Cherenkov radiation in a planarly layered waveguide in the case of polarized waves

Abstract

In the present work, we consider an isotropic planarly layered waveguide in the case of polarized waves, and calculate the Green’s function associated with a motionless unit point source in the stationary state. On the basis of the Green’s function, we approach the dynamical problem of a point source that is in uniform motion in the waveguide. The mathematical description of the transverse electric and transverse magnetic waves generated by the moving point source is given in the form of double oscillatory integrals of the time and the frequency. Then these integrals are written in terms of a large parameter $$\lambda >0$$ in order to apply the stationary phase method. This gives asymptotic formulas for the electromagnetic field as $$\lambda \rightarrow \infty $$ . The calculation of the stationary points of the phase leads us to the condition that guarantees the existence of Cherenkov radiation in the waveguide. Finally, the analysis here presented is applied to some numerical examples, which are worked with an algorithm based on the spectral parameter power series method.