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expert
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What is the Robinson theorem?
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The sum of the damping partition numbers equals 4.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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Calculate the damping times in the horizontal $( x )$ and vertical $( y )$ phase spaces, as well as in the energy/time phase space! I.10.7.22 Large Hadron Collider The Large Hadron Collider at CERN collides protons in a storage ring with $2 7 \\mathrm { k m }$ circumference. Assuming that synchrotron radiation is only emitted in the dipoles in the arcs, which have a bending radius of $2 9 0 0 \\mathrm { m }$ , calculate the following parameters for protons with an energy of $7 \\mathrm { T e V }$ : – The energy loss per turn, per particle, – The critical photon energy of synchrotron radiation, – The vertical damping time. How does these numbers compare to LEP (assuming the same circumference and dipole bending radius) at an electron energy of $1 0 0 \\mathrm { G e V ? }$ I.10.7.23 Preparation for an upgrade Petra-III is a $2 . 3 \\ \\mathrm { k m }$ circumference light source at $6 { \\mathrm { G e V } }$ and $1 \\ \\mathrm { n m }$ horizontal emittance, located at DESY in Hamburg. An upgrade based on multi-bend achromats will decrease the emittance to $1 0 \\ \\mathrm { p m }$ . Before the upgrade, the DESY team wants to test instrumentation for the new ring at low emittance.
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What is the Smith–Purcell effect equation relating wavelength λ to grating period a, electron velocity β, emission angle θ, and diffraction order m?
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? = (a/m) (1/? - cos ?), where a is grating period, m is diffraction order, ? = v/c, and ? is emission angle.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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We implemented the moments model [10] for calculation of QE for emission from semiconductors [7]. We extended the moments model to include light interference effects: $$ Q E = \\frac \\stackrel { \\infty } { \\bigcup } \\stackrel { \\bigcup } { \\int } \\stackrel { d E } { \\overbrace { E _ { g } + E _ { a } } } \\stackrel { d E } { \\overbrace { \\sqrt { E _ { a } / ( E - E _ { g } ) } } } \\stackrel { \\overbrace { d } } { \\overbrace { 2 \\int _ { E _ { g } } ^ { \\infty } d E ( E - E _ { g } ) p \\int _ { 0 } ^ { 1 } d u } } . $$ Here, $A ( \\omega )$ is the fraction of light absorbed in the photocathode film, $E _ { g }$ is the energy gap, $E _ { a }$ is the electron affinity, ‚Ñèùúî is the photon energy, $u \\ \\equiv \\ \\cos { \\theta }$ , $\\theta$ is the angle of a photo-excited electron relative to the normal of the emission surface. We define the functions $p = p ( E , \\omega )$ and $\\boldsymbol { G } = \\boldsymbol { G } ( \\omega , E , \\boldsymbol { u } )$ below. The origin of the energy axis is at the valence band maximum. The function
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What is the Smith–Purcell effect equation relating wavelength λ to grating period a, electron velocity β, emission angle θ, and diffraction order m?
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? = (a/m) (1/? - cos ?), where a is grating period, m is diffraction order, ? = v/c, and ? is emission angle.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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We implemented the moments model [4] for calculation of QE for emission from semiconductors [5]. We extended the moments model to include light interference e"ects: $$ Q E = A ( \\omega ) \\frac { \\int _ { E _ { a } } ^ { \\hbar \\omega - E _ { g } } d E E \\int _ { \\sqrt { \\frac { E _ { a } } { E } } } ^ { 1 } d u D ( E u ^ { 2 } ) u f ( \\omega , E , u ) } { 2 \\int _ { 0 } ^ { \\hbar \\omega - E _ { g } } d E E \\int _ { 0 } ^ { 1 } d u } . $$ Here, $A ( \\omega )$ is the fraction of light absorbed in the photocathode film, $E _ { g }$ is the energy gap, $E _ { a }$ is the electron a!nity, $\\hbar \\omega$ is the photon energy, $u \\equiv \\cos \\theta$ , $\\theta$ is the angle of a photo-excited electron relative to the normal of the emission surface, $D ( E u ^ { 2 } )$ is the probability of emission of an electron moving towards the emission surface with parallel kinetic energy $E u ^ { 2 }$ .
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What is the Smith–Purcell effect equation relating wavelength λ to grating period a, electron velocity β, emission angle θ, and diffraction order m?
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? = (a/m) (1/? - cos ?), where a is grating period, m is diffraction order, ? = v/c, and ? is emission angle.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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$$ \\frac { d \\langle \\theta ^ { 2 } \\rangle } { d s } = \\int _ { \\theta _ { \\mathrm { m i n } } } ^ { \\theta _ { \\mathrm { m a x } } } \\theta ^ { 2 } \\frac { d \\sigma } { d \\Omega } \\frac { N _ { \\mathrm { A } } } { A } \\rho d \\Omega , $$ where $d \\Omega \\approx \\theta d \\theta d \\phi$ can be used in the small-angle approximation. The angular cut-offs in Eq. (2) are $$ \\theta _ { \\mathrm { m i n } } = \\frac { 2 . 6 6 \\cdot 1 0 ^ { - 6 } Z ^ { 1 / 3 } } { p [ \\mathrm { G e V / c } ] } , \\theta _ { \\mathrm { m a x } } = \\frac { 0 . 1 4 } { A ^ { 1 / 3 } p [ \\mathrm { G e V / c } ] } , $$ and come from the Thomas-Fermi model [9]. Later, V. Highland [10] compared in his work Eq. (2) with the modified Molière theory from H. Bethe [2] and found inconsistencies for lower $Z$ materials. He adjusted Eq. (2) with a fitting parameter and an additional logarithmic term. G. Lynch and O. Dahl [11] fine-tuned Highland’s idea and found the final analytical expression
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What is the Smith–Purcell effect equation relating wavelength λ to grating period a, electron velocity β, emission angle θ, and diffraction order m?
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? = (a/m) (1/? - cos ?), where a is grating period, m is diffraction order, ? = v/c, and ? is emission angle.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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Eq. (3) represent the electron motion equation, Poisson’s equation, and continuity equation respectively. For simplicity we assume the initial energy modulation is a cosine modulation, so the initial condition is $$ n ( z , 0 ) = n _ { 0 } , \\quad \\eta ( z , 0 ) = \\Delta \\eta \\cos ( k z ) . $$ where $k = 2 \\pi / \\lambda _ { k }$ represents the modulation wave number. In order to solve these coupled equations, we introduce coordinate transformation and solve it in the Lagrangian coordinate system $$ t = \\tau , \\quad z = \\xi + \\int _ { 0 } ^ { \\tau } \\frac { c \\eta ( \\tau , \\xi ) } { \\gamma ^ { 2 } } d \\tau , $$ and the solutions of these equations are $$ n ( \\xi , t ) = \\frac { n _ { 0 } } { 1 - \\frac { \\omega _ { k } } { \\omega _ { b } \\gamma ^ { 2 } } \\Delta \\eta \\sin ( k \\xi ) \\sin ( \\omega _ { b } t ) } ,
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What is the Smith–Purcell effect equation relating wavelength λ to grating period a, electron velocity β, emission angle θ, and diffraction order m?
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? = (a/m) (1/? - cos ?), where a is grating period, m is diffraction order, ? = v/c, and ? is emission angle.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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The electron beam undergoes twice energy modulations in two undulators, and the laser wavenumbers are $k _ { 1 }$ and $k _ { 2 }$ , respectively. Only the dispersion parameter $R _ { 5 6 }$ is considered in dispersion (for the terahertz band, the collective effect of this process can be ignored). The energy modulation and density modulation can be approximately expressed as, $p _ { 1 } = p { + } A _ { 1 }$ sin $\\zeta + A _ { 2 } \\sin ( K \\zeta + \\phi )$ and $\\zeta _ { 1 } = \\zeta + B p _ { 1 }$ , where $\\zeta = k _ { 1 } z$ dimensionless longitudinal position, $K = k _ { 2 } / k _ { 1 }$ , $\\phi$ is the phase difference of two lasers, $B = k _ { 1 } R _ { 5 6 } \\sigma _ { E } / E _ { 0 }$ is the normalized dispersion intensity, $A _ { 1 , 2 } = \\Delta E _ { 1 , 2 } / \\sigma _ { E }$ is the normalized energy modulation amplitude. At the exit of the dispersion section, the phase space distribution becomes,
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What is the Smith–Purcell effect equation relating wavelength λ to grating period a, electron velocity β, emission angle θ, and diffraction order m?
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? = (a/m) (1/? - cos ?), where a is grating period, m is diffraction order, ? = v/c, and ? is emission angle.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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$$ where $Q _ { i }$ is dependent on the instantaneous momentum deviation $\\frac { \\Delta p _ { i } } { p _ { 0 } }$ , the nominal tune is denoted by $Q _ { 0 }$ , and $Q ^ { \\prime } , Q ^ { \\prime \\prime }$ are respectively first and second order chromaticities. We shall also assume that the momentum deviation is a consequence of a harmonic synchrotron motion, and as follows from Ref. [5], can be expressed as: $$ \\frac { \\Delta p _ { i } } { p _ { 0 } } = - \\frac { \\widehat { \\tau _ { i } } \\Omega _ { s _ { i } } } { \\eta } \\cos ( \\Omega _ { s _ { i } } t + \\varphi _ { s _ { i } } ) , $$ where $\\widehat { \\tau _ { i } }$ , $\\boldsymbol { \\Omega } _ { s _ { i } }$ and $\\varphi _ { s _ { i } }$ are respectively the amplitude, the angularbfrequency and the phase of the synchrotron motion and $\\eta$ is the slip factor, assumed to be positive above the transition energy.
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What is the Smith–Purcell effect equation relating wavelength λ to grating period a, electron velocity β, emission angle θ, and diffraction order m?
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? = (a/m) (1/? - cos ?), where a is grating period, m is diffraction order, ? = v/c, and ? is emission angle.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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$$ I ( \\Delta \\mathbf { x } ) = I _ { 0 } \\cdot \\left\\{ 1 + S ( \\theta ) \\cdot | \\mu ( \\Delta \\mathbf { x } ) | \\cdot \\cos \\left( \\frac { k } { 2 z } r ^ { 2 } \\right) \\right\\} , $$ where $\\Delta \\mathbf { x }$ denotes coordinates with respect to the center of the interference pattern, $r = | \\Delta \\mathbf { x } |$ , and $S ( \\theta )$ is the particle form factor describing how the particle scatters light as a function of the observation angle $\\theta = r / z$ . Equation (6) closely resembles Eq. (4) in form, as they both describe a system of interference fringes modulated by the radiation CCF and the form factor of the di!racting objects. However, there are subtle, albeit important di!erences. First, at variance with the Young’s scheme, interference fringes are circular. Their periodicity progressively decreases away from the center and depends on geometrical factors only. It can be shown that fringes with spatial frequency $\\mathbf { q }$ are localized at transverse displacements $\\Delta \\mathbf { x }$ with respect to center of the pattern according to the following spatial scaling [8, 10, 12]:
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What is the Smith–Purcell effect equation relating wavelength λ to grating period a, electron velocity β, emission angle θ, and diffraction order m?
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? = (a/m) (1/? - cos ?), where a is grating period, m is diffraction order, ? = v/c, and ? is emission angle.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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$$ \\frac { \\epsilon _ { x , I D s } } { \\epsilon _ { x } } = \\frac { 1 } { 1 + \\displaystyle \\frac { I _ { 2 , I D s } } { I _ { 2 , d i p . } } } $$ From Equation 1 we can also express the energy loss per turn as a function of $I _ { 2 , d i p }$ : $$ U _ { 0 } = P _ { 0 } / I \\approx \\frac { C _ { \\gamma } } { 2 \\pi } E ^ { 4 } I _ { 2 , d i p } $$ with $P _ { 0 }$ the power radiated in the nominal lattice (without IDs), $r _ { 0 }$ the classical electron radius and $C _ { \\gamma } = \\frac { 4 \\pi } { 3 } \\frac { r _ { 0 } } { ( m _ { 0 } c ^ { 2 } ) ^ { 3 } } ,$ Ôºö Now combining Equations 3, 8, and 10, the approximated expression for the emittance variation becomes :
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What is the Smith–Purcell effect equation relating wavelength λ to grating period a, electron velocity β, emission angle θ, and diffraction order m?
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? = (a/m) (1/? - cos ?), where a is grating period, m is diffraction order, ? = v/c, and ? is emission angle.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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INTRODUCTION Due to advancements in the Dielectric Laser Acceleration (DLA) technique [1], and in grating-based deflection structures [2], there is interest in an entirely grating-based compact particle accelerator. In order for this to become a reality, there are requirements for suitable diagnostics devices that are capable of single-shot bunch length measurements and beam position monitoring. Manufacturing these diagnostics from similar dielectric gratings to DLA structures is highly favourable, since the production process is reasonably fast and inexpensive. These devices would also be very compact, with dimensions in the order of millimeters or centimeters depending on the operation parameters of the beam. Bunch length diagnostic techniques generally involve either correlating the longitudinal coordinate to a transverse component of the beam through beam deflection or streaking [3], which is destructive to the bunch, or inducing the bunch to radiate and measuring the resulting spectrum, which leaves the bunch intact. As a charged particle bunch passes through an unpowered dielectric grating, it will decelerate and radiate at a wavelength equal to the period of the grating, through the radiation mechanism known as SmithPurcell Radiation (SPR) [4]. Based on this mechanism, a bunch length diagnostic has been proposed [5]. When the bunch length is smaller than the grating period, the radiation will contain significant spectral content and increase in power; by varying the longitudinal periodicity of a grating and measuring the relative radiation output, a value for the length of the bunch can be obtained. This device could also function as a Beam Position Monitor (BPM), using a double-sided grating and comparing the photon yield on each side.
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What is the Smith–Purcell effect equation relating wavelength λ to grating period a, electron velocity β, emission angle θ, and diffraction order m?
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? = (a/m) (1/? - cos ?), where a is grating period, m is diffraction order, ? = v/c, and ? is emission angle.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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The dispersion relation correspond to the free energy parabola with $m ^ { * } = m _ { e }$ . We suspect this dispersion relation arises due to the interaction of the emitted electron with the laser/plasmonic fields leading to momentum transfer in the presence of nanostructured surface non-uniformities at the center of the ASP. Interaction of emitted electron with the plasmonic EM fileds could also impart inward radial momentum to the emitted electrons potentially influencing dispersion relations. Further investigations and theoretical modelling are underway to determine the exact cause of such a dispersion and develop a better understanding of photoemission from plasmonic ASP. ACKNOWLEDGEMENT This work is supported by the NSF Center for Bright Beams under award PHY-1549132 and Department of Energy Office of Science under awards DE-SC0021092, and DE-SC0021213. C.M.P. acknowledges support from the US DOE SCGSR program. J.M was partially supported by U.S Department of Energy, Grant No. DE-SC0020144.
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What is the Smith–Purcell effect equation relating wavelength λ to grating period a, electron velocity β, emission angle θ, and diffraction order m?
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? = (a/m) (1/? - cos ?), where a is grating period, m is diffraction order, ? = v/c, and ? is emission angle.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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$$ $$ \\propto \\frac { 1 } { \\beta ^ { 2 } } \\Bigg \\{ \\ln ( 1 / \\kappa _ { \\rho } d ) \\mathrm { f o r } \\kappa _ { \\rho } d \\ll 1 , $$ The limits of equations (4), (5a) and (5b) are completely general; they set the maximum photon emission and energy loss of an electron beam coupled to an arbitrary photonic environment in either the non-retarded or retarded regimes, given only the beam properties and material composition. The key factors that determine maximal radiation are identified: intrinsic material loss (represented by $\\mathrm { I m } \\chi \\dot { } ,$ ), electron velocity $\\beta$ and impact parameter $\\kappa _ { \\rho } d$ . The metric $| \\chi | ^ { 2 } / \\mathrm { I m } \\chi$ reflects the influence of the material choice, which depends sensitively on the radiation wavelength (Fig. 1b). The electron velocity $\\beta$ also appears implicitly in the impact parameter $\\kappa _ { \\rho } d = k d / \\beta \\gamma ,$ showing that the relevant length scale is set by the relativistic velocity of the electron. The impact parameter $\\kappa _ { \\rho } d$ reflects the influence of the Lorentz contraction $d / \\gamma ;$ a well-known feature of both electron radiation and acceleration20,26,30.
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What is the Smith–Purcell effect equation relating wavelength λ to grating period a, electron velocity β, emission angle θ, and diffraction order m?
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? = (a/m) (1/? - cos ?), where a is grating period, m is diffraction order, ? = v/c, and ? is emission angle.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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Data availability. The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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What is the Smith–Purcell effect equation relating wavelength λ to grating period a, electron velocity β, emission angle θ, and diffraction order m?
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? = (a/m) (1/? - cos ?), where a is grating period, m is diffraction order, ? = v/c, and ? is emission angle.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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free-space optical elements, enabling simultaneous measurement of the spectrum and of the spatial radiation pattern. The SEM used for the experiment was a JEOL JSM-6010LA. Its energy spread at the gun exit was in the range 1.5 to $2 . 5 \\mathrm { e V }$ for the range of acceleration voltages considered in this paper. The SEM was operated in spot mode, which we controlled precisely to align the beam so that it passes tangentially to the surface near the desired area of the sample. A Nikon TU Plan Fluor 10x objective with a numerical aperture of 0.30 was used to collect light from the area of interest. The monochrome image of the radiation was taken using a Hamamatsu CCD (chargecoupled device). The spectrometer used was an Action SP-2360-2300i with a lownoise Princeton Instruments Pixis 400 CCD. A 1D grating (Au-covered single-crystalline Si: periodicity, ${ 1 4 0 } \\mathrm { n m } ;$ filling factor, $5 0 \\%$ ; patterned Si thickness, $5 3 \\pm 1 . 5 \\mathrm { { n m } }$ ; Au thickness $4 4 \\pm 1 . 5 \\mathrm { n m }$ was used as the sample in our experiment. The original nanopatterned linear silicon stamp was obtained from LightSmyth Technologies and coated using an electron beam evaporator with a $2 \\mathrm { n m T i }$ adhesion layer and $4 0 \\mathrm { n m }$ of Au at $1 0 ^ { - 7 }$ torr. The sample was mounted inside the SEM chamber to enable the alignment of free electrons to pass in close proximity to the stamps. The emitted light was coupled out of the SEM chamber to a spectrometer, while a camera was used to image the surface of the sample.
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What is the Smith–Purcell effect equation relating wavelength λ to grating period a, electron velocity β, emission angle θ, and diffraction order m?
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? = (a/m) (1/? - cos ?), where a is grating period, m is diffraction order, ? = v/c, and ? is emission angle.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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$$ P _ { \\tau } ( \\omega ) \\leq \\frac { \\varepsilon _ { 0 } \\omega \\xi _ { \\tau } } { 2 } \\int _ { V } \\mathbf { F } _ { \\mathrm { i n } } ^ { \\dagger } \\overline { { \\overline { { \\chi } } } } ^ { \\dagger } ( \\mathrm { I m } \\overline { { \\overline { { \\chi } } } } ) ^ { - 1 } \\overline { { \\overline { { \\chi } } } } \\mathbf { F } _ { \\mathrm { i n c } } \\mathrm { d } V $$ where $\\tau \\in \\{ \\mathrm { r a d } , \\log \\}$ and $\\xi _ { \\tau }$ accounts for a variable radiative efficiency $\\eta$ (defined as the ratio of radiative to total energy loss): $\\xi _ { \\mathrm { l o s s } } = 1$ and $\\xi _ { \\mathrm { r a d } } = \\eta ( 1 - \\eta ) \\le 1 / 4$ . Hereafter, we consider isotropic and non-magnetic materials (and thus a scalar susceptibility $\\chi$ ), but the generalizations to anisotropic and/or magnetic media are straightforward.
