Generalized laws of reflection and refraction from real valued boundary conditions

The generalized Snell's laws are usually derived from phase-matching condition by using harmonic inhomogeneous plane waves. The inhomogeneity makes it difficult to trace curves of energy flow. Here we show that, at a lossy interface, real valued boundary conditions are valid universally. Thus a time-dependent way is developed to directly derive generalized laws of reflection and refraction from real valued boundary conditions by using harmonic homogeneous plane waves. Finally, several novel properties of transmitted wave associated with energy losses are predicted numerically, which may be applied to experimentally test our theoretical analysis.