Investigation on the special Smith-Purcell radiation from a nano-scale rectangular metallic grating Weiwei Li, Weihao Liu, and Qika Jia Citation: AIP Advances 6, 035202 (2016); doi: 10.1063/1.4943502 View online: http://dx.doi.org/10.1063/1.4943502 View Table of Contents: http://aip.scitation.org/toc/adv/6/3 Published by the American Institute of Physics Articles you may be interested in Theory of the special Smith-Purcell radiation from a rectangular grating AIP Advances 5, 127135127135 (2015); 10.1063/1.4939538 AIP ADVANCES 6, 035202 (2016) Investigation on the special Smith-Purcell radiation from a nano-scale rectangular metallic grating Weiwei Li, Weihao Liu,a and Qika Jia National Synchrotron Radiation Laboratory, University of Science and Technology of China, Hefei, Anhui, 230029, China (Received 23 January 2016; accepted 24 February 2016; published online March 2016) The special Smith-Purcell radiation (S-SPR), which is from the radiating eigen modes of a grating, has remarkable higher intensity than the ordinary Smith-Purcell radiation Yet in previous studies, the gratings were treated as perfect conductor without considering the surface plasmon polaritons (SPPs) which are of significance for the nano-scale gratings especially in the optical region In present paper, the rigorous theoretical investigations on the S-SPR from a nano-grating with SPPs taken into consideration are carried out The dispersion relations and radiation characteristics are obtained, and the results are verified by simulations According to the analyses, the tunable light radiation can be achieved by the SSPR from a nano-grating, which offers a new prospect for developing the nanoscale light sources C 2016 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/) [http://dx.doi.org/10.1063/1.4943502] I INTRODUCTION Since the first day of its observation, Smith-Purcell radiation (SPR), occurring as electron beam passes over a periodic surface, has been one of the most attractive research topics because of its tremendous applications.1–7 The most promising application in the present time may be its potential in generating terahertz (THz) wave radiation8,9 which can not be easily obtained by other means.10–16 The metallic rectangular gratings are commonly used to get the SPR in these applications.17,18 The representative characteristic of the SPR is its famous dispersion relation: L λ = − ( − cos θ), n β (1) where λ is radiation wavelength, θ indicates radiation direction, L is the structural period, β is the ratio of beam velocity to light speed in vacuum, and n is a negative integer that indicates the harmonic order By reducing the groove width of the rectangular grating, Ref 20 recently uncovered an interesting special kind of Smith-Purcell radiation (S-SPR) which is a monochromaticity radiation with much higher intensity in specified direction The theoretical analyses of the S-SPR were carried out in Ref 21, which demonstrated that the S-SPR is exactly from the radiating eigen modes of the grating The remarkable advantages in intensity and tunability make the S-SPR a promising way for tunable wave generation and beam diagnostic In Ref 21, the metal was treated as perfect conductor, namely, the oscillating fields in metal had been ignored Based on the rigorous theory of metal, this simplification is not so accurate, especially for a nano-scale structure operating in the optical frequency region The more general properties of metal are governed by the Drude-Sommerfeld’s free-electron-gas model, according to which the metal should be described by the following dielectric function22,23: a Electronic mail: liuwhao@ustc.edu.cn 2158-3226/2016/6(3)/035202/8 6, 035202-1 © Author(s) 2016 035202-2 Li, Liu, and Jia AIP Advances 6, 035202 (2016) ε m (ω) = ε ∞ − ω2p ω2 + iγω , (2) where ε ∞ is a constant representing the influence of the metal atoms, ω p is the effective bulk plasma frequency of the free electrons in the metal, and γ indicates the electrons’ collision frequency From Eq (2) one can see that the metal is actually a dispersive (frequency dependent) dielectric medium Based on Drude’s model, the evanescent electromagnetic mode, terminologically the surface plasmon