## Abstract

Despite the recent extensive study of the nonradiating (anapole) mode in the resonant light scattering by nanoparticles, the key questions, about the dynamics of its excitation at the leading front of the incident pulse and collapse behind the trailing edge, still remain open. We answer the questions, first, by direct numerical integration of the complete set of the Maxwell equations, describing the scattering of a rectangular laser pulse by a dielectric cylinder. The simulation shows that while the excitation and the collapse periods, both have the same characteristic time-scale, the dynamics of these processes are qualitatively different. The relaxation to the steady-state scattering at the leading front is accompanied by high-amplitude oscillatory modulations of the envelope of the basic electromagnetic oscillations, while behind the trailing edge the decay of the envelope is monotonic. Then, we present the general arguments showing that this is the case for the anapole excited in any classical system. Next, we introduce a simple, exactly integrable yet accurate, physically transparent model describing the dynamics of the anapole. The model admits generalization to a broad class of resonant phenomena and may be regarded as a compliment to the commonly used Temporal Coupled-Mode Theory.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Preliminary

Despite the systematic study of the light scattering by a finite-size particle has a long history, whose excellent description is given in book [1], it still remains one of the most appealing topics of electrodynamics. Nowadays, the interest of researchers has been shifted from macro- and micro- to nanoparticles [2–5]. Recently, the attention of experts has been drawn to the anapole — a nonradiating (invisible) mode with zero scattering amplitude excited in an irradiated particle by a plane incident electromagnetic wave [6–11]. Its invisibility is explained in terms of the destructive Fano resonances [12, 13] resulting in the confinement of the associated with the anapole electromagnetic field within the scattering particle.

Note, there are two types of the field confinement in wave scattering phenomena, namely the dark states with a finite lifetime associated with the radiative losses [14] and the bound states in the continuum, whose lifetime in the ideal case of a perfect non-dissipative system is infinitely large [15]. Owing to the non-existence theorem, which stipulates that the latter cannot be realized in a 3D compact optical structure embedded in a homogeneous transparent medium, e.g., in a vacuum [16], it should be expected that anapole is the dark state.

If so, the question about the dynamics of its excitation and collapse, or, in a more
general context, about the dynamical effects at high-*Q*
resonant wave scattering by a finite obstacle arises. Nowadays, the
question has been more important than ever since the frontier of optics
lies in the range of ultrashort pulses, where these effects become
experimentally observable and may result in qualitative changes of the
scattering. Apart from the purely academic interest, the answer to this
question is very important for applications — the dynamics of the
dark states may be used for tailoring and shaping of the scattered laser
pulse, to control various nonlinear phenomena in the scatterer, etc. The
only way to clarify the case is to go beyond the steady-state scattering
describing by the conventional Mie theory [1, 17] and its various modifications and to study the
transient processes at the leading and trailing edges of the incident
laser pulse.

## 2. Outline

In our paper, we try to respond to the challenge inspecting the dynamics of the anapole mode in detail. First, we perform the direct numerical integration of the complete set of the Maxwell equations supplemented with the standard boundary conditions [1, 17] to describe the dynamics of the scattering of a rectangular laser pulse by a dielectric cylinder. The basic frequency of the pulse is selected equal to the frequency of the complete destructive interference at the Fano resonance for a dipole Mie mode, i.e., in this case, just this mode corresponds to the anapole. Based on these calculations we obtain the temporal dependence of the density of the electromagnetic energy storing in the cylinder.

The simulation shows that in agreement with the non-existence theorem, the anapole is
the dark state indeed. However, while the excitation and the collapse
periods, both have the same characteristic time-scale, the dynamics of
these processes are qualitatively different. The relaxation to the
steady-state scattering at the leading front is accompanied by
high-amplitude oscillations of the envelope of the density of the
electromagnetic energy stored in the cylinder, while behind the trailing
edge the envelope is a monotonically decaying function of time. Then, we
present the general arguments connecting the asymmetric Fano profile with
a high-*Q* Lorentzian and indicating that the anapole
excited in any classical system, no matter compact or infinite, should
have a finite lifetime, except the case when the associated Lorentzian
corresponds to the infinitely large *Q*-factor.

