Frequency-domain calculation of Smith-Purcell radiation for metallic and dielectric gratings
vversion: September 9, 2020
Frequency-domain calculation of Smith-Purcellradiation for metallic and dielectric gratings A NDRZEJ S ZCZEPKOWICZ , L EVI S CHÄCHTER , J OEL E NGLAND Institute of Experimental Physics, University of Wroclaw, Plac M. Borna 9, 50-204 Wroclaw, Poland Technion–Israel Institute of Technology, Haifa 32000, Israel SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA
Abstract:
The intensity of Smith-Purcell radiation from metallic and dielectric gratings (silicon,silica) is compared in a frequency-domain simulation. The numerical model is discussed andverified with the Frank-Tamm formula for Cherenkov radiation. For 30 keV electrons, rectangulardielectric gratings are less efficient than their metallic counterpart, by an order of magnitudefor silicon, and two orders of magnitude for silica. For all gratings studied, radiation intensityoscillates with grating tooth height due to electromagnetic resonances in the grating. 3D and 2Dnumerical models are compared. © 2020 Optical Society of America
1. Introduction
The Smith-Purcell (SP) radiation, observed for visible light in 1953 [1], has been shown to occur ina wide spectral region, from microwaves [2,3] generated using macroscopic gratings, to ultravioletradiation [4, 5] from nanogratings. One foreseen application of this effect would be a highlytunable free-electron light source [6]. SP radiation might also be used for beam diagnostics inaccelerators, for beam position monitoring [7–9] or longitudinal profile characterization [10–14].New motivation to study SP radiation comes from the development of Dielectric Laser Accelerators(DLA) [15, 16], which utilize the inverse Smith-Purcell effect. Electron beams from DLA may inturn be used to generate SP radiation in various spectral regions.The majority of experimental studies of SP radiation were carried out with metallic gratings.Some recent studies deal with dielectric gratings (eg. [6, 17]); this is caused by advances indielectric nanofabrication, improved understanding of SP emission from dielectrics, and hopethat dielectrics may in some cases outperform metals in radiation intensity [6].Calculations of SP radiation intensity from gratings have a long history. Most analyticalwork to date considers only metallic gratings (exception: a very simplified model in Ref. [18]applied to sub-THz radiation). Ref. [19] reviews some of the analytical models, and theiroutcomes are compared in Refs [20, 21]. Many of the models build upon the seminal work byToraldo di Francia [22], which treats both SP and Cherenkov radiation with the same formalism(“Cherenkovian effects”). The range of analytical methods include perturbative approaches validfor shallow gratings [23, 24] and various surface current models [19, 25–28] which are best suitedfor shallow gratings, but for high energies can also be applied to deep profiles [25]. Most of theanalytical models involve some approximations and neglect resonant cavity effects in the grating.According to Ref. [20] the results of different analytical models may differ by up to six ordersof magnitude. An exceptional position among the analytical models of SP radiation is held bythe van den BergâĂŹs model [29–31]. According to the author the model is rigorous and isapplicable to arbitrary grating profile. Although the model’s accuracy has been questioned [25],no one has explicitly shown the model to be inexact. The model does reproduce radiation intensity1 a r X i v : . [ phy s i c s . acc - ph ] S e p scillations with increasing tooth height [32], a resonant cavity effect in the grating. However,van den Berg’s approach is probably the most difficult of the SP models to apply and in the endrequires nontrivial numerical calculations [25, 29, 33].Metals are easier to deal with in analytical models than dielectrics, because with the perfectelectric conductor boundary condition it is not necessary to solve for the field inside and on theother side of the grating. Regarding dielectric gratings, Sukhikh et al. [18] report analyticalcalculation of SP radiation from a teflon grating, however with several special assumptions:geometry of an inverted lamelar grating (rectangular grating) with substrate thickness approachingzero, SP radiation only to one side, and neglect of secondary refractions (resonances within thegrating are not reproduced); the model has been applied in [18] for sub-THz radiation.In recent years an increasing number of purely numerical simulations of SP radiation werereported. Numerical simulations are equally applicable to metallic and dielectric gratings,although for a dielectric grating more time and memory resources are needed. The most commonapproach is a time-domain simulation; some recently used solvers are Lumerical FDTD [34–37]and CST [38–41]. The time of calculation is usually from hours to days on a single CPU machine.Another approach is a frequency-domain simulation, which is much faster if infinite gratings areassumed (simulation for one unit cell with periodic boundary conditions). This approach was usedin