Exceptional points in composite structures consisting of two dielectric diffraction gratings with Lorentzian line shape

Using scattering matrix formalism we derive analytical expressions for the eigenmodes of a composite structure consisting of two dielectric diffraction gratings with Lorentzian profile in reflection. Analyzing these expressions we prove formation of two distinct pairs of exceptional points, provide analytical approximations for their coordinates and by rigorous simulation demonstrate eigenmodes interchange as a result of encircling said exceptional points.


Introduction
Exceptional points (EPs) are degeneracies in non-Hermitian systems, which appear when several eigenmodes coalesce [1]. The properties of such system dramatically change in the vicinity of EPs and lead to such phenomena as enhanced optical sensing [2], loss-induced transparency [3], unidirectional transmission or reflection [4], and lasers with reversed pump dependence [5] or single-mode operation [6]. One promising feature of an EP is that adiabatically encircling an EP can result in an exchange of the eigenstate. Such behavior is expected to have applications in asymmetric mode switching [7] and on-chip non-reciprocal transmission [8] and light stopping [9].
As a rule, exceptional points are studied through analyzing the eigenvalues and eigenvectors of the proper Hamiltonian [10] or by analyzing the eigenmodes dispersion relation [11]. In this work we demonstrate the formation of EPs using ω-k x resonant approximation of Lorentzian line shape. By obtaining analytical expressions for the eigenmodes of a composite structure consisting of two dielectric diffraction gratings (DGs) with Lorentzian line shape we show the formation of two distinct EPs which can be achieved by varying the distance between said stacked gratings l. This theoretical conclusion is supported by rigorous calculation results that show eigenmodes swapping as a result of encircling said EPs in the l-k x parameter space.

ω-k x Lorentzian line shape in composite structures
A scattering matrix S relates the complex amplitudes of the plane waves incident on the diffraction structure from the superstrate u I and the substrate d I regions with the amplitudes of the transmitted T and reflected R diffraction orders [12]. For a horizontally symmetrical subwavelength DG, which allows only the 0 th reflected and transmitted diffraction orders to propagate, the S matrix takes the form:  1  1  1  1  1   ,  ,  ,  ,  ,  ,  , , x T k ω are the complex reflection and transmission coefficients of the DG for a unit-amplitude incident wave. It is worth noting that the scattering matrix in (1) does not describe the near-field effects associated with the evanescent diffraction orders of the DG.
In this paper we consider the elements of the scattering matrix (1) to be functions of the angular frequency ω and the in-plane wave component x k of the incident light. Let the DG have Lorentzian line shape profile. In this case the appropriate reflection and transmission coefficients can be approximated as follows [13]: As a DG under consideration we propose a dielectric structure shown in figure 1a. The agreement between its reflection spectra calculated using rigorous coupled-wave analysis and the one calculated using the approximations (2) confirms that said DG has Lorentzian reflection profile (figure 1b). See the caption of figure 1 for the parameters of the DG as well as the parameters for the approximation.  , , where ( ) ( ) sin x env k c n ω ω θ = , θ being the angle of incidence.  (3), we obtain the scattering matrix 2 S of the composite DG with the following reflection and transmission coefficients:

Exceptional points
According to (5) the eigenmodes of the composite structure (reflection and transmission coefficients complex poles) have the following form: . Let us choose a contour in the l-k x parameter space centering at the analytically estimated EPs coordinates (figures 2b and 2d). After encircling the EPs counterclockwise following said contours one can notice that the two rigorously calculated using RCWA complex poles corresponding to the same square root in (6) swap places (figures 2c and 2e). This interchange of eigenmodes is an intrinsic feature of EPs and proves the existence of EPs. More accurate estimate of the EPs location can be acquired by solving (7) accounting for ψ being the function of ω and θ .

Conclusion
By means of scattering matrix formalism we derived analytical expressions for the eigenmodes of a composite structure consisting of two dielectric diffraction gratings with Lorentzian profile in reflection. Using said approximations we formulate a criterion for the grating eigenmodes that if satisfied allows formation on two distinct pairs of exceptional points. Rigorous calculation results show eigenmodes interchange upon encircling said EPs.