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IPAC
|
How is alternating phase focusing applied to DLAs?
|
drift sections between grating cells lead to jumps in the synchronous phase, which can be designed to provide net focusing.
|
definition
|
Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
|
File Name:SIMULATION_OF_TAPERED_CO-PROPAGATING_STRUCTURES.pdf SIMULATION OF TAPERED CO-PROPAGATING STRUCTURES FOR DIELECTRIC LASER ACCELERATOR A. Leiva Genre∗, G. S. Mauro, D. Mascali, G. Torrisi, G. Sorbello1, INFN-LNS, Catania, Italy 1also with Dipartimento di Ingegneria Elettrica, Elettronica e Informatica, Università degli Studi di Catania, Catania, Italy A. Bacci, M. Rossetti Conti, INFN - Milano unit, Milan, Italy R. Palmeri, IREA-CNR, Naples, Italy Abstract One of the key aspects to provide on chip acceleration in Dielectric Laser Accelerators (DLA) from tens of $\\mathrm { k e V }$ up to MeV energies is the phase velocity tapering. This paper presents the simulated performance of sub-relativistic structures, based on tapered slot waveguides. We engineered channel/defect modification in order to obtain a variable phase velocity matched to the increasing velocity of the accelerated particles. Additionally, we present a hollow-core relativistic electromagnetic band gap (EGB) accelerating waveguide. In DLA structures co-propagating schemes are employed for higher efficiency and smaller footprint compared to the cross-propagating schemes. In this respect, we envisage tapered continuous copropagating structures that simultaneously allow wave launching/coupling, beam acceleration, and transverse focusing. The main figures of merit, such as the accelerating gradient, the total energy gain, and the transverse focusing/defocusing forces, are evaluated and used to guide the optimization of the channel/defect modification.
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augmentation
|
NO
| 0
|
IPAC
|
How is alternating phase focusing applied to DLAs?
|
drift sections between grating cells lead to jumps in the synchronous phase, which can be designed to provide net focusing.
|
definition
|
Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
|
Fixed Field Accelerators (FFAs) have been proposed for many applications where a large energy acceptance and rapid acceleration can be advantageous, such as muon colliders [1] and medical facilities [2–4]. A key characteristic that distinguishes FFAs from standard synchrotrons is that the accelerator parameters are a function of rigidity: this not only leads to variation in the Courant-Snyder functions, but also di!erent closed orbits for every energy, as in a cyclotron. This variation makes it di"cult to integrate FFA-style optics with other accelerator systems, as there is inevitably a mismatch in beam parameters leading to significant closedorbit distortion. The TURBO project [5] at the University of Melbourne seeks to explore some of these di"culties, with the ultimate goal of producing a beamline design to enable rapid energy switching for charged particle therapy [6]. The beam transfer line must constitute a ‘closed-dispersion arc’, with energy-independent beam position at either end. There have been several large energy acceptance beam delivery system designs using fixed fields. Where a large bending angle is required to reduce beamline size, usually strong focusing is achieved over the full range of rigidities with sextupolar and higher order multipoles [7, 8]. In BEAMLINE DESIGN Initial Considerations The design of achromatic insertions for synchrotrons is well understood, with standard schemes such as ‘missing bend’ and ‘Chasman-Green’ lattices [15] commonly employed for light sources. The first-order achromat theorem, requiring that the phase advance between the start and centre of the arc must be an odd multiple of $\\pi$ , is straightforward to achieve in a synchrotron, where the energy spread is low. In FFAs, the phase advance is in general a function of rigidity, unless the magnetic field $B$ follows the scaling law
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augmentation
|
NO
| 0
|
expert
|
How is alternating phase focusing applied to DLAs?
|
drift sections between grating cells lead to jumps in the synchronous phase, which can be designed to provide net focusing.
|
definition
|
Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
|
$$ Accordingly, in position and momentum coordinates, this reads $$ \\begin{array} { l } { \\displaystyle \\sigma _ { \\Delta P _ { z , n } } = \\sqrt [ 4 ] { \\frac { \\beta } { \\gamma } } \\sigma _ { \\Delta P _ { z } } } \\\\ { \\displaystyle \\sigma _ { \\Delta s , n } = \\sqrt [ 4 ] { \\frac { \\gamma } { \\beta } } \\sigma _ { \\Delta s } . } \\end{array} $$ Thus the adiabatic phase damping in DLAs behaves in the same way as in rf linacs. As a test of the code, we plot the long time evolution of the longitudinal emittance at zero transverse emittance for 3 different setups in Fig. 10. First, we consider a bunch matched according to Eq. (44) in linearized fields. As expected, the symplectic code preserves the emittance in linear fields. However, the linearly matched bunch shows emittance growth in the non-linear fields. Even stronger emittance increase is to be expected, when there is a mismatch of the bunch length and the energy spread (here we chose $10 \\%$ excess energy spread). The according result is obtained for the y-emittance when setting the synchronous phase into the transverse focusing regime and taking the longitudinal emittance as zero.
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4
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NO
| 1
|
expert
|
How is alternating phase focusing applied to DLAs?
|
drift sections between grating cells lead to jumps in the synchronous phase, which can be designed to provide net focusing.
|
definition
|
Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
|
$$ where $$ \\nabla _ { \\perp } \\nabla _ { \\perp } ^ { \\mathrm { T } } = \\left( { \\begin{array} { c c } { \\partial _ { x } ^ { 2 } } & { \\partial _ { x } \\partial _ { y } } \\\\ { \\partial _ { y } \\partial _ { x } } & { \\partial _ { y } ^ { 2 } } \\end{array} } \\right) $$ is the Hessian. The expansion Eq. (20) about $x _ { 0 } = 0$ , $y _ { 0 } = 0$ of Eq. (18) results in $$ \\underline { { \\vec { f } } } _ { m } ( \\Delta x , \\Delta y ) = \\frac { \\lambda _ { g z } } { 2 \\pi } \\underline { { e } } _ { m } ( 0 , 0 ) \\binom { i k _ { x } - k _ { x } ^ { 2 } \\Delta x } { - k _ { y } ^ { 2 } \\Delta y } , $$ i.e., a position independent (coherent) kick component in $x$ -direction, vanishing for $\\alpha = 0$ . Using this abstract derivation, the results of several papers proposing DLA undulators [12–14] can be recovered.
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2
|
NO
| 0
|
expert
|
How is alternating phase focusing applied to DLAs?
|
drift sections between grating cells lead to jumps in the synchronous phase, which can be designed to provide net focusing.
|
definition
|
Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
|
II. FIELDS AND KICKS IN PERIODIC STRUCTURES Usual particle tracking algorithms solve Maxwell’s equations with a predefined time step. Instead of that, we make use of the periodicity of the structure and apply only the kicks which are known not to average out a priori. The other field harmonics are neglected. The validity of this neglect depends on the effect of transients which is effectively suppressed when the structure period is matched to the beam velocity. With no loss of generality we restrict ourselves here to an infrared laser with $\\lambda _ { 0 } = 1 . 9 6 \\mu \\mathrm { m }$ and structures made of Silicon $\\left( \\varepsilon _ { r } = 1 1 . 6 3 \\right)$ . A single cell of a symmetrically driven Bragg mirror cavity structure is shown in Fig. 3. A. Analysis of the longitudinal field A coordinate system is applied such that the electron beam propagates in positive $\\mathbf { \\delta } _ { Z } .$ -direction and the $\\mathbf { Z }$ -polarized laser propagates in y-direction. The unit cell of a periodic dielectric structure has dimensions $\\lambda _ { g x }$ and $\\lambda _ { g z }$ . In order to where the underlined electric field is a phasor at the fixed frequency $\\omega = 2 \\pi c / \\lambda _ { 0 }$ of the laser, and $q$ is the charge $( q = - e$ for electrons). The variable $s$ denotes the relative position of the particle behind an arbitrarily defined reference particle moving at $z = v t$ . Thus $z$ is the absolute position in the laboratory frame, while $s$ denotes the phase shift with respect to $z$ . Due to the $z$ -periodicity, the laser field can be expanded in spatial Fourier series
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2
|
NO
| 0
|
IPAC
|
How is alternating phase focusing applied to DLAs?
|
drift sections between grating cells lead to jumps in the synchronous phase, which can be designed to provide net focusing.
|
definition
|
Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
|
The transversal components of the accelerating mode vanish at the center of the gap. Small deviations around this stability point present transversal electric fields magnitude much lower than the longitudinal field. In the scenario where the accelerator lengths are greater, transverse focusing can be achieved using ponderomotive forces or alternating phase focusing [12]. CONCLUSION AND FUTURES PERSPECTIVES A co-propagating DLA structure has been presented, optimized for the acceleration of sub-relativistic electrons in the $\\mathrm { k e V }$ kinetic energy ranges. For the first time, this structure has been modeled and simulated by using CST Microwave Studio, joining both electromagnetic and Particle-In-Cell numerical tools. Rib tapering proves to be a potential tool for tailoring slot waveguide accelerators according to the electron energy. An accelerating gradient of $G _ { z } = 0 . 2 2 7 \\mathrm { G V } \\mathrm { m } ^ { - 1 }$ was obtained for the tapered slot waveguide DLA. This value is above compared to those nowadays accelerator ones. An energy gain of $\\Delta W ~ =$ $4 . 5 \\mathrm { k e V }$ for electrons with $T _ { 0 } ~ = ~ 8 0 \\mathrm { k e V }$ was achieved. Further testing and beam manipulation are still required for both experimental and industrial applications. Further ideas such as cascade acceleration by means of tapered slot waveguides DLA could be implemented for the acceleration of sub-relativistic particles for low to mid-energy ranges. Also, a two-stage accelerator set-up (two DLAs) can be used for bunching and acceleration respectively. Next steps will include the evaluation of RF acceptance and the two-stage configuration study.
|
4
|
NO
| 1
|
expert
|
How is alternating phase focusing applied to DLAs?
|
drift sections between grating cells lead to jumps in the synchronous phase, which can be designed to provide net focusing.
|
definition
|
Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
|
$$ where $k _ { \\mathrm { y } }$ is given by Eq. (17). The tracking equations are where an explicit scheme is obtained by applying first the “kicks” and then the “pushes”. The adiabatic damping in the transverse planes is described by $$ A ^ { ( n ) } = \\frac { ( \\beta \\gamma ) ^ { ( n + 1 ) } } { ( \\beta \\gamma ) ^ { ( n ) } } = 1 + \\bigg [ \\frac { \\lambda _ { 0 } q \\mathrm { R e } \\{ e ^ { i \\varphi _ { s } } \\underline { { e } } _ { 1 } \\} } { \\beta \\gamma m _ { e } c ^ { 2 } } \\bigg ] ^ { ( n ) } . $$ Symplecticity of the scheme is confirmed by calculating $$ \\operatorname* { d e t } \\frac { \\partial ( x , x ^ { \\prime } , y , y ^ { \\prime } , \\varphi , \\delta ) ^ { ( n + 1 ) } } { \\partial ( x , x ^ { \\prime } , y , y ^ { \\prime } , \\varphi , \\delta ) ^ { ( n ) } } = A ^ { ( n ) 2 } ,
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2
|
NO
| 0
|
expert
|
How is alternating phase focusing applied to DLAs?
|
drift sections between grating cells lead to jumps in the synchronous phase, which can be designed to provide net focusing.
|
definition
|
Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
|
$$ If the beam size is significantly smaller than the aperture $( y \\ll \\beta \\gamma c / \\omega )$ , the longitudinal equation decouples and becomes the ordinary differential equation of synchrotron motion. The transverse motion becomes linear in this case, however still dependent on the longitudinal motion via $\\varphi$ . The equation of motion, $$ \\ddot { y } = \\frac { - q e _ { 1 } \\omega } { m _ { e } \\gamma ^ { 3 } \\beta c } \\sin ( \\varphi ) y , $$ is Hill’s equation, with the synchrotron angle being the focusing function. However there is a crucial difference to ordinary magnetic focusing channels. The focusing force scales as $\\gamma ^ { - 3 }$ as expected for acceleration defocusing [19], rather than with $\\gamma ^ { - 1 }$ as would be expected for a magnetic quadrupole focusing channel. The solution to Eq. (39) as function of $z$ for fixed $s = \\lambda _ { g z } \\varphi _ { s } / 2 \\pi$ , i.e., when the bunch length is significantly shorter than the period length, is $$ y = y _ { 0 } \\exp \\left( \\sqrt { \\frac { - q e _ { 1 } \\omega } { m _ { e } \\gamma ^ { 3 } \\beta ^ { 3 } c ^ { 3 } } \\sin \\varphi _ { s } z } \\right)
|
5
|
NO
| 1
|
expert
|
How is alternating phase focusing applied to DLAs?
|
drift sections between grating cells lead to jumps in the synchronous phase, which can be designed to provide net focusing.
|
definition
|
Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
|
$$ where $\\operatorname { s i n c } ( \\cdot ) = \\sin ( \\pi \\cdot ) / ( \\pi \\cdot )$ . The electric field phasor and its spatial Fourier coefficients for the structure in Fig. 3 are plotted in Fig. 4. It has a small real part, which is coincidental, and a strong first and weak second harmonic. If the round braces in Eq. (5) is non-integer, the energy gain averages to zero, if it is integer other than zero, it directly vanishes. Thus we have the phase synchronicity condition $$ \\lambda _ { g z } = m \\beta \\lambda _ { 0 } $$ and the particle’s energy gain simplifies to $$ \\begin{array} { l } { \\displaystyle \\Delta W ( x , y ; s ) = q \\lambda _ { g z } \\mathrm { R e } \\{ e ^ { 2 \\pi i \\frac { s } { \\beta \\lambda _ { 0 } } } \\underline { { e } } _ { m } ( x , y ) \\} } \\\\ { = q \\lambda _ { g z } | \\underline { { e } } _ { m } | \\cos { \\bigg ( 2 \\pi \\frac { s } { \\beta \\lambda _ { 0 } } + \\varphi _ { m } \\bigg ) } , } \\end{array}
|
augmentation
|
NO
| 0
|
expert
|
How is alternating phase focusing applied to DLAs?
