Patent Publication Number: US-10785858-B2

Title: Apparatus and methods for generating electromagnetic radiation

Description:
CROSS-REFERENCES TO RELATED APPLICATIONS 
     This application claims priority to U.S. provisional application Ser. No. 62/111,180, filed Feb. 3, 2015, entitled “NOVEL RADIATION SOURCES FROM THE INTERACTION OF ELECTRON BEAMS WITH SURFACE PLASMON SYSTEMS,” which is hereby incorporated herein by reference in its entirety. 
    
    
     GOVERNMENT SUPPORT 
     This invention was made with Government support under Grant No. W911NF-13-D-0001 awarded by the U.S. Army Research Office. The Government has certain rights in the invention. 
    
    
     BACKGROUND 
     X-rays (photon energy between about 100 eV and about 100 keV) have applications in a wide range of areas. For example, in medicine and dentistry, X-rays are used for diagnosis of broken bones and torn ligaments, detection of breast cancer, and discovery of cavities and impacted wisdom teeth. Computerized axial tomography (CAT) also uses X-rays produce cross-sectional pictures of a part of the body by sending a narrow beam of X-rays through the region of interest from many different angles and reconstructing the cross-sectional picture using computers. X-rays can also be used in elemental analysis, in which measurement of X-rays that pass through a sample allow a determination of the elements present in the sample. In business and industry, X-ray pictures of machines can be used to detect defects in a nondestructive manner. Similarly, pipelines for oil or natural gas can be examined for cracks or defective welds using X-ray photography. In the electronics industry, X-ray lithography is used to manufacture high density (micro- or even nano-scale) integrated circuits due to their short wavelengths (e.g., 0.01 nm to about 10 nm). 
     To this date, X-ray tubes are a popular X-ray source in applications such as dental radiography and X-ray computed tomography. In these tubes, electrons from a cathode collide with an anode after traversing a potential difference usually on the order of 100 kV. Radiation created by the collision generally comprises a continuous spectral background of Bremsstrahlung radiation and sharp peaks at the K-lines of the anode material. The X-rays are also emitted in all directions and the source is typically not tunable since the frequencies of the K-lines are material-specific. These limitations of X-ray tube technology translate to limitations in the resolution, contrast, and penetration depth in imaging applications. The limitations also result in longer exposure time and accordingly increased radiation dose. Moreover, the temporal resolution used for live imaging of extremely fast processes is usually beyond the reach of X-ray tubes. 
     As an alternative to X-ray tubes in some applications (e.g., elemental analysis), synchrotrons and free-electron lasers, which are usually based on large-scale accelerator facilities such as the Stanford Linear Accelerator Center (SLAC), can provide coherent X-ray beams with tunable wavelengths. However, these facilities are very expensive (e.g., on the order of billions of dollars) and are generally not accessible to everyday use. 
     A more compact approach to generate X-rays is through high harmonic generation (HHG). In this approach, an intense laser beam, usually in the infrared region (e.g., 1064 nm or 800 nm), interacts with a target (e.g., noble gas, plasma, or solid) to emit high order harmonics of the incident beam. The order of the harmonics can be greater than 200, therefore allowing generation of soft X-rays from infrared beams. However, HHG produces not only the high order harmonics in the soft X-ray region but also radiation in lower order harmonics. As a result, the energy in the particular order of harmonic of interest is generally very low and is not sufficient for most applications. 
     SUMMARY 
     Embodiments of the present invention include apparatus, systems, and methods of generating electromagnetic radiation. In one example, an apparatus includes at least one conductive layer, an electromagnetic (EM) wave source, and an electron source. The conductive layer has a thickness less than 5 nm. The electromagnetic (EM) wave source is in electromagnetic communication with the at least one conductive layer and transmits a first EM wave at a first wavelength in the at least one conductive layer so as to generate a surface plasmon polariton (SPP) field near a surface of the at least one conductive layer. The electron source propagates an electron beam at least partially in the SPP field so as to generate a second EM wave at a second wavelength less than the first wavelength. 
     In another example, a method of generating electromagnetic (EM) radiation includes illuminating a conductive layer, having a thickness less than 5 nm, with a first EM wave at a first wavelength so as to generate a surface plasmon polariton (SPP) field near a surface of the conductive layer. The method also includes propagating an electron beam at least partially in the SPP field so as to generate a second EM wave at a second wavelength less than the first wavelength. 
     In yet another example, an apparatus to generate X-ray radiation includes a dielectric layer and a graphene layer doped with a surface carrier density substantially equal to or greater than 1.5×10 13  cm −2  and disposed on the dielectric layer. The apparatus also includes a laser, in optical communication with the graphene layer, to transmit a laser beam, at a first wavelength substantially equal to or greater than 800 nm, in the graphene layer so as to generate a surface plasmon polariton (SPP) field near a surface of the graphene layer. An electron source propagates an electron beam, having an electron energy greater than 100 keV, at least partially in the SPP field so as to generate the X-ray radiation at a second wavelength less than 2.5 nm. 
     In yet another example, an apparatus includes at least one conductive layer having a thickness less than 5 nm. An electromagnetic (EM) wave source is in electromagnetic communication with the at least one conductive layer to transmit a first EM wave at a first wavelength in the at least one conductive layer so as to generate a surface plasmon polariton (SPP) field in the at least one conductive layer. An electron source is operably coupled to the at least one conductive layer to propagate an electron beam in the at least one conductive layer so as to generate a second EM wave at a second wavelength less than the first wavelength. 
     It should be appreciated that all combinations of the foregoing concepts and additional concepts discussed in greater detail below (provided such concepts are not mutually inconsistent) are contemplated as being part of the inventive subject matter disclosed herein. In particular, all combinations of claimed subject matter appearing at the end of this disclosure are contemplated as being part of the inventive subject matter disclosed herein. It should also be appreciated that terminology explicitly employed herein that also may appear in any disclosure incorporated by reference should be accorded a meaning most consistent with the particular concepts disclosed herein. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The skilled artisan will understand that the drawings primarily are for illustrative purposes and are not intended to limit the scope of the inventive subject matter described herein. The drawings are not necessarily to scale; in some instances, various aspects of the inventive subject matter disclosed herein may be shown exaggerated or enlarged in the drawings to facilitate an understanding of different features. In the drawings, like reference characters generally refer to like features (e.g., functionally similar and/or structurally similar elements). 
         FIGS. 1A-1C  illustrate a system to generate X-rays using surface plasmon polariton (SPP) fields. 
         FIG. 2  shows a graphene system having a nano-ribbon structure that can be used in the system shown in  FIGS. 1A-1C . 
         FIG. 3  shows a graphene system having a disk array structure that can be used in the system shown in  FIGS. 1A-1C . 
         FIGS. 4A-4B  show graphene systems having ring structures that can be used in the system shown in  FIGS. 1A-1C . 
         FIG. 5  shows a schematic of a system for electrostatic tuning of the Fermi energy of graphene. 
         FIGS. 6A-6C  show photon energies that can be achieved by tuning the graphene Fermi energy and the electron kinetic energy when the graphene plasmon is at a free space wavelength of 1.5 μm. 
         FIGS. 7A-7B  show frequency conversion regimes that can be achieved using the approach shown in  FIGS. 1A-1C . 
         FIG. 8  shows a schematic of a graphene-plasmon-based radiation source using a transmission electron microscope (TEM) as the electron source. 
         FIG. 9  shows a schematic of a graphene-plasmon-based radiation source using direct voltage discharge as the electron source. 
         FIG. 10  shows output frequencies as a function of discharge voltage in the system shown in  FIG. 9 . 
         FIG. 11A  shows the schematic of a radiation source using two graphene layers disposed on a dielectric substrate. 
         FIG. 11B  shows the schematic of a radiation source using two graphene layers disposed on two dielectric substrates. 
         FIG. 11C  shows the schematic of a radiation source when electrons are propagating within a graphene layer. 
         FIG. 12  shows a schematic of a radiation source using multiple electrons beams and multiple graphene layers. 
         FIG. 13  shows a schematic of a radiation source using parallel free-standing graphene layers. 
         FIG. 14  shows a schematic of a radiation source using a bundle of graphene nanotubes. 
         FIGS. 15A-15F  show the analytical and numerical results of output radiation spectra from graphene-plasmon-based radiation sources. 
         FIGS. 16A-16B  show calculated emission intensity as a function of the polar angle of the outgoing radiation (horizontal) and its energy (vertical) when electrons having energies of 3.7 MeV and 100 eV, respectively, are used in graphene-based-radiation sources. 
         FIGS. 17A-17B  show calculated emission intensity when electrons having energies of 3.7 MeV and 100 eV, respectively, are used and when the SPP has a free space wavelength of 10 μm. 
         FIGS. 18A-18B  show divergence of electron beams as a function of propagation distance within surface plasmon polaritons (SPP) fields. 
         FIGS. 19A-19F  show effects of electron beam divergence on the output radiation from graphene-plasmon-based radiation sources. 
         FIGS. 20A-20B  show ponderomotive deflection of electrons. 
         FIGS. 21A-21C  show numerical and analytical results of the radiation spectrum when a 1.5 μm SPP is used. 
         FIGS. 22A-22C  show numerical and analytical results of the radiation spectrum when a 10 μm SPP is used. 
         FIGS. 23A-23B  show full electromagnetic simulation results of output radiation when 2.3 MeV electron beams are used. 
         FIGS. 24A-24B  shows a comparison of X-ray source from a single electron interacting with a graphene SPP versus a conventional scheme. 
         FIGS. 25A-25B  show full electromagnetic simulation results of output radiation when 50 eV electron beams are used. 
         FIG. 26  shows a schematic of a system for frequency down-conversion using graphene plasmons. 
         FIG. 27  show output frequencies that can be achieved using the system shown in  FIG. 26 . 
         FIGS. 28A-28B  show schematics of a system to generate Cerenkov-like effect in graphene via hot carriers. 
         FIGS. 29A-29D, 30A-30D, and 31A-31D  show theoretical results of graphene plasmon emission from hot carriers in graphene. 
     
    
    
     DETAILED DESCRIPTION 
     Overview 
     So far, X-ray sources that can produce tunable and directional X-rays normally sacrifice compactness by requiring additional acceleration stages to bring the electron beam to extremely high energies and relativistic speeds (γ&gt;&gt;1, where γ˜(1−(v/c) 2 ) −1/2 , with v being the electron speed and c being the speed of light). These relativistic electrons then interact with an electromagnetic field that induces transverse oscillations in their trajectory, causing the electrons to emit radiation. Typically, the electromagnetic field is supplied by a counter-propagating electromagnetic wave (e.g., in nonlinear Thomson scattering or inverse Compton scattering) or by an undulator, which is a periodic structure of dipole magnets (undulator radiation). 
     In Thompson scattering or inverse Compton scattering, the energies of the emitted photons E out  and the energies of incident photons E in  are related by E out ≈4γ 2 E in . In undulators, such as SLAC, the energy of the emitted photons E out  is about 2γ 2 E in , instead of 4γ 2 E in , due to the non-propagating nature of the magnetic field. Therefore, translating laser photons (e.g., about 1 eV) into X-ray (e.g., about 40 KeV) via laser-electron interaction normally needs electron beam having an energy on the order of about 50 MeV. As another example, in free electron lasers (FELs) that use an undulator with a period of about 3 cm (functionally similar to the wavelength in Thompson scattering or inverse Compton scattering and can be translated into incident photon energy of about 4.1×10 −6  eV), it takes electron beams having electron energy of about 10 GeV (γ˜20,000) to produce X-rays of the same frequency as above. High energy electron acceleration is generally costly and bulky, thereby severely limiting the widespread use. 
     To address the limitations of existing X-ray sources such as X-ray tubes, synchrotrons, FELs, and high harmonic generation (HHG), this application describes approaches that use electron beams of modest energy and can therefore bypass the high energy electron acceleration stage altogether. X-rays are generated when electrons interact with the surface plasmon polaritons (SPPs) of two-dimensional (2D) conductive materials (e.g., graphene). SPPs in 2D conductive materials can be well confined and have high momentum. This localization of SPP fields allows more efficient energy transfer from incident photos to output photons through:
 
 E   out ≈2 n× 4γ 2   E   in   (1)
 
The factor n is the “squeezing factor” (also referred to as the confinement factor) of the electromagnetic field when it is bounded to the surface between a metal and a dielectric. For 2D conductive materials, the squeezing factor n can be more than 100 or even higher. Therefore, approaches described here make it possible for a much lower electron acceleration (e.g., about 1-5 MeV) to create the same hard-X-ray frequency (e.g., about 40 KeV).
 
By simplifying or even eliminating the high energy electron acceleration in conventional X-ray sources, apparatus and methods described herein allow the development of table-top X-ray sources that are compact, tunable, coherent, and highly directional. These X-ray sources can revolutionize many fields of science, by making high-quality X-ray beams affordable to laboratories in academia and industry. Moreover, bringing these X-ray sources into regular use in hospitals would allow for incredibly sensitive imaging techniques with unprecedented resolution deep inside a human body.
 
