Predicting the linear optical response of materials and nanostructures thanks to their complex singularities

Into the linear optical regime, optical nanostructures and materials behave as linear transfer functions. In this work, we are interested in the full understanding of the spectral and time response in terms of the singularities of the transfer function in the complex frequency plane. The singular expansion method demonstrates that the linear transfer function can be fully and exactly retrieved thanks to the complex singularities, both in the harmonic and time domains. This method applies to optical materials and permits to get accurate analytic expressions of the dielectric permittivity, for which analytical expressions must comply with the properties of complex analysis. We compare the two approaches and show that the Debye Drude Lorentz method agrees with the singularity expansion method if the Lorentz term contains an imaginary term in its numerator. We carefully study the accuracy of this novel expression of the dielectric permittivity abiding the mathematical properties of the SEM and show its excellent performances for a wide range of materials.