|
drift sections between grating cells lead to jumps in the synchronous phase, which can be designed to provide net focusing.
|
definition
|
Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
|
V. APPLICATIONS We apply our approach to similar experimental parameters as for the subrelativistic experiments at FAU Erlangen [3] and the relativistic experiments at SLAC [1,2]. Although the structures are idealized, the results are qualitatively recovered. As a next step, we show modifications and idealizations of the beam parameters, which outline the way to a microchip accelerator. A. Subrelativistic acceleration A subrelativistic DLA structure needs to be chirped in order to always fulfill the synchronicity condition (6) for the synchronous particle. The proper chirp for each cell and the synchronous velocity are obtained by iterating the two equations $$ \\Delta z ^ { ( n + 1 ) } = \\frac { q e _ { 1 } \\lambda _ { 0 } ^ { 2 } \\cos \\varphi _ { s } ^ { ( n ) } } { m _ { e } c ^ { 2 } } \\sqrt { 1 - { \\beta ^ { ( n ) } } ^ { 2 } } $$ $$ \\beta ^ { ( n + 1 ) } = \\beta ^ { ( n ) } + \\frac { \\Delta z ^ { ( n + 1 ) } } { \\lambda _ { 0 } } .
|
augmentation
|
NO
| 0
|
expert
|
How is alternating phase focusing applied to DLAs?
|
drift sections between grating cells lead to jumps in the synchronous phase, which can be designed to provide net focusing.
|
definition
|
Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
|
III. TRACKING EQUATIONS In order to study the motion of particles in the fields of periodic gratings we approximate the forces by one kick per grating period and track with the symplectic Euler method. In spite of the very high gradients in DLA structures, the energy can still be seen as an adiabatic variable, as it is the case in conventional linacs. Tracking the full time dependence of γ, as required for example in plasma accelerators, can be avoided due to the shortness of the periods. For simplicity, we restrict ourselves to $m = 1$ from this point and introduce normalized variables in the paraxial approximation $$ \\begin{array} { c c c } { { x ^ { \\prime } = \\displaystyle \\frac { p _ { x } } { p _ { z 0 } } , } } & { { \\Delta x ^ { \\prime } = \\displaystyle \\frac { \\Delta p _ { x } ( x , \\mathbf { y } , \\varphi ) } { p _ { z 0 } } , } } & { { } } \\\\ { { y ^ { \\prime } = \\displaystyle \\frac { p _ { y } } { p _ { z 0 } } , } } & { { \\Delta y ^ { \\prime } = \\displaystyle \\frac { \\Delta p _ { y } ( x , \\mathbf { y } , \\varphi ) } { p _ { z 0 } } , } } & { { } } \\\\ { { \\varphi = 2 \\pi \\displaystyle \\frac { s } { \\lambda _ { g z } } , } } & { { \\delta = \\displaystyle \\frac { W - W _ { 0 } } { W _ { 0 } } , } } & { { } } \\\\ { { \\Delta \\delta = \\displaystyle \\frac { \\Delta W ( x , \\mathbf { y } , \\varphi ) - \\Delta W ( 0 , 0 , \\varphi _ { s } ) } { W _ { 0 } } , } } & { { } } \\end{array}
|
augmentation
|
NO
| 0
|
expert
|
How is alternating phase focusing applied to DLAs?
|
drift sections between grating cells lead to jumps in the synchronous phase, which can be designed to provide net focusing.
|
definition
|
Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
|
allow the laser field to escape the structure, open boundaries in positive and negative y-direction are assumed. The energy gain of a particle in one cell is $$ \\underline { { E _ { z } } } ( x , y , z ) = \\sum _ { m = - \\infty } ^ { \\infty } \\underline { { e } } _ { m } ( x , y ) e ^ { - i m \\frac { 2 \\pi } { \\lambda _ { g z } } z } $$ $$ \\begin{array} { r l r } { { \\Delta W ( x , y ; s ) = q \\int _ { - \\lambda _ { g z } / 2 } ^ { \\lambda _ { g z } / 2 } E _ { z } ( x , y , z ; t = ( z + s ) / v ) \\mathrm { d } z } } \\\\ & { } & \\\\ & { } & { = q \\int _ { - \\lambda _ { g z } / 2 } ^ { \\lambda _ { g z } / 2 } \\mathrm { R e } \\{ \\underline { { E } } _ { z } ( x , y , z ) e ^ { i \\omega ( z + s ) / v } \\} \\mathrm { d } z , } \\end{array}
|
augmentation
|
NO
| 0
|
expert
|
How is alternating phase focusing applied to DLAs?
|
drift sections between grating cells lead to jumps in the synchronous phase, which can be designed to provide net focusing.
|
definition
|
Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
|
$$ The derivatives in $y$ -direction can be determined by the dispersion relation for the synchronous mode. We have $$ k _ { z } = \\frac { \\omega } { \\beta c } , ~ k _ { x } = \\frac { 2 \\pi } { \\beta \\lambda _ { 0 } } \\tan { \\alpha } ~ \\mathrm { a n d } ~ k = \\frac { \\omega } { c } $$ and thus $$ k _ { y } = \\pm \\sqrt { k ^ { 2 } - ( k _ { z } ^ { 2 } + k _ { x } ^ { 2 } ) } = \\pm \\frac { \\omega } { c } \\sqrt { 1 - \\frac { 1 } { \\beta ^ { 2 } } ( 1 + \\tan ^ { 2 } \\alpha ) } . $$ For a nontilt grating $( \\alpha = 0 )$ ) this is the well-known evanescent decay of the near field $k _ { y } = i \\omega / ( \\beta \\gamma c )$ . Once $k _ { x } , k _ { y }$ are determined, the fields can be found from
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augmentation
|
NO
| 0
|
IPAC
|
How is the transformer ratio defined?
|
It is the ratio of the maximum electric field behind the drive bunch to the maximum electric field within the bunch
|
Definition
|
Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
|
For the purpose of quantifying the performance of the DRFB loop in the whole system, the amplitude ratio $\\alpha _ { \\mathrm { d r f } }$ and gain $\\gamma _ { \\mathrm { d r f } }$ are introduced as follows : $$ \\alpha _ { \\mathrm { d r f } } = \\vert \\tilde { I } _ { \\mathrm { g , d r f } } / \\tilde { I } _ { \\mathrm { g , d r i v e } } \\vert , \\gamma _ { \\mathrm { d r f } } = \\alpha _ { \\mathrm { d r f } } / ( 1 - \\alpha _ { \\mathrm { d r f } } ) . $$ These definitions are the same as in Ref. [5] and $\\alpha _ { \\mathrm { d r f } }$ is often represented in dB units. A typical phasor diagram of the DRFB loop for HC is shown in Fig. 2. Each phasor is normalized to $| \\tilde { V } _ { \\mathrm { c } } |$ . The parameter set corresponds to the case shown in Fig. 4 at $( G _ { \\mathrm { d r f } } , \\phi _ { \\mathrm { d r f } } ) { = } ( 0 . 2 , { - } 3 0 )$ and $\\alpha _ { \\mathrm { d r f } }$ is estimated to be $\\mathbf { - } 3 . 2 \\mathrm { d B }$ .
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augmentation
|
NO
| 0
|
IPAC
|
How is the transformer ratio defined?
|
It is the ratio of the maximum electric field behind the drive bunch to the maximum electric field within the bunch
|
Definition
|
Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
|
$$ where $\\alpha , \\beta , \\gamma$ are the Twiss parameters of the boundary ellipse. The green curve in Fig. 7 shows the dependence of the total emittance on the cut-off threshold for the distribution in Fig.3. Indeed, the total emittance is sensitive to the distribution relative density in the range of $1 0 ^ { - 6 } - 1 0 ^ { - 4 }$ and, thus can be used to characterize the halo. It is more convenient to use the dimensionless Halo Ratio parameter $( R , t )$ defined as: $$ \\begin{array} { r } { \\left( R , t \\right) = \\left( \\sqrt { \\frac { \\varepsilon _ { b } } { \\varepsilon _ { R M S } } } , \\mathrm { t } \\right) } \\\\ { t = - \\mathrm { l o g } \\left( \\frac { f _ { m a x } } { f _ { m i n } } \\right) , } \\end{array} $$ where $f _ { m a x }$ and $f _ { m i n }$ are the maximum and minimum of the distribution function. The cut-off threshold is included explicitly in the definition to make the parameter unambiguously calculable.
|
augmentation
|
NO
| 0
|
IPAC
|
How is the transformer ratio defined?
|
It is the ratio of the maximum electric field behind the drive bunch to the maximum electric field within the bunch
|
Definition
|
Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
|
$$ Z _ { _ { L } } = - 2 Z _ { _ { c c } } \\log { \\frac { S _ { 2 1 } ^ { D U T } } { S _ { 2 1 } ^ { r e f } } } . $$ $$ Z _ { T } = - \\frac { 2 c Z _ { d d } } { \\omega \\Delta ^ { 2 } } \\log \\frac { S _ { d d 2 1 } ^ { D U T } } { S _ { d d 2 1 } ^ { r e f } } . $$ Where $\\Delta = 1 0 m m$ is the interval between the twin wires, $1 0 \\mu m$ wire radius, c is the velocity of light, the $Z _ { c c }$ and $Z _ { d d }$ are characteristic impedances for the common (cc) and the differential (dd) modes, respectively. The transmission coefficients $S _ { 2 1 } ^ { D U T }$ , $S _ { d d 2 1 } ^ { D U T } , S _ { 2 1 } ^ { r e f }$ and $S _ { d d 2 1 } ^ { r e f }$ , belong to the resistive device under test (DUT) and those for the perfectly conductive chambers (ref).
|
augmentation
|
NO
| 0
|
IPAC
|
How is the transformer ratio defined?
|
It is the ratio of the maximum electric field behind the drive bunch to the maximum electric field within the bunch
|
Definition
|
Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
|
ALGORITHM IMPLEMENTATIONS Transverse emittance is an important parameter of characterizing accelerator performance. For the main Linac, dispersion is negligible and the beam size mainly determined by the betatron-oscillation and transverse emittance. The quadrupole scanning method is one of the most commonly used method to obtain the emittance and Twiss parameter. Here profile monitor is used to measure beam size information. The transformation and relationship of Twiss parameters are shown in Eqs. (1) and (2), where $\\sigma$ is beam size at PR, and $r _ { i j }$ is the transfer matrix element for Twiss parameters. $$ { \\left( \\begin{array} { l } { { \\sigma _ { 1 } } ^ { 2 } } \\\\ { { \\sigma _ { 2 } } ^ { 2 } } \\\\ { \\vdots } \\\\ { { \\sigma _ { n } } ^ { 2 } } \\end{array} \\right) } = { \\left( \\begin{array} { l l l } { r _ { 1 1 } } & { r _ { 1 2 } } & { r _ { 1 3 } } \\\\ { r _ { 2 1 } } & { r _ { 2 2 } } & { r _ { 2 3 } } \\\\ { \\vdots } & { \\vdots } & { \\vdots } \\\\ { r _ { n 1 } } & { r _ { n 2 } } & { r _ { n 3 } } \\end{array} \\right) } \\left( { \\begin{array} { l } { \\beta \\epsilon } \\\\ { \\alpha \\epsilon } \\\\ { \\gamma \\epsilon } \\end{array} } \\right)
|
augmentation
|
NO
| 0
|
expert
|
How is the transformer ratio defined?
|
It is the ratio of the maximum electric field behind the drive bunch to the maximum electric field within the bunch
|
Definition
|
Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
|
$$ where the bunch length $l$ is $$ l = \\frac { \\frac { \\pi } { 2 } + \\sqrt { \\mathcal { R } ^ { 2 } - 1 } - 1 } { k _ { n } } . $$ Evaluating the form factor at $k = k _ { n }$ produces $$ | F ( k _ { n } ) | = \\frac { 2 \\mathcal { R } } { \\mathcal { R } ^ { 2 } + \\pi - 2 } . $$ This result leads to the accelerating gradient $E _ { \\mathrm { a c c } }$ scaling with the inverse of the transformer ratio $\\mathcal { R }$ . [1] A. Zholents, S. Baturin, D. Doran, W. Jansma, M. Kasa, A. Nassiri, P. Piot, J. Power, A. Siy, S. Sorsher, K. Suthar, W. Tan, E. Trakhtenberg, G. Waldschmidt, and J. Xu, A compact high repetition rate free-electron laser based on the Advanced Wakefield Accelerator Technology, in Proceedings of the 11th International Particle Accelerator Conference, IPAC-2020, CAEN, France (2020), https:// ipac2020.vrws.de/html/author.htm. [2] A. Zholents et al., A conceptual design of a compact wakefield accelerator for a high repetition rate multi user X-ray Free-Electron Laser Facility, in Proceedings of the 9th International Particle Accelerator Conference, IPAC’18, Vancouver, BC, Canada (JACoW Publishing, Geneva, Switzerland, 2018), pp. 1266–1268, 10.18429/ JACoW-IPAC2018-TUPMF010. [3] G. Voss and T. Weiland, The wake field acceleration mechanism, DESY Technical Report No. DESY-82-074, 1982. [4] R. J. Briggs, T. J. Fessenden, and V. K. Neil, Electron autoacceleration, in Proceedings of the 9th International Conference on the High-Energy Accelerators, Stanford, CA, 1974 (A.E.C., Washington, DC, 1975), p. 278 [5] M. Friedman, Autoacceleration of an Intense Relativistic Electron Beam, Phys. Rev. Lett. 31, 1107 (1973). [6] E. A. Perevedentsev and A. N. Skrinsky, On the use of the intense beams of large proton accelerators to excite the accelerating structure of a linear accelerator, in Proceedings of 6th All-Union Conference Charged Particle Accelerators, Dubna (Institute of Nuclear Physics, Novosibirsk, USSR, 1978), Vol. 2, p. 272; English version is available in Proceedings of the 2nd ICFA Workshop on Possibilities and Limitations of Accelerators and Detectors, Les Diablerets, Switzerland, 1979 (CERN, Geneva, Switzerland,1980), p. 61
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1
|
NO
| 0
|
IPAC
|
How is the transformer ratio defined?
|
It is the ratio of the maximum electric field behind the drive bunch to the maximum electric field within the bunch
|
Definition
|
Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
|
$$ One can then define a normalized inductance $L _ { p u }$ with only two variables as follows: $$ { \\cal L } ( N , r _ { c o r e } , F F _ { a r } , F F _ { b r } ) = N ^ { 2 } . r _ { c o r e } . { \\cal L } _ { p u } ( F F _ { a r } , F F _ { b r } ) $$ Where $L _ { p u } ( F F _ { a r } , F F _ { b r } )$ is a per-unit inductance in $\\scriptstyle { \\mathrm { H / m / t u r n } } ^ { 2 }$ with $N { = } 1$ , $r _ { c o r e } = 1 \\mathrm { m }$ . Since the analytical expression of Eq. (2) may not be applicable with sufficient accuracy for all specifications of the applications aimed in this article, the calculation of the inductance $L _ { p u } ( F F _ { a r } , F F _ { b r } )$ is performed in magneto-statics using a 2D finite element method for axisymmetric coordinates with a current of $\\scriptstyle { I = 1 \\mathrm { A } }$ . This normalized approach facilitates the robustness and efficiency of the learning process.
