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Abstract
Smith-Purcell radiation (SPR) is a kind of electromagnetic wave radiation that happens when an energetic beam of electrons passes very closely parallel to the surface of a ruled optical diffraction grating. The frequency of radiation waves varies in the upper and lower space of the grating for different electron velocity, satisfying the SPR relationship. In this study, a Fano-resonant metasurface was proposed to steer the direction of the SPR waves at the fixed resonant frequency by changing the velocity of the electron beam without varying the geometric parameters or adding extra coupling structure. The maximum emission power always locates at the resonant frequency by utilizing the integration of the Poynting vector. The relative radiated efficiency can reach to a maximum value of 91% at the frequency of 441 GHz and the efficiency curve has a dip when the direction of SPR is nearly vertical due to the high transmission. There is a great consistence of steering radiation angle from 65 degrees to 107 degrees by altering the velocity of electron beam from 0.6c to 0.95c both in analytical calculation and PIC (particle-in-cell of CST) simulation at terahertz frequencies, where c is the speed of light in vacuum. Furthermore, the destructive interference of Fano resonance between the magnetic mode and the toroidal mode shows the underlying physics of steering SPR in a fixed frequency. Our study indicates that the proposed structure can produce direction-tunable THz radiation waves at resonant frequency by varying the velocity of the electron beam, which is promising for various applications in a compact, tunable, high power millimeter wave and THz wave radiation sources.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Terahertz (THz) waves commonly refer to the electromagnetic (EM) waves between millimeter microwaves and infrared waves with frequencies ranging from 0.1 THz to 10 THz [1]. More and more researchers draw their attentions to THz waves owing to their unique properties, having great prospects for application such as security checking, biomedical analysis, wireless communication and so on [2,3]. Although THz waves have so many critical applications, tunable, compact, high emission power THz radiation source is still one of barriers to prevent the development of THz technology. There are three main ways to generate THz sources at present as follows: the photonics such as terahertz photoconductive antenna, the semiconductor devices and the vacuum electronics. Among them, vacuum electronics devices [4–6] have been considered as a popular way for THz sources due to their un-substitutability to produce relatively high emission power.
Smith-Purcell radiation (SPR) [7], which occurs when a uniform sheet of electrons passes parallel over a periodic grating structure, is one of manners to produce radiating waves from interaction between the electronic beam and evanescent wave. The frequency of SPR can be defined by [8–14],
where θ represents the emission angle of the radiation waves stimulated by the uniform sheet beam of electrons with respect to the direction of the beam, p denotes the periodicity of the grating structure, m is the radiation order, ve is the velocity of the electrons and c is the speed of the light in the vacuum. The progress from incoherent radiation to coherent radiation extends the application of SPR in microwaves, such as Orotrons [15], Traveling Wave Tube(TWT) [16], Backward-Wave Oscillators(BWOs) [17], and Smith-Purcell free electronic laser (SPFEL) [18–21]. The current concentrations of SPR mainly focus on three aspects, changing the grating structure to enhance the interaction between the beam and grating [1,13,22–26], using of new materials [27], to dynamically tune the characteristics of the radiation [9,27–32], and modulating the electronic beam to improve the radiation performance from the grating such as increasing emission power and efficiency [2,8,33–35]. Meanwhile, many experiments have been done to validate the rationality of the simulation and derivation by scanning electron microscopes(SEM), cathode electric source or other devices, such as observation of coherent Smith-Purcell radiation using continuous flat beam [36], amplification of the evanescent field by free electrons [37,38]. These researches have enhanced the output power and radiation efficiency to a certain extent. However, although utilizing the methods of adding extra coupling and collected structure such as off-axis paraboloid mirror(with ±20°collection angle), beam splitter and flat mirrors can observe radiation emitted into some ranges of angles away from the structure and collect emission energy [19], few studies have figured out the problems of variable frequencies and directions of the radiation waves with different ve without changing the original structure or adding other coupling structure.On the other hand, metasurfaces [22,26,30,39–41], consisting of 2D array of planar metallic or dielectric subwavelength building blocks or meta-atoms [30,42], have shown great prospects for practical applications due to their exceptional capability of modulating the phase, amplitude, polarization, frequency spectrum of the spatial wavefront [43]. By designing the 2D array carefully, the interaction leading to electron-beam-induced emission over a broad wavelength range [44], promotes the emission power and efficiency. Although many researchers have employed different metasurfaces structures to investigate SPR properties [30,44], they mainly concern the radiated light of fixed velocity of the electronic beam or the fixed radiation direction. Hence, how to solve the problems of variational frequencies and directions of emission waves when varying the velocity of electron beam remains no further exploration.