     In addition, the approaches of electron-SPP interaction can also be employed to create radiation in other spectral regimes, such as deep ultraviolet (UV), infrared, and Terahertz (THz), with only slight modifications. These radiation sources can have similar benefits of compactness, tunability, coherence, and directionality. 
       FIGS. 1A-1C  illustrate radiation generation based on the interaction between electrons and SPP fields of 2D conductive materials. More specifically,  FIG. 1A  shows a schematic of an apparatus to generate short-wavelength radiation.  FIG. 1B  illustrates the X-ray emission from the interaction between electrons and SPP fields created from graphene.  FIG. 1C  illustrates the X-ray radiation process shown in  FIG. 1B  via a quasi-particle model. 
     The system  100  shown in  FIG. 1A  includes a two dimensional (2D) conductive layer  110  having a thickness less than 5 nm disposed on a dielectric substrate  140 . An electromagnetic (EM) wave source  120  is in electromagnetic communication with the 2D conductive layer  110  to transmit an incident EM wave  125  toward the 2D conductive layer  110 . The interaction between the 2D conductive layer and the EM wave  120  generates a surface plasmon polariton (SPP) field near the surface (e.g., within 100 nm, with 50 nm, or within 20 nm) of the 2D conductive layer  110 . The system  100  also includes an electron source  130  to propagate an electron beam  135  at least partially in the SPP field. The interaction between the electron beam  135  and the SPP field then generates an output EM wave that has a wavelength shorter than the wavelength of the incident EM wave  125 . 
     For illustrative and non-limiting purposes only, the 2D conductive layer  110  can include graphene. Surface plasmon polaritons (SPP) in graphene (also referred to as graphene plasmons, or simply GPs) can exhibit extreme confinement of light with dynamic tunability, making them promising candidates for electrical manipulation of light on the nanoscale. Highly directional, tunable, and monochromatic radiation at high frequencies can be produced from relatively low energy electrons interacting with GPs, because strongly confined plasmons have high momentum that allows for the generation of high-energy output photons when electrons scatter off these plasmons. 
     Without being bound by any particular theory,  FIG. 1B  illustrates the mechanism behind the GP-based free-electron electromagnetic radiation source. A sheet of graphene  110  on a dielectric substrate  140  sustains a GP  101 , which can be excited by coupling a focused laser beam (not shown in  FIG. 1B ) into the graphene  110 . 
     When electrons  135  are launched parallel to the graphene  110 , subsequent interaction between electrons  135  and the GP field  101  induces transverse electron oscillations, as shown by the dotted white lines. The oscillations lead to the generation of short-wavelength, directional radiation  102 , such as X-rays. 
     Without being bound by any particular theory,  FIG. 1C  illustrates the radiation process by regarding plasmons as quasi-particles interacting with electrons. In  FIG. 1C , incoming electrons  135  “collide” with GPs  101 , scattering away the incoming electrons  134  as outgoing electrons  136  and generating output photons  102  according to fundamental rules such as the preservation of momentum and energy. This scattering process can be governed by similar fundamental rules that describe electron-photon interactions. However, the result is substantially different, because the plasmon&#39;s dispersion relation allows the plasmon to have a much higher momentum, compared to photons at the same energy. In addition, plasmons can have longitudinal field components, which are generally absent from photons. As a result, electron-plasmon scattering is distinct from the electron-photon scattering in standard Thomson/Compton effect and can open up many possibilities not achievable with regular photons. 
     Two-Dimensional Conductive Layers and SPP Fields 
     In the approach illustrated in  FIGS. 1A-1C , SPP fields  101  near the 2D conductive layer  110  function as a medium that can acquire energy from incident laser photons  125  and can then transfer the acquired energy to electrons  135  for generating short-wavelength radiations. Therefore, the properties of the SPP fields can affect the overall performance of the apparatus  100 . This section describes 2D conductive materials that can be used as the 2D conductive layer  110  to create the SPP fields  101 . 
     In general, at the interface between a metal and a dielectric (including air), there exists special electromagnetic modes called surface plasmon-polaritons (SPPs). These hybrid electron-photon states can have numerous promising applications, such as to bridge the gap between electronics and photonics, allowing high frequency communication and squeezing the photonics from micron-scale to the on-chip nano-scale. This squeezing of light can also lead to high confinement of the field to the surface, expressed in high field densities, which can be useful for enhancing many types of light-matter and light-light interactions. 
     Without being bound by any particular theory or mode of operation, the field squeezing originates from the fact that the SPP effective wavelength is reduced by a large factor (referred to as the “squeezing factor” n) relative to the wavelength in free-space (e.g., wavelength of the incident EM wave  125  that excites the SPP). This squeezing factor can be the basis for various promising features of the SPP, such as enhanced sensing and sub-wavelength microscopy. The squeezing factor n typically can be about 10-20 in regular metals. However, SPP modes in graphene can be much larger, reaching several hundreds and even more than a thousand. 
     Graphene is a two dimensional array of carbon atoms connected in a hexagonal grid. This seemingly simple material can have astonishing mechanical, electronic, and optical properties, such as high mechanical strength, high mobility, and very large absorption. One property of graphene that can be useful in the apparatus  100  shown in  FIG. 1A  is its ability to support low loss SPP modes. Graphene SPPs are supported by a single layer of atoms and can have a field confinement that is more than an order of magnitude higher than that in conventional metal-dielectric SPPs. In addition, the non-metallic structures of graphene can also sustain a higher field (electric field and/or electromagnetic field) without being ionized, therefore increasing the efficiency of this X-ray generation. 
     In the approach shown in  FIGS. 1A-1C , the SPP can function as a slowly-propagating electromagnetic undulator structure with nanometer-scale periodicity because of the large squeezing factor n. Substituting the squeezing factor n of graphene SPP (e.g., n˜500) into Equation (1) shows that the squeezing effect of graphene SPP can reduce the needed γ by more than a factor of 20, compared to conventional undulator or free electron lasers, to produce the same short-wavelength radiation. This reduction of γ is equivalent to lowering the needed acceleration voltage from about 50MV to about 2MV. This order-of-magnitude reduction of the acceleration voltage makes an X-ray source feasible on the small-lab scale, since creating electron-beams of a few MeV does not require an additional acceleration stage. Accelerator facilities around the world normally use RF electron guns producing electrons of a few MeV that are then accelerated to tens, hundreds, or even thousands and tens-of-thousands MeV. Eliminating the need for the acceleration stage can significantly simplify the design of the X-ray sources. 
     Optical excitation of SPP fields  101  through EM waves  125  from air can be enhanced by patterning the graphene. For example, a grating structure can be fabricated into the substrate  140 , deposited on top of the graphene layer  110 , or implemented as an array of graphene nano-ribbons on the substrate  140 . A graphene layer can also be implemented according to one or more of the designs shown in  FIGS. 2-4 . 
       FIG. 2  shows a graphene layer  200  having a nano-ribbon structure. The graphene layer  200  includes a plurality of graphene ribbons  210   a ,  210   b , and  210   c  cut out of a graphene plane. Each ribbon has a width w. In this configuration, plasmons can form a standing wave across the ribbon with a resonance condition given by the approximate relation w˜mλ p /2, where m is an integer and λ p =2π/q is the wavelength of plasmon from infinite graphene sheet. This means that a strong absorption of light can occur at the resonance frequency that scales as ω p ˜n s   −1/4 , where n s  is the effective electron surface density. The width w of each ribbon  210   a  to  210   c  can be from micrometers (e.g., about 10 μm, about 5 μm, about 1 μm or less) to nanometers (e.g., about 10 nm, about 50 nm, about 100 nm or more). 
       FIG. 3  shows a graphene system  300  in a disk array structure. The graphene  300  includes a plurality of disk stacks  320   a  and  320   b  (collectively referred to as disk stacks  320 ) disposed on a substrate  310 . Each disk stack  320  includes alternating graphene layers  322   a  and insulator layers  322   b . The absorption of the graphene system  300  can be tuned by tailoring the size of the disks d, their separation a, and the chemical doping in each graphene layer  322   a.    
       FIGS. 4A-4B  show schematics of graphene systems in ring structures.  FIG. 4A  shows a graphene system  401  having a concentric ring structure. The graphene system  401  includes a graphene ring  411  defining a cavity  421  that is concentric to the graphene ring  411 .  FIG. 4B  shows a graphene system  402  having a non-concentric ring structure, in which a graphene ring  421  is not concentric to a cavity  422  defined by the graphene ring  421 . This non-concentric ring structure can be easier to fabricate in practice. Plasmonic resonances of the concentric graphene system  401  and the non-concentric graphene system  402  can be tuned by changing the size of the rings. 
     Patterning graphene can also help reduce losses of SPP. Generally, plasmonics can suffer from limited propagation distances (also referred to as localization) due to short plasmon lifetimes. As an initial matter, the approach illustrated in  FIGS. 1A-1C  is different from that in other applications. In most other applications, the graphene SPPs are generated in a point with the intention that they propagate along the graphene sheet. This kind of highly localized excitation of the SPPs can be very challenging. In the approach illustrated in  FIGS. 1A-1C , a simple grating can be used for the excitation of the graphene SPPs across the entire graphene. Therefore, there is no single localized point where the SPPs are generated. Instead, the graphene SPPs are coupled to the entire graphene sheet (or at least a large area of the graphene sheet) at once. As a result, the losses of the SPPs can be significantly reduced. Alternatively, the described approaches can even work in a regime that otherwise has high losses. The issue of losses can be a bottleneck in measurements of graphene SPPs propagation, because the graphene SPP modes are themselves the carriers of information. In approaches described here, the SPPs modulate the electron. Reduction of plasmon losses also allows the use of plasmons having large squeezing factors (e.g., greater than 500). 
     Patterning the graphene can generate and couple GPs simultaneously along the entire graphene surface (e.g., through the standing wave in nano-ribbon configurations shown in  FIG. 2 ), thereby overcoming the localization of plasmons. In addition, the limitations of plasmon losses do not pose a problem in the approach illustrated in  FIGS. 1A-1C  for an additional reason. The extremely confined nature of graphene plasmons allows for efficient electron-plasmon interaction over very small distances. For example, several GP periods can be squeezed over a distance of 100 nanometers, which can be sufficient to create a plasmon wiggler. 
     The properties of GPs can be dynamically changed by electrostatic tuning of the graphene Fermi energy. The tuning of GP properties can in turn change the frequency of the output photons, therefore allowing a dynamically tunable radiation source. In addition, graphene can also be chemically doped as known in the art to further increase the dynamic range of doping. Approaches described here can use electrostatic doping, chemical doping, or both. 
       FIG. 5  shows a schematic of system for electrostatic tuning of graphene. The system  500  includes a graphene layer  510  sandwiched between two electrodes  520   a  and  520   b , which are further connected to a voltage source  530 . In addition a dielectric layer (not shown in  FIG. 5 ) can be disposed between each electrode ( 520   a  or  520   b ) and the graphene layer  510  to, for example, protect the graphene from direct contact with the electrodes  520   a/b . The doping of the graphene  510  can be dynamically adjusted by changing the output voltage of the voltage source  530  and therefore the electric field across the graphene layer  510 . Electrostatic doping can change the carrier density (electrons or holes) of graphene without implanting any external particles (also referred to as dopants) into the graphene. In contrast, chemical doping usually changes the carrier density of graphene by implanting dopants (e.g., boron or nitrogen) into the graphene. 
       FIGS. 6A-6C  show the range of photon energies that can be achieved by tuning the graphene Fermi energy and the electron kinetic energy, when a free space wavelength of 1.5 μm is used for the graphene plasmon. More specifically,  FIG. 6A  shows output photon energies when the incident electron energy is about 1 MeV to about 6 MeV.  FIG. 6B  shows output photon energies when the incident electron energy is about 30 KeV to about 1 MeV.  FIG. 6C  shows output photon energies when the incident electron energy is about 5 KeV to about 30 KeV. 
       FIGS. 6A-6C  show that for a given electron energy, the range of Fermi energies permits the tuning of the output radiation frequency by as much as 100%. For example, the output photon energy can be varies from 30 keV to over 60 keV by tuning the Fermi energy from 0.5 eV to 0.9 eV (when 6 MeV electrons are used). This wide tunability range is also seen at much lower electron energies, for example, at 30 keV that is available in transmission electron microscopy (TEM) devices. These electrons can produce UV photons from about 50 eV to about 100 eV in the same Fermi energy range of 0.5-0.9 eV. 
     The above description uses graphene as the 2D conductive layer  110  shown in  FIGS. 1A-1C  for illustrating and non-limiting purposes only. In practice, other 2D systems or even 3D systems can also be used to generate the SPP field for radiation generation. In one example, metal plasmonic systems also allow the same applications show in  FIGS. 1A-1C . The squeezing factor of metal plasmonic systems may be smaller compared to graphene plasmonics, but is still sufficient in several applications. For example, electron beams from scanning electron microscopes can have electron energy on the order to about 20 KeV and can already cause significant frequency up-conversion of infrared beams to soft x-ray regimes. 
     In another example, the 2D conductive layer  110  can include 2D metal layers (e.g., single-atom layers of metal materials such as silver), which can also support SPPs of very high squeezing factor due to the electrons behaving like a 2D electron gas. For example, a single-atom-thick silver can have higher conductivity than graphene while still having very low losses in the optical regime. 2D silver therefore can support visible SPPs that can provide higher frequencies (shorter wavelengths) to start with. 
     In yet another example, double-layer graphene sheets can be used as the 2D conductive layer  110 . Double layer graphene sheets, which include two single-atom carbon layers coupled together via van der Waals force, can have enhanced conductivity and high squeezing factors. Similar properties can also be found in other multi-layer materials such as gold, silver, and other materials with properties similar to graphene. These multi-layered structures can have their bounded electrons interacting between layers, creating properties that are generalizations of the 2D electron gas behavior of single-atom layers, such as high squeezing factor. 
     In yet another example, the 2D conductive layer  110  can include general 2D electron gas (2DEG) systems, which can exist without single-atom layers or few-atom layers. Instead, the physics of 2DEG systems can appear at the interface between bulk materials, such as in MOFSET structures. These interfaces therefore can also be used in the approaches described herein. 
     The length of the 2D conductive layer  110  in the direction of the electron motion can be just a few microns and still produce high quality radiation. This means that the structure does not have to include any space for the electron beam to move through—the penetration depth of the electrons is longer than the structure size anyway—so the structure can be solid and the electrons can just be sent directly through it. 
     The last point can be useful since it constitutes an advantage of the current approaches over conventional methods. Most electron beam-based radiation sources require electrons to travel a long distance inside a structure, e.g., to have many undulator periods. Since the electrons can pass through solid matter only to a limited distance, conventional methods typically use a vacuum channel for the electrons to pass through. This makes the sources more complicated since it requires a control over the beam spread (itself a very challenging problem). In contrast, approaches described herein only involve electron beam propagation within a small length of the sample (several microns is already enough). This can make the control over the e-beam spread much easier, and even not necessary at all in some cases. Furthermore, the distance of several microns can be even shorter than the mean-free-path of relativistic electrons in solids. The implication is that the current approach can work without any special control of the electron beam. 
     Several advantages can be derived from above discussion, including: (a) one does not need to worry that the electron-beam will destroy the sample (the energies are relatively small); (b) the exact alignment of the beam and the sample are less crucial; and (c) one can build a sandwich structure or multilayer structure by stacking many layers (dielectric-graphene-dielectric-graphene- . . . ). The structures can also be cascaded to extend the interaction length (only limited by the mean-free-path, which causes a gradual decrease in the beam velocity due to collisions). 
     Other alternative geometries are also possible, such as a sandwich structure with or without a substrate between two graphene sheets, or a stack of multiple graphene sheets with a dielectric substrate in between. 
     Electron Sources and Electron Beams 
     The electron source  130  in  FIGS. 1A-1C  is configured to provide the electron beam  135  that can emit the output radiation  102  via interaction with the SPP field  101 . Therefore, the properties of the electrons beam  135 , including electron energy, beam cross sections, and beam modes (continuous or pulsed), can directly affect the output radiation  102 . 
     The electron energy of the electron beam  135  can affect the output frequency through Equation (1).  FIGS. 7A-7B  show different frequency conversion regimes that can be achieved by the GP-based free-electron radiation sources shown in  FIGS. 1A-1C . Lines corresponding to confinement factors n=50, 180, 300 and 1,000 are shown in each diagram (n=1 is also shown for reference). 
       FIG. 7A  shows that non-relativistic electrons available from a common scanning electron microscope (SEM)—the leftmost regime—are already sufficient for hard ultraviolet and soft X-ray generation. Semi-relativistic electrons, such as those used in transmission electron microscopes (TEMs), allow the generation of soft X-rays from infrared GPs (for example, 340 eV photons from 200 keV electrons). The next regime of electron energies—modestly relativistic electrons achievable in radio frequency (RF) guns—is sufficient to generate hard X-rays, circumventing the need for additional sophisticated acceleration stages, which are necessary to produce the highly relativistic electrons (rightmost regime) usually required in most free-electron-based X-ray generation schemes. For example, 4 keV X-ray photons are attainable with 5 MeV electron beams using far-infrared (λ air =10 μm) GPs with a confinement factor of n=150. 
       FIG. 7B  shows the frequency conversion regime using non-relativistic electron energies. Frequency-doubling can be attainable with few-electron-volt electrons (for example, 2.8 eV when n=300). Several tens of volts can allow a much higher up-conversion, which can convert infrared plasmons to visible or ultraviolet wavelengths.  FIG. 7B  also presents the possibility of frequency down-conversion. 
     As described above, using SPP near 2D conductive layers can significantly reduce the electron acceleration to generate short wavelength radiation, compared to conventional free electron laser or undulators. The reduced electron energy can be readily accessible via various technologies. Examples of electron sources  130  that can provide the electron beam  135  for short-wavelength generation are described below. 
     The frequency conversion regimes shown in  FIGS. 7A-7B  can be further extended. Generation of X-ray from graphene SPPs in the UV range just tens of KeV electrons can be based on similar framework described herein. Unique graphene SPPs can exist in the UV frequency range even without doping of the graphene (i.e. using intrinsic graphene, also referred to as undoped graphene). The same theoretical framework developed herein also shows that gamma-ray photons can be emitted by graphene SPPs when placed in accelerators producing electron with energies of hundreds of MeV to tens of Ge V. As before, one can reach radiation of much higher energy with the same electron beam energy, or get the same emitted photon frequency by using less energetic electron beams. 
       FIG. 8  shows a schematic of a graphene-plasmon-based radiation source using a transmission electron microscope (TEM) as the electron source. The system  800  includes a TEM device  860  with a built-in electron source  863  and X-ray detector  862 . An arrow in  FIG. 8  indicates the place where a sample-holder  850  is inserted to support a dielectric slab  840  on which a graphene layer  810  is disposed. The built-in electron source  863  provides an electron beam  835  that propagates near the surface of the graphene  810  so as to interact with SPP fields created by, for example, a laser beam (not shown in  FIG. 8 ). The electron-SPP interaction can generate X-rays (or radiation at other wavelengths depending on, among other things, the electron energy) that are emitted within a wide angle. 
     In regular use of a TEM, the sample to be imaged is suspended by the sample-holder  850  in the path of an electron-beam  835  that moves downward along the microscope cylindrical column. Therefore, a graphene sample-holder can be constructed to mount the graphene layer  810  on the dielectric slab  840  such that the graphene layer  810  is positioned precisely near the path of the electron beam  810 . 
     In one example, the graphene sample holder can have fibers and electrical feed-throughs directed through the sample holder to give external control of the properties of the graphene layer  810  (e.g., the Fermi level), and to couple the electromagnetic field through it into the SPP mode on the surface of the graphene layer  810 . In another example, other methods such as chemical doping for doping graphene without an external applied voltage can be used, therefore simplifying the holder by removing the electric feed-through. In either case, the graphene sample holder device, when put into the path of the electron beam  835 , can create the interaction illustrated in the right panel of  FIG. 8 , where the electrons are wiggled by the SPP field, causing them to emit X-ray radiation. 
     TEMs can provide electron beams of high quality (e.g., small divergence and high velocity) so as to achieve better-than angstrom scale (10 −10  m) resolution. This high quality electron beam  835 , when used in in the system  800 , can also benefit the generation of X-rays. In general, electron beams delivered by TEMs can have electron velocity of about 0.5 to about 0.8 of the speed of light (i.e., about 0.5 c to about 0.8 c), corresponding to electron energy of about 100 KeV to about 1 MeV. According to previous discussion, these electron energies are sufficient to generate X-rays using the system  800 . In one example, the TEM  860  can provide electrons beams of about 200 to about 300 KeV. The SPP field created near the graphene layer  810  can be about 1000. Laser beams at a photon energy of about 2 eV (i.e., about 620 nm) can be used to excite the SPP field near the graphene layer  810 . With these parameters, X-ray radiation of 10 KeV, already in the hard-x-ray regime, can be readily obtained, even without any additional modifications of the TEM  860 . 
     Using TEMs as the electron source for X-ray generation based on electron-SPP interaction can have several benefits. First, TEMs are state-of-the-art instruments including a built-in electron-gun, a vacuum system, and a built-in X-ray detector that can be used to monitor the properties of the generated X-ray  802  and provide feedback control if desired. TEMs generally also have a high-quality beam control and a simple usage scheme. Second, TEMs are normally of lab size and reasonably priced (about $1M). Making small modifications (about $200K) that transform a part of this system into a coherent X-ray source would be a true revolution in X-ray sources. In particular, a TEM—unlike the very large, billion-dollar accelerator facilities—can be operated in hospitals, and in many places it already is. 
     The system  800  shown in  FIG. 8  can be modified in several ways to improve the generation of X-rays or other radiations. In one example, the graphene layer  810  can include more than one layer of graphene. Due to high level of confinement of graphene SPPs, stacking several layers of graphene-covered dielectric substrates can essentially multiply the system size to increase the output intensity. Accordingly, the electrons beam  835  can also include multiple electron beams, each of which propagates through the space defined by a pair of graphene-covered dielectric substrates. 
     In another example, the graphene layer  810  can have a length that is sufficiently long for the electrons to rearrange themselves into coherent bunches via self-amplified spontaneous emission. The length of the graphene can dependent on, for example, the current of the electron pulse and the intensity of the optical pulse that excites the SPP field. In one example, the length of the graphene can be greater than 1 μm. In another example, the length of the graphene can be greater than 5 μm. In yet another example, the length of the graphene can be greater than 10 μm. As described above, since the SPP fields are generated and coupled simultaneously over the entire graphene, potential losses due to propagation of SPP can be neglected. 
     In yet another example, the electron beam  835  can include pre-bunched electrons, i.e., a sequence of electron bunches, similar to laser beams in pulsed mode. In this case, the laser beams that are used to excite the SPP field  810  can also operate in pulsed mode and can be synchronized with the electron bunches. In other words, each pulse in the sequence of laser pulses can be synchronized with one electron bunch in the sequence of electron bunches. Since pulsed laser beam can have a higher intensity compared to continuous wave (CW) beams, the resulting SPP can also be stronger, therefore allowing more efficient generation of X-rays. 
     In addition, each bunch of electrons in the sequence of electron bunches can be micro-bunched (i.e. periodic or modulated within an electron bunch). In one example, each electron bunch in the sequence can have a micro-bunch period on the order of attoseconds, i.e. micro-bunches are separated by attoseconds within each electron bunch. This micro-bunch can help generate coherent emission from the electron-SPP interaction. In another example, the micro-bunch period can be substantially equal to one oscillation cycle of the emitted radiation. For example, the emitted radiation can be about 5 nm, which has oscillation cycles of about 1.5 attoseconds. In this case, the time interval between micro-bunches with one electron bunch can also be about 1.5 attoseconds. 
     In yet another example, the electron beam  835  can have a flat sheet configuration. In other words, the cross section of the electron beam  835  can have an elliptical shape, or even a nearly rectangular shape. The flat sheet of electrons can be substantially parallel to the graphene layer  810  when propagating through the SPP field. This flattened shape of the electron beam  835  can better match the planar shape of the SPP field above the graphene layer  810 , thereby increasing the number of electrons that can interact with the SPP field and accordingly the output energy of the output radiation  820 . 
     In yet another example, the graphene layer  810  can be doped to prevent or reduce potential damage to the graphene layer  810 . Doping the graphene layer  810  can create static charges on the graphene layer  810  and therefore repel the approaching electrons from the electron beam  835 . In fact, potential damage to the graphene layer  810  should not be an issue in the approaches described here, because the electron energy is relatively low, compared to those in conventional FELs and undulators, and further because graphene have characteristically strong structures. In addition, the high conductivity of graphene can allow for quick dissipation of accumulated charge. 
     In yet another example, dielectric materials having a large refractive index can be used to make the dielectric slab  840  that supports the graphene layer  810 . In general, a larger refractive index can result in a more confined SPP field (i.e., shorter wavelength or larger squeezing factor) near the graphene surface. In practice, example materials that can be used include, but are not limited to, silicon, silicon oxide, tantalum oxide, niobium oxide, diamond, hafnium oxide, titanium oxide, aluminum oxide, and boron nitride. 
     Other than TEM, scanning electron microscopes (SEM) can also be used as the electron source for GP-based radiation source. SEMs are normally less expensive than the TEMs and are easier to modify and control. In general, SEMs can generally provide electron beams having electron energy on the order of about 20 KeV. Due to the strong field confinement in graphene SPP (i.e. higher n), radiation in the soft-X-ray regime can be achieved. Soft-X-rays, such as those in the water window between 2.3 nm and 4.4 nm, can have useful applications in imaging live biological samples. 
     In addition, electron guns in old CRT television sets can also provide electrons having energy in the KeV range, therefore allowing the development of very cost-effective soft-X-ray source. For example, a 4 KeV acceleration, which is accessible in standard small office desk items (e.g. plasma globes) can be sufficient to create 300 eV radiation, which is a soft-X-ray that falls in the water window. 
       FIG. 9  shows a schematic of a GP-based radiation source using discharge as the electron source. The radiation source  900  includes a graphene layer  910  disposed on a substrate  940 . A pair of electrodes  930   a  and  930   b  (collectively referred to as electrodes  930 ) is disposed on the two ends of the graphene layer  910  and is further connected to a voltage source  932 . By applying a voltage across the electrodes  930 , electrons  935  can be generated via discharge (e.g., at the surface of the electrodes  930 ). These electrons  935  propagate in and interact with a SPP field  901  near the surface of the graphene layer  910  and/or within the graphene layer  910  so as to generate output emission  920 . Depending on the voltage applied across the electrodes  930 , the wavelength of the output emission  920  can span from infrared to ultraviolet (UV). The approach illustrated in  FIG. 9  is CMOS compatible, thereby allowing large-scale fabrication and widespread use. 
     Generally, the voltage applied across the electrodes  930  is on the order to tens of volt. Therefore, the electros  935  are non-relativistic. In this case, the following equation for the up conversion from the incoming photon frequency (used to excite the graphene SPP) to the emitted radiation frequency applies:
 
