1. Introduction

The generation and steering of optical beams is an important issue in nanoscale optics that has drawn great attention recently [1–4]. Though electric and optical quantum emitters along with designed nanoantenna are the main nanoscale optical sources, they lack frequency tunability. An electron beam can excite the eigenmodes of nanostructure, which will be further transformed into radiation using grating structures. In particular, Smith-Purcell radiation (SPR) can be emitted when an electron beam passes above a periodic metal grating [5–10]. This radiation comes from constructive interference of the induced oscillating dipole of free charges and imaged charges on the grating. The relationship between the emission wavelength and angle of SPR can be described by the following equation [6,10]:

Surface plasmons (SPs) are free electrons on the metal surface that perform coherent oscillation [11]. They can be excited by light using the attenuated total reflection configuration or a grating structure to match the phase velocities of light and SPs. On the other hand, the relativistic electron beam can directly excite SPs on a metal surface since the dispersion curves of the electron beam and SPs intercept each other [12,13]. Furthermore, the emission power of SPR can be enhanced at the frequency of electron-beam excited SPs on the metal substrate when it is within the emission band of SPR [6,14]. The power is enhanced because energy of the electron beam is first transferred into SPs and then reradiated through the SPR mechanism. Very recently, generation of convergent light beams by SP-manipulated SPRs on metallic chirped gratings has been demonstrated [10]. However, for one-dimensional gratings, only the emission along the direction of electron beam can be controlled. To steer both the azimuthal and polar angles of the emission pattern requires other mechanisms.

Recently, a new type of multi-color and multi-directional SPRs from sheet electron-beams driving two-dimensional sub-wavelength hole arrays has been proposed [15]. In this device, multi-color radiation is achieved by exciting different resonant modes of the holes. The multi-directional emission is obtained by designing the two-dimensional phased-array antennas such that radiation from all the holes constructively interferes in a specific direction. On the other hand, the Yagi–Uda antenna (YUA) is a directional antenna consisting of feed, reflector and directors [16–22]. The driven dipole source is placed at the center of the feed element. The YUA uses multiple elements to form an array to achieve constructive interference of radiating waves in one direction and destructive in the opposite. Recently, highly unidirectional emission of a quantum dot coupled to an optical Yagi-Uda nanoantenna (YUNA) has also been demonstrated [19]. Combining an SP-manipulated SPR mechanism and a YUNA structure will control the emission far-field pattern, though it has not yet been investigated. Electron beam size in the SP-manipulated SPR is assumed to be smaller than the width of gratings, so the above-mentioned scheme (i.e. [15]) using a sheet electron beam to generate controllable far-field pattern cannot be applied in this case. (A cold field-emission gun can be used to produce an electron source as small as 5 nm in diameter [23].) This work uses the SP-manipulated SPR with YUNA structures to generate a light beam with a designed far-field pattern and color and analyzes it by finite-difference time domain (FDTD) simulations. Dependence of the azimuthal and polar angles of emitted beam on geometry parameters of feed and directors of YUNA are examined. The far-field radiation patterns from SPR and a single dipole source placed beside the feed element are also compared. Finally, the emitted radiations with a single main lobe and multiple main lobes containing a single wavelength / multiple wavelengths are designed and investigated.

2. Simulation model and method

All the schematic diagrams of simulation structures investigated in this work are presented in Figs. 1(a)–1(g). Figure 1(a) plots the simulated structure for generation of SPR on a YUNA array. In Fig. 1(a), the substrate is an insulator-metal-insulator (IMI) structure composed of a silver (Ag) film (gray layer) sandwiched between two dielectric films with the same refractive index (blue layers) [10]. Then a buffer layer (pink layer) is added on the IMI substrate. The thicknesses of Ag, dielectric films and buffer layer are all set to 20 nm. The refractive indices (n) of the dielectric films and the buffer layer are set as 2.1 and 1.1, respectively. The YUNA array, also made of Ag, is deposited on the buffer layer. It consists of five YUNAs and each YUNA contains a feed (dark green cuboid) and three directors (green cuboid) with a common reflector (light green cuboid). Thicknesses of the reflector, feeds and directors are 60 nm. Figure 1(b) presents a top view of the YUNA array. Definitions of geometry parameters and their values are given in Table 1.