In this paper, we propose a theory of direction-tunable radiation sources at a fixed frequency without changing other geometric parameters or adding extra coupling structure. Importantly, the problems of variational frequencies and directions of emission waves when changing the velocity of the sheet electrons have been solved. A three slits metasurface structure was utilized to verify the theory. The results suggest that from the perspective of theoretical analysis, analytical simulation and PIC simulation, the direction of radiation waves can be freely modulated about 40° by changing the velocity of the electronic beam.
2. Methods and theoretical analysis
A uniform sheet of beam passes parallel over the metasurface along the x-y plane at z0=100 um and stimulates electromagnetic waves which are propagating over the metasurface and the radiated angle range of θ’ is shown in Fig. 1(a). We define three planes named 1, 2 and 3 near the metasurface, displaying in Fig. 1(a). The input, upward, downward and absorption energy flux density Sin, Sup, Sdown and Sab can be calculated by Sin=S1+S2, Sup=S1, Sdown=S3 and Sab=S2-S3, among which S1, S2, S3 are the Poynting vectors S passing through the defined planes 1, 2 and 3, respectively. The unit cell with three slits has dimensions as w=60 um, h=300 um, p=500 um, d=100 um shown in Fig. 1(b). A sharp Fano resonance peak and dip appears in the reflection spectrum under TM waves stimulating as shown in Fig. 1(b). The inserted field maps in Fig. 1(b) are the field distribution of mode patterns around the unit cell. The mode patterns are represented as the in-phase bright mode (+++) with a broad band and the antiphase dark mode (+-+) with a narrow band. The broad band bright mode has same phases at the ends of the three slits, which is like a dipole mode induced by linear polarized wave.
Fig. 1. (a) The steering electromagnetic waves are manipulated by a uniform sheet of charged particles passing parallel over the metasurface along x axis with a velocity of ve at plane z = z0 in the x-z plane. The inserted figure is the lateral view of metasurface. (b) The Fano resonance stimulated by the TM waves and the mode patterns around the unit cell.
According to the Maxwell equations shown as follows, we can figure out the expressions of the electric field and magnetic field by combining the boundary conditions,
The current density in Eq. (2) can be expressed in the form of ${\boldsymbol J}\textrm{(}x,z,t\textrm{) = }{{\boldsymbol e}_{\boldsymbol x}}q{v_e}\delta (z - {z_0})\delta (x - {v_e}t)$, where q is the charge of each structure unit length in the y direction, and ve is the velocity of the electric beam. By Fourier transformation, the current density can be reformed to frequency domain as follows,
Here kx=ke=ω/ve, where ω is the angular frequency of the radiation waves. Such current density profile can induce evanescent waves under certain conditions [30,45], which can be derived by Floquet modes as shown below,
Here, ${k_{xmn}} = {k_x} + \frac{{2m\pi }}{p},{k_{ymn}} = \frac{{2n\pi }}{p},{k_{zmn}} = \sqrt {k_0^2 - k_{xmn}^2 - k_{ymn}^2} $, are the wave numbers along x, y, z axis respectively, and m, n are integers. It is well known that if the parameters of any Floquet modes m, n is positive m or n nonzero respectively, there will be an attenuation of the evanescent fields along the z axis, leading to no far-field SPR. Therefore, in order to induce SPR, we need to define n=0 for far field radiation, which results in the wave number of y direction vanishing. Meanwhile, we introduce m= −1 and electric vector kx=ke=ω/ve as our study object and the vacuum wave vector satisfies the following equation:
By designing the period of the metasurface unit cell as