Refs. [17, 37, 42]; however, none of these papers describes the method of simulation, and it ishard to deduce how radiation energy was calculated. A description of the simulation method canbe found in papers which report frequency-domain calculations of Cherenkov radiation [43–45],but these papers do not compute radiated energy. All of the Refs. [17, 37, 42–45] use the Comsolfrequency-domain solver [46].The present work focuses on frequency-domain simulation of single-electron (“incoherent”)Smith-Purcell radiation with metallic and dielectric gratings. We start in Sect. 2 with a step-by-step description of calculation method for Smith-Purcell and Cherenkov radiation (“CherenkovianeffectsâĂŹâĂŹ [22]) using a frequency-domain numerical solver. Although simple in principle,the solution requires careful differentiation between phasors (as required by the numerical solver)and phasor densities (Fourier transforms) and proper interpretation of the well known expressionRe [ E × H ∗ ] , which is different for phasors and for phasor densities. Careful treatment leadsto the solution that is correct in absolute terms, without spurious multiplicative constants. Weverify our method by comparing the results for Cherenkov radiation with the exact analyticalFrank–Tamm formula for radiated energy [47–49].After the detailed deliberations on methodology we turn to applications. In Sect. 3 we useour frequency-domain model to compare directly radiation from gratings of fixed geometry anddifferent materials, which to our knowledge has not yet been reported in the literature, exceptfor the mentioned previously very limited model in Ref. [18], and except for a recent paper [6],which however compares theoretical upper bounds for SP radiation, not the actual computedvalues. In Sect. 4 we demonstrate that the model captures resonant effects in the grating. Whileenergy oscillations with increasing tooth height have been reported previously for metallicgratings [32, 38, 39, 50], we demonstrate them for the first time for dielectrics. Section 5 brieflycompares a numerical result from a three-dimensional (3D) and a two-dimensional (2D) model.This is an important issue, as the 3D models require large RAM memory and are more difficultto construct, so one usually starts with 2D modelling. Section 6 briefly describes radiation fromtriangular gratings, and Sect. 7 summarizes the paper.In all equations in this paper we use SI units.2 . Calculation of Smith-Purcell or Cherenkov radiation intensity with a frequency-domain solver To perform calculations using a numerical frequency-domain solver, we must carefully distinguishbetween phasors and phasor densities. In case of time-harmonic electromagnetic field we have J ( r , t ) = Re [ J ( r ) e j ω t ] = [ J ( r ) e j ω t + J ∗ ( r ) e − j ω t ] (1a) E ( r , t ) = Re [ E ( r ) e j ω t ] = [ E ( r ) e j ω t + E ∗ ( r ) e − j ω t ] (1b)and similarly for the B , D and H fields. The phasors, denoted here J ( r ) , E ( r ) . . . , are distinctfrom the temporal Fourier transforms (phasor spectral densities) J ( r , ω ) , E ( r , ω ) , . . . : J ( r , t ) = [ J ( r ) e j ω t + J ∗ ( r ) e − j ω t ] = ∫ ∞−∞ e j ω t J ( r , ω ) d ω = ∫ ∞−∞ e j ω t (cid:26) [ J ( r ) δ ( ω − ω ) + J ∗ ( r ) δ ( ω + ω )] (cid:27) d ω (2)where we use a Fourier transform convention consistent with [51, 52]. Equation (2) implies thatfor time-harmonic fields, the Fourier transforms can be expressed in terms of the correspondingphasors as J ( r , ω ) = [ J ( r ) δ ( ω − ω ) + J ∗ ( r ) δ ( ω + ω )] (3a) E ( r , ω ) = [ E ( r ) δ ( ω − ω ) + E ∗ ( r ) δ ( ω + ω )] , etc. (3b)Note the difference in units: J ( r , ω ) [ s · ( A / m )] , J ( r ) [ A / m ] , etc.Maxwell equations in the frequency domain („time-harmonic”) ∇ × E + j ω B = ∇ × H − j ω D = J (4b) ∇ · D = ρ (4c) ∇ · B = J , E , . . . in a particular context. A frequency-domain solver inengineering-oriented software like Comsol expects a phasor expression for electric current I ( r ) [A] (or current density J ( r ) [A/m ]), and outputs phasors E ( r ) [V/m] and B ( r ) [T]. The totalenergy radiated through a surface is W = ∫ ∞−∞ ∫ surface Re (cid:20) E ( r ) × H ∗ ( r ) (cid:21) · d A dt (5)where E × H ∗ is the complex Poynting vector [52] and Re [ E × H ∗ ] is the time-averaged powerflux density (cid:104) PFD (cid:105) [ W / m ] (in Comsol it is called “Power flow, time average, Poav ”). Notethat for strictly harmonic fields expression (5) is infinite.We can “cheat” the solver by entering a temporal Fourier transform for current I ( r , ω ) insteadof a phasor I ( r ) , then the solver will use the same equations (4) as for phasors to calculatethe Fourier transforms E ( r , ω ) , B ( r , ω ) , . . . Now the expression Re [ E × H ∗ ] has a different3nterpretation and a different unit [ s · W / m ] . As shown in Appendix A, the total radiatedenergy is now equal to W = ∫ ∞ ∫ surface · π · Re (cid:20) E ( r , ω ) × H ∗ ( r , ω ) (cid:21) · d A (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) dW / d ω d ω (6)where dW / d ω is the radiated energy per unit frequency. Here we use only positive frequenciesto allow comparison with experimental results. The numerical factor 4 · π depends on whichconvention for Fourier transforms is used, here it is consistent with Equations (2), (8), (9). Toobtain dW / d ω from the solver’s result, take the surface integral of (cid:104) PFD (cid:105) (Comsol:
