## Abstract

A recent study [PRB 100, 075427 (2019)], finally, demonstrated the plasmon-analog of refractive index enhancement in metal nanostructures (MNSs), which has already been studied in atomic clouds for several decades. Here, we simply utilize this phenomenon for achieving continuously-tunable enhanced Cherenkov radiation (CR) in MNSs. Beyond enabling CR from slow-moving particles, or increasing its intensity, the phenomenon can be used in continuous-tuning of the velocity cutoff of particles contributing to CR. More influentially, this allows a continuously-tunable analysis of the contributing particles as if the data is collected from many different detectors, which enables data correction. The phenomenon can also be integrated into lattice MNSs, for continuous medium tuning, where a high density of photonic states is present and the threshold for the CR can even be lifted. Additionally, vanishing absorption can heal radiation angle distortion effects caused by the metallic absorption.

## 1 Introduction

A charged particle, moving with a constant velocity in a dielectric medium, emits the well-known Cherenkov radiation (CR) [1], [2] when its velocity (*v*) exceeds the phase velocity (*ν*_{ph}=*c*/*n*(*ω*)) of light, i.e. *ν*>*ν*_{ph}, in this medium [3]. Such a condition is met for particles of energy in the order of 100 keV, which can be generated in nuclear processes and particle accelerators – also used for free-electron lasers [4], [5]. CR can be utilized for particle detection purposes [6], [7]. Velocity distribution of emitted particles can be characterized by the angle *θ*_{CR} of CR via cos *θ*_{CR}=*ν*_{ph}/*ν* [3]. Detection of slower-moving particles, however, necessitates media with much larger refractive indices.

Recent developments in the control and manufacturing of nanostructures (NSs) [8], [9], [10] enabled particle detectors based on metal nanostructures (MNSs) [11], [12], [13]. CR near a thin metal film deposited on dielectric [11] and periodic structures [14], [15], [16] of metallic nanoarrays [17], [18], [19] are very different from a medium of a uniform index. Lattices of MNSs facilitate hyperbolic metamaterials [20], [21]. In these metamaterials enhanced photonic density of states boosts the intensity of CR. Moreover, the threshold for CR radiation can be lifted in these metamaterials [22]. Such a periodic medium is experimentally demonstrated to emit CR for electron energies as low as 0.25 keV [20], [22]. Furthermore, left-handed metamaterials demonstrate a reversed CR [23], [24]. The progresses in CR metamaterials have stimulated the use of CR imaging for biological and medical applications [25].

Besides facilitating the movements in CR imaging, MNSs also provided a medium for observing plasmon analogs [10] of electromagnetically-induced transparency (EIT)-like effects [26], [27], [28], originally observed for three or more level atoms [29], [30], [31]. Fano resonances [26], [27], [32], plasmon-analog of EIT, and nonlinear response enhancement [33], [34], [35], [36] have been demonstrated in MNSs. Very recently, the plasmon-analog of refractive index enhancement was finally also demonstrated via simple analytical calculations which are supported by the exact solutions of the three-dimensional (3D) Maxwell equations [37]. Lavrinenko and colleagues demonstrated [37] that the linear response of a metal nanorod (y-aligned) to the applied electric field can be controlled, in particular it can be enhanced, via interacting with a second (perpendicular, x-aligned) nanorod, see dimers in Figure 1. This happens when the second (x-aligned) nanorod is driven with a pump which does not couple to (not excite) the first (y-aligned) nanorod directly. The interaction between the two nanorods is provided by the hotspot (near-field) of the second nanorod which relies on the intersection of the two nanorods.

In this paper, we discuss the utilization of this index control scheme to the plasmonic CR-based detectors employing metal nanoparticles. The phenomenon [37] can be used to tune (e.g. to decrease) the phase velocity,
^{[1]} of the light propagation in a particle detector continuously. We show that the phenomenon can be utilized (i) to enhance the total CR, (ii) to enable the CR from slow-moving particles, (iii) to control the wavelength of CR, and (iv) to gain control over the cutoff velocity of the particles emitting the CR. Actually, more influential than (i)–(iv) is: the continuous tuning of the detector, after it is manufactured, allows a particle velocity distribution analysis as if many detectors exist.

We consider an ensemble of metal nanorod dimers, see Figure 1, and calculate a polarization density *P*(*ω*) from the enhanced dipole moment responses of such dimers. The dipole moments of x-aligned nanorods are controlled by an x-polarized auxiliary (aux) Gaussian pulse. The x-polarized nanorod couples to the y-polarized nanorod and tunes its response to an applied (probe) E-field of polarization along the y-direction. (The y-aligned nanorod can be excited with a y-polarized field.) Thus, an x-polarized aux beam (*E*_{2}) can continuously tune the refractive index of the medium for a y-polarized CR emission. The charged particles travel through the medium in the x-direction [20], [21], [38], [39], [40]. Enhanced y-polarized CR emission and its angle, measured in the x-z plane, can be recorded using a y-polarization selector (filter). We discuss the phenomenon in the following sections in detail.

The main input of the presented scheme is neither the enhancement of the CR intensity nor bringing the velocity threshold to slower-moving particles. In current particle detectors, employing metal nanostructures, CR intensity is already enhanced due to the increased density of states [20] and the velocity threshold can even be lifted [21], [22]. None of these current detectors, however, can provide a continuous tuning, thus a continuous spectral analysis, of the particles’ velocities via CR.