the kxmn will disappear, making the emission wave propagate only along z axis [39]. However, the emission wave can impossibly be direction-tunable with p changed in the x-z plane. Mathematically, we can assume to modulate the direction of the radiation waves by changing only one variable if the other variables in the Eq. (5) are fixed, which are f, ve and p. We have found the resonant frequency is basically unchanged when varying the period length of some slits metasurface [46–48]. The proposed Fano metasurface in Fig. 1(b) has the same properties. From Fig. 1, the incident electron beam propagating along x axis can stimulate the Fano resonance with the same physical precondition (Ex≠0 and kz≠0). Then, f and p are the fixed parameters in the Eq. (5). ve is the only parameter to tune the wave vector (kxmn, kzmn) and thus the radiated angle θ. Hence, keeping f and p as a fixed constant simultaneously in Eq. (5) is crucial to the monotonous tunability of the radiated angle θ for ve. The radiated angle θ is deduced according to Eq. (1) as follows,Next, full waves simulation was applied to analyse the resonant behaviour of the metasurface (Fig. 2). As shown in Fig. 2(a), the resonant frequency near 440 GHz is nearly unchanged even varying p from 350 um to 650 um, which indicates that the range of ve intercepting from that of p shown in the rectangle box in Fig. 2(a) can be obtained between 0.6c to 0.95c according to Eq. (6). When m=−1, f0=440 GHz and p=500 um, the range of θ can be obtained as,
Moreover, Fig. 2(b) displays the dispersion curves of the metasurface with x periodicity px=500 um, showing that there is an obvious band-gap at the frequency around 440 GHz. The band-gap suggests that as for trimeric slits structure, there is no kx component but kz remains non-zero, which satisfies the precondition of Fano resonance mentioned above. The coupling interaction between adjacent unit cell can be ignored within the intercepted range of p. When electronic beam passes parallel over the metasurface, the original wave vector relationship has been changed. The wave vector of the electric beam kx=ke=ω/ve becomes the only wave vector in the x axis coinciding with the vacuum wave vector k0 because of the band-gap mentioned above. Meanwhile, the wave vector kz remains unchanged as shown in Eq. (5) so that k0 can is solely determined by the wavevector of the electron beam, resulting in the direction-tunable radiation waves in x-z plane at a fixed frequency determined by k0. In the following sections, we will employ both effective surface current full-wave simulations and rigorous PIC simulations method to validate our preconditions.
Fig. 2. (a) The contour map of the reflection spectrum characteristics of the trimeric slits metasurface with different p under the incidence of TM waves. The intercepted periodicity is boxed out with dashed line. (b) The dispersion curves of different modes at px=500 um.
3. Effective surface current simulation and PIC simulation of the direction-tunable SPR
Considering the Floquet modes mentioned in section 2, effective surface current simulation was employed to calculate Hy profile and power flow under typical values of ve at x-z plane with y=0, as shown in the form of rainbow light and arrows in the left part of Figs. 3(a)–3(f), respectively. Similarly, the PIC solver (implemented in CST) was used to simulate the beam wave interaction process. In the PIC simulation, we set 50×7 unit cells taking the same parameters as the effective surface current simulation. Different from the adopted surface current density model, the FEB (free-electron-beam) in PIC simulation is a sheet beam with rectangular cross-section. The thickness of the beam in z direction is 0.1 um and the width of the beam in y direction is the same as that of the periodic structure. The FEB has a Gaussian profile in x direction and the bunch length is 30 um. The charge quantity of the bunch is 1.6e−19 Coulomb and there are 2000 emission points in the rectangular cross-section. The Hy at the frequency of 440 GHz when ve is the same as the effective surface current simulation are shown in the right part of Figs. 3(a)–3(f). All the field are monitored at x-z plane and y=3.5p. From the zonal distribution of Hy in Fig. 3, radiation angles θ can be controlled by ve in a wide angle range and are basically in great agreement both in effective surface current and PIC simulation. Figure 3 also depicts that the direction of S is unchanged under the same ve in one wavelength region away from our metasurface. Furthermore, when ve varies from 0.6c to 0.95c, the angle of radiation waves θ changes from an acute angle to an obtuse one.