 E   out   =E   in (1+ n β)/(1− n β)  (2)
 
where n is the graphene SPP “squeezing factor” as above, and β is the normalized electron velocity, which is the velocity divided by the speed of light. Equation (2) reduces to Equation (1) when β→1, which is the relativistic limit. Although Equation (2) only describes the frequency relation along the axis of the electron beam, a more general equation can be derived in the exact same way.
 
     The output frequency of the radiation source  900  can be tunable by changing the voltage and accordingly the electron energy, i.e., β in Equation (2).  FIG. 10  shows regimes of frequency up-conversion using low voltage electrons that can be applied in an on-chip configuration (e.g., the system  900  shown in  FIG. 9 ). Several values of the squeezing factor n, including 50, 100, 300, 500, 1000, and also n=1 for comparison, are used to show possible frequency conversion. Specific examples show that frequency doubling is already reachable with a few volts (e.g., with n of 500). A couple of tens of volts can allow a much higher up-conversion, which can convert an IR plasmon to the UV range. 
     The approach illustrated in  FIG. 9  is different from conventional methods of radiation generation using graphene. Conventional methods use graphene as a photonic crystal which interacts directly with electrons to generate radiation, for example, in THz ranges. The approaches described herein uses graphene to generate the SPP field that modulates the electrons to generate radiation. In other words, the electrons generally do not interact with the graphene itself. This difference can be further illustrated by looking into the fundamental physical processes governing the interactions: conventional methods are based on the Cerenkov Effect while approaches described herein are based on the Compton Effect. 
     This difference can induce implications in several aspects. In one aspect, the emission from the radiation source shown in  FIG. 9  can be much more tunable, compared to conventional methods, since external control over the electron beam energy and the SPP frequency can be readily available. The Cerenkov-based ideas normally only have control over the electron beam energy, while a change of the photonic modes frequency requires replacing the entire structure. 
     In another aspect, the frequency conversion efficiency of approaches described herein can depend on the strength of the SPP field, which can be controlled externally and can be brought to a high level (e.g., 1 GV/m or even higher for short pulses). The efficiency of the Cerenkov-based approaches depends on the structure interaction with the electron beam, which is much weaker and cannot be externally control. 
     In yet another aspect, the emission of light  902  in approaches described herein is created by the electrons and is radiating out of the structure right away, i.e., there may be no structure-based losses involved. The radiation in the Cerenkov-based approaches is from the structure electromagnetic modes. Therefore, structure losses can reduce the intensity of the radiation. Furthermore, much of the emission power might be lost in conventional methods unless perfect coupling of this power to the outside is achieved. 
     In yet another aspect, the emission  902  in the system  900  can be substantially monochromatic because the SPP can be controlled to be monochromatic via optical excitation using laser beams. On the other hand, Cerenkov-based ideas are usually broadband. Even though a specially designed structure can partly improve the monochromatic quality of the emission, the performance can still be far away from substantially monochromatic. 
     In yet another aspect, the approaches shown in  FIG. 9  can reach a broader range of radiation frequencies (although at each frequency the emission can be substantially monochromatic), including ultraviolet. Currently the alternative methods cannot reach UV at all. Cerenkov-based graphene ideas usually only reach IR, and the photonic crystal methods can reach visible light but then require much higher voltages on the order to tens of KeV, which can be impractical for on-chip operations. 
     The electron source  130  shown in  FIGS. 1A-1C  can also use laser-based acceleration for providing the electron beam  135 . Configurations of laser-based electron acceleration include, but are not limited to, grating accelerator, Bragg and omni-guide accelerator, 2D photonic band-gap (PBG) accelerator, and 3D PBG woodpile accelerator, among others. More information of laser-based electron sources can be found in Joel England, et al., Dielectric Laser Accelerators,  Reviews of Modern Physics,  86, 1337 (2014), which is incorporated herein in its entirety. 
     GP-Based Radiation Sources Using Multiple Graphene Layers 
       FIG. 11A  shows a radiation source  1100  that uses two graphene layers  1110   a  and  1110   b  (collectively referred to as graphene layers  1110 ), each of which is disposed on a respective dielectric substrate  1140   a  and  1140   b . The graphene layers  1110  are disposed against each other to create a cavity  1145 , in which SPP fields created from the graphene layers  110  can interact with an electron beam  1135 . In one example, the cavity  1145  is filled with solid dielectric materials (e.g., silicon, silicon oxide, silicon nitride, tantalum oxide, niobium oxide, diamond, hafnium oxide, titanium oxide, aluminum oxide, or boron nitride). In another example, the cavity  1145  is simply filled with air. In yet another example, the cavity  1145  is vacuum. The distance d between the two graphene layers  110  can be less than 100 nm (e.g., less than 90 nm, less than 50 nm, less than 20 nm, less than 10 nm, or less than 5 nm) so as to allow strong interaction between the SPP fields and the electron beam  1135 . Since two graphene layers  1110   a  and  1110   b  are used, the electron beam  1135  can interact with two SPP fields. Therefore, the configuration shown in  FIG. 11A  can increase the output energy of the resulting radiation. 
     Dielectric material in the cavity  1145  would not prevent operation of the radiation source  1100  because the electron beam  1135  can generally penetrate through a few tens of microns of dielectric with almost no energy loss (and even much more if the electron beam is more energetic). Several microns of propagation can be sufficient to generate an X-ray that is substantially monochromatic (spectral width on the order of a few eV). 
       FIG. 11B  shows a radiation source  1101 , which uses two graphene layers in a sandwich configuration. The radiation source includes a dielectric layer  1115  sandwiched by two graphene layers  1111   a  and  1111   b . Alternatively, the dielectric layer  1115  can be replaced by air or vacuum. The advantage of this sandwich structure includes that the effective index n of the SPPs will then grow by a factor of almost 2, due to the high index of the dielectric layer  1115 . In practice, the radiation source  1101  can be grown on a layer-by-layer basis. In addition, a multi-layered structure can also be constructed. The multi-layered structure can include alternating layers of graphene and dielectric material, i.e. dielectric-graphene-dielectric-graphene-dielectric. 
       FIG. 11C  shows a radiation source  1102  in which a graphene layer  1112  is disposed on a dielectric substrate  1142 . An electron beam  1135  is delivered by an electron source  1132  into the graphene layer  1112  so as to interact with any SPP field within the graphene layer  1112 . An electromagnetic wave (EM) source  1122  is configured to couple an EM wave  1125  into the graphene layer  1112  to excite the SPP field. This approach can be helpful in constructing on-chip devices, at least because the electrons are moving inside the graphene layer  1112  and electron beam control can be simpler (e.g., without vacuum chamber). 
       FIG. 12  shows a schematic of a radiation source using multiple electron beams and multiple graphene layers. More specifically, the radiation source  1200  includes a plurality of graphene-substrate assemblies  1210   a ,  1210   b ,  1210   c , and  1210   d , collectively referred to as graphene-substrate assemblies  1210 . Each of the two edge assemblies  1210   a  and  1210   d  includes a graphene layer disposed on the respective substrate, while each of the two central assemblies  1210   b  and  1210   c  includes two graphene layers disposed on both sides of the respective substrate. The space defined by each pair of graphene-substrate assembly allows the passage of electron beams provided by an electron source  1230 . The electron source  1230  is configured to deliver three electron beams  1235   a ,  1235   b , and  1235   c , which are aligned with the space defined by the graphene-substrate assemblies  1210 . This configuration can increase the total amount of electrons that can interact with SPP fields and therefore increase the total output energy of the emission  1202 . 
       FIG. 13  shows a schematic of a radiation source using multiple free-standing graphene layers. The radiation source  1300  includes multiple graphene layers  1310   a ,  1310   b ,  1310   c , and  1310   d  separated by air or vacuum. Due to the high mechanical strength of graphene, free standing layers of graphene can be constructed as shown in  FIG. 13 . Three electron beams  1335   a ,  1335   b , and  1335   c  propagate in the space defined by the multiple graphene layers  1310   a  to  1310   d  and interact with SPP fields in the space to create output radiation. 
       FIG. 14  shows a schematic of a radiation source using a bundle of graphene nanotubes. The radiation source includes a nanotube bundle  1410 . Each nanotube in the nanotube bundle  1410  can be made by rolling a planar graphene layer. A plurality of electron beams  1435   a ,  1435   b , and  1435   c  are sent to the nanotube bundle  1410  for interacting with SPP fields within the nanotubes. In one example, the diameter of the electron beams  1435   a  to  1435   c  can be greater than that of the nanotubes in the nanotube bundle  1410 . In this case, each electron beam can propagate in more than one nanotube and precise alignment may not be necessary. In another example, each electron beam can have a diameter smaller than that of the nanotubes. In this case, each electron beam can be aligned to propagate through a respect nanotube in the nanotube bundle  1410  so as to increase the interaction efficiency. 
     The two schemes shown in  FIGS. 13-14  can have the advantage that the ratio of graphene (being a single-layer structure) to vacuum in the transverse cross-section is very small. Therefore, practically all of the electrons can propagate in vacuum (instead of colliding with a non-vacuum structure). 
     In one example, the systems shown in  FIGS. 11-14  use graphene of single-atom thickness. In another example, bilayer or multi-layered graphene can also be used. It is worth noting that multi-layer graphene is different from the structure discussed in the previous paragraphs with reference to  FIGS. 11-13 . Multiple layers of graphene sheets (e.g., shown in  FIG. 11B ) with dielectric separations of at least a couple of nanometers are physically coupled by the dielectric material between individual layers of graphene. Multi-layer graphene referred to in this paragraph have the quantum properties of the bound electrons directly coupled via, for example, molecular forces. 
     The substrate material or the dielectric material separating multiple graphene layers can also affect the performance of the resulting radiation sources. Silica and silicon can be used in all examples shown in  FIGS. 11-14 , but the radiations sources herein can use any dielectric, including oxides such as silica but also metal-oxides (some of them have higher n, such as tantala and niobia). Also, boron-nitride (commonly used as a graphene substrate to get very-flat, high-purity samples) can also work. Some of these substrates can make the “squeezing factor” much larger due to their high refractive index. 
     Analytical and Numerical Analysis of GP-Based Radiation 
     This section describes analytical and numerical analysis that can explain the underlying physics behind the radiation generation presented above. The analysis can offer an excellent description of both the frequency and the intensity of the radiation. The interaction between an electron and a GP can be analytically studied from a first-principles calculation of conservation laws, solving for the elastic collision of an electron of rest mass m and velocity v (normalized velocity β=v/c, Lorentz factor γ=(1−β 2 ) −1/2 ) and a plasmon of energy  ω 0  and momentum n ω 0 /c. Their relative angle of interaction is θ i , measured from the direction of the electron velocity. The output photon departing at angle θ f  has energy  ω ph  and momentum  ω ph /c, where ω ph  is given by: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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     The approximate equality, which neglects the effects of quantum recoil, can hold whenever γmc 2 &gt;&gt;n ω 0 . In the case of n=1, Equation (3) can reduce to the formula for Thomson/Compton scattering, involving the relativistic Doppler shift of the radiation due to the interaction of an electron with a photon in free space. 
     A separate derivation based on classical electrodynamics corroborates the results of the above treatment. The detailed analysis is presented below. 
     Properties of Graphene Plasmons 
     This section describes analytical expressions for the dispersion relations and the fields of electromagnetic modes sustained by a layer of graphene sandwiched between two layers of dielectric (one of them being free space in the main text). Consider a three-layer system in which Layer  1  extends from x=−∞ to x=0, Layer  2  from x=0 to x=d and Layer  3  from x=d to x=+∞, with ε 1 , ε 2 , and ε 3  being the respective permittivities of each layer. By solving Maxwell&#39;s equations and matching boundary conditions in the standard fashion, the transverse-magnetic (TM) dispersion relation can be written as: 
                     tan   ⁡     (       K   2     ⁢   d     )       =       -   i     ⁢           ɛ   1       K   1       +       ɛ   3       K   3               ɛ     2   ⁢                 K   2       +         ɛ   1     ⁢     K   2     ⁢     ɛ   3           K   1     ⁢     ɛ   2     ⁢     K   3                       (   4   )               
where Kj=(q 2 −ω 2 ε j μ 0 ) 1/2 , j=1, 2, 3, ω is the angular frequency, q=nω/c the complex propagation constant, and μ 0  is the permeability of free space, which can also be taken as the permeability of the materials. Layer  2  is also used to model a monoatomic graphene layer of surface conductivity σ s with Layer  2 , by setting ε 2 =iσ s /(ωd) and taking d→0, to obtain the dispersion relation:
 
                   1   =     i   ⁢     ω     σ   s       ⁢     (         ɛ   1       K   1       +       ɛ   3       K   3         )               (   5   )               
which, in general, can be solved numerically for q, since σ s  can have a complicated dependence on both the frequency and the wave-vector.
 
     The surface conductivity σ s  can be obtained within the random phase approximation (RPA). When the wave-vector is small enough that plasmon damping due to electron-hole excitations is not significant, a semi-classical approach that generalizes the Drude model can be used. Taking into consideration inter-band transitions derived from the Fermi golden rule, the conductivity can be written as: 
                     σ   s     =         e   2         ℏ   2     ⁢   π       ⁢     {         iE   f       ω   +     i   ⁢           ⁢     τ     -   1             +       ℏπ   4     ⁡     [       θ   ⁡     (     ℏω   -     2   ⁢     E   f         )       -       i   π     ⁢   ln   ⁢              2   ⁢     E   f       +   ℏω         2   ⁢     E   f       -   ℏω                ]         }               (   6   )               
where the low-temperature/high-doping limit (i.e., E f &gt;&gt;kT) is assumed. The first term in the above expression is the Drude conductivity, the most commonly used model for graphene conductivity to describe GPs at low frequencies. The second term captures the contribution of inter-band transitions. In the above expression, e is the electron charge, E f  is the Fermi energy, n s  is the surface carrier density, v f  is the Fermi velocity, and τ is the relaxation time that takes into account mechanisms like photon scattering and electron-electron scattering. The spatial confinement factor, defined as n=cRe(q)/ω represents the degree of spatial confinement that results from the plasmon-polariton coupling.
 
