I. INTRODUCTION

Section:

ChooseTop of pageABSTRACTI. INTRODUCTION <<II. BASIC DESIGN THEORYIII. MODE ANALYSISIV. TIME-DOMAIN SIMULATIO...V. HARMONIC GENERATIONVI. CONCLUSIONREFERENCESA free-electron laser11. D. A. G. Deacon, L. R. Elias, J. M. J. Madey, G. J. Ramian, H. A. Schwettman, and T. I. Smith, “First operation of a free-electron laser,” Phys. Rev. Lett. 38, 892–894 (1977). https://doi.org/10.1103/physrevlett.38.892 (FEL) generates laser radiation from electrons propagating in a vacuum with magnetic fields or material structures. To achieve lasing, the injected electrons experience radiation feedback in the laser structure and are collectively bunched into radiation cycles to generate stimulated emission of radiation.22. J. Gardelle, J. Labrouche, G. Marchese, J. L. Rullier, D. Villate, and J. T. Donohue, “Analysis of the beam bunching produced by a free electron laser,” Phys. Plasmas 3, 4197 (1996). https://doi.org/10.1063/1.871552 For relativistic electrons, a magnetic undulator33. Y. C. Huang, H. C. Wang, R. H. Pantell, J. Feinstein, and J. W. Lewellen, A staggered-array wiggler for far infrared, free-electron laser operation,” IEEE J. Quantum Electron. 30, 1289–1294 (1994). https://doi.org/10.1109/3.303695 is often used to induce transverse motion of the electrons and couple the electron energy to the amplification of the radiation field. Some vacuum electronic devices, such as backward wave oscillators44. H. Johnson, “Backward-wave oscillators,” Proc. IRE 43, 684–697 (1955). https://doi.org/10.1109/jrproc.1955.278054 and Cherenkov lasers,55. B. Johnson and J. Walsh, “Cherenkov infrared laser,” Nucl. Instrum. Methods Phys. Res. 237, 239–243 (1985). https://doi.org/10.1016/0168-9002(85)90355-9 usually adopt a slow-wave structure to match the longitudinal velocities between sub-relativistic electrons and a radiation wave for continuous energy transfer. A Smith–Purcell radiator,66. A. Gover, P. Dvorkis, and U. Elisha, “Angular radiation pattern of Smith–Purcell radiation,” J. Opt. Soc. Am. B 1, 723–728 (1984). https://doi.org/10.1364/josab.1.000723 driven by an electron beam above a metal grating, can also generate broadband radiations at different directions above the grating. Additional resonances, such as cavity feedback, are needed to generate narrow-line stimulated Smith–Purcell radiation.77. E. M. Marshall, P. M. Phillips, and J. E. Walsh, “Planar oratron experiments in the millimeter wavelength band,” IEEE Trans. Plasma Sci. 16, 199–205 (1988). https://doi.org/10.1109/27.3815Recently, research on laser-driven chip-size accelerators, known as dielectric laser accelerators or DLAs,8–108. J. Breuer and P. Hommelhoff, “Laser-based acceleration of nonrelativistic electrons at a dielectric structure,” Phys. Rev. Lett. 111, 134803 (2013). https://doi.org/10.1103/physrevlett.111.1348039. E. A. Peralta, K. Soong, R. J. England, E. R. Colby, Z. Wu, B. Montazeri, C. McGuinness, J. McNeur, K. J. Leedle, D. Walz, E. B. Sozer, B. Cowan, B. Schwartz, G. Travish, and R. L. Byer, “Demonstration of electron acceleration in a laser-driven dielectric microstructure,” Nature 503, 91 (2013). https://doi.org/10.1038/nature1266410. R. J. England et al., “Dielectric laser accelerators,” Rev. Mod. Phys. 86, 1337 (2014). https://doi.org/10.1103/revmodphys.86.1337 and their keV injectors1111. T. Hirano, K. E. Urbanek, A. C. Ceballos, D. S. Black, Y. Miao, R. Joel England, R. L. Byer, and K. J. Leedle, “A compact electron source for the dielectric laser accelerator,” Appl. Phys. Lett. 116, 161106 (2020). https://doi.org/10.1063/5.0003575 have attracted wide attention and inspired new opportunities for applications utilizing compact electron sources. By running the acceleration process in reverse, an accelerator chip could function as a radiation chip. It is possible that a monolithically integrated accelerator and radiator on a chip1212. C. Roques-Carmes, S. E. Kooi, Y. Yang, A. Massuda, P. D. Keathley, A. Zaidi, Y. Yang, J. D. Joannopoulos, K. K. Berggren, I. Kaminer, and M. Soljačić, “Towards integrated tunable all-silicon free-electron light sources,” Nat. Commun. 