We can make the functionality of the new scheme more visible as follows. A conventional (CR-based) particle detector, whether employing the nanoparticles or not, takes (records) the emission spectral data for the *fixed parameters* of the manufactured detector; even if the CR spectrum is enhanced and the threshold is lifted. Then, it analyses the particle velocity distribution, etc. The particle detector, we study in this paper, does the following. It can record data on the velocity distribution of particles as if one collects data on the velocity distribution of the source from many particle detectors, in principle infinite, each having different fixed parameters (e.g. indices). This way, comparing all such data sets, for example, one can make corrections on the errors of data [43] occurring due to non-uniform absorption, or other unpredictable issues. Such a game changing continuously-tunable tool/device, to the best of our knowledge, is not present in the new-generation CR-based particle detectors.

The index control mechanism we employ here provides one other significant advantage in CR-based particle velocity detection. In conventional CR-based detectors, employing periodic metal nanostructures, there exists a serious amount of loss. This severely distorts (e.g. broadens) the relation between the Cherenkov angle and the particle velocity on which the particle detection is based [14], [39]. In the index control scheme, we study here, however, the enhanced index can be achieved at vanishing/reduced absorption, see Figure 2. Vanishing/reduced absorption at different parameters can be achieved by using aux pulses operating at different frequencies.

We also present the contribution to the power of CR from different particle velocities of an ^{18}*F* emission [44] for different enhancements of the refractive index. We further calculate the angle *θ*_{CR} for particle moving at different velocities.

Here, we consider a periodic spatial distribution of the nanorod dimer centers. All dimers are aligned along a given direction. Such kinds of periodic structures can be achieved by e-beam lithography techniques [45]. Although here we consider a periodic ordering of the dimers, arbitrary distributions of such nanodimers can also be achieved in solutions when one of the nanorods of the dimer is manufactured via a magnetized metallic material [46], [47]. This can be useful for *in vitro* CR imaging [25]. Here, we do not consider the extra enhancement effects possible to appear due to periodicity, in this first demonstration of such a game-changing device. Nevertheless, periodicity of the dimers is expected to increase the CR emission via enhanced density of photonic states, a feature already appearing in their no-index-controlled counterparts [29], [30], [31]. Using lattice structures of index controlled dimers can provide not only an enhanced emission, compared to the metal nanoarrays [22], without a CR threshold, but also make their CR emission features continuously tunable.^{[2]} Although we consider the basic dimer structure Ref. [37] studies originally; nanostructures, easier to manufacture, can display similar features with such dimers.?

It is important to note that: in this work, we aim merely to provide a basic and the *first*, “proof of principle”, demonstration of an important utilization of plasmon index-enhancement (control) scheme [20] for CR applications. We present the validity of the model we use in Appendix 6, which has already been demonstrated for nanostructures of similar sizes [38]. Some small deviations from the mean polarization method may certainly appear. But, here we aim to bring a new and influential mechanism into light with basic and valid arguments (Appendix 2).

The paper is organized as follows. In Section. 2, we describe how the (auxiliary) x-polarized pulse can control the refractive index of the medium for an x-polarized light propagation. We review the results of Ref. [20], where a single frequency was considered, and generalize them to the case wherer a Gaussian control pulse is employed. In Section 3, we present the utilization of the refractive index control to the continuous spectral analysis of a radioactive source emitting charged particles. Section 4 contains our summary.

## 2 Index enhancement

Lavrinenko and colleagues [37] showed that the response of a y-aligned metal nanorod, to a field of frequency *ω*, can be enhanced when it is coupled to a perpendicular (x-aligned) nanorod which is driven by an x-polarized auxiliary pulse of the same frequency *ω*. Owing to the structure of the localized surface plasmon resonances, the x-polarized aux pulse couples (most efficiently) merely to the x-aligned nanorod, and not to the y-aligned nanorod. The y-polarized nanorod, however, couples to the plasmon mode of the x-aligned nanorod at the hotspot which appears at the intersection of the two nanorods. Ref. [37] showed that (see Figure 2) the polarization of the y-aligned nanorod – which can be driven only with a y-polarized light – responds to the incident field *E*_{1} with an enhanced polarization about a given frequency Ω=0.967*ω*_{0}. *ω*_{0} is the resonance of the nanorods. At *ω*=Ω, the polarization is enhanced with a vanishing absorption, a phenomenon also observed in atomic ensembles using aux microwave [29], [31] or optical [48], [49] pulses. Although Ref. [37] demonstrates the phenomenon for the coupling of two (identical) nanorods, as becomes apparent below, the phenomenon can be observed for the coupling of different plasmon resonances with other configurations.

The system of two coupled nanorods is treated by two coupled oscillators, as, this model is sufficient to explain almost all fundamental plasmon interaction behaviors, for example, in Fano resonances [50], [51], except second-quantized features [52]. The results are also supported with the exact solution of three-dimensional (3D) Maxwell equations, i.e. finite difference time domain (FDTD) simulations.

