Tunable optical topological transition of Cherenkov radiation

Chasing efficient methods to generate and manipulate Cherenkov radiation (CR) is always the lofty target of many scientific researchers. In this work, by combining the optical topological transition (OTT) of graphene hyperbolic metamaterials (GHMs), we study the properties and clarify the mechanisms of the OTT of CR for the first time. Specifically, we indicate two representative examples to reveal the potential of the OTT in manipulating CR properties and further applications, which correspond to the ultrafast photothermal and nonlocal properties of graphene individually. These results and conclusions pave the way for the photonic devices and applications of CR.


Introduction
CR is one of the most important physical phenomena, which is usually induced when the moving velocity of particles (e.g., free electrons) is higher than the phase velocity of light in the surrounding medium. Because of huge and broad contributions to various studies and applications, CR has won the Nobel Prize for physics in 1958.
However, the requirement for the free electrons to generate CR is quite high in normal mediums (e.g., u > c/n, where u is the velocity of free electron, c is the speed of light in free space (vacuum), and n is the (refractive index of the medium)). In recent years, this strict velocity threshold can be removed by HMMs, which can be composed of several alternative metal and dielectric layers [1,2]. Nevertheless, efficient methods to generate and manipulate CR by intrinsic and external approaches in different kinds of optical systems remain nearly untouched. For example, as indicated in this work, we can use the OTT of some optical systems to control the properties of CR by design.
Very recently, by using both the macroscopic effective medium theory (EMT) and microscopic scattering-matrix theory (SMT), we indicate the importance and uncover the mechanisms of the OTT of CR in the GHM [3]. Specifically, the ultrafast photothermal and nonlocal properties of graphene will lead to two novel effects related to OTT of CR in the GHM, which can be used in future studies and applications of CR.

Result
The proposed GHM for CR is indicated in Figure 1. By the EMT, we can get the effective properties of CR in the GHM. When the GHM locates in the elliptical state or the hyperbolic state, the required conditions (e.g., electron velocity) to generate CR and the properties of CR are quite different. Specifically, for the former case (corresponding to the high-frequency region), CR can only be induced by high-energy electrons (e.g., u > c/n), which corresponds to conventional CR. However, for the latter case (corresponding to the low-frequency region), the hyperbolic CR can be excited even with low-energy electrons (e.g., u < c/n). Interestingly, although the electron velocity is much lower, the intensity of hyperbolic CR is comparable with conventional CR, which can be observed in Figure  2. Furthermore, we can see that it exists a special frequency point, which is the boundary of the elliptical state and hyperbolic state of the GHM (roughly at 27 THz). Besides, contrary to the conventional case, the most intense position of the hyperbolic CR locates in a finite frequency point (black dashed line, roughly at 16 THz), which is nearly unchanged when the electron velocity decreases.  Next, we explain the mechanisms of the hyperbolic CR generated in the GHM. By the SMT, we can calculate the dispersion of all plasmonic modes supported in the GHM (Figure 3). And we indicate the reason why the hyperbolic CR can be induced even in low-energy electrons. With the low-energy electrons (e.g., u < v b = c/n), more than only the optical mode, the acoustic plasmon modes can be excited simultaneously. The superposition of these plasmon modes contributes to the total electric field distribution and makes the hyperbolic CR happens in the GHM (Figure 4). In other words, if the acoustic plasmon modes can be excited at some frequency points by low-energy electrons, the hyperbolic CR can be generated.  Figure 3. Dispersion of the plasmonic modes in the GHM.
In Figure 3, we note that the frequency region of the dispersion can also be divided into two parts, the boundary between these two parts is just the transition frequency of the OTT in the GHM. From Figures 2 and 3, we can find the connection between the plasmon modes and the OTT of CR in the GHM. Shortly, when the GHM locates in the hyperbolic state, only the low-energy electrons can excite the optical and acoustic plasmon modes inside the GHM, the superposition of these plasmon modes contribute to the effective field distribution of CR. Therefore, the hyperbolic CR can be generated even by low-energy electrons. However, we should notice the origin and nature of the hyperbolic CR are different from the convention CR [3].
The field distributions of two plasmon modes in elliptical state and hyperbolic state are shown in Figure 4 individually. In the first case, only the optical mode is excited, therefore, the field can hardly propagate to the bottom graphene layers in the GHM, but locates in the topmost graphene layer on the surface of the GHM. On the other hand, when we consider the plasmon mode in the hyperbolic state, the electric field propagates from the top layer to the bottom layer although the attenuations are observed clearly. When this happens, the hyperbolic CR can be generated by low-energy electrons. We also mark the directions of the wave vector (k-) and energy flux (S-) of the effective CR. By using the ultrafast photothermal and nonlocal properties of graphene, we propose that two novel effects of the OTT of CR can be induced in the GHM (Figures 5 and 6). Because of the unique and excellent properties of the hot carriers in graphene, the external optical pump can induce ultrafast OTT in the GHM, which can be used for an ultrafast CR source. For example, if we consider the frequency of CR in 30 THz, the GHM locates in the elliptical state at the beginning. However, with the increase in the temperature of the graphene electrons, the GHM states in the hyperbolic state, where the CR can be induced even by low-energy electrons. Finally, when the temperature is decreasing, the GHM will back to the elliptical state, then the hyperbolic CR is vanishing [3]. When the electron velocity is low enough, the wave vectors carried by free electrons are quite large. Therefore, the nonlocal properties of graphene can no longer be ignored. For example, as we indicated here, the Fermi velocity of graphene will bring a novel velocity threshold of the CR in the GHM [3].

Conclusion
Efficient methods for the generation and manipulation of CR are always highly desirable. In this work, we indicate that the OTT of CR in the GHM is important for the properties of CR, which is clarified clearly by both the macroscopic and microscopic analytical models. With the ultrafast photothermal and nonlocal properties of graphene, we stress that the OTT of CR should be considered in future studies and applications.