Beam-Wave Interaction in Periodic and Quasi-Periodic Structures

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A natural solution to bypass the use of bulk silicon is to leverage the superior optoelectronic properties of III—V semiconductors 18 — However, an electrically pumped all-silicon source—which would benefit from the low fabrication costs of silicon devices into VLSI 5 —has yet to be demonstrated. Free-electron sources 23 — 25 , especially in the context of Smith-Purcell SP radiation 26 , are natural candidates to address this challenge, thanks to their exceptional tunability.

However, Smith-Purcell radiation from dielectric substrates, let alone silicon, has not been utilized so far. In contrast, a related effect, the laser acceleration of particles interacting with confined modes in dielectric structures 27 , has been widely studied. More recently, there has been a growing interest in the study of incoherent 28 and coherent 29 transition radiation cathodoluminescence in dielectrics and semiconductors, with novel experimental techniques to disentangle their relative importance 30 , The lack of work in dielectrics and semiconductors for radiation generation from free electrons incident at a grazing angle 32 may originate from the fact that the SP effect was first observed in metallic gratings and was subsequently explained as the constructive interference from the periodic motion of free currents along the surface 33 , or as the motion of image charges in a perfect conductor It has recently been predicted that dielectrics, and generally low-optical-loss materials could not only emit SP radiation, but also outperform metals by judicious design of phase-matching conditions of the electron excitation with high- Q resonances 34 , Coincidentally, the prospects of all-silicon free-electron-driven sources are enhanced by the recent development of high-throughput, low bias voltage, densely integrated silicon-gated Field Emitter Arrays FEA 36 — 38 , whose performance surpasses that of their metallic counterparts Spindt-type emitters.

Similar electron energies are achievable with existing on-chip electron sources 36 , 37 , 39 — 41 and are here experimentally utilized to excite all-silicon nanogratings. In addition, we theoretically investigate the feasibility of an all-silicon radiation source, which integrates a silicon on-chip gated FEA with a silicon nanograting.

Taken together, our observation and analysis pave the way for an electrically pumped all-silicon source for potential applications in the near-infrared, and the telecommunication wavelengths. A schematic of a free-electron-driven silicon radiation source is depicted in Fig.

Beam Wave Interaction In Periodic And Quasi Periodic Structures Particle Acceleration And Detection

The interaction of free-electrons with periodic structures induces the emission of tunable radiation. Broadly tunable radiation from all-silicon nanogratings. In our current experimental setup see Fig. The output power and the incident electron beam current are experimentally measured. For each sample, the spectral efficiency normalized by the incident electron beam current is plotted for various electron kinetic energies in Fig. Our experimental data top is compared to time-domain simulation data bottom.

We notice that, at a given wavelength e. The bandwidth narrowing for slow electrons can be derived from the Smith-Purcell dispersion relation 42 and observed with adequate collection optics. For electrons in the far-field away from the structure, the generated radiation intensity scales as 34 , 43 exp - z h eff —with z being the electron-beam height above the grating surface. Therefore, we expect the output power to be considerably smaller for shorter grating periods in similar experimental conditions The incident electron current and measured output powers are reported in the colored boxes for each measurement.

Dashed lines correspond to the predicted output wavelength at normal emission direction from the SP energy-angle equation. Optical and quantum efficiencies are reported in the SI, section I. Error bars shaded area are estimated from the standard deviation of the background signal. As shown in Fig. This configuration allows electrons to pass close to the nanograting so they interact with numerous unit cells.

We use lightly doped silicon substrates, which makes the sample conductive without significantly altering its optical properties. The doping thus circumvents Coulomb repulsion from charges gathering on the surface of the sample—as would happen with electrically insulating materials.

The sample is bare silicon without any coating materials. Spectrally, spatially, and polarization-resolved modified-SEM cathodoluminescence measurement setup. A Faraday cup, mounted on the sample holder, is used to record the electron beam current I. A microscope objective collects and couples out the radiation induced by free electrons to a set of free space optics. Our setup resolves the spatial, spectral and polarization behavior of emitted cathodoluminescence; spontaneous emission from free electrons is coupled out of the SEM vacuum chamber and then goes through two arms of free space optics 1 a linear polarizer and a set of lenses focusing radiation to an optical fiber coupled to a near-infrared spectrometer and 2 a CCD camera allowing the imaging of the surface of the nanograting of which a scanning electron micrograph is shown in Fig.

