Threshold-less and focused Cherenkov radiations using sheet electron-beams to drive sub-wavelength hole arrays

Yucheng Liu; Weihao Liu; Linbo Liang; Qika Jia; Lin Wang; Yalin Lu

doi:10.1364/OE.26.034994

1. Introduction

Cherenkov radiation (CR) [1, 2], generated when velocities of charged particles are greater than the speed of light in the background, was one of the most famous discoveries in the last century and still has broad applications in fundamental physical researches [3, 4] as well as cutting-edge technologies [5–7]. Usually, particle accelerators, which are expensive and cumbersome [8], have to be used to obtain the ultra-relativistic particles for meeting the requirement of CR.

Recently, the so-called threshold-less CR [9, 10], generated by particles with any velocities, and the reversed CR, which radiates in the backward direction of moving particles [11–14], have attracted an increasing attention because of their promising prospects. For example, the threshold-less CR can be developed as low cost and integrated light sources [10] since it removes the requirement of high-energy particle accelerators. The reversed CR can be more efficient for diagnosing particles since it can easily separate the radiation and particles in experiments [15]. Yet, all the previous threshold-less and reversed CRs exclusively resorted to the matematerials [10, 16, 17], which are innately dispersive, such that CR can only be achieved in specific frequency bands. Also, the CRs in the medium are unavoidably attenuated due to the material loss, which greatly limits their applications.

In the present paper, we demonstrate a threshold-less CR in vacuo by using a sheet free-electron beam (FEB) to drive an oblique-lined sub-wavelength hole array (SHA). Unlike the conventional CR, which is incoherent radiation with a broad spectrum, the present CR is coherent radiation with specified frequencies. Its frequency can be tuned from microwave to infrared region by adjusting the structure parameters of SHA. In other words, it is a multi-color emission with shaped radiating spectrum.

In addition, by rearranging the alignment of SHA, the spatial distribution of CR can also be shaped, which is an increasing hot research topic [18–20, 22]. Different from that of the transition radiation (TR) [18, 19] and that of the Smith-Purcell radiation (SPR) [20–21], which are frequency dependent, the spectral and spatial shaping of CR in the present paper is frequency independent, namely, the radiations of all frequencies radiate in the same direction, which is very attractive for practice. Especially, by letting the sub-wavelength holes to line along a designed curve, we demonstrate that the CR can be focused at specific spots in either forward or backward directions, achieving the reversed CR in vacuo for the first time. The focused CR doesn’t have the conventional Cherenkov cone, and its intensity at the focusing spot is greatly enhanced. These newly revealed threshold-less and focused CRs may lead to a broad interest and attractive applications.

Fig. 1 (a) 3D diagram of the sheet FEB driving an oblique-lined SHA. The numbers ‘1-6’ denote different parts of the FEB, which excite different subwavelength holes. (b) The $x - z^{'}$ cross-section of the scheme, where θ denotes the radiation angle of CR relative to z^′ axis.

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2. Threshold-less Cherenkov radiation

The schematic diagram of proposed threshold-less CR is shown in Fig. 1(a). A sheet FEB flies over a straight array of rectangular sub-wavelength holes, which are etched in a conductor plate and are obliquely lined in the direction ( $z^{'}$ ) with a certain angle (α) relative to the FEB moving direction (z). As illustrated in previous literatures [23–25], each sub-wavelength hole is an independent electromagnetic resonator, which holds a series of resonant modes. As the sheet FEB skims over the SHA, the resonant modes within adjacent holes will be successively excited with a time delay of

Δ_{t} = \frac{L_{z}}{v_{e}} = \frac{L}{v_{p}},

where v_e is the velocity of the FEB in z direction,

v_{p} = v_{e} / \cos α

is the projective (effective) velocity of the FEB in z^′ direction, L_z and L are the spatial periodicities of the array in z and z^′ directions, respectively. The resonant modes will then successively radiate into space, forming a radiating sequence with the ‘effective velocity’ equal to v_p. Here each hole is a radiating unit. It is readily to find that the constructive interference (coherent radiation) from all the holes of the array will be obtained when

\frac{ω L}{v_{p}} - \frac{ω L}{c} cos θ = 2 n π,

in which ω is the angular frequency of radiation, c is the speed of light in vacuo, θ is the angle from the SHA direction (

z^{'}

) to the radiation direction as shown in Fig. 1(b), n=0, 1, 2,3,... is a non-negative integer. We note that Eq. (2) is exactly the famous SPR relation when

n \geq 1

[26]. While as n=0, it changes to

cos θ = \frac{1}{v_{p} / c} = \frac{cos α}{v_{e} / c},

which denotes the well-known CR equation [27]. Equation (3) has real solutions only when the CR requirement—the velocity of the emitter is greater than the speed of light (

v_{p} > c

)—is satisfied. It can be seen that the radiation direction of CR is independent from the frequency based on Eq. (3), namely, the waves with different frequencies radiate in the same direction. On the contary, the direction of SPR essentially depends on the frequency according to Eq. (2) [28–30].

