation of waves into the surrounding structure.
There is great current interest in the design of micro- and nano-structures in dielectric materials for
storage, channeling, amplification, compression, filtering or, in general, of light pulses. Such structures have a broad range of applications from optical communication technologies to quantum information science. Energy loss is a performance-limiting concern in the design of micro- and nano-scale components. Thus the question of how to design such components with very low radiative loss is a fundamental question.
Periodic structures are important classes of structures. In practice, these photonic crystals are structures with piecewise constant material properties. The ability of these structures to influence light propagation is achieved through variation of the period, the choice of material contrasts and through the introduction of defects.
An interesting problem is to study the cavity loss from a photonic crystal (PC) with defects. By cavity loss we mean scattering loss, that is loss due to leakage of energy from the cavity. This is in contrast to loss due to processes such as material absorption. Scattering loss (= energy leakage) from the cavity is governed by the scattering resonances associated with the cavity. Scattering resonances are solutions to the eigenvalue equation satisfied by time-harmonic solutions of the wave-equation subject to outgoing radiation conditions, imposed outside the cavity.
The scattering resonance problem is a non-selfadjoint boundary value problem having a sequence of complex eigenvalues with negative imaginary part, and corresponding resonance modes. The modes are locally square integrable but not square integrable over all space. The rate at which energy escapes from the cavity, for example measured by the rate of decay of field energy within the cavity is controlled by the resonance, k*, with largest imaginary part. The time it takes for the energy, associated with a general initial condition, localized in the defect, to radiate away is
t* = 1 / (c | Im k* |).
In practice, for example in experiments, initial conditions can be quite spectrally concentrated, and therefore the observed time-decay rate is determined by the imaginary parts of resonances whose real parts lie near the spectral support of the initial condition.
Resonances appear as peaks in the transmission coefficient as a function of wavelength, in wavelength ranges where the transmission is typically low. The real part of a scattering resonance energy corresponds to the location of the transmission peak and the imaginary part, to the transmission peak width. Thus, optimization corresponds to the sharpening of a transmission peak, by modification of the cavity.
Waves in periodic structures, e.g. electromagnetic, acoustic or elastic are governed by a wave equation with periodic coefficients. These equations have plane wave type states, parametrized by a continuous spectrum equal to the union of closed intervals called bands (photonic pass-bands). The complement of the spectrum (on the real axis) consists of the union of open intervals called gaps (photonic band-gaps). Arbitrary spatially localized states can be represented as a generalized Fourier superposition of such states. Furthermore, solutions to the initial value problem for the time-dependent wave equation in a periodic structure, disperse to zero with advancing time .
A localized defect in a periodic structure gives rise to discrete eigenvalues in the gaps. Thus, a periodic structure with a spatially localized defect may support localized time-periodic (non-decaying) states. In applications, photonic structures are often truncations of periodic structures with defects, where outside a compact set these structures have constant physical parameters.
My research interest is to study optimization methods to the scattering resonance problem in the class of piecewise constant structures, in order to systematically decrease a cavity's loss in a particular frequency range.