After verifying that the SP signal is polarized along the direction of the electron beam as was originally observed 26 and confirmed by Fig. This definition of the background relies on the assumption that the SP electric field and cathodoluminescence from the bulk are incoherent—as is usually observed for low kinetic energy electrons in semiconductors 46 —and the measured background weak incoherent cathodoluminescence in silicon 30 is polarization-independent.

Radiation from Quantum Electrons in Strong Fields

Our experimental setup also allows us to measure the optical power and quantum efficiencies of this free-electron-induced source. The power efficiencies of the experiments presented in Fig. These low power efficiencies reported in our proof-of-concept experiment naturally raise the question of the maximal power efficiency that could be achieved in a scalable on-chip device with this spontaneous emission process, which is addressed in the following section.

A fundamental metric of any system performing power conversion is its maximal power conversion efficiency. Some of the foundational principles of modern physics were used to derive such limits in the cases of, for example, heat engines Carnot efficiency 47 , based on the second principle of thermodynamics , solar cells the Shockley-Queisser limit 48 , based on fundamental considerations on recombination processes in p—n junctions.

We have recently developed a framework 34 based on optical passivity 49 that allows us to compute the maximal spontaneous radiation from free-electrons interacting with an arbitrary nanophotonic medium. We apply this formalism to the system shown in Fig. Maximal efficiency of all-silicon tunable sources in the near-infrared.

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The insets show the beam shape and alignment configuration with respect to the nanograting in this simulation. The inset shows our experimental results in a modified SEM setup as a function of the electron beam voltage. In contrast to the prevailing belief that metallic structures should yield better SP radiation efficiencies 26 , 32 , 52 , 53 , lossless dielectrics such as silicon in the near-infrared could benefit from potentially very large quality factors from nanophotonic structure designs 54 and thus become ideal for free-electron light emitting devices, in particular for narrow-bandwidth applications.

At such optical frequencies, the spatial distribution of electrons can be deformed thanks to its large dielectric susceptibility. This is in strong contrast to the case of metals—the main materials used to record Smith-Purcell radiation since its original discovery 26 —where free charges screen the field of swift electrons in vacuum with a phase delay.

Beam-Wave Interaction in Periodic and Quasi-Periodic Structures (Accelerator Physics)

Our metric, based on the fundamental constraint of optical passivity, allows us to describe the maximal power conversion efficiency of an integrated SP source and provides physical insights on how to attain this upper efficiency We compare different gated FEA designs from the literature 36 , 37 , 39 — 41 , in which intensity-voltage relations have been measured. We assume that the electron propagates at a constant speed above the nanograting, positioned in the vicinity of the anode, which is realized if its total length L G is much smaller than the millimeter scale.

The fabrication and alignment of a gated FEA would benefit from the resolution of nanofabrication techniques: assuming the electron collimated beam geometry shown in the upper inset of Fig. However, FEAs designed to achieve smaller beam diameters usually require an additional focus voltage 40 , 41 which results in leakage current in the focusing gate, thus reducing the anode current.

To illustrate the compromise between electron beam diameter and anode current, we compare FEA designs for which the electron beam diameter is measured, with state-of-the-art high-throughput silicon and Spindt-type emitters for the latter, we estimate the maximal power efficiency for several beam diameters. Results for the five FEA designs are reported in Fig.

Silicon FEAs with state-of-the-art integration densities, longevity and high current density have recently been reported: 36 , 37 thanks to their large current output, they outperform their metallic counterparts 39 with similar integration densities. The ability to leverage VLSI fabrication processes for silicon FEAs is enabling their greater scaling; here, this technological advantage of silicon FEAs finds a direct application in free-electron sources.

This perspective is particularly promising for the realization of an all-silicon free-electron-driven radiation source. The proposed device in Fig.

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However, the development of vacuum packaging techniques 55 , in addition to high-voltage DC-DC converters 56 , should ensure scalability of our approach, in addition to its compatibility with CMOS fabrication processes. In our study, the predicted maximal power efficiencies are comparable with those reported in all-silicon LED 7.

However, the measured power efficiencies are still several orders-of-magnitude below standard silicon sources.