To describe the dynamics of the anapole mode analytically, an appropriate model is required. For the time being, practically, all models for the resonant wave scattering are based on the Temporal Coupled-Mode Theory (TCMT). However, despite the governing equations of the TCMT are dynamical [18,19], to the best of our knowledge, its application to Fano resonances at light scattering by particles, so far, is reduced to the steady-state scattering solely [20, 21]. Meanwhile, the generalization of the TCMT to an essentially non-steady scattering is a tricky problem. The point is that the connection between the parameters of the TCMT and those of an actual problem in question is not straightforward. To obtain the connection in a steady-state scattering the energy balance in the form: the power released in the scatterer equals the sum of the scattered and dissipated ones is exploited. In a non-steady case, this equality may be violated dramatically, since the power either accumulated in the scatterer by excitation of the localized modes or irradiated by the collapse of these modes may give a substantial contribution to the energy balance. Therefore, the formal application of the steady-state version of the TCMT to the dynamical scattering gives rise to erroneous or even meaningless results, such as, e.g., the divergence behind the trailing edge of the incident pulse of the important parameter of the theory: the ratio of the scattered to incident power.

In the present paper, instead of the generalization of the TCMT, we propose an alternative model free from the aforementioned difficulties of the TCMT. The model is physically transparent. Its parameters are readily and unambiguously defined from the spectrum of the steady-state problem. The model is exactly integrable and exhibits an excellent agreement with the results of the direct numerical integration of the Maxwell equations. The most important feature of the model is that it admits generalization to a broad class of resonant phenomena and for this reason could be a good complement to the TMCT.

## 3. Problem formulation

To simplify the analysis we consider the 2D problem of the scattering of a linearly
polarized plane wave by an infinite right circular cylinder, whose axis is
perpendicular both to the plane of oscillations of vector **E**
of the incident wave and to the wave vector of this wave (TE polarization,
normal incidence), see the inset in Fig.
1. The steady version of the problem is exactly solvable. In this
solution, the scattered field and the field within the cylinder are
presented as the infinite series of partial multipolar waves of the
*ℓ*th order (−∞ ≤
*ℓ* ≤ ∞). The corresponding partial
fields read as follows [17]:

*iωt*]). Here

*E*

_{0}and

*H*

_{0}are, respectively, the amplitudes of the electric and magnetic fields in the incident wave, whose wave vector

**k**is aligned antiparallel to

*x*-axis; {

*X*,

_{r}*X*,

_{ϕ}*X*} denote the components of any vector

_{z}**X**in the cylindrical coordinate frame with the

*z*-axis directed along the axis of the cylinder;

*ρ*≡

*rk*;

*k*=

*ω*/

*c*is the wavenumber of the incident wave in a vacuum;

*c*stands for the speed of light; $m\equiv \sqrt{\epsilon}$ is the refractive index of the cylinder (in what follows,

*m*is supposed to be a purely real quantity — the nondissipative limit);

*ε*is its permittivity;

*J*(

_{ℓ}*z*) and ${H}_{\ell}^{(1)}(z)$ stand for the Bessel and Hankel functions of the first kind, respectively, and the prime denotes the derivative with respect to the entire argument of the function. The cylinder is regarded nonmagnetic so that its permeability

*μ*equals unity.

Thus, all individual information about a specific case of the wave scattering is hidden
in the values of the refractive index *m* and the
scattering coefficients *a _{ℓ}*,

*d*, given by the expressions:

_{ℓ}*N*(

_{ℓ}*z*) are the Neumann functions;

*q*=

*kR*stands for the

*size parameter*and

*R*is the radius of the cylinder. Note, |

*d*| = 1 at

_{ℓ}*m*= 1 thanks to the identity

*J*(

_{ℓ}*z*)

*N′*(

_{ℓ}*z*) −

*J′*(

_{ℓ}*z*)

*N*(

_{ℓ}*z*) ≡ 2/(

*πz*). Then, the departure of |

*d*| from unity could be a quantitative measure of the rate of the

_{ℓ}**E**field enhancement (suppression) within the particle. The same role plays the departure of

*m*|

*d*| for the field

_{ℓ}**H**, see Eqs. (1), (2).