 

Fig. 1. (a) Simulation structure for generation of SPR from a YUNA array deposited on an IMI substrate. The electron bunch moves along the +x direction under the IMI substrate and directly below the feed elements to excite SP on the IMI substrate. (b) Top view of (a). Definitions of geometry parameters and their values are listed in Table 1. (c) Simulation structure for x-polarized plane wave normally incident to a single feed element deposited on an IMI substrate. (d) Simulation structure for SP-manipulated SPR on an IMI substrate with the feed elements as gratings. (e) Simulation structure for a dipole current density source placed at the side of the feed element of a single YUNA on an IMI substrate. (f) and (g) Simulation structures for generating two beams, using two groups of YUNA arrays, and for three beams, using three groups of YUNA arrays.

Download Full Size | PDF

Table 1. Definitions and values of geometry parameters used in simulations

View Table

This work uses the FDTD program MEEP (v1.3) [24]. The simulations are performed in Cartesian x-y-z coordinate system with the grid size in all directions set as 4 nm. Perfectly matched layers (PMLs) are applied to enclose the whole simulation region. The Drude model of Ag is utilized with the high-frequency dielectric constant, plasma frequency and collision frequency set as 5.0, $1.4433 \times {10^{16}}$ rad/s and $1.4995 \times {10^{14}}$ rad/s, respectively [25]. In the simulation, the electron bunch is represented by a dipole source which moves in its path at 0.328 times the velocity of light (i.e. 30 keV, using the “change-sources!” function in MEEP). Far-field patterns are obtained by near-to-far-field transformation in the frequency domain. Electron bunches move along the + x direction under the IMI substrate (and directly below the feed elements) with the energy of 30 keV. In this configuration, frequency of the excited SPs on the IMI substrate and the subsequent SPR is $\textrm{5}.607 \times {10^{14}}$ Hz (i.e. the vacuum wavelength (${\lambda _0}$) is 535 nm) [10]. Notably, the vacuum wavelength of SPR can be manipulated by changing the refraction index of the dielectric film and the electron bunch energy [10].

3. Results and discussion

Because SPR mechanisms are strongly related to the radiating dipoles formed on the gratings, the dipole mode of a single feed element on an IMI substrate excited by a normally incident x-polarized plane wave is examined first. Figure 1(c) plots the simulation structure with W_F = 40 nm and changed L_F (see Table 1). Figure 2(a) presents the excited spectra (the amplitude of an excited x-component electric (Ex) field versus wavelength, measured at the yellow point in Fig. 1(c)) for different values of L_F. In Fig. 2(a), two groups of resonant modes are observed. The long-wavelength and short-wavelength resonant modes are the dipole-like mode and the quadrupole-like mode, respectively. Figures 2(b) and 2(c) plot the simulated time-averaged electric field energy (${{\vec{\textrm E}}^ \ast } \cdot {\vec{\textrm D}}\,/\,2$, D-energy) density in the x-z plane, cut at the center of the feed element in the y direction, and in the x-y plane, at the top surface of the feed element, respectively. Here L_F = 80 nm at ${\lambda _0} = 478\;\textrm{nm}$, shown as the second peak of the green line in Fig. 2(a). Figures 2(b) and 2(c) show that the pattern around the feed element is an electric dipole. However, some energy is also trapped in the IMI substrate, so it is referred to as the dipole-like mode in this work. Figure 2(d) presents the simulated peak wavelength of the dipole-like mode as a function of L_F. Based on Fig. 2(d), for the most resonant wavelength of dipole-like mode being at ${\lambda _0} = 535\;\textrm{nm}$, L_F should be designed at 112 nm. Notably, because of the wide bandwidth of the dipole-like mode, it still can be excited by the incident plane wave at ${\lambda _0} = 535\;\textrm{nm}$ with other values of L_F (e.g. L_F can be the values between 40 nm and 120 nm). (The dipole localized SP mode of a small particle generally has a small quality factor and a wide absorption spectrum, see page 186 of [11].)

 

Fig. 2. (a) Excited Ex field spectra for different values of L_F. (b) and (c) Simulated time-averaged electric field energy density in the x-z plane and the x-y plane, respectively, for L_F = 80 nm at ${\lambda _0} = 478\;\textrm{nm}$. (d) Simulated peak wavelength of a dipole-like mode as a function of L_F.

