Transforming terahertz plasmonics within subwavelength hole arrays into enhanced terahertz mission via Smith-Purcell effect

Zijia Yu; Yucheng Liu; Weihao Liu

doi:10.1364/OE.389266

1. Introduction

Terahertz electromagnetic wave is one of the most attractive topics thanks to its tremendous application prospects in wide region of scientific and technological fields [1,2]. Efficient terahertz emission sources are essential for various applications [3–5]. The sub-wavelength hole arrays (SHAs), functioned by surface plasmonics (SPs), have tremendous application prospects in the terahertz region, and have attracted an increasing interest [6–10]. In previous studies, the SHAs generally operated as passive devices, which can not generate terahertz emission. Using free-electrons to interact with the SPs within nano scale SHAs has been demonstrated to be an effective way for coherent light emission in the visible light region [11,12]. Yet, the interaction between free-electrons with terahertz SPs on sub-millimeter SHAs still has not been investigated specifically and unitized in practices.

Recent researches illustrated that using free-electrons to excite one dimensional (1D) or two-dimensional (2D) arrays of sub-millimeter sub-wavelength holes can generate coherent and steerable terahertz emissions via modified forms of Smith-Purcell radiation (SPR) [13] and Cherenkov radiation (CR) [14]. Each RSH is an independent resonant and radiative cavity with a series of resonant modes. The coherent radiation is achieved when the radiations from all RSHs in the array constructively interfere. Yet, in their schemes, the SHAs were etched in metal, which is treated as perfect electric conductor (PEC), in other word, the SPs had not been taken into consideration.

In the present paper, we propose and analyze the case that the PEC is replaced by semiconductors or metamaterials with terahertz SPs taken into account. We will show, by detailed theoretical analyses and simulations, that the electromagnetic properties of the RSH will change significantly. These SPs modes can then be transformed into coherent and enhanced terahertz emission via the modified SPR effect. Compared with the PEC case, the radiation intensity can be increased by more than an order of magnitude, offering a promising way for efficient terahertz wave emission. The paper is organized as follows: section II describes the model and its operating mechanism; section III analyzes the properties of plasmonic modes within the RSH; section IV shows the terahertz emission properties of the proposed scheme; section V concludes the paper.

2. Model description

The diagram of proposed scheme is shown in Fig. 1. A uniformly moving free-electron beam (FEB) parallelly skims over an array of rectangular sub-wavelength holes (RSHs), which are periodically etched in a metamaterial or semiconductor plate with SPs taken into consideration. The z-directional width of the hole is much less than the z-directional periodicity ($d\;<<\;L$), such that the coupling between adjacent holes can be ignored, and each sub-wavelength hole is an independent resonator with specific resonant modes. As the FEB skims over the array, the SPs modes within each RSH (hole-modes) and on the material-air interface of the plate between every two adjacent RSHs (surface-modes), will be successively induced. These hole-modes and surface-modes are then diffracted into up and down half-spaces through the apertures of the holes due to the abrupt changes of boundaries. The diffractions from all hole-modes or surface-modes will constructively interfere, forming two sets of superimposed coherent radiation, when [15,16]

(1)$$\frac{\omega_{h,s}}{v_e}L-\frac{\omega_{h,s}}{c}L\cos \theta={{k}_{z0}}L-k_0L\cos \theta =2n\pi,$$

in which $\omega _{h,s}$ is the radiation frequency (the subscripts ‘h’ and ‘s’ denote that from the hole-mode and the surface-mode, respectively), $v_e$ and c are respectively the electron velocity and the speed of light, L is the periodical distance between two adjacent RSHs, n is a positive integer, and $\theta$ denotes the radiation angle relative to the electron velocity. We note that Eq. (1) is a modification of the famous SPR relation [17]. Yet, here the radiation frequency $\omega _{h,s}$ is specified, determined by the hole-mode and the surface-mode, which is different from the conventional SPR with frequency covering a certain spectral band [18].