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What is the Smith–Purcell effect equation relating wavelength λ to grating period a, electron velocity β, emission angle θ, and diffraction order m?
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? = (a/m) (1/? - cos ?), where a is grating period, m is diffraction order, ? = v/c, and ? is emission angle.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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In closing, we have theoretically derived and experimentally probed a universal upper limit to the energy loss and photon emission from free electrons. The limit depends crucially on the impact parameter $\\kappa _ { \\rho } d$ , but not on any other detail of the geometry. Hence, our limit applies even to the most complex metamaterials and metasurfaces, given only their constituents. Surprisingly, in the near field, slow electrons promise stronger radiation than relativistic ones. The limit predicts a divergent radiation rate as the material loss rate goes to zero, and we show that BIC resonances enable such staggering enhancements. This is relevant for the generation of coherent Smith–Purcell radiation14,34,35. The long lifetime, spectral selectivity and large field enhancement near a BIC can strongly bunch electrons, allowing them to radiate coherently at the same desired frequency, potentially enabling low-threshold Smith–Purcell freeelectron lasers. The combination of this mechanism and the optimal velocity prediction reveals prospects of low-voltage yet high-power free-electron radiation sources. In addition, our findings demonstrate a simple guiding principle to maximize the signal-to-noise ratio for electron energy-loss spectroscopy through an optimal choice of electron velocity, enabling improved spectral resolution. The predicted slow-electron-efficient regime still calls for direct experimental validation. We suggest that field-emitter-integrated free-electron devices (for example, ref. 10) are ideal to confirm the prediction due to the achievable small electron–structure separation and high electron beam quality at relatively large currents. Alternatively, the microwave or terahertz frequencies could be suitable spectral ranges for verifying the slow-electron-efficient regime, where the subwavelength separation requirement is more achievable.
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What is the Smith–Purcell effect equation relating wavelength λ to grating period a, electron velocity β, emission angle θ, and diffraction order m?
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? = (a/m) (1/? - cos ?), where a is grating period, m is diffraction order, ? = v/c, and ? is emission angle.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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The limit in equation (4) can be further simplified by removing the shape dependence of $V$ , since the integrand is positive and is thus bounded above by the same integral for any enclosing structure. A scatterer separated from the electron by a minimum distance $d$ can be enclosed within a larger concentric hollow cylinder sector of inner radius $d$ and outer radius $\\infty$ ‚Äã. For such a sector (height $L$ and opening azimuthal angle $\\scriptstyle \\psi \\in ( 0 , 2 \\pi ] )$ , equation (4) can be further simplified, leading to a general closed-form shape-independent limit (see Supplementary Section 2) that highlights the pivotal role of the impact parameter $\\kappa _ { \\rho } d$ : $$ T _ { \\tau } ( \\omega ) \\leq \\frac { \\alpha \\xi _ { \\tau } } { 2 \\pi c } \\frac { | \\chi | ^ { 2 } } { \\mathrm { I m } \\chi } \\frac { L \\psi } { \\beta ^ { 2 } } [ ( \\kappa _ { \\rho } d ) K _ { 0 } ( \\kappa _ { \\rho } d ) K _ { 1 } ( \\kappa _ { \\rho } d ) ]
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What is the Smith–Purcell effect equation relating wavelength λ to grating period a, electron velocity β, emission angle θ, and diffraction order m?
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? = (a/m) (1/? - cos ?), where a is grating period, m is diffraction order, ? = v/c, and ? is emission angle.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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The grating limit (equation (6)) exhibits the same asymptotics as equations (5a) and (5b), thereby reinforcing the optimal-velocity predictions of Fig. 1c. The $( \\beta , k \\dot { d } )$ dependence of $\\mathcal { G }$ (see Fig. 2a) shows that slow (fast) electrons maximize Smith–Purcell radiation in the small (large) separation regime. We verify the limit predictions by comparison with numerical simulations: at small separations (Fig. 2b), radiation and energy loss peak at velocity $\\beta \\approx 0 . 1 5$ , consistent with the limit maximum; at large separations (Fig. 2c), both the limit and the numerical results grow monotonically with $\\beta$ . The derived upper limit also applies to Cherenkov and transition radiation, as well as bulk loss in electron energy-loss spectroscopy. For these scenarios where electrons enter material bulk, a subtlety arises for the field divergence along the electron’s trajectory $( \\rho = 0$ in equation (2)) within a potentially lossy medium. This divergence, however, can be regularized by introducing natural, system-specific momentum cutoffs26, which then directly permits the application of our theory (see Supplementary Section 6). Meanwhile, there exist additional competing interaction processes (for example, electrons colliding with individual atoms). However, they typically occur at much smaller length scales.
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What is the Smith–Purcell effect equation relating wavelength λ to grating period a, electron velocity β, emission angle θ, and diffraction order m?
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? = (a/m) (1/? - cos ?), where a is grating period, m is diffraction order, ? = v/c, and ? is emission angle.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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Author contributions Y.Y., O.D.M., I.K. and M.S. conceived the project. Y.Y. developed the analytical models and numerical calculations. A.M. prepared the sample under study. Y.Y., A.M., C.R.-C., S.E.K. and I.K. performed the experiment. Y.Y., T.C. and O.D.M. analysed the asymptotics and bulk loss of the limit. S.G.J., J.D.J., O.D.M., I.K. and M.S. supervised the project. Y.Y. wrote the manuscript with input from all authors. Competing interests The authors declare no competing interests. Additional information Supplementary information is available for this paper at https://doi.org/10.1038/ s41567-018-0180-2. Reprints and permissions information is available at www.nature.com/reprints. Correspondence and requests for materials should be addressed to Y.Y. or O.D.M. or I.K. Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Methods Fourier transform convention. Throughout the paper, we adopt the following Fourier transform conventions $$ f ( \\omega ) \\triangleq \\int f ( t ) \\mathrm { e } ^ { i \\omega t } \\mathrm { d } t , f ( t ) \\triangleq \\frac { 1 } { 2 \\pi } \\int f ( \\omega ) \\mathrm { e } ^ { - i \\omega t } \\mathrm { d } \\omega $$ $$
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What is the Smith–Purcell effect equation relating wavelength λ to grating period a, electron velocity β, emission angle θ, and diffraction order m?
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? = (a/m) (1/? - cos ?), where a is grating period, m is diffraction order, ? = v/c, and ? is emission angle.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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Next, we specialize in the canonical Smith–Purcell set-up illustrated in Fig. 1e inset. This set-up warrants a particularly close study, given its prominent historical and practical role in free-electron radiation. Aside from the shape-independent limit (equations (5a) and (5b)), we can find a sharper limit (in per unit length for periodic structure) specifically for Smith–Purcell radiation using rectangular gratings of filling factor $\\varLambda$ (see Supplementary Section 3) $$ \\frac { \\mathrm { d } { \\varGamma } _ { \\tau } ( \\omega ) } { \\mathrm { d } x } \\leq \\frac { \\alpha \\xi _ { \\tau } } { 2 \\pi c } \\frac { | \\boldsymbol \\chi | ^ { 2 } } { \\mathrm { I m } \\chi } \\varLambda \\mathcal { G } ( \\beta , k d ) $$ The function ${ \\mathcal { G } } ( { \\boldsymbol { \\beta } } , k d )$ is an azimuthal integral (see Supplementary Section 3) over the Meijer G-function $G _ { 1 , 3 } ^ { 3 , 0 }$ (ref. $^ { 3 2 } )$ that arises in the radial integration of the modified Bessel functions $K _ { n }$ . We emphasize that equation (6) is a specific case of equation (4) for grating structures without any approximations and thus can be readily generalized to multi-material scenarios (see Supplementary equation (37)).
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augmentation
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NO
| 0
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expert
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What is the bunch distribution at SHINE?
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Two horn current profile
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Summary
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Beam_performance_of_the_SHINE_dechirper.pdf
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This paper begins by reviewing the dechirper parameters for a small metallic pipe. The wakefield effects are studied with an ultra-short electron bunch in the Shanghai high repetition rate XFEL and extreme light facility (SHINE). Then, the process in dechirper is studied analytically and verified by numerical simulations. The longitudinal wakefield generated by the corrugated structure was adopted as the dechirper was calculated and simulated for SHINE. The following chapter focuses on the transverse wakefield, discussing emittance dilution effects on different arrangements of two orthogonally oriented dechirpers, mainly for the case of an on-axis beam. We also propose using ‘fourdechirpers’ as a novel approach for controlling the beam emittance dilution effect during dechirping and compare it with the conventional scheme. We then close with a brief conclusion. 2 Background The layout of the SHINE linac and the beam parameters along the beamline are shown in Fig. 1. The electron beam generated by the VHF gun is accelerated to $1 0 0 \\ \\mathrm { M e V }$ before entering the laser heater, to suppress potential subsequent microbunching by increasing the uncorrelated energy spread of the beam. The overall linac system consists of four accelerating sections (L1, L2, L3 and L4) and two bunch compressors (BC1 and BC2). A higher harmonic linearizer (HL), which works in the deceleration phase, is also placed upstream of BC1 for the purpose of linear compression.
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1
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Yes
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expert
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What is the bunch distribution at SHINE?
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Two horn current profile
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Summary
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Beam_performance_of_the_SHINE_dechirper.pdf
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$$ \\epsilon _ { \\mathrm { f } } / \\epsilon _ { 0 } = ( \\langle \\gamma _ { \\mathrm { f } } \\rangle \\langle \\beta _ { \\mathrm { f } } \\rangle - \\langle \\alpha _ { \\mathrm { f } } \\rangle ^ { 2 } ) ^ { 1 / 2 } , $$ where the subscripts f o represent the final (original) situation. Then, the other plane is also suitable. In contrast to the conventional FODO design, we utilized the two- and four-dechirpers, including the quadrupole wake generated in the corrugated structure. This destroys the periodicity in the Twiss parameters and causes the changes in the projected emittance in the final. The projected emittance growth for different magnet lengths $L$ and magnet strengths $K$ of the quadrupole magnets is demonstrated in Fig. 10. However, the minimum projected emittances in the two- and four-dechirpers are $0 . 0 1 7 5 \\%$ and $0 . 0 0 2 3 8 \\%$ , respectively. These values are comparatively small. The projected four-dechirper emittance still performs better. In the case of the symmetric feature in the FODO structure, the conclusion reached for the $x$ -direction does not apply to the $y$ -direction. The optimized working point (red dot) is selected for the quadrupole magnets, where the function $\\beta _ { x }$ is minimal and shows the best performance in terms of the emittance growth. The above calculation confirms the validity of dividing the dechirpers into four.
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1
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Yes
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expert
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What is the bunch distribution at SHINE?
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Two horn current profile
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Summary
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Beam_performance_of_the_SHINE_dechirper.pdf
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$$ After calculating the inverse Fourier transformation, the distance s between the test and driving particles yields the longitudinal wake at the origin of $s = 0 ^ { + }$ , according to $w _ { \\mathrm { l } } \\sim e ^ { \\sqrt { s / s _ { 0 1 } } }$ . The relationship between the longitudinal point wake and the distance about the test charge behind the driving charge is expressed as Eq. (8) [19]. $$ w _ { 1 } ( s ) = - \\frac { \\pi Z _ { 0 } c } { 1 6 a ^ { 2 } } e ^ { - \\sqrt { s / s _ { 0 1 } } } . $$ The wakefield of the short bunch is obtained by convoluting the wake with the bunch shape $\\lambda ( s )$ . For a pencil beam, the original wake $w _ { 0 }$ on the axis becomes [19] $$ w _ { 0 } = \\frac { \\pi Z _ { 0 } c } { 1 6 a ^ { 2 } } . $$ Equation (8) shows that the required dechirper length $L$ is determined by the half-gap for a given dechirper strength. The half-gap $a = 1 \\mathrm { m m }$ was chosen as the baseline for the following reasons. On the one hand, it ensures that a sufficiently large proportion of the beam is included in the clear region for beam propagation, an essential requirement for controlling beam loss in a superconducting linac. On the other hands, the aperture size is constrained by the transverse emittance dilution effect, which is discussed in the following section. In [26], it is assumed that the corrugation dimensions are no greater than the gap size, i.e., $t , p \\leq a$ . This ensures that the structure is ‘steeply corrugated,’ such that short-range wakes can be neglected.
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1
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Yes
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expert
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What is the bunch distribution at SHINE?
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Two horn current profile
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Summary
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Beam_performance_of_the_SHINE_dechirper.pdf
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As described in Eq. (567), the distance factor is affected by the dechirper parameters, especially by the ratio $t / p$ . The wakefields induced by the Gaussian bunch with different $t /$ $p$ values are shown in Fig. 3. Over the initial $2 0 ~ { \\mu \\mathrm { m } }$ , all the induced wakefields have the same slope coefficient and differ mainly in terms of the maximal chirp. As $t / p$ increases, the wakefield decreases progressively until it settles when $t / p$ reaches 0.5. Therefore, $t / p = 0 . 5$ is selected for SHINE as the dechirper parameter for which deviations are tolerable. Equation (1) is suitable only for dechirpers with a flat geometry, with corrugations in the $y -$ and $z$ -directions and with $x$ extending to infinity horizontally. However, in practice, it assumes the presence of a resistive wall in the $x$ - direction, as defined by the width $w$ . The wake calculated in the time domain by the wakefield solver ECHO2D [22] is adopted to simulate the actual situation. This is expressed as a sum of discrete modes, with odd mode numbers m corresponding to the horizontal mode wavenumbers $k x =$ $m \\pi / \\nu$ $( m = 1 , 3 , 5 . . . )$ . To obtain the exact simulated wakefield, it has been verified that $w \\gg a$ should be satisfied, and that more than one mode contribute to the impedance of the structure [17].
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2
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Yes
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expert
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What is the bunch distribution at SHINE?
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Two horn current profile
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Summary
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Beam_performance_of_the_SHINE_dechirper.pdf
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The $\\beta$ functions for both models are plotted in Fig. 9. In [28], the emittance growth caused by the quadrupole wakefield is fully compensated only if $\\beta _ { x } = \\beta _ { y }$ . In practice, however, the beta functions always fluctuate, and the beam suffers from the residual quadrupole wakefield. For a period FODO cell, the difference between the maximum and minimum $\\beta$ values is given by Eq. (18) [29], where $K$ and $L$ denote the quadrupole strength and length separately, and $l$ the length for each separated dechirper (5 and $2 . 5 \\mathrm { ~ m ~ }$ for the two- and four-dechirpers, respectively). $$ \\beta _ { \\mathrm { m a x } } - \\beta _ { \\mathrm { m i n } } = { 4 l } / { \\sqrt { 6 4 - K ^ { 2 } L ^ { 2 } l ^ { 2 } } } . $$ Figure 11 compares the $\\beta$ functions for two- and fourdechirpers by scanning the magnet strength $K$ and the magnet length $L$ . The minimum differences are $2 . 5 0 \\mathrm { ~ m ~ }$ and 1.25 for the two- and four-dechirpers, respectively. This implies a better compensation of the quadrupole wakefieldinduced emittance growth in the four-dechirper schemes. In practical engineering applications, four-dechirpers are considered appropriate to simplify the system.
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1
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Yes
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expert
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What is the bunch distribution at SHINE?
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Two horn current profile
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Summary
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Beam_performance_of_the_SHINE_dechirper.pdf
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$$ where $f ( q ) = n / d$ , and with $$ \\begin{array} { l } { n = q [ \\cosh [ q ( 2 a - y - y _ { 0 } ) ] - 2 \\cosh [ q ( y - y _ { 0 } ) ] } \\\\ { \\qquad + \\cosh [ q ( 2 a + y + y _ { 0 } ) ] ] } \\\\ { \\qquad - i k \\zeta [ \\sinh [ q ( 2 a - y - y _ { 0 } ) ] + \\sinh [ q ( 2 a + y + y _ { 0 } ) ] ] , } \\end{array} $$ $$ d = [ q \\mathrm { s e c h } ( q a ) - i k \\zeta \\mathrm { c s c h } ( q a ) ] [ q \\mathrm { c s c h } ( q a ) - i k \\zeta \\mathrm { s e c h } ( q a ) ] . $$ We use $( x _ { 0 } , \\ y _ { 0 } )$ and $( x , y )$ to represent the driving and testing particles, respectively. In Eq. (1), the surface impedance is written as $\\zeta$ , which is related to the wavenumber $k$ . In Ref. [18], the impedance at large $k$ is expanded, keeping terms to leading order $1 / k$ and to the next order $1 / k ^ { 3 / 2 }$ . Then, we compare this equation to the round case for a disk-loaded structure in a round geometry, with the same expression in parameters as rectangular plates. The longitudinal impedances at large $q$ for the round and rectangular plates are given by [18–21]
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augmentation
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Yes
| 0
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expert
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What is the bunch distribution at SHINE?
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Two horn current profile
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Summary
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Beam_performance_of_the_SHINE_dechirper.pdf
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$$ \\begin{array} { l } { \\displaystyle { Z _ { \\mathrm { { r } } } ( k ) = \\frac { 4 i } { k c a ^ { 2 } } \\left[ 1 + \\frac { 1 + i } { \\sqrt { 2 k S _ { 0 \\mathrm { { r } } } } } \\right] ^ { - 1 } , } } \\\\ { \\displaystyle { Z _ { \\mathrm { { l } } } ( k ) = \\frac { 4 i } { k c a ^ { 2 } } \\left[ 1 + \\frac { 1 + i } { \\sqrt { 2 k S _ { 0 \\mathrm { { l } } } } } \\right] ^ { - 1 } . } } \\end{array} $$ The distance scale factors $S _ { \\mathrm { 0 r } }$ and $S _ { 0 1 }$ for the round and flat are strongly influenced by the dechirper parameters: $$ \\begin{array} { l } { { \\displaystyle S _ { 0 \\mathrm { r } } = \\frac { a ^ { 2 } t } { 2 \\pi \\alpha ^ { 2 } p ^ { 2 } } , } } \\\\ { { \\displaystyle \\alpha ( x ) = 1 - 0 . 4 6 5 \\sqrt { ( x ) } - 0 . 0 7 0 ( x ) , } } \\\\ { { \\displaystyle S _ { 0 1 } = 9 S _ { 0 \\mathrm { r } } / 4 . } } \\end{array}
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augmentation
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Yes
| 0
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expert
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What is the bunch distribution at SHINE?
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Two horn current profile
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Summary
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Beam_performance_of_the_SHINE_dechirper.pdf
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$$ Expanding the surface impedance $\\zeta ~ [ 1 8 ]$ in the first two orders, the short-range vertical dipole and quadrupole wakes near the axis are given by [19] $$ \\begin{array} { r l } & { w _ { y \\mathrm { d } } \\approx \\displaystyle \\frac { Z _ { 0 } \\mathrm { c } \\pi ^ { 3 } } { 6 4 a ^ { 4 } } { \\mathit { s } } _ { 0 \\mathrm { d } } \\bigg [ 1 - \\big ( 1 + \\sqrt { { s } / { s } _ { 0 \\mathrm { d } } } \\big ) e ^ { \\sqrt { { s } / { s } _ { 0 \\mathrm { d } } } } \\bigg ] , } \\\\ & { w _ { y \\mathrm { q } } \\approx \\displaystyle \\frac { Z _ { 0 } \\mathrm { c } \\pi ^ { 3 } } { 6 4 a ^ { 4 } } { \\mathit { s } } _ { 0 \\mathrm { q } } \\bigg [ 1 - \\big ( 1 + \\sqrt { { s } / { s } _ { 0 \\mathrm { q } } } \\big ) e ^ { \\sqrt { { s } / { s } _ { 0 \\mathrm { q } } } } \\bigg ] , } \\\\ & { { \\mathit { s } } _ { 0 \\mathrm { d } } = s _ { 0 \\mathrm { r } } \\bigg ( \\displaystyle \\frac { 1 5 } { 1 4 } \\bigg ) ^ { 2 } , { \\mathit { s } } _ { 0 \\mathrm { q } } = s _ { 0 \\mathrm { r } } \\bigg ( \\displaystyle \\frac { 1 5 } { 1 6 } \\bigg ) ^ { 2 } . } \\end{array}
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augmentation
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Yes
| 0
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expert
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What is the bunch distribution at SHINE?
