Dispersive 2D Cherenkov radiation on a dielectric nano-film

We report a modified two-dimensional Cherenkov radiation, which occurs on a high-index dielectric nano-film driven by uniformly moving electron-beam. It is essentially different from the ordinary Cherenkov radiation in that, in the nondispersive medium, it shows unique dispersion characteristics—the waves with higher frequencies radiate at larger Cherenkov angles. Its radiation frequency and direction are essentially determined by structure parameters as well as the beam-velocity. By means of fully electromagnetic simulations and theoretical analyses, we explored the mechanism and requirements of this radiation. This new Cherenkov radiation may lead to promising applications in a broad range of fields.


Supplement
In this supplement, analytic derivations of the two-dimensional dispersive Cherenkov radiation on the dielectric nano-film together with its dispersion relations are presented.
The diagram is shown in Figure A1. The whole space can be divided into three regions as shown in the inset. We will solve the Maxwell equations in different regions respectively, and then apply boundary conditions to get electromagnetic fields in all regions. Figure A1. Schematic diagram.
According to the Maxwell equations, all other field components can be expressed in terms of the y-directional electric field component Ey and magnetic field component Hy, which will be considered at first.
In the upper half space (region-I shown in the inset of the figure), Fourier transformations of E I y and H I y components satisfy the following non-homogeneous equation in the Cartesian coordinate system shown in the figure [A1-A2] , ω is angular frequency, c is light speed in vacuum, ky is y-directional wave-number, ε0 and μ0 are respectively permittivity and permeability of the vacuum, ρ(ω) and Jz(ω) are Fourier transformations of charge density ρ and current density J, respectively. For particles with charge quantity of q and velocity of v, ρ and J can be expressed as: where δ is the Dirac delta function, x0 denotes the beam position as shown in the figure. Substituting Eq. (A2) into Eqs. (A1), and applying the Fourier transformation method, Ey and Hy components in region-I can be solved. The solutions are composed by two parts: the special solution which represents the incident wave from particles and the general solution signifying reflection waves from the structure. Thus Ey and Hy fields in region-I can be expressed as: Substituting Eqs. (A3~A5) to Eq. (A6), expressions of all field components in three regions can be obtained.
At boundaries of the nano-film, tangential electromagnetic fields satisfy following boundary matching conditions [A1,A3] : 00 00 00 00 , Substituting all tangential field components into Eq. (A7) and carrying out numerical calculations, the coefficients A1,2,3,4 and B1,2,3,4 can be solved, indicating that fields in all regions are obtained. When considering the 2D CR in the nano-film, we only need to get A2,3 and B2,3.
We note that in above equations all coefficients are determined by ω and ky.
Considering that ky can be expressed as tan yz kk   , radiation fields essentially dependent on ω and θ. One of them is preset, we can get dependence of field on the other. Thus we can obtain the radiation intensity of the 2D CR depending on ω and θ, respectively. The numerical calculation results are shown in the following.
Calculated contour maps of the fields (Ey component) with different frequencies on the nano-film are shown in Figure A2, in which all parameters follow that given in Figure 2 of the main text. One can see that the radiation direction changes with frequency, agreeing well with simulation results presented in Figure 2 of the main text.
It should be noted that all the calculated fields are in the frequency domain, such that the time evolution of particles skimming over the nano-film has not been illustrated.  Figure A3, where all parameters follow that given in Figure 3 of the main text. One can see that the radiation direction changes gradually with beam-energy, being in well agreement with simulation results given in Figure 3 of the main text. Figure A3. Calculated contour maps of the Ey field on the nano-film for different beam-energies Now we consider the dispersion equation which governs wave propagating along the nano-film. We should solve the homogeneous equations (without particles) in all regions [A5] . Actually we only need to let the special solutions in Eq. (A3) to be zero,