10, 3176 (2019). https://doi.org/10.1038/s41467-019-11070-7 could be realized with good efficiency in the near future. Currently, high-brightness keV electrons are already available from an electron microscope. An electron microscope equipped with a built-in FEL chip can be a powerful pump–probe tool for material research. However, a tiny electron emitter can only generate a small electron current. For instance, the average temporal separation of two adjacent electrons in a nano-ampere current from a DLA or a transmission electron microscope (TEM) is about ∼100 ps. At 50 keV, the spatial separation of the two electrons is about a centimeter. If the radiation chip has a longitudinal length of the order of a millimeter, there is only one electron driving the chip at a time. To generate laser-like radiation in one electron transit, it requires strong coupling between the electron and the radiation mode. In this scenario, stimulated emission from many gradually bunched electrons in a conventional FEL is irrelevant to the radiation process.In the optical frequencies, metal is lossy, making dielectric the choice for most optical components. A conventional Fabry–Perot resonator with a many-wavelength length is too long to provide any optical feedback to an electron in one transit. However, in a dielectric grating with nano-periodicity, an electron with an extended Coulomb field could resonantly excite and amplify the distributed optical feedback from individual grating grooves with little time delay. Although Cherenkov radiation in a photonic crystal13,1413. C. Luo, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Cherenkov radiation in photonic crystals,” Sci 299, 368–371 (2003). https://doi.org/10.1126/science.107954914. T. Denis, M. W. van Dijk, J. H. H. Lee, R. van der Meer, A. Strooisma, P. J. M. van der Slot, W. L. Vos, and K. J. Boller, “Coherent Cherenkov radiation and laser oscillation in a photonic crystal,” Phys. Rev. A 94, 053852 (2016). https://doi.org/10.1103/physreva.94.053852 or metamaterial1515. F. Liu, L. Xiao, Y. Ye, M. Wang, K. Cui, X. Feng, W. Zhang, and Y. Huang, “Integrated Cherenkov radiation emitter eliminating the electron velocity threshold,” Nat. Photonics 11, 289–293 (2017). https://doi.org/10.1038/nphoton.2017.45 and Smith–Purcell radiation from metallic nano-grating16,1716. Y. Yang, A. Massuda, C. Roques-Carmes, S. E. Kooi, T. Christensen, S. G. Johnson, J. D. Joannopoulos, O. D. Miller, I. Kaminer, and M. Soljačić, “Maximal spontaneous photon emission and energy loss from free electrons,” Nat. Phys. 14, 894–899 (2018). https://doi.org/10.1038/s41567-018-0180-217. Y. Ye, F. Liu, M. Wang, L. Tai, K. Cui, X. Feng, W. Zhang, and Y. Huang, “Deep-ultraviolet smith–Purcell radiation,” Optica 6, 592–597 (2019). https://doi.org/10.1364/optica.6.000592 or dielectric–metal hybrid structures18,1918. Z. Rezaei and B. Farokhi, “Start current and growth rate in Smith–Purcell free-electron laser with dielectric-loaded cylindrical grating,” J. Theor. Appl. Phys. 14, 149–158 (2020). https://doi.org/10.1007/s40094-019-00358-019. M. Cao, W. Liu, Y. Wang, and K. Li, “Dispersion characteristics of three-dimensional dielectric-loaded grating for terahertz Smith-Purcell radiation,” Phys. Plasmas 21, 023116 (2014). https://doi.org/10.1063/1.4866157 have been studied in the past, this study utilizes a much simpler dielectric-grating waveguide to maximize the electron-wave coupling and build up narrow-line radiation in a single electron transit. The highly directional and focused wave from the waveguide output can be a major advantage for a downstream application.