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Two horn current profile
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Summary
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Beam_performance_of_the_SHINE_dechirper.pdf
|
We next simply consider the quadrupole wake, where the beam is on-axis $( \\mathrm { y } _ { \\mathrm { c } } = 0 )$ . The transfer matrices for the focusing and defocusing quadrupole are given in Eq. (16), where $L$ is the length of the corrugated structure [25]. $$ \\begin{array} { r } { \\boldsymbol { R } _ { \\mathrm { f } } = \\left[ \\begin{array} { c c } { \\cos k _ { \\mathrm { q } } L } & { \\frac { \\sin k _ { \\mathrm { q } } L } { k _ { \\mathrm { q } } } } \\\\ { - k _ { \\mathrm { q } } \\sin k _ { \\mathrm { q } } L } & { \\cos k _ { \\mathrm { q } } L } \\end{array} \\right] , \\boldsymbol { R } _ { \\mathrm { d } } = \\left[ \\begin{array} { c c } { \\cosh k _ { \\mathrm { q } } L } & { \\frac { \\sin k _ { \\mathrm { q } } L } { k _ { \\mathrm { q } } } } \\\\ { k _ { \\mathrm { q } } \\sinh k _ { \\mathrm { q } } L } & { \\cosh k _ { \\mathrm { q } } L } \\end{array} \\right] . } \\end{array}
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augmentation
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Yes
| 0
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IPAC
|
What is the highest energy gain observed in a plasma wakefield accelerator?
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44 GeV
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Reasoning
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Blumenfeld_et_al._-_2007_-_Energy_doubling_of_42_GeV_electrons_in_a_metre-scale_plasma_wakefield_accelerator.pdf
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INTRODUCTION Particle accelerators are among the grandest machines of the twentieth century because of their contributions to medicine, materials development, renewable energy, and the many fields of high-energy physics and life sciences, with roughly a third of all Nobel Prizes in physics being related to the use or advancements of particle accelerators. However, conventional accelerators are costly due to the size required to accelerate electrons to high energy. Dielectric breakdown in the RF cavities of conventional linear accelerators limits the accelerating gradient to $E _ { z } < 5 0 \\mathrm { M e V / m }$ [1]. Circular accelerators also face major drawbacks for accelerating electrons, since energy loss due to synchrotron radiation scales with the relativistic factor to the fourth power $( \\gamma _ { b } ^ { 4 } )$ . Both limitations are overcome by increasing the size of the machine to reach higher energies. Plasma wakefield acceleration (PWFA) possesses much higher accelerating gradients with some experiments demonstrating $E _ { z } > 1 0 0 \\mathrm { G e V } / \\mathrm { m } \\ [ 2$ , 3]. This suggests that PWFA can decrease the size of accelerators from the kilometer scale to the meter scale.
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augmentation
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NO
| 0
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IPAC
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What is the highest energy gain observed in a plasma wakefield accelerator?
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44 GeV
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Reasoning
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Blumenfeld_et_al._-_2007_-_Energy_doubling_of_42_GeV_electrons_in_a_metre-scale_plasma_wakefield_accelerator.pdf
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The Livingston plot, shown in Figure 1, illustrates how the progress in achieving the energy frontier has been enabled by the history of invention in accelerator science and technology. One can clearly see that over several decades, there has been an exponential growth in the maximum attained energy. But the exponential growth in maximum achieved energy was made possible by the development of different accelerator technologies (for example Electrostatics, Cyclotrons, Linacs, Synchrotrons, Colliders). As often occurs in any technological field, new accelerating technologies often replaced each other once the previous technology had reached its full potential and saturates its evolution curve [1]. In more recent decades, represented by the LHC collider, the exponential energy growth has started slowing down again. This suggests that existing acceleration technologies have reached their maximum potential and further advancements would require the development of new accelerator concepts, possibly based on more compact and cost-effective methods. Promising emerging techniques, such as laser-driven and beam-driven plasma acceleration, have the potential to reestablish the exponential trend in the energy growth depicted by the Livingston plot. Plasma wakefield accelerator relies on a coherent charge density oscillation in a plasma to provide the accelerating force. The plasma oscillation is driven by an externally injected short pulse, which can be either a laser (LWFA [3]) or an electron beam (PWFA [4]), which blows outward the electrons in an ionized gas (plasma), leaving behind a region of positive charge, as shown in Figure 2. Along the axis where the beam propagates, the electric field causes a trailing pulse of electrons injected near the rear of the bubble to feel a very strong forward acceleration. Gradient up to $1 0 0 \\mathrm { ~ G V / m }$ have been observed in several experiments [5]. Difficulties in the plasma scheme arise from the small scales involved (sub-mm transverse diameter), required micrometer tolerances and stability which may cause beam quality degradation with respect to conventional accelerators. But in recent time the plasma generated beam quality has advanced sufficiently to reach the requirements for driving a Free Electron Laser (FEL). There have been several reports of pilot free-electron lasing in plasma-based accelerators: one relying on LWFA by a team in China [6] and one relying on PWFA by the EuPRAXIA team in Frascati [7,8], Italy. Another experiment run by a French and German team has also recently confirmed seeding of the FEL process in a LWFA plasma accelerator [9].
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augmentation
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NO
| 0
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IPAC
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What is the highest energy gain observed in a plasma wakefield accelerator?
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44 GeV
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Reasoning
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Blumenfeld_et_al._-_2007_-_Energy_doubling_of_42_GeV_electrons_in_a_metre-scale_plasma_wakefield_accelerator.pdf
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A PLASMA TARGET FOR HIGH-EFFICIENCY ACCELERATION Using the same input bunch parameters and a half-metre plasma with $n _ { e } \\sim \\bar { 8 \\times 1 0 ^ { 1 5 } } \\mathrm { c m } ^ { - 3 }$ there is the prospect of achieving energy gains of at least $0 . 5 \\mathrm { G e V }$ . With this motivation, a discharge plasma cell was designed around a $5 0 0 \\mathrm { m m }$ sapphire tube with a $1 . 7 \\mathrm { m m }$ inner diameter and $4 . 3 \\mathrm { m m }$ outer diameter, and it was characterised in DESY’s ADVANCE Lab [23]. An image of a plasma formed in this capillary is shown in Fig. 3 (a). A mixture of $9 7 \\%$ Ar, $3 \\%$ $\\mathrm { H } _ { 2 }$ gas was fed into the cell from a buffer held at 9.25 mbar via a mass flow controller at a rate of $0 . 1 4 \\mathrm { m b a r l { s } ^ { - 1 } }$ . To aid with reproducible plasma generation, a ‘glow discharge’ was used. This is a low-ionisation-state plasma maintained by a constant applied voltage of $3 . 2 \\mathrm { k V }$ . To produce the desired plasma densities for acceleration a $2 0 \\mathrm { k V }$ , microsecond duration voltage pulse was applied across the existing low-density plasma. An example current trace, which had a typical amplitude of $2 9 0 \\mathrm { A }$ , is shown in Fig. 3(b).
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augmentation
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NO
| 0
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IPAC
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What is the highest energy gain observed in a plasma wakefield accelerator?
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44 GeV
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Reasoning
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Blumenfeld_et_al._-_2007_-_Energy_doubling_of_42_GeV_electrons_in_a_metre-scale_plasma_wakefield_accelerator.pdf
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THE AWAKE EXPERIMENT AWAKE is an R&D experiment at CERN with the aim to develop proton-driven based plasma wakefield acceleration. The wakefields are driven by highly-relativistic ${ \\mathrm { 4 0 0 G e V } }$ , relativistic factor $\\gamma _ { p + } \\sim 4 2 7 )$ and energetic $( > 1 9 \\mathrm { k J } )$ proton bunches, supplied by the CERN Super Proton Synchrotron (SPS). Since these proton bunches are longer than the plasma wavelength $\\lambda _ { p e }$ (where $\\lambda _ { p e } = 2 \\pi c / \\omega _ { p e }$ , with $\\omega _ { p e } = \\sqrt { n _ { p e } e ^ { 2 } / \\epsilon _ { 0 } m _ { e } }$ is the plasma electron frequency, $c$ the speed of light, $m _ { e }$ the electron mass, $e$ the electron charge, $\\epsilon _ { 0 }$ the vacuum permittivity and $n _ { p e }$ the plasma electron density) and less dense than the plasma $\\hat { ( n _ { b } < \\sim 1 0 ^ { - 3 } n _ { p e } }$ where $n _ { b }$ is the bunch density), the proton bunches have to be self-modulated to excite wakefields with $\\mathrm { G V / m }$ amplitudes [1, 2]; this requires $n _ { p e } > 1 0 ^ { 1 4 } \\mathrm { c m } ^ { - 3 }$ (corresponding to $\\lambda _ { p e } < 3 \\mathrm { m m } )$ ). The plasma is created by laser ionisation (pulse length: ${ \\sim } 1 0 0$ fs, energy per pulse: $\\sim 1 0 0 \\mathrm { m J }$ , central wavelength: $8 0 0 \\mathrm { n m }$ ) of rubidium vapour [3, 4]. It is $1 0 \\mathrm { m }$ long, with a radius $> \\sim 1 \\mathrm { m m }$ . Seeded proton bunch selfmodulation and subsequent wakefield growth, as well as the acceleration of externally injected witness electrons was demonstrated in AWAKE Run 1 [5–7].
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augmentation
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NO
| 0
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IPAC
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What is the highest energy gain observed in a plasma wakefield accelerator?
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44 GeV
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Reasoning
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Blumenfeld_et_al._-_2007_-_Energy_doubling_of_42_GeV_electrons_in_a_metre-scale_plasma_wakefield_accelerator.pdf
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File Name:ION-ION_COLLISIONS_IN_PLASMA_WAKEFIELD_ACCELERATORS.pdf ION-ION COLLISIONS IN PLASMA WAKEFIELD ACCELERATORS M.Yadav‚àó, K. Letko, J.B. Rosenzweig University of California, Los Angeles, California, USA Abstract The plasma wakefield accelerator, with acceleration gradients ranging from $\\mathrm { G e V / m }$ to $\\mathrm { T e V } / \\mathrm { m }$ , holds promise for propelling particles to high energies in linear colliders. This results in exceptionally bright beams characterized by intense ion-derived focusing, leading to the collapse of plasma ions. Our study extends prior research focused on electron acceleration by investigating ion-ion collisions, studying different collision models and emphasizing the near-equilibrium state post-ion collapse using the OSIRIS Particle -in-cell (PIC) code. Notably, our findings reveal that parametric excitations arising from plasma non-uniformity have an insignificant impact on phase space diffusion, a crucial insight for optimizing linear colliders. INTRODUCTION Plasma wakefield acceleration (PWFA) employs waves in a plasma medium chosen to naturally avoid breakdown issues. These waves are excited by an intense drive beam to accelerate a trailing beam. PWFAs have already demonstrated acceleration gradients exceeding $5 0 \\mathrm { G e V / m }$ . To explore the suitability of the PWFA for linear collider applications, proposals [1, 2] have been put forward that analyze the use of an afterburner at the end of a conventional linear collider injector, with the goal of doubling the beam energy [3].
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augmentation
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NO
| 0
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IPAC
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What is the highest energy gain observed in a plasma wakefield accelerator?
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44 GeV
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Reasoning
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Blumenfeld_et_al._-_2007_-_Energy_doubling_of_42_GeV_electrons_in_a_metre-scale_plasma_wakefield_accelerator.pdf
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File Name:PROGRESS_TOWARDS_HIGH-QUALITY,_HIGH-REPETITION-RATE.pdf PROGRESS TOWARDS HIGH-QUALITY, HIGH-REPETITION-RATE PLASMA ACCELERATION AT FLASHForward J. C. Wood∗,1, L. Boulton1, J. Beinortaite˙1,2, J. Björklund Svensson1, G. Boyle1, J. Cowley3, A. Ferran Pousa1, B. Foster1,2, M. J. Garland1, P. González-Caminal1, M. Huck1, H. Jones1, A. Kanekar1, C. A. Lindstrøm,1,4, G. Loisch1, T. Long1, S. M. Mewes1, J. Osterhoff1, F. Peña1, S. Schröder1, M. Thévenet1, S. Wesch1, M. Wing1,2 and R. D’Arcy1,3 1Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany 2University College London, United Kingdom 3 University of Oxford, United Kingdom 4 University of Oslo, Norway Abstract Plasma-wakefield acceleration represents an exciting route towards reducing the footprint of future high-energy electron accelerators by accelerating bunches in fields exceeding ${ \\mathrm { ~ 1 ~ G V / m } }$ . One such technique employs a doublebunch structure where the trailing bunch is accelerated in the field of a high-amplitude plasma-density wake driven by the leading bunch. A future particle collider or photon science facility incorporating plasma accelerators will be required to accelerate up to millions of bunches per second with high energy efficiency while preserving the brightness of the accelerating bunch. This contribution presents the latest progress towards these goals at FLASHForward (DESY). INTRODUCTION Electron-bunch-driven plasma wakefield accelerators (PWFAs) [1, 2] have the potential to greatly extend the energy reach of existing and future electron accelerators in a compact footprint by boosting the energy of bunches in fields $> 1 \\mathrm { G V } \\mathrm { m } ^ { - 1 }$ . A short, relativistic electron bunch of density $n _ { b }$ travelling through an underdense plasma of density $n _ { e } \\ll n _ { b }$ will expel all nearby plasma electrons, driving a fully-cavitated plasma wake that travels at close to the speed of light [3, 4]. The heavier plasma ions barely move over short timescales, providing linear focussing fields that can preserve bunch quality [5], and a strong longitudinal field providing rapid, phase-locked acceleration for a trailing bunch. By shaping the trailing bunch, the wakefield can be loaded to preserve the energy spread of the entire trailing bunch, while simultaneously transferring energy from the driver to the trailing bunch with high efficiency [6, 7].
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augmentation
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NO
| 0
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IPAC
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What is the highest energy gain observed in a plasma wakefield accelerator?
|
44 GeV
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Reasoning
|
Blumenfeld_et_al._-_2007_-_Energy_doubling_of_42_GeV_electrons_in_a_metre-scale_plasma_wakefield_accelerator.pdf
|
INTRODUCTION To perform physics precision studies or discover physics beyond the Standard Model, high-energy colliders such as the existing Large Hadron Collider (LHC), the past Large Electron-Positron (LEP) or the Future Circular Collider (FCC) [1, 2] are desireable. However, limitations such as speed or radio-frequency characteristics create barriers to achieving higher physics goals, with gradient limits typically in the order of $1 0 0 { \\mathrm { M V / m } }$ due to surface breakdown, arcing, cavity damage, or wakefield effects [3, 4]. In the ’80s-’90s, Tajima and Dawson proposed laser wakefield acceleration (LWFA) where laser pulses were used as wakefield drivers [5]. To further overcome the limits of the existing techniques and achieve acceleration gradients on the order of $\\mathrm { T V / m }$ and beyond, alternative methods based on solid-state plasma wakefield were also proposed [6, 7] Taking into account that solid-state structures can have a density of conduction electrons 4-5 orders of magnitude higher compared to gaseous plasma medium [8], preionised solid-state targets might offer a way to create inhomogeneous structured plasmas, able to sustain ultra-high acceleration gradients [9, 10]. CNT array-based nanostructures can create a structured non-homogeneous plasma with a density modulation wavelength of several $\\mu \\mathrm { m }$ which can be tailored to optimize the acceleration gradient and the confinement of particles [11].
|
augmentation
|
NO
| 0
|
IPAC
|
What is the highest energy gain observed in a plasma wakefield accelerator?
|
44 GeV
|
Reasoning
|
Blumenfeld_et_al._-_2007_-_Energy_doubling_of_42_GeV_electrons_in_a_metre-scale_plasma_wakefield_accelerator.pdf
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LASER-PLASMA ACCELERATOR Recent demonstrations of ${ \\sim } 1 \\mu \\mathrm { C }$ electron acceleration from kilo-joule laser OMEGA EP [2] and stable generation of ${ \\sim } 2 . 2 \\mathrm { p C }$ electron acceleration at $2 . 5 \\mathrm { H z }$ with $1 7 0 \\mathrm { m J }$ Ti-Sapphire laser [3] in the MeV range indicate promise of MeV range laser wakefield accelerators for application in various fields. We consider employing the supersonic gas jet target [4] used in both experiments [2, 3] along with ARCO Hybrid Ti-Saphh laser from Amplitude [5] or Quark 30/45 from THALES [6] to drive a laser plasma accelerator with mean electron energy of $2 0 \\mathrm { M e V }$ , total charge of $1 2 \\mathrm { - } 2 2 \\mathrm { p C }$ and geometric emittance $< 3 3 \\mu \\mathrm { m }$ mrad and beam divergence of less than $5 ^ { \\circ }$ . Following similar approach to Ref. [7], we estimate the desired laser and gas-target parameters for laser wakefield acceleration [8] and the corresponding anticipated plasma and electron beam parameters in Table 1. We note that $\\leq 1 \\%$ of the electron beam charge with energy spread $\\leq 1 0 ^ { - 3 }$ transmits through the collimator (Fig. 1) to be accelerated in the cryomodules.
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augmentation
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NO
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IPAC
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What is the highest energy gain observed in a plasma wakefield accelerator?
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44 GeV
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Reasoning
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Blumenfeld_et_al._-_2007_-_Energy_doubling_of_42_GeV_electrons_in_a_metre-scale_plasma_wakefield_accelerator.pdf
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In order to gain better insight into the experimental results, we conducted 3D-PIC simulations using the OSIRIS code. We chose OSIRIS based on its ability to handle highly nonlinear and kinetic processes that occur during high-intensity particle and laser interactions with the plasma. As the relativistic beam propagates, it expels all plasma electrons out of its way and thus generates in its wake a positively charged cavity. The fields in this cavity, also known as wakefields, reach values of $1 0 0 { \\mathrm { M V / m } }$ if the gas density is in the range 1013‚àí1014 cm‚àí3 as shown in Fig. 4. A significant challenge with PWFA is accelerating a beam while keeping energy spread and emittance growth small even for a longer propagation length. We investigate propagation of high-intensity charged particle beams in plasma. The simulation box size was $- 8 k _ { p } ^ { - 1 }$ in the transverse direction and $1 0 0 k _ { p } ^ { - 1 }$ in longitudinal direction and 8 macroparticles per cell. The code used a static window approach, where the simulation box moves at the speed of light, and the pulse is initialized near the leftmost edge of the window. OSIRIS also incorporates the ability to launch EM waves into the simulation, either by initializing the EM field of the simulation box accordingly or by injecting them from the simulation boundaries. The mapping of trapped electrons and accelerating fields throughout the ionized gas was constantly simulated.
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augmentation
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IPAC
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What is the highest energy gain observed in a plasma wakefield accelerator?