Download Full Size | PDF

Next, the emission properties of SP-manipulated SPR on the IMI substrate with feed elements only, acting as gratings, are examined (see Fig. 1(d) with five feeds, L_F = 112 nm and W_F = 40 nm). Figures 3(a), 3(b) and 3(c) plot snapshots of simulated D-energy density contours around the feed elements in the x-z plane (cut at the y position of the electron beam) for P = 136, 176 and 240 nm, respectively (other geometry parameters are listed in Table 1, ${\lambda _0} = 535\;\textrm{nm}$). Based on Eq. (1), the corresponding emission angles, ${\theta _0}$, are $\textrm{ - 6}{2^\textrm{o}}$, ${0^\textrm{o}}$ and ${55^\textrm{o}}$, respectively. Figures 3(a)–3(c) show that the simulated emission angles agree with the calculated ones. They also demonstrate that the emission angles in the x direction (i.e. the electron beam’s moving direction) can be controlled by changing the grating period. Figure 3(d) presents the same snapshot of D-energy density contours in the y-z plane at the center of the third feed element in x direction for P = 176 nm. Figure 3(d) shows that the radiation lacks directionality in the y-z plane. Therefore, the emission angle perpendicular to the electron beam’s movement cannot be manipulated by the one-dimensional gating structure.

 

Fig. 3. (a) – (c) Snapshots of simulated electric energy density contours around the feed elements in x-z plane for P = 136, 176 and 240 nm, respectively (L_F = 112 nm and W_F = 40 nm). (d) The same simulated contour as in (b) except for in the y-z plane (cut at the center of the third feed element in x direction).

Download Full Size | PDF

The far-field radiation patterns of SP-manipulated SPR on the YUNA array are subsequently examined (see Fig. 1(a) and Table 1). Here the YUNA array is designed such that the radiation will emit upward by the gratings (i.e. the feed elements with P = 176 nm) and turn to the + y direction by adding the director elements along + y direction. With this configuration, the azimuthal angle ($\phi$) and polar angle ($\theta$) of the far-field pattern should be $\phi = {90^\textrm{o}}$ and ${0^\textrm{o}} < \theta < {90^\textrm{o}}$. (The rectangular (x, y, z) and spherical ($r,\,\theta ,\,\phi$) coordinate systems are plotted in the inset of Fig. 4(a).) The effects of L_F and L_D on the far-field radiation pattern are investigated, and other parameters are fixed, as listed in Table 1. Figure 4(a) plots the simulated polar angle as a function of L_F for L_D = 48 nm, 88 nm and L_F. Figure 4(a) shows that $\theta$ generally decreases gradually as L_F increases. With these three values of L_D, the polar angle is between 40° and 60°. Figure 4(a) also shows that the second lobe (${0^\textrm{o}} < \theta < {10^\textrm{o}}$, the dashed lines) emerges for L_D = 88 nm and L_D = L_F when L_F is larger than 96 nm. Figures 4(b)–4(f) present the simulated far-field radiation power patterns in polar coordinates, normalized to the maximum radiation power, for (L_F, L_D) equal to (48 nm, 88 nm), (96 nm, 48 nm), (152 nm, 48 nm), (152 nm, 88 nm) and (128 nm, 128 nm), respectively. These are marked as 1, 2, 3, 4 and 5, respectively in Fig. 4(a). Figures 4(b)–4(f) show that the $\phi$ angle of the far-field pattern is about 90°, which agrees with the design in Fig. 1(a). They also show that the $\theta$ angle can be controlled by changing the values of L_F and L_D. Furthermore, the second lobe of the radiation pattern is clearly shown in Figs. 4(d), 4(e) and 4(f). This lobe has a small $\theta$ angle, indicating it is emitted vertically. In Figs. 4(e) and 4(f), the observed radiation power of the second lobe is even larger than that of the first lobe. In these cases, the YUNA array cannot effectively direct the radiation in the y direction, so the original vertically-emitted SPR pattern (due to P = 176 nm) is displayed. Furthermore, the simulation results also reveal that, for L_D = 48 nm and 88 nm, the SPR has the maximum emission power at about L_F = 96 nm. This length is slightly less than 112 nm, which is the required feed element length for the peak of the excited spectrum of dipole-like mode at ${\lambda _0} = 535\;\textrm{nm}$. This is ascribed to that the near-field coupling between two adjacent feed elements enables the dipole-like mode to be red-shifted [11].