Poav ) andmultiply by 4 · π ; the result is in [ J · s ] (for phasors it would be [ J / s ] ). Depending on the softwareused, the “cheated” solver may signal wrong units. This can be resolved by multiplying theexpression for current by an arbitrary frequency range ∆ ω , for example by unit angular frequency ∆ ω = (cid:104) PFD (cid:105) by ( ∆ ω ) .The expression Re [ E × H ∗ ] is discussed in many electrodynamics texts for phasors E ( r ) , H ( r ) , but its interpretation for transforms E ( r , ω ) , H ( r , ω ) and the formula (6) cannot easily befound in textbooks. To calculate power using a phasor-based frequency-domain code, one needsto compare directly equations (5) and (6). Formulas similar to (6) appear in some papers dealingwith analytical models of S-P radiation [24, 53, 54], but may be not evident to users of numericalfrequency-domain codes (for example, this issue is not addressed in tutorials and manuals ofComsol [46]). Ref. [55] presents a heuristic argument for energy computation in a frequencydomain solver, which however leads to results that are too small by a factor of 4. Adding toconfusion, the expression Re (cid:2) E ( r , ω ) × H ∗ ( r , ω ) (cid:3) is sometimes called “Poynting vector in thefrequency domain” [53], which can easily be misunderstood as the Fourier transform of thePoynting vector (these are distinct quantities with different units, see also Appendix A). Anothersource of confusion is that energy density may be defined on either ω ∈ (−∞ , ∞) or ω ∈ ( , ∞) –the two definitions differ by factor 2. Incorrect numerical factors can also arise when rivalingFourier transform conventions from different papers are confused. All these problems call forsome verification of calculated radiation intensities. This issue will be addressed in Sect. 2.4. To calculate Smith-Purcell radiation or Cherenkov radiation, we consider a point charge (− e ) moving in the ˆ z direction with constant velocity v = β c . For a particle at z = t = J ( r , t ) = (− e ) v δ ( x ) δ ( y ) δ ( z − v t ) ˆ z = (− e ) δ ( x ) δ ( y ) δ ( z / v − t ) ˆ z . (7)After integration over the transverse coordinates x , y , we obtain the expression for the current I ( z , t ) = (− e ) δ ( z / v − t ) = (− e ) π ∫ ∞−∞ e − j ω ( z / v − t ) d ω = ∫ ∞−∞ (cid:26) (− e ) π e − j ( ω / v ) z (cid:27) e j ω t d ω, (8)so the temporal Fourier transform of I ( z , t ) is I ( z , ω ) = (− e ) π e − j ( ω / v ) z . (9)This expression is input to the solver as the “edge current”. Note that the current is a functionof the longitudinal coordinate and not all of the frequency-domain solvers will allow for this4ependence on position. The current is Floquet-periodic: I ( z + a , ω ) = I ( z , ω ) exp (− j k F a ) , (10)with the Floquet vector k F = ω / v . For a uniform or a periodic medium, the same spatialperiodicity in the frequency domain is followed by the fields: E ( x , y , z + a , ω ) = E ( x , y , z , ω ) exp (− j k F a ) , (11) H ( x , y , z + a , ω ) = H ( x , y , z , ω ) exp (− j k F a ) . (12)For Cherenkov radiation in a uniform medium a is arbitrary. For a non-uniform medium theFloquet periodicity occurs if the refractive index is periodic in z (eg. an infinite grating) – in thiscase a must be equal to the period. For non-periodic systems (eg. a finite grating) Eqs. (11–12) donot hold. However, they should hold in an approximate sense near the center of a finite structurewith many periods. In this paper we consider infinite gratings, periodic in the z coordinate. The described model isvalid for arbitrary grating profile. We choose to focus on simple grating profiles: rectangular,and in Sect. 6 – triangular. As expected for numerical models, there is no additional difficulty ifany other grating profile is considered, as long as the profile is well resolved by the calculationmesh (for 3D models computer memory limits may come into play).Figure 1 shows the basic grating and beam configuration assumed in this paper. The electriccharge moving in the ˆ z direction at a distance d from a grating is the source of SP radiation. In the3D model the source is a point charge e , and all radiation at a given frequency ω is collected. The2D model corresponds to the source being a line charge with charge density e / ∆ y , and radiationcollected from a corresponding strip of width ∆ y . In the latter case the resulting radiated energydepends on the arbitrary transverse length ∆ y – this issue will be addressed in Sect. 5.Figures 2 and 3 show the details of the model. For infinite gratings only one unit cell is neededin the calculation. The boundaries of the calculation domain in the ± ˆ z directions are connected bythe Floquet boundary condition (11–12). Perfectly Matched Layers (PMLs) attenuate radiation inthe transverse directions. The calculation domain consists of vacuum, characterized by relativepermittivity = 1, and the grating