104/105 學年度第二學期
休假研究期滿報告書
休假研究計畫名稱:
電漿子超穎材料與電子束互作用機制模擬研究
單位: 光電科學與工程學系
姓名: 藍 永 強
職稱: 教授
中華民國 106 年 9 月 18 日
Contents
1. Temperature tunability of surface plasmon enhanced Smith-Purcell terahertz radiation for semiconductor-based grating ………. 1
2. Generation of convergent light beams by using surface plasmon locked
Smith-Purcell radiation ………... 18
3. Appendix ……… 33
Temperature tunability of surface plasmon enhanced Smith-Purcell terahertz radiation for semiconductor-based grating
Abstract
In this work, the terahertz (THz) Smith-Purcell radiations (SPRs) for the relativistic electron bunch passing over an indium antimonide (InSb)-based substrate with a subwavelength grating under various temperatures of substrate are investigated by FDTD simulations and theoretical analyses. The explored SPR is locked and enhanced at a certain emission wavelength with the emission angle still following the wavelength-angle relation of the traditional SPR. This wavelength agrees with the (vacuum) wavelength of surface plasmons (SPs) at the air-InSb interface excited by the electron bunch. The enhancement of SPR at this wavelength is attributed to the energy from electron concentrated in the excited SPs and then transformed into radiation via the SPR mechanism. When the temperature of InSb increases, the emission wavelength of the enhanced SPR decreases along with the emission angles increasing gradually. This work demonstrates that the emission wavelength and angle of the enhanced SPR from the InSb grating can be manipulated by the temperature of InSb. The temperature tunability of SP-enhanced SPR has potential applications in the fields of optical beam steering and metamaterial light source.
Introduction
Strong demand of terahertz (THz) applications has attracted great attention in the development of compact and tunable light source. Especially, high-efficiency generation of THz light source is one of the most important issues and still remains to be overcome for the field of imaging and diagnostics [1, 2]. As an electron beam moves over a metallic surface, the surface plasmon (SP) on the surface can be excited.
Subsequently, SP can be transformed into radiation modes by the designed structures such as slits, grooves and periodic gratings on the surface [3-6]. However, only the SPs whose operating frequencies close to the intrinsic plasma frequency of the metal can be efficiently excited. Hence the radiation frequencies are limited in optical and ultraviolet regions for most used noble metal [7].
An electron beam passing over a metallic grating can also generate Smith-Purcell radiation (SPR). This type of radiation comes from the oscillation of charges and imaged charges on the grating which induces the periodically changing current density on the grating [8-12]. The emission frequency of SPR ranges from microwave to visible depending on the grating period and the electron velocity. Very recently, SPR that are manipulated by excitation of SPs and mimimic-SPs has been proposed and demonstrated [9, 10, 12-14]. Especially, Ref. 10 investigates the condition that
both the frequencies of SP and mimic-SP are out of the radiation band of SPR. The SPR is not enhanced and the radiations from SPR, SP and mimic-SP are all observed.
References 13 and 14 propose and demonstrate amplification and manipulation of SPR by excitation of SPs on Ag film. And in Refs. 9 and 12, the generation of enhanced coherent THz SPR by excitation of mimic-SPs are proposed and explored.
The indium antimonide (InSb)–dielectric interface can also support propagation of SPs under illumination of terahertz light source [15]. The optical property of InSb at THz region is similar to that of noble metals in the optical region [16–18].
Furthermore, the optical properties of InSb can be tailored by changing its temperature [19], by applying external magnetic field on it [20] and by doping impurity into it [21]. Recently, many InSb-based tunable optoelectronic components based on controlling their temperatures have been studied, such as tunable photonic crystals [19], infrared photo-detectors [22], thermally controlled metamaterial [23–26], data storage [27], and subwavelength resolution [28]. Considering the above features of InSb, the enhanced and tunable THz light source can be implemented by applying the SPR with an InSb-based grating.
In this work, generating wavelength-adjustable narrow-band THz radiation via using SP-enhanced SPR on the InSb-based gratings and by controlling the temperature of substrate is proposed and investigated by FDTD simulations and theoretical analyses. This type of THz source with temperature-controlled emission wavelength and angle based on SP-enhanced SPR has never been proposed and investigated. The mechanism of concentration and enhancement of SPR at certain emission wavelengths and angles via excitation of SP is also elucidated by examining the effects of substrate temperature on the spectra of SP and SPR. The SPR on the InSb substrate provides a flexible device to verify the proposed SP-enhanced SPR mechanism. Here the manipulation of substrate temperature can be achieved by adding a heater under the substrate. In addition, part of emitted electrons may be absorbed in the substrate and hence increase the temperature of the investigated system [29].
Results
Figure 1a presents the schematic diagram of SPR emitting from a grating. Figure 1b plots the simulation model in this work. The grating is formed of an InSb substrate with carving ten periodic subwavelength grooves on it. An electron bunch moves above the proposed structure. In Fig. 1b, the period (L), width (a) and depth (h) of the groove are 45 um, 22.5 um and 22.5 um, respectively. The distance (R) between the observation points and the center of the grating is 700 um. (How the geometrical parameters affect the simulation results are analyzed in the section of Method and Materials.) The simulation domain occupies an area of 1600
800 um2. Thesimulation method and detailed settings are given in the section of Method and Materials. The temperature-dependent dielectric constants of InSb are also shown in Method and Materials.
To prove our concept, the behavior of SPs (excited by an electron bunch) propagating on the surface of InSb substrate is examined first. Figure 2a plots the dispersion curves of SPs at the air-InSb interface under various temperatures of InSb.