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1
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NO
| 0
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IPAC
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How is the transformer ratio defined?
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It is the ratio of the maximum electric field behind the drive bunch to the maximum electric field within the bunch
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Definition
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Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
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Inductance value estimation The main geometric dimensions of an air-core inductor are shown in Fig.1. The design variables are the three geometric dimensions 𝑟௖௢௥௘, 𝑎௖௢௜௟, 𝑏௖௢௜௟ and the number of turns 𝑁. To calculate the inductance $L ( N , r _ { c o r e } , a _ { c o i l } , b _ { c o i l } )$ , a method of normalizing the dimensional variables has been adopted. A simplified analytical expression of $L$ [3] was previously used to establish a suitable normalization base: $$ \\begin{array} { r } { L ( N , r _ { c o r e } , a _ { c o i l } , b _ { c o i l } ) = k _ { L } N ^ { 2 } \\frac { ( 2 . r _ { c o r e } + a _ { c o i l } ) ^ { 2 } } { 6 . r _ { c o r e } + 1 3 . a _ { c o i l } + 9 . b _ { c o i l } } } \\end{array}
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1
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NO
| 0
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IPAC
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How is the transformer ratio defined?
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It is the ratio of the maximum electric field behind the drive bunch to the maximum electric field within the bunch
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Definition
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Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
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$$ V " _ { c j } = V _ { j } \\left( 1 + \\Gamma _ { j } \\right) n _ { j } = 2 \\alpha _ { j } \\sqrt { \\alpha _ { 0 } / \\alpha _ { j } } e ^ { i \\Psi } \\cos \\Psi V _ { j } $$ Then, if such voltage is transported to the $\\mathbf { k }$ -th load via the multiplication by the turn ratio $1 / n _ { k }$ , one has exactly the voltage $\\boldsymbol { V } _ { j k } ^ { r }$ induced on the $\\mathbf { k }$ -th load by the j-th feed $$ V _ { j k } ^ { r } = \\frac { { V ^ { \\prime } } _ { c j } } { n _ { k } } = 2 \\alpha _ { j } \\sqrt { \\alpha _ { k } / \\alpha _ { j } } e ^ { i \\Psi } \\cos \\Psi V _ { j } $$ Therefore, the overall voltage on the $\\mathbf { k }$ -th feed is given by
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1
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NO
| 0
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expert
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How is the transformer ratio defined?
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It is the ratio of the maximum electric field behind the drive bunch to the maximum electric field within the bunch
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Definition
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Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
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Table: Caption: TABLE I. Parameters and variables used throughout the paper. Body: <html><body><table><tr><td colspan="2">Parameter</td></tr><tr><td>K</td><td>Wakefield loss factor</td></tr><tr><td>βg</td><td>Normalized group velocity</td></tr><tr><td>vg</td><td>Group velocity</td></tr><tr><td>α</td><td>Attenuation constant</td></tr><tr><td>Q</td><td>Quality factor</td></tr><tr><td>0</td><td>Electrical conductivity</td></tr><tr><td></td><td>Corrugation spacing parameter</td></tr><tr><td>S</td><td>Corrugation sidewall parameter</td></tr><tr><td>a</td><td>Corrugation minor radius</td></tr><tr><td>t</td><td>Corrugation tooth width</td></tr><tr><td>g</td><td>Corrugation vacuum gap</td></tr><tr><td>d</td><td>Corrugation depth</td></tr><tr><td>p</td><td>Corrugation period</td></tr><tr><td>rt</td><td>Corrugation tooth radius</td></tr><tr><td>rg</td><td>Corrugation vacuum gap radius</td></tr><tr><td>L</td><td>Corrugated waveguide length</td></tr><tr><td>F</td><td>Bunch form factor</td></tr><tr><td>q0</td><td>Drive bunch charge</td></tr><tr><td>fr</td><td>Bunch repetition rate</td></tr><tr><td>T</td><td>rf pulse decay time constant</td></tr><tr><td>8</td><td>Skin depth</td></tr><tr><td>Pf</td><td>rf pulse power envelope</td></tr><tr><td>P</td><td>Instantaneous rf pulse power</td></tr><tr><td>Eacc</td><td>Accelerating field</td></tr><tr><td>Emax</td><td>Peak surface E field</td></tr><tr><td>Hmax</td><td>Peaksurface H field</td></tr><tr><td>Qdiss</td><td>Energy dissipation</td></tr><tr><td>Pd</td><td>Power dissipation distribution</td></tr><tr><td>W</td><td>Average thermal power density</td></tr><tr><td>△T</td><td>Transient temperature rise</td></tr><tr><td>C</td><td>Speed of light</td></tr><tr><td>Z0</td><td>Impedance of free space</td></tr><tr><td>8</td><td>Initial beam energy</td></tr><tr><td>R</td><td>Transformer ratio</td></tr></table></body></html> In the parametric analysis that follows, the corrugation dimensions are expressed in terms of the normalized spacing parameter $\\xi$ and sidewall parameter $\\zeta$ defined as $$ \\begin{array} { l } { \ { \\xi = \\frac { g - t } { p } } , } \\\\ { \ { \\zeta = \\frac { g + t } { p } } . } \\end{array} $$ The spacing parameter $\\xi$ determines the spacing between the corrugation teeth and ranges from $^ { - 1 }$ to 1 for the minimum and maximum radii profiles, where positive values of $\\xi$ result in spacing greater than the tooth width and vice versa for negative values. The sidewall parameter $\\zeta$ controls the sidewall angle of the unequal radii profile, where $\\zeta < 1$ leads to tapered sidewalls and $\\zeta > 1$ leads to undercut sidewalls. These dependencies are illustrated in Fig. 4.
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2
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NO
| 0
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IPAC
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How is the transformer ratio defined?
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It is the ratio of the maximum electric field behind the drive bunch to the maximum electric field within the bunch
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Definition
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Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
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$$ Where $\\alpha _ { 0 } : = { \\cal Q } _ { \\scriptscriptstyle L } / { \\cal Q } _ { 0 } , \\alpha _ { \\scriptscriptstyle k } : = \\alpha _ { 0 } \\beta _ { \\scriptscriptstyle k } , \\Psi : = \\mathrm { a t a n } ( - { \\mathrm Q } _ { \\scriptscriptstyle L } \\delta ) .$ The reflection coefficient at AA’ is given by $\\Gamma _ { _ { A d ^ { \\prime } } } = \\Gamma _ { \\boldsymbol { k } } = 2 \\alpha _ { \\boldsymbol { k } } e ^ { i \\Psi } \\cos \\Psi - 1$ and the reflected voltage is given by $$ V _ { k k } ^ { r } = \\sqrt { 2 R P _ { k } } e ^ { i \\phi _ { k } } \\left( 2 \\alpha _ { k } e ^ { i \\Psi } \\cos \\Psi - 1 \\right) $$ Now, we need to add all the transmitted voltages from the other feeds. For such a purpose, let us suppose that the $\\mathrm { j }$ -th feed is active and the other ones are simply replaced by the dummy loads. In this case the cavity voltage is given by the transmitted voltage due to the $\\mathrm { j }$ -th feed transported to the secondary via the multiplication by the turn ratio nj.
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1
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NO
| 0
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expert
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How is the transformer ratio defined?
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It is the ratio of the maximum electric field behind the drive bunch to the maximum electric field within the bunch
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Definition
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Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
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File Name:Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf Design of a cylindrical corrugated waveguide for a collinear wakefield accelerator A. Siy ,1,2,\\* N. Behdad,1 J. Booske,1 G. Waldschmidt,2 and A. Zholents 2,† 1University of Wisconsin, Madison, Wisconsin 53715, USA 2Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439, USA (Received 30 May 2022; accepted 7 November 2022; published 7 December 2022) We present the design of a cylindrical corrugated waveguide for use in the A-STAR accelerator under development at Argonne National Laboratory. A-STAR is a high gradient, high bunch repetition rate collinear wakefield accelerator that uses a $1 - \\mathrm { m m }$ inner radius corrugated waveguide to produce a $9 0 \\ \\mathrm { M V } \\mathrm { m } ^ { - 1 }$ , 180-GHz accelerating field when driven by a $1 0 \\mathrm { - n C }$ drive bunch. To select a corrugation geometry for A-STAR, we analyze three types of corrugation profiles in the overmoded regime with $a / \\lambda$ ranging from 0.53 to 0.67, where $a$ is the minor radius of the corrugated waveguide and $\\lambda$ is the free-space wavelength. We find that the corrugation geometry that optimizes the accelerator performance is a rounded profile with vertical sidewalls and a corrugation period $p \\ll a$ . Trade-offs between the peak surface fields and thermal loading are presented along with calculations of pulse heating and steady-state power dissipation. In addition to the $\\mathrm { T M } _ { 0 1 }$ accelerating mode, properties of the $\\mathrm { H E M } _ { 1 1 }$ mode and contributions from higher order modes are discussed.
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augmentation
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NO
| 0
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expert
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How is the transformer ratio defined?
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It is the ratio of the maximum electric field behind the drive bunch to the maximum electric field within the bunch
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Definition
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Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
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$$ E _ { z , n } ( s \\to \\infty ) = 2 \\kappa _ { n } q _ { 0 } \\mathrm { R e } \\{ e ^ { j k _ { n } s } F ( k _ { n } ) \\} $$ Expanding the real part $$ \\begin{array} { r } { E _ { z , n } ( s \\infty ) = 2 \\kappa _ { n } q _ { 0 } [ \\cos ( k _ { n } s ) \\mathrm { R e } \\{ F ( k _ { n } ) \\} } \\\\ { - \\sin ( k _ { n } s ) \\mathrm { I m } \\{ F ( k _ { n } ) \\} ] . \\qquad } \\end{array} $$ Since we are interested in the maximum value of the longitudinal accelerating field, we define $E _ { \\mathrm { a c c } }$ as the amplitude of $E _ { z , n } \\big ( s \\infty \\big )$ : $$ E _ { \\mathrm { a c c } } = 2 \\kappa q _ { 0 } \\sqrt { \\mathrm { R e } \\{ F ( k _ { n } ) \\} ^ { 2 } + \\mathrm { I m } \\{ F ( k _ { n } ) \\} ^ { 2 } }
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augmentation
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NO
| 0
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expert
|
How is the transformer ratio defined?
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It is the ratio of the maximum electric field behind the drive bunch to the maximum electric field within the bunch
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Definition
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Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
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IV. ELECTROMAGNETIC PARAMETERS Each synchronous eigenmode solution of the periodic structure is characterized by a wakefield loss factor $\\kappa$ , group velocity $v _ { g } ,$ and attenuation constant $\\alpha$ . These parameters determine how the electron beam interacts with the given mode as well as the propagation characteristics of the corresponding wakefield. In this section, equations for the electromagnetic parameters are defined and applied to the structures found in Sec. III. The results are plotted against the corrugation spacing parameter $\\xi$ and period $p$ at each of the three aperture ratios to show how the wave propagation and beam interaction depend on the corrugation geometry and frequency. The loss factor $\\kappa$ describes the energy coupled from a charged particle to the structure and is defined as [16]: $$ \\kappa = \\frac { V ^ { 2 } / U } { 4 ( 1 - \\beta _ { g } ) p } , $$ where $V$ is the induced voltage, $U$ is the stored energy in the unit cell, and $\\beta _ { g }$ is the normalized group velocity $v _ { g } / c$ . The induced voltage is calculated from the time harmonic electric field of the synchronous mode with angular frequency $\\omega$ as
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augmentation
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NO
| 0
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expert
|
How is the transformer ratio defined?