region, characterized by arbitrary complex relative permittivity (cid:15) .Note that “top” PMLs are vacuum, while the “bottom” PMLs are grating material, see Figs. 2(b),3(b). This means that we model thick gratings (this case occurs more often in the experiments). Ifthe bottom PML is changed to vacuum, a thin grating is modelled, reflections from the back sideof the grating occur, and guided mode resonances occur for certain frequencies (see Ref. [56] andreferences therein), with major impact on the radiation intensity. The 3D model is more realisticthan a 2D model, but requires large RAM memory (10–20 GB); the computation time for a singlefrequency is of the order of minutes on a single CPU machine. The 2D model requires much lessmemory and the computation time is of the order of seconds. In the 3D simulation, radiation at agiven frequency ω is collected from all transverse directions, by calculating the surface integralin Eq. (6) over the inner boundary of the PML (violet cylindrical strip in Fig. 3(a)). In the 2Dmodel, for a given frequency ω , the integration is carried out over the planar inner PML boundary(Fig. 2(b)).We performed all calculations using the frequency-domain solver in the Comsol simulationsoftware environment [46], which utilizes the Finite Element Method (FEM). The FEM mesh isshown in Fig. 3(d). We tried various refinements of the mesh and obtained slight scatter of thecomputed results. On the basis of these trials we believe that the numerical values presented in thispaper are accurate up to ± a) metallic or dielectric gratingvacuum (b) 3D model (c) 2D model Fig. 1. (a) Basic grating and beam configuration assumed in this paper: grating period a =
300 nm, tooth height h =
200 nm, fill factor F = . β = .
328 correspondingto 30 keV electrons, impact parameter d =
100 nm. We focus on the radiation in firstspectral order m =
1. For θ = ◦ the radiation wavelength is λ ⊥ =
914 nm. (b) A 3Dmodel with a moving point charge; the grating is infinite in the y and z directions. (c)A 2D model, invariant in the y direction and infinite in the z direction, with a movingline charge (flat beam pulse). a) P M L P M L vacuumgratingmaterialelectriccurrent radiationbackward radiation(for dielectrics) Floquetboundarycondition vacuuminfinite gratingelectron path (b)
Fig. 2. (a) A 2D infinite grating is modelled as a (b) 2D unit cell with the Floquetboundary conditions. along z to account for the contraction of the electromagnetic field of the electron. We noteone numerical peculiarity: in Comsol, for 3D models, the default iterative solver fails to find asolution within one hour, but if the direct solver is chosen, the solution is found within minutes. At the end of Sect. 2.1 we pointed out several pitfalls that may lead to incorrect multiplicativefactors in calculations of SP radiation intensity. This doesn’t matter if only relative intensityis needed or one aims at order-of-magnitude estimates, but if accurate results are needed, it isbest to initially test the used model against some rigorous analytical result. We propose a novelapproach to this problem. We take advantage of the fact that the numerical model describedabove is the same for SP and for Cherenkov radiation – the only difference is in the distributionof relative permittivity in space, (cid:15) ( r ) . So for a moment we change the relative permittivity of thevacuum region to (cid:15) and calculate the radiated energy in the uniform medium – the Cherenkovradiation, shown in Fig. 4, and compare the radiation intensity with the Frank–Tamm formula,Eq. (31) in Appendix B. The results are shown in Table 1. The ratio of the numerical radiatedenergy d W num / dz d ω to the analytical value is close to 1, within 10%. The slight discrepancycan be reduced by refining the FEM mesh (within accessible computer memory). The parametersin the last row in the table do not fulfil the Cherenkov radiation condition (see Appendix B); inthis case the calculated energy is 20 orders of magnitude lower, on the level of numerical noise ofthe calculation – this is the expected result. We also checked that the computed energy is linearin ω . These results demonstrate that our model is exact, with no spurious multiplicative factors.7 a) PML PMLPML PML gratingmaterialvacuum electric current (b)(c) (d)
Fig. 3. A 3D infinite grating is modelled as a 3D unit cell with the Floquet boundaryconditions. (a) Perfectly matched layers (PMLs). (b) Electron path. (c) A pair ofsurfaces with Floquet boundary condition. (d) An example mesh used by a finiteelement method solver (Comsol).Fig. 4. Cherenkov radiation – visualization of the Fourier transform of the electric field,Re [ E z ( r , ω )] , for ω = π · · s − , β = . n = d W num dz d ω (cid:44) d W analytical dz d ωβ = . n = . β = . n = . β = . n = β = . n = Table 1. Cherenkov radiation – verification of the numerical results against theFrank-Tamm formula for one frequency, ω = π · · s − . Here we comparethe numerical value d W num / dz d ω against the analytical value d W analytical / dz d ω = ( e / π ) µ ω ( − / β n ) (see Appendix B); the second column shows the ratio of thesetwo values.