The wavevector of excited SP (k ) can be described as sp
2 / 1
d s
d sp s
ε ε
ε ε c
k ω , (1)
where ω is the angular frequency and c is the speed of light; εd (εs) denotes the relative permittivity of dielectric (the relative permittivity of InSb in THz region, see Method and Materials) (here εd=1). As the electron bunch passes over the InSb substrate, ω and k of the excited SP can be determined from the intersections of sp dispersion curves of SP and electron bunch as shown in Fig. 2a. Figure 2a reveals that the working points (ω and k of SP) change with the slope of dispersion curve of sp electron bunch, which is determined by [30]
c v β γ12
1 , (2)
where β denotes the relativistic factor and v and c are the speeds of electron and light, respectively;
eV (eV)
γ E 6
10 511 . 1 0
and E is energy of incident electron bunch.
When the frequency of SPs at the air-InSb interface excited by electron bunch is within the emission frequency band of SPR, the intensity of SPR at this frequency will be enhanced (this phenomenon will be discussed later). Figure 2a also exhibits that increasing the energy of electron will decrease the wavevector of excited SPs (i.e.
increase the wavelength of SPs). Furthermore, the frequency of excited SPs increases with temperature of InSb for a fixed electron energy, also shown in Fig. 2a. Hence, it is easier to observe and manipulate the SP-enhanced SPR by using the InSb substrate at THz region. Figures 2b, 2c and 2d plot the simulated contours of z-component magnetic field (Hz) for SPs at the air-InSb interface excited by the electron bunch with E = 30 keV, 40 keV and 50 keV, respectively, at temperatures (T) of 290 K, 300 K and 310 K. The calculated wavevectors of SPs in Figs. 2b – 2c and the correspondent frequencies agree with those in Fig. 2a well.
For an electron bunch passing over a metallic grating, the relation between the emission (vacuum) wavelength and angle of SPR can be express as [10, 31]
1sin n
L , (3)
where λ is the emission wavelength and β is the relativistic factor; θ denotes the observation angle measured from the direction normal to the grating surface (θ0 (θ0) for forward (backward) emission); L is the period of grating and n denotes the order of SPR. Equation (3) originates from the constructive interference of radiation with emission wavelength λ emitted at an angle θ from two successive grooves of the grating initiated by the electron bunch passing above the grating with velocity βc . This SPR can also be viewed as the electron-excited SPR. Equation (3) reveals that the emission wavelengths strongly depend on the period of grating. For generating the light source in THz region, L is chosen as 45 um in this study.
When SP on the substrate excited by electron bunch passes through the periodic gratings, its energy will be transformed into radiation emitted by the gratings. The wavevector-match condition for this process can be described as
) sin(
) (ω/c θ nk
ksp G , (4)
where ksp denotes the wavevector of excited SP; kG 2π/L is the reciprocal
wavevector of gratings; ω/cklight is the wavevector of emission light, and n represents the diffraction order. Reorganizing Eq. (4) will yield Eq. (3) (i.e. SPR and this SP radiation have the same emission wavelength-angle relationship). Actually, the emission mechanisms of SPR and SP radiation are the same. Both of them originate from the constructive interference of radiation emitted from two successive grooves of the grating due to the oscillating current density on the grating. The only difference is that the oscillating current densities are induced by electron bunch in SPR and by SP in SP radiation. Therefore, SP radiation can be viewed as the SP-excited SPR.
Furthermore, Eq. (4) also reveals that SP-excited SPR is angle-independent for different emission angle coming from diffraction order n. Besides, mimic-SP is also a kind of surface wave on the metallic surface with periodic structures and its property is similar to that of SP. The electron excited mimic-SP can also be transformed into radiation via periodic gratings. Hence, this process can also be viewed as the mimic SP-excited SPR.
Next, the SPR from a perfect electric conductor (PEC) grating is investigated. It is considered as a standard SPR and the simulated results of SPR for an InSb grating will be compared with those of this standard SPR. Figure 3a presents the simulated
contours of Fourier spectra of Hz fields as functions of emission wavelength and angle at the observation points (R = 700 um) for the radiation emitted from a PEC grating with E = 40 keV. As a comparison, the curves of wavelength versus angle for n = 1 and 2 of Eq. (3) are also plotted in Fig. 3a. Here the index n is the order of SPR (see Eq. (3)). It is the order of diffraction of the gratings. n = 1 and 2 correspond to the first and second, respectively, orders of SPR. Figure 3a indicates that the major part of the radiation is the first-order SPR which is a continuous spectrum with the wavelength between 75 um and 165 um and the angle from 90o to -90o. The second-order SPR is also observed in Fig. 3a but with the field amplitude being much smaller than that of the first-order SPR. Moreover, Fig. 3a also displays a weak radiation that is emitted in all direction with an unchanged wavelength. This type of radiation originates from the excitation of mimic-SP by electron bunch and then transformation of the mimic-SP into radiation by the grating. Figure 3b presents the dispersive curves of mimic-SP (obtained from FDTD simulation, see Method and Materials) and electron bunch with E = 40 keV. The intersection of these two dispersive curves gives the angular frequency (and its vacuum wavelength) of mimic-SP (here the angular frequency ω = 1.111013 rad/s and the vacuum wavelength λ = 170 um), which confirms that the angle-independent radiation in Fig. 3a comes from the excitation of mimic-SP.
Subsequently, the SPRs emitting from an InSb grating are explored. Figure 4a plots the simulated contours of Fourier spectra of Hz fields versus emission wavelength and angle at the observation points for the radiation emitted from an InSb grating with temperature T = 300 K and E = 40 keV. Here the wavelength-angle relations of the traditional SPR for n = 1 and 2 are also presented in Fig. 4a. Figure 4a shows that the relationship between wavelength and angle of the radiation still satisfies Eq. (3). However, the amount of radiation is concentrated at the wavelength
λ = 120 um (i.e. the angular frequency ω = 1.571013 rad/s) and θ = -10o. This specific wavelength (angular frequency) is consistent with that of the SPs at the air-InSb interface excited by electron bunch with E = 40 keV (see Fig. 2a, T = 300 K).