Fig. 1. Schematics of the proposed coherent Smith-Purcell radiation (SPR) transformed from the electron-beam-excited SPs modes on the material surface and within the subwavelength holes. Here ‘SM’ and ‘HM’ denote surface-mode and hole-mode, respectively.

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As is known that the resonant frequency of a RSH within the PEC plate can be expressed as [13]: ${{f}_{l,m,p}}=\frac {c}{2}\sqrt {{{(\frac {l}{h})}^{2}}+{{(\frac {m}{w})}^{2}}+{{(\frac {p}{d})}^{2}}}$, 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. When the SPs effects of the material are significant, the resonant properties of the RSH will change significantly, detailed analyses of which will be given in the next section

3. Analyses of the hole-modes and surface-modes

3.1 Analytic derivations

When the SPs are taken into consideration, the relative permittivity (dielectric function) of the plate can be approximated by using the modified Drude model [19]:

(2)$${\varepsilon_r(\omega)}=1-\frac{{\omega_{p}^2}}{{{\omega^2}+i\gamma_e\omega}},$$

in which $\omega _{p}$ denotes the effective electric plasma frequency within the material, $\omega$ the operating frequency, $\gamma _e$ the electric loss [20,21]. In practices, $\varepsilon _r$ is a complex number with negative real part, such that the electromagnetic fields can penetrate into the material and decay exponentially, which is the physical base of SPs [22].

We first consider the hole-modes within the RSH, the diagram of which is shown in Fig. 2(a). To get the resonant properties of RSH, the Maxwell equations should be solved in and around the RSH, which is usually a demanding task. In previous studies, the effective dielectric theory (EDT) [23,24] was resorted to approximately deal with the properties of RSHs. Unfortunately, the EDT can not consider the fields in the longitudinal (x) direction, which essentially defines the resonant modes within the RSH. In the present paper, we use the dielectric waveguide theory [25], which first treats the RSH to be an infinite long rectangular dielectric waveguide (RDW) in x direction and then use the resonant conditions to get the plasmonic modes within the RSH. The cross-section of RDW can be divided into nine regions according to the boundaries, see Fig. 2(b). To completely obtain the fields within and around the RDW, one needs to solve Maxwell equations in all regions according to the boundary conditions. Also, there is neither TE nor TM mode exists in the RDW as that in the PEC waveguides: both the longitudinal electric ($E_x$) and magnetic ($H_x$) fields exist in the waveguide, which makes the problem be more complex. To simplify the analysis, we resort to the following two approximations: 1) The fields in four corners (shaded regions in the figure) can be ignored since they are relatively weak in compared with that in other regions. Thus, considering the symmetry of the structure, we only need to handle the fields in three regions (region-I, region-II, and region-III) as shown in the figure. 2) The fields are approximated as the superposition of transverse-magnetic modes in z direction ($TM_z$ modes with $E_z$, $E_x$, $E_y$, $H_x$, $H_y$) and that in y direction ($TM_y$ modes with $E_y$, $E_x$, $E_z$, $H_x$, $H_z$), both of which can be considered independently [25]. In the present paper, we only investigated $TM_z$ modes in detail, which can be excited by FEB moving in z direction.

Fig. 2. (a) 3D diagram of the RSH. (b) The y-z cross section of the RSH and the partition of regions according to the boundaries.