polaritons (SPPs), will present on the metal surface in the optical region The physical base of SPPs are the collective oscillations of the conduction electrons in metal In past years, SPPs have attracted extensive interests for their great promising applications, especially with the advancement of the nano-scale fabrication techniques.24–26 When the SPPs are considered in the metallic grating, the electromagnetic properties as well as the radiation properties of the grating will greatly be changed The SPR from the metallic surface with SPPs taken into consideration was preliminarily analyzed in Ref 27, which showed that the radiation intensity can be enhanced in specific directions Years later, Ref 28 analytically predicted a large emission rate from a chain of nano aluminum spheres via the SPR effect Ref 29 simulated the SPR from a rectangular metallic grating, and it showed the radiations were composed by three primary components: the ordinary SPR, the original SPPs, and the so-called mimic-SPPs Most recently, Refs 30 and 31 demonstrated that the SPPs on a metallic nanowire-array can be transformed into radiation via an exotic Smith-Purcell effect In mechanism, it was the collective radiation from the SPPs modes in each unit of the array The radiation power density can be significantly higher than the ordinary SPR All those results illustrate the copious interesting phenomena when the SPPs are considered In present paper, we will carry out the rigorous theoretical investigation on the S-SPR from the rectangular metallic grating with the SPPs taken into consideration More interesting results will be obtained, which indicate new potential applications The paper is organized as follows: Section II presents the derivations of the theoretical formulas, Section III shows the calculation results and simulation verifications, and Section IV gives the conclusion II FORMULATION OF THE PROBLEM The schematic diagram of the problem that we are going to study is shown in Fig An electron beam passes over a metallic grating with rectangular groves When the SPPs are considered, the fields will penetrate into the metallic grating, which means the whole space will be filled with the oscillation fields This will greatly increase the complexity of the problem as will be shown To handle the problem, we apply the mode matching method and divide the whole space into four regions according to the boundaries as shown in the figure The fields in different regions can be obtained by solving the homogeneous or non-homogeneous Maxwell equations In each region, all electromagnetic field components can be expressed as functions of the z-directional electric field component Ez , which should be obtained first In the upper half space FIG Schematic diagram of the problem 035202-3 Li, Liu, and Jia AIP Advances 6, 035202 (2016) (region I), Ez component satisfies the following non-homogeneous equation in the Cartesian coordinate system32: ∂ Ez ∂ρ ∂ Ez ∂ Ez ∂ Jz + − = + µ0 , ε ∂z ∂t ∂ x2 ∂z c2 ∂t (3) where c is the light speed in vacuum, ε and µ0 are respectively the permittivity and permeability of the vacuum, ρ and Jz respectively denote the charge density and current density For the line electron beam with quantity of q and velocity of v, they can be expressed as: ρ = qδ(x − x 0)δ(z − vt), (4) Jz = qv δ(x − x 0)δ(z − vt), (5) where x denotes the x-directional position of the electron beam as shown in the figure With the help of Fourier transformation method, Eq (3) can be solved The solutions are composed by two parts: the special solution which represents the incident wave from the electron beam and the general solution signifying the diffraction wave from the grating To satisfy the periodic boundary conditions of the grating, the diffraction wave should be expanded into infinite space harmonics Then the total field in region I can be expressed as: I I,i I,r Eω, z = Eω, z + Eω, z ∞ j kI −qk c − j k c |x−x0|− j k z z I + An x n e−k x n x e− j k z n z e = 4πωε ωε n=−∞ I I,i I,r H = H + H ω, y ω, y ω, y ∞ q I |x−x |− − j k c j kz z + = s e An e−k x n x e− j k z n z 4π n=−∞ (6) where k c2 = k02 − k z2 = ( ωc )2 − ( ωv )2, k xI n = k z2n − k 02, k z n = k z + 2nπ L , s = −1 for x > x