From Eq. (3), we find that the CR in vacuo can be achieved if $v_{p} = v_{e} / \cos α > c$ , which is realizable since the effective velocity v_p of the emitters can be much greater than the real velocity of FEB and even greater than the speed of light in vacuo. Furthermore, the CR can be generated by the FEB with any velocities when α is close to $90^{o}$ , indicating that the threshold-less CR will be obtained. Note that here the CR is achieved, essentially because the sheet FEB is used. Within the sheet FEB, the electrons at different transverse positions in y direction move side by side and excite different subwavelength holes of the array, see Fig. 1(a). Thus, the effective velocity v_p is actually the collective result achieved by electrons at different transverse positions within the sheet FEB. It essentially differs from the single particle model used in conventional SPR and CR, where all electrons are functionally identical and a single electron has to excite different periods (radiators) successively, such that the effective velocity is always less than the speed of light in vacuo.

Fig. 2 (a) Simulation obtained snapshot in time domain of the field contour map (E_z component) within the $x - z^{'}$ cross-section. Here ‘BP’ denotes the position of the FEB in z^′ direction. (b) Radiation spectrum detected in the space and field distributions within the holes for each peak frequency. (c) Snapshots of the E_z field contour maps in the frequency domain for three peak frequencies.

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As mentioned, the present CR is emitted from the resonant modes within the sub-wavelength holes excited by FEB. And its radiation frequency is exactly the resonant frequency of the hole, indicating that it has a series of definite frequencies rather than a broad spectrum as that in conventional CRs. In light of the shapes and boundary conditions of sub-wavelength holes, the frequencies of resonant modes can be approximately expressed as:

f_{l, m, p} = \frac{c}{2} \sqrt{{(\frac{l}{h})}^{2} + {(\frac{m}{w})}^{2} + {(\frac{p}{d})}^{2}},

in which h, w and d are the hole parameters shown in Fig. 1. l, m and p are non-negative integers signifying the mode orders. In the present paper, we set the hole width (d) to be much less than the operation wavelength (

d < < λ

), such that the field variation in z direction could be ignored (p = 0) [31], and the resonant frequencies only depend on l and m. Considering the boundaries in both x and y directions, we find that m should be positive (

m \geq 1

) and l can be zero (

l \geq 0

).

To verify the above analyses, we perform the fully electromagnetic simulations by using the finite-difference-time-domain (FDTD) based particle-in-cell code [32]. Here the conductor plate is treated as perfect-electric-conductor (PEC). The structure parameters of the SHA are set to be: $d = 0.01$ mm, $w = 0.5$ mm, $h = 0.5$ mm, and $L = 0.3$ mm. The oblique angle of the SHA is $α = 85^{o}$ and the number of holes in the array is 25. The FEB is a sheet beam with rectangular cross-section: the beam widths in x and y directions are 0.1 mm and 7 mm, respectively. Note that the y-directional width of the sheet FEB should be large enough to cover all the subwavelength holes of the array. The FEB has a Gaussian profile in z direction (the bunch-length is 0.5 mm), and the electron energy is 5 keV, which is much less than that used in conventional CRs. Figure 2(a) shows the simulated (time-domain) snapshot of contour map of the electric field (E_z component) in $x - z^{'}$ cross section. It illustrates that the radiation is generated in both upper and down half-spaces of the SHA, and the radiation angle (θ) is $59^{o}$ , satisfying the CR relation of Eq. (3). It indicates that CR is achieved in vacuo as predicted. The frequency spectrum of the radiating fields in the space is detected in Fig. 2(b), which shows that the radiation has three peak frequencies: 0.3 THz, 0.38 THz, and 0.56 THz, namely, it is a multicolor coherent CR rather than a non-coherent CR with broad-band spectrum. The electric field distributions within the holes are shown as the insets of the figure, which denotes that 0.3 THz, 0.38 THz, and 0.56 THz respectively signify the resonant modes of $(m, l) = (1, 0), (1, 1), (1, 2)$ in Eq. (4). Figure 2(c) illustrates the contour maps of the E_z field in frequency domain for three frequencies. We can see that the radiation angles of different frequencies are the same ( $θ = 59^{o}$ ), which is a typical feature that distinguishes CR from SPR.