3. Comparison of radiation from metallic and dielectric gratings
We assume grating and beam configuration described in Fig. 1(a–b), a 3D model, and calculatethe SP radiation emitted within the frequency range 2 π · . < ω < π · . . ◦ < θ < . ◦ ; the corresponding wavelength range is 907 nm < λ <
921 nm; seethe SP formula in Fig. 1(a).To simulate a perfect conductor, we set the relative permittivity to (cid:15) = − + i – largenegative real part and zero imaginary part. This is based on the observation that for opticalfrequencies good conductors have a large negative real part of (cid:15) (eg. (cid:15) ( Cu ) = − . + . i , (cid:15) ( Au ) = − . + . i at λ =
914 nm [58, 59]), while the imaginary part of (cid:15) describes energydissipation and should vanish for a perfect conductor. This corresponds to large imaginary indexof refraction n = i , yielding almost instantaneous decay of the field inside the material (skindepth = 0 . λ = 1.5 nm). We verified numerically that for S–P radiation from metallic gratingsin the considered frequency range, the bulk condition (cid:15) = − + i yields the same resultas the surface PEC condition (Perfect Electric Conductor, ˆ n × E = ); this would not be validin the sub-THz frequency range [18]. While the boundary condition is computationally moreefficient, we choose to use the bulk condition, so that exactly the same numerical model can beapplied both to metallic and dielectric gratings âĂŞ- only the gratingâĂŹs relative permittivity (cid:15) is changed.The resulting SP radiation for five materials is shown in Table 2. Metallic gratings outperformdielectric gratings by 1–2 orders of magnitude in terms of radiation intensity under the conditionsstudied (simple rectangular grating, 30 keV electrons, radiation wavelength ∼ λ =
914 nm) Radiated energy W Energy radiated intothe grating W in Real part (cid:15) Imaginarypart (cid:15) Copper − .
85 1.361 4 . · − J 0Gold − .
36 1.462 4 . · − J 0Perfect conductor − . · − J 0Fused silica 2.107 0 4 . · − J 8 . · − JSilicon 13.32 0.03099 3 . · − J 6 . · − J Table 2. SP radiation from gratings of different materials, emitted perpendicularto the grating within the frequency range 2 π · . < ω < π · . . ◦ < θ < . ◦ , per electron per grating period, forthe grating geometry and beam parameters from Fig. 1(a–b). Relative permittivity istaken from Refs. [58–61].
4. Energy oscillations with grating tooth height – grating resonances
Results of the previous section were for a grating with a fixed tooth height. Does the intensity ofSP radiation change if the tooth height is changed? The answer is shown in Fig. 5. SP radiationintensity W oscillates with increasing tooth height h (the impact parameter d is kept constant,see Fig. 1). The effect is known for metallic gratings, it has been experimentally observed inthe millimeter-wave spectral region using a special metallic grating setup with variable toothheight [32]. The effect has also been shown for metallic gratings in an analytical calculationbased on van den Berg’s model [32] and in numerical time-domain models [32, 38, 39, 50].Here we confirm oscillations of W ( h ) for metallic gratings in a frequency-domain simulation; inaddition, we demonstrate that the oscillations also occur for dielectric gratings, see Fig. 5(b–c).An interesting result is that for all materials, the optimum tooth height is close to one quarter ofthe radiation wavelength in the perpendicular direction.The W ( h ) oscillations are of resonant nature. Similar effects for the more general phenomenonof diffraction radiation [63] have been predicted for the motion of charged particles near anopen metallic or dielectric resonator [64, 65] or an array of metallic resonators [66]; enhancedradiation occurs for frequencies close to one of the resonant frequencies of the cavity. In caseof the oscillatory effect shown in Fig. 5, the cavity is formed between adjacent grating teeth.The essence of the effect is already captured by a 2D model, see Fig. 6. The green insets showthe analogy with the elementary one-dimensional theory of clarinets or organ pipes, where theresonant lengths of a cavity which is open at one end, for a fixed wavelength λ , are equal to λ + m · λ (vertical dashed lines in Fig. 6; see also Ref. [38]). This simple one-dimensionalreasoning explains the essence of the effect, but does not explain the slight shift of the calculatedradiation maxima W ( h ) towards smaller h . Note that in the acoustic case this model is alsoapproximate. The simple model does not work in case of dielectric cavities – the period of W ( h ) oscillations in Fig. 5(b–c) has no obvious relation to the vacuum wavelength or the wavelengthinside the dielectric; maybe one could deduce the effective wavelength on the grounds of effectivemode index theory.The intensity oscillations for a metallic grating can alternatively be explained by consideringthe groove cavity as a transmission line which is short-circuited at one end (bottom of the groove;impedance = 0) and open-circuited at the other end (top of the groove). If in the groove only10 a) Metallic grating, 3D model (b)
