J. Eur. Opt. Society-Rapid Publ. 21, 5( 2025) 39
coupling between the resonator modes and the ports are fitted from experimental or numerical data. There are exceptions for certain geometries. For instance, for microring filters, the coupling issue may accurately reduce to calculating the interaction between guided modes in bent waveguides with a Hermitian theoretical framework [ 11 ].
In this work, we concentrate on the first equation and its theoretical derivation. In a recent study [ 10 ], by trying to establish a rigorous foundation for the CMT using electromagnetic QNM theory, an alternative evolution equation, distinct from the one proposed in the CMT, was derived directly from Maxwell equations. This new equation, termed“ exact” Maxwell Evolution( EME) equation, shares some similarities with the classical CMT evolution equation but also shows notable differences. A key distinction is in the nature of the driving term: in the EME equation, the excitation is proportional to the temporal derivative of the incident field, whereas in the CMT evolution equation, it is proportional to the incident field itself. This unexpected result was validated through a numerical test, which remains unchallenged in this study. Additionally, it was supported by a mathematical demonstration.
Here, we highlight that the demonstration lacks from mathematical rigor and propose a new demonstration that is both direct and rigorous. Importantly, the new demonstration also helps clarifying the key difference between a driving force proportional to the incident field and one proportional to its temporal derivative.
The manuscript is organized as follows. Section 2 helps the reader in contextualizing the topic by reintroducing the origin of the double integral necessary for deriving the EME equation. Section 3 highlights several aspects of the demonstration in Section 2 of [ 10 ] which lack mathematical rigor. Section 4 provides a direct and explicit demonstration of the EME equation, which is further supported in Appendix A by a more general demonstration with minimum assumptions on the incident field. Section 5 clarifies the approximation carried out by assuming a driving force directly proportional to the incident field. Section 6 concludes the work.
2 Background
We start by considering recent results obtained for QNM-expansions in the spectral domain for harmonic fields. Electromagnetic QNMs, labeled by the integer m, are source-free solutions of Maxwell equations, r E~ m ¼ �i x~ m l 0 H~ m, r H~ m ¼ i x~ m e ðx~ Þ E~ m, which satisfy the outgoing-wave condition for | r |? 1 [ 3, 4 ]. Here, E~ m and H~ m represent the QNM electric and magnetic fields, respectively, e denotes the possibly dispersive permittivity tensors. The QNM fields decay exponentially over time, so that Im ðx~ m Þ < 0 in the exp( �ixt) notation. From this point forward, we assume the QNMs are normalized, which is a crucial step in any QNM theory. There are several methods available for normalizing electromagnetic QNMs, all of which are thoroughly discussed in [ 12 ]. In this work, we utilize the so-called PML normalization approach, where PML stands for Perfectly Matched Layer.
In the scattered-field formulation, the total field at frequency x is decomposed as a sum of the background field
[ E b( r, x), H b( r, x)] exp( �ixt) and the field [ E s( r, x), H s( r, x)] exp( �ixt) scattered by the nanoresonator. The scattered field can be expanded in an infinite set of normalized QNMs and PML modes [ 12, 13 ]
½ E s ðr; xÞ; H s ðr; xÞŠ ¼ X a mðxÞ E~ m ðrÞ; H~ m ðrÞ m; ð1Þ
where a m( x) denotes the excitation coefficient of the mth modes Z. Note that the normalized modes satisfy
E~ @ xe m E~ m � H~ @ xl m 0
H~ m d 3 r ¼ 1 [ 12 ]. @ x
@ x The scattered and normalized mode fields thus have different units.
The QNM framework based on a complex-mapping regularization provides a unique expression for the excitation coefficient in equation( 1) [ 13 ]
Z x a m ðxÞ ¼ eðrÞE b ðr; xÞ E~ m d 3 r; ð2Þ x~ m � x V r
for the case of nondispersive materials under consideration here. An equivalent EME equation for dispersive systems with Drude-Lorentz permittivities is derived in [ 13 ]. In this case, the driving force consists of two components: one proportional to the temporal derivative of the excitation field, and the other to the excitation field itself. Numerical tests conducted on a silver bowtie antenna, as shown in Figure 2b of [ 13 ], strongly validate the robustness of the analysis. However, it would be valuable to expand the study further, for example, by conducting additional numerical tests for different geometries. It would be also valuable to generalize the theory to materials with arbitrary dispersion.
Equation( 2) corresponds to equation( 3) in [ 10 ]. a m( x) essentially represents an overlap integral between the QNM and the background field. De specifies the permittivity variation used for the scattered-field formulation, given by De( r) = e R( r) �e b( r), where e R( r) and e b( r) denote the permittivities of the resonator and the background, respectively. In general, De( r) is non-zero within a compact volume V R( r) thatdefines the resonator in the scatteredfield formulation.
We now consider that the background fieldisanoptical pulse, E b( r, t), i. e. a wave packet that can be Fourier transformed E b ðr; xÞ ¼ ð2pÞ R �1 1 E �1 bðr; tÞ exp ðixtÞdt. Driven by the incident pulse, the resonator scatters a time-dependent electric field, E s( r, t). Every infinitesimal frequency component of the background field E b( r, x) dx gives rise
Pto an infinitesimal harmonic scattered field dE s ðr; xÞ ¼ m a mðxÞ E~ m ðrÞdx, and the scattered field in the time domain is obtained by summing � up all the frequency components, E S ðr; tÞ ¼ Re R 1 dE �1 sðr; xÞ exp ð�ixtÞÞ. The latter is conveniently expressed with a QNM expansion [ 3, 13 ]
ðr; tÞ ¼ Re X b m mðÞ t E~ m ðrÞ
; ð3Þ with
b m ðÞ¼ t
Z 1
�1
E S
Z x exp ð�ixtÞ eðrÞE b ðr; xÞ E~ m d 3 r dx: x~ m � x V r ð4Þ