Here, we first introduce a few theoretical guidelines for designing the proposed nano-chip FEL, and then employ mode-expansion theory and simulation code to confirm the parameters for a realistic design. Finally, we perform time-domain simulations to understand the radiation mechanism and device performance for a single-electron excited nano-grating waveguide. Before we conclude this paper, we demonstrate numerically coherent harmonic generation from the nano-grating waveguide by driving it with a periodic pulse train of single electrons from a DLA operating at a sub-harmonic frequency. Finally, we summarize the study of this paper.

II. BASIC DESIGN THEORY

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. BASIC DESIGN THEORY <<III. MODE ANALYSISIV. TIME-DOMAIN SIMULATIO...V. HARMONIC GENERATIONVI. CONCLUSIONREFERENCESFigure 1 depicts the proposed dielectric-grating waveguide in which an electron propagates at a distance lip, called the impact parameter, above the grating surface. The structure is a corrugated dielectric film on a dielectric substrate. Although the substrate is not essential for wave guiding, a thick enough substrate is often necessary to support a sub-micrometer thick laser waveguide. Without loss of generality, the surface corrugation is assumed to have a rectangular shape with a period of Λg and depth of tg. The smooth film layer under the grating has thickness tf. In this work, we assume that the grating and the film layer are of the same optical material. To guide the radiation, the grating waveguide has a refractive index nf, which is larger than that of substrate ns. In the vacuum region, there will be Smith–Purcell radiation, which will be shown below as correlated with the radiation inside the waveguide. The two-dimensional (2D) structure in Fig. 1(a) is first adopted for our theoretical analysis. To plan a real experiment, we perform numerical simulations to further study the finite-width three-dimensional (3D) structure in Fig. 1(b). Figure 1(c) shows the scanning-electron-microscopy images of a fabricated silicon grating for an ongoing feasibility study. In our design, the grating period is set at 300 nm with 155 nm grating linewidth, 160 nm groove height, and 145 nm groove width. The width of the grating is 4 µm. We investigated both deep reactive ion etching and focused ion beam for fabricating the grating. The latter shows superior quality for our fabricated sample. In the image, the successfully fabricated grating has a 300.6-nm period, 152.6-nm linewidth, 205.4-nm groove height, and 146.5-nm groove width. The radiation wave is guided in the grating film above a substrate. On the grating surface, the electron’s kinetic energy is transferred to the evanescent field of the transverse magnetic (TM) waveguide mode when the electron velocity matches the mode field’s phase velocity. Before presenting a more detailed numerical study, we first list below a few physics laws to estimate the parameters to generate radiation with a desired wavelength.For the radiation to be guided in the film layer, the incidence angle θ of the guided wave at the film and substrate interface must be larger than the critical angle of total internal reflection θc orFor a silicon grating of nf = 3.4 on a glass substrate of ns = 1.5, the condition for total internal reflection is θ > 26°. Assuming that the surface corrugation is a small perturbation to the radiation modes in the film layer, the velocity matching to ensure continuous energy transfer from the electron to the radiation field is primarily governed by the Cherenkov condition, given bywhere βe = ve/c0 is the electron velocity ve normalized to the vacuum speed of light c0 and ϕ = 90° − θ is the Cherenkov angle. From (1) and (2), one obtains the range of speed of the electron for a guided Cherenkov radiation in the film layerThe condition βe > 1/nf is the Cherenkov threshold in a bulk dielectric of refractive index = nf. For a silicon film (nf = 3.4) on a glass substrate (ns = 1.5), the electron-energy range to generate the guided radiation, according to Eq. (3), is between 24 and 175 keV.In the vacuum region, the surface field of the guided mode synchronously propagates with the electron along z. The synchronous field is evanescent with an exponential decay constant α satisfying the dispersion relationship(ω/c0)2=(ω/ve)2−α2,(4)where ω is the radiation frequency and c0 is the speed of light in a vacuum. This dispersion relationship defines a mode-field depth above the grating surface, given bywhere the Lorentz factor of the electron γ is about unity for cases subject to (3) and λ0 is the