Si grating, 3D model (c)
SiO grating, 3D model Fig. 5. SP radiation from gratings of different materials, in the angular range 88 . ◦ <θ < . ◦ , per electron per grating period, for grating geometry shown in Fig. 1(a–b)(3D model). Solid line – numerical calculation in the frequency domain. For themetallic grating the result is compared with an analytical calculation based on Ref. [28]– dashed line. etallic grating, 2D model Fig. 6. SP radiation from gratings of different tooth heights h , in the angular range88 . ◦ < θ < . ◦ , per electron per grating period, for grating geometry shown inFig. 1(a,c) (2D model) with ∆ y = F = . F = . the TEM mode is considered (transmission line approximation), the first resonance will occurwhen h = λ / W ( h ) . The explanation can be found in the originalpaper by Brownell, Walsh and Doucas: “in deep tooth profiles several facets may form a cavity andlimit the field modes when the wavelength is comparable to or longer than the cavity dimensions.The model described here is best suited for shallow gratings where cavity behavior is negligiblebut can be applied to deep profiles if the energy is sufficiently high so that the wavelength ismuch smaller than any cavity” [25].
5. Comparison of 3D and 2D results for the Smith-Purcell radiation
As was shown in the previous section, certain effects in the SP radiation can be captured wellwithin a 2D model, which is much simpler to construct than a 3D model. What about radiationintensity: can we estimate it from a 2D model? Comparison of the numbers in Figs. 5(a) and6 reveals a difference in energy by 3 orders of magnitude. One should not expect agreement,because the two physical situations are different: single electron vs. an infinite line charge withlinear density ρ = e / ∆ y . The latter contains an arbitrary parameter ∆ y , and the final calculatedenergy W is inversely proportional to this parameter ( W ∝ ρ ∆ y = ( e / ∆ y ) ∆ y = e / ∆ y – we aregrateful to Urs Häusler for pointing this out). For Fig. 6 the parameter ∆ y was arbitrarily chosento equal 1 nm (radiation is generated by a line of charge with a transverse charge density of e / W is collected from a longitudinal strip of the grating of width 1 nm), a length12ot connected with the characteristic length scales of the considered grating/beam configuration.It appears reasonable to replace ∆ y = ∆ y = λ ⊥ , which is one of the characteristiclengths of the system. The result is shown in Table 3. The 2D result now predicts the order ofGeometry of the model Radiated energy W
3D model 3 . · − J2D model, ∆ y = λ ⊥ =
914 nm 2 . · − J2D model, ∆ y = . · − J Table 3. Comparison of 3D and 2D SP radiation in the angular range 88 . ◦ < θ < . ◦ for a metallic grating with beam/grating configuration of Fig. 1. magnitude of the 3D result. It remains an open question whether this trick would work equallywell for other beam/grating configurations.
6. Metallic triangular grating (3D)
In previous sections we considered rectangular gratings. In this section we briefly considera 3D model of a grating with a triangular profile shown in Fig. 7. This is motivated by the (a)(b) (c)
Fig. 7. (a) An infinite triangular grating, 3D model. All parameters are the same as inFig. 1, but h is varied. (b) The unit cell for h =
40 nm and (c) h =
320 nm. In all casesthe grating period and the impact parameter are a =
300 nm, d =
100 nm. expectation that the surface current model of Ref. [28] should work well for such a configuration,provided that the grating is shallow, h (cid:28) a , so that resonant cavities do not form. We decided totest this hypothesis. In Table 4 we compare our 3D results with the analytical result based onformulas from Ref. [28]. Similarly as in Fig. 5(a), we obtain an order of magnitude agreement.However, contrary to our expectations, the agreement gets worse as the tooth height is decreased.13rating tooth height h W , numerical model W (cid:48) , analytical model Ratio W (cid:48) / W
40 nm 1 . · − J 5 . · − J 4.080 nm 3 . · − J 1 . · − J 3.5160 nm 4 . · − J 1 . · − J 2.4320 nm 6 . · − J 6 . · − J 1.1
Table 4. SP radiation from metallic triangular gratings, in the angular range 88 . ◦ <θ < . ◦ , per electron per grating period, for grating geometry shown in Fig. 7. Wecompare our numerical results with the analytical model of Ref. [28] We believe that this points either to the inexactness of surface current model even for shallowgratings, or to some multiplicative factor issue as discussed at the end of Sect. 2.1. This remainsan open question, we have only checked that it is not the frequency range issue (−∞ , ∞) vs. ( , ∞) ,because Eqs (14.60) and (14.70) in Ref. [48], which are the starting point in Ref. [25], are for ω ∈ ( , ∞) , same as assumed in our calculations.