Compared to SPR from the PEC grating, the radiation intensities from the InSb grating are largely increased at λ = 120 um but reduced at the other wavelengths of the SPR band. Figure 4b plots the Fourier spectra of Hz field as a function of wavelength at θ = -10o (R = 700 um) for an InSb grating with various values of height (h) and a PEC grating with h = 22.5 um. Figure 4b displays that at λ = 120 um, the radiation from an InSb grating with h = 10 um has the largest Hz-field amplitude and the amplitude decreases with the increase in h. And for h = 10 um (h = 22.5 um) at λ = 120 um, the amplitude for an InSb grating is about six (four) times of that for a PEC grating.
Figures 4a and 4b imply that the SPR is enhanced at the specific emission wavelength and angle by excitation of SP. Changing the wavelength of the excited SPs within the SPR band will change the emission wavelength and angle of the enhanced SPR. The mechanism of SP-enhanced SPR is explained below. As the SP (mimic-SP) on the substrate is excited by electron bunch, the SP (mimic-SP) and electron have the same velocities. When the frequency of the electron excited SP (mimic-SP) is within the radiation band of electron-excited SPR, the radiations from electron-excited SPR and SP-excited SPR (mimic SP-excited SPR) are in phase and hence SPR is enhanced by the SP (mimic-SP). Conversely, if the frequency of the SP (mimic-SP) is out of the radiation band of SPR, the electron-excited SPR and SP-excited SPR (mimic SP-excited SPR) are two independent emission processes. In brief, the SP-enhanced SPR is attributed to that, when the frequency of SP is within the radiation band of SPR, the energy from electron concentrated in the excited SPs and then transformed into radiation via SPR mechanism. Notably, the SP-enhanced SPR can occur from THz to visible light strongly depending on the period of grating and plasma frequency of the substrate material. If InSb is replaced by the noble metal such as silver and gold, SP-enhanced SPR will move to visible light and infrared regions. (Of course, the grating period need to be reduced such that the radiation band of SPR is also moved to visible light and infrared regions.)
The angle-independent radiation from the excitation of mimic-SP with λ = 200 um is also observed in Fig. 4a. To verify it, the dispersive curves of mimic-SP on the InSb grating are also examined. Figure 4c plots the dispersive curves of mimic-SP (also obtained from FDTD simulation) at T = 290, 300 and 310 K and electron energy of 40 keV. Figure 4c exhibits that at T = 300 K, the dispersion curves of mimic-SP and electron bunch intersect each other at ω = 9.421012 rad/s (i.e. λ = 200 um), which confirms that the observed angle-independent radiation in Fig. 4a is ascribed to excitation of mimic-SP on the InSb grating. Moreover, Fig. 4c also indicates that the frequency (vacuum wavelength) of 40 keV-excited mimic-SP on the InSb grating increases (decreases) with the increase of T.
Finally, the effects of InSb’s temperatures on the enhanced SPR are examined.
The temperature will affect the carrier density (see Eq. (6)) and the plasma frequency in InSb. And the frequency of SP on InSb substrate excited by the electron bunch changes with InSb’s plasma frequency. Figures 5a, 5b, 5c and 5d present the simulated contours of Fourier spectra of Hz fields as functions of emission wavelength and angle at the observation points for the radiation emitted from an InSb grating with temperatures T = 290 K, 300 K, 310 K and 320 K, respectively, and E = 40 keV. (Note that, Fig. 4a is redrawn in Fig. 5b.) (Here these temperatures are chosen because we are only interested in the SP-enhanced SPR around room temperature and
the temperature dependent carrier density formula (see Eq. (6), based on Ref. 19) is only applicable to temperature between 260 K and 330 K.) Figures 5a - 5d show that the emission wavelengths of the enhanced SPR decrease as the temperature increases along with the emission angles increasing from -30o to 30o gradually. The emission wavelengths of the enhanced SPR at various temperatures are in accordance with the vacuum wavelengths of excited SPs on the InSb grating at the correspondent temperatures as shown in Fig. 2a. The relationship between the emission wavelength of enhanced SPR and the emission angle for each T also follows the prediction of Eq.
(3). In addition, Figs. 5a – 5d also exhibit that the emission wavelength of the angle-independent radiation owing to excitation of mimic-SPs decreases with the increase of temperature, which is consistent with the dispersion relations shown in Fig.
4c. These results demonstrate that the emission wavelength and angle of the enhanced SPR from an InSb grating can be manipulated by the temperature of InSb.
Figures 5a - 5d also exhibit a non-radiative band just below the emission wavelength of SP-enhanced SPR. For example, at T = 300 K (Fig. 5b), the wavelength of the non-radiative band is around 115 um (its angular frequency is about 1.61013 rad/s, slightly larger than the frequency of the cross point, see the green line in Fig. 2a). This non-radiative band originates from SP on the InSb substrate with extremely large wavevectors also excited by the electron bunch. However, this kind of SP belongs to the so-called bonding mode (i.e. non-radiative mode). Its energy is strongly confined at the InSb-air interface and cannot be transformed into radiation.
As a result, the emission spectrum displays a non-radiative band. Below the wavelength of non-radiative band, SP cannot be excited by the electron bunch any more. The observed weaker radiation in Figs. 5a – 5d comes from the original non-enhanced SPR. Because the plasma frequency and the SP frequency of InSb increase with the temperature of InSb, the wavelengths of SP-enhanced SPR and the non-radiative band decrease with the increase of temperature.
In our work, the structure and material parameters are designed such that the frequency of the SP (mimic-SP) is within (out of) the radiation band of electron excited SPR (T = 290 K ~ 320 K, E = 40 keV). Therefore, the strong SP-enhanced SPR and the weak angle-independent mimic SP-excited SPR are observed in Figs. 5a – 5d. At T = 270 K, the frequency of the SP is out of the radiation band of SPR.