The destructive Fano resonances correspond to the roots of the equation *F _{ℓ}*(

*m*,

*q*) = 0 when

*a*vanishes. Since

_{ℓ}*d*never vanishes, see Eq. (5), it means that at the points of the destructive resonances the resonant multipole does not contribute anything to the scattering field outside the cylinder, but still has a finite field amplitude within the cylinder, i.e., becomes a nonradiating mode.

_{ℓ}For definiteness, let us inspect the case *ℓ* = ±1 and *m* = 4. This choice corresponds to the anapole mode discussed by Miroshnichenko *et al* [6]. The first root of the equation *F*_{1}(4*q*) = 0 is *q* ≈ 1.04.

To study the dynamical behavior of the scattering problem we employ a numerical
integration of the complete set of Maxwell’s equations with the
conventional boundary conditions [1,17] — the
continuity of the tangential components of **E** and
**H** at the surface of the cylinder and the radiation condition
for the scattered wave at infinity. To this end, our own code has been
developed. In the code, the Finite-Difference Time-Domain Method and the
standard Yee Algorithm [22,23] have been
used. The code has passed careful tests against the asymptotical
convergence of the generated time-dependent field patterns to the exact
analytical steady-state solution in a broad domain of variations of the
problem parameters.

In the simulation, the incident wave may be regarded as practically monochromatic,
since *ωT* ≫ 1, where *T* is
the pulse duration. We employ a rectangular laser pulse so that the
amplitude of the incident wave is zero at *t* < 0,
a constant at 0 ≤ *t* ≤ *T*,
and zero at *t* > *T*. We suppose
that *T* is also much larger than the inverse linewidth of
the dipole resonant mode (see below). In this case, the field dynamics at
the leading edge of the pulse makes it possible to study the transient to
the steady-state scattering, when the incident field is abruptly switched
on. The dynamics at the trailing edge exhibits the opposite process, when
the incident wave is abruptly switched off. Finally, it is convenient to
normalize the spatial scale on *R* and to introduce the
dimensionless time *θ* =
*ω*_{0}*t* and frequency
Ω =
*ω*/*ω*_{0}, where
*ω*_{0} is the eigenfrequency of the
resonant dipole mode |*d*_{1}|
corresponding to the maximum of its resonance line
(*q*_{max} ≈ 0.92), so that
Max{|*d*_{1}(Ω)|}
= |*d*_{1}(1)|, see Fig. 1.

The basic frequency of the incident pulse is selected so that *a*_{1}(Ω) = 0. For the given value of *m* = 4 it results in Ω ≈ 1.14 (*q* ≈ 1.04), see Fig. 1.

To quantitate the transient processes, we calculate the total electromagnetic energy
*W*(*θ*) stored per unit length of
the cylinder at a given moment of time. Neglecting the dispersion of the
permittivity we may write *W*(*θ*)
in the following simple form [24]:

**E**(

*θ*) and

**H**(

*θ*) are

*real*quantities.

Naturally, apart from the dipolar mode, the plane incident wave excites the entire spectrum of other multipoles. However, the calculations based on the exact solution [17] show that at the selected values of the problem parameters the contribution of the dipolar mode to 〈*W*_{ste}〉 is overwhelming. Here 〈*W*_{ste}〉 designates *W*(*θ*) averaged over the period of the incident wave oscillations for the steady-state scattering.

## 4. Numerics

Plot *W*(*θ*) normalized on 〈*W*_{ste}〉 obtained as a result of the numerical integration of the Maxwell equations is shown in Fig. 2. Time *θ* begins to be counted from the moment when the leading front of the incident pulse hits the surface of the cylinder. In the introduced dimensionless units the duration of the pulse *τ* = *Tω*_{0} = 550. It is seen clearly that a collapse of the anapole begins immediately after the incident pulse is over. The collapse is the solid evidence that, in agreement with the non-existence theorem, the anapole is a dark state and cannot exist without the incident pulse.