vacuum wavelength of the radiation. To have enough coupling between the electron and the mode field, one usually sets an impact parameter lip∼ h. For a keV electron with βe ∼ 0.5, the impact parameter lip is approximately one tenth of the radiation wavelength.The surface corrugation is meant to provide distributed optical feedback to the resonant mode, so that narrow-band coherent radiation can be established through a single electron transit. In practice, to avoid incident electrons charging the structure, the surface of the grating waveguide can be coated with a metal layer much thinner than the skin depth. Consider each grating tooth along y as a short transmission line terminated with a perfect conductor. The impedance seen by an incidence wave has a period of half wavelength2020. D. K. Cheng, Field and Wave Electromagnetics, 2nd ed. (Addison-Wesley, New York, 1989), p. 454. in y. To have the maximum impedance contrast for the grating,2121. A. Szczepkowicz, L. Schächter, and R. J. England, “Frequency domain calculation of Smith–Purcell radiation for metallic and dielectric gratings,” App. Opt. 59, 11146–11155 (2020). https://doi.org/10.1364/ao.409585 the depth of the corrugation, tg, can be a quarter wavelength of the mode field in y, given bytg=λy4=λ04nf⁡sin⁡ϕ=βeλ04βe2nf2−1,(6)where the Cherenkov condition, Eq. (2), has been used to derive the expression as a function of the normalized electron velocity βe.Given a Cherenkov angle ϕ, a thick film tf could contain high-order transverse modes with slow group velocities and weak fields. It is therefore desirable to excite a fundamental mode in the dielectric film. To have a single-mode waveguide at a design wavelength λ0, the thickness of the waveguide film must satisfy the condition2222. B. E. A. Saleh, and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, Hoboken, NJ, 2007), p. 302.Supposing the design wavelength is 1.5 µm, the single-mode film thickness is tfnf = 3.4 (silicon film) and ns = 1.5 (glass substrate).A grating structure can provide two types of distributed-feedback resonance to an electromagnetic wave. The first type is Bragg resonance,2323. W. H. Bragg and W. L. Bragg, “The reflection of X-rays by crystals,” Proc. R. Soc. London, Ser. A 88, 428–438 (1913). https://doi.org/10.1038/091477b0 and the second one is backward-wave oscillation.2424. H. Johnson, “Backward-wave oscillators,” Proc. IRE 43, 684–697 (1955). https://doi.org/10.1109/jrproc.1955.278054 The Bragg resonance establishes a standing wave with two counter-propagating components in the grating waveguide. For instance, highly stable and useful single-frequency distributed-feedback diode lasers2525. G. P. Agrawal, N. K. Dutta, and N. K. Dutta, N. K. Dutta, Long-wavelength Semiconductor Lasers (Van Nostrand Reinhold, New York, 1986), pp. 287–332. https://doi.org/10.1007/978-94-011-6994-3_7 are based on Bragg resonance in a semiconductor gain waveguide. In a backward-wave oscillator, a backward-wave mode has a group velocity in the direction opposite the electron propagation. With single-electron excitation, it is likely that the co-propagating component of the Bragg mode will have a lower lasing threshold, although gain competition from the backward-wave mode is possible. As a first-order design for the nano-chip FEL, we aim to build up the low-threshold Bragg mode.The first-order Bragg resonance in the grating is given by the conditionwhere kz = (k0nf) × cos ϕ = 2πnf/λ0 × cos ϕ is the propagation constant of the wave along the electron axis z. Physically, it means the roundtrip reflection phase of the electromagnetic field over a grating period is 2π. This condition allows each period of the grating to form a small resonator with a half-wavelength length. From Eqs. (2) and (8), the grating period depends on the speed of the electron βe, given byIt is straightforward to show that the Bragg condition along the axial direction is the same as the Littrow-grating diffraction condition2626. E. Hecht, Optics, 3rd ed. (Addison-Wesley, New York, 1998), p. 470. for which the diffraction angle of an incident wave is the same as the incident angle of the wave. This is illustrated in Fig. 1(a), wherein each bouncing ray has two arrows to denote counter-propagating zigzag components oscillating inside the structure.Equations (3), (5)–(7), and (9) offer a set of theoretical guidelines to perform the first-order design for a nano-chip FEL structure prior to numerical optimizations. We begin by supposing that the available electron energy is 50 keV, corresponding to an electron speed of βe = 0.41. We further assume a target laser wavelength of