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44 GeV
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Reasoning
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Blumenfeld_et_al._-_2007_-_Energy_doubling_of_42_GeV_electrons_in_a_metre-scale_plasma_wakefield_accelerator.pdf
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File Name:AUTOMATED_EMITTANCEAND_ENERGYGAINOPTIMIZATIONFOR.pdf AUTOMATED EMITTANCE AND ENERGY GAIN OPTIMIZATION FOR PLASMA WAKEFIELD ACCELERATION M. Stobbe‚àó, R. Holtzapple Department of Physics, California Polytechnic State University, San Luis Obispo, CA, USA A. Knetsch, D. Storey SLAC National Accelerator Laboratory, Menlo Park, CA, USA Abstract At the Facility for Advanced Accelerator Experimental Tests (FACET-II) accelerator, a pair of $1 0 \\mathrm { G e V }$ high-current electron beams is used to investigate Plasma Wakefield Acceleration (PWFA) in plasmas of different lengths. While PWFA has achieved astonishingly high accelerating gradients of tens of $\\mathrm { G e V / m }$ , matching the electron beam into the plasma wake is necessary to achieve a beam quality required for precise tuning of future high energy linear accelerators. The purpose of this study was to explore how start-to-end simulations could be used to optimize two important measures of beam quality, namely maximizing energy gain and minimizing transverse emittance growth in a $2 \\mathrm { c m }$ long plasma. These two beam parameters were investigated with an in-depth model of the FACET-II accelerator using numerical optimization. The results presented in the paper demonstrate the importance of utilizing beam-transport simulations in tandem with particle-in-cell simulations and provide insight into optimizing these two important beam parameters without the need to devote significant accelerator physics time tuning the FACET-II accelerator.
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IPAC
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What is the highest energy gain observed in a plasma wakefield accelerator?
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44 GeV
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Blumenfeld_et_al._-_2007_-_Energy_doubling_of_42_GeV_electrons_in_a_metre-scale_plasma_wakefield_accelerator.pdf
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INTRODUCTION In recent years charged particle acceleration using solidstate nanostructured plasmas has attracted attention as a novel method of achieving ultra-high acceleration gradients, beam manipulation and gamma- or $\\mathrm { \\Delta X }$ -ray generation $[ 1 -$ 10]. In this context, PIC simulations have shown that the excitation of carbon nanotube (CNT) or graphene based multi-channel structures using either a driver beam [1,6] or a laser pulse [2,9] might achieve electric wakefield amplitudes in the order of $\\mathrm { T V / m }$ or beyond. In this paper, we specifically investigate laser wakefield acceleration (LWFA) in multi-hollow nanostructured high density plasmas. In this setup, a single, short, high-intensity pulse drives the wakefield. To excite a wakefield, the laser pulse length $L$ must be in the order of the plasma wavelength $\\lambda _ { p }$ (see for example [11]): $L \\simeq \\lambda _ { p } = 2 \\pi / \\omega _ { p } =$ $2 \\pi \\sqrt { m _ { e } \\varepsilon _ { 0 } / ( n _ { e } e ^ { 2 } ) }$ , where $\\omega _ { p }$ is the angular frequency of the plasma, $\\scriptstyle { \\varepsilon _ { 0 } }$ the vacuum permittivity, $e$ the elementary electric charge, $m _ { e }$ the electron rest mass and $n _ { e }$ the plasma density. This ensures that the excitations created by the ponderomotive force are in phase with the pulse group velocity.
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What is the highest energy gain observed in a plasma wakefield accelerator?
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44 GeV
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Blumenfeld_et_al._-_2007_-_Energy_doubling_of_42_GeV_electrons_in_a_metre-scale_plasma_wakefield_accelerator.pdf
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The beam has geometric transverse emittances of $\\varepsilon _ { x } = 9 . 5 \\times 1 0 ^ { - 1 0 } \\mathrm { m }$ and $\\varepsilon _ { y } = 1 . 2 \\times 1 0 ^ { - 1 0 } \\mathrm { m }$ . It is focused with a quadrupole doublet to a spot with $1 0 \\mu \\mathrm { m }$ radius at the entrance of the plasma. With this beam energy, bunch length and spot size, the corresponding power density is $3 \\times 1 0 ^ { 2 0 } \\mathrm { W } \\mathrm { c m } ^ { - 2 }$ . Plasma generation. A column of lithium vapour with a density of $2 . 7 \\times 1 0 ^ { 1 7 } \\mathrm { c m } ^ { - 3 }$ is produced in a heat-pipe oven21. The lithium vapour is confined by a helium buffer gas, which is in turn separated from the beam-line vacuum by a $5 0 \\mathrm { - } \\mu \\mathrm { m }$ -thick beryllium window upstream and by a $7 5 \\mathrm { - } \\mu \\mathrm { m }$ -thick beryllium window downstream. Lithium was chosen because of the low ionization potential of its first electron $( 5 . 4 \\mathrm { e V } )$ and the relatively high potential for its two subsequent electrons (76 and $1 2 2 \\mathrm { e V }$ ). In the present experiments the transverse electric field of the ultrashort electron pulses is large enough to field-ionize the first lithium electron over a timescale shorter than the bunch duration. The ADK theory for field ionization22 indicates that full ionization occurs in the volume surrounding the pulse in which the electric field exceeds ${ \\sim } 6 \\mathrm { G V m } ^ { - 1 }$ .
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What is the highest energy gain observed in a plasma wakefield accelerator?
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44 GeV
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Reasoning
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Blumenfeld_et_al._-_2007_-_Energy_doubling_of_42_GeV_electrons_in_a_metre-scale_plasma_wakefield_accelerator.pdf
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It has been demonstrated that PWFA can be optimized with large datasets of accelerator measurements [3], which suggests that a search for an optimum could be automated [4]. At laser-driven plasma wakefield accelerators, Bayesian optimization was already applied successfully [5–7]. The objective of this work is to examine how simulations using a computational model of the FACET-II beamline could serve as a guide for experimental efforts at increasing the energy gain of PWFA while maintaining or reducing transverse emittance growth, a figure of merit for beam quality. A start-to-end model of the beamline was developed and used in a numerical optimization schema to determine the final focusing quadrupole magnet strengths which would best optimize these two beam characteristics. During the PWFA process, the energy gain is calculated by quantifying the difference between the mean energy of the particle beam at the beginning and end of the plasma process. The mean energy gain divided by the length of the plasma $( L _ { \\mathrm { p l a s m a } } )$ , gives the acceleration gradient, or the amount of energy the particles gained on average per meter of travel: INTRODUCTION A relatively new method of providing high accelerating gradients for charged particles in the Accelerator Physics community is known as Plasma Wakefield Acceleration (PWFA). This technique, which uses strong electromagnetic fields generated in plasma, has demonstrated accelerating gradients of over $1 0 \\mathrm { G e V / m }$ [1] which is orders of magnitude larger than traditional radiofrequency (RF) technology. The wake field, excited by a drive electron beam transfers energy to the witness electron beam trailing in the back of the wake. One of the challenges in the development of PWFA is that the plasma wake not only provides strong longitudinal fields which accelerate charged particles but it also makes strong transverse fields that can lead to deterioration of beam quality. The ability to sustain good beam quality and high accelerating gradients is a vital concern that we hope to address.
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What is the highest energy gain observed in a plasma wakefield accelerator?
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44 GeV
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Reasoning
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Blumenfeld_et_al._-_2007_-_Energy_doubling_of_42_GeV_electrons_in_a_metre-scale_plasma_wakefield_accelerator.pdf
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Recent plasma wakefield accelerator experiments have shown high-gradient acceleration of electrons using a 10-cm-long plasma11. To obtain energy gains of interest to high-energy physics, these high gradients must be extended over metre-scale plasmas. Such an extension transitions the plasma wakefield accelerator from a regime in which the drive beam has no time to distort, deplete or go unstable, to a regime in which it is significantly depleted in energy, deformed owing to combined effects of diffraction and multiple transverse oscillations, and possibly goes unstable because of the electron-hose instability16. This work is in this latter regime. A schematic of the experimental set-up is shown in Fig. 1. In the present work carried out at the Final Focus Test Beam facility at SLAC, the nominally 50-femtosecond-long electron beam containing $1 . 8 \\\\times { { 1 0 } ^ { 1 0 } }$ particles is focused to a spot size of ${ \\\\sim } 1 0 \\\\mu \\\\mathrm { m }$ at the entrance of an $8 5 \\\\mathrm { - c m }$ -long column of lithium vapour with a density $n _ { \\\\mathrm { e } }$ of $2 . 7 \\\\times 1 0 ^ { 1 7 } \\\\mathrm { c m } ^ { - 3 }$ . The nominally $4 2 \\\\mathrm { G e V }$ beam has a correlated energy spread of approximately $1 . 5 \\\\mathrm { G e V }$ , with electrons in the front of the beam at higher energies than those at the back. The beam exiting the plasma traverses a metre-long dipole magnet, which disperses the beam electrons according to their energy. The transverse distribution of the dispersed electrons is measured at two distances (planes 1 and 2 in Fig. 1) downstream of the dipole magnet to distinguish the energy changes of the electrons from their possible transverse deflection due to the plasma.
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What is the highest energy gain observed in a plasma wakefield accelerator?
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44 GeV
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Blumenfeld_et_al._-_2007_-_Energy_doubling_of_42_GeV_electrons_in_a_metre-scale_plasma_wakefield_accelerator.pdf
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In a plasma wakefield accelerator large-amplitude electric fields result from space-charge waves excited by the passage of an ultrarelativistic electron beam through a plasma12. A fully ionized plasma can be formed in a neutral vapour when the radial electric field of the electron beam exceeds the field ionization threshold13. The ionization occurs in a narrow region in the front of the beam. This ionization front produces a plasma that has a radius much larger than the beam itself. If the beam density exceeds the plasma density, the plasma electrons are expelled from the volume of the electron pulse, leaving a column of more massive ions behind14. Subsequently, the expelled plasma electrons are pulled back (by the ions) to the beam axis behind the pulse, overshoot, and set up a space-charge oscillation or wake. The longitudinal field of this wake varies continuously along the pulse, decelerating its core but accelerating the particles in the back. The ion column also provides a focusing force15 that guides the beam over many diffraction lengths, allowing an efficient transfer of the beam energy to the wake. This force also causes the transverse size of the beam to oscillate as it propagates through the plasma—the socalled betatron oscillations (see Supplementary Movie 1).
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What is the highest energy gain observed in a plasma wakefield accelerator?
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44 GeV
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Reasoning
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Blumenfeld_et_al._-_2007_-_Energy_doubling_of_42_GeV_electrons_in_a_metre-scale_plasma_wakefield_accelerator.pdf
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Thus, the full ionization extends over a radius of more than $1 0 0 \\mu \\mathrm { m }$ and ionization begins far earlier than the peak of the bunch current. Because the ionization region extends over a radius larger than the plasma collisionless skin depth $c / \\omega _ { \\mathrm { p } } ,$ where $\\omega _ { \\mathrm { p } } = ( n _ { \\mathrm { e } } e ^ { 2 } / \\varepsilon _ { 0 } m _ { \\mathrm { e } } ) ^ { 1 / 2 }$ is the plasma angular frequency; $e$ is the charge on the electron, $\\scriptstyle { \\varepsilon _ { 0 } }$ is the permittivity of free space and $m _ { \\mathrm { e } }$ is the mass of the electron), the wake is similar to that in a preformed plasma. Energy measurement. The energy spectrometer consists of a dipole magnet that disperses the electrons vertically according to their momentum $\\boldsymbol { p }$ . The dispersion can be closely approximated by a deflection at the centre of the magnet: $\\theta _ { 1 } = e \\int B \\mathrm { d } L / p$ . Using the measured dispersion, its integrated magnetic flux density #BdL was calculated to be $1 . 2 \\mathrm { T m }$ . In general, all particles in a pulse leave the plasma from a well-defined spot, but with a non-negligible exit angle $\\theta _ { 0 }$ . To discriminate between a vertical exit angle and the deflection by the magnet, the particle distribution is measured at two planes, $8 6 \\mathrm { c m }$ and $1 8 6 \\mathrm { c m }$ downstream of the centre of the dipole (Fig. 1).
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NO
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expert
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What is the highest energy gain observed in a plasma wakefield accelerator?
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44 GeV
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Reasoning
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Blumenfeld_et_al._-_2007_-_Energy_doubling_of_42_GeV_electrons_in_a_metre-scale_plasma_wakefield_accelerator.pdf
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The images have been corrected at the level of a few per cent for the nonuniform collection efficiency of the optics. Pixel-to-pixel variations in the CCD offset and a common mode have been subtracted; the signal from X-rays that hit the CCD directly has been eliminated. Simulations. The simulations were done using the quasi-static, three-dimensional, particle-in-cell code called QuickPIC. The three-dimensional computational grid forms a box xy $z ( 2 4 0 \\\\mu \\\\mathrm { m } \\\\times 2 4 0 \\\\mu \\\\mathrm { m } \\\\times 2 6 0 \\\\mu \\\\mathrm { m } )$ in size whose axial coordinate is z-ct. Therefore, the simulation window moves at the speed of light, which is very close to the beam speed in the $z$ direction. The number of grid points is $2 5 6 \\\\times 2 5 6 \\\\times 5 1 2$ respectively. The beam is initialized so that in vacuum, it would focus $1 5 \\\\mathrm { c m }$ beyond the start of the lithium vapour with a $1 0 \\\\mu \\\\mathrm { m }$ root-mean-square spot size. The longitudinal current profile is extracted from the unique LiTrack simulation that matches the experimentally measured beam spectrum produced by the SLAC accelerator. The resulting current profile approximates a gaussian $( \\\\sigma _ { z } \\\\approx 1 5 \\\\mu \\\\mathrm { m } )$ with a small tail. We use 8.4 million particles for the beam and $2 . 6 \\\\times 1 0 ^ { 5 }$ particles for each ‘slice’ of lithium. In the quasi-static approximation, as the entire beam moves through a slice of gas, the lithium ionizes, the resulting plasma evolves transversely and, to account for the axial motion, the charge on each particle is suitably changed. The resulting plasma forces are stored for each slice and are then used to advance the momentum and position of each beam electron. The beam electrons are advanced every $1 . 0 \\\\mathrm { m m }$ , which is 1/26th of a betatron wavelength for 42 GeV electrons in the flat density region. The simulations were done on the Apple X-serve Dawson Cluster at UCLA.
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NO
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expert
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What is the impact parameter κρd, and why is it pivotal in determining maximal radiation?
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??d = kd/(??), representing the normalized electron–structure separation; it dictates whether fast or slow electrons maximize radiation in far- or near-field regimes.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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We perform quantitative experimental measurement of Smith– Purcell radiation to directly probe the upper limit. Figure 3a shows our experimental set-up (see Methods and Supplementary Section 7 for details). A one-dimensional (1D) $5 0 \\%$ -filling-factor grating (Au-covered single-crystalline Si)—the quintessential Smith– Purcell set-up—is chosen as a sample, and shown by scanning electron microscope (SEM) images in Fig. $^ { 3 \\mathrm { b } , \\mathrm { c } }$ Free electrons pass above and impinge onto the sample at a grazing angle of $1 . 5 ^ { \\circ }$ under 10 to $2 0 \\mathrm { k V }$ acceleration voltages. Figure 3d depicts our measurements of first-order $m = 1$ Smith– Purcell radiation appearing at wavelengths between 500 and $7 5 0 \\mathrm { n m }$ . In quantitative agreement with equation (1) evaluated at the normal emission angle (solid lines), the measured radiation spectra (dots) blueshift with increasing electron velocity. Notably, we experimentally obtain the absolute intensity of the collected radiation via a calibration measurement (see Supplementary Section 7). The upper limits (equation (4)) for the surface-normal emission wavelengths $\\left( \\lambda = a / \\beta \\right)$ are evaluated at the centre of the interaction region (height ${ \\approx } 1 4 0 \\mathrm { n m }$ $( k d \\approx 1 . 5 )$ , varying with beam energy), and is shown with shading in Fig. 3d to account for the thickness uncertainty $( \\pm 1 . 5 \\mathrm { n m } )$ . The envelope spanned by the measurement peaks follows the upper-limit lineshape across the visible spectrum: both the theoretical limit and the measured intensities peak near $5 5 0 \\mathrm { n m }$ and decrease in a commensurate manner for other wavelengths. This lineshape originates from two competing factors. At shorter wavelengths, the material factor $| \\chi | ^ { 2 } / \\mathrm { I m } \\Dot { \\chi }$ decreases significantly for both Au and Si (see Fig. 1c), which accounts for the reduced radiation intensity. At longer wavelengths, the major constraint becomes the less efficient interaction between the electrons and the structure, as the electron-beam diameters increase for the reduced brightness of the electron gun (tungsten) at lower acceleration voltages (see Supplementary Section 7). These pieces of experimental evidence for the upper limit are at $k d \\approx 1 . 5$ (estimated from a geometrical raytracing model; see Supplementary Section 7), where fast electrons are still preferred (Fig. 2a). To further confirm our theory, we also conduct a near-infrared Smith–Purcell experiment (Supplementary Section 8) at $k d \\approx 1$ , where the envelope lineshape of the emission spectra again follows our prediction. We also obtain complementary supporting evidence (extracted from a recent work10) for our slowelectron-efficient prediction (see Supplementary Section 9).
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What is the impact parameter κρd, and why is it pivotal in determining maximal radiation?
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??d = kd/(??), representing the normalized electron–structure separation; it dictates whether fast or slow electrons maximize radiation in far- or near-field regimes.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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Finally, we turn our attention to an ostensible peculiarity of the limits: equation (4) evidently diverges for lossless materials $( \\mathrm { I m } \\chi \\to 0 ) \\dot { { \\frac { . } { . } } }$ ), seemingly providing little insight. On the contrary, this divergence suggests the existence of a mechanism capable of strongly enhancing Smith–Purcell radiation. Indeed, by exploiting high-Q resonances near bound states in the continuum (BICs)13 in photonic crystal slabs, we find that Smith–Purcell radiation can be enhanced by orders of magnitude, when specific frequency-, phaseand polarization-matching conditions are met. A 1D silicon $\\left( \\chi = 1 1 . 2 5 \\right)$ -on-insulator $\\mathrm { \\ S i O } _ { 2 } ,$ $\\chi = 1 . 0 7 \\$ ) grating interacting with a sheet electron beam illustrates the core conceptual idea most clearly. The transverse electric (TE) $( E _ { x } , H _ { y } , E _ { z } )$ band structure (lowest two bands labelled $\\mathrm { T E } _ { 0 }$ and $\\mathrm { T E } _ { 1 . } ^ { \\cdot }$ ), matched polarization for a sheet electron beam (supplementary equation S41b)), is depicted in Fig. 4b along the $\\Gamma { \\mathrm { - } } \\mathrm { X }$ direction. Folded electron wavevectors, $k _ { \\nu } = \\omega / \\nu ,$ are overlaid for two distinct velocities (blue and green). Strong electron–photon interactions are possible when the electron and photon dispersions intersect: for instance, $k _ { \\nu }$ and the $\\mathrm { T E } _ { 0 }$ band intersect (grey circles) below the air light cone (light yellow shading). However, these intersections are largely impractical: the $\\mathrm { T E } _ { 0 }$ band is evanescent in the air region, precluding free-space radiation. Still, analogous ideas, employing similar partially guided modes, such as spoof plasmons33, have been explored for generating electron-enabled guided waves34,35.
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expert
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What is the impact parameter κρd, and why is it pivotal in determining maximal radiation?
