Year: 2024
Type: article
Abstract: The Mathieu equation occurs naturally in the description of vibrations or in the propagation of waves in media with a time-periodic refractive index. It is known to lead to exponential parametric instability in some regions of the parameter space. However, even in the stable region the matrix that propagates the initial conditions forward in time is non-normal, and therefore, it can result in transient amplification. By optimizing over initial conditions as well as initial time we show that significant transient amplifications can be obtained, going beyond the one simply stemming from adiabatic invariance. Moreover, we explore the monodromy matrix in more depth by studying its $\ensuremath{\epsilon}$ pseudospectra and Petermann factors, demonstrating that is the degree of non-normality of this matrix that determines the global amplifying features. In the context of wave propagation in time-varying media, this transient behavior allows us to display arbitrary amplification of the wave amplitude that is not due to exponential parametric instability.
Language: English
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
Funder: Athens County Foundation
OpenAlex ID: W4403640607