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In the coordinates shown in the figure, the fundamental $TM_z$ mode within the RDW (region-II) can be expressed as:

(3)$$E_x^{II} ={-} j{A_2}{k_x}{k_z}\cos ({k_y}y)\sinh ({k_z}z){e^{ - j{k_x}x}}$$

(4)$$E_y^{II} = {A_2}{k_z}{k_y}\sin ({k_y}y)\cosh ({k_z}z){e^{ - j{k_x}x}}$$

(5)$$E_z^{II} = {A_2}({\omega ^2}{\mu _0}{\varepsilon _0} + {k_z}^2)\cos ({k_y}y)\cosh ({k_z}z){e^{ - j{k_x}x}}$$

(6)$$H_x^{II} ={-} j{A_2}\omega {\varepsilon _0}{k_y}\sin ({k_y}y)\cosh ({k_z}z){e^{ - j{k_x}x}}$$

(7)$$H_y^{II} ={-} {A_2}\omega {\varepsilon _0}{k_x}\cos ({k_y}y)\cosh ({k_z}z){e^{ - j{k_x}x}}$$

where $k_x^2+k_y^2+k_z^2={{k}_{0}}^{2}$, $k_0=\omega /c$ is the wave number in the vacuum, $k_x$, $k_y$, $k_z$ are wave numbers in x, y, z directions, respectively. $A_2$, together with $A_1$ and $A_3$ in the following, are coefficients to be determined by boundary conditions. Note that the fields are standing waves in both y and z directions due to the reflections at sidewalls of RDW. In z direction, the fields are written as decaying wave due the SPs effects.

Considering the boundary conditions, the fields in region-I can be expressed as:

(8)$$E_x^I = j{A_1}{k_x}k_{_z}^I\cos ({k_y}y){e^{ - k_{_z}^Iz}}{e^{ - j{k_x}x}}$$

(9)$$E_y^I = {A_1}k_{_z}^I{k_y}\sin ({k_y}y){e^{ - k_{_z}^Iz}}{e^{ - j{k_x}x}}$$

(10)$$E_z^I = {A_1}({\omega ^2}{\mu _0}{\varepsilon _0}{\varepsilon _r} + {k_z^I}^2)\cos ({k_y}y){e^{ - k_{_z}^Iz}}{e^{ - j{k_x}x}}$$

(11)$$H_x^I ={-} j{A_1}\omega {\varepsilon _0}{\varepsilon _r}{k_y}\sin ({k_y}y){e^{ - k_{_z}^Iz}}{e^{ - j{k_x}x}}$$

(12)$$H_y^I ={-} {A_1}\omega {\varepsilon _0}{\varepsilon _r}{k_x}\cos ({k_y}y){e^{ - k_{_z}^Iz}}{e^{ - j{k_x}x}}$$

where $k_x^2+k_y^2+{k_z^I}^2={{\varepsilon }_{r}}{{k}_{0}}^{2}$, and ${\varepsilon }_{r}$ is the relative dielectric function of the material expressed by Eq. (2). Note that the fields decay exponentially (with a factor of ${e}^{-k_{z}^{I}z}$) in the material.

Likewise, the fields in region-III can be expressed as:

(13)$$E_x^{III} ={-} j{A_3}{k_x}{k_z}{e^{ - k_y^{III}y}}\sinh ({k_z}z){e^{ - j{k_x}x}}$$

(14)$$E_y^{III} = {A_3}{k_z}k_y^{III}{e^{ - k_y^{III}y}}\cosh ({k_z}z){e^{ - j{k_x}x}}$$

(15)$$E_z^{III} = {A_3}({\omega ^2}{\mu _0}{\varepsilon _0}{\varepsilon _r} + {k_z}^2){e^{ - k_y^{III}y}}\cosh ({k_z}z){e^{ - j{k_x}x}}$$

(16)$$H_x^{III} ={-} j{A_3}\omega {\varepsilon _0}{\varepsilon _r}k_y^{III}{e^{ - k_y^{III}y}}\cosh ({k_z}z){e^{ - j{k_x}x}}$$

(17)$$H_y^{III} ={-} {A_3}\omega {\varepsilon _0}{\varepsilon _r}{k_x}{e^{ - k_y^{III}y}}\cosh ({k_z}z){e^{ - j{k_x}x}}$$

where $k_x^2+{k_y^{III}}^2+{k_z}^2={{\varepsilon }_{r}}{{k}_{0}}^{2}$.