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??d = kd/(??), representing the normalized electron–structure separation; it dictates whether fast or slow electrons maximize radiation in far- or near-field regimes.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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free-space optical elements, enabling simultaneous measurement of the spectrum and of the spatial radiation pattern. The SEM used for the experiment was a JEOL JSM-6010LA. Its energy spread at the gun exit was in the range 1.5 to $2 . 5 \\mathrm { e V }$ for the range of acceleration voltages considered in this paper. The SEM was operated in spot mode, which we controlled precisely to align the beam so that it passes tangentially to the surface near the desired area of the sample. A Nikon TU Plan Fluor 10x objective with a numerical aperture of 0.30 was used to collect light from the area of interest. The monochrome image of the radiation was taken using a Hamamatsu CCD (chargecoupled device). The spectrometer used was an Action SP-2360-2300i with a lownoise Princeton Instruments Pixis 400 CCD. A 1D grating (Au-covered single-crystalline Si: periodicity, ${ 1 4 0 } \\mathrm { n m } ;$ filling factor, $5 0 \\%$ ; patterned Si thickness, $5 3 \\pm 1 . 5 \\mathrm { { n m } }$ ; Au thickness $4 4 \\pm 1 . 5 \\mathrm { n m }$ was used as the sample in our experiment. The original nanopatterned linear silicon stamp was obtained from LightSmyth Technologies and coated using an electron beam evaporator with a $2 \\mathrm { n m T i }$ adhesion layer and $4 0 \\mathrm { n m }$ of Au at $1 0 ^ { - 7 }$ torr. The sample was mounted inside the SEM chamber to enable the alignment of free electrons to pass in close proximity to the stamps. The emitted light was coupled out of the SEM chamber to a spectrometer, while a camera was used to image the surface of the sample.
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expert
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What is the impact parameter κρd, and why is it pivotal in determining maximal radiation?
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??d = kd/(??), representing the normalized electron–structure separation; it dictates whether fast or slow electrons maximize radiation in far- or near-field regimes.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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In closing, we have theoretically derived and experimentally probed a universal upper limit to the energy loss and photon emission from free electrons. The limit depends crucially on the impact parameter $\\kappa _ { \\rho } d$ , but not on any other detail of the geometry. Hence, our limit applies even to the most complex metamaterials and metasurfaces, given only their constituents. Surprisingly, in the near field, slow electrons promise stronger radiation than relativistic ones. The limit predicts a divergent radiation rate as the material loss rate goes to zero, and we show that BIC resonances enable such staggering enhancements. This is relevant for the generation of coherent Smith–Purcell radiation14,34,35. The long lifetime, spectral selectivity and large field enhancement near a BIC can strongly bunch electrons, allowing them to radiate coherently at the same desired frequency, potentially enabling low-threshold Smith–Purcell freeelectron lasers. The combination of this mechanism and the optimal velocity prediction reveals prospects of low-voltage yet high-power free-electron radiation sources. In addition, our findings demonstrate a simple guiding principle to maximize the signal-to-noise ratio for electron energy-loss spectroscopy through an optimal choice of electron velocity, enabling improved spectral resolution. The predicted slow-electron-efficient regime still calls for direct experimental validation. We suggest that field-emitter-integrated free-electron devices (for example, ref. 10) are ideal to confirm the prediction due to the achievable small electron–structure separation and high electron beam quality at relatively large currents. Alternatively, the microwave or terahertz frequencies could be suitable spectral ranges for verifying the slow-electron-efficient regime, where the subwavelength separation requirement is more achievable.
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Yes
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expert
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What is the impact parameter κρd, and why is it pivotal in determining maximal radiation?
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??d = kd/(??), representing the normalized electron–structure separation; it dictates whether fast or slow electrons maximize radiation in far- or near-field regimes.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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$$ $$ \\propto \\frac { 1 } { \\beta ^ { 2 } } \\Bigg \\{ \\ln ( 1 / \\kappa _ { \\rho } d ) \\mathrm { f o r } \\kappa _ { \\rho } d \\ll 1 , $$ The limits of equations (4), (5a) and (5b) are completely general; they set the maximum photon emission and energy loss of an electron beam coupled to an arbitrary photonic environment in either the non-retarded or retarded regimes, given only the beam properties and material composition. The key factors that determine maximal radiation are identified: intrinsic material loss (represented by $\\mathrm { I m } \\chi \\dot { } ,$ ), electron velocity $\\beta$ and impact parameter $\\kappa _ { \\rho } d$ . The metric $| \\chi | ^ { 2 } / \\mathrm { I m } \\chi$ reflects the influence of the material choice, which depends sensitively on the radiation wavelength (Fig. 1b). The electron velocity $\\beta$ also appears implicitly in the impact parameter $\\kappa _ { \\rho } d = k d / \\beta \\gamma ,$ showing that the relevant length scale is set by the relativistic velocity of the electron. The impact parameter $\\kappa _ { \\rho } d$ reflects the influence of the Lorentz contraction $d / \\gamma ;$ a well-known feature of both electron radiation and acceleration20,26,30.
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expert
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What is the impact parameter κρd, and why is it pivotal in determining maximal radiation?
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??d = kd/(??), representing the normalized electron–structure separation; it dictates whether fast or slow electrons maximize radiation in far- or near-field regimes.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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A surprising feature of the limits in equations (4), (5a) and (5b) is their prediction for optimal electron velocities. As shown in Fig. 1c, when electrons are in the far field of the structure $( \\kappa _ { \\rho } d \\gg 1 )$ , stronger photon emission and energy loss are achieved by faster electrons—a well-known result. On the contrary, if electrons are in the near field $( \\kappa _ { \\rho } d \\ll 1 )$ , slower electrons are optimal. This contrasting behaviour is evident in the asymptotics of equation (5b), where the $1 / \\beta ^ { 2 }$ or $\\mathrm { e } ^ { - 2 \\kappa _ { \\rho } d }$ dependence is dominant at short or large separations. Physically, the optimal velocities are determined by the incident-field properties (equation (2)): slow electrons generate stronger near-field amplitudes although they are more evanescent (Supplementary Section 2). There has been recent interest in using low-energy electrons for Cherenkov10 and Smith–Purcell31 radiation; our prediction that they can be optimal at subwavelength interaction distances underscores the substantial technological potential of non-relativistic free-electron radiation sources. The tightness of the limit (equations (4), (5a) and (5b)) is demonstrated by comparison with full-wave numerical calculations (see Methods) in Fig. 1d,e. Two scenarios are considered: in Fig. 1d, an electron traverses the centre of an annular Au bowtie antenna and undergoes antenna-enabled transition radiation $( \\eta \\approx 0 . 0 7 \\% )$ , while, in Fig. 1e, an electron traverses a Au grating, undergoing Smith–Purcell radiation $( \\eta \\approx 0 . 9 \\% )$ . In both cases, the numerical results closely trail the upper limit at the considered wavelengths, showing that the limits can be approached or even attained with modest effort.
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expert
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What is the impact parameter κρd, and why is it pivotal in determining maximal radiation?
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??d = kd/(??), representing the normalized electron–structure separation; it dictates whether fast or slow electrons maximize radiation in far- or near-field regimes.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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File Name:Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf Maximal spontaneous photon emission and energy loss from free electrons Yi Yang $\\textcircled { 1 0 } 1 \\star$ , Aviram Massuda1, Charles Roques-Carmes $\\oplus 1$ , Steven E. Kooi $\\oplus 2$ , Thomas Christensen1, Steven G. Johnson1, John D. Joannopoulos1,2, Owen D. Miller $\\textcircled { 1 0 } 3 \\star$ , Ido Kaminer $\\textcircled { 1 0 } 1 , 4 \\star$ and Marin Soljačić1 Free-electron radiation such as Cerenkov1, Smith–Purcell2 and transition radiation3,4 can be greatly affected by structured optical environments, as has been demonstrated in a variety of polaritonic5,6, photonic-crystal7 and metamaterial8–10 systems. However, the amount of radiation that can ultimately be extracted from free electrons near an arbitrary material structure has remained elusive. Here we derive a fundamental upper limit to the spontaneous photon emission and energy loss of free electrons, regardless of geometry, which illuminates the effects of material properties and electron velocities. We obtain experimental evidence for our theory with quantitative measurements of Smith–Purcell radiation. Our framework allows us to make two predictions. One is a new regime of radiation operation—at subwavelength separations, slower (non-relativistic) electrons can achieve stronger radiation than fast (relativistic) electrons. The other is a divergence of the emission probability in the limit of lossless materials. We further reveal that such divergences can be approached by coupling free electrons to photonic bound states in the continuum11–13. Our findings suggest that compact and efficient free-electron radiation sources from microwaves to the soft X-ray regime may be achievable without requiring ultrahigh accelerating voltages.
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What is the impact parameter κρd, and why is it pivotal in determining maximal radiation?
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??d = kd/(??), representing the normalized electron–structure separation; it dictates whether fast or slow electrons maximize radiation in far- or near-field regimes.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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The Smith–Purcell effect epitomizes the potential of free-electron radiation. Consider an electron at velocity $\\beta = \\nu / c$ traversing a structure with periodicity $a$ ; it generates far-field radiation at wavelength $\\lambda$ and polar angle $\\theta$ , dictated by2 $$ \\lambda = \\frac { a } { m } \\left( \\frac { 1 } { \\beta } - \\mathrm { c o s } \\theta \\right) $$ where $m$ is the integer diffraction order. The absence of a minimum velocity in equation (1) offers prospects for threshold-free and spectrally tunable light sources, spanning from microwave and terahertz14–16, across visible17–19, and towards $\\bar { \\mathrm { X - r a y } } ^ { 2 0 }$ frequencies. In stark contrast to the simple momentum-conservation determination of wavelength and angle, there is no unified yet simple analytical equation for the radiation intensity. Previous theories offer explicit solutions only under strong assumptions (for example, assuming perfect conductors or employing effective medium descriptions) or for simple, symmetric geometries21–23. Consequently, heavily numerical strategies are often an unavoidable resort24,25. In general, the inherent complexity of the interactions between electrons and photonic media have prevented a more general understanding of how pronounced spontaneous electron radiation can ultimately be for arbitrary structures, and consequently, how to design the maximum enhancement for free-electron light-emitting devices.
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What is the impact parameter κρd, and why is it pivotal in determining maximal radiation?
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??d = kd/(??), representing the normalized electron–structure separation; it dictates whether fast or slow electrons maximize radiation in far- or near-field regimes.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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We begin our analysis by considering an electron (charge $- e$ ) of constant velocity $\\nu \\hat { \\mathbf { x } }$ traversing a generic scatterer (plasmonic or dielectric, finite or extended) of arbitrary size and material composition, as in Fig. 1a. The free current density of the electron, ${ \\bf \\dot { J } } ( { \\bf r } , t ) = - { \\hat { \\bf x } } e \\nu \\delta ( y ) { \\bf \\ddot { \\delta } } ( z ) \\delta ( x - \\nu t )$ , generates a frequency-dependent $( { \\bf e } ^ { - i \\omega t }$ convention) incident field26 $$ \\mathbf { E } _ { \\mathrm { i n c } } ( \\mathbf { r } , \\omega ) = \\frac { e \\kappa _ { \\rho } \\mathrm { e } ^ { i k _ { \\nu } x } } { 2 \\pi \\omega \\varepsilon _ { 0 } } [ \\hat { \\mathbf { x } } i \\kappa _ { \\rho } K _ { 0 } ( \\kappa _ { \\rho } \\rho ) - \\hat { \\mathbf { \\rho } } \\hat { \\mathbf { e } } k _ { \\nu } K _ { 1 } ( \\kappa _ { \\rho } \\rho ) ]
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What is the impact parameter κρd, and why is it pivotal in determining maximal radiation?
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??d = kd/(??), representing the normalized electron–structure separation; it dictates whether fast or slow electrons maximize radiation in far- or near-field regimes.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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$$ written in cylindrical coordinates $( x , \\rho , \\psi )$ ; here, $K _ { n }$ is the modified Bessel function of the second kind, $k _ { \\nu } = \\omega / \\nu$ and $k _ { \\rho } = \\sqrt { k _ { \\nu } ^ { 2 } - k ^ { 2 } } =$ k/βγ $\\scriptstyle ( k = \\omega / c$ , free-space wavevector; $\\gamma = 1 / \\sqrt { 1 - \\beta ^ { 2 } }$ , Lorentz factor). Hence, the photon emission and energy loss of free electrons can be treated as a scattering problem: the electromagnetic fields $\\mathbf { F } _ { \\mathrm { i n c } } { = } ( \\mathbf { E } _ { \\mathrm { i n c } } , \\ Z _ { 0 } \\mathbf { H } _ { \\mathrm { i n c } } ) ^ { \\mathrm { T } }$ (for free-space impedance $Z _ { 0 } ^ { \\mathrm { ~ \\tiny ~ { ~ \\chi ~ } ~ } }$ are incident on a photonic medium with material susceptibility $\\overline { { \\overline { { \\chi } } } }$ (a $6 { \\times } 6$ tensor for a general medium), causing both absorption and far-field scattering—that is, photon emission—that together comprise electron energy loss (Fig. 1a).
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What is the impact parameter κρd, and why is it pivotal in determining maximal radiation?
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??d = kd/(??), representing the normalized electron–structure separation; it dictates whether fast or slow electrons maximize radiation in far- or near-field regimes.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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As recently shown in refs 27–29, for a generic electromagnetic scattering problem, passivity—the condition that polarization currents do no net work—constrains the maximum optical response from a given incident field. Consider three power quantities derived from $\\mathbf { F } _ { \\mathrm { i n c } }$ and the total field F within the scatterer volume $V !$ the total power lost by the electron, $P _ { \\mathrm { l o s s } } = - ( 1 / 2 ) \\mathrm { R e } \\int _ { \\mathrm { V } } \\mathbf { J } ^ { * } \\cdot \\mathbf { E d } V = ( \\epsilon _ { 0 } \\omega / 2 ) \\mathrm { I m } \\hat { \\int _ { V } } \\mathbf { F } _ { \\mathrm { ~ i n c } } ^ { \\dagger } \\overline { { \\chi } } \\mathbf { F d } V ,$ the power absorbed by the medium, $P _ { \\mathrm { { a b s } } } \\mathrm { { = } } \\left( \\epsilon _ { 0 } \\omega / 2 \\right) \\mathrm { I m } \\stackrel { \\cdot } { \\int } _ { V } \\mathbf { F } ^ { \\dagger } \\overline { { \\chi } } \\mathbf { F } \\mathrm { { d } } V ,$ and their difference, the power radiated to the far field, $P _ { \\mathrm { r a d } } { = } P _ { \\mathrm { l o s s } } { - } P _ { \\mathrm { a b s } }$ . Treating $\\mathbf { F }$ as an independent variable, the total loss $P _ { \\mathrm { l o s s } }$ is a linear function of $\\mathbf { F }$ , whereas the fraction that is dissipated is a quadratic function of F. Passivity requires non-negative radiated power, represented by the inequality $P _ { \\mathrm { a b s } } { < } P _ { \\mathrm { l o s s } } ,$ which in this framework is therefore a convex constraint on any response function. Constrained maximization (see Supplementary Section 1) of the energy-loss and photon-emission power quantities, $P _ { \\mathrm { l o s s } }$ and $P _ { \\mathrm { r a d } } ,$ directly yields the limits
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What is the impact parameter κρd, and why is it pivotal in determining maximal radiation?
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??d = kd/(??), representing the normalized electron–structure separation; it dictates whether fast or slow electrons maximize radiation in far- or near-field regimes.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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$$ P _ { \\tau } ( \\omega ) \\leq \\frac { \\varepsilon _ { 0 } \\omega \\xi _ { \\tau } } { 2 } \\int _ { V } \\mathbf { F } _ { \\mathrm { i n } } ^ { \\dagger } \\overline { { \\overline { { \\chi } } } } ^ { \\dagger } ( \\mathrm { I m } \\overline { { \\overline { { \\chi } } } } ) ^ { - 1 } \\overline { { \\overline { { \\chi } } } } \\mathbf { F } _ { \\mathrm { i n c } } \\mathrm { d } V $$ where $\\tau \\in \\{ \\mathrm { r a d } , \\log \\}$ and $\\xi _ { \\tau }$ accounts for a variable radiative efficiency $\\eta$ (defined as the ratio of radiative to total energy loss): $\\xi _ { \\mathrm { l o s s } } = 1$ and $\\xi _ { \\mathrm { r a d } } = \\eta ( 1 - \\eta ) \\le 1 / 4$ . Hereafter, we consider isotropic and non-magnetic materials (and thus a scalar susceptibility $\\chi$ ), but the generalizations to anisotropic and/or magnetic media are straightforward.
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What is the impact parameter κρd, and why is it pivotal in determining maximal radiation?
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??d = kd/(??), representing the normalized electron–structure separation; it dictates whether fast or slow electrons maximize radiation in far- or near-field regimes.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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Next, we specialize in the canonical Smith–Purcell set-up illustrated in Fig. 1e inset. This set-up warrants a particularly close study, given its prominent historical and practical role in free-electron radiation. Aside from the shape-independent limit (equations (5a) and (5b)), we can find a sharper limit (in per unit length for periodic structure) specifically for Smith–Purcell radiation using rectangular gratings of filling factor $\\varLambda$ (see Supplementary Section 3) $$ \\frac { \\mathrm { d } { \\varGamma } _ { \\tau } ( \\omega ) } { \\mathrm { d } x } \\leq \\frac { \\alpha \\xi _ { \\tau } } { 2 \\pi c } \\frac { | \\boldsymbol \\chi | ^ { 2 } } { \\mathrm { I m } \\chi } \\varLambda \\mathcal { G } ( \\beta , k d ) $$ The function ${ \\mathcal { G } } ( { \\boldsymbol { \\beta } } , k d )$ is an azimuthal integral (see Supplementary Section 3) over the Meijer G-function $G _ { 1 , 3 } ^ { 3 , 0 }$ (ref. $^ { 3 2 } )$ that arises in the radial integration of the modified Bessel functions $K _ { n }$ . We emphasize that equation (6) is a specific case of equation (4) for grating structures without any approximations and thus can be readily generalized to multi-material scenarios (see Supplementary equation (37)).
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What is the impact parameter κρd, and why is it pivotal in determining maximal radiation?
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??d = kd/(??), representing the normalized electron–structure separation; it dictates whether fast or slow electrons maximize radiation in far- or near-field regimes.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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The grating limit (equation (6)) exhibits the same asymptotics as equations (5a) and (5b), thereby reinforcing the optimal-velocity predictions of Fig. 1c. The $( \\beta , k \\dot { d } )$ dependence of $\\mathcal { G }$ (see Fig. 2a) shows that slow (fast) electrons maximize Smith–Purcell radiation in the small (large) separation regime. We verify the limit predictions by comparison with numerical simulations: at small separations (Fig. 2b), radiation and energy loss peak at velocity $\\beta \\approx 0 . 1 5$ , consistent with the limit maximum; at large separations (Fig. 2c), both the limit and the numerical results grow monotonically with $\\beta$ . The derived upper limit also applies to Cherenkov and transition radiation, as well as bulk loss in electron energy-loss spectroscopy. For these scenarios where electrons enter material bulk, a subtlety arises for the field divergence along the electron’s trajectory $( \\rho = 0$ in equation (2)) within a potentially lossy medium. This divergence, however, can be regularized by introducing natural, system-specific momentum cutoffs26, which then directly permits the application of our theory (see Supplementary Section 6). Meanwhile, there exist additional competing interaction processes (for example, electrons colliding with individual atoms). However, they typically occur at much smaller length scales.
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What is the impact parameter κρd, and why is it pivotal in determining maximal radiation?