The polarizations (plasmon oscillations) of the plasmon modes of the two perpendicular nanorods can be described with two coupled oscillators:

where *γ*_{1}=*γ*_{1}=0.026*ω*_{0}, *ω*_{1}=*ω*_{2}=*ω*_{0} and *g*=0.06*ω*_{0} [37] are the damping rates, resonances, and the coupling between the two plasmon oscillations of the two nanorod, respectively.^{[3]}*ω*_{0} is the resonance of both nanorods. The second nanorod is driven by the field *x*_{1}/*F*_{1} is investigated, in Ref. [37], when a y-polarized electric field *ω*. In such a case an analytical solution for the plasmon oscillation *x*_{1}(*t*)=*x*_{1}*e*^{−}^{iωt}

can be obtained with *x*_{1} is plotted in Figure 2 [37] for *ϕ*=*μ*/2. Polarization density, (dipole moment)/volume [53], of the medium to a y-polarized field is given by

where *f* is the dimensionless oscillator strength determined by the density of the dimers. We calculate *f* as follows. The polarizibility of a nanorod (ellipsoid), in units of volume, can be calculated analytically [54], [55] as *α*(*ω*)=*ν*(*ε*(*ω*)–*ε _{h}*)/[

*ε*+

_{h}*r*(

*ε*(

*ω*)–

*ε*)] with

_{h}*r*=(1–

*e*

^{2})/

*e*

^{2}{–1+

*ln*[(1+

*e*)/(1–

*e*)]/2

*e*}. Here,

*ν*is the volume of the nanorod and

*e*

^{2}=1-

*b*

^{2}/

*L*

^{2}.

We use *L*=30 nm for the length and *b*=10 nm the width of the ellipsoid as considered in Ref. [37]. Polarization density for applied E-field (without enhancement) can be obtained via *χ*(*ω*)=*P*/*E*=*ρα*, where *ρ* is the number density of such nanorods, which we set *ρν*=0.02. Here, we also use the experimental dielectric function *ε*(*ω*) of silver [37], [56] (*γ*_{1}=*γ*_{2}=0.026*ω*_{0}) and obtain *f*=0.23 in Eq. (4) by setting *χ*(*ω*=*ω*_{0}, *g*=0)=*f*/(*γ*_{1}/*ω*_{0}) equal to *χ*(*ω*_{0})=*ρα*(*ω*_{0}). Actually, for a “proof of principle" demonstration of the benefits of the index-enhancement scheme for particle detectors, or CR imaging, such a tidy choice for *ε*(ω) is not necessary.

At one point, we differentiate from the scheme of Ref. [37], i.e. given in Figure 2, a little. Because, the E-field induced via CR is not like probing the y-polarization oscillations (in the first nanorod) with a relatively small *E*_{1} field. The CR is emitted spontaneously when the particle velocity along a direction exceeds the phase velocity of light. That is, we cannot simply consider sending a y-polarized probe pulse at frequency *ω*. For this reason, we use an aux pump pulse of Gaussian shape in the frequency domain of spectral width Δ*ω*=0.02*ω*, in Figure 3B, and Δ*ω*=0.0052*ω* in Figure 3, i.e. *x*_{1} by simply solving Eqs. (1–2) in the frequency domain *x*_{1}(*t*)=∑_{ω}x_{1}(*ω*)*e*^{−}* ^{iωt}*.

^{[4]}

Figure 3A.i and B.i plot the real and imaginary parts of the susceptibility of a dielectric medium composed of such silver dimers with density *ρν*=0.02, when the x-polarized pulse is off and on, respectively. The carrier frequency of the aux pulse, of width Δ*ω*=0.02*ω*_{0}, is chosen to coincide with the frequency in Figure 2 where index enhancement with vanishing absorption appears Ω=0.967*ω*_{0}. In Figure 4A.i, we use a sharper Δ*ω*=0.005*ω*_{0} aux pulse and we decrease (tune) the *E*_{2}/*E*_{1} ratio in Figure 4B.i. We observe that the index of such a medium can be “continuously-tuned” order of magnitude at a narrow frequency range, if desired, at which one looks after observing tunable CR. Figure 3A.ii, B.ii, 4A.ii and B.ii are the CR intensities for particles moving at different speeds, possible via such a tuning. (The calculations are carried out in Section 3.)

The ratio *E*_{2}/*E*_{1} is a quantity that can be measured in an experiment. One can note that in an experiment, one does not need to tune the *E*_{2}/*E*_{1} ratio. The *E*_{2}/*E*_{1} ratio is, rather, needed for the analysis of the particles’ velocity distribution, see Appendix 1 for some detailed discussion.

We remark that tuning the *E*_{2}/*E*_{1} ratio between *E*_{2}=100*E*_{1} and *E*_{2}=10*E*_{1}, in Figure 4A.i and B.i, does not change the frequency where imaginary part of the dielectric function vanishes, i.e. *ω*=Ω=0.967*ω*_{0}. This enables the continuous-tuning of the refractive index by circumventing the metallic losses causing the distortion in Cherenkov angle relation [14], [39].

In Figure 2, one may appreciate that index enhancement is observable in a wider frequency range, of order *γ*_{1,2}. This allows one to choose a considerably wider range of frequencies for the index enhancement in processes where increased absorption is tolerable, i.e. when a smaller (partially absorbed) CR signal is sufficient where otherwise particle velocity is much below the cutoff.

## 3 Control of the CR

We calculate the total number of photons emitted via CR, per unit traveling distance, using the well-developed formula

generalized to a dispersive medium [57], [58], [59], [60]. Here, the constraint *ν*/(*c*/*n*(*ω*))>1 is modified as *β*^{2}Re[*ε*(*ω*)]>1 [57], [58] for an absorbing medium. In Appendix 2, we present the refractive index for dielectric media composed of nanoparticles of sizes much smaller than the radiation wavelength. Such a material can be treated as an effective, homogeneous medium, so transition radiation can be neglected [39], [61], [62]. Because such media are shown to be homogenized to a very good approximation [63], [64]. Central region of the medium, given in Figure 1, can be unfilled to allow for unobstructed propagation of charged particles [39].

In Figures 3A.ii, B.ii, 4A.ii and B.ii, we plot the number of photons CR emitted at different frequencies by particles traveling at different speeds for the choice of a pump phase *ϕ*=*π*/2 [37].

