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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 ℓth order (−∞ ≤ ℓ ≤ ∞). The corresponding partial fields read as follows [17]:
within the cylinder and outside it (the temporal dependence of the fields is exp[−iωt]). Here E0 and H0 are, respectively, the amplitudes of the electric and magnetic fields in the incident wave, whose wave vector k is aligned antiparallel to x-axis; {Xr, Xϕ, Xz} denote the components of any vector 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; 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; Jℓ(z) and 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.
Fig. 1 The resonance lines |a1| and |d1| for the exact solution of the steady scattering problem (solid brown and dashed blue lines, respectively) and the fitting of |d1(Ω)| with the line for a harmonic oscillator (black dot-dashed line). Both profiles — for |d1| (actual and fitted) have the same linewidths (FWHM) and are normalized on the corresponding maximal values; Ω ≈ 1.14 (marked with the vertical line) is the anapole eigenfrequency. The inset shows the mutual orientation of the cylinder, coordinate frame, and the plane TE-polarized incident wave.
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 aℓ, dℓ, given by the expressions:
where Nℓ(z) are the Neumann functions; q = kR stands for the size parameter and R is the radius of the cylinder. Note, |dℓ| = 1 at m = 1 thanks to the identity Jℓ(z)N′ℓ(z) − J′ℓ(z)Nℓ(z) ≡ 2/(πz). Then, the departure of |dℓ| from unity could be a quantitative measure of the rate of the E field enhancement (suppression) within the particle. The same role plays the departure of m|dℓ| for the field H, see Eqs. (1), (2).The destructive Fano resonances correspond to the roots of the equation Fℓ(m, q) = 0 when aℓ vanishes. Since dℓ 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.
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 F1(4q) = 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 = ω0t and frequency Ω = ω/ω0, where ω0 is the eigenfrequency of the resonant dipole mode |d1| corresponding to the maximum of its resonance line (qmax ≈ 0.92), so that Max{|d1(Ω)|} = |d1(1)|, see Fig. 1.
The basic frequency of the incident pulse is selected so that a1(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]:
where 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 〈Wste〉 is overwhelming. Here 〈Wste〉 designates W(θ) averaged over the period of the incident wave oscillations for the steady-state scattering.
4. Numerics
Plot W(θ) normalized on 〈Wste〉 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 = Tω0 = 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.
Fig. 2 Temporal dependence of the normalized instant total energy stored in the irradiated cylinder, obtained by direct numerical integration of the complete set of the Maxwell equations (magenta) and the one for a simple harmonic oscillator model (blue) at Ω ≈ 1.14. The envelope of a rectangular incident (driving) pulse in a.u. is shown in green; θ = ω0t, see the text for details. The inset shows the tails of the decay processes, where two more curves for the actual problem with Ω ≈ 1.07 (dashed orange) and Ω = 1 (dotted black) are added.
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 aℓ and dℓ are connected by a certain identity, which for the problem in question has the following form:
Here is the scattering coefficient obtained from aℓ, given by Eqs. (5)–(7) in the limit m → ∞. It correspond to aℓ for the same cylinder made of a hypothetical material — Perfect Electric Conductor.Importantly, that the asymmetric Fano profile for |aℓ|2 is obtained owing to the contribution of the two terms in the right-hand-side (RHS) of Eq. (9). As for |dℓ|2 solely, it has a usual bell-like shape, well approximated by the conventional Lorentzian. Specifically, for the problem in question, it reads
See Fig. 1, where the profile |d1(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 |d1(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 a1 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 is a function of a single variable q, while d1 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 d1 solely. As it just has been said, this equation is nothing but the well-known equation for the forced harmonic oscillations:
where subscripts 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) = dθ(0) = 0, while A(θ) = A0[H(θ) − H(θ − τ)], where A0 = const and H(z) is the Heaviside step function. This problem is exactly integrable. At 0 ≤ θ ≤ τ its solution is Here γ = Γ/2 and .At θ > τ it is
where d(τ) and dθ(τ) are calculated according to Eq. (12).The total energy of the oscillator W is (1/2)[(Re dθ)2 + (Re d)2]. The dependence W(θ)/〈Wste〉 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 〈Wste〉 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 |d1|. 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 Ωdrive ≈ 1.14 the free oscillations with the eigenfrequency Ω0 = 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 |Ωdrive − 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 〈Wste〉. Then, the excess of the energy is irradiated from the cylinder, 〈W(θ)〉 becomes smaller than 〈Wste〉, and so on.
The simulations of the actual problem with other values of Ωdrive show, that when Ωdrive 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 Ωdrive = 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 〈Wste〉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.
Fig. 3 Smoothed temporal dependence of the normalized instant total energy stored in the irradiated cylinder obtained by direct numerical integration of the complete set of the Maxwell equations (magenta, solid) and the one for a simple harmonic oscillator model (blue, dashed) at three different values of Ω (indicated in the plot), θ = ω0t, see the text for details.
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 〈Wste〉, 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.
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