λ0 = 1.5 µm for a silicon-on-glass grating waveguide. From Eq. (5), the impact parameter of the electron is calculated to be 98 nm. The quarter-wave grating depth, tg, calculated from Eq. (6), is 160 nm. To excite single-mode radiation at 1.5 µm in the film layer, Eq. (7) gives a maximum film thickness of tf = 246 nm. From Eq. (9), the grating period matched to the Bragg resonance is 308 nm. Note that the chosen film thickness is only 1.5 times the groove depth of the grating, which challenges the perturbation assumption made for the first-order design. However, increasing the waveguide thickness tf could weaken the coupling between the electron and the mode field. For what follows, we round those estimated parameters and use them for a more detailed numerical analysis. Table I lists the chosen parameters used for our numerical studies for a nano-chip FEL emitting at 1.5 µm.

TABLE I. The first-order design parameters for a 1.5-µm nano-chip FEL with a silicon (nf = 3.4) grating waveguide on a glass substrate (ns = 1.5).

Design wavelength (µm)Electron energy (keV)Grating period Λg (nm)Grating depth tg (nm)Film thickness tf (nm)Impact parameter lip (nm)1.550310160240100

III. MODE ANALYSIS

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. BASIC DESIGN THEORYIII. MODE ANALYSIS <<IV. TIME-DOMAIN SIMULATIO...V. HARMONIC GENERATIONVI. CONCLUSIONREFERENCESThe dispersion of the grating waveguide determines the radiation frequency subject to the velocity matching between the electron and the guided mode field. To find the dispersion of a periodic structure, most theories consider steady-state loss-free eigenmodes in an infinitely long periodic waveguide satisfying the Floquet theory.2727. L. Schachter, Beam-wave Interaction in Periodic and Quasi-Periodic Structures, 2nd ed. (Springer-Verlag, Berlin Heidelberg, 2011), p. 236. To account for the transient radiation excited by a single electron, we adopt the mode-expansion formulism developed by Peng, Tamir, and Bertoni2828. S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguide,” IEEE Trans. Microwave Theory Tech. 23, 123–133 (1975). https://doi.org/10.1109/tmtt.1975.1128513 (the PTB model) in which both steady-state and leaky modes are included without assuming the grating grooves providing a small perturbation in a 2D dielectric waveguide. With the design parameters in Table I, Fig. 2(a) shows the dispersion curves calculated from the PTB model for the TM modes in the proposed grating-waveguide structure. The guided-mode curves are in the region between the light lines of the film and the substrate. The Bragg resonances, marked with colored squares, are located at the intercepting points of the dispersion curves and the vertical line kz/kg = 0.5. It is seen that the fundamental Bragg mode has a resonant frequency at 0.2009 PHz or a vacuum wavelength of almost 1.5 µm, which is very close to the design value of 1.5 µm. The first Bragg point is intercepted by the electron line with a slope associated with 50.79-keV energy, which is just 1.8% higher than the design value of 50 keV. All the steady-state mode curves have a zero slope at the Bragg resonances, where the group velocity of a resonating standing wave is zero. The red-dashed lines are the dispersion curves of the low-loss modes found in the PTB model. However, the imaginary part of their kz is only 10−10–10−18 of the real part. Some branch of them has a negative slope. For instance, the 50.79-keV electron line intercepts the red-dashed curves at frequencies 0.2028 and 0.2463 PHz, capable of exciting backward radiations inside the grating waveguide.As a comparison, Fig. 2(b) plots the dispersion curves of the TM modes calculated by the simulation code COMSOL.2929. COMSOL – Software for Multiphysics Simulations, www.comsol.com Figures 2(a) and 2(b) show that the first three Bragg frequencies agree with each other by 1%–2%. In (b), the first Bragg mode is intercepted by a 50.73-keV electron line, which matches very well with the 50.79 electron line in (a). The insets in Fig. 2(b) show the TM-mode-field (Hx) patterns at the Bragg resonances, indicating strong wave guiding in the silicon-grating region. Unlike the PTB model, COMSOL does not provide solutions for those low-loss mode curves shown as red dashed curves in Fig. 2(a). In our simulation, COMSOL seems to always treat the propagation constant as a real number when applying the Floquet boundary condition to a periodic structure. On the other hand, the PTB model specifically defines a loss coefficient to account for the complex nature of kz and gives mode curves with different propagation losses.