Thus, the anapole does not exchange energy with the environment at the steady-state scattering solely. When the leading edge of the incident pulse hits the cylinder the incident power is pumped into the anapole quite effectively. However, the closer the anapole mode to its steady-state the narrower the energy exchange channel between the anapole and the environment — the broad channel transforms into a bottleneck, which, eventually, is closed entirely at the steady-state scattering. Just behind the trailing edge of the incident pulse the channel abruptly becomes broadly open and the stored in the anapole mode electromagnetic energy rapidly gets out owing to the radiative losses.

## 5. Discussion

To understand the underlying physics of this dynamics we have to employ the results by Tribelsky *et al* [25]. According to them, scattering coefficients *a _{ℓ}* and

*d*are connected by a certain identity, which for the problem in question has the following form:

_{ℓ}*a*, given by Eqs. (5)–(7) in the limit

_{ℓ}*m*→ ∞. It correspond to

*a*for the same cylinder made of a hypothetical material —

_{ℓ}*P*erfect

*E*lectric

*C*onductor.

Importantly, that the asymmetric Fano profile for
|*a _{ℓ}*|

^{2}is obtained owing to the contribution of the

*two*terms in the right-hand-side (RHS) of Eq. (9). As for |

*d*|

_{ℓ}^{2}solely, it has a usual bell-like shape, well approximated by the conventional Lorentzian. Specifically, for the problem in question, it reads

*d*

_{1}(Ω)| following from Eq. (10) is compared with the actual one obtained from the exact solution of the scattering problem. The numerical value of Γ there is selected equal to the half-maximum linewidth (FWHM) in the actual profile. It gives rise to Γ ≈ 0.10.

Thus, in the frequency-domain the lineshape of
|*d*_{1}(Ω)|^{2} is
in a quite reasonable agreement with the simple Eq. (10) up to Ω ≈ 1.14
(the anapole eigenfrequency) and even a bit beyond this point. Then, in
the time-domain the corresponding dynamics should agree with that for the
system exhibiting in the frequency-domain the Lorentzian lineshape, that
is to say, with forced oscillations of a *single* damped
harmonic oscillator. If this is the case, the physical grounds for the
dynamics observed in the numerics become absolutely clear. Indeed, the
vanishing of *a*_{1} at Ω ≈ 1.14 in
the steady-state scattering means that the two terms in the RHS of Eq. (9) cancel each other. Note
that ${a}_{1}^{(\text{PEC})}$ is a function of a single variable
*q*, while *d*_{1} depends on
*q* and *mq*, see Eq. (5). At *m* ≫ 1
product *mq* is a fast variable relative to
*q*. Since Ω is proportional to *q*,
it means, in the Ω-domain the first term in the RHS of Eq. (9) has a broad line, while
the second has sharp. In other words, we have a sum of
low-*Q* and high-*Q* modes. At a sharp
variation of the amplitude of the drive, the response of the
low-*Q* mode is fast, whereas for the
high-*Q* mode it is slow. Therefore, during the transient,
the balance between the two modes is violated, the complete cancellation
does not take place, and the anapole begins to receive (radiate)
electromagnetic energy from (to) the environment.

The aforementioned arguments are very general and may be applied to a broad class of
problems exhibiting a superposition of weakly coupled
high-*Q* and low-*Q* resonant modes. To
describe the problem in question, let us introduce a simple model which
incorporates all these key features of the phenomenon.