 

Fig. 4. (a) Simulated far-field polar angles as a function of L_F for L_D = 48 nm, 88 nm and L_F (P = 176 nm). Inset: Rectangular (x, y, z) and spherical ($r,\,\theta ,\,\phi$) coordinate systems. (b) – (f) Simulated far-field radiation power contours in polar coordinates (normalized to the maximum radiation power) for (L_F, L_D) equal to (48 nm, 88 nm), (96 nm, 48 nm), (152 nm, 48 nm), (152 nm, 88 nm) and (128 nm, 128 nm), respectively. These are marked as 1, 2, 3, 4 and 5, respectively in (a).

Download Full Size | PDF

To further demonstrate that the emission feature perpendicular to the direction of e-beam movement is manipulated by YUNAs, the far-field radiation pattern of SPR on a YUNA array (P = 176 nm, Fig. 1(a)) is also compared with that of a dipole current density source placed at the side of the feed element of a single YUNA. This is shown in Fig. 1(e), L_F = 96 nm and L_D = 88nm, where the source is a monochromatic source with vacuum wavelength of 535 nm. Figures 5(a) and 5(b) plot simulated far-field radiation power patterns for the structures of Fig. 1(a) and 1(e), respectively. Figures 5(a) and 5(b) show that both SPR and the dipole source emit radiation into the far field with the same $\theta$ angle, which is determined by the structure of YUNA. However, the radiation from SPR has a narrower distribution of $\phi$ angle around $\phi = {90^\textrm{o}}$ because the period of the YUNA array is designed for upward emission based on Eq. (1).

 

Fig. 5. (a) and (b) Simulated far-field radiation contours in polar coordinates (normalized to the maximum radiation power) for structures of Figs. 1(a) and 1(e), respectively (L_F = 96 nm and L_D = 88nm for both Figs. 1(a) and 1(e), and P = 176 nm in Fig. 1(a)). (c) Semi-analytical model of (b) with the monochromatic line dipole sources F, D1, D2 and D3. (d) Simulated far-field radiation contours in polar coordinates (also normalized to the maximum radiation power) for the semi-analytical model.

Download Full Size | PDF

The far-field radiation pattern of a single YUNA is also examined by a semi-analytical model. In this model, we measure the simulated amplitudes and phase differences of the x-component electric fields (Ex) at the edge of feed and director elements excited by the dipole source (see Fig. 1(e)). Then the feed and director elements are replaced by four monochromatic (${\lambda _0} = 535\;nm$) x-polarized line dipole sources with the measured amplitudes and phase differences. This is shown in Fig. 5(c), with the amplitudes and phases (amp, ${\varphi _0}$) of the monochromatic line dipole sources F, D1, D2 and D3 (see Fig. 5(c)) being (1.0, 0.0), (0.2442, 0.2748π), (0.1163, 0.7328π) and (0.1047, 1.3435π), respectively. They are placed at the positions of feed and director elements and 10 nm above the buffer layer. The sizes of F are 96 nm, 40 nm and 12 nm in x, y and z, respectively, directions. The sizes of D1, D2 and D3 are 88 nm, 40 nm and 12 nm in x, y and z, respectively, directions. Here the reflector element is retained in the model. Figure 5(d) presents the simulated far-field radiation power pattern of the proposed model. Comparing Fig. 5(b) with Fig. 5(d) shows that the far-field patterns of a single YUNA excited by a dipole source and the semi-analytic model are very similar. This demonstrates that the far-field pattern of a single YUNA originates from constructive interference of radiating waves from the line dipole source that are induced on the feed and director elements and with different amplitudes and phase differences.