IV. TIME-DOMAIN SIMULATION FOR RADIATION GENERATION

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. BASIC DESIGN THEORYIII. MODE ANALYSISIV. TIME-DOMAIN SIMULATIO... <<V. HARMONIC GENERATIONVI. CONCLUSIONREFERENCESIn practice, a grating structure has a finite length and width, as shown in Fig. 1(b). The calculations in Fig. 2 only consider an infinitely long 2D periodic structure without a width along x. In the following, we use the 3D time-domain simulation code, CST Studio Suite,3030. CST Studio Suite, CST. www.cst.com to study a practical case for the proposed nano-chip FEL. To compare with the previous 2D calculations, we set a grating width of 4 µm in the range of x = ±2 µm so that the width is about 10 times the 1.5-µm wavelength for the first Bragg resonant mode. Therefore, the reflection feedback from the x boundaries cannot reach a single electron traversing along z. The 3D structure consists of 100 grating periods along z, having a total length of 31 µm. The rest of the design parameters are listed in Table I, except that we compare below the radiation characteristics for a waveguide grating with tf = 240 nm and a bulk grating with tf = ∞. The transit time for a 50-keV electron along the 31-µm long grating is a quarter of a picosecond. For a beam current less than 0.6 µA, on average, there is at most one electron exciting the structure at a time.Figure 3 shows the calculated TM-field (Hx) patterns on the y–z plane (cut at x = 0) for the grating structures with (a) tf = ∞ and (b) tf = 240 nm at 0.2 and 0.6 ps after the electron is injected from the left edge of the structure. The colored dots are the locations of the field probes installed in the simulation for presenting the signals in Fig. 4. The structure in (a) is a bulk grating with no waveguiding in the dielectric layer, whereas the structure in (b) is a grating waveguide confining the radiation in the film layer. It is seen from (a-1) that, at 0.2 ps, a Cherenkov radiation cone following the electron is extended into the whole dielectric region under the grating; at the same time, Smith–Purcell radiation appears in both the vacuum and dielectric regions. In (a-2), radiations dissipate into the whole space after the electron leaves the grating. In (b-1), a single electron excites the strongly confined radiation inside the grating waveguide. The enlarged field pattern in the inset clearly shows the characteristic Bragg resonance with Λg = λz/2, where λz is the longitudinal wavelength of the radiation mode. After the electron exits the structure, the radiation field stored inside the waveguide starts to ring down over time. Figure 3(b-2) shows the field patterns recorded at 0.6 ps, indicating emission of quasi-coherent radiation with well-defined wavefronts in both the forward and backward directions. Figure 3(c) is the ring-down of the Hx field at the downstream output of the waveguide, which starts to emit at 0.25 ps when the electron just exits the structure and rolls off over a period of about 1 ps.Figure 4 shows the Fourier spectra of the TM radiation field, Hx(f), detected by the field probes installed at the colored dots for (a) the bulk grating and (b) the waveguide grating. On the y–z plane, the (0, 0, 0) origin of the coordinate system is denoted as O in the insets, located at the top left edge of the first grating tooth. A 50-keV electron is injected along z at the coordinates (0, 0.1, 0) in units of µm. For the waveguide grating, the film–glass interface is located at (0, 0, −0.4). Since the structure contains 100 grating periods with 310-nm periodicity, the downstream end of the structure is located at (0, 0, 31). The amplitudes of all the curves are normalized to the peak amplitude of the cyan curve in Fig. 4(a-1), which is the Fourier transform of Hx detected at the downstream output point of the silicon bulk grating (0, −0.28, 31). Also, in (a-1), the orange curve is the signal recorded at (0, −0.28, 15.73) or in the dielectric slightly below the longitudinal center of the bulk grating. Both curves in (a-1) are broadband. The signal below the grating (orange curve) is stronger and modulated with weak resonances from the surface grating. Figure 4(a-2) shows