Therefore, both the electron-excited SPR (with a continuous emission band) and SP-excited SPR (which is angle independent and with a fixed wavelength) are observed with the peak field amplitudes being much smaller than those in Figs. 5a – 5d (i.e. at T = 270 K, the SPR is not enhanced by excited SP). These results further verify the proposed mechanism of SP-enhanced SPR and its temperature tunability.
This work also provides a possible solution to conduct an experiment for the
proposed design. The InSb grating structure with various parameters can be fabricated on an InSb wafer by using the chemical vapor deposition and followed by the reactive ion etching to remove unwanted parts of InSb. The field emission gun of scanning electron microscope (SEM) can provide the electron beam. All of the components are mounted in a vacuum chamber. The emitted THz radiation at different positions can be collected by using THz fibers and delivered to a THz spectrometer [32]. The temperature tunability of the SP-enhanced SPR has potential applications in the fields of optical beam steering [33, 34] and metamaterial light source [35].
Conclusion
The SP-enhanced SPRs for an electron bunch passing over an InSb-based substrate with a subwavelength grating under various temperatures of substrate are investigated by FDTD simulations and theoretical analyses. The SPR is locked and enhanced at a certain emission wavelength with the emission angle still satisfying the wavelength-angle relation of the traditional SPR, Eq. (3). This wavelength agrees with the (vacuum) wavelength of SPs at the air-InSb interface excited by the electron bunch. The enhancement of SPR at this wavelength is attributed to the energy from electron concentrated in the excited SPs and then transformed into radiation via the SPR mechanism. When the temperature of InSb increases, the emission wavelength of the enhanced SPR decreases along with the emission angle increasing gradually. This work demonstrates that the emission wavelength and angle of the enhanced SPR from the InSb grating can be manipulated by the temperature of InSb. The temperature tunability of SP-enhanced SPR has potential applications in the fields of optical beam steering and metamaterial light source.
Methods and Materials
Generally, in THz regime, the relative permittivity of InSb can be described as [16, 19]
ε ω / ω jγ
εs p2 2 , (5)
where εs and ε are the frequency-dependent relative permittivity and
high-frequency relative permittivity, respectively, of InSb; ωp denote the plasma frequency of InSb; ω represents the angular frequency, and γ is the damping constant. In Eq. (5), ωp is given by Ne2/ ε0m* where N, e, ε0 and m* denote the intrinsic carrier density, electron charge, free space permittivity, and effective mass, respectively. Comparing to the noble metal in visible regime, the plasma frequency of semiconductor is very sensitive to the temperature variation in THz
regime (0.1 – 10 THz, i.e., 3000 – 30 um). The relationship between carrier density (in unit of cm-3) and temperature can be expressed as [19]
) 2 / 26 . 0 exp(
10 76 .
5 20T3/2 k T
N B , (6)
where k is the Boltzmann constant and T represents the temperature in Kelvin. B In this work, the FDTD based commercial electromagnetic software Lumerical is used in the simulation. The two-dimensional simulation is performed in the Cartesian x–y coordinate system. Here, the electron bunch is represented by a series of dipoles with phase delay that is related to the electron velocity. The dimensions of uniform grid cells in x and y directions are both set as 0.25 um, which is enough for investigating both the radiative THz light and SPs [36]. The surrounding boundaries of the simulation model are perfectly matched layers (PMLs). Additionally, the simulation time is 40000 fs, which is long enough to record all the scattered signals from the grating at the measured positions.
To obtain the dispersion curves of mimic-SP of a grating by the FDTD method, we first put a line source (by setting the x-component electric field (Ex)) with a Gaussian temporal pulse in one groove of the grating and perform the simulation.
Then, we take the temporal Fourier transform of the measured time-domain Ex field in another groove to acquire its frequency spectrum. The peak frequencies in the spectrum are the eigenfrequencies of the mimic-SP. Next, we use the same line source but with a sinusoidal temporal function of one eigenfrequency and perform the simulation again. Finally, the wavevector of this eigenfrequency is obtained by taking the spatial Fourier transform of the measured instant Ex field along the x direction on top surface of the grating. All the pairs of eigenfrequencies and the corresponding wavevectors constitute the dispersion curve of mimic-SP of the grating.
In this work, the period of groove (L) is chosen as 45 um based on the fact that the frequency of SP on the InSb substrate around room temperature is within the radiation band of SPR to demonstrate the temperature tunability of SP-enhanced SPR.
Hence, it cannot be chosen arbitrarily. The width of groove (a) is set as 22.5 um. The width of groove does not affect the radiation band of SPR. The emission wavelength and angle for SP-enhanced SPR are almost unchanged with the groove width. The groove width only affects the field amplitude of radiation. The depth of groove (h) is set as 22.5 um. The effect of groove depth on the Fourier spectra of Hz field is discussed in Fig. 4(b). At λ = 120 um and θ = -10o (the emission wavelength and angle of SP-enhanced SPR), the radiation for h = 10 um has the largest Hz-field amplitude and the amplitude decreases with the increase in h. However, the groove depth does not affect the radiation band of SPR and the phenomenon of SP-enhanced SPR. Finally, the distance between observed points and gratings (R) needs to be large
enough for measuring the far field of emission. In this work, R = 700 um satisfies this requirement.
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Figure 1|Schematic diagram and simulation model of proposed InSb-based grating for generating SP-enhanced SPR. (a) Schematic diagram. The yellow waves symbolize the excitedSPs as electron bunch passes over the InSb grating. (b) Two-dimensional simulation model.