and s = for x < x In Eq (6), the E x component was omitted for simplification as it will not be used in the mode matching method Note that if k z n > k 0, k xI n is real, namely, the diffraction waves are evanescent in x direction and can not radiate into space; while if k z n < k0, k xI n is imaginary, which means the diffraction waves can propagate in x direction It’s easy to find that only the negative harmonic waves (n < 0) can radiate into upper space The radiation wavelength and direction should satisfy the SPR relation of Eq (1) For the fields in regions II and IV, the homogeneous equation of Eq (3) should be solved according to their own boundary conditions Here we only consider the fundamental modes, which is reasonable since the gap width is much less than the wavelength Then the fields in regions II and IV can respectively be expressed as: ) cos k II z k IIx ( II z II j k II x x − C e− j k x x E = B e 1 ω, z II d/2 ωε IIr ε cos k z ) sin k II z − j k zII ( II z II j k II x x + C e− j k x x Eω, , = B e 1 x IIε II ωε cos k z d/2 r ( ) cos k zII z II j k II x − j k II x Hω, y = B1e x + C1e x cos k zIId/2 (7) ( ) ek zIV(d/2−|z |) + ek zIV(|z |−L+d/2) k IV x IV j k IV x − j k IV x x x Eω, z = B2e − C2e IV ωε IV + ek z (d−L) r ε0 IV ) ek z (d/2−|z |) − ek zIV(|z |−L+d/2) j k zIVsign (z) ( IV j k IV x − j k IV x Eω, x x B2e + C2e x = IV −ωε IV + ek z (d−L) r ε0 ( ) ek zIV(d/2−|z |) + ek zIV(|z |−L+d/2) IV IV j k IV x x + C e− j k x x Hω, y = B2e IV + ek z (d−L) (8) and 035202-4 Li, Liu, and Jia AIP Advances 6, 035202 (2016) II2 IV IV2 IV2 II In above equations, ε IIr k02 = k II2 x +k z , ε r k = k x +k z , ε r is the dielectric constant of the medium filled in region II (in following calculations its value is unit), and ε IV r = ε m is the dielectric function of metal expressed by Eq (2) Following the case of region I, fields in region III can be written as: III III Hω, = Dn ek x n x− j k z n z y n (9) j k III III III Dn III x n ek x n x− j k z n z Eω, z = − ε r ε 0ω n III2 III2 in which ε III r k = k x n +k zn In all previous equations, the An s, B, C, and D n s are the coefficients to be determined by the boundary conditions Now we consider the boundary conditions At the boundary of adjacent gaps z = ±d/2, the continuity conditions for the tangential fields in regions II and IV can be expressed as: H II = H yIV z=±d/2 y z=±d/2 IV E II x z=±d/2 = E x z=±d/2 (10) Substituting Eqs (7), (8) into Eqs (10) we can get IV kzIV − ek z (d−L) k zII II tan k d/2 = , z k IV(d−L) ε IIr ε IV r 1+e z (11) which defines the relation between frequency and k xI I So it is actually the dispersion equation concerning the waves propagating in the x direction along the interfaces of the metal and the gaps.33 At the boundaries of x=0 and x=-h, the following mode matching conditions can be used34: , |z| < d/2 , d/2 < |z| < L/2 x=0 , |z| < d/2, E II I,i I,r ω, z Eω, + E = z ω, z x=0 x=0 E IV ω, z I,r II H I,i + H = Hω, ω, y x=0 y ω, y x=0 x=0 (12) x=0 and II Eω, z x=−h , |z| < d/2 III Eω, = z x=−h E IV ω, z x=−h , d/2 < |z| < L/2 II H III ω, y x=−h = Hω, y x=−h , |z| < d/2 (13) Substituting Eqs (6)-(9) into Eqs (12) and (13) and doing complicate but straightforward arithmetical operation, coefficients (An s) of the diffraction waves can be solved The radiation power can then be obtained Following Ref 35, the radiation power per solid angle is ( ) dIn Zq2 n2sin2θ 2x |Rn | exp − = , (14) dΩ 2L 2ε (1/ β − cos2θ)3 λe where Rn is the radiation factor of the grating given by: |Rn |2 = An e j k c x0 q 4π (15) Letting the incident waves vanish in Eq (12), we can get the dispersion equation of the structure: j M1, n N1, n j tan k IIx s n = j M2, n N1, n + N2 n n M2, n N1, n − tan k IIx s N2 N2 , (16) 035202-5 Li, Liu, and Jia AIP Advances 6, 035202 (2016) where M1, n = Mn/ j k Ix n L , N1, n = d sin c (k z nd ) N2 = cos k zIId/2 , d/2 cos k zII zdz, (17) −d/2 and FIG Dispersion curves (a) and radiation spectra (b) versus the grove depth h, here L=300 nm, d=20 nm Dispersion curves (c) and radiation spectra (d) versus the grove width d, here L=300 nm, h=30 nm The shaded regions of the figure indicate the surface modes, and curves above the light lines signify the radiating