Further simulations show that the electron energy of the FEB can be reduced by increasing α to be closer to $90^{o}$ , indicating that the threshold-less CR can be obtained in this scheme. According to Eq. (4), the frequencies of resonant modes, also being the radiation frequencies, will be changed by adjusting structure parameters of the holes. We note that the frequency can be tuned from microwave to infrared region by varying the holes, namely, the threshold-less CR can be achieved in such a broad spectrum range, much larger than the threshold-less CRs obtained by metamaterials. Note that in the visible light region the surface plasmons on the conductor plate should be taken into consideration, and will change the boundary conditions of the holes and resonant characteristics [33].

Now we would like to compare the present CR with available CRs and SPRs with similar configurations considered in previous literatures. In [34], the CR in vacuo was obtained by using a uniformly moving (v_e) charged filament to obliquely (with angle α) impinge upon a dielectric screen, on which a surface current with phase velocity $v_{p} = v_{e} / \cos α$ is generated. When the phase velocity is greater than the speed of light, the CR is generated in vacuo, with direction determined by Eq. (3). Although it shares the same formula with the present CR, its forming mechanism, together with the radiating characteristic, is essentially different. In the scheme of [34], the CR is physically formed by the combination (constructive interference) of a series of transition radiation, which occurs as charged particles pass through different mediums. The effective ‘radiating sources’ are the surface currents on the screen, and its radiating characteristics cannot be controlled. In contrast, the CR in the present paper is formed by the constructive interference of diffraction radiations from an array of subwavelength holes under the excitation of the FEB. And the ‘radiating sources’ are resonant modes within sub-wavelength holes. More notably, it can be effectively manipulated in the frequency and direction by changing the sub-wavelength holes. In other words, both the spectrum and spatial distributions of radiation can be shaped, which will be illustrated in the next section.

Another phenomenon that needs to be compared is the conical effect in SPR [35–37], which is generated when the electrons move along an oblique line with a certain angle (α) from the periodic direction (z) of grating. It looks similarly to the scheme of the present CR in appearance. Yet, its mechanisms are essentially different. In the conical effect of SPR, which is governed by the aforementioned single particle model, the electron velocity v_e, together with its projected components in both z and y directions (expressed by $v_{e} \cos α$ and $v_{e} \sin α$ , respectively), is always less than the speed of light in vacuo. Thus, the CR cannot be obtained in any directions, only the SPR can be generated. It is worth noting that a fundamental feature that distinguishes SPR from CR is the requirement of periodic structure. The periodic structure (such as grating and SHA), which is expressed by the periodicity L in Eq. (2), is indispensable for any SPR. Thus, the SPR will not be generated if the subwavelength holes are not arranged periodically (with the same distance) along z^′ direction in the present scheme. In contrast, the CR does not needs a period structure according to Eq. (3), and it can still be generated when the subwavelength holes are not periodically arranged in z^′ direction, which is verified by our simulations.

3. Focused Cherenkov radiation

In conventional CRs and the threshold-less CR illustrated above, the electromagnetic emission from all radiating units along the FEB-path will simultaneously reach a conical wave-front, composing the Cherenkov cone [27]. In many practical applications, the lenses or other optical components have to be used to focus the radiation fields of CR on a target. The losses of these optical components are unavoidable. In this section, we will show that the CR obtained above can be effectively manipulated and be focused at specific spots in the space by re-arranging SHA.

As that in all other CRs, the focused CR will be achieved when the radiation from all the sub-wavelength holes in the array simultaneously reach the focusing points. This process can be schematically illustrated in Fig. 3(a), based on the geometric relation shown in which the requirement of the focused CR can be expressed as:

T = T_{1} + T_{2} = \frac{z}{v_{e}} + \frac{\sqrt{{(y - Y_{0})}^{2} + {(Z_{0} - z)}^{2} + H_{0}^{2}}}{c} = c o n s t,

where (0, y, z) is the coordinates of holes, (H₀, Y₀, Z₀) is the coordinate of focusing point in the upper half-space of the SHA, T₁ is the time for the FEB to fly before exciting the hole, T₂ is the time duration of radiation from the hole to the focusing point. Performing mathematical transformation on Eq. (5), we could get the equation of the curve, along which the sub-wavelength holes should be lined:

{(y - Y_{0})}^{2} - \frac{{(z - z_{c})}^{2}}{M^{2}} = F .