Next, emission with controllable multiple beams by using a few groups of YUNA arrays is investigated. The simulated structures for generating two and three beams are plotted in Fig. 1(f), using two groups of YUNA arrays, and in 1(g), using three groups of YUNA arrays, respectively. Here we design the period of each group of YUNA arrays such that the distances of $\phi$ angle in the far-field pattern between the emitted beams are as large as possible. Furthermore, the number of YUNAs (N) and the value of L_F in each group of YUNA arrays are adjusted to enable the emission powers of the beams to be close to each other. Figure 6(a) presents the simulated far-field radiation power pattern for the generation of two beams with P = 136 nm (216 nm), N = 6 (4), L_F = 96 nm (72 nm) and L_D = 48 nm (48 nm) in the first (second) group of YUNA array (see Fig. 1(f), and other geometry parameters are also listed in Table 1). The distance between the two groups of YUNA arrays, L_G, is set to 600 nm to avoid cross interference of the emission beams. These parameters will make the upward-emitted beam from the first (second) group tilt into both -x and + y (+x and –y) directions. Figure 6(a) shows two distinct beams with $\textrm{(}\phi ,\textrm{ }\theta \textrm{)}$ equal to (141°, 51°) for group 1, and (321°, 51°) for group 2. This agrees with the design of two main emitted beams having the difference of $\phi$ angle of about 180°. Weak side lobes of the emitted beams are also observed in Fig. 6(a), coming from interference of the two emitted beams.

 

Fig. 6. (a) and (b) Simulated far-field radiation power contours (normalized to the maximum radiation power) for structures of Figs. 1(f) and 1(g), respectively. In (a), the geometrical parameters for the first (second) group of YUNA are: P = 136 nm (216 nm), N = 6 (4), L_F = 96 nm (72 nm) and L_D = 48 nm (48 nm). L_G = 600 nm. In (b), the geometric parameters for the first (second, third) group are: P = 176 nm (136 nm, 240 nm) and L_F = 72 nm (96 nm, 72 nm), where L_D = 48 nm for all groups of arrays, L_G = 400 nm.

Download Full Size | PDF

Figure 6(b) plots the simulated far-field radiation power pattern generating three beams by using three groups of YUNA arrays (Fig. 1(g)). Geometric parameters for the first (second, third) group are: P = 176 nm (136 nm, 240 nm) and L_F = 72 nm (96 nm, 72 nm), where L_D = 48 nm for all groups of arrays, L_G = 400 nm, and other geometry are as given in Table 1. The design methodology is the same as that in Fig. 6(a). Figure 6(b) shows three main emitted beams in the far field with $\textrm{(}\phi ,\textrm{ }\theta \textrm{)}$ equal to (267°, 45°) from group 1, (147°, 54°) from group 2 and (24°, 54°) from group 3. This is also in accord with the design. There are also weak side lobes of emitted beams in the far-field pattern due to interference between the emitted beams. Figures 6(a) and 6(b) both demonstrate that the emission of multiple beams with demanded azimuthal and polar angles can be devised by utilizing the YUNA arrays with tuning the lengths of feed and director components and the period of arrays.

Finally, emission of red, green and blue beams by a single electron bunch is illustrated, also using three groups of YUNA arrays. The simulated structure is the same as Fig. 1(g) except that the refractive indices of the dielectric films in the second and third regions are changed to 2.6 and 1.7, respectively. According to [10], the vacuum wavelengths of e-bunch excited SPs and subsequent SP-manipulated SPRs in regions 1, 2 and 3 are 535 nm (green light), 633 nm (red light) and 457 nm (blue light), respectively. The geometric parameters for the first group stay the same (i.e. P = 176 nm and L_F = 72 nm) so that the green light emits upward and tilt to the –y direction. In the second (third) group, P and L_F are changed to 160 nm and 80 nm (200 nm and 80 nm), respectively. The other geometry parameters are the same as in Fig. 6(b). This design makes the red and blue beams emit upward and tilt to -x and + x directions, at about −65° and 50° respective to the + z direction, based on Eq. (1), and tilt to the + y direction. Figures 7(a), 7(b) and 7(c) plot the simulated far-field radiation power patterns for green, red and blue lights, respectively. Figures 7(a)–7(c) display that the green, red and blue beams in the far field are concentrated at $\textrm{(}\phi ,\textrm{ }\theta \textrm{)}$ equal to (261°, 42°), (162°, 48°) and (39°, 54°), respectively, which also matches the design. Notably, the measured maximum radiation powers of red and blue light are about one fourth and double, respectively, that of green light. However, in each region the radiation power increases with the number of YUNAs. Adjusting the number of YUNAs and the value of L_F in each group of YUNA arrays can make the emission powers of red, green and blue lights to be close to each other.