7. Summary and conclusions
We constructed a numerical frequency-domain model useful for quick calculations of Smith-Purcell radiation intensity from a single particle, verified its accuracy, and discussed someconcrete applications. We were somewhat conservative in the choice of grating geometry(rectangular and triangular gratings of uniform material). This was a deliberate choice whichfacilitated comparison with the older literature. The paper in large part deals with the method,but also presents new results regarding comparison of SP radiation from metallic and dielectricgratings; the possible numerous other applications are left for future work. The main results ofthe paper can be summarized as follows:1. A frequency-domain numerical model offers quick calculations of SP radiation intensity(seconds for 2D models, minutes for 3D models, for one frequency, on a single CPUmachine).2. While it is relatively easy to calculate SP radiation in arbitrary units, obtaining anaccurate result in concrete units with no spurious multiplicative constants requires carefuldifferentiation between phasors (quantities expected by the numerical solver) and phasordensities (temporal Fourier transforms). After this issue is properly taken care of, themodel is simple, easy to implement, and its accuracy is limited only by the spatial meshdensity, possibly limited by the amount of RAM computer memory accessible to the user.3. The accuracy of the numerical model can be conveniently checked against the analyticalresult: the Frank-Tamm formula.4. For 30 keV electrons dielectric gratings are less efficient in terms of intensity of SPradiation than their metallic counterpart, by an order of magnitude for silicon, and twoorders of magnitude for silica.5. The described numerical model captures resonant effects in the grating which are notaccounted for in some analytical models. In particular, both for metallic and dielectricgratings, SP radiation intensity oscillates with grating tooth height. Both for metals anddielectrics, the optimum tooth height for maximal SP radiation perpendicular to the gratingis close to a quarter of the radiation wavelength.14. For dielectric gratings, more SP radiation enters the grating bulk than is radiated outwardinto the vacuum.
Funding
This work was supported by the Gordon and Betty Moore Foundation (grant no. GBMF4744)and the U.S. Department of Energy, Office of Science (grant nos. DE-AC02-76SF00515 andDE-SC0009914). LS was supported by the Israel Science Foundation.
Acknowledgments
We are grateful to Avi Gover for pointing us to the older literature on the Smith-Purcell effectand for helpful comments; to Yen-Chieh Huang, Koby Scheuer, Naama Cohen and Urs Häuslerfor inspiring discussions.A.S. is grateful to the Wroclaw Centre for Networking and Supercomputing for granting accessto the Platon U3 computing infrastructure.
Disclosures
The authors declare no conflicts of interest.
A. Energy spectral density – derivation of Eq. (6)
We will show that ∫ ∞−∞ E ( r , t ) × H ( r , t ) dt = ∫ ∞ · π · Re (cid:20) E ( r , ω ) × H ∗ ( r , ω ) (cid:21) d ω (13)which is equivalent to Eq. (6). This formula is a version of the Parseval’s theorem useful forfrequency-domain electromagnetic calculations; it shows the connection between the energydistribution in time and energy distribution in frequency. All equations are consistent with theFourier transform convention of Eq. (2).First we determine the Fourier transform of the Poynting vector. S ( r , t ) = E ( r , t ) × H ( r , t ) = ∫ ∞−∞ E ( r , ω (cid:48) ) e j ω (cid:48) t d ω (cid:48) × ∫ ∞−∞ H ( r , ω (cid:48)(cid:48) ) e j ω (cid:48)(cid:48) t d ω (cid:48)(cid:48) (14) = ∫ ∞−∞ ∫ ∞−∞ e j ( ω (cid:48) + ω (cid:48)(cid:48) ) t E ( r , ω (cid:48) ) × H ( r , ω (cid:48)(cid:48) ) d ω (cid:48)(cid:48) d ω (cid:48) (15)After