Here z_c, M, and F are respectively expressed as: $z_{c} = \frac{T c β^{3} - Z_{0} β^{4}}{{(1 - β^{2})}^{2}}$ , $M = \frac{β}{\sqrt{1 - β^{2}}}$ , and $F = {(T c)}^{2} - Z_{0}^{2} - H_{0}^{2} + {(T c β^{2} - Z_{0} β^{3})}^{2} {(1 - β^{2})}^{3}$ , in which $β = v_{e} / c$ . Equation (6) signifies a hyperbolic curve, which means that, in order to get the focused spot in the space, the sub-wavelength holes should be lined hyperbolically on the conductor plate. According to Eq. (6), the shape of the curve depends on four parameters: H₀, Y₀, Z₀, and T, all of which can be prescribed in practices. In other words, the focusing point can be at any designed positions in the upper space of SHA. Notably, when the focusing spot is at reverse side of the FEB, the radiation direction of CR will be in the backward direction of particle velocity, namely, the reversed CR will be achieved, see Fig. 3(b). Also, we note that these focused CRs do not have the Cherenkov cone as that in the conventional CR, and they haven’t definite Cherenkov-angles either since the radiation directions of different units in the array differ from each other.

Fig. 3 (a) 3D diagram of sheet FEB driving a SHA, which is lined along the curve $f (y, z) = 0$ . (b) is the same as (a) except for reversed CR. (c-f) Simulated field intensity ( $| E |$ ) distributions in the focusing plane. (c) and (d) illustrate the forward CRs. (e) and (f) illustrate the backward radiation cases. In (c) and (e), the focusing spots are in the middle of the plane. While in (d) and (f), the focusing spots deviate from the middle of the plane. Here ‘P’ is the focusing point of radiation in the upper half-space.

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Figure 3(c-f) illustrate simulated distributions of electric field intensity ( $| E |$ ) in the focusing planes $z = Z_{0}$ for both forward and backward CRs. Here the parameters of sub-wavelength holes and of the FEB follow that given in Fig. 2. We note that the focusing spots at various designed points in both forward and backward directions are effectively achieved, according with predictions. We note that field intensities at focusing spots are much higher than that of the un-focused CRs.

It is worth to compare the focused (or steered) CR in the present scheme with the shaping of spatial and spectral of radiations previously studied in [18–21]. [18, 19] applied the holographic plasmonic metasurfaces to steer the direction of TR and [20] used complex periodic and aperiodic gratings to shape the spatial and spectral distributions of SPR. In all these schemes, the radiation is frequency dependent—waves with different frequencies radiate in different directions. On the contrary, the CR in the present paper is frequency independent according to Eqs. (3) and (5), which will lead to many attractive applications which can not be achieved by other means. In addition, the TR and SPR in mentioned literature are incoherent radiation since they have a broad spectrum, differing from the coherent radiation of the present scheme.

4. Discussion and conclusion

In this section, we would like to make a discussion about the realization and application of the thresholdless and focused CRs achieved in the present paper. First, we discuss the realization of sheet electron beams, which are of essential for aforementioned CRs. The sheet FEB used in the present paper is actually a sheet-shaped electron bunch. It can be generated from a cathode with a flat rectangular emitting surface, together with four focusing electrodes, which had been approximately achieved in experiments by several groups [38–40].

As for the applications of the thresholdless CR, it is very attractive for developing compact and high-power light sources since it uses low energy free-electron. Also, it is a directional radiation with all the frequencies being in the same direction, which means that it can be developed as a multi-color and directional light source.

Then, we would like to give an example of using the focused CR illustrated in the present paper in practical applications. In the terahertz near-field imaging microscopy, the reflection mirrors and lenses have to be used to focus the terahertz wave on a small tip, which determines the resolution of the imaging [41]. Using the focused radiation, the reflection mirrors or lenses, which usually have great loss in the terahertz region, can be avoided. It is quite helpful in practices.

In summary, we have demonstrated a coherent and threshold-less Cherenkov radiation in vacuo by using a sheet free-electron beam to drive an oblique-lined sub-wavelength hole array. In addition, we have shown a focused Cherenkov radiation by re-arranging the sub-wavelength holes. These radiations may lead to broad applications in developing the integrated light sources from the microwave to infrared region.

Funding

National Natural Science Foundation of China (61471332, 51627901, U1632150, 11675178); Chinese Universities Scientific Fund (WK2310000059).

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Threshold-less and focused Cherenkov radiations using sheet electron-beams to drive sub-wavelength hole arrays

Abstract

1. Introduction

2. Threshold-less Cherenkov radiation

3. Focused Cherenkov radiation

4. Discussion and conclusion

Funding

References

Cited By

Figures (3)

Equations (6)

Optics Express