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It is the ratio of the maximum electric field behind the drive bunch to the maximum electric field within the bunch
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Definition
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Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
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$$ W = \\frac { E _ { \\mathrm { a c c } } ^ { 2 } f _ { r } } { 8 \\pi a \\kappa } . $$ Referring to the plot for $\\kappa$ in Fig. 7, the power dissipation density is reduced by minimizing the corrugation period $p$ and maximizing the spacing parameter $\\xi$ . For structures with $p / a \\lesssim 0 . 5$ , the power dissipation density decreases with an increasing aperture ratio. This results in a trade-off between minimizing the peak surface fields and minimizing the thermal loading of the CWG, where choosing a larger aperture ratio (higher $\\mathrm { T M } _ { 0 1 }$ frequency) results in higher peak fields but less thermal power dissipation. Using $\\kappa _ { \\mathrm { m a x } }$ from Eq. (10) in Eq. (16), we obtain the lower bound of the energy dissipation density as $$ \\frac { d Q _ { \\mathrm { d i s s } } ( z \\infty ) } { d S } \\geq \\frac { E _ { \\mathrm { a c c } } ^ { 2 } a } { 4 Z _ { 0 } c } .
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augmentation
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NO
| 0
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expert
|
How is the transformer ratio defined?
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It is the ratio of the maximum electric field behind the drive bunch to the maximum electric field within the bunch
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Definition
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Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
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VI. THERMAL LOADING Thermal loading of the corrugated waveguide places a limit on the maximum repetition rate $f _ { r }$ of the accelerator, where $f _ { r }$ is the number of bunches injected into the structure per second. The thermal loading depends on the electromagnetic properties of the $\\mathrm { T M } _ { 0 1 }$ mode as well as the length of the corrugated waveguide and the conductivity of the wall material. Achieving a high repetition rate requires active cooling of the structure as well as an optimally designed corrugation profile. Here we focus on designing a corrugation that minimizes the steady-state thermal load and transient pulse heating. The thermally induced stresses due to temperature gradients in the wall pose additional design considerations which are discussed further in [26]. Because the group velocity $v _ { g }$ of the electromagnetic wave is less than the electron bunch velocity, the length of the rf pulse behind the bunch grows as it traverses the structure. This causes the thermal energy density deposited in the CWG wall to increase along the direction of propagation. At a distance $z$ from the beginning of the CWG, the field strength of the rf pulse induced by the electron bunch entering at time $t = 0$ is
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augmentation
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NO
| 0
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expert
|
How is the transformer ratio defined?
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It is the ratio of the maximum electric field behind the drive bunch to the maximum electric field within the bunch
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Definition
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Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
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$$ where $$ x ^ { \\prime } = \\frac { x } { \\hat { a } } , \\qquad y ^ { \\prime } = \\frac { y } { \\hat { a } } , \\qquad z ^ { \\prime } = \\frac { z } { \\hat { a } } , \\qquad \\omega ^ { \\prime } = \\frac { \\omega } { \\hat { a } } . $$ Scaling the fields by $\\hat { a } ^ { - 3 / 2 }$ keeps the stored energy $U ^ { \\prime }$ of the scaled structure same as that of the unscaled structure $U$ , which is seen by integrating the total energy in the fields: $$ \\begin{array} { l } { { \\displaystyle U ^ { \\prime } = \\iiint \\frac { \\epsilon _ { 0 } } { 2 } | \\hat { a } ^ { - 3 / 2 } E ( x ^ { \\prime } , y ^ { \\prime } , z ^ { \\prime } ) | ^ { 2 } } } \\\\ { { \\displaystyle \\quad + \\frac { \\mu _ { 0 } } { 2 } | \\hat { a } ^ { - 3 / 2 } H ( x ^ { \\prime } , y ^ { \\prime } , z ^ { \\prime } ) | ^ { 2 } d x d y d z } } \\\\ { { \\displaystyle = \\iiint \\frac { \\epsilon _ { 0 } } { 2 } | E ( x ^ { \\prime } , y ^ { \\prime } , z ^ { \\prime } ) | ^ { 2 } } } \\\\ { { \\displaystyle \\quad + \\frac { \\mu _ { 0 } } { 2 } | H ( x ^ { \\prime } , y ^ { \\prime } , z ^ { \\prime } ) | ^ { 2 } d x ^ { \\prime } d y ^ { \\prime } d z ^ { \\prime } = U , } } \\end{array}
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augmentation
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NO
| 0
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expert
|
How is the transformer ratio defined?
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It is the ratio of the maximum electric field behind the drive bunch to the maximum electric field within the bunch
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Definition
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Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
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$$ Q = \\frac { \\omega U } { P _ { d } } , $$ where $U$ is stored energy and $P _ { d }$ is the power dissipated in the cavity walls. The power dissipation density per unit area is $$ \\frac { d P _ { d } } { d A } = \\frac { 1 } { 2 } \\sqrt { \\frac { \\omega \\mu } { 2 \\sigma } } | { \\cal H } | ^ { 2 } . $$ In the scaled structure, the power dissipation and resulting quality factor become $$ P _ { d } ^ { \\prime } = \\hat { a } ^ { - 3 / 2 } P _ { d } , \\qquad Q ^ { \\prime } = \\hat { a } ^ { 1 / 2 } Q , $$ leading to the scaled attenuation constant from Eq. (9) $$ \\alpha ^ { \\prime } = \\hat { a } ^ { - 3 / 2 } \\alpha . $$ Scaling of the attenuation constant $\\alpha$ with conductivity is accomplished by multiplying $\\alpha$ by $\\sqrt { \\sigma / \\sigma ^ { \\prime } }$ where $\\sigma$ is the conductivity of the unscaled structure and $\\sigma ^ { \\prime }$ is the conductivity of the scaled structure.
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augmentation
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NO
| 0
|
expert
|
How is the transformer ratio defined?
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It is the ratio of the maximum electric field behind the drive bunch to the maximum electric field within the bunch
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Definition
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Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
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$$ \\frac { E _ { \\mathrm { m a x } } ^ { 3 0 } t _ { p } ^ { 5 } } { \\mathrm { B D R } } = \\mathrm { c o n s t . } $$ From a design perspective, reducing the BDR is achieved by reducing the peak surface fields and the pulse length. Calculation of the absolute threshold value of the fields that induce breakdown in sub-THz structures is an active area of research [19,22,23] and reliable models have not yet been developed. The modified Poynting vector introduced in [21] has been used to predict rf breakdown in structures operating up to $3 0 ~ \\mathrm { G H z }$ , but there are limited data for its applicability at higher frequencies. For this reason, the BDR and maximum gradient of the CWA must ultimately be determined experimentally. For the purpose of optimization, we choose the peak surface fields $E _ { \\mathrm { m a x } }$ and $H _ { \\mathrm { m a x } }$ as figures of merit which should be minimized to increase the attainable accelerating gradient. Since some evidence suggests that pulse heating is of fundamental importance to the initiation of rf breakdown in high frequency accelerating structures [24], we give additional weight to the minimization of the peak magnetic surface field. This choice leads to a higher overall thermal efficiency which will be discussed further in Sec. VI.
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augmentation
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NO
| 0
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expert
|
How is the transformer ratio defined?
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It is the ratio of the maximum electric field behind the drive bunch to the maximum electric field within the bunch
|
Definition
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Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
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$$ and the Fourier transform of the step function $$ \\mathcal { F } \\{ \\theta ( t ) \\} = \\pi \\biggl ( \\frac { 1 } { j \\pi \\omega } + \\delta ( \\omega ) \\biggr ) , $$ the wake impedance becomes $$ \\begin{array} { r } { Z _ { n | | } ( \\omega ) = \\kappa _ { n } \\Bigg [ \\pi [ \\delta ( \\omega - \\omega _ { n } ) + \\delta ( \\omega + \\omega _ { n } ) ] } \\\\ { - j \\Bigg ( \\cfrac { 1 } { ( \\omega - \\omega _ { n } ) } + \\frac { 1 } { ( \\omega + \\omega _ { n } ) } \\Bigg ) \\Bigg ] . } \\end{array} $$ Using $Z _ { n | | } ( \\omega )$ in Eq. (B12) and evaluating the integral yields $$ P _ { w , n } = \\frac { \\kappa _ { n } c } { 2 } ( | I ( \\omega _ { n } ) | ^ { 2 } + | I ( - \\omega _ { n } ) | ^ { 2 } ) .
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augmentation
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NO
| 0
|
expert
|
How is the transformer ratio defined?
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It is the ratio of the maximum electric field behind the drive bunch to the maximum electric field within the bunch
|
Definition
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Design_of_a_cylindrical_corrugated_waveguide.pdf.pdf
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$$ \\begin{array} { l } { \\displaystyle P ^ { 1 / 2 } ( z , t ) = \\sqrt { \\frac { 2 \\kappa q _ { 0 } ^ { 2 } | F | ^ { 2 } v _ { g } } { 1 - \\beta _ { g } } } e ^ { \\frac { - \\alpha ( v _ { g } t - \\beta _ { g } z ) } { 1 - \\beta _ { g } } } \\cos { \\left[ \\omega \\left( t - \\frac { z } { c } \\right) \\right] } } \\\\ { \\displaystyle \\times \\Pi \\bigg ( \\frac { 2 v _ { g } t - z ( 1 + \\beta _ { g } ) } { 2 z ( 1 - \\beta _ { g } ) } \\bigg ) . } \\end{array} $$ Here the field strength is defined in units of $\\sqrt { \\mathrm { W } }$ for consistency with the units provided by CST simulation, $F$ is the bunch form factor derived in Appendix B, $q _ { 0 }$ is the drive bunch charge, and $\\Pi ( x )$ is the rectangular window function.
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augmentation
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NO
| 0
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expert
|
Is it accurate to only consider the first harmonic?
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the higher nonsynchronous harmonics average out. It can be shown, that their second order (ponderomotive) contribution is also small.
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reasoning
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Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
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$$ V = q e _ { 1 } \\left[ \\frac { \\lambda _ { g z } } { 2 \\pi } \\cosh \\left( \\frac { \\omega y } { \\beta \\gamma c } \\right) \\sin \\left( \\frac { 2 \\pi s } { \\lambda _ { g z } } \\right) - s \\cos \\varphi _ { s } \\right] . $$ This potential and its adiabatic change with $\\beta$ is illustrated in Fig. 8. The full 6D Hamiltonian reads $$ H = \\frac { 1 } { 2 m _ { e } \\gamma } ( p _ { x } ^ { 2 } + p _ { y } ^ { 2 } + \\Delta P _ { z } ^ { 2 } ) + V , $$ where $\\Delta p _ { z } / \\gamma$ was replaced with $\\Delta P _ { z }$ . The coupled equations of motion are $$ \\begin{array} { r l } & { \\ddot { x } = 0 } \\\\ & { \\ddot { y } = - \\frac { q e _ { 1 } } { m _ { e } \\gamma ^ { 2 } } \\mathrm { s i n h } \\Bigg ( \\frac { \\omega y } { \\beta \\gamma c } \\Bigg ) \\mathrm { s i n } \\Bigg ( \\frac { 2 \\pi s } { \\lambda _ { g z } } \\Bigg ) } \\\\ & { \\ddot { s } = \\frac { q e _ { 1 } } { m _ { e } \\gamma ^ { 3 } } \\Bigg ( \\mathrm { c o s h } \\Bigg ( \\frac { \\omega y } { \\beta \\gamma c } \\Bigg ) \\mathrm { c o s } \\Bigg ( \\frac { 2 \\pi s } { \\lambda _ { g z } } \\Bigg ) - \\mathrm { c o s } \\varphi _ { s } \\Bigg ) . } \\end{array}
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4
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Yes
| 1
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expert
|
Is it accurate to only consider the first harmonic?
|
the higher nonsynchronous harmonics average out. It can be shown, that their second order (ponderomotive) contribution is also small.
|
reasoning
|
Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
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$$ V = q e _ { 1 } \\frac { \\lambda _ { 0 } } { 2 \\pi } \\mathrm { c o s h } \\left( \\frac { \\omega \\tan \\alpha } { c } y \\right) \\cos { \\left[ \\frac { \\omega } { c } ( \\Delta s + x \\tan \\alpha ) \\right] } . $$ The equations of motion become $$ { \\ddot { x } } = { \\frac { q e _ { 1 } } { m _ { e } \\gamma } } \\tan ( \\alpha ) \\cosh \\left[ { \\frac { \\omega \\tan \\alpha } { c } } y \\right] \\sin \\left[ { \\frac { \\omega } { c } } \\left( \\Delta s + x \\tan \\alpha \\right) \\right] $$ $$ { \\ddot { y } } = { \\frac { - i k _ { y } \\lambda _ { g z } q e _ { 1 } } { 2 \\pi m _ { e } \\gamma } } \\sinh \\left[ { \\frac { \\omega \\tan \\alpha } { c } } y \\right] \\cos \\left[ { \\frac { \\omega } { c } } \\left( \\Delta s + x \\tan \\alpha \\right) \\right]
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4
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Yes
| 1
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expert
|
Is it accurate to only consider the first harmonic?
|
the higher nonsynchronous harmonics average out. It can be shown, that their second order (ponderomotive) contribution is also small.
|
reasoning
|
Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
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V. APPLICATIONS We apply our approach to similar experimental parameters as for the subrelativistic experiments at FAU Erlangen [3] and the relativistic experiments at SLAC [1,2]. Although the structures are idealized, the results are qualitatively recovered. As a next step, we show modifications and idealizations of the beam parameters, which outline the way to a microchip accelerator. A. Subrelativistic acceleration A subrelativistic DLA structure needs to be chirped in order to always fulfill the synchronicity condition (6) for the synchronous particle. The proper chirp for each cell and the synchronous velocity are obtained by iterating the two equations $$ \\Delta z ^ { ( n + 1 ) } = \\frac { q e _ { 1 } \\lambda _ { 0 } ^ { 2 } \\cos \\varphi _ { s } ^ { ( n ) } } { m _ { e } c ^ { 2 } } \\sqrt { 1 - { \\beta ^ { ( n ) } } ^ { 2 } } $$ $$ \\beta ^ { ( n + 1 ) } = \\beta ^ { ( n ) } + \\frac { \\Delta z ^ { ( n + 1 ) } } { \\lambda _ { 0 } } .