the Smith–Purcell radiation in the vacuum region, detected at the upstream point (0, 7.5, 0) and downstream point (0, 7.5, 31) as only a few percent of the Smith–Purcell radiation immediately below the dielectric grating. Note that while (a-1) and (a-2) have been plotted in the same vertical range, the amplitude of Hx(f) in (a-2) has been multiplied by 10 to enhance visibility of the detected vacuum radiation.In Fig. 4(b-1), the forward radiation at the downstream output point of the waveguide grating, detected at (0, −0.28, 31), has a narrow Bragg resonance at 0.2043 PHz, which matches the theoretical and COMSOL predictions. In Fig. 4(b-2), the backward radiation detected at (0, −0.28, 0) from the waveguide grating is about three times weaker at 0.2 PHz and contains a few satellite peaks at slightly higher frequencies, consistent with the additionally marked resonances in Fig. 2(a). Compared with the radiation from the bulk grating driven by a 50-keV electron, the spectral amplitude of the narrow-line radiation emitted from the grating waveguide is ten times higher. A stronger signal in the forward direction also signifies an amplification gain from the co-propagating electron.When the radiation modes resonantly build up inside the waveguide, some radiation can transmit through the grating and become useful radiations in the vacuum region for applications. Consider the grating formula for a dielectric transmission gratingΛgλ0nf⁡sin⁡θ−Λgλ0sinθd=m,(10)where θd is the diffraction angle above the grating and m is the diffraction order number. Choosing θ = 90° − ϕ with ϕ being the Cherenkov angle and using Eq. (2) in Eq. (10) result insinθd=1βe−mλ0Λg,(11)which is simply the Smith–Purcell radiation angle3131. D. Li, K. Imasaki, Z. Yang, and G.-S. Park, “Three-dimensional simulation of super-radiant Smith–Purcell radiation,” Appl. Phys. Lett. 88, 201501 (2006). https://doi.org/10.1063/1.2204750 above a grating for a given electron velocity, grating period, and radiation wavelength.Smith–Purcell radiation above a grating is usually broadband, having different wavelength components emitting along different directions. For the diffraction angle θd to exist in (11), the wavelength λ0 must fall into the rangeΛgm1βe−1<λ0<Λgm1βe+1.(12)For βe = 0.41 (50 keV) and Λg = 310 nm, the wavelength ranges of the Smith–Purcell radiation described by Eq. (12) are 446 nm 00m = 1 and 2, respectively. Since the narrow-band radiations of the waveguide modes could also emerge as the Smith–Purcell radiation, one would expect a broad radiation spectrum with sharp peaks in the vacuum region. Indeed, Fig. 5 shows a few narrow lines in the Hx spectrum detected at the longitudinal center above the grating waveguide [probe coordinates = (0, 7.5, 15.5)]. Again, the amplitude of Hx(f) in the plot is normalized to the peak value of the cyan curve in Fig. 3(a). When compared with the TM-field spectrum above the bulk grating in Fig. 3(b), the radiation in Fig. 4 is more intense and spikier, containing a strong peak from the leaked Bragg mode at 0.2 PHz.The quasi-coherent radiations emitted from the waveguide ends have the highest spectral brightness. To estimate the efficiency of the useful radiation from the waveguide, we extract the field data vs time from the CST simulation and integrate its power density over a ring-down time of 1 ps across the waveguide aperture 0.24 × 6 µm2. The radiation energy in the forward direction is about 2.7 aJ and that in the backward direction is about 0.51 aJ. The much higher forward radiation energy again implies an amplification gain for the synchronous radiation field co-propagating with the moving electron. Given the injection energy of 8 fJ for a 50-keV electron, the conversion efficiency for the laser-like energy is about 4 × 10−4. Given a design radiation wavelength in a fixed-length single-mode waveguide, a straightforward way to further increase the radiation efficiency is to reduce the impact parameter and thereby increase the electron-wave coupling, according to Eq. (5). As will be shown below, periodic excitation of single electrons can further increase the radiation efficiency due to constructive superposition of the radiation fields.