Figure 2|Dispersion curves and excited SPs at the air-InSb interface. (a) Blue (green, red) line: Dispersion curve of SP at the air-InSb interface with T = 310 K (300 K, 290 K). Black line (dark grey, light gray): Dispersion curve of electron bunch with E = 50 keV (40 keV, 30 keV). (b), (c), and (d) Simulated Hz field contours of SPs excited by electron bunch passing over the InSb substrate under various temperatures (i.e., 290 K, 300 K, and 310K) with E = 30 keV, 40 keV, and 50 keV, respectively.
Figure 3|Simulated results for an electron bunch passing over a PEC grating. (a) Simulated contours of Fourier spectra of Hz fields as functions of emission wavelength and angle at the observation points. Red solid and dashed lines:
wavelength-angle relation of traditional SPR for n = 1 and 2, respectively. (b) Dispersive curves of mimic-SP (black line) and electron bunch (red line). E = 40 keV and h = 22.5 um.
Figure 4| Simulated results for an electron bunch passing over an InSb grating.
(a) Simulated contours of Fourier spectra of Hz fields versus emission wavelength and angle at the observation points (h = 22.5 um). Red solid (dashed) line also denotes the wavelength-angle relation of traditional SPR for n = 1 (n = 2). (b) Simulated Fourier spectra of Hz field as a function of wavelength at θ = -10o (R = 700 um) for an InSb grating with various values of h and a PEC grating with h = 22.5 um. (c) Dispersive curves of mimic SP with various temperatures (red, green and blue lines) and electron bunch with E = 40 keV (black line). h = 22.5 um.
Figure 5| Simulated results for temperature effects on SP-enhanced SPR on an InSb grating. (a), (b), (c), and (d) Simulated contours of Fourier spectra of Hz fields versus emission wavelength and angle at the observation points with investigated temperatures of 290 K, 300 K, 310K, and 320K, respectively. E = 40 keV and h = 22.5 um.
Generation of convergent light beams by using surface plasmon locked Smith-Purcell radiation
Abstract
An electron bunch passing through a periodic metal grating can emit Smith-Purcell radiation (SPR). Recently, it has been found that SPR can be locked and enhanced at some emission wavelength and angle by excitation of surface plasmon (SP) on the metal substrate. In this work, the generation of a convergent light beam via using the SP-locked SPR is proposed and investigated by computer simulations. The proposed structure is composed of an insulator-metal-insulator (IMI) substrate with chirped gratings on the substrate. The chirped gratings are designed such that a convergent beam containing a single wavelength is formed directly above the gratings when an electron bunch passes beneath the substrate. The wavelength of the convergent beam changes with the refractive index of dielectric layer of the IMI structure, which is determined by the frequency of SP on the IMI substrate excited by the electron bunch. Moreover, reversing the direction of electron bunch will make the emitted light from the proposed structure to switch from a convergent beam to a divergent beam. Finally, the formation of a convergent beam containing red, green and blue lights just above the chirped gratings is also demonstrated. This work offers potential applications in the fields of optical imaging, optical beam steering, holography, microdisplay, cryptography and light source.
Introduction
Optical beam steering is an important issue in nanoscale imaging [1]. Recently, plasmonic lens that uses a metal film with a nanoslit surrounding by a few pairs of chirped nanogrooves at the output surface to achieve nanoscale beam focusing has been proposed and demonstrated [2-4]. In this device, the incident beam will couple into surface plasmons (SPs) in the nanoslit (i.e. a metal-insulator-metal (MIM) structure) [5]. At the output surface, SPs will be transformed into radiation and scattered into the nanogrooves which will also be transformed into radiation with different phases and amplitudes. Both the radiations from the nanoslit and nanogrooves shape the output beam. Very recently, the side-illuminated plasmonic lens that strongly depends on the transverse propagation of SPs in the MIM waveguide is also considered [6]. However, all of the plasmonic lenses suffer from some drawbacks such as lacking of frequency tunability and incapability to generate a focusing beam with multiple frequencies.
As an electron bunch pass above a periodic metal grating, it is capable of generating far-field radiation owing to the coherent oscillation of free and imaged charges on the grating,
i.e. the so-called Smith-Purcell radiation (SPR) [7-13]. Because the electron bunch carries wide frequency information, the emission frequency of SPR can range from millimeter to visible light which is determined by the grating period. Especially, the SPR-based terahertz (THz) light source has drawn a lot of attention in recent research [14-17]. Very recently, it has been found that SPR can be locked and enhanced at some emission wavelength and angle by excitation of SPs on the substrate [18, 19]. This phenomenon is attributed to that the energy from electron concentrated in the excited SPs and then transformed into radiation via SPR mechanism.
SPs on an insulator-metal-insulator (IMI) structure can also be excited by an electron bunch passing above it [5, 20, 21]. The frequency of SPs that is determined from the intersection points of the dispersive curves of SP and electron bunch can be altered by changing the relative permittivity of the insulator. Combining the frequency adjustable IMI structure and SP-locked SPR with a specific emission angle, the nanoscale convergent beam containing a single wavelength or multiple wavelengths can be yielded. However, this kind of device has never been explored before. In this work, the SP-locked SPR with chirped gratings on an IMI substrate for generation of a convergent beam is proposed and investigated by finite-difference time domain (FDTD) simulations. First, a single-wavelength convergent beam emitted from an electron bunch passing beneath the proposed structure is examined.
Next, reversing the electron bunch moving direction to make the beam become divergent is also demonstrated. Finally, the formation of a convergent beam composed of red, green and blue lights by using the proposed structure is investigated.