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4
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Yes
| 1
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expert
|
Is it accurate to only consider the first harmonic?
|
the higher nonsynchronous harmonics average out. It can be shown, that their second order (ponderomotive) contribution is also small.
|
reasoning
|
Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
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$$ describes the acceleration ramp, where the synchronous phase $\\varphi _ { \\mathrm { s } }$ can be chosen arbitrarily \in each grating cell. The variables $e _ { 1 } , \\lambda _ { g z } , W _ { 0 } , \\beta , \\gamma , \\varphi _ { \\mathrm { s } }$ and all variables \in Eq. (24) are stored as arrays indexed by the grating cell number. The kicks are obtained using Eqs. (7), (18), (19), and (13) and read $$ \\begin{array} { r l } & { \\Delta x ^ { \\prime } = - \ \\frac { q \\lambda _ { 0 } } { p _ { z 0 } c } \\tan ( \\alpha ) \\cosh ( i k _ { y } y ) \\mathrm { R e } \\{ \\underline { { e } } _ { 1 } e ^ { i \\varphi + i \\frac { 2 \\pi x } { \\lambda _ { g x } } } \\} } \\\\ & { \\Delta y ^ { \\prime } = \ \\frac { - i k _ { y } \\lambda _ { 0 } ^ { 2 } q \\beta } { 2 \\pi p _ { z 0 } c } \\mathrm { s i n h } ( i k _ { y } y ) \\mathrm { I m } \\{ \\underline { { e } } _ { 1 } e ^ { i \\varphi + i \\frac { 2 \\pi x } { \\lambda _ { g x } } } \\} } \\\\ & { \\Delta \\delta = \ \\frac { q \\lambda _ { g z } } { \\gamma m _ { e } c ^ { 2 } } \\mathrm { R e } \\{ \\underline { { e } } _ { 1 } ( \\mathrm { c o s h } ( i k _ { y } y ) e ^ { i \\varphi + i \\frac { 2 \\pi x } { \\lambda _ { g x } } } - e ^ { i \\varphi _ { s } } ) \\} , } \\end{array}
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5
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Yes
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expert
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Is it accurate to only consider the first harmonic?
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the higher nonsynchronous harmonics average out. It can be shown, that their second order (ponderomotive) contribution is also small.
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reasoning
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Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
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$$ \\underline { { e } } _ { m } ( x , y ) = \\underline { { e } } _ { m } ( 0 , 0 ) \\cosh ( i k _ { y } y ) e ^ { i k _ { x } x } , $$ where $\\lambda _ { g x } = \\lambda _ { g z } /$ tan $\\alpha$ . A map of the energy gain and transverse kicks for the grating in Fig. 6 can be seen in Fig. 7 for a grating tilt angle $\\alpha = 3 0$ deg. The results labeled numerical are obtained by line integration [Eq. (7)] of the electric field simulated with CST MWS [16] and the analytical results correspond to Eq. (18). The transverse kicks are obtained by Eq. (13) as $$ \\begin{array} { r l r } { { \\vec { \\underline { { f } } } _ { m } ( x , y ) = \\underline { { e } } _ { m } ( 0 , 0 ) \\cosh ( i k _ { y } y ) e ^ { i k _ { x } x } i m \\tan \\alpha \\vec { e } _ { x } } } \\\\ & { } & { + \\underline { { e } } _ { m } ( 0 , 0 ) \\sinh ( i k _ { y } y ) e ^ { i k _ { x } x } ( i k _ { y } \\lambda _ { g z } / 2 \\pi ) \\vec { e } _ { y } } \\end{array}
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4
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Yes
| 1
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expert
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Is it accurate to only consider the first harmonic?
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the higher nonsynchronous harmonics average out. It can be shown, that their second order (ponderomotive) contribution is also small.
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reasoning
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Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
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$$ where $W _ { 0 } = \\gamma m _ { e } c ^ { 2 }$ and $p _ { z 0 } = \\beta \\gamma m _ { e } c$ . The particle at the synchronous phase $\\varphi _ { \\mathrm { s } }$ has $\\Delta \\delta = 0$ , i.e., its energy gain is entirely described by the acceleration ramp. The energy gain $\\Delta W$ is given by Eq. (7) and thus the energy gain of the synchronous particle is $$ \\Delta W ( 0 , 0 , \\varphi _ { \\mathrm { s } } ) = q \\lambda _ { g z } \\mathrm { R e } \\{ e ^ { i \\varphi _ { \\mathrm { s } } } \\underline { { e } } _ { 1 } \\} , $$ where we write $e _ { 1 } = \\underline { { e } } _ { 1 } ( x = 0 , y = 0 )$ for brevity. Note that the synchronous phase and the phase of each particle always refer to the laser phase. The sum of the kicks $$ W ( N ) = W _ { \\mathrm { i n i t } } + \\sum _ { n = 1 } ^ { N } \\Delta W ^ { ( n ) } ( 0 , 0 , \\varphi _ { \\mathrm { s } } ^ { ( n ) } )
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5
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Yes
| 1
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expert
|
Is it accurate to only consider the first harmonic?
|
the higher nonsynchronous harmonics average out. It can be shown, that their second order (ponderomotive) contribution is also small.
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reasoning
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Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
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$$ and are depicted as arrows in Fig. 7. For the numerical results, the gradient is determined by finite differences in MATLAB [8]. Note that $- i k _ { y } \\in \\mathbb { R } ^ { + }$ , i.e., the kick in $x$ -direction is in phase with the acceleration while the kick in $y$ -direction is 90 degrees shifted. For a particle that is only slightly displaced from the beam axis by $\\Delta \\vec { x } = ( \\Delta x , \\Delta y )$ , the kick can be written as two-dimensional Taylor expansion $$ \\begin{array} { l } { \\displaystyle \\vec { \\underline { { f } } } _ { m } ( x , y ) = \\vec { \\underline { { f } } } _ { m } ( x _ { 0 } , y _ { 0 } ) + ( \\nabla _ { \\perp } \\underline { { \\vec { f } } } _ { m } ( x _ { 0 } , y _ { 0 } ) ) \\Delta \\vec { x } + O ( | | \\Delta \\vec { x } | | ^ { 2 } ) } \\\\ { \\displaystyle = \\frac { \\lambda _ { g z } } { 2 \\pi } ( \\nabla _ { \\perp } \\underline { { e } } _ { m } ( x _ { 0 } , y _ { 0 } ) } \\\\ { \\displaystyle \\quad \\quad + ( \\nabla _ { \\perp } \\nabla _ { \\perp } ^ { \\mathrm { T } } ) \\underline { { e } } _ { m } ( x _ { 0 } , y _ { 0 } ) \\Delta \\vec { x } ) + O ( | | \\Delta \\vec { x } | | ^ { 2 } ) , \\quad ( 2 0 } \\end{array}
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augmentation
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Yes
| 0
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expert
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Is it accurate to only consider the first harmonic?
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the higher nonsynchronous harmonics average out. It can be shown, that their second order (ponderomotive) contribution is also small.
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reasoning
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Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
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$$ \\sigma _ { \\Delta W } = \\frac { c _ { 0 } } { \\lambda _ { 0 } } \\sqrt { - 2 \\pi \\lambda _ { g z } m _ { e } \\gamma ^ { 3 } q e _ { 1 } \\sin \\varphi _ { s } } \\sigma _ { \\Delta s } . $$ For a slow change of the potential and filling the bucket only up to a small fraction, the phase space area given by $\\pi \\sigma _ { \\Delta \\varphi } \\sigma _ { \\Delta W }$ is conserved. Moreover, using Eq. (44) a normalized bunch length and energy spread can be written as [19] $$ \\begin{array} { l } { \\displaystyle \\sigma _ { \\Delta W , n } = \\frac { 1 } { \\sqrt [ 4 ] { \\beta ^ { 3 } \\gamma ^ { 3 } } } \\sigma _ { \\Delta _ { W } } } \\\\ { \\displaystyle \\sigma _ { \\Delta \\varphi , n } = \\sqrt [ 4 ] { \\beta ^ { 3 } \\gamma ^ { 3 } } \\sigma _ { \\Delta \\varphi } . } \\end{array}
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augmentation
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Yes
| 0
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expert
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Is it accurate to only consider the first harmonic?
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the higher nonsynchronous harmonics average out. It can be shown, that their second order (ponderomotive) contribution is also small.
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reasoning
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Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
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Although the experimentally demonstrated gradients in DLA structures are very promising, there are still crucial challenges to create a miniaturized DLA-based particle accelerator. So far the experimentally achieved gradients could only be used to increase the beam’s energy spread and not for coherent acceleration. Moreover, the interaction length with present DLA structures is limited to the Rayleigh range (see Appendix) of the incident electron beam. For low energy electrons, due to the high gradient, the acceleration defocusing even leads to interaction distances significantly shorter than the Rayleigh range. Thus, in order to use DLA for a real accelerator, focusing schemes have to be developed. One option would be alternating phase focusing (APF) as outlined in Fig. 1. Here, drift sections between grating cells lead to jumps in the synchronous phase, which can be designed to provide net focusing. Such schemes can be a way to increase the interaction length in DLAs and make an accelerator on a microchip feasible. A challenge in the creation of a DLA based optical accelerator is related to the complex 3D beam dynamics in DLA structures, which has not been treated systematically in the existing literature yet. In order to facilitate front to end simulations and identify optimized DLA structures, we employ a simple and efficient numerical tracking scheme, which does not require a large amount of computing power, it runs in MATLAB [8] on an ordinary PC.
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augmentation
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Yes
| 0
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expert
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Is it accurate to only consider the first harmonic?
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the higher nonsynchronous harmonics average out. It can be shown, that their second order (ponderomotive) contribution is also small.
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reasoning
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Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
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We plan to achieve such phase jumps by inserting drift sections as already outlined in Fig. 1. Other options are to modify the accelerating Fourier coefficient in each cell, e.g., by phase masking within the structure or by active phase control of individual parts of the laser pulse. In general, we believe that this paper gives a beam dynamics foundation on which DLA structures providing stable long distance beam transport schemes can be developed. ACKNOWLEDGMENTS The authors wish to thank Ingo Hofmann for proofreading the manuscript. This work is funded by the Gordon and Betty Moore Foundation (Grant No. GBMF4744 to Stanford) and the German Federal Ministry of Education and Research (Grant No. FKZ:05K16RDB). APPENDIX: RAYLEIGH RANGE FORLIGHT AND PARTICLE BEAMS The Rayleigh range for a particle beam can be defined in the same way as for a light beam. The envelope of an externally focused beam is $$ w = w _ { 0 } \\sqrt { 1 + \\left( \\frac { z } { z _ { 0 } } \\right) ^ { 2 } } . $$ Inserting into the envelope equation $$ w ^ { \\prime \\prime } = w ^ { - 3 } $$
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augmentation
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Yes
| 0
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expert
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Is it accurate to only consider the first harmonic?
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the higher nonsynchronous harmonics average out. It can be shown, that their second order (ponderomotive) contribution is also small.
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reasoning
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Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
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$$ \\mathbf { M } = \\langle \\vec { r } \\vec { r } ^ { T } \\rangle , $$ where the average is taken component-wise. In the absence of nonlinearities, particular emittances are conserved. That is in the case of coupling only the 6D emittance given by $$ \\varepsilon _ { 6 D } = \\sqrt { \\operatorname* { d e t } { \\bf M } } . $$ In case of decoupled planes, the determinants of the diagonal blocks (the emittances of the respective plane) are conserved individually. They read $$ \\begin{array} { l } { \\displaystyle \\varepsilon _ { x , n } = \\frac { 1 } { m _ { e } c } \\sqrt { \\operatorname* { d e t } \\mathbf { M } _ { 1 } } , } \\\\ { \\displaystyle \\varepsilon _ { y , n } = \\frac { 1 } { m _ { e } c } \\sqrt { \\operatorname* { d e t } \\mathbf { M } _ { 2 } } , } \\\\ { \\displaystyle \\varepsilon _ { z , n } = \\frac { 1 } { e } \\sqrt { \\operatorname* { d e t } \\mathbf { M } _ { 3 } } , } \\end{array}
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augmentation
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Yes
| 0
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expert
|
Is it accurate to only consider the first harmonic?
|
the higher nonsynchronous harmonics average out. It can be shown, that their second order (ponderomotive) contribution is also small.
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reasoning
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Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
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$$ where $$ \\nabla _ { \\perp } \\nabla _ { \\perp } ^ { \\mathrm { T } } = \\left( { \\begin{array} { c c } { \\partial _ { x } ^ { 2 } } & { \\partial _ { x } \\partial _ { y } } \\\\ { \\partial _ { y } \\partial _ { x } } & { \\partial _ { y } ^ { 2 } } \\end{array} } \\right) $$ is the Hessian. The expansion Eq. (20) about $x _ { 0 } = 0$ , $y _ { 0 } = 0$ of Eq. (18) results in $$ \\underline { { \\vec { f } } } _ { m } ( \\Delta x , \\Delta y ) = \\frac { \\lambda _ { g z } } { 2 \\pi } \\underline { { e } } _ { m } ( 0 , 0 ) \\binom { i k _ { x } - k _ { x } ^ { 2 } \\Delta x } { - k _ { y } ^ { 2 } \\Delta y } , $$ i.e., a position independent (coherent) kick component in $x$ -direction, vanishing for $\\alpha = 0$ . Using this abstract derivation, the results of several papers proposing DLA undulators [12–14] can be recovered.
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augmentation
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Yes
| 0
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expert
|
Is it accurate to only consider the first harmonic?
|
the higher nonsynchronous harmonics average out. It can be shown, that their second order (ponderomotive) contribution is also small.
