A detailed procedure based on an analytical transfer matrix method is presented to solve bound-state problems. The derivation is strict and complete. The energy eigenvalues for an arbitrary one-dimensional potential can be obtained by the method. The anharmonic oscillator potential and the rational potential are two important examples. Checked by numerical techniques, the results for the two potentials by the present method are proven to be exact and reliable.
This paper investigates the photon tunneling and transmittance resonance through a multi-layer structure including a left-handed material（LHM）. An analytical expression for the transmittance in a five-layer structure is given by the analytical transfer matrix method. The transmittance is studied as a function of the refractive index and the width of the LHM layer. The perfect photon tunneling results from the multi-layer structure, especially from the relation between the magnitude of the refractive index and the width of the LHM layer and those of the adjoining layers. Photons may tunnel through a much greater distance in this structure. Transmittance resonance happens, the peaks and valleys appear periodically at the resonance thickness. For an LHM with inherent losses, the perfect transmittance is suppressed.