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??d = kd/(??), representing the normalized electron–structure separation; it dictates whether fast or slow electrons maximize radiation in far- or near-field regimes.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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To overcome this deficiency, we theoretically propose a new mechanism for enhanced Smith–Purcell radiation: coupling of electrons with $\\mathrm { B I C } s ^ { 1 3 }$ . The latter have the extreme quality factors of guided modes but are, crucially, embedded in the radiation continuum, guaranteeing any resulting Smith–Purcell radiation into the far field. By choosing appropriate velocities $\\beta = a / m \\lambda$ ( $m$ being any integer; $\\lambda$ being the BIC wavelength) such that the electron line (blue or green) intersects the $\\mathrm { T E } _ { 1 }$ mode at the BIC (red square in Fig. 4b), the strong enhancements of a guided mode can be achieved in tandem with the radiative coupling of a continuum resonance. In Fig. 4c, the incident fields of electrons and the field profile of the BIC indicate their large modal overlaps. The BIC field profile shows complete confinement without radiation, unlike conventional multipolar radiation modes (see Supplementary Fig. 9). The Q values of the resonances are also provided near a symmetry-protected $\\mathrm { B I C ^ { 1 3 } }$ at the $\\Gamma$ point. Figure 4d,e demonstrates the velocity tunability of BIC-enhanced radiation—as the phase matching approaches the BIC, a divergent radiation rate is achieved.
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What is the impact parameter κρd, and why is it pivotal in determining maximal radiation?
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??d = kd/(??), representing the normalized electron–structure separation; it dictates whether fast or slow electrons maximize radiation in far- or near-field regimes.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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The BIC-enhancement mechanism is entirely accordant with our upper limits. Practically, silicon has non-zero loss across the visible and near-infrared wavelengths. For example, for a period of $a = 6 7 6 \\mathrm { n m }$ , the optimally enhanced radiation wavelength is $\\approx 1 { , } 0 5 0 \\mathrm { n m }$ , at which $\\chi _ { \\mathrm { s i } } \\approx 1 1 . 2 5 + 0 . 0 0 1 \\mathrm { i }$ (ref. 36). For an electron– structure separation of $3 0 0 \\mathrm { n m }$ , we theoretically show in Fig. 4f the strong radiation enhancements ${ > } 3$ orders of magnitude) attainable by BIC-enhanced coupling. The upper limit (shaded region; 2D analogue of equation (4); see Supplementary Section 10) attains extremely large values due to the minute material loss $( | \\chi | ^ { 2 } / \\mathrm { I m } \\chi \\approx 1 0 ^ { 5 } )$ ; nevertheless, BIC-enhanced coupling enables the radiation intensity to closely approach this limit at several resonant velocities. In the presence of an absorptive channel, the maximum enhancement occurs at a small offset from the BIC where the $Q$ -matching condition (see Supplementary Section 11) is satisfied (that is, equal absorptive and radiative rates of the resonances).
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What is the impact parameter κρd, and why is it pivotal in determining maximal radiation?
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??d = kd/(??), representing the normalized electron–structure separation; it dictates whether fast or slow electrons maximize radiation in far- or near-field regimes.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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Author contributions Y.Y., O.D.M., I.K. and M.S. conceived the project. Y.Y. developed the analytical models and numerical calculations. A.M. prepared the sample under study. Y.Y., A.M., C.R.-C., S.E.K. and I.K. performed the experiment. Y.Y., T.C. and O.D.M. analysed the asymptotics and bulk loss of the limit. S.G.J., J.D.J., O.D.M., I.K. and M.S. supervised the project. Y.Y. wrote the manuscript with input from all authors. Competing interests The authors declare no competing interests. Additional information Supplementary information is available for this paper at https://doi.org/10.1038/ s41567-018-0180-2. Reprints and permissions information is available at www.nature.com/reprints. Correspondence and requests for materials should be addressed to Y.Y. or O.D.M. or I.K. Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Methods Fourier transform convention. Throughout the paper, we adopt the following Fourier transform conventions $$ f ( \\omega ) \\triangleq \\int f ( t ) \\mathrm { e } ^ { i \\omega t } \\mathrm { d } t , f ( t ) \\triangleq \\frac { 1 } { 2 \\pi } \\int f ( \\omega ) \\mathrm { e } ^ { - i \\omega t } \\mathrm { d } \\omega $$ $$
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What is the impact parameter κρd, and why is it pivotal in determining maximal radiation?
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??d = kd/(??), representing the normalized electron–structure separation; it dictates whether fast or slow electrons maximize radiation in far- or near-field regimes.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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g ( \\mathbf { k } ) \\triangleq \\int f ( \\mathbf { r } ) \\mathrm { e } ^ { - i \\mathbf { k } \\cdot \\mathbf { r } } \\mathrm { d } \\mathbf { r } , g ( \\mathbf { r } ) \\triangleq \\frac { 1 } { \\left( 2 \\pi \\right) ^ { 3 } } \\int g ( \\mathbf { k } ) \\mathrm { e } ^ { i \\mathbf { k } \\cdot \\mathbf { r } } \\mathrm { d } \\mathbf { k } $$ Numerical methods. The photonic band structure in Fig. 4b is calculated via the eigenfrequency calculation in COMSOL Multiphysics. Numerical radiation intensities (Figs. 1d,e, 2b,c, 3d and 4d–f) are obtained via the frequency-domain calculation in the radiofrequency module in COMSOL Multiphysics. A surface (for 3D problems) or line (for 2D problems) integral on the Poynting vector is calculated to extract the radiation intensity at each frequency. Experimental set-up and sample fabrication. Our experimental set-up comprises a conventional SEM with the sample mounted perpendicular to the stage. A microscope objective was placed on the SEM stage to collect and image the light emission from the surface. The collected light was then sent through a series of
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What is the impact parameter κρd, and why is it pivotal in determining maximal radiation?
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??d = kd/(??), representing the normalized electron–structure separation; it dictates whether fast or slow electrons maximize radiation in far- or near-field regimes.
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Definition
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Maximal_spontaneous_photon_emission_and_energy_loss_from_free_electrons.pdf
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Data availability. The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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What is the inverse design approach employed for optimizing the radiator structure?
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A gradient-based photonic inverse design using the Adam optimizer to tailor dielectric distributions within unit cells under symmetry constraints to maximize radiation power.
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hermann-et-al-2022-inverse-designed-narrowband-thz-radiator-for-ultrarelativistic-electrons.pdf
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Michelson Interferometer and THz Detector. For the spectrum measurements, we installed a Michelson interferometer outside the vacuum chamber. The THz pulse was first sent through an in-vacuum lens made of PMMA with a diameter of $2 5 \\ \\mathrm { m m }$ and a focal length of $1 0 0 ~ \\mathrm { { m m } }$ . The lens collimates radiation in the vertical plane, but it does not map the entire radiation of the $4 5 \\ \\mathrm { m m }$ long structure onto the detector. The angular acceptance in the horizontal plane is calculated via ray tracing (see Figure 3). A fused silica vacuum window with about $5 0 \\%$ transmission for the design wavelength of the structure $( 9 0 0 ~ \\mu \\mathrm { m } )$ is used as extraction port. The beam splitter is made of $3 . 5 \\ \\mathrm { m m }$ -thick plano‚àíplano high-resistivity float-zone silicon (HRFZ-Si) manufactured by TYDEX. It provides a splitting ratio of $5 4 / 4 6$ for wavelengths ranging from 0.1 to $1 ~ \\mathrm { m m }$ . Translating one of the mirrors of the interferometer allowed us to measure the first-order autocorrelation, from which the power spectrum is obtained via Fourier transform.
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What is the inverse design approach employed for optimizing the radiator structure?
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A gradient-based photonic inverse design using the Adam optimizer to tailor dielectric distributions within unit cells under symmetry constraints to maximize radiation power.
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Definition
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hermann-et-al-2022-inverse-designed-narrowband-thz-radiator-for-ultrarelativistic-electrons.pdf
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ACS PHOTONICS READ Quasi-BIC Modes in All-Dielectric Slotted Nanoantennas for Enhanced $\\mathbf { E r ^ { 3 + } }$ Emission Boris Kalinic, Giovanni Mattei, et al.JANUARY 18, 2023 ACS PHOTONICS READ Get More Suggestions >
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What is the inverse design approach employed for optimizing the radiator structure?
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A gradient-based photonic inverse design using the Adam optimizer to tailor dielectric distributions within unit cells under symmetry constraints to maximize radiation power.
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Definition
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hermann-et-al-2022-inverse-designed-narrowband-thz-radiator-for-ultrarelativistic-electrons.pdf
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A typical autocorrelation measurement for a charge of 9.4 pC is depicted in Figure 2b. The shape of the autocorrelation is not perfectly symmetric in amplitude and stage position. The amplitude asymmetry could be a result of a nonlinear detector response (onset of saturation). This is in agreement with the slight deviation of the pulse energy from the quadratic fit (Figure 4). Since the length of only one arm is changed and the radiation might not be perfectly collimated, the position scan of the mirror is not creating a perfectly symmetric autocorrelation signal. AUTHOR INFORMATION Corresponding Author Rasmus Ischebeck − Paul Scherrer Institut, 5232 Villigen, PSI, Switzerland; $\\circledcirc$ orcid.org/0000-0002-5612-5828; Email: rasmus.ischebeck@psi.ch Authors Benedikt Hermann − Paul Scherrer Institut, 5232 Villigen, PSI, Switzerland; Institute of Applied Physics, University of Bern, 3012 Bern, Switzerland; Galatea Laboratory, Ecole Polytechnique Fédérale de Lausanne (EPFL), 2000 Neuchâtel, Switzerland; $\\circledcirc$ orcid.org/0000-0001-9766-3270 Urs Haeusler − Department Physik, Friedrich-AlexanderUniversität Erlangen-Nürnberg (FAU), 91058 Erlangen, Germany; Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom; $\\circledcirc$ orcid.org/0000- 0002-6818-0576 Gyanendra Yadav − Department of Physics, University of Liverpool, Liverpool L69 7ZE, United Kingdom; Cockcroft Institute, Warrington WA4 4AD, United Kingdom Adrian Kirchner − Department Physik, Friedrich-AlexanderUniversität Erlangen-Nürnberg (FAU), 91058 Erlangen, Germany
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What is the inverse design approach employed for optimizing the radiator structure?
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A gradient-based photonic inverse design using the Adam optimizer to tailor dielectric distributions within unit cells under symmetry constraints to maximize radiation power.
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Definition
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hermann-et-al-2022-inverse-designed-narrowband-thz-radiator-for-ultrarelativistic-electrons.pdf
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The objective function $G$ , quantifying the performance of a design $\\phi ,$ is given by the line integral of the Poynting vector $\\begin{array} { r } { { \\bf S } ( x , y ) = \\mathrm { R e } \\left\\{ \\frac { 1 } { 2 } { \\bf E } \\times { \\bf H } ^ { * } \\right\\} } \\end{array}$ in the $x$ -direction along the length of one period, evaluated at a point $x _ { S }$ outside the design region: $$ G ( \\phi ) = \\int _ { 0 } ^ { a } S _ { x } ( x _ { S } , y ) \\mathrm { d } y $$ The optimization problem can then be stated as $$ \\begin{array} { r l } & { \\operatorname* { m a x } _ { \\phi } G ( \\phi ) \\quad \\mathrm { s u b j e c t t o } \\quad \\nabla \\times \\mathbf { E } = - i \\omega \\mu \\mathbf { H } \\quad \\mathrm { a n d } } \\\\ & { \\quad \\nabla \\times \\epsilon ( \\phi ) ^ { - 1 } \\nabla \\times \\mathbf { H } - \\omega ^ { 2 } \\mu \\mathbf { H } = \\nabla \\times \\epsilon ( \\phi ) ^ { - 1 } \\mathbf { J } } \\end{array}
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What is the inverse design approach employed for optimizing the radiator structure?
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A gradient-based photonic inverse design using the Adam optimizer to tailor dielectric distributions within unit cells under symmetry constraints to maximize radiation power.
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Definition
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hermann-et-al-2022-inverse-designed-narrowband-thz-radiator-for-ultrarelativistic-electrons.pdf
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An in-vacuum PMMA lens with a diameter of $2 5 \\mathrm { ~ m m }$ collimated parts of the emitted radiation. A Michelson interferometer was used to measure the first-order autocorrelation of the electromagnetic pulse and to obtain its power spectrum via Fourier transform (Figure 2b and Methods). The measured spectrum is centered around 881 $\\mu \\mathrm { m }$ $\\left( 0 . 3 4 ~ \\mathrm { T H z } \\right)$ and has a full width at half-maximum of ${ \\sim } 9 \\%$ (Figure 3). DISCUSSION The observed spectrum agrees well with a 3D finite-differences time-domain (FDTD) simulation of the experiment (Figure 3). In contrast, a finite-differences frequency-domain (FDFD) simulation reveals that the design can in principal emit even more narrowbandly, originating from the high mode density inside the Fabry‚àíPerot cavity formed by the two distributed Bragg reflectors on both sides of the electron channel. The difference between the two simulations can be explained by their distinct grid resolutions. The FDFD simulation considers only a single period of the structure with periodic boundaries, corresponding to an infinitely long structure. Hence, the cell size is small, allowing to use a high grid resolution. The timedomain simulation, on the other hand, calculates the electromagnetic field of the entire 50-period-long structure for each time step. This high memory requirement comes at the cost of a lower spatial resolution. Since the experiment was similarly limited by the fabrication resolution of $1 4 0 \\ \\mu \\mathrm { m } ,$ the FDTD simulation reproduced the measured spectrum much better. We also note that potential absorption losses in the structure can reduce its quality factor and broaden the radiation spectrum. Due to the small contribution from $\\varepsilon ^ { \\prime \\prime } =$ $0 . 0 8 , ^ { 2 4 }$ absorption effects were not considered here but would dominate at higher quality factors.
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What is the inverse design approach employed for optimizing the radiator structure?
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A gradient-based photonic inverse design using the Adam optimizer to tailor dielectric distributions within unit cells under symmetry constraints to maximize radiation power.
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Definition
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hermann-et-al-2022-inverse-designed-narrowband-thz-radiator-for-ultrarelativistic-electrons.pdf
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We drove the structure with electron bunches with a duration of approximately 30 fs (RMS), which is much shorter than the resonant wavelength corresponding to a period of 3 ps. Hence, we expect to see the coherent addition of radiated fields. To experimentally verify this, we varied the bunch charge. Figure 4 shows the detected pulse energy for six bunch charge settings ranging from $0 ~ \\mathrm { p C }$ to $1 1 . 8 ~ \\mathrm { p C }$ The scaling is well approximated by a quadratic fit, which confirms the expected coherent enhancement of the $\\mathrm { T H z }$ pulse energy.14 We observe a slight deviation for the highest charge measurement from the quadratic fit, which might be a result of detector saturation (see Methods). We note that the quadratic scaling would enable $\\mathrm { T H z }$ pulse energies orders of magnitude larger by driving the structure at higher bunch charges. The THz pulse emitted perpendicular to the Smith-Purcell radiator possesses a pulse-front tilt of close to $4 5 ^ { \\circ }$ since it is driven by ultrarelativistic electrons. Depending on the length of the radiator and the application, the tilt can be compensated for with a diffraction grating.
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What is the inverse design approach employed for optimizing the radiator structure?
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A gradient-based photonic inverse design using the Adam optimizer to tailor dielectric distributions within unit cells under symmetry constraints to maximize radiation power.
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Definition
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hermann-et-al-2022-inverse-designed-narrowband-thz-radiator-for-ultrarelativistic-electrons.pdf
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During and after our experiments, the structure did not show any signs of performance degradation or visible damage. It was used continuously for eight hours with a bunch charge of approximately $1 0 ~ \\mathrm { p C }$ at a pulse repetition rate of $1 \\ \\mathrm { H z }$ . CONCLUSION The here-presented beam-synchronous radiation source can be added to the beamline of an FEL to enrich capabilities for pump‚àíprobe experiments. For ultrarelativistic electrons, a second beamline may be used to compensate for the longer path length of the THz radiation and achieve simultaneous arrival with the X-ray radiation created in the undulator of the FEL (Figure 5a). Smith‚àíPurcell radiation represents a costefficient alternative to the broadband generation of THz by optical rectification, which requires an external laser system and precise synchronization to the accelerator. Our inverse design approach to Smith‚àíPurcell emitters can produce beamsynchronous narrowband THz radiation, which could propel pump‚àíprobe studies with THz excitations in solids, for instance, resonant control of strongly correlated electron systems, high-temperature superconductors, or vibrational modes of crystal lattices (phonons).28,29 Further improvement of our THz structure can be achieved by higher fabrication accuracy and the use of a fully 3Doptimized geometry with a higher quality factor, resulting in more narrowband emission and higher pulse energy. Moreover, the inverse design suite could be extended to composite structures of more than one material, which could provide extra stability for complicated 3D designs. In the case of highly resonant structures, materials with low absorption, for example, polytetrafluoroethylene (PTFE),24 are a necessity. The measured THz pulse energy can be increased by a factor of almost 300 by raising the driving bunch charge from the used $1 1 . 8 ~ \\mathrm { p C }$ up to the $2 0 0 ~ \\mathrm { p C }$ available at SwissFEL. Whether the currently used material can withstand such high fields and radiation remains to be investigated. Combining 3D optimization, longer structures, larger collection optics, and higher bunch charges will result in a THz pulse energy multiple orders of magnitude larger than observed in the presented experiment $( 0 . 6 ~ \\mathrm { p J } )$ .
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Expert
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What is the inverse design approach employed for optimizing the radiator structure?
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A gradient-based photonic inverse design using the Adam optimizer to tailor dielectric distributions within unit cells under symmetry constraints to maximize radiation power.
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Definition
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hermann-et-al-2022-inverse-designed-narrowband-thz-radiator-for-ultrarelativistic-electrons.pdf
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The second difficulty arises from the long-range evanescent waves of ultrarelativistic electrons. The spectral density of the electric field of a line charge decays with $\\bar { \\exp ( - \\kappa | x | ) }$ , where $\\kappa =$ $2 \\pi / \\beta \\gamma \\lambda$ , with $\\beta \\approx 1$ and $\\gamma \\approx 6 0 0 0$ for $E = 3 . 2$ GeV.34 This means the evanescent waves will reach the boundaries of our simulation cell in the $x$ -direction. Generalized perfectly matched layers $\\left( \\mathrm { P M L s } \\right) ^ { 3 5 }$ are chosen, such that they can absorb both propagating and evanescent waves. A detailed look at Figure 1b reveals that our implementation of generalized PMLs is not fully capable of absorbing evanescent waves. Hence, we make use of symmetry to further reduce the effect of evanescent waves at the boundaries of the simulation cell. Note that the evanescent electric field for $\\beta \\approx 1$ is almost entirely polarized along the transverse direction $x .$ . This means if the simulation cell is mirror symmetric with respect to the electron channel, antiperiodic boundaries can be applied after the PMLs to cancel out the effect of evanescent waves at the boundaries. This turned out to work well for us, although the structure is not mirror symmetric with respect to the electron axis.
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Yes
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Expert
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What is the inverse design approach employed for optimizing the radiator structure?
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A gradient-based photonic inverse design using the Adam optimizer to tailor dielectric distributions within unit cells under symmetry constraints to maximize radiation power.