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reasoning
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Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
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$$ If the beam size is significantly smaller than the aperture $( y \\ll \\beta \\gamma c / \\omega )$ , the longitudinal equation decouples and becomes the ordinary differential equation of synchrotron motion. The transverse motion becomes linear in this case, however still dependent on the longitudinal motion via $\\varphi$ . The equation of motion, $$ \\ddot { y } = \\frac { - q e _ { 1 } \\omega } { m _ { e } \\gamma ^ { 3 } \\beta c } \\sin ( \\varphi ) y , $$ is Hill’s equation, with the synchrotron angle being the focusing function. However there is a crucial difference to ordinary magnetic focusing channels. The focusing force scales as $\\gamma ^ { - 3 }$ as expected for acceleration defocusing [19], rather than with $\\gamma ^ { - 1 }$ as would be expected for a magnetic quadrupole focusing channel. The solution to Eq. (39) as function of $z$ for fixed $s = \\lambda _ { g z } \\varphi _ { s } / 2 \\pi$ , i.e., when the bunch length is significantly shorter than the period length, is $$ y = y _ { 0 } \\exp \\left( \\sqrt { \\frac { - q e _ { 1 } \\omega } { m _ { e } \\gamma ^ { 3 } \\beta ^ { 3 } c ^ { 3 } } \\sin \\varphi _ { s } z } \\right)
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augmentation
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Yes
| 0
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expert
|
Is it accurate to only consider the first harmonic?
|
the higher nonsynchronous harmonics average out. It can be shown, that their second order (ponderomotive) contribution is also small.
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reasoning
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Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
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With no loss of generality, we restrict ourselves to symmetric grating structures driven from both lateral sides. This makes sure that the axis of symmetry is in the center and the fields have a cosh profile. In the case of nonsymmetric structures or nonsymmetric driving, the fields will have an exponential or an off-axis cosh profile. However, single driver systems can be combined with Bragg mirrors in order to obtain a good approximation to an on-axis cosh profile with a single side driver (see again Fig. 1). Another option is to reshape the structure as e.g., presented in [4], or, just to accept the asymmetry which then leads to a smaller effective aperture. Furthermore, we restrict ourselves to linear dielectrics. Driving the dielectric into its nonlinear regime is discussed in [10] (experimental), whereas the theoretical reader [11] particularly covers quantum aspects of high fields. As it is usually done, e.g., for the synchrotron motion in ion synchrotrons, we take the limit from the tracking difference equations to differential equations. Since the three-dimensional kick must be irrotational due to the Panofsky-Wenzel theorem, it can be derived from a scalar potential. This potential directly allows to determine the 6D Hamiltonian which completely describes the single particle dynamics analytically.
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augmentation
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Yes
| 0
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expert
|
Is it accurate to only consider the first harmonic?
|
the higher nonsynchronous harmonics average out. It can be shown, that their second order (ponderomotive) contribution is also small.
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reasoning
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Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
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C. Dynamics in tilted gratings Finally, we address the tilted grating with the same laser parameters and a bunched electron beam with parameters $\\varepsilon _ { x } = \\varepsilon _ { y } = 1 \\ \\mathrm { n m } , \\sigma _ { x } = 1 \\ \\mu \\mathrm { m } , \\sigma _ { y } = 0 . 4 \\mu \\mathrm { m } .$ $\\sigma _ { z } = 3 0 ~ \\mathrm { n m }$ , $\\sigma _ { W } = 1 0 ~ \\mathrm { k e V }$ and a focusing angle of 5 mrad in the y-direction. The grating tilt angle is 70 degrees and again $| \\underline { { e } } _ { 1 } | = 1 \\ \\mathrm { G V / m }$ . Figure 19 shows the evolution of the phase space in all three planes. Evaluating Eq. (54), one finds $\\lambda _ { u } \\approx 1 6 0 \\lambda _ { 0 }$ , i.e., half an oscillation period in the $\\mathbf { \\boldsymbol { x } }$ -direction in the displayed 80 grating cells. As visible in Fig. 19, the horizontal and longitudinal phase spaces are correlated. The projections of the energy spectrum can be seen in Fig. 20 together with the particle loss, which takes place at the physical aperture in y-direction at $\\pm 4 0 0 ~ \\mathrm { n m }$ . Unlike the straight grating with relativistic particles, the tilted grating creates a defocusing force in the y-direction which significantly decreases the Rayleigh range. The energy spread shows a breathing mode, similar to the quadrupole modes in the synchrotron motion. However, since the synchrotron motion is practically frozen due to the high $\\gamma$ , this mode arises entirely due to the correlation with the $\\mathbf { \\boldsymbol { x } }$ -plane. Excluding the defocusing by setting $\\varepsilon _ { y } = 0$ , two coherent oscillation periods are displayed in Fig. 21.
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augmentation
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Yes
| 0
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expert
|
Is it accurate to only consider the first harmonic?
|
the higher nonsynchronous harmonics average out. It can be shown, that their second order (ponderomotive) contribution is also small.
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reasoning
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Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
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$$ where the energy momentum differential $\\Delta p _ { \\| } = \\Delta W / ( \\beta c )$ was applied. Moreover, if the phase-synchronicity condition [Eq. (6)] is fulfilled, the kick becomes $$ \\begin{array} { l } { { \\displaystyle \\Delta \\vec { p } _ { \\perp } ( x , y ; s ) = - \\frac { \\lambda _ { g z } ^ { 2 } } { 2 \\pi m } q \\frac { 1 } { \\beta c } \\nabla _ { \\perp } \\mathrm { I m } \\{ e ^ { 2 \\pi i \\frac { s } { \\beta \\lambda _ { 0 } } } \\underline { { e } } _ { m } ( x , y ) \\} } } \\\\ { { \\displaystyle \\ = - \\frac { \\lambda _ { g z } } { m } q \\frac { 1 } { \\beta c } \\mathrm { I m } \\{ e ^ { 2 \\pi i \\frac { s } { \\beta \\lambda _ { 0 } } } \\underline { { \\vec { f } } } _ { m } ( x , y ) \\} , } } \\end{array}
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augmentation
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Yes
| 0
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expert
|
Is it accurate to only consider the first harmonic?
|
the higher nonsynchronous harmonics average out. It can be shown, that their second order (ponderomotive) contribution is also small.
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reasoning
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Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
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Both the numerical and the analytical approach can be generalized from ordinary DLA gratings to tilted DLA gratings, which have been proposed as deflectors or laser driven undulators [12–14]. Such a grating is depicted in Fig. 2. However, since our code does not include the radiation fields, a dedicated code as, e.g., [15] can be used to treat the dynamics self-consistently. The analytical kicks reported here can serve as input quantities. Our approach aims at maximal simplicity such that studies of fundamental questions, as, e.g., transverse focusing and deflection, are quickly possible. The paper is organized as follows. Section II presents the determination of the longitudinal and transverse fields and kicks in a single grating period. Here we use CST Studio Suite [16] to calculate the longitudinal kick at the center of the structure. The dependence on the transverse coordinates as well as the transverse kicks are modeled analytically. In Sec. III we present a symplectic 6D tracking method based on one kick per grating period. Analytical descriptions of the coupled longitudinal and transverse beam dynamics as well as the full 6D Hamiltonian are given in Sec. IV. Simplifications and beam matching in linearized fields are also discussed in this section. In Sec. V we address the three crucial examples: subrelativistic acceleration, relativistic acceleration, and deflection by means of DLA gratings. The paper concludes with a summary and an outlook to DLA focusing channels in Sec. VI.
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augmentation
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Yes
| 0
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expert
|
Is it accurate to only consider the first harmonic?
|
the higher nonsynchronous harmonics average out. It can be shown, that their second order (ponderomotive) contribution is also small.
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reasoning
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Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
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$$ in the usual units of $\\textrm { m }$ rad and eVs, respectively. The analysis of emittance coupling by means of the eigen-emittances $$ \\varepsilon _ { \\mathrm { e i g } , i } = \\mathrm { e i g s } ( \\mathbf { J } \\mathbf { M } ) , $$ where $\\mathbf { J }$ is the symplectic matrix, is also possible with our code, however beyond the scope of this paper. IV. CONTINUOUS EQUATIONS OF MOTION In order to address the continuous motion in DLA structures we employ positions and momentum as canonically conjugate variables in all directions. The transformation for the energy is $\\Delta p _ { z } = \\Delta W / ( \\beta c )$ . We address the flat and the tilted grating separately and assume for simplicity $\\lvert \\underline { { e } } _ { 1 } \\rvert$ to be constant for all cells and $\\arg ( \\underline { { e } } _ { 1 } ) = 0$ . A. Flat grating Hamilton’s equations can be written as $$ \\dot { x } = \\frac { p _ { x } } { m _ { e } \\gamma }
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augmentation
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Yes
| 0
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expert
|
Is it accurate to only consider the first harmonic?
|
the higher nonsynchronous harmonics average out. It can be shown, that their second order (ponderomotive) contribution is also small.
|
reasoning
|
Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
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For an adiabatic Hamiltonian and if stable orbits exist, a matched locally Gaussian distribution is given by $$ f = C e ^ { - H / \\langle H \\rangle } $$ and a locally elliptic (Hofmann-Pedersen [23]) matched distribution is given by $$ f = C { \\sqrt { H _ { \\operatorname* { m a x } } - H } } . $$ The normalization constant $C$ is determined by integration. Note that in the case of nonperiodic motion $f$ will not be integrable. Thus, we can only write a matched distribution for the longitudinal plane if $\\varphi _ { s } \\in [ \\pi / 2 , \\pi ]$ and for the transverse plane if $\\varphi _ { s } \\in [ \\pi , 3 / 2 \\pi ]$ . The Hamiltonian is not time independent, however its dependence on $\\beta$ and $\\gamma$ is adiabatic. Thus, if $\\varphi _ { s }$ is changing at most adiabatically, the distribution will deform such that the emittance increase is bounded, i.e., also the emittance remains an adiabatic invariant. First, we consider the longitudinal plane and linearized fields. For a given bunch length $\\sigma _ { \\Delta s }$ the matched energy spread is
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augmentation
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Yes
| 0
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expert
|
Is it accurate to only consider the first harmonic?
|
the higher nonsynchronous harmonics average out. It can be shown, that their second order (ponderomotive) contribution is also small.
|
reasoning
|
Beam_dynamics_analysis_of_dielectric_laser_acceleration.pdf
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For subrelativistic accelerators, the grating needs to be chirped in period length in order to always fulfill Eq. (6) on the energy ramp. The change of period length is given by the energy velocity differential $$ \\frac { \\Delta z } { \\lambda _ { g z } } = \\frac { 1 } { \\beta ^ { 2 } \\gamma ^ { 2 } } \\frac { \\Delta W } { W } $$ and is in the range of ${ \\lesssim } 1 \\%$ for $W _ { \\mathrm { k i n } } = 3 0 ~ \\mathrm { k e V }$ and $\\Delta W / \\lambda _ { g z } = 1 \\mathrm { G e V / m }$ . The thus created “quasiperiodic" gratings can be seen in good approximation as periodic, however, phase drifts have to be compensated in the structure design [18]. B. Analysis of the transverse field The transverse field probed by a rigidly moving charge can be obtained using the Panofsky-Wenzel theorem [9], which holds for either vanishing fields at infinity or periodic boundary conditions as $$ \\begin{array} { l } { \\nabla ^ { \\prime } \\times \\Delta \\vec { p } ( \\vec { r } _ { \\perp } , s ) = \\displaystyle \\int _ { - T / 2 } ^ { T / 2 } \\mathrm { d } t [ \\nabla \\times \\vec { F } ( \\vec { r } _ { \\perp } , z , t ) ] _ { z = v t - s } } \\\\ { = \\vec { B } | _ { - T / 2 } ^ { T / 2 } = 0 . } \\end{array}
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augmentation
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Yes
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IPAC
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What are insertion devices?
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They are periodic magnetic structures, such as undulators and wigglers, placed in straight sections to generate radiation.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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C-BAND INJECTOR DESIGN We have designed two injectors. One design is plug-free and is shown in Fig. 2. This design focuses on the tuning and high gradient testing of the RF cavity without a high QE photocathode or plug [2]. The design is very similar to the plug-insert design, as shown by the RF parameters in Table 2 . The design uses a solid Cu backplane and thus a Cu photocathode instead of a plug, and has less symmetry due to being an earlier design. A picture of the fabricated injector before brazing is shown in Table: Caption: Table 1: Klystron Conditioning Results for Three Pulse Widths Body: <html><body><table><tr><td>Freq. (Hz)</td><td>1.00 μs</td><td>1.25 μs</td><td>1.50 μs</td></tr><tr><td>1</td><td>36 MW</td><td>36 MW</td><td>36 MW</td></tr><tr><td>20</td><td>NA</td><td>36 MW</td><td>36MW</td></tr><tr><td>40</td><td>NA</td><td>25 MW</td><td>20 MW</td></tr></table></body></html> Plug Insert Design Overview The second photoinjector design incorporates a removable plug for incorporating high QE photocathodes. Two views of the plug-insert injector are shown in Figs. 3 and 4. The design has several important features to symmetrize the RF fields for emittance reduction. There are three symmetry stubs in the cathode cell and in the full cell, respectively, designed for minimizing the dipole RF content induced by the RF coupling slot at the top in each cell. In the cathode cell, on the outer faces of the two symmetry stubs on the horizontal plane, laser pipes are opened. This design allows the laser, when reflected by the photocathode film, to exit the cavity without being scattered inside the cathode cell. The key beam parameters for the plug free and plug insert design are compared in Table 2.