modes 035202-6 Li, Liu, and Jia k IIx Mn = II ε r cos k zIId/2 + 2k IV x ) ( IV ε IV + ek z (d−L) r AIP Advances 6, 035202 (2016) d/2 e j k z n z cos k zIId/2 dz −d/2 L/2( e k zIV(d/2−|z |) +e k zIV(|z |−L+d/2) (18) ) cos (k z n z) dz d/2 It is a complex transcendental equation which should be solved by the numerical method The detailed calculations will be given in the next section For the case that the SPPs are not considered, IV we only need to let ε III r , ε r → −∞ III DISPERSION RELATIONS AND RADIATION PROPERTIES The electromagnetic dispersion properties of the grating essentially determines the radiation characteristics According to Ref 21, there are two kinds of eigen modes that can be supported by the grating: the surface eigen modes and radiating eigen modes To get the S-SPR which has much higher intensity than the ordinary SPR, the radiating eigen modes should be effectively excited by the electron beam whereas the surface eigen modes should not be excited It is essentially different from the ordinary SPR, in which the electron beam principally interacts with the surface wave.18,19 Here we will show that when the SPPs are taken into consideration for a nano-grating, both the dispersion and radiation properties of the grating will be changed Based on the equations derived in Section II, the dispersion curves and radiation spectra of the grating can be calculated Figure shows the calculation results for the gratings with different structure parameters Here the dielectric parameters of the metal are set as: ω p = 1.39 × 1016rad/s, γ = 3.2 × 1013 H z, and ε ∞ = 4.5, indicating that the silver grating is considered The exciting beam energy is 100 keV One can observe that the radiating eigen modes are effectively excited, and the S-SPRs with obvious peak frequencies are generated The radiation frequencies change gradually with the width ‘d’ or depth ‘h’ of the groves (increase with d while decrease with h) The radiation from the first order radiating modes can be tuned from 500 THz to 700 THz And frequency of the second order radiating modes can extend to be higher than 900 THz In other words, the frequency can be tuned from infrared to ultraviolet region by adjusting the structure parameters For comparisons, the cases without considering the SPPs are also calculated The results show that neither surface modes nor radiating modes can be excited by the electron beam, and the radiation intensity is significantly lower than that with SPPs as shown in the figure This is because that there is no wave can be supported by the grating in this frequency region when the SPPs are not considered In other words, under the influences of the SPPs, both the dispersion curves and the radiation spectra are significantly changed Then we exam the electrical tunability of the radiation According to the dispersion curves obtained above, we choose the structure parameters to be L=300 nm, h=20 nm, d=20 nm, and the beam energy changes from 60 keV to 100 keV The calculated radiation spectra are shown in Fig FIG Radiation spectra versus the beam voltage, here L=300 nm, h=20 nm, and d=20 nm 035202-7 Li, Liu, and Jia AIP Advances 6, 035202 (2016) FIG Comparisons for the theoretical and simulation results of the radiation frequency depending on (a) h, (b) d, and (c) beam energy Other structure parameters are the same as that given in Fig One can observe that the radiation frequency increases with the beam energy, and the tuning range is from 600 THz to 700 THz To verify the theoretical results, we perform the simulations by applying the fully electromagnetic particle-in-cell code.36,37 The simulation model and conditions follow that given in Refs 20 and 30, in which the simulation obtained radiation snapshots and spectra are presented The comparisons for the theoretical and simulation results of the radiation frequencies are shown in Fig Here only the first order radiating modes are presented One can observe that their results are reasonably agreed with each other IV CONCLUSION The investigation on the special Smith-Purcell radiation from the rectangular metallic nano grating with surface plasmon polaritons taken into 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