When there is no aux field, particle velocities down to *ν*=0.55*c* is possible to emit a y-polarized CR, see Figure 3A.ii, within the dimer density we choose. When the aux field is turned on, by contrast, emission of a y-polarized CR from particles moving at speeds down to *ν*=0.1*c* becomes possible^{[5]},^{[6]}, see Figure 3B.ii, via enhancement of the dielectric susceptibility *χ*=*P*/*E*, see Figure 3B.i. What more important is: the velocity threshold can be tuned *continuously* by the intensity of the aux Gaussian pulse, for example, compare Figure 4A.ii and B.ii, after the detector is manufactured. The spectrum of the CR can also be tuned via tuning the spectral width and carrier frequency of the aux pulse even after the dimer ensemble is manufactured. The frequency where absorption vanishes, i.e. *ω*=Ω=0.967*ω*_{0}, does not change between *E*_{2}=100*E*_{1} and *E*_{2}=10*E*_{1} values. So, an aux pulse of carrier frequency *ω _{c}*=Ω=0.967

*ω*

_{0}works in the absorption-free regime in the continuous tuning of the aux pulse (∝|

*E*

_{2}|

^{2}) intensity.

In Figures 2–4, we consider the phase *ϕ*=*π*/2 for the aux pulse. The position of a dimer with respect to the aux pulse, however, alters the position-dependent phase at which the dimer operates. In Figure 5, we plot the CR intensities for dimers operating at different pump phases *ϕ*. When the periodicity of the dimers is not manufactured to match certain ratios of the aux pulse wavelength; a CR, possible to be emitted, is likely to feel a *ϕ*-averaged index.

In Figure 2 we present the enhancement scheme for *ϕ*=*π*/2 [37] and consider the enhancement around *ω*=Ω≃0.967*ω*_{0}. We remark that when *ϕ* assigns other values, the enhancement scheme, presented in Figure 2, changes dramatically. At the frequency *ω*=Ω≃0.967*ω*_{0}, a value we choose referring to Figure 2, may not correspond to an enhanced susceptibility. In Figure 5, at *ϕ*/*π*=0.5, i.e. *ϕ*=*π*/2, all particle velocities *ν*=0.1…0.3 generate CR. For a particle of velocity, for example, *ν*=0.1*c*, however, one cannot observe the CR in the phase range *ϕ*=0…0.18*π*. Because in this phase range, according to Eq. (4), the refractive index is not that high to suffice for creating a CR for a particle of velocity *ν*=0.1*c*. For higher velocity values, for example, *ν*=0.2*c*, the refractive index, at *ω*=Ω≃0.967*ω*_{0}, becomes sufficient to generate CR emission in the phase range *ϕ*=0.04…0.96*π* via Eqs (4) and (5). We also underline that the relative-phase, a single dimer operates, depends also on the *E*_{1}-field CR produces, i.e. not merely on the position of the dimer.

In Figure 6, we plot the CR emission at different particle velocities by considering the energy distribution of a ^{18}*F* [44], an isotope widely used, for example, in imaging applications [25], [65]. In Figure 6B, we also perform an average over the phase *ϕ* considering a (general) periodic structure whose periodicity does not match the aux pulse wavelength. We also note that dimers (~30 nm) are much smaller compared to the CR emission, at optical wavelengths, for example ~500 nm, which overlaps many of such dimers in 3D.

In Figure 7, we plot the CR angle *θ*_{CR} from slower-moving particles depend on the pump-phase dramatically, comparing the results for *ϕ*=0.50*π* with *ϕ*=0.05*π*.^{[7]} Therefore, for periodic structures, where even a wavelength contains hundreds of nanodimers, one needs to consider an average over the phase (dimer positions) *ϕ*, before calculating the CR angle relations. This may require calibration with a known source before carrying out the actual experiment. Therefore, beyond observing CR from slow-moving particles and CR intensity enhancement, *θ*_{CR} analysis is better to be held with periodic structures where a *ϕ*-average can be considerable for optical CR.

*Measurement scheme* – We note that the propagation of the aux-pulse-controlled CR emission, which is y-polarized, can be only in the x-z plane. As the incident particles are propagating along the x-direction, the controllable (continuously-tunable) CR emission is tilted along the z-direction via a CR emission angle *θ*_{CR}. In order to observe the distribution of the CR angle, one can filter-in (collect) the y-polarized CR emission. At this stage, there appears a complicating effect: high-velocity particles can also emit y-polarized CR whose origin is not related with the y-polarized nanorod holding a controllable index, for example, due to the background solution itself. This complication (actually any such effect leading to aux pulse-invariant CR emission) can be traced out from the CR via the index control phenomenon as follows. CR angle measurement can be carried out (calibrated) for a turned off auxiliary (y-polarized index-controlling) pulse and all other particle spectroscopies can be carried out referring this calibration point.

## 4 Summary

In summary, we show that index enhancement in plasmonic metamaterials [37] can be utilized to gain control over CR. An auxiliary x-polarized Gaussian pulse controls the refractive index of the medium for a y-polarized Cherenkov emission. Beyond enhancing the overall CR and enabling radiation from much slower particles; such a setup enables spectral analysis of the particle velocities as if there exist many different particle detectors. Because, different intensity values of the auxiliary pulse operates different particle detector media and this allows the data correction [43]. Moreover, vanishing absorption heals the distorting effects on the Cherenkov angle relation which are caused by the absorption of metal nanostructures.

In addition, the frequency width of the CR can also be tuned with the width of the auxiliary pulse. The utility of the technique, we introduce here, can be well appreciated by considering that: the phenomenon of CR has implementations in particle detection and medical imaging.