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augmentation
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NO
| 0
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expert
|
What are insertion devices?
|
They are periodic magnetic structures, such as undulators and wigglers, placed in straight sections to generate radiation.
|
Definition
|
Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ If the phase of the radiation wave advances by $\\pi$ between $A$ and $B$ , the electromagnetic field of the radiation adds coherently3. The light moves on a straight line $\\overline { { A B } }$ that is slightly shorter than the sinusoidal electron trajectory $\\widetilde { A B }$ $$ { \\frac { \\lambda } { 2 c } } = { \\frac { \\widetilde { A B } } { v } } - { \\frac { \\overline { { A B } } } { c } } . $$ The electron travels on a sinusoidal arc of length $\\widetilde { A B }$ that can be calculated as $$ \\begin{array} { r l } { \\overrightarrow { A B } } & { = \\displaystyle \\int _ { 0 } ^ { \\infty } \\sqrt { 1 + \\left( \\frac { \\mathrm { d } x } { \\mathrm { d } \\xi } \\right) ^ { 2 } } \\mathrm { d } z } \\\\ & { \\approx \\displaystyle \\int _ { 0 } ^ { \\infty } \\left( 1 + \\frac { 1 } { 2 } \\left( \\frac { \\mathrm { d } x } { \\mathrm { d } \\xi } \\right) ^ { 2 } \\right) \\mathrm { d } z } \\\\ & { = \\displaystyle \\int _ { 0 } ^ { \\infty } \\left( 1 - \\frac { K ^ { 2 } } { 2 \\sqrt { 3 \\xi _ { 0 } ^ { 2 } \\gamma ^ { 2 } } } \\mathrm { e } ^ { \\mathrm { i } \\xi } \\mathrm { d } \\xi \\right) \\mathrm { d } z } \\\\ & { = \\displaystyle \\int _ { 0 } ^ { \\infty } \\left( 1 - \\frac { K ^ { 2 } } { 2 \\sqrt { 3 \\xi _ { 0 } ^ { 2 } \\gamma ^ { 2 } } } \\mathrm { e } ^ { \\mathrm { i } \\xi } \\mathrm { d } \\xi \\right) \\mathrm { d } z } \\\\ & { = \\displaystyle \\frac { \\lambda _ { 0 } } { 2 } \\left( 1 + \\frac { K ^ { 2 } } { 4 ( 3 \\xi _ { 0 } ^ { 2 } \\gamma ^ { 2 } ) } \\right) } \\\\ & { \\approx \\displaystyle \\frac { \\lambda _ { 0 } } { 2 } \\left( 1 + \\frac { K ^ { 2 } } { 4 \\gamma ^ { 2 } } \\right) . } \\end{array}
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They are periodic magnetic structures, such as undulators and wigglers, placed in straight sections to generate radiation.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ j _ { x } + j _ { y } + j _ { z } = 4 . $$ This means that the damping is not uniformly distributed along the three sub-spaces of the phase space (horizontal, vertical and longitudinal), but it is split according to specific partition numbers. These partition numbers are determined by the accelerator lattice, which gives the designers of accelerators some freedom to optimize the damping times. I.10.4 Diffraction limited storage rings The pursuit of higher brilliance and coherence is a driving force in the development of synchrotrons. As we have seen above, while the emission of synchrotron radiation reduces the transverse emittance of the beams in an electron synchrotron, the quantum nature of the radiation imposes a limit on how small the beam will become, and thus set a ceiling on the achievable brilliance. The source size of the $\\mathrm { \\Delta X }$ -ray beam is given by the electron beam size in the undulators. We have seen in Section I.10.3.4 that the vertical emittance is typically significantly smaller than the horizontal emittance. The vertical beam size is indeed typically so small that the X-ray beams are diffraction-limited in this dimension.
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They are periodic magnetic structures, such as undulators and wigglers, placed in straight sections to generate radiation.
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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 $$ ( A B + B C ) - ( A C ^ { \\prime } ) = n \\lambda
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They are periodic magnetic structures, such as undulators and wigglers, placed in straight sections to generate radiation.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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d) . . . requires the rotation of the sample around three orthogonal axes I.10.7.52 Undulator radiation Derive the formula for the fundamental wavelength of undulator radiation emitted at a small angle $\\theta$ : $$ \\lambda = \\frac { \\lambda _ { u } } { 2 \\gamma ^ { 2 } } \\left( 1 + \\frac { K ^ { 2 } } { 2 } + \\gamma ^ { 2 } \\theta ^ { 2 } \\right) $$ from the condition of constructive interference of the radiation emitted by consecutive undulator periods! I.10.7.53 Binding energies In which atom are the core electrons most strongly bound to the nucleus? a) Neon b) Copper c) Lithium d) Osmium e) Helium $f$ ) Iron g) Sodium $h$ ) Gold What about the valence electrons? I.10.7.54 Electron and X-Ray diffraction In comparison to diffractive imaging using electrons, X-ray diffraction. . $a$ ). . . has the advantage that the sample does not need to be in vacuum b). . . gives a stronger diffraction signal for all crystal sizes $c$ ). . . generates the same signal for all atoms in the crystal What are the consequences for the optimum sample thickness for electron diffraction in comparison to X-ray diffraction?
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They are periodic magnetic structures, such as undulators and wigglers, placed in straight sections to generate radiation.
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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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They are periodic magnetic structures, such as undulators and wigglers, placed in straight sections to generate radiation.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ Equation (I.10.16) becomes $$ \\begin{array} { r c l } { { \\displaystyle \\frac { \\lambda } { 2 c } } } & { { = } } & { { \\displaystyle \\frac { \\lambda _ { u } } { 2 \\beta c } \\left( 1 + \\frac { K ^ { 2 } } { 4 \\gamma ^ { 2 } } \\right) - \\frac { \\lambda _ { u } } { 2 c } } } \\\\ { { \\Longrightarrow } } & { { \\lambda } } & { { = } } & { { \\displaystyle \\frac { \\lambda _ { u } } { 2 \\gamma ^ { 2 } } ( 1 + K ^ { 2 } / 2 ) , } } \\end{array} $$ where we have used $\\beta = \\textstyle \\sqrt { 1 - \\gamma ^ { - 2 } } \\approx 1 - \\frac { 1 } { 2 } \\gamma ^ { - 2 }$ for $\\gamma \\gg 1$ . Radiation emitted at this wavelength adds up coherently in the forward direction. More generally, the radiation adds up coherently at all odd harmonics $n = 2 m - 1 , m \\in \\mathbb { N }$
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They are periodic magnetic structures, such as undulators and wigglers, placed in straight sections to generate radiation.
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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– The Lorentz factor $\\gamma$ , – The magnetic field to bend the beam (assume for simplicity that the ring consists of a uniform dipole field), – The critical energy of the synchrotron radiation, – The energy emitted by each electron through synchrotron radiation in one turn. I.10.7.7 Future Circular Collider: FCC-hh Particle physicists are evaluating the potential of building a future circular collider, which aims at colliding two proton beams with $5 0 0 \\mathrm { m A }$ current each and $1 0 0 \\mathrm { T e V }$ particle energy (FCC-hh). The protons would be circulating in a storage ring with $1 0 0 \\mathrm { k m }$ circumference, guided by superconducting magnets. The dipoles aim at a field of $1 6 \\mathrm { T }$ Calculate: – The Lorentz factor $\\gamma$ , – The critical energy of the synchrotron radiation, – The total power emitted by both beams through synchrotron radiation. I.10.7.8 Simple storage ring Let’s build a very simple synchrotron! Consider a storage ring that is located at the (magnetic) North Pole of the Earth. Assume that the Earth’s magnetic field of $5 0 \\mu \\mathrm { T }$ is used to confine electrons to a circular orbit, and ignore the need for focusing magnets. As a particle source, we will use the electron gun of an old TV, which accelerates the particles with a DC voltage of $2 5 \\mathrm { k V }$ (we will ignore the requirement of an injection system).
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They are periodic magnetic structures, such as undulators and wigglers, placed in straight sections to generate radiation.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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I.10.7.45 Crystals Which of the following are crystalline (more than one answer is may be correct)? $a$ ) The glass on the screen of my mobile phone $b$ ) The sapphire glass on an expensive watch c) Asbestos d) Icing sugar e) Sapphire $f$ ) Fused silica g) Snowflakes $h$ ) Paracetamol (Acetaminophen) powder in capsules $i$ ) The DNA in my body $k$ ) A diamond l) Viruses Why are crystals important for diffractive imaging? How is the X-ray diffraction from quartz different from that of fused silica? I.10.7.46 Absorption and diffraction A scientist wants to record a diffraction pattern of a silicon crystal at a photon energy of $8 \\mathrm { k e V . }$ What is the optimum thickness of the crystal, that maximizes the intensity of the diffracted spot? Hint: you can find the mass absorption coefficient of silicon on page 1-41 (page 49 in the PDF) of the X-Ray Data Booklet, and the density on page 5-5 (page 153). I.10.7.47 Detectors Name two or more advantages of semiconductor detectors, as compared to Röntgen’s photographic plates! I.10.7.48 X-ray absorption What is the dominant process for X-Ray absorption of – 10 keV photons – 1 MeV photons – 100 MeV photons
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They are periodic magnetic structures, such as undulators and wigglers, placed in straight sections to generate radiation.
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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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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ which is Bragg’s law. Note that contrary to the diffraction on a two-dimensional surface, which is often considered fir visible light, $\\mathrm { \\Delta } \\mathrm { X }$ -rays diffract on a three-dimensional crystal lattice. In this case, not only the exit angle matters, but also the incoming angle must fulfill the resonance condition! X-ray diffraction is one of the key techniques to resolve molecular structure in samples that can be crystallized. In the following section, we will look at different applications of synchrotron radiation in science, medicine and industry. I.10.6 Applications of synchrotron radiation Synchrotron radiation is used in a wide range of scientific and industrial applications, and over 60 synchrotron radiation sources are operating around the world. New facilities are under construction, reflecting the growing demand in research and industrial applications. I.10.6.1 Diffraction Coherent diffraction on crystals has been used before the emergence of synchrotrons, at the time enabled by X-ray tubes. The renowned Photo 51, recorded by Rosalind Franklin and her student Raymond Gosling, found its way (through dubious ways) into the hands of James Watson and Francis Crick, who used it to decipher the double helix structure of DNA (see Fig. I.10.12). Why do scientists use diffraction in place of imaging to determine the structure of molecules? Would it not be easier to simply magnify the X-ray image onto a detector, as we do in transmission electron microscopes?
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They are periodic magnetic structures, such as undulators and wigglers, placed in straight sections to generate radiation.
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ 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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They are periodic magnetic structures, such as undulators and wigglers, placed in straight sections to generate radiation.
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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– Vacuum system: as a result of the smaller inner bore of the magnets, the vacuum chamber diameter needs to be reduced to a point where a conventional pumping system becomes difficult to implement. A key enabling technology is the use of a distributed getter pump system, where the entire vacuum chamber is coated with a non-evaporable getter (NEG); – Generation of hard X-rays: the strong field in longitudinal gradient bends, peaking at 4. . . 6 Tesla, results in very hard X-rays, up to a photon energy of $8 0 \\mathrm { k e V }$ ; – Momentum compaction factor: when designing a magnetic lattice that employs LGBs and reverse bends, one can achieve a situation where a higher-energy particle takes a shorter path. This can then result in a negative momentum compaction factor of the ring (like in proton synchrotrons below transition energy). The Paul Scherrer Institut is upgrading its storage ring in the year 2024, making use of the principles outlined in this section [5]. I.10.5 Interaction of $\\mathbf { X }$ -rays with matter In the subsequent sections, we will look at the interaction of X-rays with matter, and the use of X-rays for experiments. To understand the processes that lead to absorption, scattering, and diffraction, we will proceed in three steps, and look at the interaction of $\\mathrm { \\Delta X }$ -rays with:
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They are periodic magnetic structures, such as undulators and wigglers, placed in straight sections to generate radiation.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ After the emission of a photon, the action of our single electron is $$ \\begin{array} { r c l } { { J _ { y } ^ { \\prime } } } & { { = } } & { { \\displaystyle \\frac { 1 } { 2 } \\gamma _ { y } y ^ { 2 } + \\alpha _ { y } y p _ { y } \\left( 1 - \\frac { d p } { P _ { 0 } } \\right) + \\frac { 1 } { 2 } \\beta _ { y } p _ { y } ^ { 2 } \\left( 1 - \\frac { d p } { P _ { 0 } } \\right) ^ { 2 } } } \\\\ { { } } & { { } } & { { } } \\\\ { { } } & { { = } } & { { \\displaystyle \\frac { 1 } { 2 } \\gamma _ { y } y ^ { 2 } + \\alpha _ { y } y p _ { y } - \\alpha _ { y } y p _ { y } \\frac { d p } { P _ { 0 } } + \\frac { 1 } { 2 } \\beta _ { y } p _ { y } ^ { 2 } - 2 \\cdot \\frac { 1 } { 2 } \\beta _ { y } p _ { y } ^ { 2 } \\frac { d p } { P _ { 0 } } + \\frac { 1 } { 2 } \\beta p _ { y } ^ { 2 } \\left( \\frac { d p } { P _ { 0 } } \\right) ^ { 2 } . } } \\end{array}
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They are periodic magnetic structures, such as undulators and wigglers, placed in straight sections to generate radiation.
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Definition
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ 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 $$ \\begin{array} { r c l } { { F _ { n } ( K ) } } & { { = } } & { { \\displaystyle \\frac { n ^ { 2 } K ^ { 2 } } { ( 1 + K ^ { 2 } / 2 ) ^ { 2 } } \\left( J _ { ( n + 1 ) / 2 } ( Y ) - J _ { ( n - 1 ) / 2 } ( Y ) \\right) ^ { 2 } } } \\\\ { { } } & { { } } & { { } } \\\\ { { Y } } & { { = } } & { { \\displaystyle \\frac { n K ^ { 2 } } { 4 ( 1 + K ^ { 2 } / 2 ) } , } } \\end{array}
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They are periodic magnetic structures, such as undulators and wigglers, placed in straight sections to generate radiation.