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Nonetheless, unlike solid-state devices, our proposal of an all-silicon integrated radiation source can leverage the advantages specific to free-electron devices to increase its power conversion efficiency, such as 1 the possibility of partially recovering the energy of electrons having interacted with the structure with a depressed collector—as with a field emitter integrated in traveling-wave tubes 57 , 2 the engineering of the electron beam spatial distribution in order to increase the coupling between the beam and the nanograting see Fig.

This engineering is enabled by the precise alignment of the electron beam with the nanograting on chip, allowed by the nanoscale resolution of CMOS fabrication techniques; or 3 pre-bunching the electron beam to facilitate stimulated emission 58 , The latter could be achieved with an ultrafast FEA, such as out-of-plane silicon 60 and in-plane plasmonic 61 , 62 ultrafast FEAs that have recently been demonstrated.

These advantages, as well as the scheme of coupling to high- Q photonic modes, makes silicon a particularly interesting platform for free-electron light sources. As silicon exhibits similar dielectric properties in the mid-infrared, we expect similar maximal power efficiencies for that wavelength range also, thus enabling applications in mid-infrared sensing; this wavelength range could also benefit from narrower bandwidths, with lower-energy electron sources The possibility of achieving stimulated emission on chip is particularly exciting, as few experiments have observed coherent emission from electrons interacting with passive systems 64 , Achieving stimulated emission on chip with an all-silicon structure would solve probably one of the most resilient problems of modern engineering and physics: the demonstration of an electrically pumped all-silicon laser.

Adding this feature to a SP source integrated with an FEA would result in an ultra-compact device comprising 1 an on-chip free-electron source pumping 2 a nanograting, acting as an SP source whose generated radiation is 3 spatially spectrally shaped for instance, potentially replacing bulky focusing optics at the output of the light source. We have also provided a theoretical analysis of a scalable, all-silicon electron-beam light source, consisting of a silicon field emitter integrated with a silicon nanograting. In this framework we have computed the maximal power efficiency and output power that can be achieved by phase-matching the electron beam velocity with a high- Q photonic mode of the periodic structure We have also proposed several directions to further enhance the power efficiency by taking advantage of the features of free-electron sources, building upon early demonstrations of free-electron driven radiation on-chip with silicon 67 and molybdenum Spindt-type field emitters 68 , Our theoretical framework 34 is readily transferrable to other wavelength ranges and materials for which free-electron sources could also be technologically interesting, for instance in the far-infrared and THz regimes Our work suggests that free-electron-driven light-matter interactions may offer a viable way to generate light in all-silicon structures.

Techniques in both electron beam physics and nanophotonic design could be leveraged to make the interaction of the electrons with the structure more efficient. Our experimental setup is based on a modified Scanning Electron Microscope SEM and is comprised of a set of adequate free space optical components to spectrally, spatially and polarization-resolve the emission of photons from free-electrons interacting with a nanophotonic structure see Fig. We mounted a periodic sample close to parallel to the electron beam direction inside the vacuum chamber of the SEM in order to send electrons at a grazing angle with respect to the grating plane.

The SEM was operated in spot mode, in which we control precisely to position the beam so that it passes parallel to the surface near the desired area of the sample. The spectrometer used was an Acton SP—i with a linear InGaAs photodiode detector array with detection range 0. Monochrome images of the radiation were collected with a Hamamatsu CCD, in order to align the optical setup and spatially resolve the observed radiation. The beam currents were measured using Keithley picoammeter connected to a Faraday Cup mounted on a SEM sample holder.

We performed time-domain simulations in order to estimate the power spectrum of photons emitted by electrons propagating at a constant height h above an all-silicon periodic structure.

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We designed this simulation setup in order to confirm our experimental results. We then fit the simulated power spectrum to the measured one with a single fitting parameter.

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The electron beam is represented by a discrete set of dipoles with polarization along the electron direction. Each dipole is turned on at the time when the electron is flying by its position.

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We computed the scattered spectrum from this array of dipoles via a near-to-far-field transformation. We verified that the far-field spectrum converges when the number of dipoles was large enough. Zhao, D. The nonresonant perturbation theory based field measurement and tuning of a linac accelerating structure. Ang Wen Cheng, Tong De Chun, Gu The nonresonant perturbation theory based field measurement and tuning of a linac accelerating structure.

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