Results
Figure 1(a) plots the schematic diagram of a convergent beam containing a single wavelength generated by the SP-locked SPR with chirped gratings. In Fig. 1(a), the substrate is an IMI structure formed of a silver (Ag) film (the gray layer) sandwiched between two dielectric films with the same refractive index (the purple layers). The thicknesses of the silver film and the dielectric films are all 20 nm. A buffer layer (the light blue layer) with a refractive index closed to air (here set to 1.1) and 20 nm thick is added for SPs not to be destroyed by the gratings. The perfect electrical conductor (PEC) chirped gratings are deposited on the buffer layer with the grating period l changed (along the positive x-direction) from 185 to 125 nm, from 216 to 148 nm and from 256 to 174 nm for the designed radiation wavelengths of 457 (blue), 535 (green) and 633 nm (red), respectively. In our structure, every two grooves have the same period. And the depth (H) and width of the groove are equal to 60 nm and l/2, respectively. The detailed values of l for each wavelength are listed in Table 1 (see Method and Materials). (The design methodology is to adjust the period of groove according to its position and required emission angle.) For SP-locked SPR with emission wavelengths of 633 nm, 535 nm and 457 nm, the refractive indices (n) of the dielectric films
are set as 2.6, 2.1 and 1.7, respectively. The total lengths of the IMI structure and the buffer layer are both 10 μm. (The chirped gratings are placed at the center of the substrate.)
The electron bunch moves along the positive x-direction and is beneath the IMI structure with the distance between the lower dielectric layer and the electron bunch to be 10 nm.
(Hence, the SPR is TM polarized.) The electron energy (E) is 30 keV (i.e. the relativistic factor of electron = 0.328). The observation points lie on the circumference of upper semicircle of radius R = 4 μm and distributed for every 5° as presented in Fig. 1(b). The convergent spot is designed directly above the gratings (i.e. R = 4 μm and = 0°, see Fig.
1(b)). And the background material is assumed to be air. The simulation method and detailed settings are given in the section of Method and Materials. The plasma and collision frequencies of Drude model of Ag are also shown in Method and Materials. Notably, the silver film can be replaced by other noble metals such as gold or aluminum which relies on the designed radiation wavelength band of the convergent beam. And the real metal gratings can be substituted for the PEC gratings. (Our simulation results indicate that using Ag gratings instead of PEC grating doesn’t affect the convergent beam formation.)
The property of SPs on the IMI substrate of Fig. 1(a) excited by an electron bunch is examined first. Figure 2(a) plots the dispersive curves of SPs on the IMI structure for three different refractive indices of the dielectric layers, electron bunch with E = 30 keV (obtained from k v and v is the electron bunch velocity) and light line in vacuum. In Fig.
2(a), the cross points of the dispersive curves provide the frequencies and wavevectors of the SPs excited by the electron bunch. Figure 2(a) also shows that the frequency of the electron-excited SPs decreases with increasing the refractive index of dielectric layer. For an electron bunch passing beneath a metallic grating, the relationship between the emission wavelength and angle of SPR can be express as [10]
) sin (β 1 θ m
λ l (1)
where is the emission wavelength and is the relativistic factor; denotes the observation angle measured from the direction normal to the grating surface ( > 0 ( < 0) for forward (backward) emission, see Fig. 1(b)); l is the period of grating and m denotes the order of SPR. Equation (1) reveals that SPR emits a continuous spectrum with the emission angle ranging from -90° to 90°. Conversely for SP-locked SPR, the emission intensity is locked and enhanced at the frequency of SP with the emission wavelength-angle relationship still satisfying Eq. (1). Figures 2(b) and 2(c) present the simulated contours of absolute value of z-component magnetic field (|Hz|) for traditional SPR (electron bunch moving under uniform PEC gratings with l = 228 nm) and SP-locked SPR (electron bunch moving under the structure of Fig. 1 except for uniform PEC gratings with l = 228 nm), respectively. (In both
cases, the distance between the electron bunch and the structure is 10 nm.) Figure 2(b) displays that traditional SPR emits at all directions of the upper semicircle. However, Fig. 2(c) exhibits that the emission for SP-locked SPR is enhanced and concentrated around 45° with
= 535 nm (the refractive index of dielectric layers is 2.1).
Next, the generation of a convergent beam with a specific emission wavelength by using the SP-locked SPR is investigated. Figures 3(a), 3(b) and 3(c) plot the simulated contours of Fourier spectra of Hz fields as functions of emission wavelength and angle at the observation points (see Fig. 1(b)) for the emission wavelength designed at 457 nm, 535 nm and 633 nm, respectively. (The geometrical parameters for the chirped gratings are given in Sec. 2.) Figures 3(a), 3(b) and 3(c) show that the convergent spots are formed in the wavelength-angle plane at the designed wavelengths. And the full width at half maximum (FWHM) of emission angles (emission wavelengths) of the convergent spots for the designed wavelengths of 457 nm, 535 nm and 633 nm are 6.8° (16 nm), 8.4° (20 nm) and 9.9° (26 nm), respectively. (Since the two-dimensional simulation is performed, the real pattern in the x-z plane (R = 4 μm) will be a line rather than a spot.) Figures 3(a) – 3(c) also display that the intensity of Hz field at the convergent spot decreases with increase of emission wavelength. These results are attributed to that for a longer wavelength, the grating period is larger and hence the grating has fewer grooves. And the larger the peak field intensity at the convergent spot is, the smaller the FWHM of angle becomes. Therefore, the FWHM of angle increases with the emission wavelength, as Figs. 3(a)-3(c) show. Figure 3(d) presents the same contours but for the SPs-locked SPR with uniform gratings designed for =535 nm (i.e. the non-convergent case, Fig. 2(c), l = 228 nm, emission concentrated at = 45°). Comparing Fig. 3(d) with Figs. 3(a) - 3(c) indicates that the Hz-field intensity at the convergent spot for the chirped gratings is about two times as large as that for the uniform gratings, which is ascribed to that the chirped gratings can emit radiation into the same place.