     In the limit of a large confinement factor (i.e., q 2 &gt;&gt;ω 2 ε j μ 0 ), the dispersion relation Equation (5) can be well approximated by: 
                   q   ≈     i   ⁢       ω   ⁡     (       ɛ   1     +     ɛ   3       )         σ   s                 (   7   )               
which shows that the propagation constant, and hence the confinement factor, can be enhanced by the presence of a dielectric layer above or below the graphene. In the electrostatic limit, inter-band transitions may be ignored.
 
     An analytical expression for the plasmon group velocity may be derived from Equation (5) by first differentiating the propagation constant to obtain: 
                       (       ∂   q       ∂   ω       )       -   1       =       q   ⁡     (         ɛ   1       K   1   3       +       ɛ   3       K   3   3         )                     i   ⁢           ⁢     σ   s       ω     ⁢     (         1     σ   s       ⁢       ∂     σ   s         ∂   ω         -     1   ω       )       +                   ωμ   0     ⁡     (         ɛ   1   2       K   1   3       +       ɛ   3   2       K   3   3         )       +     (         1     K   1       ⁢       ∂     ɛ   1         ∂   ω         +       1     K   3       ⁢       ∂     ɛ   3         ∂   ω           )                       (   8   )               
and then evaluating the above equation at ω=ω 0 , where ω 0  is the plasmon frequency. When the confinement factor is large, and losses are negligible so that surface conductivity σ s =iσ si , the group velocity of a GP may be approximated by the analytical expression:
 
                     v   g     ≈     c               (     1   -         ω   0       σ   si       ⁢       ∂     σ   si         ∂   ω           )     ⁢       c   ⁡     (       ɛ   1     +     ɛ   3       )         σ   si         +                     σ   si         ɛ   0     ⁢   c       ⁢         ɛ   1   2     +     ɛ   3   2           (       ɛ   1     +     ɛ   3       )     2         +           ω   0     ⁢   c       σ   si       ⁢     (         ∂     ɛ   1         ∂   ω       +       ∂     ɛ   3         ∂   ω         )                         (   9   )               
where all variables are evaluated at ω=ω 0 . The contribution of the substrate&#39;s material dispersion—captured by the third term in the denominator—can be ignored when:
 
                       ω   0     ⁡     (         ∂     ɛ   1         ∂   ω       +       ∂     ɛ   3         ∂   ω         )       ⪡       ɛ   1     +     ɛ   3               (   10   )               
This is a condition that can be obtained by comparing the first and third terms in the denominator of Equation (9). In one example, the graphene can have SiO 2  as a substrate and free space on the other side, and the a free space wavelength of 1.5 μm can be used. SiO 2  has a chromatic dispersion d(ε/ε 0 ) 1/2 /dλ=−0.011783 μm −1 . The equation may be rearranged to give ω 0 dε/dω=0.051ε 0 &lt;&lt;ε 1,2 , which satisfies Equation (10).
 
     Equation (9) may be simplified even further in the case of large confinement factors, for which one usually has σ si &lt;&lt;ε 0 c˜1/120π, allowing the second term in the denominator of Equation (10) to be dropped without affecting the accuracy of Equation (10) significantly. In one examples, E f =0.66 eV and ε Si =1.4446, giving a confinement factor of n=180 at free space wavelength 1.5 μm. For these parameters, the surface conductivity is found to be σ s =8.18×10 −9 +i4.56×10 −5  S, according to the RPA approach. 
     To summarize, in the limit of large confinement factors and negligible material dispersion of the substrate, the group and phase velocities of a GP may be approximated by the analytical expressions: 
                       v   ph     ≈       σ   si       (       ɛ   1     +     ɛ   3       )         ,       v   g     ≈         (     1   -         ω   0       σ   si       ⁢       ∂     σ   si         ∂   ω           )       -   1       ⁢       σ   si       (       ɛ   1     +     ɛ   3       )                   (   11   )               
where all variables are evaluated at ω=ω 0 . Since the electrostatic limit for the surface conductivity (i.e., the Drude model conductivity) is not assumed, the above expressions also hold for larger plasmon energies.
 
     Electromagnetic Fields of Graphene Plasmons 
     An electromagnetic solution of the system is in general polychromatic and involves an integral over multiple frequencies subject to the RPA dispersion relation q=q(ω) obtained above. For a pair of counter-propagating, pulsed TM modes, the electric and magnetic fields in the free space portion x&gt;0 are: 
                       E   x     =     Re   ⁡     (     ∫     d   ⁢           ⁢   ω   ⁢         F   ⁡     (   ω   )       ⁢     q   ⁡     (   ω   )           iK   ⁡     (   ω   )         ⁢       exp   ⁡     [       -     K   ⁡     (   ω   )         ⁢   x     ]       ·     {       exp   ⁡     [     i   ⁡     (         q   ⁡     (   ω   )       ⁢     (     z   +     z   1       )       -     ω   ⁢           ⁢   t       )       ]       +     exp   ⁡     [     i   ⁡     (         q   ⁡     (   ω   )       ⁢     (     z   -     z   1       )       +     ω   ⁢           ⁢   t       )       ]         }           )         ⁢     
     ⁢       E   z     =     -     Re   ⁡     (     ∫     d   ⁢           ⁢   ω   ⁢           ⁢     F   ⁡     (   ω   )       ⁢       exp   ⁡     [       -     K   ⁡     (   ω   )         ⁢   x     ]       ·     {       exp   ⁡     [     i   ⁡     (         q   ⁡     (   ω   )       ⁢     (     z   +     z   1       )       -     ω   ⁢           ⁢   t       )       ]       -     exp   ⁡     [     i   ⁡     (         q   ⁡     (   ω   )       ⁢     (     z   -     z   1       )       +     ω   ⁢           ⁢   t       )       ]         }           )           ⁢     
     ⁢       H   y     =     Re   ⁡     (     ∫     d   ⁢           ⁢   ω   ⁢         F   ⁡     (   ω   )       ⁢   ω   ⁢           ⁢     ɛ   0         iK   ⁡     (   ω   )         ⁢       exp   ⁡     [       -     K   ⁡     (   ω   )         ⁢   x     ]       ·     {       exp   ⁡     [     i   ⁡     (         q   ⁡     (   ω   )       ⁢     (     z   +     z   1       )       -     ω   ⁢           ⁢   t       )       ]       -     exp   ⁡     [     i   ⁡     (         q   ⁡     (   ω   )       ⁢     (     z   -     z   1       )       +     ω   ⁢           ⁢   t       )       ]         }           )                 (   12   )               
where F(ω) is the complex spectral distribution, ε 0  is the permittivity of free space, z i &gt;0 is the initial pulse position of the backward-propagating pulse, and the frequency dependence of each component is explicitly shown. Subscripts denoting layer have been omitted for convenience. A large confinement factor normally implies a very small group velocity v g  (e.g., vg=2×10 5  m/s for confinement factor n=300 and a substrate of SiO 2  of refractive index 1.4446 at free space wavelength 1.5 μm), which can be negligible compared to the speed of free electrons from standard electron microscopes and DC electron guns. Hence, the counter-propagating pulses practically approximate a stationary, standing wave grating.
 
     When the GP pulse duration is large, a simplified form for Equation (12) can be: 
                     E   x     =         E     0   ⁢           ⁢   s       2     ⁢               [         exp   ⁡     (       ξ   -     -       K     0   ⁢           ⁢   r       ⁢   x       )       ⁢     cos   ⁡     (       ψ   -     -       K     0   ⁢           ⁢   s       ⁢   x       )         +       exp   ⁡     (       ξ   +     -       K     0   ⁢   s       ⁢   x       )       ⁢     cos   ⁡     (       ψ   +     -       K     0   ⁢   i       ⁢   x       )           ]     ⁢     
     ⁢     E   z       =       -       E     0   ⁢           ⁢   s         2   ⁢            q   0          2           ⁢               {         exp   ⁡     (       ξ   -     -       K     0   ⁢           ⁢   r       ⁢   x       )       ·     [         (         -     q     0   ⁢   s         ⁢     K     0   ⁢           ⁢   i         +       q     0   ⁢           ⁢   i       ⁢     K     0   ⁢   r           )     ⁢     cos   ⁡     (       ψ   -     -       K     0   ⁢           ⁢   s       ⁢   x       )         -       (         q     0   ⁢   r       ⁢     K     0   ⁢           ⁢   r         +       q     0   ⁢   s       ⁢     K     0   ⁢   i           )     ⁢   sin   ⁢           ⁢     (       ψ   -     -       K     0   ⁢           ⁢   i       ⁢   x       )         ]       -       exp   ⁡     (       ξ   +     -       K     0   ⁢   s       ⁢   x       )       ·     [         (         q     0   ⁢           ⁢   r       ⁢     K     0   ⁢   i         +       q     0   ⁢   i       ⁢     K     0   ⁢           ⁢   r           )     ⁢     cos   ⁡     (       ψ   +     -       K     0   ⁢           ⁢   i       ⁢   x       )         -       (         q     0   ⁢   r       ⁢     K     0   ⁢   r         +       q     0   ⁢   i       ⁢     K     0   ⁢   i           )     ⁢     sin   ⁡     (       ψ   +     -       K     0   ⁢   i       ⁢   x       )           ]         }     ⁢     
     ⁢     H   y       =           ω   0     ⁢     ɛ   0     ⁢     E     0   ⁢   s           2   ⁢            q   0          2         ⁢     {         exp   ⁡     (       ξ   -     -       K     0   ⁢           ⁢   r       ⁢   x       )       ⁡     [         q     0   ⁢   r       ⁢     cos   ⁡     (       ψ   -     -       K     0   ⁢           ⁢   i       ⁢   x       )         +       q     0   ⁢   i       ⁢     sin   ⁡     (       ψ   -     -       K     0   ⁢   i       ⁢   x       )           ]       -       exp   ⁡     (       ξ   +     -       K     0   ⁢   r       ⁢   x       )       ⁢           [         q     0   ⁢   r       ⁢     cos   ⁡     (       ψ   +     -       K     0   ⁢           ⁢   r       ⁢   x       )         +       q     0   ⁢   i       ⁢     sin   ⁡     (       ψ   +     -       K     0   ⁢   i       ⁢   x       )           ]     }                                 (   13   )               
where the subscript “0” in K and q denotes the wave-vector at the central frequency ω 0  and ξ ± =−((z∓z i )/v g ±t) 2 /2T 0   2 , ψ ± =q 0k (z∓z i )±ω 0 t+ψ 0± , q 0 =q(ω 0 ), K 0 =K(ω 0 ) and E 0s  is the peak electric field amplitude on the graphene sheet. The additional subscripts “r” and “i” on q 0  and K 0  refer to the associated variable&#39;s real and imaginary parts respectively.
 
     The physical meaning of q 0  may be understood by considering its real and imaginary parts separately: The real part q 0r  is related to the plasmon phase velocity through the confinement factor n, giving v ph =c/n. The imaginary part q 0i  is related to the plasmon attenuation. T 0  is the pulse duration associated with the number of spatial cycles N z  and temporal cycles N t  in the intensity full-width-half-maximum (FWHM) of the plasmon Gaussian pulse as: 
                     T   0     =           N   t       ω   0       ⁢     π       ln   ⁢           ⁢   2           =         N   z       ω   0       ⁢     π       ln   ⁢           ⁢   2         ⁢       v   ph       v   g                   (   14   )               
Note that T 0  can also be related to the spatial extent L by T 0 =L/nv g .
 
     Electrodynamics in Graphene Plasmons 
     This section describes analytical expressions approximating the dynamics of a charged particle (e.g., an electron) interacting with a GP, based on the results from the previous section. The motion of an electron in an electromagnetic field is governed by the Newton-Lorentz equation of motion: 
                       d   ⁢     p   _       dt     =     Q   ⁡     (       E   _     +       v   _     ×     B   _         )               (   15   )               
where {right arrow over (p)} is the momentum of the electron, m is its rest mass, Q=−e is its charge, {right arrow over (v)} is its velocity and γ=(1−(v/c) 2 ) −1/2  is the Lorentz factor. For the fields described in Equation (12), Equation (15) becomes:
 
     
       
         
           
             
               
                 
                   
                     
                       
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         γ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
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                     = 
                     
                       
                         Q 
                         
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                           ⁢ 
                           
                               
                           
                           ⁢ 
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                         ( 
                         
                           
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                     0 
                   
                   ⁢ 
                   
                     
 
                   
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                       dt 
                     
                     = 
                     
                       
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                           ⁢ 
                           
                               
                           
                           ⁢ 
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                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     For the purposes of simplifying Equation (16), it can be assumed that: a) transverse velocity modulations are small enough so γ˜(1−(v z /c) 2 ) −1/2  and x˜x 0  throughout the interaction; b) longitudinal velocity modulations are negligible so z˜z 0 +v z0 t and γ is approximately constant throughout the interaction, and c) q 0 =q 0r , which can be made possible by pumping the plasmon along the entire range of interaction; along z (e.g., via a grating). The subscript “0” refers to the respective variables at initial time. 
     Then Equation (16) may be analytically evaluated to give: 
                           β   x     ≈         QE   0       m   ⁢           ⁢   c   ⁢           ⁢     ω   0     ⁢     γ   0         ⁡     [           1   -       β   ph     ⁢     β     z   ⁢           ⁢   0                 β     z   ⁢           ⁢   0       /     β   ph       -   1       ⁢     exp   ⁡     (     ξ     0   -       )       ⁢     sin   ⁡     (     ω_t   +     ψ     0   -     ′       )         +         1   +       β   ph     ⁢     β     z   ⁢           ⁢   0                 β     z   ⁢           ⁢   0       /     β   ph       +   1       ⁢     exp   ⁡     (     ξ     0   +       )       ⁢     sin   ⁡     (         ω   +     ⁢   t     +     ψ     0   +     ′       )           ]         ⁢     
     ⁢       β   z     ≈           QE   0       m   ⁢           ⁢   c   ⁢           ⁢     ω   0     ⁢     γ   0   3         ⁢         h     0   ⁢   i         q     0   ⁢   r         ⁡     [         -       exp   ⁡     (     ξ     0   -       )             β   z0     /     β   ph       -   1         ⁢     cos   ⁡     (     ω_t   +     ψ     0   -     ′       )         +         exp   ⁡     (     ξ     0   +       )             β   z0     /     β   ph       -   1       ⁢     cos   ⁡     (         ω   +     ⁢   t     +     ψ     0   +     ′       )           ]         +     β     z   ⁢           ⁢   0           ⁢     
     ⁢           ⁢     ω   ±       =       ω   0     ⁡     (     1   ±       β     z   ⁢           ⁢   0       /     β   ph         )         ⁢     
     ⁢           ⁢       E   0     =       E     0   ⁢   s       ⁢       exp   ⁡     (       -     K       0   ⁢   r     ⁢                 ⁢     x   0       )       /   2         ⁢     
     ⁢           ⁢       ξ     0   ±       =         -       [         (         β     z   ⁢           ⁢   0       /     β   g       ±   1     )     ⁢   t     +       (       z   0     ∓     z   i       )     /     v   g         ]     2       /   2     ⁢     T   0   2                 (   17   )               
β g =v g /c, and β ph =v ph /c. Note that in the case of a large confinement factor n, the last expression gives β ph =v ph /c˜1/n, resulting in ω ± =ω 0 (1±n β z0 ). Ψ′ 0±  is used to abstract away the phase constants that do not contribute in our case to the resulting radiation.
 
     The resulting oscillations in x and z are: 
                       δ   ⁢           ⁢   x     ≈         -     QE   0         m   ⁢           ⁢     ω   0   2     ⁢     γ   0         ⁡     [           1   -       β   ph     ⁢     β     z   ⁢           ⁢   0               (         β     z   ⁢           ⁢   0       /     β   ph       -   1     )     2       ⁢     exp   ⁡     (     ξ     0   -       )       ⁢     cos   ⁡     (     ω_t   +     ψ     0   -     ′       )         +         1   +       β   ph     ⁢     β     z   ⁢           ⁢   0               (         β     z   ⁢           ⁢   0       /     β   ph       +   1     )     2       ⁢     exp   ⁡     (     ξ     0   +       )       ⁢     cos   ⁡     (         ω   +     ⁢   t     +     ψ     0   +     ′       )           ]         ⁢     
     ⁢       δ   ⁢           ⁢   z     ≈         QE   0       m   ⁢           ⁢     ω   0   2     ⁢     γ   0   3         ⁢         h     0   ⁢   i         q     0   ⁢   r         ⁡     [         -       exp   ⁡     (     ξ     0   -       )           (         β     z   ⁢           ⁢   0       /     β   ph       +   1     )     2         ⁢     sin   ⁡     (     ω_t   +     ψ     0   -     ′       )         +         exp   ⁡     (     ξ     0   +       )           (         β     z   ⁢           ⁢   0       /     β   ph       +   1     )     2       ⁢     sin   ⁡     (         ω   +     ⁢   t     +     ψ     0   +     ′       )           ]                   (   18   )               
Here δx and δz are the oscillating components of the electron displacements in x and z respectively.
 
