J. Eur. Opt. Society-Rapid Publ. 21, 5( 2025) 41
t 0. As the second integral in this equation is defined, see equation( 10), the integral on the left-hand side, R 1 x~ m
�1 x~ m � x exp ð ix ð t0 � tÞÞdx, is also divergent.
4 Demonstration of equation( 7)
In this Section, we provide a demonstration of equation( 7) that does not rely on complex analysis. By introducing the
spectral overlap Om ðxÞ ¼ R V r
eðrÞE b ðr; xÞ E~ m d 3 r, equation( 2) becomes �ixa m ðxÞ ¼ �i x~ m a m ðxÞþ ixO m ðxÞ: ð11Þ
We take the Fourier transform of equation( 11) to move in the temporal domain
Z 1
�1
�ixa m ðxÞ exp ð�ixtÞdx
Z 1
¼�i x~ m a m ðxÞ exp ð�ixtÞdx
þ
Z 1
�1
�1
ixO m ðxÞ exp ð�ixtÞdx: ð12Þ
To obtain equation( 12), we need two assumptions on the driving pulse
the Lebesgue integral of xa m( x) isfinite, i. e.
Z 1
�1 jxa m ðxÞjdx < 1: ð13aÞ
the Lebesgue integral of xO m( x) isfinite, i. e.
Z 1
�1 xO m ðxÞ dx < 1: ð13bÞ
R 1
Let us consider the left-hand term: �ixa �1 m
R 1 ðxÞ exp ð�ixtÞdx. It is equal to a �1 mðxÞ d dt
R exp ð�ixtÞdx ¼ d 1 a dt �1 mðxÞ exp ð�ixtÞdx ¼ d b dt mðÞ t.
We can do exactly the same for the last term of equation
R 1
( 12), ix O �1 m ðxÞ exp ð�ixtÞdx ¼� d O dt mðÞ t. Then,
directly from equation( 12), wehave
db m ðÞ t
¼�i x~ m b dt m ðÞ� t d dt O mðtÞ; ð14Þ
which corresponds to the main result obtained for non-dispersive materials in [ 10 ], specifically the EME evolution. It is straightforward to verify that the expression for b m( t) provided in equation( 7) satisfies equation( 14). This completes the proof of equation( 7).
In Appendix A, we present an alternative demonstration based on less stringent( possibly R minimal) assumptions
1 regarding the driving pulse: j a �1 mðxÞjdx < 1 and O m ðxÞ dx < 1.
R 1 �1
5 Clarification of the origin of the temporal derivative in the driving force
This section emphasizes the approximation made in classical CMT, where the driving force is assumed to be proportional to the incident field rather than its time derivative.
To clarify the approximation introduced in the CMT, we remember that a derivative in the real temporal domain amounts to multiply by the conjugated variable in the frequency domain in Fourier( or Laplace) analysis. We thus intuitively multiply the expression of the excitation coefficient by x~ m = x to remove the x-dependence in the numerator of equation( 2). We obtain an approximate expression of the excitation coefficient denoted by a ðappÞ m a ðappÞ m ðxÞ ¼ x~ m x a m ¼ x~ m
O x~ m � x m ðxÞ: ð15Þ
We expect the approximate expression to be most accurate when the frequency of the monochromatic incident field is close to the QNM resonance frequency. As demonstrated in Section 4, the temporal R excitation coefficient is given by b ðappÞ t m ðÞ¼i t x~ O � � m �1 mðt 0 Þexp i x~ m t 0 � t dt 0. This leads by differentiation to
db ðappÞ m dt
¼�i x~ m b ðappÞ ðÞþi t x~ m O m ðÞ t; m ð16Þ in contrast to the EME equation db m
¼�i x~ m b dt m ðÞ� t dO m
: ð17Þ dt
Equation( 16) aligns with the usual approach in CMT, here the driving force is proportional to an overlap integral between the incident field( rather than its derivative) and the QNM field, with a prefactor of x~ m as expected from dimensional analysis.
Next, we evaluate the accuracy of equation( 16) for a 1D Fabry-Perot resonator in air. The resonator is illuminated at normal incidence by a 10-fs plane-wave Gaussian pulse( Fig. 1a). The central frequency of the pulse corresponds to the real part of the eigenfrequency of one of the QNMs( QNM 1 in Fig. 1b).
Figure 1c compares | b 1 | and | b 2 |, computed using the approximate and EME equations. For each QNM, O m( t) is obtained by simply computing the overlap between the QNM electric field E~ m ðrÞ and the Gaussian pulse E b( r, t): O m ðÞ¼ t R V r
eðrÞE b ðr; tÞ E~ m ðrÞd 3 r. Thevalues of | b 1 | and | b 2 | are then determined by solving the differential equations( 16) and( 17). Significant differences are observed primarily for the off-resonant QNM 2, for which the resonance frequency differs from the pulse central frequency – the approximation made in equation( 15) by multiplying by the excitation coefficient by x~ m = x becomes inaccurate. The reverse occurs when we adjust the central frequency of the incident pulse to match Reð x~ 2
Þ, implying the QNM field distribution does not play any role in the difference reported. Additionally, note that in classical CMT, the coupling coefficient is typically fitted, and more precise predictions than those derived from equation( 17) are possible, as discussed in Section 3 of [ 10 ].