Subsequently, the effect of reversing the electron bunch moving direction on beam convergence is also explored. Figure 4(a) plots the simulated contours of Fourier spectra of Hz fields versus emission wavelength and angle at the observation points with the electron bunch moving along the negative x-direction. (The geometrical parameters of the device are the same as those in Fig. 3(b)). Figure 4(b) presents the field intensities of Hz fields at the observation points (see Fig. 1 (b)) as a function of emission angle at =535 nm for electron bunch moving along the positive x-direction (blue, a convergent SPR) and negative x-direction (red, a divergent SPR). Figures 4(a) and 4(b) illustrate that, when the moving direction of electron bunch is reversed, the SP-locked SPR from the proposed structure becomes divergent with the emission angle ranging from -65° to 65° for =535 nm.
According to Eq. (1), reversing the electron moving direction will change the sign of emission angle. As a result, the original design for converging beam becomes to diverge it.
Furthermore, since the emission energy of the divergent beam is distributed over a large
angular range, its peak field intensity is much smaller than that of the convergent beam in the convergent spot. Figures 4(a) and 4(b) imply that a light emitted from the proposed structure can be switched from a convergent beam to a divergent beam by only reversing the electron bunch moving direction.
Finally, the generation of a convergent beam containing multiple wavelengths by using the SP-locked SPR is examined. Figure 5(a) plots the schematic diagram for this goal involving red, green and blue lights by using an IMI substrate with chirped gratings on it. In Fig. 5(a), the IMI substrate is divided into three sections with the refractive indices of dielectric layer, from left to right (i.e. along the x-direction), equal to 1.7 (blue light, 457 nm), 2.1 (green light, 535 nm) and 2.6 (red light, 633 nm). To form a convergent spot at = 0°
and R = 6 μm (in this case, the convergent spot is designed at R = 6 μm), the chirped gratings need to be designed such that different lights emitted from different positions all propagate toward the convergent spot. The detailed geometrical parameters of the chirped gratings for this objective are given in Table 2 (see Method and Materials). Figure 5(b) presents the simulated contours of Fourier spectra of Hz fields as functions of emission wavelength and angle at the observation points (R = 6 μm). Figure 5(b) shows that the lights of 457 nm, 535 nm and 633 nm are all converged at = 0° with the FWHM of angles to be 13.2°, 15.7° and 16.4°, respectively (the FWHM of wavelengths for the three designed wavelength are equal to 21 nm, 34 nm and 30 nm, respectively). In our studies, the FWHM of angles of the multi-wavelength spot is larger than that of the single-wavelength spot. These results are also due to the multi-color design having fewer grooves of the chirped gratings as well as a larger value of R and hence smaller field intensity at the convergent spot than the single-color design.
As we mentioned above, the intensity of Hz field is inversely proportional to the wavelength.
Moreover, the intensity of Hz field decreases with increasing the distance between the gratings and the convergent spot. These two factors are considered in design of geometrical parameters (Table. 2). Therefore, the intensities of the three wavelengths are almost the same in Fig 5(b). However, the intensity for each wavelength component is about one third of that for the single-wavelength cases in Fig 3. Notably, the convergent spot position (R) can be manipulated by changing the whole sizes of gratings and each groove period.
The power conversion efficiency for the proposed device is also calculated. Here we consider the proposed structure with five uniform gratings designed for the vertical emission (i.e. 0o). The power conversion efficiency of SPR is defined as
P0
PSPR
, where PSPR denotes the total Poynting power of SPR integrated over all the simulation time and over the whole x-space measured at 4000 nm above the electron bunch and P0 is the same Poynting power except that the structure is removed and measured at 20 nm above the electron bunch (i.e. the total available power of electron bunch in its entire path). The calculated power conversion efficiencies for blue (l =
150 nm), green (l = 176 nm) and red (l = 208 nm) lights are 1.34 %, 1.58 % and 1.15
%, respectively. Moreover, the power conversion efficiency increases linearly with the number of gratings. This work offers potential applications in the fields of optical imaging, optical beam steering, holography, microdisplay, cryptography and light source.
Conclusion
In conclusion, the generation of a convergent beam via using the mechanism of SP-locked SPR is proposed and investigated by FDTD simulations. The proposed structure is composed of an IMI substrate with chirped gratings on the substrate. Based on the relationship of emission wavelength and angle of SPR, the chirped gratings are designed such that a convergent beam containing a single wavelength is formed directly above the gratings (R = 4 um and = 0°) as an electron bunch passes beneath the substrate. The wavelength of the convergent beam changes with the refractive index of dielectric layer of the IMI structure, which is determined by the frequency of SP on the IMI substrate excited by the electron bunch. The FWHM of angle for all investigated wavelengths is smaller than 10° at R = 4 um.
Moreover, reversing the direction of electron bunch will make the emitted light to switch from a convergent beam to a divergent beam. Finally, the formation of a convergent beam containing red, green and blue lights just above the chirped gratings (i.e. at R = 6 um and
= 0° with the maximum angle’s FWHM to be 16.4°) is also demonstrated. To realize it, the IMI substrate is divided into three sections and each section emits one color by adjusting the refractive index of the dielectric layers. And the chirped gratings are designed such that different lights emitted from different positions all propagate toward the convergent spot. This work offers potential applications in the fields of optical imaging, optical beam steering, holography, microdisplay, cryptography and light source.
Methods and Materials
The FDTD program Lumerical is utilized in the simulation [22]. The two-dimensional simulation is performed in the Cartesian x–y coordinate system. The dimensions of uniform grid cells in x and y directions are both set as 2 nm. The entire region is enclosed by perfectly matched layers (PMLs). The plasma and collision frequencies of Drude model of Ag are 1.44331016 rad/s and 1.49951014 rad/s, respectively [23]. In this work, the electron bunch is represented by a series of dipoles in its path with phase delay that is related to the electron velocity. The movement of electron bunch is achieved by sequentially switching on and off the dipoles [24].