The tangential field components should satisfy the following continuous conditions at the boundaries of the RDW:

(18)$$E_{x}^{I}{{|}_{z=\frac{d}{2}}}=E_{x}^{II}{{|}_{z=\frac{d}{2}}}$$

(19)$$H_{y}^{I}{{|}_{z=\frac{d}{2}}}=H_{y}^{II}{{|}_{z=\frac{d}{2}}}$$

(20)$$E_{z}^{III}{{|}_{y=\frac{w}{2}}}=E_{z}^{II}{{|}_{y=\frac{w}{2}}}$$

(21)$$H_{x}^{III}{{|}_{y=\frac{w}{2}}}=H_{x}^{II}{{|}_{y=\frac{w}{2}}}.$$

Note that the boundary conditions of other four tangential field components, including $E_{y}$, $H_{x}$ at $z=\frac {d}{2}$ and $E_{x}$, $H_{z}$ at $y=\frac {w}{2}$, should also be satisfied. Yet in the case of $d\;<<\;w$, the relations of $H_{x}\;<\;H_{y}$ and $E_{x}\;<\;E_{z}$ are satisfied, such that the ignorance of these boundary conditions will not significantly affect the results in practices. For the case of $w\;<\;d$, the $TM_y$ mode will be dominant in practices, which can be handled by the same method.

Substituting field expressions into the boundary conditions of Eqs. (18)–(21), with complicated but straightforward mathematical operations, we can get the dispersion equations of the $TM_z$ modes in RDW:

(22)$$\begin{aligned}\left\{ \begin{array}{l} - {k_z}\tanh ({k_z}\frac{d}{2}) = \frac{{k_{_z}^I}}{{{\varepsilon _r}}}\\ \frac{{{k_y}}}{{{\omega ^2}{\mu _0}{\varepsilon _0} + {k_z}^2}}\tan ({k_y}\frac{w}{2}) = \frac{{{\varepsilon _r}k_y^{III}}}{{{\omega ^2}{\mu _0}{\varepsilon _0}{\varepsilon _r} + {k_z}^2}} \end{array} \right. \end{aligned}$$

These are two interconnected equations with variables of $\omega$, $k_z$, and $k_x$. Jointly solving them, we can get the dependencies of $k_z$ and $k_x$ on $\omega$, which determine the dispersion properties of RDW.

These dispersion equations govern the wave propagating in RDW and determines the relation between frequency and longitudinal propagation constant $k_x$. For the RSH, the waves in x direction will be reflected at both ends of the holes due to the abrupt changes of boundaries, which composes the resonant modes (hole-modes) within the RSH. To satisfy these boundary conditions, $k_x$ can be approximated as $k_x=g\pi /h$ (g is a non-negative integer indicating the mode order), which is exactly the Fabry-Pérot resonant condition of RSH [26,27]. The resonant frequencies can then be obtained by letting $k_x=n\pi /h$ in the dispersion equations.

Besides the resonant modes within the RSH, the SPs mode on the material surface (surface-mode) will also be excited by the incident fields of the FEB, and are govern by the following dispersion equation [28]:

(23)$${{\varepsilon }_{r}}{{k}_{1}}\left( 1+{{e}^{2{{k}_{m}}h}} \right)={{k}_{r}}\left( 1-{{e}^{2{{k}_{m}}h}} \right).$$

Here ${{k}_{1}}=\sqrt {k{{_{zp}^{{}}}^{2}}-k_{0}^{2}}$, $k_{zp}=\omega /v_p$, $v_p$ is the phase velocity of the SPs propagating in the z-direction, ${{k}_{r}}=\sqrt {{{k}_{zp}}^{2}-{{\varepsilon }_{r}}k_{0}^{2}}$. In order to excite the surface-mode, the velocity of the electron beam should match the phase velocity of the surface mode $v_e=v_p$ [29], which leads to $k_{zp}=k_{z0}$. In other word, the frequency of the surface-mode can be tuned by changing the electron velocity (energy).