## 6. The model

The simplest problem exhibiting the asymmetric Fano profile is forced vibrations of *two* weakly-coupled harmonic oscillators [26]. However, our concern is the field *within* the cylinder. To describe its dynamics it suffices to have a model governing equation for the coefficient *d*_{1} solely. As it just has been said, this equation is nothing but the well-known equation for the forced harmonic oscillations:

*θ*,

*θθ*designate the corresponding derivatives and subscript 1 is dropped to simplify the notations. To describe the numerics discussed above, Eq. (11) should be supplemented with the initial conditions

*d*(0) =

*d*(0) = 0, while

_{θ}*A*(

*θ*) =

*A*

_{0}[

*H*(

*θ*) −

*H*(

*θ*−

*τ*)], where

*A*

_{0}=

*const*and

*H*(

*z*) is the Heaviside step function. This problem is exactly integrable. At 0 ≤

*θ*≤

*τ*its solution is

*γ*= Γ/2 and ${\mathrm{\Omega}}_{0}=\sqrt{1-{\gamma}^{2}}$.

At *θ* > *τ* it is

*d*(

*τ*) and

*d*(

_{θ}*τ*) are calculated according to Eq. (12).

The total energy of the oscillator *W* is (1/2)[(Re *d _{θ}*)

^{2}+ (Re

*d*)

^{2}]. The dependence

*W*(

*θ*)/〈

*W*

_{ste}〉 for the actual problem, superimposed on the corresponding dependence for the oscillator calculated with the help of Eqs. (12)–(13) is shown in Fig. 2. For the latter 〈

*W*

_{ste}〉 also designates the average of

*W*(

*θ*) over the period of the asymptotical steady-state forced oscillations.

The agreement between the two plots is quite impressive (the smaller amplitude of the high-frequency oscillations of *W*(*θ*) in the actual problem relative to those for the oscillator is an artifact related to the sampling of the huge database of the numerics in order to present it graphically). A little difference is observed just in the decay rate. It is hardly noticeable in the linear scale, but more pronounced in the logarithmic one, see the inset in Fig. 2: While the model exhibits the pure exponential decay, in the actual problem the decay gradually departures from the exponential law, becoming less rapid. This behavior of the decay rate is not a specific feature of the anapole mode — it is generic for the problem, see two more curves in the inset: for Ω = 1 (the maximum of the resonance line) and for Ω ≈ 1.07 (*q* ≈ 0.98), lying just in between Ω = 1 and Ω ≈ 1.14. We attribute the slowing down of the decay rate at large time in the actual problem relative to that in the model to the contributions of the multipoles with *ℓ* ≠ 1, which our single-mode approximation does not include. As it has been mentioned above, at the steady-state scattering the amplitudes of these modes are small compared to |*d*_{1}|. However, since the *Q*-factor increases sharply with an increase in *ℓ* [27–29], the higher multipoles decay slower than the dipole mode (anapole), and, despite the smallness of the initial amplitudes, gradually their contribution becomes overwhelming.

Note also, in principle our single-mode model may be extended so that the contribution of some other modes could be taken into account too. The applicability of the model is based on the possibility to approximate the actual lineshape of a given mode by a Lorentzian profile. The approximation is good, provided Ω is close to the maximum of the corresponding actual line. If a resonance line of another mode is so close to the given one that its Lorentzian approximation is still valid in the region of the overlap of the two modes, the dynamic of this mode is described by the same Eq. (11) but, naturally, with different values of the parameters. If this is not the case, the overlap occurs in a far wing of the second mode, where its contribution is small and may be neglected.

The proposed model makes it possible to explain in a very simple manner even more
subtle features of the observed dynamics of the actual problem and get a
deeper insight into it. Specifically, when the drive is abruptly switched
on, in addition to the forced oscillations with the frequency
Ω_{drive} ≈ 1.14 the free oscillations with the
eigenfrequency Ω_{0} = 1 are excited. Since there
is a mismatch between the two frequencies, see Fig. 1, it results in beats in the energy profile
with the frequency approximately equal to
|Ω_{drive} − 1|. The beats decay
with the characteristic timescale ∼ 1/Γ. It gives rise to
an oscillatory behavior of the envelope of the energy density stored in
the cylinder as a function of time. The same is true for the average (over
the period of the fast underlying oscillations) value of the energy
density 〈*W*(*θ*)〉.
Initially the energy “overfills” the cylinder and reaches
a pronounced peak, whose value is considerably larger than
〈*W*_{ste}〉. Then, the excess of
the energy is irradiated from the cylinder,
〈*W*(*θ*)〉 becomes
smaller than 〈*W*_{ste}〉, and so
on.

