Determining the response of optical systems in both time and harmonic domains with the singularity expansion method

. Physical systems are characterized by their transfer operators in the harmonic domain. These operators are usually locally approximated as rational functions or pole expansions. We generalize this result and introduce the Multiple-Order Singularity Expansion Method (MOSEM) which offers an exact description of linear systems in terms of their singularities and Laurent series coefficients or zeros. The interest of this approach is first illustrated by the simple but fundamental case of a dispersive Fabry-Perot cavity, where it provides an analytical expression of the reflected field in both the time and harmonic domains. In a second step, we show that this method must be applied for defining the complex expression of the dielectric permittivity that describes the physical response of a system (the material) to an excitation field. This rigorous expression of the permittivity is shown to provide highly accurate results for a broad range of materials.


Introduction
The description of linear optical systems can be achieved by providing a local approximation of their transfer function in the harmonic domain [1,2].These local expressions can be generalized into singularity expansion method (SEM), which is rigorously derived to offer an exact description of physical systems in terms of all their singularities (along with their order) and their associated Laurent series coefficients [3,4].Poles of order 1 are usually sufficient.In addition, the distribution of the poles and Laurent series coefficients are constrained by the Hermitian symmetry and causality principle by which all physical systems must abide, yielding: In these expressions,   is the residue of  associated with the pole   ,   is the constant, non-resonant term and   () is the resonant term expressed as a sum of generalized Lorentz function.Let us stress that the poles and residues come in pairs: (  , −  ̅̅̅̅), (  , −  ̅ ), and that their imaginary parts (  ) are negative.Eqs.(1-3) must numerically be truncated but converges fast with only a few dozens of poles.

Case of a dispersive Fabry-Perot cavity
We consider the reflection coefficient () of a gold layer, i.e. a dispersive slab, which has a known analytical expression obtained via the Airy model or the thin-film formalism [5].As depicted in Fig. (2), this exact expression can be replaced by the truncated MOSEM with little error by considering 78 pairs of poles.Among these pairs of poles, more than 50 are background poles which can be merged into one virtual pair of poles while keeping an accurate expression as shown in Fig. (3).Finally, we compare the analytical temporal expression obtained by inverse Laplace transform of the truncated MOSEM to the result of a CST simulation.Results displayed in Fig. (4) show that the singularity expansion can be used to highlight the temporal dynamics with a good accuracy.Fig. 1.Reflected field at a distance  = −1µ from the left interface of a slab of gold, using MOSEM with 20 pairs of poles and a virtual pole compared with a CST simulation, for a sinusoidal incident field at normal incidence.

Analytical model of the permittivity
The derivation of rigorous and analytical expressions of the permittivity of dispersive media over wide spectral windows is an important challenge for numerical methods [6][7][8].4) compared to experimental data over a spectral window ranging from 100nm to 1600nm.
Our approach to address this important question is to show that, as the dielectric permittivity is a transfer function, this problem must be solved in the framework The singularity expansion can be written in a form that This generalized Drude-Lorentz form can then be fitted to match experimental data over the real frequency axis.We do this for the permittivity of gold () [9] and compare the results to experimental data between 100 and 1600nm in Fig. (5).

Conclusion
The singularity expansion method offers a description of optical systems through accurate analytical expressions which involve a small, discrete set of poles of the system.The poles are physically constrained and can be linked to the shape of the responses or used to retrieve the temporal dynamics.In addition, we show, with the case of the permittivity, that it is a natural description of transfer functions that includes and generalizes existing models.The studied cases involved 1D functions, but the method can be extended to multi-port operators such as the Smatrix, or anisotropic permittivity and permeability.

Fig. 1 .
Fig. 1.Test of convergence of the singularity expansion method in the case of a metallic and dispersive, compared to the thinfilm formalism (dotted line).

Fig. 1 .
Fig. 1.Real and imaginary part of the permittivity of gold obtained with the fitted Drude-Lorentz form of MOSEM from Eq. (4) compared to experimental data over a spectral window ranging from 100nm to 1600nm.