3.2 Numerical calculations and simulations

In this section, we carry out numerical calculations based on the equations obtained above, and perform simulations on the SPs modes excited by a FEB within the RSH. In simulations, the fully electromagnetic particle-in-cell code [30] is used, and the electron-energy of the FEB is set to be 40 kV. The simulated field spectrum detected at the aperture of the RSH is given in Fig. 3(a), which shows two peak frequencies: 0.117 THz and 0.184 THz. Here the hole parameters are set to be: d=0.06 mm, w=0.4 mm, and h=0.3 mm. The parameters of the material are set to be: $\omega _{p}=3.14\times 10^{12}$, $\gamma _{e}=1\times 10^{10}$, following that given in [31]. The field contour maps of these two frequencies illustrated by the insets show that they denote the two lowest hole-modes with $g$=0 and $g$=1, respectively. For comparison, the simulations on the PEC hole with the same size and shape have also been performed, the result of which is shown in the figure. We note that the frequencies of the hole-modes decrease remarkable when the SPs are taken into consideration (the frequency of the PEC-hole-mode is about 0.37 THz). More importantly, the field intensity within the hole is greatly enhanced by the SPs effect. The simulated field spectrum, together with the field contour map, detected on the surface of the material is presented in Fig. 3(b), which shows that the frequency of the surface-mode is about 0.35 THz and its fields are largely confined nearby the surface of the material—a typical feature of the SPs on a planar surface.

Fig. 3. (a) Simulated field spectrum detected at the aperture of the RSH. The insets show the contour maps of $E_z$ field at the frequencies of 0.117 THz and 0.184 THz. The case of the PEC-hole is shown for comparison. (b) Simulated field spectrum detected on the material surface and the contour maps of $E_z$ field at the peak frequency.

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The calculated frequencies of the first (g=0) and the second (g=1) hole-modes within the RSH versus the structure parameters are shown in Fig. 4. The frequencies decrease as the transverse width (w) increases, similarly to that in PEC holes. Yet, they increase as the longitudinal width (d) increases, contrary to the case of PEC holes. It is due to the coupling of the surface fields on the parallel sidewalls (along y direction) of the RSH [23]. We note that for the first mode, the theoretical results agree well with the simulated ones. Yet, for the second mode, the calculated frequencies are obviously higher than the simulated ones. It can be explained that in calculating the frequency of the second mode, the assumption of $k_x=\pi /h$ is used, by which we assume that the fields are totally reflected at two apertures of the RSH in x direction. In fact, the fields will extend to the outsides of the RSH as shown in the inset of Fig. 3(a), which physically increases the hole depth h. Thus, we enlarge the value of h (by multiplying a factor of 1.25) to get the modified frequency of the second hole-mode, see the dash lines in the figure, which agree well with the simulated results.

Fig. 4. Calculated and simulated frequencies of the first hole-mode (a) and of the second hole-mode (b) versus w and d. The dash lines in the subplot (b) denote the modified results. The black, red, and blue lines denote the cases of w=0.4 mm, 0.6 mm, and 0.8 mm, respectively.

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4. Coherent radiation from the plasmonics modes within the array of RSH

In this section, we calculate and simulate the SPR from the array of hole-modes and of surface-modes excited by electron beam. The total radiation energy from the array can be expressed as [32]:

(24)$$W_t=W_0{{\left| \sum_{n=1}^{M}{{{e}^{{-}j{{k}_{z0}}nL+jk_{0}nL\cos \theta}}} \right|}^{2}}, \\$$

in which $W_0$ is the radiation energy from the SPs mode within a single period. $M$ is the number of periods in the array. We note that $W_t={M}^2W_0$ when the SPR relation Eq. (1) is satisfied, indicating that the radiation from all SPs modes in the array is coherent.