The simulations of the actual problem with other values of Ω_{drive} show, that when Ω_{drive} approaches unity, the period of the beats increases. Since their damping rate, determined by Γ, remains fixed, it results in suppression of the amplitude of the beats. Eventually, at Ω_{drive} = 1 the beats disappear and 〈*W*(*θ*)〉 becomes a smooth, monotonic function.

In contrast, at the trailing edge of the pulse, when the drive is switched off instantly, just the free oscillations with Ω = 1 are excited, see Eq. (13). Thus, the maximal power irradiated by the cylinder corresponds to the very beginning of the decay process, then it gradually decreases in the course of time, so that 〈*W*(*θ*)〉 is a monotonically decreasing function at any frequency of the drive.

To illustrate these features of the problem and to show the agreement of the dynamics
of the developed model with those of the actual problem at different
values of the drive frequency, the corresponding dynamics at three values
of Ω equals the ones in the inset of Fig. 2 are presented in Fig. 3. To make the demonstration of the accuracy of
the model more pronounced in Fig. 3
the high-frequency oscillations of
*W*(*θ*) are suppressed by applying
to the dataset the frequency filter from the *National Instruments
Digital Filter toolkit* with the *Q*-factor equal
to 3. The corresponding quantity is designated as
〈*W*(*θ*)〉. At all
values of Ω this quantity is normalized on
〈*W*_{ste}〉_{Ω≈1.14}
— the value of the average stored energy for the steady-state
oscillations at the anapole frequency, i.e., the normalizing factor for
all curves is one and the same and equal to that used in Fig. 2. In other words, since the vertical scale for
the model is adjusted to that for the actual problem at Ω
≈ 1.14, we do not have any degree of freedom to change the scales
of the curves at Ω ≈ 1.07 and Ω = 1
— they are fixed by the solutions of Eq. (11) at the corresponding value of
Ω.

It is worth presenting some numerical estimates. As it is seen from Fig. 2, both transient processes (at the leading and trailing edges of the pulse) last about 50 dimensionless units. It lies within the experimentally resolved timescales even if the resonant anapole frequency corresponds to the visible range of the spectrum, to say nothing about the far IR and radio domains. Regarding the possible experimental verification of the discussed phenomena, in our view, the most straightforward experimental evidence may be obtained at the microwave carrier frequency of the incident pulse with the setup similar to the one described in our publication [30].

## 7. Conclusion

As a résumé, we may say the following:

- The anapole cannot exist as a stand-alone excitation — it is the essentially nonequilibrium dark state and may exist if and only if the host particle is irradiated by a continuous wave.
- In transient processes, the anapole becomes a usual radiative mode exchanging energy with the environment.
- The main features of the dynamics of the excitation and collapse of the anapole may be explained within the framework of the toy model of the forced vibrations of a simple harmonic oscillator.
- The model does not have any fitting parameters, since the eigenfrequency of the oscillator and its damping factor are unambiguously determined, respectively, by the resonant frequency and the linewidth of the corresponding resonant mode describing the field within the scattering particle. Nonetheless, the model exhibits high accuracy, provides a detailed quantitative description of transient processes, and can be generalized to a broad class of resonant phenomena exhibiting a superposition of weakly-coupled high-
*Q*and low-*Q*resonant modes, see, e.g., [31]. - Despite the characteristic time-scale for the transient processes during the excitation and the collapse of the anapole are the same, the dynamics of these processes are completely different. The excitation is accompanied by the strong modulations of the amplitude of the electromagnetic field excited in the particle owing to beats caused by the mismatch between the frequency of the incident wave and the eigenfrequency of the resonant mode. In contrast, the decay of the anapole amplitude behind the trailing front of the incident pulse develops in a monotonic manner.

