Without loss of generality, we consider a swift charged particle of velocity \(\overline{v}\) travelling parallel to the interface of an isotropic background medium (with a refractive index \(n_{a}\)) and a uniaxial birefringent crystal (with ordinary and extraordinary refractive indices denoted as \(n_{o}\) and \(n_{e}\), respectively). The distance between the particle trajectory and the interface is \(y_{0} = 200\) nm. The optical axis of birefringent crystal is orientated parallel to the interface, as shown in Fig. 1a. In this configuration, DSWs exist if \(n_{e} > n_{a} > n_{o}\) [41,42,43,44,45]. To satisfy this condition, we choose Si3N4 as the isotropic background medium and YVO4 as the birefringent crystal. The refractive indices of studied materials are \(n_{a} = 2.04\), \(n_{o} = 1.99\) and \(n_{e} = 2.22\). In general, DSWs are highly directional and exist only within small angular regions in four quadrants (i.e., \(\theta_{d}\), \(\pi - \theta_{d}\), \(\pi + \theta_{d}\), \(2\pi - \theta_{d}\)), where \(\theta_{d} \in \left[ {29.43^\circ , \;30.15^\circ } \right]\) is the angle between the phase velocity of DSWs and the optical axis of the YVO4 crystal. In this study, we focus our discussions on the behaviors of DSWs in the first quadrant. Owing to the mirror symmetry, our results also apply to DSWs in other three quadrants.
We begin our analysis by studying the excitation condition of DSWs by the swift charged particle. Denote \(\theta_{q}\) as the angle between the optical axis and the particle trajectory (Fig. 1a). We find that the excitation of Dyakonov surface modes requires \(\theta_{q}\) fulfilling the following condition (Additional file 1: Section S4):
$$\theta_{q} = \theta_{d} \pm \cos^{ - 1} \left( {\frac{\omega }{{k_{d} v}}} \right),$$
(1)
where \(\omega\) is the angular frequency; \(k_{d}\) is the magnitude of in-plane wavevector of the Dyakonov surface mode. Unlike conventional Cherenkov photons which can be produced by charged particles travelling along any direction (i.e. regardless of \(\theta_{q}\)), Dyakonov surface modes can be excited only for some specific \(\theta_{q}\).
The radiation field pattern from the swift charged particle is very susceptible to the magnitude and direction of the particle velocity in our platform. To illustrate this point and reveal the impact of DSWs, we plot in Fig. 1b–d radiation field patterns for three circumstances. When Eq. (1) is rigorously satisfied, the radiation mode is a superposition of the conventional Cherenkov photons and DSWs, as shown in Fig. 1b. In this case, the radiation field can penetrate deeply into the YVO4 crystal. Such a large penetration length results from the weak longitudinal confinement of DSWs on the interface [17]. In addition, surface Dyakonov–Cherenkov radiation also features an extremely asymmetric field pattern in the transverse plane (Additional file 1: Fig. S5). On the other hand, when the magnitude or direction of the particle velocity changes slightly such that Eq. (1) is no longer satisfied, the swift charged particle emits only conventional Cherenkov photons. As a result, the radiation field decays rapidly in the YVO4 crystal (Fig. 1c, d) and the field pattern becomes symmetric in the transverse plane (Additional file 1: Fig. S5). These results clearly demonstrate that the excitation of DSWs relies heavily on the particle trajectory. This property provides an effective way to determine simultaneously the particle velocity and trajectory for high-energy particles whose directions are oriented parallel to the surface of the birefringent crystal, through the direct measurement of the radiation field pattern.
Excitation of DSWs modifies not only the near-field pattern, but also the energy loss of the swift particle. As shown in Fig. 2, the power spectral density is quite susceptible to the particle velocity and trajectory, and increases dramatically when Eq. (1) is satisfied. To explore the underlying physical mechanism, we divide the total radiation power into two parts, i.e., the radiation loss \(G_{{{\text{ph}}}}\) through the emission of free-space Cherenkov photons and the radiation loss \(G_{{{\text{DSW}}}}\) through the excitation of DSWs.