     In the above treatment, the assumption of a relatively narrow-band GP allows the neglecting of chromatic changes in group and phase velocity in going from Equations (12) to Equation (13). Propagation losses can also be neglected from Equation (16) to Equation (17). Such approximations are justified when the confinement factor is large, in which case the group velocity tends to be negligible compared to the free electron velocity, so the GP propagates negligibly during the GP-electron interaction and both loss and pulse-broadening can be ignored. 
     Radiation from GP-Electron Interaction 
     This section describes analytical expressions approximating the spectral intensity as a function of output photon frequency, polar angle, and azimuthal angle, when an electron interacts with a GP. Although the radiation spectrum for a free electron wiggled by electromagnetic fields in free space was studied before, the analysis here of electron-plasmon scattering generalizes the electron-photon scattering to regimes of n&gt;1 and arbitrary dispersion relations, including those describing surface plasmon polaritons. This approach allows for the study of the previously unexplored regime of extreme electromagnetic field confinement (n&gt;&gt;1). Such high levels of field confinement affect the physics of the problem significantly through implications such as a very high plasmon momentum, a phase and group velocity far below the speed of light, and a ratio of magnetic to electric field that is much smaller than in typical waveguide systems and in vacuum. In addition, the graphene plasmons—contrary to traditional Thomson scattering configurations—have electric fields whose z-components (E z ) can be comparable to the x-components (E x ) in the vicinity of the electron beam. These factors motivate a new formulation of the scattering problem that in fact applies to physical systems beyond plasmons in graphene, including other surface plasmon polaritons such as those in silver and gold, layered systems of metal-dielectric containing plasmon modes. 
     The single-sided spectral intensity of the radiation emitted by a charged particle bunch, based on a Fourier transform of radiation fields obtained via the Lienard-Wiechert potentials: 
                         d   2     ⁢   I       d   ⁢           ⁢   ω   ⁢           ⁢   d   ⁢           ⁢   Ω       =         ω   2       16   ⁢     π   3     ⁢     ɛ   0     ⁢   c       ⁢              ∫     -   ∞     ∞     ⁢       n   ^     ×       ∑     j   =   1     N     ⁢       Q   j     ⁢       β   _     j     ⁢     exp   ⁡     [     i   ⁢           ⁢     ω   ⁡     (     t   -         n   ^     ·       r   _     j       ⁢     /     ⁢   c       )         ]       ⁢   dt                2               (   19   )               
where {circumflex over (n)}={circumflex over (x)}cos ϕ+ŷsin ϕ+{circumflex over (z)}cos θ is the unit vector pointing in the direction of observation, ε 0  is the permittivity of free space, N is the number of particles in the bunch, and {right arrow over (r)} j  is the position of each of the charged particles. A Taylor expansion of the exponential factor gives:
 
                     exp   ⁡     [     i   ⁢           ⁢     ω   ⁡     (     t   -       n   ^     ·         r   _     j     /   c         )         ]       ≈       (     1   -     i   ⁢       ω   ⁢           ⁢     n   x     ⁢   δ   ⁢           ⁢     x   j       c         )     ⁢     (     1   -     i   ⁢       ω   ⁢           ⁢     n   z     ⁢   δ   ⁢           ⁢     z   j       c         )     ⁢     exp   ⁡     [       i   ⁢           ⁢     ω   ⁡     (     1   -       β     z   ⁢           ⁢   0   ⁢   j       ⁢     n   x         )       ⁢   t     +   …     ]                 (   20   )               
where the ellipsis in the argument of the exponential abstracts away constant phase terms.
 
     After substituting Equation (18), (17) and (20) into (19) and further simplification, for a single charged particle: 
     
       
         
           
             
               
                 
                   
                       
                   
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                           32 
                           ⁢ 
                           
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                         [ 
                         
                           
                             
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                   ( 
                   21 
                   ) 
                 
               
             
             
               
                 
                   
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                                   2 
                                 
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                   ( 
                   22 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
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                       ± 
                     
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                                         ⁢ 
                                         
                                             
                                         
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                                     ⁢ 
                                     
                                         
                                     
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                                   g 
                                 
                               
                               ± 
                               1 
                             
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                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     Equations (21) to (23) hold when losses are negligible, but make no assumption about the size of the confinement factor besides n≥1. Equations (21)-(23) apply to the interaction between any charged particle and a surface plasmon of arbitrary group and phase velocity, where the transverse velocity oscillations of the particle are small compared to the charged particle&#39;s longitudinal velocity component. These results thus apply to physical systems beyond plasmons in graphene, including other surface plasmons such as those in silver and gold, and layered systems of metal-dielectric containing plasmon modes. In addition, although electrons are used as an example, the above results apply to any charged particle when the corresponding values for charge and rest mass are used in Q and m respectively. 
     For a group of N charged particles of the same species having a distribution W(x,y), a replacement can be made in Equation (21), where it is assumed that the particles radiate in a completely incoherent fashion.
 
 E   0   2   →NE   0s   2 ∫ 0   ∞   W ( x,y )exp(−2 K   0r   x ) dx   (24)
 
Note that the exponential factor in the integrand arises from the exponential decay of the GP fields away from the surface, highlighting the importance of working with flat, low-emittance electron beams traveling as close as possible to the graphene surface. This can be especially important when n is large.
 
     If a uniform random distribution of N charged particles (of the same species) is considered extending from x=x 1  to x=x 2  (0&lt;x 1 &lt;x 2 ), the replacement becomes: 
                     E   0   2     →       N     Δ   ⁢           ⁢   x       ⁢       E     0   ⁢   s     2       4   ⁢           ⁢     K     0   ⁢   r           ⁢         exp   ⁡     (       -   2     ⁢           ⁢     K     0   ⁢   r       ⁢     x   1       )       -     exp   ⁡     (       -   2     ⁢           ⁢     K     0   ⁢   r       ⁢     x   2       )         2               (   25   )               
where Δx=x 1 −x 2 .
 
       FIGS. 15A-15F  compares the results of analytical theory with that of the exact numerical simulation over a range of output angles. More specifically,  FIGS. 15A-15C  show results from exact numerical simulations, while  FIGS. 15D-15F  show results of analytical theory. Excellent agreement are achieved in the case of 3.7 MeV ( FIGS. 15A and 15D ) and in the case of 100 eV ( FIGS. 15C and 15F ). In these cases, the electromagnetic field intensity is low enough that the electron is not deflected away from the GP by radiation pressure. The interaction in  FIG. 15B  is prematurely terminated due to electron deflection by the GP radiation pressure, explaining the lower output intensity in  FIG. 15B  compared to that in  FIG. 15E . The spectral shape and bandwidth of the output radiation are not adversely affected by the ponderomotive deflection. 
     Owing to the high field enhancement of the GPs, fields on the order of several GV/m can be achievable from conventional continuous-wave (CW) lasers of several Watts, or pulsed lasers in the pJ-nJ range. Ultra-short laser pulses may allow access to even larger electric field strengths, thereby further enhancing output intensity. The use of pulses can benefit from synchronizing the arrival of the photon pulse with that of the electron pulse. 
     Assuming that the incident radiation excites a standing wave comprising counter-propagating GP modes—one of which co-propagates with the electrons—the output peak frequency as a function of device parameters and output angle θ is: 
                     ω     ph   ±       =       ω   ±       1   -     β   ⁢           ⁢   cos   ⁢           ⁢   θ                 (   26   )               
where ω ± =ω 0 (1±nβ) and ω 0  is the central angular frequency of the driving laser. In Equation (26), ω ph+  is due to electron interaction with the counter-propagating GP, whereas ω ph−  is due to interaction with the co-propagating GP. Note that the rightmost expression in Equation (1) reduces to ω ph+ (ω ph− ) when θ i =π (θ i =0).
 
     The spectrum of the emitted radiation as a function of its frequency ω, azimuthal angle ϕ and polar angle θ, making the assumption of high confinement factors n&gt;&gt;1 to achieve a completely analytical result: 
                                 d   2     ⁢   I       d   ⁢           ⁢   ω   ⁢           ⁢   d   ⁢           ⁢   Ω       ≈     ∫       dxdyW   ⁡     (     x   ,   y     )       ⁢         Q   4     ⁢     E     0   ⁢   s     2     ⁢     exp   ⁡     (       -   2     ⁢           ⁢   Kx     )       ⁢     L   2         32   ⁢           ⁢     π   2     ⁢     ɛ   0     ⁢     c   5     ⁢     m   2     ⁢     n   2         ⁢           ⁢       (     ω       ω   0     ⁢   γ       )     2                 ×           [           ⁢         U   +     ⁢           ⁢     exp   (           ⁢     ς   +     )       +       U   ⁢   _exp     ⁢     (     ς   -     )         ]           (   27   )                       ⁢   Where                                           ⁢       U   ±     =       ⁢     {     1   +       sin   2     ⁢   θ   ×     [           ω   2       ω   ±   2       ⁢     β   2     ⁢     cos   2     ⁢   ϕ     +     (       1     γ   4       -       cos   2     ⁢   ϕ       )                                     ⁢       (     1   +       ω     ω   ±       ⁢   βcos   ⁢           ⁢   θ       )     2     ]     }     ⁢       (       ω   0       ω   ±       )     2     ⁢       (     β   ±     β   g       )       -   2               ⁢     
     ⁢           ⁢   And           (   28   )                       ⁢       ς   ±     =     -           L   2     ⁡     [       -     ω   ±       +     ω   ⁡     (     1   -     β   ⁢           ⁢   cos   ⁢           ⁢   θ       )         ]       2         c   2     ⁢         n   2     ⁡     (     β   ±     β   g       )       2                     (   29   )               
where ε 0  is the permittivity of free space, L is the spatial extent (intensity FWHM) of the GP, E 0s  is the peak electric field amplitude on the graphene, β g  is the GP group velocity normalized to c, K≈nω 0 /c is the GP out-of-plane wavevector, Q is the electron charge (although the theory holds for any charged particle), and W(x,y) is the electron distribution in the beam (x is the distance from the graphene, as in  FIG. 1B ).
 
     The first and second terms between the square brackets of Equation (27) correspond to spectral peaks associated with the counter-propagating (ω ph+ ) and co-propagating (ω ph− ) parts of the standing wave, respectively.  FIGS. 16A-16B  show the emission intensity as a function of the polar angle of the outgoing radiation (horizontal) and its energy (vertical) when electrons having energy of 3.7 MeV and 100 eV, respectively, are used. The double-peak phenomenon described in this paragraph is also captured in the figures. In  FIGS. 16A-16B , the GP has a temporal frequency of ω 0 /2π=2×10 14  Hz (λ air =1.5 μm), in a graphene sheet that is electrostatically gated, or chemically doped, to have a carrier density of n s =3.2×10 13  cm −2  (Fermi level of E F =0.66 eV). This gives a GP spatial period of 8.33 nm, corresponding to a spatial confinement factor n (the ratio of the free-space wavelength to the GP wavelength) of 180. The graphene sheet is several micrometers in length, the interaction length being determined by the spatial size of the laser exciting the GP, which is 1.5 μm long (FWHM). 
     More specifically,  FIG. 16A  shows highly directional hard X-ray (20 keV) generation from 3.7 MeV electrons, which may be obtained readily from a compact RF electron gun. This level of electron energy requirement obviates the need for further electron acceleration, for which huge facilities (for example, synchrotrons) are necessary. In addition, this scheme does not require the bulky and heavy neutron shielding (which would add to the cost and complexity of the equipment and installation) that is necessary when electron energies above 10 MeV are used, as is often the case when X-rays are produced from free electrons in a Thomson or Compton scattering process. 
       FIG. 16B  illustrates a different regime of operation, but based on the same physical mechanism, in which electrons with a kinetic energy of only 100 eV (a non-relativistic kinetic energy that can even be produced with an on-chip electron source) generate visible and ultraviolet photons at on-axis peak energies of 2.16 eV (0.32% spread) and 3.85 eV (0.2% spread). The lack of radiative directionality can be due to the lack of relativistic angular confinement when non-relativistic electrons are used. 
       FIGS. 17A-17B  show the emission intensity when electrons having energy of 3.7 MeV and 100 eV, respectively, are used and when the SPP has a free space wavelength of 10 μm. The main difference in radiation output—compared to the λ=1.5 μm case for the same confinement factor—lies in the output photon energy, which is smaller for a given electron energy due to the larger spatial period of the surface plasmons. More specifically, in  FIG. 17A , it can be seen that highly-directional, monoenergetic (0.23% FWHM energy spread), few-keV X-rays are generated by 3.7 MeV electrons, which may be obtained readily from a compact RF electron gun. In  FIG. 17B , 100 eV electrons now generate near/mid-infrared photons at on-axis peak energies of 0.58 eV (0.2% energy spread) and 0.32 eV (0.3% energy spread). As before, the lack of radiative directionality in the 100 eV case is an inevitable result of the lack of relativistic angular confinement when non-relativistic electrons are used. 
     The resulting 20 keV photons in  FIG. 16A  are highly directional and monoenergetic, with an on-axis full-width at half-maximum (FWHM) energy spread of 0.25% and an angular spread of less than 10 mrad. The effect of electron beam divergence is discussed below. 
     Space Charge and Electron Beam Divergence 
     This section examines the effect of space charge, i.e., inter-electron repulsion, and electron beam divergence on the output of the GP radiation source. To this end, regular circular beams and electron beams with elliptical cross-sections are used. These elliptical, or “flat”, charged-particle beams are of general scientific interest as they can transport large amounts of beam currents at reduced intrinsic space-charge forces and energies compared to their cylindrical counterparts. Elliptical electron beams can also couple efficiently to the highly-confined graphene plasmons, which occupy a relatively large area in the y-z plane, but can decay rapidly in the x-dimension. 
     The elliptical charged-particle beam has semi-axes X in the x-dimension and Y in the y-dimension and travels in the z-direction with the beam axis oriented along the z-axis (see inset of  FIG. 18A ). Assuming a uniform distribution, the electrostatic potential of such a charged-particle beam in its rest frame is given by: 
                     Φ   ′     =         -     ρ   ′         2   ⁢     ɛ   0         ⁢     (           x   2     ⁢   X     +       y   2     ⁢   Y         X   +   Y       )               (   30   )               
where ρ′ is the charge density in the rest frame (primes are used to denote rest frame variables throughout this section). A beam current of I in the lab frame gives a lab frame charge density of ρ=I/(πXYv), where v is the speed of the charged particles in the z-direction, and a corresponding rest frame charge density of ρ′=ρ/γ, where γ is the relativistic Lorentz factor. According to the Newton-Lorentz equation, the resulting electromagnetic force in the lab frame gives the second-order differential equation for the evolution of the beam semi-axes:
 
                         d   2     ⁢   X       dz   2       =           d   2     ⁢   Y       dz   2       =       2   ⁢   C       X   +   Y                 (   31   )               
where C=QI/(2πmε 0 γ 3 v 3 ), Q and m are the charge and rest mass respectively of each particle, and z is the position along the beam in the z-direction, z=0 being the point of zero beam divergence (i.e. the focal plane of the charged particle beam), where X=X 0 , Y=Y 0 , and dX/dz=dY/dz=0. Note that the factor of γ 3  in the denominator of C implies that the effect of space charge diminishes rapidly as the charged particles become more and more relativistic.
 
     Equation (31) is accurate as long as the transverse velocity is small compared to the longitudinal velocity, and the transverse beam distribution remains approximately uniform. Equation (31) can be solved to get: 
                   z   =           X   0     +     Y   0           2   ⁢   C         ⁢       ∫   0         ln   ⁡     (       2   ⁢   X     -     X   0     +     Y   0       )       -     ln   ⁡     (       X   0     +     Y   0       )             ⁢       e     t   2       ⁢   dt                 (   32   )               
which is an implicit expression for X as a function of z. The beam divergence angle is:
 
                     θ   d     =       arctan   ⁡     (       ⅆ   X       ⅆ   z       )       =     arctan   ⁡     (       2   ⁢       C   ⁢   ln     ⁡     (         2   ⁢   X     -     X   0     +     Y   0           X   0     +     Y   0         )           )                 (   33   )               
The corresponding value of Y is given by: Y=X−X 0 +Y 0 .
 