the change of variable in the inner integral, ω = ω (cid:48) + ω (cid:48)(cid:48) , d ω = d ω (cid:48)(cid:48) , we obtain S ( r , t ) = ∫ ∞−∞ ∫ ∞−∞ e j ω t E ( r , ω (cid:48) ) × H ( r , ω − ω (cid:48) ) d ω d ω (cid:48) (16) = ∫ ∞−∞ e j ω t ∫ ∞−∞ E ( r , ω (cid:48) ) × H ( r , ω − ω (cid:48) ) d ω (cid:48) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) S ( r , ω ) d ω (17)This shows that the Fourier transform of the Poynting vector S ( r , ω ) is the convolution oftransforms of the fields (convolution theorem).From the definition of the Poynting vector and Eq. (17), the total energy radiated through asurface is W = ∫ ∞−∞ ∫ surface S ( r , t ) · d A dt (18) = ∫ ∞−∞ ∫ surface ∫ ∞−∞ e j ω t ∫ ∞−∞ E ( r , ω (cid:48) ) × H ( r , ω − ω (cid:48) ) d ω (cid:48) d ω · d A dt (19)15imilarly as in Ref. [48], chapter 14.5, we simplify the formula by noting a representation ofthe delta function, ∫ ∞−∞ e j ω t dt = πδ ( ω ) , and noting that H ( r , − ω (cid:48) ) = H ∗ ( r , ω (cid:48) ) (a property oftransforms of real functions): W = ∫ surface π ∫ ∞−∞ E ( r , ω (cid:48) ) × H ∗ ( r , ω (cid:48) ) d ω (cid:48) · d A (20)In experimental investigations, a useful, measurable quantity is the distribution of energy inpositive frequencies, so we reduce the domain of integration over frequencies: W = ∫ surface π ∫ ∞ [ E ( r , ω (cid:48) ) × H ∗ ( r , ω (cid:48) ) + E ∗ ( r , ω (cid:48) ) × H ( r , ω (cid:48) )] d ω (cid:48) · d A (21) = ∫ surface · π ∫ ∞ Re (cid:20) E ( r , ω (cid:48) ) × H ∗ ( r , ω (cid:48) ) (cid:21) d ω (cid:48) · d A (22)The factor is for easy comparison with the energy formula for phasors (5). B. Derivation of the Frank-Tamm formula from the potentials
For additional verification of Eq. (6), we will use it to derive the Frank–Tamm formula. Thepotentials for a point charge (− e ) moving with velocity β c in a uniform medium characterized bythe refractive index n , can be expressed as A r = , A φ = , A z ( r , z , ω ) = (− e ) µ ( π ) K (cid:18) j ω c r (cid:113) n − β − (cid:19) exp (cid:18) − j ωβ c z (cid:19) , (23) Φ ( r , z , ω ) = cn β A z ( r , z , ω ) , (24)see Ref. [52], Eqs (2.1.36–38), (2.4.20). K is a modified Bessel function of the second kind. Insubsequent calculations, to determine the energy flow in the far field, we use the first term of theasymptotic expansion of K :K ( j ξ ) (cid:39) (cid:114) π ξ e − j π / e − j ξ for ξ (cid:29) r and length ∆ z surrounding the electron trajectory.According to Eq. (6), the energy radiated through this surface is ∆ W = ∫ ∞ (cid:26) · π · ( π r ∆ z ) ˆ r · Re (cid:20) E ( r , ω ) × µ B ∗ ( r , ω ) (cid:21) (cid:27) d ω (26)The fields are determined from the potentials in the usual manner – the equations for transformsare the same as for phasors: E ( r , ω ) = − j ω A ( r , ω ) − ∇ Φ ( r , ω ) = − j ω A z ˆ z − cn β (cid:18) ˆ r ∂ A z ∂ r + ˆ z ∂ A z ∂ z (cid:19) (27) B ∗ ( r , ω ) = ∇ × A ∗ ( r , ω ) = − ˆ φ ∂ A ∗ z ∂ r (28)Here the scalar potential was eliminated using Eq. (24), and the fields are all expressed as afunction of the longitudinal component of the vector potential. Derivatives of A z are computedusing Eqs. (23) and (25). We obtainˆ r · [ E ( r , ω ) × µ − B ∗ ( r , ω )] = µ − (cid:20) − j ω + cn β (cid:18) j ωβ c (cid:19)(cid:21) A z (cid:20) − r + j ω c (cid:113) n − β − (cid:21) A ∗ z (29)16he real part of this expression depends on whether the Cherenkov radiation condition is fulfilled( β c > c / n ):ˆ r · Re [ E ( r , ω ) × µ − B ∗ ( r , ω )] = ω (cid:18) − n β (cid:19) µ e π r for n > / β n < / β (30)Insertion of this expression into Eq. (26) yields the energy radiated by the charge per unit travelledlength – the Frank–Tamm formula [47–49]: ∆ W / ∆ z = ∫ ∞ e µ π ω (cid:18) − n β (cid:19) for n > / β n < / β d ω. (31) References
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