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Definition
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hermann-et-al-2022-inverse-designed-narrowband-thz-radiator-for-ultrarelativistic-electrons.pdf
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Simulations. The 3D frequency-domain simulation was performed in COMSOL, based on the finite element method. The simulation cell, as shown in the lower right inset of Figure 1c, consists of a single unit cell of the grating, with a height of 4 mm and periodic boundaries along the electron propagation direction. An optional phase shift at the boundaries in longitudinal direction enables simulations for nonperpendicular Smith-Purcell emission, $\\lambda \\neq a$ . Perfectly matched layers are applied in all remaining, transverse directions. The electron beam $\\stackrel { \\prime } { E } = 3 . 2 \\mathrm { G e V } _ { \\mathrm { ; } }$ ， $Q = e$ ) had a Gaussian shape of width $\\sigma _ { x } =$ $\\sigma _ { z } = 5 0 \\ \\mu \\mathrm { m }$ in the transverse direction. The 3D time-domain simulation of the full structure, as shown in Figure 1c with the connecting filaments at the top and bottom, was performed in CST Studio Suite 2021. A single electron bunch $\\left( E = 3 \\mathrm { G e V } \\right)$ with Gaussian charge distribution was assumed. Its width in the transverse direction was $\\sigma _ { x } = \\sigma _ { z } =$ $0 . 1 \\ \\mathrm { m m }$ and in the longitudinal direction $\\sigma _ { y } = 0 . 2 ~ \\mathrm { m m }$ with cutoff length $0 . 4 \\mathrm { ~ m m }$ . The simulation was performed for a longer bunch length than the experimental bunch length due to computational resource limitations for smaller mesh cell resolutions. Nevertheless, we expect this approximation to yield a realistic emission spectrum, since the simulated bunch length is still substantially shorter than the central wavelength. A convergence test showed that a hexahedral mesh with a minimum cell size of $1 5 \\ \\mu \\mathrm { m }$ was sufficient. To imitate free space, perfectly matched layers and open-space boundary conditions were applied, where a $\\lambda / 2$ thick layer of vacuum was added after the dielectric structure. The radiation spectrum was then obtained via far-field approximations at multiple frequencies.
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Yes
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Expert
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What is the inverse design approach employed for optimizing the radiator structure?
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A gradient-based photonic inverse design using the Adam optimizer to tailor dielectric distributions within unit cells under symmetry constraints to maximize radiation power.
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Definition
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hermann-et-al-2022-inverse-designed-narrowband-thz-radiator-for-ultrarelativistic-electrons.pdf
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Accelerator Setup. The experiments used $1 0 ~ \\mathrm { p C }$ electron bunches from the $3 . 2 \\mathrm { G e V }$ Athos beamline of SwissFEL27 operated at a pulse repetition rate of $1 \\ \\mathrm { H z }$ to keep particle losses during alignment at a tolerable level. The standard bunch charge at SwissFEL is $2 0 0 \\mathrm { p C }$ at a repetition rate of 100 $\\mathrm { H z }$ . For the low charge working point, the aperture and intensity of the cathode laser are reduced. The normalized emittance of the electron beam with a charge of $9 . 5 \\ \\mathsf { p C }$ was $1 1 0 ~ \\mathrm { { \\ n m } }$ rad in both planes and was measured with a quadrupole scan in the injector at a beam energy of 150 MeV.36 For the experiment, we scanned the charge from 0 to 11.8 pC by adjusting the intensity of the cathode laser, which results in a slight emittance degradation and mismatch of the transverse beam parameters. This is due to charge density changes in the space charge dominated gun region. Nevertheless, the beam size remained small enough for full transmission through the THz Smith‚àíPurcell structure.
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augmentation
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Yes
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Expert
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What is the inverse design approach employed for optimizing the radiator structure?
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A gradient-based photonic inverse design using the Adam optimizer to tailor dielectric distributions within unit cells under symmetry constraints to maximize radiation power.
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Definition
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hermann-et-al-2022-inverse-designed-narrowband-thz-radiator-for-ultrarelativistic-electrons.pdf
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A bunch length of 30 fs (RMS) was measured for similar machine settings in a separate shift with a transverse deflecting cavity (TDC) in the Aramis beamline of the accelerator. Therefore, we expect the longitudinal dimension of the electron beam at the ACHIP chamber to be on the order of $1 0 \\ \\mu \\mathrm m ,$ , almost 2 orders of magnitude shorter than the period of the structure and radiated wavelength. The transverse beam size at the interaction point was $3 0 \\mu \\mathrm { m }$ in the horizontal and $4 0 \\ \\mu \\mathrm m$ in the vertical direction (for a charge of $9 . 5 \\ \\mathrm { p C } ,$ ), as measured with a scintillating YAG screen imaged with an out-of-vacuum microscope onto a CCD camera. After position and angular alignment of the structure using an in-vacuum hexapod, the beam could be transmitted without substantial losses through the $2 7 2 \\ \\mu \\mathrm { m }$ wide channel of the THz generating structure. Structure Fabrication. The structure was fabricated with a commercial PMMA stereolithography device Formlabs Form 2. The resolution of the device is $1 4 0 \\ \\mu \\mathrm { { m } , }$ , which provides subwavelength feature sizes for the geometry with a periodicity of $9 0 0 \\mu \\mathrm { m }$ . The height of the structure $( 6 ~ \\mathrm { { m m } ) }$ was limited by the stability of the structure rods during the fabrication process. The high temperature resin used for this study can be heated to $2 3 5 ~ ^ { \\circ } \\mathrm { C }$ . A sufficiently low outgassing rate for the installation at SwissFEL was achieved after baking the device for $s \\mathrm { ~ h ~ }$ under vacuum conditions at $1 7 5 ~ ^ { \\circ } \\mathrm { C } . ^ { 2 4 }$ Thanks to the rapid improvements in SLA technology and other free-form manufacturing techniques, the geometry could certainly be fabricated also at shorter wavelengths and higher resolution for future experiments. An increased manufacturing quality is required to achieve an even narrower emission bandwidth.
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augmentation
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Yes
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Expert
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What is the inverse design approach employed for optimizing the radiator structure?
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A gradient-based photonic inverse design using the Adam optimizer to tailor dielectric distributions within unit cells under symmetry constraints to maximize radiation power.
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Definition
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hermann-et-al-2022-inverse-designed-narrowband-thz-radiator-for-ultrarelativistic-electrons.pdf
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The geometric acceptance angle of the Michelson interferometer $\\Delta \\theta$ in the plane of the electron beam and the THz radiation defines the accepted bandwidth of the setup. According to the Smith‚àíPurcell relation (eq 1), it is given by $$ \\Delta \\lambda = a \\sin \\theta \\Delta \\theta $$ Around the orthogonal direction $( \\theta \\ : = \\ : 9 0 ^ { \\circ } ,$ ), the accepted bandwidth covers the measured spectrum (Figure 3). We calculated the acceptance with ray tracing including the size of the emitting structure and the apertures of the collimating lens $( 2 5 ~ \\mathrm { m m } )$ and the detector ( $\\cdot 1 2 \\ \\mathrm { m m } ,$ ). A Schottky diode (ACST, Type 3DL 12C LS2500 A2) was used as THz detector, sensitive from 300 to $4 0 0 0 \\ \\mu \\mathrm { { m } }$ . The manufacturer indicates a responsivity of $1 2 0 ~ \\mathrm { V / W }$ at $9 0 0 \\ \\mu \\mathrm { { m } , }$ , which we used to estimate the energy deposited on the detector. The signal from the detector is transmitted via a $2 0 \\mathrm { m }$ long coaxial cable to an oscilloscope outside of the accelerator bunker. For absolute pulse energy measurements, the detector setup including absorption in cables and the vacuum window should be characterized with a calibrated $\\mathrm { T H z }$ source. We calculated the pulse energy for different charges (Figure 4) by averaging over all shots during the oscillating autocorrelation measurement.
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augmentation
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Yes
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expert
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What is the main goal of the UA9 experiment?
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Demonstrate the feasibility of a crystal-based collimation for high-energy hadron machines.
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Fact
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CpFM_paper.pdf
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In Fig. 7 the angular scan of the UA9 crystal-1 during a proton run is shown. It is displayed both by the BLMs and the CpFM (CpFM position is such that both the bars intercept the whole channeled beam when the crystal is in the optimal channeling position). The first and the last angular regions (angle $< - 2 7 0 0$ μrad and angle $> - 2 4 0 0 ~ \\mathrm { \\mu r a d } )$ correspond to the amorphous orientation. As expected, here the loss rate registered by the BLMs is maximum while the CpFM signal rate is minimum. After the first amorphous region, the channeling peak appears (around angle of $- 2 6 2 0 \\ \\mu \\mathrm { r a d } )$ as a maximum in the CpFM rate and as minimum in the BLMs rate. Just after, only in the BLMs signal, the volume reflection area is clearly visible. In this angular region the particles experience a deflection to the opposite side with respect to the planar channeling deflection. For this reason volume reflection is not detectable by the CpFM except as a slight reduction in the background counts. Although the BLMs rate profile is an effective instrument for the estimation of the best channeling orientation, it is based on beam losses and is generally less sensitive than the CpFM rate profile which on the contrary measures directly the presence of channeled protons.
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expert
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What is the main goal of the UA9 experiment?
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Demonstrate the feasibility of a crystal-based collimation for high-energy hadron machines.
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Fact
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CpFM_paper.pdf
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In order to fully characterize this collimation system, it is essential to steadily monitor the flux of the halo particles deflected by the crystal towards the absorber. Typical crystal-extracted fluxes range from $1 0 ^ { 5 }$ up to $1 0 ^ { 7 }$ protons/s (i.e. from 1 up to 200 protons per SPS revolution) and about $1 0 ^ { 5 }$ ions/s (1–3 ions per SPS revolution). Such a low flux does not allow to use the standard SPS instrumentation, for example BCTs (Beam Current Transformer [8]) which are optimized for higher fluxes ${ \\mathrm { ( > ~ } } 1 0 ^ { 9 }$ protons/s). For this reason, the Cherenkov detector for proton Flux Measurement (CpFM) was designed and developed. 2. The Cherenkov detector for proton flux measurement The CpFM detector has been devised as an ultra-fast proton flux monitor. It has to provide measurements of the extracted beam directly inside the beam pipe vacuum, discriminating the signals coming from different proton bunches in case of multi-bunch beams, with a 25 ns bunch spacing. It is also able to stand and to detect very low ion fluxes (1–3 ions per turn). The sensitive part of the detector is located in the beam pipe vacuum in order to avoid the interaction of the protons with the vacuum–air interface, hence preserving the resolution on the flux measurement. All the design choices are explained in detail in [9].
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NO
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expert
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What is the main goal of the UA9 experiment?
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Demonstrate the feasibility of a crystal-based collimation for high-energy hadron machines.
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Fact
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CpFM_paper.pdf
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Finally, particular attention has to be paid to the shape of the distributions in Fig. 9(b). They are not Gaussian. For such a high fluxes, this cannot depend on the detector resolution, at least for the CpFM 2 channel which has the better efficiency. This can be demonstrated deriving the CpFM 2 resolution for an incident and constant flux of 180 protons per turn, as measured by the BCTDC. From the single ion distributions in Fig. 6(a), the resolution with respect to a single incident lead ion can be computed. The CpFM 2 resolution for 180 protons is easily derived scaling the ion resolution by the factor $\\sqrt { ( 6 7 2 4 / 1 8 0 ) }$ . It corresponds to $9 \\%$ . Thus, if the number of protons extracted by the crystal was constant and equal to 180 (distributed according a Gaussian distribution centered in 180), the CpFM 2 signal would be characterized by a narrower peak having 15 protons $\\sigma$ . The beam extracted by the crystal is therefore not constant on the time scale probed by the CpFM. There are several possible reasons for this: the diffusion dynamics of the halo beam, goniometer instabilities or orbit instabilities. The CpFM detector offers an interesting chance to address this issue at the μs scale but the current data acquisition electronics of the detector represent a limit. The CpFM detector is indeed able to accept only 1/1000 SPS trigger, since the data acquisition electronics are not fast enough $( < 1 \\mathrm { k H z } )$ . Faster electronics, matching the revolution frequency of the machine, could strengthen the detector capability in studying the impact of the listed factors on the crystal halo extraction.
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expert
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What is the main goal of the UA9 experiment?
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Demonstrate the feasibility of a crystal-based collimation for high-energy hadron machines.
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Fact
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CpFM_paper.pdf
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File Name:CpFM_paper.pdf Commissioning and operation of the Cherenkov detector for proton Flux Measurement of the UA9 experiment F.M. Addesa a,∗, D. Breton d, L. Burmistrov d, G. Cavoto a,b, V. Chaumat d, S. Dubos d, L. Esposito c, F. Galluccio e, M. Garattini c,g, F. Iacoangeli a, J. Maalmi d, D. Mirarchi c, S. Montesano c, A. Natochii $^ { \\mathrm { c , d , f } }$ , V. Puill d, R. Rossi c, W. Scandale c, A. Stocchi d a INFN - Istituto Nazionale di Fisica Nucleare - sezione di Roma, Piazzale A. Moro 2, Roma, Italy b Università degli Studi di Roma "La Sapienza" -Department of Physics, Piazzale A. Moro 2, Roma, Italy c CERN, European Organization for Nuclear Research, Geneva 23, CH 1211, Switzerland d LAL - Laboratoire de l’Accélérateur Linéaire - Université Paris-Sud 11, Centre Scientifique d’Orsay, B.P. 34, Orsay Cedex, F-91898, France e INFN Sezione di Napoli, Complesso Universitario di Monte Sant’Angelo, Via Cintia, 80126 Napoli, Italy f Taras Shevchenko National University of Kyiv (TSNUK), 60 Volodymyrska Street, City of Kyiv, 01033, Ukraine g Imperial College, London, United Kingdom A R T I C L E I N F O Keywords: Beam instrumentation (beam flux monitors,
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NO
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expert
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What is the main goal of the UA9 experiment?
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Demonstrate the feasibility of a crystal-based collimation for high-energy hadron machines.
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Fact
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CpFM_paper.pdf
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1. Introduction The primary goal of the UA9 experiment [1] is to demonstrate the feasibility of a crystal-based halo collimation as a promising and better alternative to the standard multi-stage collimation system for high-energy hadron machines. The main installation of the experiment is located in the Long Straight Section 5 (LSS5) of the CERN Super Proton Synchrotron (SPS) and consists of a crystal-assisted collimation prototype. It is made by preexisting optical elements of the SPS and new installations including three goniometers to operate different crystal types used as primary collimators, one dedicated movable absorber, several scrapers, detectors and beam loss monitors (BLMs) to study the interaction of the crystal with the beam halo [3]. A schematic of the layout of the experiment is shown in Fig. 1. The main process investigated is the so-called planar channeling: particles impinging on a crystal having a small angle (less than $\\theta _ { c }$ , called the critical angle for channeling) with respect to the lattice planes are forced by the atomic potential to move between the crystal planes. If the crystal is bent, the trapped particles follow the bending and are deflected correspondingly. When an optimized crystal intercepts the beam halo to act as collimator, about $8 0 \\%$ of the particles are channeled, coherently deflected and then dumped on the absorber (see Fig. 1), effectively reducing the beam losses in the sensitive areas of the accelerator [4–7].
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expert
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What is the main goal of the UA9 experiment?
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Demonstrate the feasibility of a crystal-based collimation for high-energy hadron machines.
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Fact
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CpFM_paper.pdf
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beam profile monitors) Cherenkov detectors In-vacuum detectors High-energy particle accelerators A B S T R A C T The UA9 Experiment at CERN-SPS investigates channeling processes in bent silicon crystals with the aim to manipulate hadron beams. Monitoring and characterization of channeled beams in the high energy accelerators environment ideally requires in-vacuum and radiation hard detectors. For this purpose the Cherenkov detector for proton Flux Measurement (CpFM) was designed and developed. It is based on thin fused silica bars in the beam pipe vacuum which intercept charged particles and generate Cherenkov light. The first version of the CpFM is installed since 2015 in the crystal-assisted collimation setup of the UA9 experiment. In this paper the procedures to make the detector operational and fully integrated in the UA9 setup are described. The most important standard operations of the detector are presented. They have been used to commission and characterize the detector, providing moreover the measurement of the integrated channeled beam profile and several functionality tests as the determination of the crystal bending angle. The calibration has been performed with Lead $\\left( \\mathrm { P b } \\right)$ and Xenon (Xe) beams and the results are applied to the flux measurement discussed here in detail.
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IPAC
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What is the natural emittance in a storage ring?
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It is the equilibrium emittance where damping and quantum excitation balance out.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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At the state art of laser-optical technologies, it is possible to create an optical resonator with a power incident on mirrors not exceeding several tens of $\\mathrm { k W }$ . Under these conditions the scattered photons beam intensity will not exceed of about 1012 phot/s. Considering this circumstance, it would be expedient to weaken the requirements for the beam separatrix and symmetrize the ring lattice. The NESTOR storage ring circumference is equal to $\\mathrm { C } = 1 5 . 4 1 8 \\mathrm { m }$ , harmonics number is $\\mathrm { h } = 3 6$ under RF frequency of $\\mathrm { f R F } = 2 7 9 7 \\mathrm { M H z }$ . The simplest option for upgrading the ring with minimal alterations when switching to a frequency of $2 8 5 6 \\mathrm { M H z }$ would be to increase the RF harmonics number to 37. To match the ring perimeter with new RF frequency it is needed to enlarge the ring circumference of about $1 2 \\mathrm { c m }$ . It may be performed by the lengthening of both ring long straight sections by $6 \\mathrm { c m }$ . The injection path of the ring remains unchanged.
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augmentation
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NO
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IPAC
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What is the natural emittance in a storage ring?
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It is the equilibrium emittance where damping and quantum excitation balance out.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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File Name:A_NOVEL_METHOD_TO_SUPPRESS_THE_EMITTANCE_VARIATION_IN.pdf A NOVEL METHOD TO SUPPRESS THE EMITTANCE VARIATION IN EXTREMELY LOW EMITTANCE LIGHT SOURCE STORAGE RINGS\\* K. Soutome†1, T. Hiraiwa, H. Tanaka, RIKEN SPring-8 Center, Sayo, Japan 1also at JASRI, Sayo, Japan Abstract We propose a novel method to suppress the emittance variation caused by the opening and closing of the gap of insertion devices (IDs) in extremely low emittance light source storage rings. The core idea is to leak a small amount of dispersion into the straight section where IDs are installed and optimize its value so that the radiation excitation and damping caused by IDs are balanced. The proposed method is passive and applicable to any light source storage ring, and the emittance variation is potentially expected to be less than $1 \\%$ by carefully optimizing the dispersion leakage. INTRODUCTION In modern light source storage rings, a multi-bend achromat (MBA) lattice [1, 2] is adopted to achieve extremely small emittance values of a few hundred pmrad or less. The straight sections, where insertion devices (IDs) are installed, are designed to be dispersion-free not to degrade the source size or the brilliance due to the finite energy spread of the electron beam. The gap of IDs can be freely changed according to users' needs (independent tuning). In such extremely low emittance storage rings, the radiation from bending magnets is generally weak, in contrast to the third-generation light source storage rings, and the ID gap change can cause a variation of the energy loss and hence a non-negligible effect on the emittance. This emittance variation during the independent tuning of IDs will be one of major obstacles for conducting precise experiments in extremely low emittance light source storage rings [3-5].
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expert
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What is the natural emittance in a storage ring?