The simulated field spectrum detected above the array is given in Fig. 5, which shows two peak radiating frequencies: 0.184 THz and 0.346 THz. They match the frequencies of the second hole-mode and the surface-mode, respectively, indicating that the radiation is emitted from the SPs modes. The simulated contour maps of $E_z$ component at these two frequencies are illustrated in Fig. 6(b) and Fig. 6(c), which show that the radiation directions of these modes are $\theta =86.7^o$ and $\theta =100^o$, respectively, agreeing with the theoretical results based on Eq. (1). In other word, the SPs modes are transformed into coherent SPR as predicted. Note that the first hole-mode with frequency of 0.117 THz has not been detected in the radiation field, which is because that it does not satisfy the SPR relation and the fields can not effectively radiate as illustrated in Fig. 6(a). For comparison, the simulations for the case of SHAs within a PEC plate are also performed, the radiation spectrum of which is shown in Fig. 5. We note that the radiation field intensities from the SPs modes are more than 4 times higher, indicating the radiation power will be enhanced by more than an order of magnitude.

Fig. 5. Simulated radiation field spectrum detected above the SHA. The results of the PEC case is shown for comparison.

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Fig. 6. Simulated contour maps of the $E_z$ field for (a) 0.117 THz, (b) 0.184 THz, and (c) 0.346 THz, respectively.

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In the following, we would like to discuss the tunability and efficiency of SPR from the proposed scheme. As mentioned previously, the frequency of the surface-mode can be effectively tuned by changing the velocity (energy) of the electron beam, then by which the frequency of SPR can also be tuned. In contrast, the frequency of the hole-mode is independent from the electron energy as indicated in Eq. (23). Thus, the frequency of SPR from the hole-mode can only be tuned by varying the structure parameters of RSH as illustrated in Fig. 4. According to Eq. (24), the radiation energy of SPR is proportional to the square of the number of periods in the array ($M^2$). Thus, the efficiency will also be proportional to $M^2$ provided that the energy of the electron beam is kept unchanged, namely, the more periods, the higher efficiency. It can be understood that the electron beam can interact with more plasmonic modes on the SHAs with more periods. It should be noted that the electron beam will lose kinetic energy and will be decelerated as it generates SPR, which is the so-called radiation loss. If $M$ is too large, the electron will lose substantial energy and will be decelerated obviously, such that the radiation intensity will be lowered and the radiation direction will also be changed according to Eq. (1). In other word, the maximum efficiency of SPR is limited by the radiation loss. In practices, in order to keep the radiation intensity together with a desirable spectral purity at specified directions, the efficiency of SPR should be generally less than 5$\%$.

5. Conclusion

We illustrated an enhanced terahertz radiation by using free-electron beams to excite an array of rectangular sub-wavelength holes (RSHs) with terahertz plasmonics. The radiative plasmonic modes within each RSH are successively excited by the electron beam, and then collectively generate coherent radiation due to the constructive interference, composing a special kind of Smith-Purcell radiation. Compared with the case without surface plasmonics, the plasmonic modes have much higher field intensity, which leads to a remarkably enhanced radiating intensity: the enhancement is more than an order of magnitude. Theoretical analyses of the plasmonic modes within RSH and of the radiating properties were performed and verified by simulations. The proposed scheme affords an efficient way for developing terahertz radiating sources.

Funding

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

Disclosures

The authors declare no conflicts of interest.

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Transforming terahertz plasmonics within subwavelength hole arrays into enhanced terahertz mission via Smith-Purcell effect

Abstract

1. Introduction

2. Model description

3. Analyses of the hole-modes and surface-modes

3.1 Analytic derivations

3.2 Numerical calculations and simulations

4. Coherent radiation from the plasmonics modes within the array of RSH

5. Conclusion

Funding

Disclosures

References

Cited By

Figures (6)

Equations (24)

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