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J. Eur. Opt. Society-Rapid Publ. 21, 5( 2025)
Figure 1. Comparison of the EME and approximate equations for a 1D Fabry-Perot resonator illuminated by a short 10-fs pulse.( a) Sketch of the geometry.( b) QNMs of the resonator in the complex frequency plane, calculated using the MAN software. The circle size indicates the excitation level of the QNMs. The real part of the eigenfrequency for QNM 1 corresponds to the central frequency of the incident pulse.( c) The values of b 1 and b 2 obtained from both the EME and approximate equations.( d) A comparison of the totalfield reconstruction using the EME and approximate equations with 250 QNMs against full-wave numerical results from COMSOL.
Temporal excitation coefficients, being abstract quantities, cannot be directly measured. Figure 1d offers a comparison using measurable quantities, such as the total field. The two panels in Figure 1d show the reconstructed total field: first, inside the resonator for a fixed time t = 65 fs, and second, at a fixed coordinate x = 150nm as a function of time. For the reconstruction, 250 QNMs are used in the expansion of equation( 1). TheEMEequation( 17) yields a reconstruction that perfectly matches reference data from COMSOL time-domain simulations, while the approximate equation( 16) exhibits significant deviations.
It is important to note that the driving pulse is ultrashort and has a broad spectral range. Furthermore, all the QNMs have low quality factors, making them easily excited even when the central frequency of the driving pulse deviates significantly from the QNM resonance frequency. This is why we used 250 QNMs in the reconstruction shown in Figure 1d. With such a large number of QNMs, the maximum difference between the black curves of equation( 17) and the COMSOL data is less than 0.005 for both curves in Figure 1d, providing a strong quantitative validation of the approach.
6 Conclusion
The derivations of the EME equation in Section 4 and Appendix A are straightforward and do not depend on complex analysis or generalized functions, providing a solid foundation for the equation across a broad range of incident pulses.
The numerical tests in Section 5 are intentionally carried out on a simple 1D geometry, commonly used with CMT, which emphasizes the validity of the EME equation. This is achieved by using a driving term