     Varying the parameter X in Equation (32) and then inverting z=z(X) to X=X(z) can get the solutions for X(z), which also gives Y(z) and θ d (z) from Equation (33). In this way, the divergence angle and the semi-axes as a function of z along the charged-particle beam can be plotted, as shown in  FIGS. 18A-18B , for electron beams of kinetic energies 3.7 MeV (panel a) and 100 eV (panel b). 
     As can be seen from the  FIGS. 18A-18B , the large Lorentz factor of the relativistic 3.7 MeV electrons permits an even larger current to be used without causing the beam to diverge significantly over the interaction distance. The divergence angle of the 100 eV electron beam remains reasonably small over the interaction region, but additional beam-focusing stages may probably be needed for larger currents or longer interaction distances. 
     When X-X 0 &lt;&lt;(X 0 +Y 0 )/2, as is the case in the plots of  FIGS. 18A-18B , Equations (32) and (33) can be simplified via Taylor expansions to obtain analytical expressions of X, Y and θ d  as functions of z: 
     
       
         
           
             
               
                 
                   
                     
                       X 
                       ≈ 
                       
                         
                           
                             C 
                             
                               
                                 X 
                                 0 
                               
                               + 
                               
                                 Y 
                                 0 
                               
                             
                           
                           ⁢ 
                           
                             z 
                             2 
                           
                         
                         + 
                         
                           X 
                           0 
                         
                       
                     
                     , 
                     
                       
 
                     
                     ⁢ 
                     
                       Y 
                       ≈ 
                       
                         
                           
                             C 
                             
                               
                                 X 
                                 0 
                               
                               + 
                               
                                 Y 
                                 0 
                               
                             
                           
                           ⁢ 
                           
                             z 
                             2 
                           
                         
                         + 
                         
                           Y 
                           0 
                         
                       
                     
                     , 
                     and 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     θ 
                     d 
                   
                   ≈ 
                   
                     
                       2 
                       ⁢ 
                       Cz 
                     
                     
                       
                         X 
                         0 
                       
                       + 
                       
                         Y 
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
     Equation (34) holds for θ d &lt;&lt;1. The appearance of Y 0  in the denominator of terms in Equation (34) shows that, for a given X 0 , a more elliptical charged-particle beam profile can ameliorate the beam expansion and divergence due to space charge. The approximations in Equation (34) are useful analytical expressions for modeling the propagation of elliptical charged-particle beams. 
     The divergence of the electron beam (e.g., due to space charge and energy spread of the source) can be accounted for by performing multi-particle numerical simulations for beams with angular divergences of 0.1° and 1° relative to the z axis as shown in  FIGS. 19A-19F . The angular divergences can be modeled by introducing a corresponding Gaussian spread for the momenta of each particle in the x, y and z directions. 10 4  macro-particles are used in each simulation. The electrons interact with one another through the electromagnetic fields they produce, with Coulomb repulsion being the most significant contributor to the interaction. The results show variations of peak intensity within an order of magnitude, but no significant change to bandwidth or peak frequency: Comparing the case with 0.1° divergence ( FIG. 19B ) to the ideal case ( FIG. 19A ) for the 3.7 MeV electron beam, a decrease in peak photon intensity of ˜60% is observed. Still, the energy spread remains small (increasing from 0.25% to 0.4%) and the shift in peak frequency is negligible. 
     For the 100 eV electron beam, a 0.1° divergence ( FIG. 19E ) can cause the peak photon intensity to decrease by ˜20%, whereas the energy spread is practically unaffected. This shows that, for either regime of parameters, the scheme is still viable when a small but non-negligible energy spread exists in the electron beam. However, as observed from  FIG. 19C  and  FIG. 19F , increasing the beam divergence to 1° may cause the radiation output to deteriorate for both relativistic and non-relativistic cases, demonstrating the importance of controlling the electron beam divergence for the efficient operation of the scheme. 
     Ponderomotive Deflection of Electrons 
     In deriving Equation (27), it is assumed, first, that transverse and longitudinal electron velocity modulations are small enough that γ is approximately constant throughout the interaction and, second, that the beam centroid is displaced negligibly in the transverse direction, both of which are very good approximations in most cases of interest. Details of the derivation are already provided above, where the general problem of radiation scattered by electrons interacting with GP modes of arbitrary n (not just n&gt;&gt;1) is addressed. In addition, an expression is also derived below for the threshold beyond which our approximations break down due to ponderomotive deflection. 
     An advantage of a GP&#39;s large confinement factor in our scheme is to generate photons of relatively high energy with electrons of relatively low energy. When the relativistic mass of an electron is very small, however, the electron may be readily deflected away from the graphene surface by radiation pressure: the time-averaged ponderomotive force that pushes charged particles from regions of higher intensity to regions of lower intensity. This deflection potentially shortens the GP-electron interaction, resulting in lower output power than if the electron experienced an undeflected trajectory. 
       FIGS. 20A-20B  show ponderomotive deflection of electrons, pushing them away from the graphene surface.  FIG. 20A  shows the electric field threshold for significant ponderomotive deflection as a function of electron energy. Each red cross corresponds to a line in  FIG. 20B , where the trajectory of a 100 eV electron 1 nm away from the graphene surface (n=180) is plotted for different values of peak electric field amplitude at the graphene surface E 0s  (the value in the labels). For reference, the GP field is displayed in the background. 
     An important implication of the results in  FIGS. 20A-20B  is that for strong electric fields the distance of interaction is limited by the ponderomotive force, in addition to limitations imposed by the graphene size and the electron beam divergence. For small electron beam energies (less than a few hundred eVs), the ponderomotive force becomes the dominant factor limiting the interaction length. This practically limits the amplitudes of useful GPs in cases of low-energy electron beams. Nevertheless, the onset of significant radiation pressure for electron energies around 50 keV is already 20 GV/m, which is about the graphene breakdown field strength. This implies that the constraints imposed by the ponderomotive force are already negligible at the upper end of scanning electron microscope energies, and become even more negligible at higher electron energies (e.g., on the scale of transmission electron microscope and radiofrequency gun energies). 
     In the interest of maximizing output spectral intensity, it is desirable to have as large an E 0  as possible. However, too large an E 0  may cause the electron to significantly deviate from its intended trajectory, resulting in a smaller effective interaction duration. One way to overcome the problem of ponderomotive deflection without having to decrease the GP intensity can use a symmetric configuration of graphene-coated dielectric slabs (i.e., a slab waveguide configuration), in which the electrons are confined to the minimum of an intensity well formed by surface plasmon-polaritons above and below the electron bunch. Recent advances in creating graphene heterostructures might make this configuration desirable for a GP-base radiation source device. 
       FIGS. 21A-21C  show numerical and analytical results of the radiation spectrum.  FIG. 21A  shows numerically (circles) and analytically (solid lines) computed radiation intensities in units of photons per second per steradian per 1% bandwidth (BW) for 3.7 MeV electrons with a peak electric field amplitude of E 0s =3 GVm −1  on the graphene surface.  FIG. 21B  shows the radiation spectrum when 100 eV electrons with E 0s =0.3 GVm −1  are used.  FIG. 21C  shows the radiation spectrum when 100 eV electrons with E 0s =30 MVm −1  are used. The radiation spectra correspond to an average current of 100 μA. The electron beam is centered 5 nm from the graphene sheet and has a transverse distribution of standard deviation 10 nm. All GP parameters are the same as in  FIGS. 16A-16B . The different colors represent measurements from different angles. 
       FIGS. 22A-22C  show results corresponding to those in  FIGS. 21A-21C , but with a GP free space wavelength of λ=10 μm since most GP experiments so far have been performed at this wavelength.  FIGS. 22A-22C  show an excellent agreement between numerically and analytically computed radiation intensities in the regime for which ponderomotive scattering is negligible. The effect of ponderomotive scattering—which decreases the effective interaction length—is responsible for the discrepancy between analytical and numerical results in  FIG. 21B  and  FIG. 22B . Throughout this section, the graphene parameters correspond to a confinement factor of n=180 (obtained for E f =0.1 eV), a plasmon group velocity of 0.00184 c, and a surface conductivity of σ s =2.25×10 −8 +i4.55×10 −5  S, as obtained within the RPA. 
     Full Electromagnetic Simulation 
     This section describes full electromagnetic simulations that also include the electrons dynamics. The presented results are for two particular set of parameters that both lead to hard X-ray radiation. Both options are simulated for an electron beam going parallel to the side of a graphene sheet placed on a silicon substrate. 
       FIGS. 23A-23B  show radiation spectrum when electron energy at 2.3 MeV, λ air =2 μm, squeezing factor n=580, and doping of 0.6 eV are used.  FIG. 23A  shows a cross section plot that can emphasize the narrowness of the peak, indicating that the output emission from GP-based radiation sources is highly monochromatic.  FIG. 23B  shows that the spectrum peak is centered at 21 KeV then gradually shifts for larger angles. 
       FIGS. 24A-24B  shows a comparison of X-ray source from a single electron interacting with a graphene SPP versus a conventional scheme. The conventional scheme includes a field of the same frequency and the same peak amplitude, interacting over the same distance. In order to achieve X-ray energy of 10 KeV in both cases, it is assumed that the electrons in the conventional scheme have somehow been accelerated to 16.7 MeV. Surprisingly, even without accounting for the acceleration stage, there are additional inherent advantages of GP-based scheme. First, GP-based scheme can have lower energy consumption. The SPP is a surface wave hence a field of the same amplitude is confined to smaller regime, resulting in less total energy. Also, the electrons energy is lower since γ is smaller. Second, the output radiation in the GP-based scheme is monochromatic with the spectral width of the generated X-ray being smaller. Third, the output radiation from the GP-based scheme is also coherent because the SPP confinement might lead to self-amplified stimulated emission due to the feedback from the X-rays causing self-synchronization of the electrons. Fourth, the output radiation from the GP-based scheme has a wider angular spread. A well-known technical limit of the conventional scheme is that the X-ray emission is parallel to the electron-beam. The intensity and energy of the X-ray drop quickly at larger angles. The graphene SPP scheme creates radiation in larger angles, and even perpendicular to the electron-beam. This can considerably simplifies technical considerations in separating the X-ray beam from the electron beam. 
       FIGS. 25A-25B  show radiation spectrum when electron energy at 50 eV, λ air =2 μm, squeezing factor n=580, and doping of 0.6 eV are used.  FIG. 25A  shows a cross section plot that can emphasize the narrowness of the peak, indicating that the output emission from GP-based radiation sources is highly monochromatic.  FIG. 25B  shows that the spectrum peak is centered at 5.7 eV then gradually shifts for larger angles. 
     Frequency Down-Conversion and THz Generation 
     This section describes a frequency down-conversion scheme to generate compact, coherent, and tunable terahertz light. Demand for terahertz sources is being driven by their usefulness in many areas of science and technology, ranging from material characterization to biological analyses and imaging applications. Free-electron methods of terahertz generation are typically implemented in large accelerator installations, making compact alternatives desirable. 
     Approaches described in this section use a configuration in which light co-propagates with the electron. The phase velocity of the light can be slower than the speed of light in vacuum, which may be achieved with the cladding mode of a dielectric waveguide (e.g., cylindrical, rectangular, planar etc.) or using a surface plasmon polariton with a squeezing factor n&gt;1 (phase velocity of the SPP is then c/n). The field in the waveguide may be oscillating at optical or infrared frequencies (technically, any frequency is possible). 
       FIG. 26  shows a schematic of a system for frequency down-conversion using graphene SPP fields. The system  2600  includes a pair of graphene layers  2610   a  and  2610   b , each of which is disposed on a respective substrate. The two graphene layers are disposed against each other such that a SPP field  2601  exists within the space between the two graphene layers  2610   a  and  2610   b . An electron source (e.g., a DC or RF electron gun) delivers an electron beam  2635  into the SPP field  2601  to co-propagate with the SPP field. Since the squeezing factor of the SPP field can significantly reduce the phase velocity of light in the space between the two graphene layers  2610   a  and  2610   b , the electron beam  2635  can therefore propagate at a speed comparable to the phase velocity of light in the same space, thereby achieving velocity matching. The interaction between the electron beam  2635  and the SPP field  2601  generates the output emission  2602 , which can have a longer wavelength compared to the optical beam (not shown in  FIG. 26 ) that excites the SPP field  2601 . 
     The output frequency may be tuned by adjusting the energy of the input electron pulse. Down-converted radiation is collected in the forward direction. The on-axis output frequency v is given by:
 
 v=v   0 (1− nβ   0 )/(1−β 0 )  (35)
 
where v 0  is the frequency of the electromagnetic wave that excites the SPP field and β 0  is the initial speed of the electron in the +z direction.
 
       FIG. 27  shows the output photon energy as a function of electron kinetic energy for the co-propagating configuration, for various values of n. Initial photon energy is 1.55 eV (corresponding to a wavelength of 0.8 μm). Clearly, down-conversion is possible when the initial electron velocity closely matches the phase velocity of the co-propagating electromagnetic wave. The input electron pulse may be relativistic or non-relativistic, depending on the phase velocity of the chosen mode (i.e. it is possible to design the structure to use either relativistic or non-relativistic electrons). To achieve coherence, the electron pulse may be pre-bunched such that each bunch is of a length much smaller than the emission wavelength. Techniques that enhance emission output for the frequency up-conversion scheme in previous sections, such the using of a stack structure, may also be applied here. 
     Electrons Beam Oblique to 2D Systems 
     In previous sections, electrons are generally propagating substantially parallel to graphene layers. In contrast, this section describes the situations in which electrons are propagating at an oblique angle with respect to the graphene layers or photonic crystals. 
     The interaction of electron beams launched perpendicularly (or with some angle) onto a layered structure can have several promising applications for the creation of new sources of radiation. This type of radiation is generally referred to as transition radiation. Transition radiation is a form of electromagnetic radiation emitted when a charged particle passes through inhomogeneous media, such as a boundary between two different media. This is in contrast to Cerenkov radiation. The emitted radiation is the homogeneous difference between the two inhomogeneous solutions of Maxwell&#39;s equations of the electric and magnetic fields of the moving particle in each medium separately. In other words, since the electric field of the particle is different in each medium, the particle has to “shake off” the difference of energy when it crosses the boundary. 
     The total energy loss of a charged particle on the transition depends on its Lorentz factor γ=E/mc 2  and is mostly directed forward, peaking at an angle of the order of 1/γ relative to the particle&#39;s path. The intensity of the emitted radiation is roughly proportional to the particle&#39;s energy E. The characteristics of transition radiation make it suitable for particle discrimination, particularly of electrons and hadrons in the momentum range between 1 GeV/c and 100 GeV/c. The transition radiation photons produced by electrons have wavelengths in the X-ray range, with energies typically in the range from 5 to 15 keV. 
     Conventional transition radiation systems are normally based on bulky and expensive systems, thereby limiting the usefulness and widespread adoption. However, with new materials, new fabrication methods, and new theoretical techniques from nano-photonics, there are a lot of new possibilities to make revolutionary applications. One such application can be a table-top x-ray source based on the principle of transition radiation that can be made possible 
     Coherent Light Generation and Light-Matter Interaction in IR-Visible-UV Regime Using Resonant Transition Radiation 
     In this regime strong effects on the emitted photons can emerge from the theory of photonic crystals. A variety of different multilayer structures (isotropic photonic crystal, anisotropic photonic crystal, or metamaterials, etc.) can be used. Creating a resonance in the emitted spectrum can produce monochromatic radiation, and can create a new way to generate coherent light. In one example, using one dimensional photonic crystal angular selective behavior can be achieved. With this property, beam steering of created IR-visible-UV light can be achieved. In another example, a laser can be created from the multilayer structure, where there is no need for a gain material—the electron beam can be used instead of or in addition to gain. 
     Resonant Transition Radiation Near Plasma Frequency Regime 
     In this regime, the effective dielectric constant of materials can drop below 1 to zero, and even to negative values. This opens up many possibilities—usually considered unique to metamaterials—that can now be realized here. For example, metamaterials with refractive index less than 1 (or negative) can be used to make very thin absorbers, electrically small resonators, phase compensators, and improved electrically small antennas. These might be used for an enhanced slowing down of the electron, for controlling its velocity, energy spread, or even its wave function. Since the transition radiation spectrum is broadband, the light generated in that frequency regime can see a system that is very different from visible light in photonic crystals. This can lead to a new state of matter and many new applications, including slow light, light trapping, nanoscale resonators and possibly light cloaking. 
     X-Ray and Soft-X-Ray Generation 
     The transition radiation from a stack of very thin layers (several nanometers to several tens of nanometers) can cause an electron beam to emit x-ray. This does not require a highly relativistic electron beam. Moderately relativistic electron beams (several hundreds of KeVs to several MeVs), even with slower electrons over several tens of KeVs) can still produce x-ray in this way. Significant improvements in fabrication methods in recent years now allow for the fabrication of such stacked structures. Structures in higher dimensions (2D and 3D photonic crystals, and metallic photonic crystal) can be even more suitable for x-ray generation. The resulting radiation can be emitted at a wavelength that is close to the layer thickness divided by γ—the effect of γ may not be significant here, because it is close to 1. Still, the radiation is in the x-ray thanks to the layers being very thin. 
     In the past, the limitations on fabrication methods allowed only for thick layers, in turn requiring very energetic electron beams to achieve radiation in the x-ray regime. The possibility of making very thin layers allows X-ray generation without high energy electron beams. It is worth noting that previously, very large scale (and expensive) electron acceleration systems were needed in order to accelerate electrons to MeV or GeV energies and produce X-ray radiation. However, if electron energy can be reduced to tens or hundreds of KeVs, it would be much easier and cheaper to generate such electrons. Consequently, the system size cost for an x-ray source would be significantly reduced. 
     Multiple 2DEG Layers 
     By Placing a Graphene Sheet (or Several Sheets) in Between Each of the Layers, or by placing other metallic layers that support surface plasmons, one can increase the efficiency of the transition radiation. The result is producing higher intensity radiation. For most materials the transition radiation becomes smaller when the layer thickness is smaller than the formation length. This limit can disappear when the surface of the layer supports surface plasmons. These surface plasmons can enhance the transition radiation, so that even very thin layers (thinner than the formation length) can still cause significant transition radiation to be emitted. This can potentially reduce the size and cost of an x-ray source even more. 
     This approach can also operate with 2DEG systems on the interface between different materials other than graphene layers. There are several other scenarios where the physics of 2DEG is found. For example, the interface between BaTiO 3  and LaAlO 3 , or the interface between lanthanum aluminate (LaAlO 3 ) and strontium titanate (SrTiO 3 ) can be used as 2DEG systems. In another example, layers of ferromagnetic materials can also be used to construct 2DEG. 
     The multiple 2DEG layer structure can include a couple of tens of dielectric (or metallic) layers. A higher number of layers can generally improve the result such as increasing the output intensity and/or improving the monochromatic quality. 
     The multiple 2DEG layer structure can be further improved by adding small holes within the stack of layers. If the holes are smaller than the wavelength, they normally do not affect the emission of radiation, while the electrons can pass through them. In this way, the electrons can propagate through a longer distance in the stack structure before they slow down and stop emitting radiation. A longer penetration depth (also a longer mean free path) can allow more layers to take part in the radiation emission. 
     Cerenkov-Like Effect 
     This section describes graphene-based devices that emits radiation through a Cerenkov-like effect, induced from current flowing through the graphene sheet (suspended on dielectric or not). This approach does not require any external source of electromagnetic radiation, and is therefore highly attractive for on-chip CMOS compatible applications. 
     This approach can achieve direct coupling between electric current and SPPs in graphene. These SPP can be coupled to radiation modes in several ways, including creating defects on graphene, making a grating (1D or 2D) on graphene, making a grating (2D or 2D) from graphene (by patterning the graphene sheet), modulating the voltage applied on graphene to create a periodic refractive index that can allow tunable control of the radiation, fabricating almost any photonic crystal (any periodic dielectric structure) as the substrate of the graphene, specially designed photonic crystal that has high density of states at a particular frequency above the light cone, which can be achieved by employing one or more unique band structure properties such as van-Hove singularities, flat bands around Dirac points, or super-collimation contours. 
     To improve the efficiency of the effect, the electric current can be configured to include electrons that have the smaller velocity spread (i.e., more uniform velocity distribution). This is possible to graphene due to its Dirac cone band structure. In addition, the graphene can be doped to have high enough mobility so that the phase velocity of the graphene SPP can be lower than the velocity of the electrons. This can be seen by comparing the “squeezing factor” n from above, which has to be larger than the ratio between the speed of light and the electron velocity. A proper design of the electron current can create electrons moving at the Fermi velocity, which can be 300 times slower than the speed of light. This means that n&gt;300 can already create the desired effect. Such values of n are achievable as shown in above sections. 
     The radiation can be emitted in four possible regimes, each requiring a different kind of structure. For example, Terahertz radiation can be created without doping the graphene. Infrared radiation can be achieved by doping the graphene. Visible light can be created by high doing of graphene, while UV light can be created based on additional plasmonic range in the UV region. 
     The phenomenon of a Cerenkov-like coupling between electron current and SPPs in graphene can be the first occurrence of Cerenkov radiation from bounded electrons in nature. This is bound to lead to more attractive applications based on the same phenomenon, since it bridges the gap between photonics and electronics. 
     A related effect exists in existing methods, in which a periodic structure interacts with flowing electrons. The difference between this existing idea and the approach described herein is that the existing idea is based on a Smith-Purcell radiation, and does not use the SPP modes of the system, which can be important for an efficient process. 
     The electron beam can be sent in the air/vacuum near the graphene sample. It can be beneficial for the free electron beam to pass very close to the sample (on the order of nanometers—similar to the wavelength of the graphene SPP). The advantage of this technique is that the velocity of the electron beam can be fully controlled and does not depend on graphene properties. 
     Since the Cerenkov-like effect can directly couple DC current to light (in the form of plasmons), it can have several other applications, including measurement the distribution of velocities in the graphene, measurement the conductivity, integrating optics with electronics for on-chip photonic capabilities, feedback effects where external light (coupled to plasmons) changes the properties of the plasmon excitations to influence the current (inverse Cerenkov) that can accelerate the electrons and also change the resistivity. 
     The same approach can be implemented in other 2DEG systems or even in other plasmonic systems. Notice that even in regular plasmonic systems, the Cerenkov-like generation of plasmons was never studied nor used to any of the applications we proposed here. 
     Quantum Čerenkov Effect from Hot Carriers in Graphene 
     Achieving ultrafast conversion of electrical to optical signals at the nanoscale using plasmonics can be a long-standing goal, due to its potential to revolutionize electronics and allow ultrafast communication and signal processing. Plasmonic systems can combine the benefits of high frequencies (10 14− 10 15  Hz) with those of small spatial scales, thus avoiding the limitation of conventional photonic systems, by using the strong field confinement of plasmons. However, the realization of plasmonic sources that are electrically pumped, power efficient, and compatible with current device fabrication processes (e.g. CMOS), can be challenging. 
     This section describes that under proper conditions charge carriers propagating within graphene can efficiently excite GPs, through a 2D Čerenkov emission process. Graphene can provide a platform, on which the flow of charge alone can be sufficient for Čerenkov radiation, thereby eliminating the need for accelerated charge particles in vacuum chambers and opening up a new platform for the study of ČE and its applications, especially as a novel plasmonic source. Unlike other types of plasmon excitations, the 2D ČE can manifest as a plasmonic shock wave, analogous to the conventional ČE that creates shockwaves in a 3D medium. On a quantum mechanical level, this shockwave can be reflected in the wavefunction of a single graphene plasmon emitted from a single hot carrier. 
     The mechanism of 2D ČE can benefit from two characteristics of graphene. On the one hand, hot charge carriers moving with high velocities 
             (       up   ⁢           ⁢   to   ⁢             ⁢             ⁢   the   ⁢           ⁢   Fermi   ⁢             ⁢             ⁢   velocity   ⁢           ⁢     v   f       ≈       10   6     ⁢     m   s         )         
are considered possible, even in relatively large sheets of graphene (10 μm and more). On the other hand, plasmons in graphene can have an exceptionally slow phase velocity, down to a few hundred times slower than the speed of light. Consequently, velocity matching between charge carriers and plasmons can be possible, allowing the emission of GPs from electrical excitations (hot carriers) at very high rates. This can pave the way to new devices utilizing the ČE on the nanoscale, a prospect made even more attractive by the dynamic tunability of the Fermi level of graphene. For a wide range of parameters, the emission rate of GPs can be significantly higher than the rates previously found for photons or phonons, suggesting that taking advantage of the ČE allows near-perfect energy conversion from electrical energy to plasmons.
 
