Resonant State Spectral expansions including bound-state contributions

. Extensions of the Resonant State or (Quasi-normal mode) formalism are shown to permit uniﬁed descriptions of the true ‘bound’ states within the framework of the ‘leaky’ mode formulation of basis sets. We review recent progress in the famed ‘normalization / regularization’ problems of the QNMs modes as well as the necessity of including ‘background’ contributions whose expressions can be determined through response function sum-rules.


Introduction
Resonant states, also known as Quasi-Normal Modes, play a prominent role in the modeling photonic systems and devices due to their ability to compactly parameterize and characterize open systems that radiate energy into their environment.We review and extend recent progress on the use of resonant states methods to derive the spectral expansions of the S -matrix response functions [1], and the means by which this methodology can determine both the RS 'normalization' and the necessary inclusion of 'background' terms in RS spectral expansions.We find that 'true' guided modes and the leaky RSs can be described in a unified framework.

Spectral S-matrix formulation of 1D scattering
We carry out this discussion in a concrete setting by considering the S -matrix of a solvable 1D scatterer at arbitrary incidence angles as shown in the schematic of fig. 1 An advantage of working on this system is that its Smatrix is expressed in terms of the coefficients, C (±) L,R , of a two component partial wave basis , of a 2 × 2 : where, r p and t p are respectively the reflection and transmission coefficients, with the index p designating the polarization states (TE,TM).Modern analytic expressions for r p and t p are still usually based on electromagnetic Fresnel coefficients and are generalizations of the famous Airy functions that he derived not long after Fresnel first derived his famous coefficients two centuries ago, but such expressions have to be reevaluated at each and every frequency and are thereby poorly adapted to extracting the spectral expansions.We found it expedient to directly construct spectral expansions by combining S -matrix symmetry properties with an RS analysis based entirely on electromagnetic eigenvalue information to obtain: where for brevity q(p, e, ) is a combined multipole (p, j)-RS index( ), and κ (+) q , R p , and S (bkg) p , are respectively RS eigenvalues, residues, and background contributions, all of which are in general frequency independent functions of the multipole numbers, (p, j), RS indices ( ), the scatterer constitutive parameters (ε s , µ s ), together with the incident field parameter, α ≡ sin θ i , β ≡ √ 1 − α 2 = cos θ i .The spectral formulas of eq. ( 2) precisely reproduce the same numerical results as the original Airy formulas, but their spectral form provides access to additional informa-, 04023 (2023) tion that allow time domain analysis and modal descriptions based on the RS wave functions, including Q-factors and complex effective mode volume, V eff .
The results of an off-normal incidence calculation for scattering by a slab using eq.( 2), are shown in fig. 2. The positions of κ p, j, are plotted as blue(even) and red(odd) dots with the y axis being treated as the imaginary axis of a complex plane, which shows that they contain physically meaningful information since their real parts indicate the frequencies of maximum transmission while their imaginary determine the Q-factors of the transmission resonances.Perhaps even more important is the fact that the exact predictions of the reflection and transmission coefficients in fig. 2

Normalization for both 'leaky' and 'bound' states
Since energy in resonant states is lost to the outside world, they have sometimes been referred to as 'leaky' modes, and their correct 'normalization' has been the subject of much debate.One can show however that associating normalization to the response functions poles, R p , amounts to assigning appropriate values of the generalized 'inner product', Ψ (+) . For a scattering system without medium losses, this inner product amounts to integrating a complex 'Lagrangian energy density', over all x.
Even though the electric and magnetic fields of RSs both diverge in the far-field, their contributions to Lg q (x) exactly cancel outside the slab [3], rendering the normalization integral trivial, as illustrated by plotting its real and imaginary contributions in fig. 3.As one can see in this example, Re Lg q (x) is null everywhere for the lossless medium resonant states, while Im Lg q (x) is non-zero (and uniform) only inside the slab.
Equation (2) remains valid even for evanescent illumination, i.e. α > 1, but for such incident field parameters, the RS eigenvalues now all lie on the real axis (Im{κ q } = 0), and the associated complex valued RS eigen-functions then describe 'bound' guided modes.It is conventional to normalize such 'true' modes using the Poynting-vector associated with their electromagnetic fields, but they fit equally well into the same schema as the 'Leaky' RS modes as we can see by again plotting Lg q (x) in fig. 4 for these 'bound' guided modes, where the energy contributions are now real-valued everywhere, characterized by evanescent in the regions outside the 'scatterer'.

Conclusion
A unified framework for describing both 'Leaky' and 'true' propagative modes allows ready access to a large body of physically pertinent information for both understanding and designing photonic systems.

Figure 1 .
Figure 1.Schematic of an S -matrix representation for a mirror symmetric 1D infinite 'slab' with constitutive parameters, ε s , µ s immersed in a homogeneous background medium with incoming partial waves, Φ (−) L,R impinging on the surface at an incident angle θ i from the left(L) or right(R), which are 'transformed' into outgoing waves, Φ (+) L,R , by scattering.
was only possible due to the determination of residue factors, R p , and background contributions, S (bkg) p .

Figure 2 .
Figure 2. Reflection and transmission coefficients of a high index (ε s = 9) 1D slab at off-normal incidence on a lossless scatterer superposed on complex plane positions of RS eigenvalues, κ q .