## Acknowledgments

Contribution of the authors MG, YLC and MET are 25%, 25% and 50%, respectively. We thank Mehmet Yildiz for instructive discussions. YLC would like to acknowledge the support of Ministry of Science and Technology (MOST) of Taiwan. MET gratefully thanks Atac Imamoglu, Manuela Weber-Semler and Mehmet Erbudak for their hospitality. MET and MG acknowledge support from TUBITAK 1001 No: 117F118, Funder Id: http://dx.doi.org/10.13039/501100004410. MET acknowledges support from TUBA GEBIP 2017.

# Appendix

## 1 The *E*_{2}/*E*_{1} ratio

There are two ways to perform an *E*_{2}/*E*_{1} arrangement.

Before using the manufactured medium with the auxiliary pulse, one can learn the properties of the medium without the aux pulse. One needs to perform this only once after manufacturing the detector. One can use an already known source for the detector calibration and obtain the curve in Figure 3A.ii, for example, by recording at a specific wavelength. Then, in the same calibration process, one can turn on the aux Gaussian pulse, for a trial *E*_{2}, and records the new *E*_{1}. This way, one can gain an initial information on the CR response/characteristics of the device before using it with unknown velocity particles.

Actually, tuning (knowing) *E*_{2}/*E*_{1} before an experiment is not necessary. Because the purpose of this new device is to provide a continuous spectral analysis on the velocity distribution of the particles, for example, with known (measured) *E*_{2}/*E*_{1} values. *E*_{2} and *E*_{1} (*I*_{2} and *I*_{1}) are measured in an experiment with an unknown radioactive source. Hence, one already records the continuous data (parameter) of *E*_{2}/*E*_{1}. One can also compare them with the data already obtained from the calibration with the known source, even though this is not necessary. Besides the continuous spectral analysis of the unknown source – which is the true aim of the proposed device – if one aims to tune the *E*_{2}/*E*_{1} ratio, for example, wants to operate the detector at a specific regime, the experimentalist can actually do this via a continuous measurement of the *E*_{2}/*E*_{1} ratio, for example, at a specific wavelength.

## 2 Effective polarizibility and CR

CR in particle detectors, employing nanoparticles, is a well-developed formalism which has been studied for a decade. The effective medium/polarization methods, which rely on calculating the average electric/polarization fields [17], [18], [19], [20], [21], [39], [40], [66], [67], [68], [69], have already been demonstrated to work well in nanowire/nanoparticle arrays in such parameter regimes where the size of the nanostructures are much smaller than the CR wavelength.

These methods are able to treat also the periodicity of the wires, for example, by inserting the Floquet mode solutions (due to periodicity) into polarization/electric field averages. While many recent works concentrate on effective permittivity of periodic nanoparticle structures, owing to the unique optical features of such metamaterials, Ref. [38] presents a clear demonstration of the effective polarization for the arbitrary distribution of, for example, silicon, nanoparticles in a medium where the theoretical treatment fits the experimental data. Thus, although we consider a periodic structure of the silver dimers here, arbitrarily distributions of such dimers, for example, in solutions, can also be utilized as index-controlled continuously-tunable particle detectors. Dimers can be aligned using magnetized metallic material [46], [47].

We calculate the average polarization of the medium, for example, in a wavelength size, which contains many of such nanoparticles. The maximum length of a dimer is *L*=30 nm, which is much smaller than an optical wavelength. The mean separation between the centers of the dimers is crudely *ρν*=0.02 of such dimers. We note that this, i.e. *ρν* =0.02, is an arbitrary value we assign in the manuscript, where a denser/lighter ensemble would be possible. The volume of a dimer is approximately 30×30×10=8100 nm^{3}. The volume of a typical wavelength is V* _{λ}*=500

^{3}nm

^{3}. Hence, in a single wavelength volume there exists 310 of such dimers. This value can be increased/decreased via other choices for the density of dimers. That is, for 310 of such dimers in a wavelength, assigning an effective polarization (even for arbitrarily-ordered dimers) is rather like defining a polarization density in the books of electromagnetic theory [53], [70].

Actually, for the validity of the simple treatments of CR in such media, the critical issue is a concept related with the “length of formation” [38], [71], [72]. A Cherenkov wave takes on the character of a radiation, that is no longer tied to its source, after the particle travels the formation length [38], [71], [72] which is order of a wavelength or less. That is, the condition for CR is (i) material to be extended for some degree (i.e. formation length) for the CR to be formed, and (ii) in the packed nanoparticle system it must be provided that nanoparticle dimensions are much shorter than the propagation wavelength. Satisfying these conditions, a packed nanoparticle system can be treated as an effective medium in the context of formation [13]. Thus, for CR to emit, the medium has to extend at least a formation length; i.e. a single nanoparticle alone smaller than the formation length cannot emit the CR and a continuation of such particles is necessary. After this condition satisfied, an effective medium treatment regarding the wavelength of the virtual photons to be emitted, for example, 310 dimers in a wavelength volume, can be considered.

On top of all these discussions, there will certainly appear complicating effects. These, however, are certainly out of the scope of this work which aims to present the audience a basic (as the first) demonstration of the implementation of the index-enhancement phenomenon to particle detectors allowing continuously-tunable spectral analysis – not targeting a rigorous simulation, revealing all effects, in an already conducted experiment.