Note that if nonlinear effects with the threshold lying beyond
〈*W*_{ste}〉, but below the peak
value of 〈*W*(*θ*)〉,
are incorporated into the problem, the pronounced peak of the field
intensity in the cylinder at the leading front of the incident pulse may
be effectively employed to design new nanodevices generating ultrashort
electromagnetic pulses (an actual nanolaser associated with a single
nanoparticle).

In a more broad context, we should stress that transient processes at resonant Mie’s scattering are accompanied with dramatic changes in the topological structure of the electromagnetic field within the scattering particle and its near field zone, whose detailed discussion will be presented elsewhere. In addition to the purely academic interest, it might find plenty of applications in technologies, such as telecommunications; data storage, and processing; optical computers; etc. We believe our results may stimulate further study in this appealing field.

## Funding

Grant of the President of the Russian Federation (MK-2761.2019.2); Russian Foundation for Basic Research (17-02-00401); Russian Science Foundation (19-72-30012); MEPhI Academic Excellence Project (02.a03.21.0005).

## Acknowledgments

The work of S.E.S. was supported by the grant of the President of the Russian Federation; M.I.T. acknowledges the financial support of the Russian Foundation for Basic Research for the analytical study, the MEPhI Academic Excellence Project (agreement with the Ministry of Education and Science of the Russian Federation of August 27, 2013) for the modeling of the resonant light scattering and the contribution of Russian Science Foundation for the computer simulation.

## References

**1. **M. Kerker, *The Scattering of Light and Other Electromagnetic Radiation* (Elsevier Science, 2013).

**2. **L. Novotny and B. Hecht, *Principles of nano-optics* (Cambridge University, 2006). [CrossRef]

**3. **V. Klimov, *Nanoplasmonics* (Pan Stanford, 2014). [CrossRef]

**4. **M. Rahmani, A. E. Miroshnichenko, D. Y. Lei, B. Luk’yanchuk, M. I. Tribelsky, A. I. Kuznetsov, Y. S. Kivshar, Y. Francescato, V. Giannini, and M. Hong *et al.*, “Beyond the hybridization effects in plasmonic nanoclusters: diffraction-induced enhanced absorption and scattering,” Small **10**, 576–583 (2014). [CrossRef]

**5. **M. I. Tribelsky and B. S. Luk’yanchuk, “Light scattering by small particles and their light heating: New aspects of the old problems in fundamentals of laser,” in *Fundamentals of Laser-Asssted Micro- and Nanotechnologies*,(Springer, 2014), Springer Series in Materials Science, vol. 195, pp. 125–146. [CrossRef]

**6. **A. E. Miroshnichenko, A. B. Evlyukhin, Y. F. Yu, R. M. Bakker, A. Chipouline, A. I. Kuznetsov, B. Luk’yanchuk, B. N. Chichkov, and Y. S. Kivshar, “Nonradiating anapole modes in dielectric nanoparticles,” Nat. Commun. **6**, 8069 (2015). [CrossRef] [PubMed]

**7. **J. Gongora, A. Miroshnichenko, Y. Kivshar, and A. Fratalocchi, “Anapole nanolasers for mode-locking and ultrafast pulse generation,” Nat. Commun. **8**, 15535 (2017). [CrossRef]

**8. **B. Luk’yanchuk, R. Paniagua-Domínguez, A. I. Kuznetsov, A. E. Miroshnichenko, and Y. S. Kivshar, “Hybrid anapole modes of high-index dielectric nanoparticles,” Phys. Rev. A **95**, 063820 (2017). [CrossRef]

**9. **N. Papasimakis, V. Fedotov, V. Savinov, T. Raybould, and N. Zheludev, “Electromagnetic toroidal excitations in matter and free space,” Nat. Mater. **15**, 263–271 (2016). [CrossRef] [PubMed]

**10. **L. Wei, Z. Xi, N. Bhattacharya, and H. P. Urbach, “Excitation of the radiationless anapole mode,” Optica **3**, 799–802 (2016). [CrossRef]

