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.

Figure 1:

Silver nanoparticle dimers, aligned in the x and y directions, are illuminated with an x-polarized auxiliary Gaussian pulse of frequency Ω.

The x-aligned silver nanoparticles interact with the incident field very strongly to produce nm-size hotspots and yields an enhancement in the polarization of the y-aligned nanoparticle via strong interaction between the two nanoparticles. A slow-moving charged particle traveling in the medium [20], [21], [38], [39], [40] emits a y-polarized CR via the index enhancement. Polarization density when the auxiliary pulse is off and on are given in Figure 3A.i and B.i, respectively.

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 (E₂) 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.

Figure 2:

Enhanced oscillation strength [37] of the y-aligned nanorod in the presence of an auxiliary pulse E₂, oscillating at the same frequency with the probe pulse E₁(t)=E₁e^−iωt.

Without the enhancement scheme, maximum x₁ attains a value of 0.2 only. At Ω=0.967ω₀ the polarization is enhanced with vanishing absorption. The phase of the aux pump is fixed to ϕ=π/2, in Eq. (3), as in Ref. [37].

We also present the contribution to the power of CR from different particle velocities of an ¹⁸F 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 E₁ with an enhanced polarization about a given frequency Ω=0.967ω₀. ω₀ 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 γ₁=γ₁=0.026ω₀, ω₁=ω₂=ω₀ and g=0.06ω₀ [37] are the damping rates, resonances, and the coupling between the two plasmon oscillations of the two nanorod, respectively.^[3]ω₀ is the resonance of both nanorods. The second nanorod is driven by the field F˜2(t)=F2e−iϕe−iωt and the response of the x₁/F₁ is investigated, in Ref. [37], when a y-polarized electric field F˜1(t)=F1e−iωt is applied on the first nanorod oscillating with the same frequency ω. In such a case an analytical solution for the plasmon oscillation x₁(t)=x₁e^−iωt

can be obtained with δi=ωi2−ω2−iγiω.x₁ is plotted in Figure 2 [37] for F2/ω02=1,F1/ω02=0.01 and 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_h)] with r=(1–e²)/e²{–1+ ln[(1+e)/(1–e)]/2e}. Here, ν is the volume of the nanorod and e²=1-b²/L².

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] (γ₁=γ₂=0.026ω₀) and obtain f=0.23 in Eq. (4) by setting χ(ω=ω₀, g=0)=f/(γ₁/ω₀) equal to χ(ω₀)=ρα(ω₀). 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 E₁ 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 Δω=0.02ω, in Figure 3B, and Δω=0.0052ω in Figure 3, i.e. F˜2=∑ωF2e−iϕexp[−(ω−Ω)2/(Δω)2]e−iωt. We determine the response x₁ by simply solving Eqs. (1–2) in the frequency domain x₁(t)=∑_ωx₁(ω)e^−iωt.^[4]

Figure 3:

Susceptibility and Cherenkov radiation intensity without and with an auxiliary Gaussian pulse.

(A.i) and (B.i) plot the dielectric susceptibility χ of the medium (polarization density) without and with the auxiliary Gaussian pulse of width Δω=0.02ω₀, respectively. (B.i) Susceptibility is largely enhanced compared to the bare susceptibility of the metal nanodimer (A.i). (A.ii) and (B.ii) plot the intensity of the CR Eq. (5) for charged particles moving at different speeds β=ν/c. The electric field strength of the auxiliary pulse is assigned to be E₂=100E₁, with E₁ is the electric field of the Cherenkov emission. Enhanced susceptibility enables the radiation of particles moving at much smaller speeds. We set ϕ=π/2, in Eq. (4), similar to Ref. [37].

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 Δω=0.02ω₀, is chosen to coincide with the frequency in Figure 2 where index enhancement with vanishing absorption appears Ω=0.967ω₀. In Figure 4A.i, we use a sharper Δω=0.005ω₀ aux pulse and we decrease (tune) the E₂/E₁ 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 E₂/E₁ is a quantity that can be measured in an experiment. One can note that in an experiment, one does not need to tune the E₂/E₁ ratio. The E₂/E₁ ratio is, rather, needed for the analysis of the particles’ velocity distribution, see Appendix 1 for some detailed discussion.

We remark that tuning the E₂/E₁ ratio between E₂=100E₁ and E₂=10E₁, in Figure 4A.i and B.i, does not change the frequency where imaginary part of the dielectric function vanishes, i.e. ω=Ω=0.967ω₀. 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 β²Re[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=0.55c 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=0.1c 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=0.967ω₀, does not change between E₂=100E₁ and E₂=10E₁ values. So, an aux pulse of carrier frequency ω_c=Ω=0.967ω₀ works in the absorption-free regime in the continuous tuning of the aux pulse (∝|E₂|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.

Figure 4:

Tuning the velocity threshold of Cherenkov radiation.

(A.i) and (B.i) Enhancement of the polarization density with a narrower Δω=0.005ω₀ auxiliary pulse for two different (pump) electric field strengths, E₂=100E₁ and E₂=10E₁, respectively. (A.ii) and (B.ii) demonstrates the corresponding emission intensity. Radiation can be made in a (tunable) narrower spectrum via tuning the pump spectral width. Contribution from smaller particle velocities can also be tuned via the pump electric field E₂. We set ϕ=π/2, in Eq. (4), similar to Ref. [37].

Figure 5:

Change of the emission intensity of charged particles at the enhancement frequency, ω=Ω=0.967ω₀, for different phases ϕ of the auxiliary Gaussian pulse (or, equivalently, for different positions of the metal nanodimer).

Different pump phases result in different refractive indices. Aux pulse width is set to Δω=0.005ω₀.

In Figure 2 we present the enhancement scheme for ϕ=π/2 [37] and consider the enhancement around ω=Ω≃0.967ω₀. We remark that when ϕ assigns other values, the enhancement scheme, presented in Figure 2, changes dramatically. At the frequency ω=Ω≃0.967ω₀, 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=0.1c, however, one cannot observe the CR in the phase range ϕ=0…0.18π. 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=0.1c. For higher velocity values, for example, ν=0.2c, the refractive index, at ω=Ω≃0.967ω₀, becomes sufficient to generate CR emission in the phase range ϕ=0.04…0.96π via Eqs (4) and (5). We also underline that the relative-phase, a single dimer operates, depends also on the E₁-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 ¹⁸F [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.

Figure 6:

Cherenkov emission from 18F.

(A) Velocity distribution of ¹⁸F emission, a material widely used in biological imaging [25], [65]. (B) Radiated power of the particles emitted by ¹⁸F, at the enhancement frequency ω=Ω=0.967ω₀, for different field strengths E₂ of the auxiliary Gaussian pulse.

Distribution depicted in (A) is used in the calculation and an average is considered over the phase ϕ of the auxiliary pulse (or, equivalently, over the positions of the nanodimers.) Aux pulse width is set to Δω=0.005ω₀.

Figure 7:

Direction of the CR θ_Chrkv with respect to the velocity of the charged particle for different values of the phase ϕ of the Gaussian pulse.

Carrier frequency and the spectral width of the aux pulse are set to ω_c=Ω=0.0967ω₀ and Δω=0.005ω₀, respectively.

In Figure 7, we plot the CR angle cosθCR=Re[ϵ(ω)]/β|ϵ(ω)| in a dispersive medium [57], [58] emitted from a charged particle moving at different velocities. We observe that θ_CR from slower-moving particles depend on the pump-phase dramatically, comparing the results for ϕ=0.50π with ϕ=0.05π.^[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.