 

Fig. 7. (a) – (c) Simulated far-field radiation power contours for green, red and blue light, respectively. For each light, the power is normalized to its maximum radiation power. The structure is the same as Fig. 1(g) and Fig. 6(b) except that refractive indices of the dielectric films, P and L_F in the second (third) region are 2.6 (1.7), 160 nm (200 nm) and 80 nm (80 nm), respectively.

Download Full Size | PDF

The power conversion efficiency for the proposed device can be calculated by $\eta \; = \;{P_{SPR}}/{P_0}$, where P_SPR is the total Poynting power of SPR integrated over all the simulation time and over the whole surfaces enclosing the upper simulation regions (measured at the edges of perfectly-matched-layer boundaries), and P₀ is the total available power of electron bunch in its entire path (i.e. the same Poynting power except that the structure is removed and measured at 20 nm above the electron bunch) [10]. The calculated power conversion efficiencies for SPR with feed only (Fig. 1(d)) and with a YUNA array (Fig. 1(a) and Fig. 4(c) where SPR has the maximum emission power) are 2.42% and 2.33%, respectively. Furthermore, the power conversion efficiency increases linearly with the number of gratings.

It is worth noting that, for specific wavelengths, the number of beams and their far-field emission angles (i.e. $\textrm{(}\phi ,\textrm{ }\theta \textrm{)}$) on demand can be controlled by designing the geometry parameters of YUNA arrays. First, the number of beams is determined by the number of groups of YUNAs. Second, the emission wavelength can be manipulated by altering the refractive indices of the dielectric films in the IMI substrate. After the wavelength is decided, emission angles of SPR along (noted as ${\alpha _{/{/}}}$) and perpendicular to (noted as ${\alpha _ \bot }$) the direction of electron movement can be controlled by changing the grating period (based on Eq. (1)) and the geometry parameters of YUNA arrays (based on the simulated results, Fig. 4(a)), respectively. Finally, combining ${\alpha _{/{/}}}$, ${\alpha _ \bot }$ and the position of YUNA array (i.e. it is located on the left side or right side of the electron beam, Figs. 1(f) and 1(g)) can roughly determine the far-field emission angles. For $\phi = {90^\textrm{o}}$ and ${270^\textrm{o}}$, using ${\alpha _{/{/}}} = {0^\textrm{o}}$ (upward emission), $\theta$ is decided mainly by ${\alpha _ \bot }$ (i.e. the beam tilts to + y direction or -y direction). For other cases, $\theta$ and $\phi$ roughly depend on ${\alpha _{/{/}}}$ and ${\alpha _ \bot }$, respectively. (${\alpha _{/{/}}}$ makes the beam emit upward and tilt to + x direction or –x direction and then ${\alpha _ \bot }$ makes the beam further tilt to + y direction or –y direction.) Actually, the far-field emission angles in Figs. 6 and 7 are designed by this rule. The proposed mechanism may have potential applications for light sources, optical imaging, optical beam steering, holography, microdisplay and cryptography.

4. Conclusion

This paper proposes the generation of light beams with steerable far-field patterns by SP-manipulated SPR on Yagi-Uda nanoantenna and investigates it by FDTD simulations. Oscillating dipoles on the grating structures (feed elements) that are induced by SPs excited by electron beams are driving sources of the nanoantenna. The feed, reflector and director elements together direct the radiation in a specific direction to the far field. Emission of SPR along and perpendicular to the e-beam movement direction can be manipulated by designing grating period and Yagi-Uda nanoantenna structure, respectively. Furthermore, the emission of multiple beams with demanded far-field $\textrm{(}\phi ,\textrm{ }\theta \textrm{)}$ angles can be achieve using different groups of Yagi-Uda nanoantenna arrays. The emission wavelength can be changed by varying the electron beam energy and the refractive index of dielectric layer of the IMI substrate. Accordingly, multi-wavelength beams with different far-field patterns can also be generated at the same time by multiple groups of Yagi-Uda nanoantenna arrays on different IMI substrates. This work can be applied in the fields of light sources, optical imaging, optical beam steering, holography, microdisplay and cryptography.