Trapped modes in particles with a negative refractive index

The natural oscillations of the electromagnetic field in a particle made from left-handed metamaterial, where both permittivity and permeability are negative, are considered. Based on the exact solution of the sourceless Maxwell equations, it is shown that due to the opposite directions of the phase and group velocities in the metamaterial, natural oscillations in such particles decay exponentially at infinity, that is, these natural oscillations can be considered as trapped modes with a finite energy. The manifestation of such modes in experiments with Bessel beams is also discussed.