## References

[1] Cherenkov PA. Visible emission of clean liquids by action of *γ* radiation. Doklady Akademii Nauk SSSR 1934;2:252.Search in Google Scholar

[2] Frank I, Tamm I. Coherent Visible Radiation of Fast Electrons Passing Through Matter. In: Bolotovskii B.M., Frenkel V.Y., Peierls R., eds. Selected Papers. Berlin, Heidelberg, Springer, 1991.10.1007/978-3-642-74626-0Search in Google Scholar

[3] Landau LD, Pitaevskii LP, Lifshitz EM. Electrodynamics of continuous media. Oxford, UK, Butterworth-Heinemann, 1994:8. https://www.elsevier.com/books/electrodynamics-of-continuous-media/landau/978-0-08-057060-0Search in Google Scholar

[4] Fisch EE, Walsh JE. Operation of the sapphire Cerenkov laser. Appl Phys Lett 1992;60:1298–300.10.1063/1.107322Search in Google Scholar

[5] Owens IJ, Brownell JH. Compact superradiant Čerenkov source. J Appl Phys 2005;97:104915.10.1063/1.1901840Search in Google Scholar

[6] Ahmed SN. Physics and engineering of radiation detection. San Diego, CA, USA, Elsevier, 2014.Search in Google Scholar

[7] IceCube Collaboration, Aartsen MG, Abbasi R, et al. Evidence for high-energy extraterrestrial neutrinos at the icecube detector. Science 2013;342:1242856.10.1126/science.1242856Search in Google Scholar PubMed

[8] Yeatman EM, Gramling HM, Wang EN. Introduction to the special topic on nanomanufacturing. Microsyst Nanoeng 2017;3:17079.10.1038/micronano.2017.79Search in Google Scholar PubMed PubMed Central

[9] Xu Y, Ji D, Song H, et al. Light-matter interaction within extreme dimensions: From nanomanufacturing to applications. Adv Opt Mater 2018;6:1800444.10.1002/adom.201800444Search in Google Scholar

[10] Tame MS, McEnery KR, Ö¨zdemir ŞK, Lee J, Maier SA, Kim MS. Quantum plasmonics. Nat Phys 2013;9:329–40.10.1038/nphys2615Search in Google Scholar

[11] Liu S, Zhang P, Liu W, et al. Surface polariton Cherenkov light radiation source. Phys Rev Lett 2012;109:153902.10.1103/PhysRevLett.109.153902Search in Google Scholar PubMed

[12] Georgescu I. Čerenkov radiation: Light from ripples. Nat Phys 2012;8:704.10.1038/nphys2447Search in Google Scholar

[13] Zhao T, Zhong R, Hu M, et al. Cherenkov radiation via surface plasmon polaritons excitation by an electron beam in a layered metal-dielectric structure. Eur Phys J D 2015;69:120.10.1140/epjd/e2015-50829-9Search in Google Scholar

[14] Lin X, Easo S, Shen Y, et al. Controlling Cherenkov angles with resonance transition radiation. Nat Phys 2018;14:816–21.10.1364/CLEO_QELS.2018.FM4J.6Search in Google Scholar

[15] Luo C, Ibanescu M, Johnson SG, Joannopoulos JD. Cerenkov radiation in photonic crystals. Science 2003;299:368–71.10.1126/science.1079549Search in Google Scholar PubMed

[16] Liu W. Dispersive 2d Cherenkov radiation on a dielectric nano-film. Sci Rep 2017;7:5787.10.1038/s41598-017-06176-1Search in Google Scholar PubMed PubMed Central

[17] Vorobev VV, Tyukhtin AV. Nondivergent Cherenkov radiation in a wire metamaterial. Phys Rev Lett 2012;108:184801.10.1103/PhysRevLett.108.184801Search in Google Scholar PubMed

[18] Tyukhtin AV, Vorobev VV, Galyamin SN. Radiation excited by a charged-particle bunch on a planar periodic wire structure. Phys Rev Special Top-Ac Beams 2014;17:122802.10.1103/PhysRevSTAB.17.122802Search in Google Scholar

[19] So J-K, Won J-H, Sattorov MA, et al. Cerenkov radiation in metallic metamaterials. Appl Phys Lett 2010;97:151107.10.1063/1.3492846Search in Google Scholar

[20] Fernandes DE, Maslovski SI, Silveirinha MG. Cherenkov emission in a nanowire material. Phys Rev B 2012:85;155107. (Also see the references therein.)10.1103/PhysRevB.85.155107Search in Google Scholar

[21] Silveirinha MG. Metamaterials: a low-energy Cherenkov glow. Nat Photonics 2017;11:269–71.10.1038/nphoton.2017.59Search in Google Scholar

[22] Liu F, Xiao L, Ye Y, et al. Integrated Cherenkov radiation emitter eliminating the electron velocity threshold. Nat Photonics 2017;11:289–92.10.1038/nphoton.2017.45Search in Google Scholar

[23] Lu J, Grzegorczyk TM, Zhang Y, et al. Čerenkov radiation in materials with negative permittivity and permeability. Opt Express 2003;11:723–34.10.1364/OE.11.000723Search in Google Scholar PubMed

[24] Xi S, Chen H, Jiang T, et al. Experimental verification of reversed Cherenkov radiation in left-handed metamaterial. Phys Rev Lett 2009;103:194801.10.1103/PhysRevLett.103.194801Search in Google Scholar PubMed

[25] Shaffer TM, Pratt EC, Grimm J. Utilizing the power of Cerenkov light with nanotechnology. Nature Nanotechnol 2017;12:106–17.10.1038/nnano.2016.301Search in Google Scholar PubMed PubMed Central