Fig. 3. The direction-tunable radiation waves by varying ve. The typical values of ve in (a)-(f) are 0.66c, 0.7c, 0.73c, 0.8c, 0.85c and 0.9c, respectively. The left panel of each sub-figure is the distribution of Hy (rainbow light) and power flow (red and black arrows) in effective surface current simulation and the corresponding contour maps of Hy captured at the x-z plane with y=3.5p in PIC simulation are displayed at the right panel in each sub-figure. The amplitudes of the magnetic field have already been normalized by the maximum magnetic amplitude of each velocity, respectively.
For the purpose of further numerical analysis, the curves of the relationship between ve and θ at the frequency of 440 GHz both in theory based on the Eq. (7), analytical simulation based on the direction of S and in PIC simulation based on the direction of Hy are plotted in Fig. 4, which shows a great consistence between theoretical calculation, effective surface current simulation and PIC simulation when ve is larger than 0.68c. Meanwhile, there is certain discrepancies between theoretical calculation, effective surface current simulation and PIC simulation when ve is smaller than 0.68c, which can be attributed to the relatively small period, leading to the increasing of coupling interaction between adjacent unit cells. The kx will be influenced by such coupling interaction and k0 will be changed simultaneously, leading to the slight radiation angle deviation from corresponding theoretical value. All in all, radiated angle θ can be modulated in a range of approximately 40 degrees from 65° to 107° when ve changes from 0.6c to 0.95c.
Fig. 4. Quantitative analysis of the relationship between radiation angle θ and electron velocity ve. There is a great consistence among the data from theoretical calculation (the red solid line), effective surface current simulation (the blue dot) and PIC simulation (the green triangle dot).
In order to further analyse the radiated energy performance of the voltage-controlling direction THz waves, Fig. 5 and Fig. 6 show the energy flux densitiy and the radiated efficiency by varying ve from 0.6c to 0.95c according to Eq. (6). Figure 5 illustrates that although ve ranges from 0.6c to 0.95c, there is a sole peak at the frequency near 440 GHz for Sin, Sup, Sdown, Sab within a broad frequency range. Herein, a fluctuation constant named κ1=(fmax-fmin)/Δf ≈0.03077 is defined to describe the fluctuation of the frequency at which the peak occurs in the contour map. fmax and fmin represent the maximum and minimum value of the peak frequency. The fluctuation of the peak frequency mainly concentrates on ve<0.68c, which is the same as simulation errors of analytical and PIC simulation with theoretical calculation in Fig. 4. When ve>0.68c, the peak frequencies under different ve remain basically unchanged. The minor κ1 indicates that the radiation energy mainly concentrates at the frequency near 440 GHz, which is also the ‘cold’ resonant frequency of the metasurface. As for emission efficiency, two methods are generally considered. One is the absolute efficiency normalized by the kinetic energy of electric beam and a formula of the relationship between ve and the kinetic energy from the perspective of restricted theory of relativity displayed as follows,
Fig. 5. The functions of energy flux density versus ve from 0.6c to 0.95c. (a)-(d) the contour map of energy flux density Sin, Sup, Sdown and Sab, respectively.
Here We is the kinetic energy of the electrons, and $m_{oc}^2$ is the energy of static electron which is equal to 0.511 MeV numerically. The other relative radiated efficiency is normalized by formulas mentioned in Ref. [30]: Sup/Sin, Sdown/Sin, Sab/Sin. Here we considered the latter scheme. Figure 6(a) and Fig. 6(b) show the contour map of Sup/Sin and Sdown/Sin and reveal that the energy mainly concentrate on the frequency near 440 GHz which we focus on because the proposed direction-tunable THz radiation source is at a fixed frequency of 440 GHz. Figure 6(c) illustrates the curves of the relative radiated efficiency at the frequency of 440 GHz. There is an obvious peak of Sdown/Sin and dip of Sup/Sin when ve is near 0.75c. This is because kx is almost zero at this moment and the radiation waves are almost determined by kz, propagating near vertical angle. It also signifies that most of energy passed through the metasurface so that there is an increasing of Sdown and Sab but the Sup decreases a lot. Figure 6(c) also displays the corresponding data shown as purple line from C-aperture structure in Ref. [30], where 0.47 and 0.05 are the optimal value of Sup/Sin=Sdown/Sin, and Sab/Sin, respectively. Figure 6(d) shows the maximum value of Sup/Sin. The result illustrates the trimeric slits metasurface have a greater performance than that of C-aperture structure for the relatively high Sup/Sin and low Sdown/Sin.