Figure 2b, c clarify quantitatively the respective contributions of \(G_{{{\text{ph}}}}\) and \(G_{{{\text{DSW}}}}\) to the total power spectral density at λ = 0.635 µm. Our results show that \(G_{{{\text{ph}}}}\) and \(G_{{{\text{DSW}}}}\) display distinctly different responses to the particle velocity and trajectory. On the one hand, \(G_{{{\text{ph}}}}\) (as denoted as the straight dashed line) increases smoothly as the particle velocity increases (Fig. 2b) while at the same time remains almost a constant over a broad angular band (Fig. 2c). Such a behavior makes particle detection with traditional Cherenkov photons difficult. On the other hand, \(G_{{{\text{DSW}}}}\) is much more sensitive to small variations in particle velocity/trajectory and acquires a nonzero value only when v and θq strictly satisfy the condition given by Eq. (1) (see Fig. 2a and the bulges in Fig. 2b, c). The velocity range for nonzero \(G_{{{\text{DSW}}}}\) is generally smaller than 0.02c, e.g., from 0.502c to 0.498c, 0.596c to 0.604c, 0.694c to 0.706c, and 0.792c to 0.808c for \(\theta_{q} = 40.9^\circ\), \(64.8^\circ\), \(75.1^\circ\), and \(81.8^\circ\), respectively; the angular band for nonzero \(G_{{{\text{DSW}}}}\) is generally smaller than 1°, e.g., from 40.4° to 41.4°, 64.3° to 65.3°, 74.6° to 75.6°, and 81.3° to 82.3° for \(v = 0.5c\), \(0.6c\), \(0.7c\), and \(0.8c\), respectively. θq,max (i.e., θq at the maximum energy loss) shows excellent agreement with θq fulfilling Eq. (1). The enhanced sensitivity in energy loss offers an alternative approach to measure simultaneously the particle velocity and trajectory. We also reveal that the maximum achievable photon number \(N_{{{\text{DSW}}}} \left( {\theta_{{q,{\text{max}}}} } \right)\) of DSW generated per unit length of the particle path is greater than that \(N_{{{\text{ph}}}}\) of free-space Cherenkov radiation in the low-speed regime (i.e., 0.49c < v < 0.7c), while \(N_{{{\text{ph}}}}\) dominates the total photon number in the high-speed regime (i.e., 0.7c < v < c) (Fig. 2d). The consideration of realistic chromatic dispersion of materials will not affect our findings (Additional file 1: Figs. S9, S10).
The negligible chromatic dispersion and small dissipation loss of our structure can greatly enhance the photon extraction by means of DSWs, facilitating particle detection in the far field. To highlight this point, we compare the power flow density of surface Dyakonov–Cherenkov radiation with that of surface-polariton Cherenkov radiation [46]. In our comparison, the surface-polariton Cherenkov radiation is investiageted in the plasmonic system made of Aluminum Oxide (Al2O3) and Gold (Au), such that the corresponding SPPs have a propagation constant identical to that of DSWs at a wavelength of \(\lambda = 0.641{ }\) µm (Additional file 1: Fig. S8). Meanwhile, the distance from the particle trajectory and the interface is set as y0 = 0.09δ (with δ as the penetration depth of the surface mode in the superstrate) such that the interaction strengths between the swift charge and surface waves are the same for both configurations. Here, the dielectric constants of Al2O3 and Au are taken from the experimental data [47, 48]. Figure 3 demonstrates that DSWs are more strongly excited than conventional SPPs despite of their weak confinement along the longitudinal direction. First, the photon extraction efficiency of DSWs is much higher (i.e. the power flow density is an orders of magnitude larger) than that of conventional SPPs at \(l = 0\) μm, as shown in Fig. 3c. Second, DSWs have a much longer propagation distance and remain detectable in the far field, e.g. the Poynting power of DSWs attenuates less than 5% over a distance of 10 μm, while conventional SPPs have already faded away over such a distance. On the other hand, owing to the long penetration depth δDSW (22 times larger than that of SPPs at 0.641 μm), the excitation efficiency of DSWs is robust against the variation of y0, e.g. from 0 to 200 nm (Fig. 3b). In sharp contrast, the swift particle cannot efficiently excite SPPs when y0 > δSPP = 100 nm.
Finally, to demonstrate that surface Dyakonov–Cherenkov radiation can be used for the particle detection, we plot θq,max (i.e. θq at the maximum energy loss) as a function of the velocity/momentum of particles. As shown in Fig. 4, measuring θq,max not only determines the particle velocity, but also offers a possible route for the particle discrimination, e.g. θq,max for electron, pion, kaon and proton are 90.45°, 89.55°, 80.41° and 54.35°, respectively, at the momentum of 0.6 GeV/c. Such a strong variation in θq,max indicates that our configuration provides a high detection sensitivity when applied to particle detection.