[26] Luk’yanchuk B, Zheludev NI, Maier SA, et al. The Fano resonance in plasmonic nanostructures and metamaterials. Nat Mater 2010;9:707–15.10.1038/nmat2810Search in Google Scholar PubMed

[27] Limonov MF, Rybin MV, Poddubny PN, Kivshar YS. Fano resonances in photonics. Nat Photonics 2017;11:543–54.10.1038/nphoton.2017.142Search in Google Scholar

[28] Peng B, Özdemir ŞK, Chen W, Nori F, Yang L. What is and what is not electromagnetically induced transparency in whispering-gallery microcavities. Nat Commun 2014;5:5082.10.1038/ncomms6082Search in Google Scholar PubMed

[29] Fleischhauer M, Keitel CH, Scully MO, Su C, Ulrich BT, Zhu SY. Resonantly enhanced refractive index without absorption via atomic coherence. Phys Rev A 1992;46:1468–87.10.1103/PhysRevA.46.1468Search in Google Scholar

[30] Fleischhauer M, Imamoglu A, Marangos JP. Electromagnetically induced transparency: optics in coherent media. Rev Mod Phys 2005;77:633.10.1103/RevModPhys.77.633Search in Google Scholar

[31] Scully MO, Zubairy MS. Quantum optics. New York: Cambridge University Press, 1997.10.1017/CBO9780511813993Search in Google Scholar

[32] Panaro S, Nazir A, Liberale C, et al. Dark to bright mode conversion on dipolar nanoantennas: a symmetry-breaking approach. ACS Photonics 2014;1:310–4.10.1021/ph500044wSearch in Google Scholar

[33] Butet J, Martin OJF. Fano resonances in the nonlinear optical response of coupled plasmonic nanostructures. Opt Express 2014;22:29693–707.10.1364/OE.22.029693Search in Google Scholar PubMed

[34] Singh SK, Abak MK, Tasgin ME. Enhancement of four-wave mixing via interference of multiple plasmonic conversion paths. Phys Rev B 2016;93:035410.10.1103/PhysRevB.93.035410Search in Google Scholar

[35] Taşgın ME, Bek A, Postacı S. Fano resonances in the linear and nonlinear plasmonic response. In: Fano resonances in optics and microwaves. Cham, Switzerland, Springer Nature Switzerland AG, 2018.10.1007/978-3-319-99731-5_1Search in Google Scholar

[36] Postaci S, Yildiz BC, Bek A, Tasgin ME. Silent enhancement of sers signal without increasing hot spot intensities. Nanophotonics 2018;7:1687–95.10.1515/nanoph-2018-0089Search in Google Scholar

[37] Panahpour A, Mahmoodpoor A, Lavrinenko AV. Refraction enhancement in plasmonics by coherent control of plasmon resonances. Phys Rev B 2019;100:075427.10.1103/PhysRevB.100.075427Search in Google Scholar

[38] Yurtsever A, Couillard M, Muller DA. Formation of guided cherenkov radiation in silicon-based nanocomposites. Phys Rev Lett 2008;100:217402.10.1103/PhysRevLett.100.217402Search in Google Scholar PubMed

[39] Ginis V, Danckaert J, Veretennicoff I, Tassin P. Controlling cherenkov radiation with transformation-optical metamaterials. Phys Rev Lett 2014;113:167402.10.1103/PhysRevLett.113.167402Search in Google Scholar PubMed

[40] Morgado TA, Fernandes DE, Silveirinha MG. Analytical solution for the stopping power of the cherenkov radiation in a uniaxial nanowire material. Photonics 2015;2:702–18.10.3390/photonics2020702Search in Google Scholar

[41] Carusotto I, Artoni M, La Rocca GC, Bassani F. Slow group velocity and Cherenkov radiation. Phys Rev Lett 2001;87:064801.10.1103/PhysRevLett.87.064801Search in Google Scholar PubMed

[42] Artoni M, Bassani F, Carusotto I, La Rocca GG. The regime of electromagnetically induced transparency in optically dense media: from atoms to excitons. Brazilian J Phys 2002;32:275–83.10.1590/S0103-97332002000200007Search in Google Scholar

[43] Tanny S, Nicholas Sperling N, Parsai EI. Correction factor measurements for multiple detectors used in small field dosimetry on the varian edge radiosurgery system. Med Phys 2015;42:5370–6.10.1118/1.4928602Search in Google Scholar PubMed

[44] Levin CS, Hoffman EJ. Calculation of positron range and its effect on the fundamental limit of positron emission tomography system spatial resolution. Phys Med Biol 1999;44:781–99.10.1088/0031-9155/44/3/019Search in Google Scholar PubMed

[45] Li K, Li J, Reardon C, et al. High speed e-beam writing for large area photonic nanostructures – a choice of parameters. Sci Rep 2016;6:32945.10.1038/srep32945Search in Google Scholar PubMed PubMed Central

[46] Anderson SD, Gwenin VV, Gwenin CD. Magnetic functionalized nanoparticles for biomedical, drug delivery and imaging applications. Nanoscale Res Lett 2019;14:1–16.10.1186/s11671-019-3019-6Search in Google Scholar PubMed PubMed Central

[47] Alrasheed S, Di E. Giant magnetic field enhancement in hybridized mim structures. IEEE Photonics Technol Lett 2017;29:2151–4.10.1109/LPT.2017.2765402Search in Google Scholar

[48] Yavuz DD. Refractive index enhancement in a far-off resonant atomic system. Phys Rev Lett 2005;95:223601.10.1103/PhysRevLett.95.223601Search in Google Scholar PubMed