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It is the equilibrium emittance where damping and quantum excitation balance out.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ Figure I.10.2 shows the development of the peak brilliance of $\\mathrm { \\Delta X }$ -ray sources during the last century. Scientists working in synchrotron radiation facilities have gotten accustomed to an extremely high flux, as well as an excellent stability of their X-ray source. The flux is controlled on the permille level, and the position stability is measured in micrometers. Synchrotron radiation was first observed on April 24, 1947 by Herb Pollock, Robert Langmuir, Frank Elder, and Anatole Gurewitsch, when they saw a gleam of bluish-white light emerging from the transparent vacuum tube of their new $7 0 \\mathrm { M e V }$ electron synchrotron at General Electric’s Research Laboratory in Schenectady, New York1. It was first considered a nuisance because it caused the particles to lose energy, but it was recognised in the 1960s as radiation with exceptional properties that overcame the shortcomings of X-ray tubes. Furthermore, it was discovered that the emission of radiation improved the emittance of the beams in electron storage rings, and additional series of dipole magnets were installed at the Cambridge Electron Accelerator (CEA) at Harvard University to provide additional damping of betatron and synchrotron oscillations. The evolution of synchrotron sources has proceeded in four generations, where each new generation made use of unique new features in science and engineering to increase the coherent flux available to experiments:
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expert
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What is the natural emittance in a storage ring?
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It is the equilibrium emittance where damping and quantum excitation balance out.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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I.10.3.5 Some observations I.10.3.5.1 Dependence of damping times on particle energy and type As you can see in Equation I.10.9, the radiation power emitted by a charged particle circulating in a storage ring is inversely proportional to the fourth power of its mass, for a given energy. This fundamental relationship has profound implications for the damping times observed in electron versus proton accelerators. Electron storage rings have typically damping times on the order of tens of milliseconds. Protons, in contrast, typically emit a negligible amount of synchrotron radiation at the same energy. Consequently, the damping times of proton accelerators extend much longer, often on the order of days. In these cases, damping may typically be neglected, and the beam emittance remains constant for stored beams. I.10.3.5.2 Top-up injection Radiation damping, a distinctive feature in electron accelerators, facilitates an innovative operational mode known as top-up injection. In this mode, rather than filling the storage ring once and then gradually losing beam current due to scattering and other losses, the bunches stored in the accelerator continually or periodically receive additional charges. New particles are injected close to the existing bunches in phase space. Due to the presence of radiation damping, these freshly injected particles rapidly lose their excess emittance through the emission of synchrotron radiation, thereby reducing their oscillations around the ideal orbit. Consequently, they are effectively ‘sucked into’ the main beam, seamlessly integrating with the stored bunches.
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expert
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What is the natural emittance in a storage ring?
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It is the equilibrium emittance where damping and quantum excitation balance out.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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The term diffraction-limited refers to a system, typically in optics or imaging, where the resolution or image detail is primarily restricted by the fundamental diffraction of light rather than by imperfections or aberrations in the source, or in imaging components. In such a system, the performance reaches the theoretical physical limit dictated by the wave nature of the radiation. Achieving diffraction-limited performance means maximizing image sharpness and detail by minimizing all other sources of distortion or blurring to the extent that diffraction becomes the overriding factor in limiting resolution. The horizontal emittance, conversely, is typically an order of magnitude larger. The X-ray beams are thus not diffraction-limited in this dimension. Diffraction-limited storage rings (DLSRs) overcome these constraints by minimizing the horizontal emittance to a level such that horizontal and vertical beam sizes in the undulators are similar. The diffraction limit of the X-ray beam thus becomes the defining factor for the source size, which leads to beams that are transversely fully coherent. This increased coherence translates into improved resolution and contrast in experimental techniques like X-ray imaging and scattering. The implications of achieving diffraction-limited performance are profound. The significantly improved coherence of the X-ray beams allow scientists to use the full beam for diffraction experiments, opening doors to previously intractable scientific questions. We will now see how this is achieved, and discuss briefly the challenges for design, construction and operation.
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augmentation
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NO
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expert
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What is the natural emittance in a storage ring?
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It is the equilibrium emittance where damping and quantum excitation balance out.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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File Name:Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation.pdf Chapter I.10 Synchrotron radiation Rasmus Ischebeck Paul Scherrer Institut, Villigen, Switzerland Electrons circulating in a storage ring emit synchrotron radiation. The spectrum of this powerful radiation spans from the far infrared to the $\\boldsymbol { \\mathrm { \\Sigma } } _ { \\mathrm { X } }$ -rays. Synchrotron radiation has evolved from being a mere byproduct of particle acceleration to a powerful tool leveraged in diverse scientific and engineering fields. Indeed, synchrotrons are the most brilliant X-ray sources on Earth, and they find use in a wide range of fields in research. In this chapter, we will look at the generation of radiation of charged particles in an accelerator, at the influence of this on the beam dynamics, and on the physics behind applications of synchrotron radiation for research. I.10.1 Introduction It is difficult to overstate the importance of X-rays for medicine, research and industry. Already a few years after their discovery by Wilhelm Conrad Röntgen, their ability to penetrate matter established Xrays as an important diagnostics tool in medicine. Experiments with $\\mathrm { \\Delta X }$ -rays have come a long way since the inception of the first $\\mathrm { \\Delta } X$ -ray tubes. The short wavelength of $\\mathrm { \\Delta X }$ -rays allowed Rosalynd Franklin and Raymond Gosling to take diffraction images that would lead to the discovery of the structure of DNA. Today, X-ray diffraction is an indispensable tool in structural biology and in pharmaceutical research. Industrial applications of X-rays range from cargo inspection to sterilization and crack detection.
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augmentation
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NO
| 0
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expert
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What is the natural emittance in a storage ring?
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It is the equilibrium emittance where damping and quantum excitation balance out.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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How would you measure this radiation? I.10.7.27 Superconducting undulators What is the advantage of using undulators made with superconducting coils, in comparison to permanentmagnet arrays? What are drawbacks? I.10.7.28 In-vacuum undulators What are the advantages of using in-vacuum undulators? What are possible difficulties? I.10.7.29 Instrumentation How would you measure the vertical emittance in a storage ring? I.10.7.30 Top-up operation What are the advantages of top-up operation? What difficulties have to be overcome to establish top-up in a storage ring? (give one advantage and one difficulty for 1P.; give one more advantage and one more difficulty for $\\begin{array} { r } { 1 \\mathrm { P } . \\star . } \\end{array}$ .) I.10.7.31 Fundamental limits The SLS 2.0, a diffraction limited storage ring, aims for an electron energy of $2 . 4 \\mathrm { G e V }$ and an emittance of $1 2 6 \\mathrm { p m }$ . How far is this away from the de Broglie emittance, i.e. the minimum emittance given by the uncertainty principle? I.10.7.32 Applications Why are synchrotrons important for science? I.10.7.33 Applications What applications for industry are there to synchrotrons? I.10.7.34 Orbit correction Which devices are used to measure and correct the orbit inside a synchrotron?
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What is the natural emittance in a storage ring?
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It is the equilibrium emittance where damping and quantum excitation balance out.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ respectively. The solution to Maxwell’s equations for this time-varying charge and current density can be found by using the wave equation for the electromagnetic potentials. In the Lorentz gauge, this wave equation reads $$ \\vec { \\nabla } ^ { 2 } \\Phi - \\frac { 1 } { c ^ { 2 } } \\frac { \\partial ^ { 2 } \\Phi } { \\partial t ^ { 2 } } = - \\frac { e } { \\varepsilon _ { 0 } } \\vec { \\nabla } ^ { 2 } \\vec { A } - \\frac { 1 } { c ^ { 2 } } \\frac { \\partial ^ { 2 } \\vec { A } } { \\partial t ^ { 2 } } = - \\mu _ { 0 } \\vec { j } . $$ The general solutions for the potentials given by time-varying charge and current densities can be found by integrating over time and space $$ \\Phi ( \\vec { x } , t ) = \\frac { 1 } { 4 \\pi \\varepsilon _ { 0 } } \\int d ^ { 3 } \\vec { x } ^ { \\prime } \\int d t ^ { \\prime } \\frac { \\rho ( \\vec { x } ^ { \\prime } , t ) } { | \\vec { x } - \\vec { x } ^ { \\prime } | } \\delta \\left( t ^ { \\prime } + \\frac { \\vec { x } - \\vec { x } ^ { \\prime } } { c } - t \\right)
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What is the natural emittance in a storage ring?
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It is the equilibrium emittance where damping and quantum excitation balance out.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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I.10.7.55 Practical applications of synchrotron radiation The Italian Light Source Elettra is a 3rd generation synchrotron source with $2 5 9 \\mathrm { m }$ circumference, and can operate at beam energies of either $2 . 0 \\mathrm { G e V }$ or $2 . 4 \\mathrm { G e V } ,$ with beam currents of $3 1 0 \\mathrm { m A }$ and $1 6 0 \\mathrm { m A }$ , respectively. The Machine Director is feeling thirsty, and would like to use Elettra to make a splendid espresso. By assuming that all radiation emitted as SR from the dipole magnets can be converted into heat, calculate how much time is needed for the $2 . 0 \\mathrm { G e V }$ beam to heat up the espresso water from $2 0 ^ { \\circ } \\mathrm { C }$ to $8 8 ^ { \\circ } \\mathrm { C }$ . One espresso is $3 0 \\mathrm { m L }$ . The radius of curvature in the dipoles is $5 . 5 \\mathrm { m }$ . Neglect potential insertion devices! Hint: the specific heat capacity of water is $\\begin{array} { r } { c _ { w } = 4 . 1 8 6 \\frac { \\mathrm { ~ J ~ } } { \\mathrm { g K } } } \\end{array}$
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What is the natural emittance in a storage ring?
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It is the equilibrium emittance where damping and quantum excitation balance out.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ Computing the photon flux $\\dot { N } _ { \\gamma }$ for an undulator is even more elaborate than the calculation for a single dipole, and we just cite the result [2] $$ { \\dot { N } } _ { \\gamma } = 1 . 4 3 \\cdot 1 0 ^ { 1 4 } N I _ { b } Q _ { n } ( K ) , $$ where $$ Q _ { n } ( K ) = \\frac { 1 + K ^ { 2 } / 2 } { n } F _ { n } ( K ) . $$ We denote the harmonic number by $n = 2 m - 1$ with $m \\in \\mathbb { N }$ , the number of periods in the undulator by $N$ , the beam current in A by $I _ { b }$ , the undulator parameter by $K$ , and $F _ { n } ( K )$ is given by $$ \\begin{array} { r c l } { { F _ { n } ( K ) } } & { { = } } & { { \\displaystyle \\frac { n ^ { 2 } K ^ { 2 } } { ( 1 + K ^ { 2 } / 2 ) ^ { 2 } } \\left( J _ { ( n + 1 ) / 2 } ( Y ) - J _ { ( n - 1 ) / 2 } ( Y ) \\right) ^ { 2 } } } \\\\ { { } } & { { } } & { { } } \\\\ { { Y } } & { { = } } & { { \\displaystyle \\frac { n K ^ { 2 } } { 4 ( 1 + K ^ { 2 } / 2 ) } , } } \\end{array}
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What is the natural emittance in a storage ring?
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It is the equilibrium emittance where damping and quantum excitation balance out.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ The difference to Equation I.10.13 is small for $k _ { u } y \\ll 1$ and will be neglected in the following. Helical undulators have a magnetic field on the axis $$ \\begin{array} { r } { \\vec { B } ( z ) = \\vec { u } _ { x } B _ { 0 } \\cos ( k _ { u } z ) - \\vec { u } _ { y } B _ { 0 } \\sin ( k _ { u } z ) . } \\end{array} $$ A rigorous analytic discussion of helical undulators is somewhat easier since the longitudinal component of the electron velocity $v _ { z } = \\beta _ { z } c$ is constant. Planar undulators, however, are much more common in synchrotron radiation facilities, therefore we will continue our discussion using a magnetic field according to Equation I.10.13. The magnetic field exerts a force on the electron $$ m _ { e } \\gamma \\frac { \\mathrm { d } \\vec { v } } { \\mathrm { d } t } = \\vec { F } = - e \\vec { v } \\times \\vec { B }
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What is the natural emittance in a storage ring?
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It is the equilibrium emittance where damping and quantum excitation balance out.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ which is Bragg’s law. Note that contrary to the diffraction on a two-dimensional surface, which is often considered fir visible light, $\\mathrm { \\Delta } \\mathrm { X }$ -rays diffract on a three-dimensional crystal lattice. In this case, not only the exit angle matters, but also the incoming angle must fulfill the resonance condition! X-ray diffraction is one of the key techniques to resolve molecular structure in samples that can be crystallized. In the following section, we will look at different applications of synchrotron radiation in science, medicine and industry. I.10.6 Applications of synchrotron radiation Synchrotron radiation is used in a wide range of scientific and industrial applications, and over 60 synchrotron radiation sources are operating around the world. New facilities are under construction, reflecting the growing demand in research and industrial applications. I.10.6.1 Diffraction Coherent diffraction on crystals has been used before the emergence of synchrotrons, at the time enabled by X-ray tubes. The renowned Photo 51, recorded by Rosalind Franklin and her student Raymond Gosling, found its way (through dubious ways) into the hands of James Watson and Francis Crick, who used it to decipher the double helix structure of DNA (see Fig. I.10.12). Why do scientists use diffraction in place of imaging to determine the structure of molecules? Would it not be easier to simply magnify the X-ray image onto a detector, as we do in transmission electron microscopes?
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What is the natural emittance in a storage ring?
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It is the equilibrium emittance where damping and quantum excitation balance out.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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I.10.7.16 Critical energy For the electron beam of the previous exercise, calculate the critical photon energy $\\varepsilon _ { c }$ that is emitted by the superbends with $B = 6 \\mathrm { \\ : T }$ and draw a sketch of the radiation spectrum. What is the useful photon energy range for experiments, assuming that the spectral intensity should be within $1 \\%$ of the maximum value? I.10.7.17 Critical frequency What do we understand by critical frequency? a) The frequency $\\omega _ { c }$ at which a storage ring becomes unstable b) The frequency of the photons coming from an undulator c) The frequency $\\omega _ { c }$ at which the integrated spectral density of photons with $\\omega < \\omega _ { c }$ is $50 \\%$ of the total energy radiated d) The revolution frequency of the electrons in a synchrotron e) The frequency $\\omega _ { c }$ where the highest spectral density of photos is emitted f) The frequency $\\omega _ { c }$ at which critical components fail I.10.7.18 Undulator radiation Assume an undulator of $1 8 ~ \\mathrm { m m }$ period and $5 . 4 \\mathrm { ~ m ~ }$ length. The pole tip field is $B _ { t } = 1 . 5 \\ : \\mathrm { T } _ { : }$ , and the gap can be varied between 10 and $2 0 \\mathrm { m m }$ .
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What is the natural emittance in a storage ring?
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It is the equilibrium emittance where damping and quantum excitation balance out.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ where a dimensionless undulator parameter has been introduced, $$ K = \\frac { e B _ { 0 } } { m _ { e } c k _ { u } } . $$ The electron follows a sinusoidal trajectory $$ x ( z ) = - \\frac { K } { k _ { u } \\gamma \\beta _ { z } } \\sin ( k _ { u } z ) . $$ Synchrotron radiation is emitted by relativistic electrons in a cone with opening angle of approximately $\\frac { 1 } { \\gamma }$ (Equation I.10.7). In an undulator, the maximum angle of the particle velocity with respect to the undulator axis $\\begin{array} { r } { \\alpha = \\arctan ( \\frac { v _ { x } } { v _ { z } } ) } \\end{array}$ is always smaller than the opening angle of the radiation, therefore the radiation field may add coherently. Consider two photons emitted by a single electron at the points $A$ and $B$ , which are one half undulator period apart (see Fig. I.10.5) $$ \\overline { { A B } } = \\frac { \\lambda _ { u } } { 2 } .
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What is the natural emittance in a storage ring?
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It is the equilibrium emittance where damping and quantum excitation balance out.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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– Auger electrons: similarly to fluorescence, this effect starts with the ionization or excitation of an inner-shell electron due to the interaction with the X-ray photon. This leaves a vacancy in the inner shell, which is then filled with an outer-shell electron. However, instead of releasing the excess energy as a photon, the energy is transferred non-radiatively to another outer-shell electron. This transfer of energy gives the second electron enough energy to be ejected from the atom, resulting in the emission of what is known as an Auger electron. These processes are summarized in Fig. I.10.10. Inelastic processes always lead to an energy deposition in the material, often leading to radiation damage, which limits the exposure time in many X-ray experiments. I.10.5.3 Crystal diffraction Imagine many atoms, arranged in a regular lattice, illuminated by a coherent $\\mathrm { \\Delta X }$ -ray source. The elastic scattering on the electron clouds of these atoms will add constructively if all individual waves are in phase. This situation is shown in Fig. I.10.11. Considering a distance $d$ between the crystal planes, and referring to the notation in this figure, we get constructive interference when $$ ( A B + B C ) - ( A C ^ { \\prime } ) = n \\lambda
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expert
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What is the natural emittance in a storage ring?
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It is the equilibrium emittance where damping and quantum excitation balance out.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ \\vec { B } ( 0 , 0 , z ) = \\vec { u } _ { y } B _ { 0 } \\sin ( k _ { u } z ) , $$ where $k _ { u } = 2 \\pi / \\lambda _ { u }$ with $\\lambda _ { u }$ the period of the magnetic field, $B _ { 0 }$ is the maximum field and $\\vec { u } _ { y }$ is the unit vector in $y$ direction. Due to the Maxwell equations, the curl and divergence of the static magnetic field vanish in vacuum, $\\vec { \\nabla } \\times \\vec { B } = 0$ and $\\vec { \\nabla } \\cdot \\vec { B } = 0$ . Thus, the field acquires a $z$ component for $y \\ne 0$ $$ \\begin{array} { r c l } { { { \\cal B } _ { x } } } & { { = } } & { { 0 } } \\\\ { { { \\cal B } _ { y } } } & { { = } } & { { { \\cal B } _ { 0 } \\cosh ( k _ { u } y ) \\sin ( k _ { u } z ) } } \\\\ { { { \\cal B } _ { z } } } & { { = } } & { { { \\cal B } _ { 0 } \\sinh ( k _ { u } y ) \\cos ( k _ { u } z ) . } } \\end{array}
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expert
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What is the natural emittance in a storage ring?
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It is the equilibrium emittance where damping and quantum excitation balance out.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ and $$ \\vec { A } ( \\vec { x } , t ) = \\frac { 1 } { 4 \\pi \\varepsilon _ { 0 } C ^ { 2 } } \\int d ^ { 3 } \\vec { x } ^ { \\prime } \\int d t ^ { \\prime } \\frac { \\vec { j } ( \\vec { x } ^ { \\prime } , t ) } { | \\vec { x } - \\vec { x } ^ { \\prime } | } \\delta \\left( t ^ { \\prime } + \\frac { \\vec { x } - \\vec { x } ^ { \\prime } } { c } - t \\right) . $$ Solving the wave equation in this most general sense is quite elaborate. The derivation can be found in Jackson [1], Chapter 6. Here, we just cite the result: the intensity of the radiation per solid angle $d \\Omega$ and per frequency interval $d \\omega$ is given by $$ \\frac { d ^ { 3 } I } { d \\Omega d \\omega } = \\frac { e ^ { 2 } } { 1 6 \\pi ^ { 3 } \\varepsilon _ { 0 } c } \\left( \\frac { 2 \\omega \\rho } { 3 c \\gamma ^ { 2 } } \\right) ^ { 2 } \\left( 1 + \\gamma ^ { 2 } \\vartheta ^ { 2 } \\right) ^ { 2 } \\left[ K _ { 2 / 3 } ^ { 2 } ( \\xi ) + \\frac { \\gamma ^ { 2 } \\vartheta ^ { 2 } } { 1 + \\gamma ^ { 2 } \\vartheta ^ { 2 } } K _ { 1 / 3 } ^ { 2 } ( \\xi ) \\right] ,
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