When Reflection and Transmission Amplitude Coefficients Exceed Unity: Evanescent Waves, Resonances, and Optical Modes

The reflection and transmission of propagating harmonic waves in linear optical systems have been widely discussed in the literature and are generally well understood. In passive systems involving only propagating waves, energy conservation constrains measurable quantities such as reflectance and transmittance, as well as the corresponding amplitude coefficients. However, when evanescent or damped waves are present, the reflection and transmission coefficients of the field amplitude may exhibit atypical behaviour, with the real or imaginary parts being much larger than one or even unbounded. In this study, we analyse this phenomenon in several simple optical configurations, including dielectric-metal interfaces, surface-plasmon-resonance prism couplers, plasmonic microcavities, step-index slab waveguides and prism-coupled waveguides. We demonstrate that these large amplitude coefficients naturally arise from extending reflection and transmission coefficients to evanescent or damped waves. On the other hand, they are not directly associated with measurable reflectance or transmittance and, therefore, do not conflict with energy conservation. Instead, they provide a useful formal description of resonant field enhancement, modal excitation and coupling processes in plasmonic and dielectric optical systems.