**11. **B. S. Luk’yanchuk, N. V. Voshchinnikov, R. Paniagua-Domínguez, and A. I. Kuznetsov, “Optimum forward light scattering by spherical and spheroidal dielectric nanoparticles with high refractive index,” ACS Photon. **2**, 993–999 (2015). [CrossRef]

**12. **U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. **124**, 1866–1878 (1961). [CrossRef]

**13. **M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, “Fano resonances in photonics,” Nat. Photon. **11**, 543 (2017). [CrossRef]

**14. **C. W. Hsu, B. G. DeLacy, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Theoretical criteria for scattering dark states in nanostructured particles,” Nano Lett. **14**, 2783–2788 (2014). [CrossRef] [PubMed]

**15. **C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. **1**, 16048 (2016). [CrossRef]

**16. **M. G. Silveirinha, “Trapping light in open plasmonic nanostructures,” Phys. Rev. A **89**, 023813 (2014). [CrossRef]

**17. **C. F. Bohren and D. R. Huffman, “*Absorption and scattering of light by small particles*,” (Wiley, 2008), chap. 8.4.

**18. **W. H. Louisell, *Coupled mode and parametric electronics* (Wiley, 1960).

**19. **H. A. Haus, *Waves and fields in optoelectronics* (Prentice-Hall, 1984).

**20. **Z. Ruan and S. Fan, “Temporal coupled-mode theory for fano resonance in light scattering by a single obstacle,” J. Phys. Chem. C **114**, 7324–7329 (2010). [CrossRef]

**21. **Y. Fang and Z. Ruan, “Temporal coupled-mode theory for light scattering and absorption by nanostructures,” in *Fano Resonances in Optics and Microwaves*, (Springer, 2018), pp. 157–183. [CrossRef]

**22. **A. Taflove and S. C. Hagness, *Computational electrodynamics: the finite-difference time-domain method* (Artech house, 2005).

**23. **K. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Transactions on antennas propagation **14**, 302–307 (1966). [CrossRef]

**24. **L. D. Landau and E. Lifshitz, *Course of Theoretical Physics. Vol. 8: Electrodynamics of Continuous Media* (Pergamon, 1984).

**25. **M. I. Tribelsky and A. E. Miroshnichenko, “Giant in-particle field concentration and fano resonances at light scattering by high-refractive-index particles,” Phys. Rev. A **93**, 053837 (2016). [CrossRef]

**26. **Y. S. Joe, A. M. Satanin, and C. S. Kim, “Classical analogy of fano resonances,” Phys. Scripta **74**, 259–266 (2006). [CrossRef]

**27. **B. Luk’yanchuk, M. Tribel’skii, and V. Ternovskii, “Light scattering at nanoparticles close to plasmon resonance frequencies,” J. Opt. Tech. **73**, 371–377 (2006). [CrossRef]

**28. **B. Luk’yanchuk, M. Tribelsky, V. Ternovsky, Z. Wang, M. Hong, L. Shi, and T. Chong, “Peculiarities of light scattering by nanoparticles and nanowires near plasmon resonance frequencies in weakly dissipating materials,” J. Opt. A: Pure Appl. Opt. **9**, S294–S300 (2007). [CrossRef]

**29. **B. Luk’yanchuk, Z. Wang, M. Tribelsky, V. Ternovsky, M. Hong, and T. Chong, “Peculiarities of light scattering by nanoparticles and nanowires near plasmon resonance frequencies,” J. Phys. Conf. Ser. **59**, 234–239 (2007). [CrossRef]

**30. **P. Kapitanova, V. Ternovski, A. Miroshnichenko, N. Pavlov, P. Belov, Y. Kivshar, and M. Tribelsky, “Giant field enhancement in high-index dielectric subwavelength particles,” Sci. Rep. **7**, 731 (2017). [CrossRef] [PubMed]

**31. **M. I. Tribelsky and A. E. Miroshnichenko, “Dynamic fano resonances: From toy model to resonant mie scattering,” arXiv preprint arXiv:1809.02474 (2018).