     In addition, contrary to expectations, plasmons can be created at energies above 2E f —thus exceeding energies attainable by photon emission—resulting in a plasmon spectrum that can extend from terahertz to near infrared frequencies and possibly into the visible range. 
     Furthermore, tuning the Fermi energy by external voltage can control the parameters (direction and frequency) of enhanced emission. This tunability also reveals regimes of backward GP emission, and regimes of forward GP emission with low angular spread; emphasizing the uniqueness of ČE from hot carriers flowing in graphene. 
     GP emission can also result from intraband transitions that are made possible by plasmonic losses. These kinds of transitions can become significant, and might help explain several phenomena observed in graphene devices, such as current saturation, high frequency radiation spectrum from graphene, and the black body radiation spectrum that seems to relate to extraordinary high electron temperatures. 
     Conventional studies, which generally focus on cases of classical free charge particles moving outside graphene, have revealed strong Čerenkov-related GP emission resulting from the charge particle-plasmon coupling. In contrast, this work focuses on the study of charge carriers inside graphene, as illustrated in  FIGS. 28A-28B . 
     A quantum theory of ČE in graphene is developed. Analysis of this system gives rise to a variety of novel Čerenkov-induced plasmonic phenomena. The conventional threshold of the ČE in either 2D or 3D (v&gt;v p ) may seem unattainable for charge carriers in graphene, because they are limited by the Fermi velocity v≤v f , which is smaller than the GP phase velocity v f &lt;v p , as shown by the random phase approximation calculations. However, quantum effects can come into play to enable these charge carriers to surpass the actual ČE threshold. Specifically, the actual ČE threshold for free electrons can be shifted from its classically-predicted value by the quantum recoil of electrons upon photon emission. Because of this shift, the actual ČE velocity threshold can in fact lie below the velocity of charge carriers in graphene, contrary to the conventional predictions. At the core of the modification of the quantum ČE is the linearity of the charge carrier energy-momentum relation (Dirac cone). Consequently, a careful choice of parameters (e.g. Fermi energy, hot carrier energy) allows the ČE threshold to be attained—resulting in significant enhancements and high efficiencies of energy conversion from electrical to plasmonic excitation. 
     The quantum ČE can be described as a spontaneous emission process of a charge carrier emitting into GPs, calculated by Fermi&#39;s golden rule. The matrix elements can be obtained from the light-matter interaction term in the graphene Hamiltonian, illustrated by a diagram like  FIG. 1B . To model the GPs, the random phase approximation can be used to combine with a frequency-dependent phenomenological lifetime to account for additional loss mechanisms such as optical phonons and scattering from impurities in the sample (assuming graphene mobility of μ=2000 cm 2 /Vsec). This approach can give good agreement with experimental results. 
       FIGS. 28A-28B  show a system  2800  including a graphene layer  2810  disposed on a substrate  2840 . The graphene layer  2810  includes hot carriers  2830  flowing within the graphene material. The graphene layer  2810  is in the yz plane, and the charge carrier  2830  is moving in the z direction. 
     For the case of low-loss GPs, the calculation reduces to the following integral: 
     
       
         
           
             
               
                 
                   Γ 
                   = 
                   
                     
                       
                         2 
                         ⁢ 
                         π 
                       
                       ℏ 
                     
                     ⁢ 
                     
                       
                         ∫ 
                         
                           - 
                           ∞ 
                         
                         ∞ 
                       
                       ⁢ 
                       
                         
                           
                              
                             
                               M 
                               
                                 
                                   k 
                                   i 
                                 
                                 → 
                                 
                                   
                                     k 
                                     f 
                                   
                                   + 
                                   q 
                                 
                               
                             
                              
                           
                           2 
                         
                         ⁢ 
                         
                           δ 
                           ⁡ 
                           
                             ( 
                             
                               
                                 E 
                                 
                                   k 
                                   i 
                                 
                               
                               - 
                               
                                 ℏω 
                                 ⁡ 
                                 
                                   ( 
                                   q 
                                   ) 
                                 
                               
                               - 
                               
                                 E 
                                 
                                   k 
                                   f 
                                 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           
                             
                               d 
                               2 
                             
                             ⁢ 
                             q 
                           
                           
                             
                               
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                                 ) 
                               
                               2 
                             
                             / 
                             A 
                           
                         
                         ⁢ 
                         
                           
                             
                               d 
                               2 
                             
                             ⁢ 
                             
                               k 
                               f 
                             
                           
                           
                             
                               
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                                 ) 
                               
                               2 
                             
                             / 
                             A 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
             
               
                 
                   
                     M 
                     
                       
                         k 
                         i 
                       
                       → 
                       
                         
                           k 
                           f 
                         
                         + 
                         q 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           q 
                           e 
                         
                         ⁡ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             π 
                           
                           ) 
                         
                       
                       2 
                     
                     ⁢ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             q 
                             y 
                           
                           + 
                           
                             k 
                             fy 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       δ 
                       ⁡ 
                       
                         ( 
                         
                           
                             k 
                             iz 
                           
                           - 
                           
                             q 
                             z 
                           
                           - 
                           
                             k 
                             fz 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       v 
                       f 
                     
                     ⁢ 
                     
                       
                         
                           
                             ℏ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             q 
                           
                           
                             
                               ɛ 
                               0 
                             
                             ⁢ 
                             
                               
                                 ω 
                                 ~ 
                               
                               ⁡ 
                               
                                 ( 
                                 q 
                                 ) 
                               
                             
                             ⁢ 
                             
                               A 
                               3 
                             
                           
                         
                       
                       · 
                       
                         [ 
                         SP 
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
     Where M k     i     →k     f     +q  is the matrix element, A is the surface area used for normalization, q e  is the electric charge, ε 0  is the vacuum permittivity, [SP] is the spinor-polarization coupling term, and {tilde over (ω)}(q) is the GP dispersion-based energy normalization term ({tilde over (ω)}(q)= ϵ   r ω·v p /v g , using the group velocity v g =∂ω/∂q). 
     The GP momentum q=(q y , q z ) satisfies ω 2 /v p   2 =q y   2 +q z   2 , with the phase velocity v p =v p (ω) or v p (q) obtained from the plasmon dispersion relation as v p =ω/q. The momenta of the incoming (outgoing) charge carrier k i =(k iy , k iz ) (k f =(k fy , k fz )) correspond to energies E k     i    (E k     f   ) according to the conical momentum-energy relation E k   2 =   2 v f   2 (k y   2 +k z   2 ). The charge velocity is v=E k /| k|, which equals a constant (v f ). The only approximation in Equations (36) and (37) comes from the standard assumption of high GP confinement (free space wavelength/GP wavelength&gt;&gt;1). Substituting Equation (36) into (37) obtain (denoting E i =E k     i   ): 
     
       
         
           
             
               
                 
                   Γ 
                   = 
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       ∞ 
                     
                     ⁢ 
                     
                       
                         
                           α 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           c 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ℏ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               v 
                               g 
                             
                             ⁡ 
                             
                               ( 
                               q 
                               ) 
                             
                           
                         
                         
                           
                             
                               ϵ 
                               _ 
                             
                             r 
                           
                           ⁢ 
                           
                             
                               
                                 v 
                                 p 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 q 
                                 ) 
                               
                             
                             / 
                             
                               v 
                               f 
                               2 
                             
                           
                         
                       
                       ⁢ 
                       δ 
                       ⁢ 
                       
                         ( 
                         
                           
                             q 
                             y 
                           
                           + 
                           
                             k 
                             fy 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         δ 
                         ⁡ 
                         
                           ( 
                           
                             
                               k 
                               iz 
                             
                             - 
                             
                               q 
                               z 
                             
                             - 
                             
                               k 
                               fz 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         δ 
                         ⁡ 
                         
                           ( 
                           
                             
                               E 
                               i 
                             
                             - 
                             
                               ℏω 
                               ⁡ 
                               
                                 ( 
                                 q 
                                 ) 
                               
                             
                             - 
                             
                               E 
                               
                                 k 
                                 f 
                               
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                            
                           SP 
                            
                         
                         2 
                       
                       ⁢ 
                       
                         d 
                         2 
                       
                       ⁢ 
                       
                         qd 
                         2 
                       
                       ⁢ 
                       
                         k 
                         f 
                       
                     
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
     
     Where 
             α   ⁡     (     ≈     1   137       )           
is the fine structure constant, c is the speed of light, and  ϵ   r  is the relative substrate permittivity obtained by averaging the permittivity on both sides of the graphene. Assume  ϵ   r =2.5 for all the figures. Because material dispersion can be neglected, all spectral features can be uniquely attributed to the GP dispersion and its interaction with charge carriers and not to any frequency dependence of the dielectrics.
 
     It can be further defined that the angle φ for the outgoing charge and θ for the GP, both relative to the z axis, which is the direction of the incoming charge. This notation allows simplification of the spinor-polarization coupling term [SP] for charge carriers inside graphene to |SP| 2 =cos 2 (θ−φ/2) or |SP| 2 =sin 2 (θ−φ/2) for intraband or interband transitions respectively. The delta functions in Equation (38) can restrict the emission to two angles θ=±θ Č  (a clear signature of the ČE), and so we simplify the rate of emission to: 
     
       
         
           
             
               
                 
                   
                     cos 
                     ⁡ 
                     
                       ( 
                       
                         θ 
                         
                           C 
                           ⋓ 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         v 
                         p 
                       
                       
                         v 
                         f 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         1 
                         - 
                         
                           
                             ℏω 
                             
                               2 
                               ⁢ 
                               
                                 E 
                                 i 
                               
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               1 
                               - 
                               
                                 
                                   v 
                                   f 
                                   2 
                                 
                                 
                                   v 
                                   p 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   
                     39 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     Γ 
                     ω 
                   
                   = 
                   
                     
                       
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ac 
                         
                         
                           
                             v 
                             f 
                           
                           ⁢ 
                           
                             
                               ϵ 
                               _ 
                             
                             r 
                           
                         
                       
                       ⁢ 
                       
                         
                            
                           
                             1 
                             - 
                             
                               
                                 ℏω 
                                 
                                   2 
                                   ⁢ 
                                   
                                     E 
                                     i 
                                   
                                 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     
                                       
                                         v 
                                         f 
                                       
                                       
                                         v 
                                         p 
                                       
                                     
                                     ⁢ 
                                     
                                       cos 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           θ 
                                           
                                             C 
                                             ⋓ 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                            
                         
                         
                            
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 θ 
                                 
                                   C 
                                   ⋓ 
                                 
                               
                               ) 
                             
                           
                            
                         
                       
                     
                     = 
                     
                       
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ac 
                         
                         
                           
                             v 
                             f 
                           
                           ⁢ 
                           
                             
                               ϵ 
                               _ 
                             
                             r 
                           
                         
                       
                       ⁢ 
                       
                          
                         
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 θ 
                                 
                                   C 
                                   ⋓ 
                                 
                               
                               ) 
                             
                           
                           
                             1 
                             - 
                             
                               
                                 v 
                                 p 
                                 2 
                               
                               / 
                               
                                 v 
                                 f 
                                 2 
                               
                             
                           
                         
                          
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     39 
                     ⁢ 
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
     By setting  →0 in the above expressions, one can recover the classical 2D ČE, including the Čerenkov angle cos(θ Č )=v p /v, that can also be obtained from a purely classical electromagnetic calculation. However, while charge particles outside of graphene satisfy  ω&lt;&lt;E i , making the classical approximation almost always exact, the charges flowing inside graphene can have much lower energies because they are massless. Consequently, the introduced   terms in the ČE expression modifies the conventional velocity threshold significantly, allowing ČE to occur for lower charge velocities. e.g., while the conventional ČE requires charge velocity above the GP phase velocity (v&gt;v p ), Equation (39a) allows ČE below it, and specifically requires the velocity of charge carriers in graphene (v=v f ) to reside between 
               v   p     &gt;     v   f     &gt;       v   p     ⁢            1   -       2   ⁢           ⁢     E   i         ℏ   ⁢           ⁢   ω              .             
Physically, the latter case involves interband transitions made possible when graphene is properly doped: when the charge carriers are hot electrons (holes) interband ČE requires negatively (positively) doped graphene.
 