V. HARMONIC GENERATION

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ChooseTop of pageABSTRACTI. INTRODUCTIONII. BASIC DESIGN THEORYIII. MODE ANALYSISIV. TIME-DOMAIN SIMULATIO...V. HARMONIC GENERATION <<VI. CONCLUSIONREFERENCESIn electron radiation generation, when the electron bunch length is significantly shorter than the radiation wavelength, all the radiation fields of the electrons can add up coherently and the radiation power is proportional to the square of the number of electrons in the bunch. Such intense radiation is dubbed as electron superradiance.32,3332. A. Gover, “Superradiant and stimulated-superradiant emission in prebunched electron-beam radiators. I. Formulation,” Phys. Rev. Spec. Top.–Accel. Beams 8, 030701 (2005). https://doi.org/10.1103/physrevstab.8.03070133. A. Gover, R. Ianconescu, A. Friedman, C. Emma, C. Sudar, P. Musumeci, and C. Pellegrini, “Superradiant and stimulated-superradiant emission of bunched electron beams,” Rev. Mod. Phys. 91, 035003-1–035003-45 (2019). https://doi.org/10.1103/revmodphys.91.035003 Furthermore, if there are Np electron bunches repeating at a sub-harmonic of the radiation frequency, the spectral power at the radiation frequency is further enhanced by a factor of Np2 and the radiation linewidth is reduced by Np times due to the constructive interference of the radiation fields from the periodic electron bunches. Such harmonic generation is advantageous in generating high-frequency radiation from a low-frequency accelerator. The envisaged DLA is to produce a periodic electron pulse train at optical frequencies. Usually, a large accelerator cell driven by a long-wavelength laser can ease the structure fabrication and deliver more electrons. Assume that a 50-keV DLA generates a beam with one electron in each optical cycle and the optical cycle repeats at 0.1 PHz (a DLA driven by a 3-µm laser). By using the CST simulation code, we study the radiation of such an electron train injected into the proposed nano-chip FEL with the first Bragg resonance at 0.2 PHz. The design parameters for the structure are the same as those listed in Table I.In the CST model, we individually inject 25 electrons at 0.1 PHz into the structure with an impact parameter of 100 nm. The first electron is injected along z at t = 0 at the coordinates (0, 0.1, 0). The last electron exits the downstream end of the grating at 0.5 ps at the coordinates (0, 0.1, 31). Figure 6(a) shows the field pattern of Hx recorded at 0.2 ps, wherein the periodic array of the 0.1-PHz electrons is seen to occupy roughly the first three-quarter of the grating section and generates a guided field pattern with a period of 4Λg (inset). However, the low-threshold Bragg mode at 0.2 PHz can be resonantly built up from the coherent excitation of the 0.1-PHz electron train.3434. L.-H. Yu, M. Babzien, I. Ben-Zvi, L. F. DiMauro, A. Doyuran, W. Graves, E. Johnson, S. Krinsky, R. Malone, I. Pogorelsky, J. Skaritka, G. Rakowsky, L. Solomon, X. J. Wang, M. Woodle, V. Yakimenko, S. G. Biedron, J. N. Galayda, E. Gluskin, J. Jagger, V. Sajaev, and I. Vasserman, “High-gain harmonic-generation free-electron laser,” Science 289