Analyticity, causality, energy conservation and the sign of the imaginary part of the permittivity and permeability

In this paper, we have applied the concepts of Poynting Theorem and causality to investigate the sign of the imaginary part of permittivity or permeability. For a passive, symmetric and reciprocal medium, it is shown that the conservation of energy is automatically satisfied. For a nonsymmetric and/or nonreciprocal medium, we can also show that the conservation of energy still applies provided correct constitutive relations are employed for describing characteristics of EM waves propagation. The resonance model we proposed to fit the dispersion curve in the plasmon resonance region shows that the complex permittivity and permeability are analytic in the upper half plane as required by causality. Based upon the analysis we made in this paper, it can be concluded that neither Poynting Theorem nor causality requires the imaginary parts of permittivity and permeability to be positive simultaneously.