       FIGS. 29A-29D  and  FIGS. 30A-30D  show interband ČE that indeed occurs for charge velocities below the conventional velocity threshold. 
       FIG. 29A  illustrate possible transitions, including interband transition and intraband transition in graphene energy diagrams.  FIG. 29B  shows mapping of GP emission rate as a function of frequency and angle. Most of the GP emission around the dashed blue curves that are exactly found by the Čerenkov angle.  FIG. 29C  shows spectrum of the ČE GP emission process, with the red regime marking the area of high losses, the vertical dotted red line dividing between interband to intraband transitions, and the thick orange line marking the spectral cutoff due to the Fermi sea beyond which all states are occupied.  FIG. 29D  shows explanations of the GP emission with the quantum ČE. The red curve shows the GP phase velocity, with its thickness illustrating the GP loss. The blue regime shows the range of allowed velocities according to the quantum ČE. Enhanced GP emission occurs in the frequencies for which the red curve crosses the blue regime, either directly or due to the curve thickness. All figures are presented in normalized units except for the angle shown in degrees. 
       FIGS. 30A-30D  also illustrate GP emission from hot carriers. Caption same as  FIG. 2 . The green dots in  FIG. 30B  show the GPs can be coupled out, as light, with the size illustrating the strength of the coupling. 
       FIGS. 31A-31D  illustrate GP emission from hot carriers, in which most of the emission occurs in the forward direction with a relatively low angular spread. The green dot shows that GPs a particular frequency can be coupled out as light. 
     The inequalities can be satisfied in two spectral windows simultaneously for the same charge carrier, due to the frequency dependence of the GP phase velocity (shown by the intersection of the red curve with the blue regime in  FIG. 29D ). Moreover, part of the radiation (or even most of it, as in  FIGS. 29A-29D ) can be emitted backward, which is considered impossible for ČE in conventional materials. 
     Several spectral cutoffs appear in  FIGS. 29C, 30C, and 31C , as seen by the range of non-vanishing blue spectrum. These can be found by substituting θ Č =0 in Equation (39a), leading to  ω cutoff =2E i /(1±v f /v p ), exactly matching the points where the red curve in  FIGS. 29D, 30D, and 31D  crosses the border of the blue regime. The upper most frequency cutoff marked by the thick orange line in  FIGS. 29-31  occurs at  ω=E i +E f  due to the interband transition being limited by the Fermi sea of excited states. This implies that GP emission from electrical excitation can be more energetic than photon emission from a similar process (that is limited already by  ω≲2E f ). Finite temperature will broaden all cutoffs by the expected Fermi-Dirac distribution. However, for most frequencies, the GP losses are a more significant source of broadening. 
     To incorporate the GP losses (as we do in all the figures), the matrix elements calculation can be modified by including the imaginary part of the GP wavevector q I =q I (ω), derived independently for each point of the GP dispersion curve. This is equivalent to replacing the delta functions in Equation (38) by Lorentzians with 1/γ width, defining γ(ω)=q R (ω)/q I (ω). The calculation can be done partly analytically yielding: 
     
       
         
           
             
               
                 
                   
                     Γ 
                     
                       ω 
                       , 
                       θ 
                     
                   
                   = 
                   
                     
                       ac 
                       
                         
                           π 
                           2 
                         
                         ⁢ 
                         
                           
                             ϵ 
                             _ 
                           
                           r 
                         
                         ⁢ 
                         
                           
                             v 
                             p 
                           
                           ⁡ 
                           
                             ( 
                             ω 
                             ) 
                           
                         
                       
                     
                     ⁢ 
                     
                        
                       
                         
                           
                             E 
                             i 
                           
                           
                             ℏ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ω 
                           
                         
                         - 
                         1 
                       
                        
                     
                     ⁢ 
                     
                       
                         ∫ 
                         0 
                         
                           2 
                           ⁢ 
                           π 
                         
                       
                       ⁢ 
                       
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         φ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           { 
                           
                             
                               
                                 
                                   
                                     
                                       
                                         cos 
                                         2 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           θ 
                                           - 
                                           
                                             φ 
                                             / 
                                             2 
                                           
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     intraband 
                                     ⁢ 
                                     
                                       
                                           
                                       
                                       ⁢ 
                                       
                                           
                                       
                                     
                                     ⁢ 
                                     transition 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         sin 
                                         2 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           θ 
                                           - 
                                           
                                             φ 
                                             / 
                                             2 
                                           
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
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     The immediate effect of the GP losses can be the broadening of the spectral features, as shown in  FIGS. 29C, 30C, and 31C . Still, the complete analytic theory of Equations (37) and (38) can matches very well with the exact graphene ČE (e.g., regimes of enhanced emission agree with Equation (39a), as marked in  FIGS. 29B, 30B, and 31B  by blue dashed curves). The presence of GP loss also opens up a new regime of quasi-ČE that takes place when the charge velocity is very close to the Čerenkov threshold but does not exceed it. The addition of Lorentzian broadening then closes the gap, creating significant non-zero matrix elements that can lead to intraband GP emission ( FIGS. 31A-31D ). This GP emission occurs even for hot electrons (holes) in positively (negatively) doped graphene, with the only change in  FIGS. 31A-31D  being that the upper frequency cutoff is instead shifted to  ω≤E i −E f  (eliminating all interband transitions). The dip in the spectrum at the boundary between interband and intraband transitions ( FIG. 31C ) follows from the charge carriers density of states being zero at the tip of the Dirac cone. 
     The interband ČE in  FIGS. 31A-31D  shows the possibility of emission of relatively high frequency GPs, even reaching near-infrared and visible frequencies. These are interband transitions as in  FIGS. 29-30  thus limited by  ω≤E i +E f . This limit can get to a few eVs because E i  is controlled externally by the mechanism creating the hot carriers (e.g., p-n junction, tunneling current in a heterostructure, STM tip, ballistic transport in graphene with high drain-source voltage, photoexcitation). The existence of GPs can be at near-infrared frequencies. The only fundamental limitation can be the energy at which the graphene dispersion ceases to be conical (˜1 eV from the Dirac point). Even then, equations presented here are only modified by changing the dispersion relations of the charge carrier and the GP, and therefore the graphene ČE should appear for E i  as high as ˜3 eV. The equations here are still valid since they are written for a general dispersion relation, with v p (ω) and γ(ω) as parameters, thus the basic predictions of the equations and the ČE features we describe will continue to hold regardless of the precise plasmon dispersion. For example, an alternative way of calculating GP dispersion, giving larger GP phase velocities at high frequencies—this will lead to more efficient GP emission, as well as another intraband regime that can occur without being mediated by the GP loss. 
     There exist several possible avenues for the observation of the quantum ČE in GPs, having to do with schemes for exciting hot carriers. For example, apart from photoexcitation, hot carriers have been excited from tunneling current in a heterostructure, and by a biased STM tip, therefore, GPs with the spectral features presented here ( FIGS. 29C, 30C, and 31C ) should be achievable in all these systems. 
     In case the hot carriers are directional, measurement of the GP Čerenkov angle (e.g.  FIGS. 29B, 30B, and 31B ) should also be possible. This might be achieved by strong drain-source voltage applied on a graphene p-n junction, or in other graphene devices showing ballistic transport. Another intriguing approach could be exciting the hot carriers and measuring the generated ČE with the Photon-Induced Near-Field Electron Microscopy, which might allow the visualization of the temporal dynamics of the Čerenkov emission. This approach can be especially exciting since the temporal dynamics of the ČE is expected to appear in the form of a plasmonic shockwave (as the conventional ČE appears as a shockwave of light). 
     Hot carriers generated from a tunneling current or p-n junction may have a wide energy distribution (instead of a single E i ). The ČE spectrum corresponding to an arbitrary hot carrier excitation energy distribution is readily computed by integrating over a weighted distribution of ČE spectra for monoenergetic hot carriers. The conversion efficiency remains high even when the carriers energy distribution is broad, as implied by the high ČE efficiencies for the representative values of E i  studied here ( FIGS. 29-31  all show rates on the order of Γ˜1). This high conversion efficiency over a broad range of E i  owes itself to the low phase velocity and high confinement of graphene plasmons over a wide frequency range. 
     The ČE emission of GPs can be coupled out as free-space photons by creating a grating or nanoribbons—fabricated in the graphene, in the substrate, or in a layer above it—with two arbitrarily-chosen examples marked by the green dots in  FIGS. 30B and 31B . Careful design of the coupling mechanism can restrict the emission to pre-defined frequencies and angles, with further optimization needed for efficient coupling. This clearly indicates that the GP emission, although usually considered as merely a virtual process, can be in fact completely real in some regimes, with the very tangible consequences of light emission in terahertz, infrared or possibly visible frequencies. Such novel sources of light could have promising applications due to graphene&#39;s dynamic tunability and small footprint (due to the small scale of GPs). Moreover, near perfect conversion efficiency of electrical energy into photonic energy might be achievable due to the ČE emission rate dominating all other scattering processes. In addition, unlike plasmonic materials such as silver and gold, graphene can be CMOS compatible. 
     The hot carrier lifetime due to GP emission in doped graphene is defined by the inverse of the total rate of GP emission, and can therefore be exceptionally short (down to a few fs). Such short lifetimes are in general agreement with previous research on the subject that have shown electron-electron scattering as the dominant cooling process of hot carriers, unless hot carriers of relatively high energies (E i ≈2E f  and above) are involved. In this latter case, one can expect single-particle excitations to prevail over the contribution of the plasmonic resonances. This is also in agreement with the fact that plasmons with high energies and momenta (in the electron-hole continuum, pink areas in  FIGS. 29-31 ) are very lossy. Additional factors that keep the ČE from attaining near-perfect conversion efficiency include other scattering processes like acoustic and optical phonon scattering. Due to the relatively long lifetime from acoustic phonon scattering (hundreds of fs to several ps), however, any deterioration due to this effect is not likely to be significant. Scattering by optical phonons can be more significant for hot carriers above 0.2 eV, but its contribution can be still about an order of magnitude smaller in our regime of interest. 
     The high rates of GP emission also conform to research of the reverse process—of plasmons enhancing and controlling the emission of hot carriers—that is also found to be particularly strong in graphene. This might reveal new relations between ČE to other novel ideas of graphene-based radiation sources that are based on different physical principles. 
     It is also worth noting that Čerenkov-like plasmon excitations from hot carriers can be found in other condensed matter systems such as a 2D electron gas at the interface of semiconductors. Long before the discovery of graphene, such systems have demonstrated very high Fermi velocities (even higher than graphene&#39;s), while also supporting meV plasmons that can have slow phase velocities, partly due to the higher refractive indices possible in such low frequencies. The ČE coupling, therefore, can also be found in materials other than graphene. In many cases, the coupling of hot carriers to bulk plasmons is even considered as part of the self-energy of the carriers, although the plasmons are then considered as virtual particles in the process. Nonetheless, graphene can offer opportunities where the Čerenkov velocity matching can occur at relatively high frequencies, with plasmons that have relatively low losses. These differences can make the efficiency of the graphene ČE very high. 
     CONCLUSION 
     While various inventive embodiments have been described and illustrated herein, those of ordinary skill in the art will readily envision a variety of other means and/or structures for performing the function and/or obtaining the results and/or one or more of the advantages described herein, and each of such variations and/or modifications is deemed to be within the scope of the inventive embodiments described herein. More generally, those skilled in the art will readily appreciate that all parameters, dimensions, materials, and configurations described herein are meant to be exemplary and that the actual parameters, dimensions, materials, and/or configurations will depend upon the specific application or applications for which the inventive teachings is/are used. Those skilled in the art will recognize, or be able to ascertain using no more than routine experimentation, many equivalents to the specific inventive embodiments described herein. It is, therefore, to be understood that the foregoing embodiments are presented by way of example only and that, within the scope of the appended claims and equivalents thereto, inventive embodiments may be practiced otherwise than as specifically described and claimed. Inventive embodiments of the present disclosure are directed to each individual feature, system, article, material, kit, and/or method described herein. In addition, any combination of two or more such features, systems, articles, materials, kits, and/or methods, if such features, systems, articles, materials, kits, and/or methods are not mutually inconsistent, is included within the inventive scope of the present disclosure. 
     The above-described embodiments can be implemented in any of numerous ways. For example, embodiments of designing and making the technology disclosed herein may be implemented using hardware, software or a combination thereof. When implemented in software, the software code can be executed on any suitable processor or collection of processors, whether provided in a single computer or distributed among multiple computers. 
     Further, it should be appreciated that a computer may be embodied in any of a number of forms, such as a rack-mounted computer, a desktop computer, a laptop computer, or a tablet computer. Additionally, a computer may be embedded in a device not generally regarded as a computer but with suitable processing capabilities, including a Personal Digital Assistant (PDA), a smart phone or any other suitable portable or fixed electronic device. 
     Also, a computer may have one or more input and output devices. These devices can be used, among other things, to present a user interface. Examples of output devices that can be used to provide a user interface include printers or display screens for visual presentation of output and speakers or other sound generating devices for audible presentation of output. Examples of input devices that can be used for a user interface include keyboards, and pointing devices, such as mice, touch pads, and digitizing tablets. As another example, a computer may receive input information through speech recognition or in other audible format. 
     Such computers may be interconnected by one or more networks in any suitable form, including a local area network or a wide area network, such as an enterprise network, and intelligent network (IN) or the Internet. Such networks may be based on any suitable technology and may operate according to any suitable protocol and may include wireless networks, wired networks or fiber optic networks. 
     The various methods or processes (outlined herein may be coded as software that is executable on one or more processors that employ any one of a variety of operating systems or platforms. Additionally, such software may be written using any of a number of suitable programming languages and/or programming or scripting tools, and also may be compiled as executable machine language code or intermediate code that is executed on a framework or virtual machine. 
     In this respect, various inventive concepts may be embodied as a computer readable storage medium (or multiple computer readable storage media) (e.g., a computer memory, one or more floppy discs, compact discs, optical discs, magnetic tapes, flash memories, circuit configurations in Field Programmable Gate Arrays or other semiconductor devices, or other non-transitory medium or tangible computer storage medium) encoded with one or more programs that, when executed on one or more computers or other processors, perform methods that implement the various embodiments of the invention discussed above. The computer readable medium or media can be transportable, such that the program or programs stored thereon can be loaded onto one or more different computers or other processors to implement various aspects of the present invention as discussed above. 
     The terms “program” or “software” are used herein in a generic sense to refer to any type of computer code or set of computer-executable instructions that can be employed to program a computer or other processor to implement various aspects of embodiments as discussed above. Additionally, it should be appreciated that according to one aspect, one or more computer programs that when executed perform methods of the present invention need not reside on a single computer or processor, but may be distributed in a modular fashion amongst a number of different computers or processors to implement various aspects of the present invention. 
     Computer-executable instructions may be in many forms, such as program modules, executed by one or more computers or other devices. Generally, program modules include routines, programs, objects, components, data structures, etc. that perform particular tasks or implement particular abstract data types. Typically the functionality of the program modules may be combined or distributed as desired in various embodiments. 
     Also, data structures may be stored in computer-readable media in any suitable form. For simplicity of illustration, data structures may be shown to have fields that are related through location in the data structure. Such relationships may likewise be achieved by assigning storage for the fields with locations in a computer-readable medium that convey relationship between the fields. However, any suitable mechanism may be used to establish a relationship between information in fields of a data structure, including through the use of pointers, tags or other mechanisms that establish relationship between data elements. 
     Also, various inventive concepts may be embodied as one or more methods, of which an example has been provided. The acts performed as part of the method may be ordered in any suitable way. Accordingly, embodiments may be constructed in which acts are performed in an order different than illustrated, which may include performing some acts simultaneously, even though shown as sequential acts in illustrative embodiments. 
     All definitions, as defined and used herein, should be understood to control over dictionary definitions, definitions in documents incorporated by reference, and/or ordinary meanings of the defined terms. 
     The indefinite articles “a” and “an,” as used herein in the specification and in the claims, unless clearly indicated to the contrary, should be understood to mean “at least one.” 
     The phrase “and/or,” as used herein in the specification and in the claims, should be understood to mean “either or both” of the elements so conjoined, i.e., elements that are conjunctively present in some cases and disjunctively present in other cases. Multiple elements listed with “and/or” should be construed in the same fashion, i.e., “one or more” of the elements so conjoined. Other elements may optionally be present other than the elements specifically identified by the “and/or” clause, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, a reference to “A and/or B”, when used in conjunction with open-ended language such as “comprising” can refer, in one embodiment, to A only (optionally including elements other than B); in another embodiment, to B only (optionally including elements other than A); in yet another embodiment, to both A and B (optionally including other elements); etc. 
     As used herein in the specification and in the claims, “or” should be understood to have the same meaning as “and/or” as defined above. For example, when separating items in a list, “or” or “and/or” shall be interpreted as being inclusive, i.e., the inclusion of at least one, but also including more than one, of a number or list of elements, and, optionally, additional unlisted items. Only terms clearly indicated to the contrary, such as “only one of” or “exactly one of,” or, when used in the claims, “consisting of,” will refer to the inclusion of exactly one element of a number or list of elements. In general, the term “or” as used herein shall only be interpreted as indicating exclusive alternatives (i.e., “one or the other but not both”) when preceded by terms of exclusivity, such as “either,” “one of,” “only one of,” or “exactly one of” “Consisting essentially of,” when used in the claims, shall have its ordinary meaning as used in the field of patent law. 
     As used herein in the specification and in the claims, the phrase “at least one,” in reference to a list of one or more elements, should be understood to mean at least one element selected from any one or more of the elements in the list of elements, but not necessarily including at least one of each and every element specifically listed within the list of elements and not excluding any combinations of elements in the list of elements. This definition also allows that elements may optionally be present other than the elements specifically identified within the list of elements to which the phrase “at least one” refers, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, “at least one of A and B” (or, equivalently, “at least one of A or B,” or, equivalently “at least one of A and/or B”) can refer, in one embodiment, to at least one, optionally including more than one, A, with no B present (and optionally including elements other than B); in another embodiment, to at least one, optionally including more than one, B, with no A present (and optionally including elements other than A); in yet another embodiment, to at least one, optionally including more than one, A, and at least one, optionally including more than one, B (and optionally including other elements); etc. 
     In the claims, as well as in the specification above, all transitional phrases such as “comprising,” “including,” “carrying,” “having,” “containing,” “involving,” “holding,” “composed of,” and the like are to be understood to be open-ended, i.e., to mean including but not limited to. Only the transitional phrases “consisting of” and “consisting essentially of” shall be closed or semi-closed transitional phrases, respectively, as set forth in the United States Patent Office Manual of Patent Examining Procedures, Section 2111.03.