The designed periods of chirped gratings (l) for generation of a convergent beam containing a single wavelength and involving the red, green and blue lights are listed in Table 1 and 2, respectively. The tendencies of the grating periods in the x position for Table 1 and 2 are plotted in Fig. 6(a) and 6(b), respectively.
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Table 1. Designed periods of chirped gratings for generation of a convergent beam containing a single wavelength.
Wavelength Designed values (l) of groove period (along positive x-direction) (nm)
457 nm 185, 182, 177, 173, 169, 164, 160, 156, 152, 148, 145, 142, 139, 136, 134, 132, 130, 129, 127, 126, 125 (total 21 values)
535 nm 216, 210, 204, 198, 192, 187, 181, 176, 171, 166, 163, 159, 156, 154, 151, 149, 147 (total 17 values)
633 nm 256, 248, 240, 232, 223, 215, 208, 201, 195, 190, 186, 182, 179, 176, 174 (total 15 values)
Table 2 Designed periods of chirped gratings for generation of a convergent beam involving red, green and blue lights.
Region Designed values (l) of groove period (along positive x-direction) (nm)
Blue region 188, 186, 183, 181, 178, 175, 172, 169, 167, 164 (the first section, total 10 values)
Green region 185, 181, 177, 174, 171, 168 (the second section, total 6 values)
Red region 193, 190, 187, 184, 182, 180, 178, 176, 174, 173, 172 (the third section, total 11 values)
Figure 1|Schematic diagram and simulation model for generation of convergent beams. (a) Schematic diagram of a convergent beam generated by SP-locked SPR with chirped gratings on an IMI substrate. The electron bunch passes beneath the IMI substrate to excite SP. (b) Two-dimensional simulation model. The observation points lie on the circumference of upper semicircle of radius R = 4 μm and are distributed for every 5°.
Figure 2|Dispersion curves of SP and traditional and SP-locked SPRs. (a) Dispersion curves of SP on IMI substrate for refractive indices (n) equal to 2.6 (red), 2.1 (green) and 1.7 (blue), electron bunch with E = 30 keV (black) and light line in vacuum (pink). (b) Simulated contours of |Hz| of traditional SPR for electron bunch moving under uniform PEC gratings with l = 228 nm. (c) Simulated contours of |Hz| of SP-locked SPR for electron bunch moving under proposed structure in Fig. 1 except for uniform PEC gratings with l = 228 nm. In (b) and (c), the distance between electron bunch and structure is 10 nm.
Figure 3|Simulated results for generation of a convergent beam with a specific wavelength. (a) – (c) Simulated contours of Fourier spectra of Hz fields as functions of emission wavelength and angle at the observation points for SPR emitted from the proposed structure in Fig. 1 with refractive indices of dielectric layers equal to 1.7 (
= 457 nm), 2.1 ( = 535 nm), and 2.6 ( = 633 nm), respectively. Upper insets in (a) ((b), c(c)): Hz-field intensity versus emission angle at = 457 nm ( = 535 nm, =
636 nm). Right insets in (a) – (c): Hz-field intensity versus emission wavelength at
= 0°. The detailed values of l for chirped gratings are listed in Table 1. (d) The same simulated contours as in (a) – (c) except for uniform PEC gratings with l = 228 nm (i.e. Fig. 2(c)). Upper and right insets in (d): Hz-field intensity versus emission angle at = 535 nm and Hz-field intensity versus emission wavelength at = 45°, respectively.
Figure 4|Simulated results for reversing the electron bunch moving direction. (a) The same simulated contours and insets as in Fig. 3(b) except for reversing the electron bunch moving direction (i.e. along the negative x-direction). (b) Hz-field intensities as a function of emission angle at = 535 nm for electron bunch moving along the positive x-direction (blue, from Fig. 3(b)) and negative x-direction (red, from Fig. 4(a)).
Figure 5|Schematic diagram and simulated results for generating a convergent beam containing multiple wavelengths. (a) Schematic diagram for a convergent beam involving red, green and blue lights generated by SP-locked SPR with chirped gratings on an IMI substrate. The detailed values of l of chirped gratings for each color are given in Table 2. (b) Simulated contours of Fourier spectra of Hz fields as functions of emission wavelength and angle at the observation points for SPR emitted from the structure in (a). Upper and right insets in (b): Hz-field intensity versus emission angle at = 457 nm (blue), 535 nm (green) and 633 nm (right) and Hz-field intensity versus emission wavelength at = 0° (black), respectively.
Figure 6|Tendencies of the grating periods. The tendencies of the grating periods (l) in x position for generation of a convergent beam (a) containing a single wavelength (Table 1) and (b) involving the red, green and blue lights (Table 2).
Appendix
In addition to the results mentioned above, four journal papers listed below are published:
1. B. H. Cheng, H. W. Chen, Y. J. Jen, Y. C. Lan*, and D. P. Tsai*, “Tunable tapered waveguide for efficient compression of light to graphene surface plasmons,”
Scientific Reports 6, 28799 (2016).
2. Y. C. Lai, B. H. Cheng, Y. C. Lan*, and D. P. Tsai*, “Plasmonic Archimedean spiral modes on concentric metal ring gratings,” Optics Express 24, pp.
15021-15028 (2016).
3. B. H. Cheng, Y. S. Ye1, Y. C. Lan* and D. P. Tsai*, “Temperature tunability of surface plasmon enhanced Smith-Purcell terahertz radiation for semiconductor- based grating,” Scientific Reports 7, 6443 (2017).
4. Y. C. Lai, T. C. Kuang, B. H. Cheng, Y. C. Lan* and D. P. Tsai*, Generation of convergent light beams by using surface plasmon locked Smith-Purcell radiation, , Scientific Reports 7, 11096 (2017).