[49] Proite NA, Unks BE, Green JT, Yavuz DD. Refractive index enhancement with vanishing absorption in an atomic vapor. Phys Rev Lett 2008;101:147401.10.1103/PhysRevLett.101.147401Search in Google Scholar PubMed

[50] Wu X, Gray SK, Pelton M. Quantum-dot-induced transparency in a nanoscale plasmonic resonator. Opt Express 2010;18:23633–45.10.1364/OE.18.023633Search in Google Scholar PubMed

[51] Lovera A, Gallinet B, Nordlander P, Martin OFJ. Mechanisms of Fano resonances in coupled plasmonic systems. Acs Nano 2013;7:4527–36.10.1021/nn401175jSearch in Google Scholar PubMed

[52] Finazzi M, Ciccacci F. Plasmon-photon interaction in metal nanoparticles: Second-quantization perturbative approach. Phys Rev B 2012;86:035428.10.1103/PhysRevB.86.035428Search in Google Scholar

[53] Griffiths DJ. Introduction to Electrodynamics. 2005.10.1016/B978-1-85573-953-6.50026-XSearch in Google Scholar

[54] Bohren CF, Huffman DR. Absorption and scattering of light by small particles. John Wiley & Sons, 2008.Search in Google Scholar

[55] Ngom M. Novel approach to the study of surface plasmon resonance and field enhancement properties of Noble metal nanostructures. Ph.D. thesis, The University of Michigan, 2009.Search in Google Scholar

[56] Hohenester U, Trügler A. Mnpbem – a matlab toolbox for the simulation of plasmonic nanoparticles. Comput Phys Commun 2012;183:370–81.10.1016/j.cpc.2011.09.009Search in Google Scholar

[57] Grichine VM. On the energy-angle distribution of Cherenkov radiation in an absorbing medium. Nucl Instrum Meth Phys Res A: Accelerators, Spectrometers, Detectors and Associated Equipment 2002;482:629–33.10.1016/S0168-9002(01)01927-1Search in Google Scholar

[58] Saffouri MH. Treatment of Čerenkov radiation from electric and magnetic charges in dispersive and dissipative media. Il Nuovo Cimento D 1984;3:589–622.10.1007/BF02457771Search in Google Scholar

[59] Fermi E. The ionization loss of energy in gases and in condensed materials. Phys Rev 1940;57:485.10.1103/PhysRev.57.485Search in Google Scholar

[60] Sternheimer RM. The energy loss of a fast charged particle by Čˇerenkov radiation. Phys Rev 1953;91:256.10.1103/PhysRev.91.256Search in Google Scholar

[61] Duan Z, Wu B-I, Xi S, Chen H, Chen M. Research progress in reversed Cherenkov radiation in double-negative metamaterials. Prog Electromagn Res 2009;90:75–87.10.2528/PIER08121604Search in Google Scholar

[62] Galyamin SN, Tyukhtin AV, Kanareykin A, Schoessow P. Reversed cherenkov-transition radiation by a charge crossing a left-handed medium boundary. Phys Rev Lett 2009;103:194802.10.1103/PhysRevLett.103.194802Search in Google Scholar PubMed

[63] Căbuz AI, Nicolet A, Zolla F, Felbacq D, Bouchitté G. Homogenization of nonlocal wire metamaterial via a renormalization approach. J Opt Soc Am B 2011;28:1275–82.10.1364/JOSAB.28.001275Search in Google Scholar

[64] Căbuz AI, Felbacq D, Cassagne D. Homogenization of negative-index composite metamaterials: a two-step approach. Phys Rev Lett 2007;98:037403.10.1103/PhysRevLett.98.037403Search in Google Scholar PubMed

[65] Spinelli AE, D’Ambrosio D, Calderan L, Marengo M, Sbarbati A, Boschi F. Cerenkov radiation allows in vivo optical imaging of positron emitting radiotracers. Phys Med Biol 2009;55:483–95.10.1088/0031-9155/55/2/010Search in Google Scholar PubMed

[66] Silveirinha MG. Nonlocal homogenization model for a periodic array of *E*-negative rods. Phys Rev E 2006;73:046612.10.1103/PhysRevE.73.046612Search in Google Scholar
PubMed

[67] Silveirinha MG, Fernandes CA. Homogenization of 3-d-connected and nonconnected wire metamaterials. IEEE Transactions Microw Theory Techniques 2005;53:1418–30.10.1109/TMTT.2005.845128Search in Google Scholar

[68] Tyukhtin AV, Doilnitsina EG. Effective permittivity of a metamaterial from coated wires. J Phys D Appl Phys 2011;44:265401.10.1088/0022-3727/44/26/265401Search in Google Scholar

[69] Nefedov IS, Viitanen AJ, Tretyakov SA. Propagating and evanescent modes in two-dimensional wire media. Phys Rev E 2005;71:046612.10.1103/PhysRevE.71.046612Search in Google Scholar PubMed

[70] Jackson JD. Classical electrodynamics, 3rd edition. New York, NY, USA, John Wiley & Sons, 1999.Search in Google Scholar

[71] Zolotorev MS, McDonald KT. Classical radiation processes in the Weizsacker-Williams approximation. arXiv Preprint Physics 2000;0003096.Search in Google Scholar

[72] Klein S. Suppression of bremsstrahlung and pair production due to environmental factors. Rev Mod Phys 1999;71:1501.10.1103/RevModPhys.71.1501Search in Google Scholar

**Received:**2020-02-02

**Revised:**2020-03-25

**Accepted:**2020-03-30

**Published Online:**2020-04-30

© 2020 Mehmet Emre Tasgin et al., published by De Gruyter, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.