Fig. 6. (a) The contour map of Sup/Sin. (b) The contour map of Sdown/Sin. (c) The relative radiated efficiency at the frequency of 440 GHz. (d) The maximum value of Sup/Sin near 440 GHz.
4. Theoretical explanation of the required stable Fano-resonance with a varying period for the direction-tunable SPR
The trimeric slits metasurface supports Fano resonance and is able to steer the radiation direction of a fixed frequency waves by the voltage exerted on the electron beam. However, the underlying physics is still indispensable, which is the main object in the following discussion part. Toroidal mode in three stripes structure (the complementary structure of the slits) contributes the robustness of the Fano resonance with varying period experimentally [48]. According to the Babinet theory, the three slits structure has a magnetic toroidal mode contributes to the robust Fano resonance. However, it is much easier to calculate the corresponding scattering power of each multipole moment in stripes than magnetic multipole moments if the current distribution in the three stripes structure shown as Eq. (10) can be achieved [49],
Table 1. Current multipoles and corresponding far-field scattering power
As shown in Fig. 7, we plot the scattering power curves of each multipole moment versus frequency for five typical p values from 350 um to 650 um in Fig. 2. As we all know, Fano resonance occurs when a discrete localized state couples with a continuum of state [51]. Ip from electric dipolar moment has the largest radiated intensity in the full frequency band, which is the dominant multipole determining the macroscopic reflection and transmission properties of the structure. The scattered power Ip can be recognized as a wide spectrum and the scattered power IT from toroidal dipolar moment as a narrow spectrum. A great constructive interference between electric and toroidal dipoles interferes at the resonant frequency around 440 GHz. The frequency of destructive interference remains almost unchanged even p has changed in a wide range, which is consistent with the variation of the resonant frequency of Fig. 2(a). Herein, another fluctuation constant κ2=(Fmax-Fmin)/ΔF≈0.0263 is defined to describe the fluctuation of constructive interference frequency when p varying from 400 to 650 um. Fmax and Fmin represent the maximum and minimum of the frequency of interference, respectively. The minor κ2 indicates the low fluctuation of interference frequency. Numerically, κ1≈κ2, the fluctuation of interference frequency is in great consistence with that of radiation power peak frequency. The conditions of the constructive interference stemmed from the electric dipolar moment and toroidal dipolar moment are invariant. Finally, according to the Babinet principle, the Fano resonance of the three slits structure in reflection is caused by the constructive interference between the magnetic mode and the magnetic toroidal mode.
Fig. 7. The full multipolar decomposition of the first contributing five multipole moments in the far field as a function of frequency under five special p values of the three rods structure intercepted from 350 um to 650 um: electric dipole (Ip-black solid line), magnetic dipole (Im-red solid line), toroidal dipole (IT-blue solid line), electric quadrupole (Ieq-green solid line), magnetic quadrupole (Imq-purple solid line)
5. Conclusion
In conclusion, we employed a Fano-resonant metasurface to achieve the direction-tunable SPR at a fixed emission frequency. The metasurface is composed of trimeric slits unit cells, supporting Fano resonances originating from the constructive interference between magnetic mode and magnetic toroidal mode. The resonant frequency locating in the THz domain is basically unchanged when varying p. The direction of emission waves can be modulated in a range of about 40 degrees via changing ve (voltage-controlling) without adjusting the geometric parameters, as verified via: theoretical calculation, effective surface current simulation and PIC simulation. The problems of variable frequencies and directions of the radiation waves with different ve has been solved, and we believe that our findings may provide useful guidelines for the design and manufacture of compact, tunable, and high emission power millimeter wave and THz wave radiation sources.
Funding
Guangzhou Science and Technology Program (202102020566); National Natural Science Foundation of China (11965009, 62075084); Natural Science Foundation of Guangxi Province (2018GXNSFAA281193, 2018JJA170010); Fundamental Research Funds for the Central Universities (21620415); Guangdong Basic and Applied Basic Research Foundation (2020A1515010615); Innovation Project of Guangxi Graduate Education (YCSW2021183).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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