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ with a horizontal damping time $$ \\tau _ { x } = \\frac { 2 } { j _ { x } } \\frac { E _ { 0 } } { U _ { 0 } } T _ { 0 } . $$ All effects related to the dispersion are summarized in the horizontal partition number $j _ { x }$ $$ j _ { x } = 1 - { \\frac { I _ { 4 } } { I _ { 2 } } } . $$ The second synchrotron radiation integral is defined in Equation I.10.12. For the sake of completeness, we now define all five synchrotron radiation integrals $$ \\begin{array} { r c l } { { I _ { 1 } } } & { { = } } & { { \\displaystyle \\oint \\frac { \\eta _ { x } } { \\rho } d s } } \\\\ { { } } & { { } } & { { } } \\\\ { { I _ { 2 } } } & { { = } } & { { \\displaystyle \\oint \\frac { 1 } { \\rho ^ { 2 } } d s } } \\\\ { { } } & { { } } & { { } } \\\\ { { I _ { 3 } } } & { { = } } & { { \\displaystyle \\oint \\frac { 1 } { | \\rho | ^ { 3 } } d s } } \\\\ { { } } & { { } } & { { } } \\\\ { { I _ { 4 } } } & { { = } } & { { \\displaystyle \\oint \\frac { \\eta _ { x } } { \\rho } \\left( \\frac { 1 } { \\rho ^ { 2 } } + 2 k _ { 1 } \\right) d s , \\qquad k _ { 1 } = \\frac { e } { P _ { 0 } } \\frac { \\partial B _ { y } } { \\partial x } } } \\\\ { { } } & { { } } & { { } } \\\\ { { I _ { 5 } } } & { { = } } & { { \\displaystyle \\oint \\frac { \\hat { \\mathcal { H } } _ { x } } { | \\rho | ^ { 3 } } d s , ~ \\qquad \\mathcal { H } _ { x } = \\gamma _ { x } \\eta _ { x } ^ { 2 } + 2 \\alpha _ { x } \\eta _ { x } \\eta _ { \\rho x } + \\beta _ { x } \\eta _ { p x } ^ { 2 } . } } \\end{array}
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They are periodic magnetic structures, such as undulators and wigglers, placed in straight sections to generate radiation.
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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I.10.7.35 Instrumentation How would you measure the bunch length in a synchrotron? I.10.7.36 Instrumentation How would you measure the stability of the orbit in a storage ring? I.10.7.37 Detection What possibilities exist to detect X-Rays? How has the development of X-ray detectors influenced experiments at synchrotron sources? I.10.7.38 Monochromators What dispersive element is used to monochromatize X-Rays? What differences exist to monochromators for visible light? I.10.7.39 Refractive index The passage of electromagnetic radiation can be described classically by an index of refraction. What are the properties of the index of refraction of most materials at X-ray wavelengths? I.10.7.40 DLSRs How do longitudinal gradient bends contribute towards the goal of achieving a lower horizontal emittance in a diffraction limited storage ring? I.10.7.41 Diffraction limited storage rings Which of the following methods are employed to reduce the horizontal emittance in the DLSR SLS 2.0? $a$ ) Minimize the dispersion in areas of large dipole fields $b$ ) Maximize coupling between horizontal and vertical plane $c$ ) Increase the beam pipe diameter to reduce wake fields $d$ ) Alternate between insertion devices with horizontal and vertical polarization I.10.7.42 Globatron Enrico Fermi proposed the Globatron, a storage ring for protons suspended in space around the earth. This would have $5 \\mathrm { P e V }$ proton beams in a ring with $8 0 0 0 { \\mathrm { k m } }$ radius (Fig. I.10.18). Calculate:
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applications in Section I.10.6. The total radiated power per particle, obtained by integrating over the spectrum, is $$ P _ { \\gamma } = \\frac { e ^ { 2 } c } { 6 \\pi \\varepsilon _ { 0 } } \\frac { \\beta ^ { 4 } \\gamma ^ { 4 } } { \\rho ^ { 2 } } . $$ The energy lost by a particle on a circular orbit, i.e. in an accelerator consisting only of dipole magnets, is $$ U _ { 0 } = \\frac { e ^ { 2 } \\beta ^ { 4 } \\gamma ^ { 4 } } { 3 \\varepsilon _ { 0 } \\rho } , $$ where we have used $T = 2 \\pi \\rho / c$ , assuming $v \\approx c$ . Of course, real accelerators contain also other types of magnets. The energy lost per turn for a particle in an arbitrary accelerator lattice can be calculated by the following ring integral $$ U _ { 0 } = \\frac { C _ { \\gamma } } { 2 \\pi } E _ { \\mathrm { n o m } } ^ { 4 } \\oint \\frac { 1 } { \\rho ^ { 2 } } d s ,
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Ischebeck_-_2024_-_I.10_—_Synchrotron_radiation
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$$ where $\\vartheta$ is the angle at which the photon is scattered. I.10.5.2 Scattering of $\\mathbf { X }$ -rays on atoms In the case of photon energies less than a few keV, the wavelength is longer than the size of the atom. The scattering is then coherent, i.e., the phases of the scattered waves from different parts of the electron cloud add up constructively. The electric field amplitude of the scattered wave is then proportional to the total number of electrons in the atom $Z$ , and the scattered intensity is proportional to $Z ^ { 2 }$ . The total cross section is then $$ \\sigma = Z ^ { 2 } \\sigma _ { T } . $$ The $Z ^ { 2 }$ dependence makes the scattering cross section for heavier atoms much larger compared to lighter ones, significantly influencing how $\\mathrm { \\Delta X }$ -rays are used in science and medicine. When we increase the photon energy, the wavelength becomes smaller than the size of the electron cloud of an atom, and decoherence between the scattered waves reduces the scattering cross section. As an approximation, the cross-section drops off as $1 / E _ { \\gamma } ^ { 2 }$ . The precise drop-off can be described by the atomic form factor $f ^ { 0 }$ , which depends on both the scattering angle and the photon wavelength. It can be parametrized as
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$$ where $J$ is the Bessel function of the first kind. As $K$ increases, the higher harmonics play a more signicificant role, but the fundamental harmonic always has the highest flux. I.10.3 Effects of the emission of radiation on beam dynamics In this section, we will delve deeper into the interplay between the radiation emission and the ensuing dynamics of the beam. The treatment closely follows the book by Wolski [4]. First, we will explore the energy transfer that occurs when an electron emits a photon. Following this, we will make a coordinate transformation to the more beneficial action and angle variables, providing a clearer perspective on the underlying mechanisms. We will then proceed to compute the ensemble average to calculate the implications on the emittance of the beam. A noteworthy observation will emerge from our analysis: the emittance decreases exponentially, plateauing at a limit dictated by the fundamental principles of quantum mechanics. This revelation underscores the intricate ties between quantum mechanics and relativistic beam dynamics, shedding light on the broader consequences of radiation emission in storage rings. In the following sections, we will make use of Hamiltonian mechanics. Those not familiar with this matter are invited to watch two introductory videos: "Hamiltonian formalism $1 ^ { \\dag 6 }$ and "Hamiltonian formalism 2"7.
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What are the filter settings for the DSCR screens?
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The screens use two filters with 10% and 1% attenuation, that give attenuations of 10%, 1%, and 0.1%.
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[ScreenUpgrade]RevSciInst_94_073301(2023).pdf
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The evaluation of the beam size resolution of the screen profile monitors is usually first performed in a laboratory, using calibrated optical targets, and then sometimes checked in the beam itself. However, the in-beam checks can be difficult to execute properly without damaging the scintillators. The simplest way to evaluate the beam size resolution of the screen would be to focus the electron beam to a single point on the screen and record the resulting profile size on the monitor. However, this approach has the disadvantage that the tightly focused beam may damage the screen, degrading its performance and reliability. This article demonstrates a gentler way of determining the resolution of the screen while they are in use. II. MEASUREMENT SETUP The diagnostic screens (DSCRs) used at SwissFEL were developed to use a Scheimpflug geometry to minimize the effect of COTR generated from microbunching that may be present in the SwissFEL electron bunch on the profile measurement.10,14 An optical setup was developed to cover the large range of intensities that the scintillating effect would generate under various measurement regimes of the electron beam. The optical setup uses a Nikon $2 0 0 \\mathrm { m m } \\mathrm { f } / 4$ ED-IF AF Micro lens set at a working distance of $2 5 0 ~ \\mathrm { m m }$ from the scintillating screen. The Ce:YAG scintillating screen is in vacuum, with the light propagating through a $1 5 ^ { \\circ }$ Scheimpflug geometry, through a sapphire vacuum viewport, a mirror, and then into the lens, with a PCO.edge 5.5 camera behind it. The scintillating light has a wavelength range from about 500 to $7 0 0 \\mathrm { n m }$ , with a maximum at $5 5 0 \\mathrm { n m }$ . The design keeps the camera gain at a constant level to maximize the signal-to-noise ratio of the camera electronics and introduces a $1 \\%$ or $1 0 \\%$ neutral density (ND) filter about $2 0 ~ \\mathrm { m m }$ before the lens along the optical path to reduce the intensity of the scintillator light going into the camera. This gives the system the ability to observe the image at $1 0 0 \\%$ , $1 0 \\%$ , $1 \\%$ , or $0 . 1 \\%$ transmission, depending on which combination of ND filters we insert, if any at all. The original chosen filters were Kodak filter foils. The thin foils were thought to have a minimal lensing effect on the optical setup due to their thinness. The optical components are centered on the optical path axis of the scintillated light. The preliminary measurements in an optical laboratory showed that the optical system should have a resolution of about $1 4 \\mu \\mathrm m$ . This diagnostic screen setup was tested at the SwissFEL test facility with a tightly focused, low-charge electron beam and showed a resolution of about $1 6 \\mu \\mathrm m$ .10 A schematic drawing of the setup is shown in Fig. 1.
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What are the filter settings for the DSCR screens?
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The screens use two filters with 10% and 1% attenuation, that give attenuations of 10%, 1%, and 0.1%.
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[ScreenUpgrade]RevSciInst_94_073301(2023).pdf
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However, the first experimental use of these devices after installation to evaluate the emittance of the electron beam at SwissFEL showed that the resolution was significantly worse and was found to be between 20 and $4 0 \\ \\mu \\mathrm { m }$ when using the filters.15–17 An investigation of the filters showed that the thin film deforms and degrades over time and adds a blurring effect, spoiling the optical system resolution. The design was changed to use higher quality ND filters that are made of glass but would affect the optical path. This necessitated the addition of a motorized stage for the lens and camera system to be able to re-focus for every new filter setting, as shown in Fig. 2. This new setup differs from the original in that it allows for changes of the optical path to control the focusing and in the quality of the filters. All other components are unchanged. After the optimization of the camera and lens position to find the best focus for each ND filter combination, we set the control system for the DSCRs so that the proper position would be set for every ND filter combination and resulting transmission. We obtained the screen resolution by measuring the beam size for different electron beam energies, while keeping the same emittance and optics. The measured beam size has two contributions: (1) the true beam size, which can be expressed as $\\sqrt { \\beta \\varepsilon / \\gamma } , ^ { 1 8 }$ where $\\beta$ is the beta-function of the electron beam, $\\pmb { \\varepsilon }$ is the norma/lized emittance, and $\\gamma$ is the relativistic Lorentz factor of the electron beam (with $\\gamma$ being the electron beam energy divided by $m _ { e } c ^ { 2 }$ ), and (2) the screen resolution $\\sigma _ { \\mathrm { s c r } }$ . Since the two components are, in principle, not correlated, the square of the measured beam size can be expressed as the sum of the square of each of the two contributions,
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What are the filter settings for the DSCR screens?
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The screens use two filters with 10% and 1% attenuation, that give attenuations of 10%, 1%, and 0.1%.
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[ScreenUpgrade]RevSciInst_94_073301(2023).pdf
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$$ \\sigma _ { t o t } ^ { 2 } = \\sigma _ { s c r } ^ { 2 } + \\frac { \\beta \\varepsilon } { \\gamma } . $$ It is clear from the above equation that by measuring the electron beam sizes $\\sigma _ { \\mathrm { t o t } }$ for different electron beam energies $\\gamma _ { : }$ , one can reconstruct the screen resolution $\\sigma _ { \\mathrm { s c r } }$ as well as the product of the emittance and beta function $\\beta \\varepsilon$ . This method is inspired by similar ones where certain beam or lattice parameters are varied to obtain the screen resolution.15–17,19,20 It is an implementation of the approach proposed in Ref. 17 for a location without dispersion. III. RESULTS We used the standard $2 0 0 ~ \\mathrm { p C }$ beam and changed the electron beam energy at the end of the SwissFEL linac on the Aramis beamline from 3 to $6 \\ : \\mathrm { G e V }$ . The measurement of the performance of the new system was directly compared with that of the old setup with foils by putting both sets of filters in one optical box and using both for each electron beam energy setting. We recorded ten images for each electron beam energy and filter. The beam size for each image was obtained by fitting a Gaussian function to the image projection. We then fit Eq. (1) to the measured beam sizes to reconstruct the screen resolution and the product of emittance and beta function. Figure 3 shows the single-shot images for different settings. Figure 4 displays the vertical beam sizes averaged over ten shots and the calculated fits under different conditions. As shown in Fig. 4, the measured beam sizes are significantly larger with the foil filter when compared to the glass filter, indicating a worse screen resolution.
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What are the filter settings for the DSCR screens?
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The screens use two filters with 10% and 1% attenuation, that give attenuations of 10%, 1%, and 0.1%.
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[ScreenUpgrade]RevSciInst_94_073301(2023).pdf
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File Name:[ScreenUpgrade]RevSciInst_94_073301(2023).pdf Testing high-resolution transverse profile monitors by measuring the dependence of the electron beam size on the beam energy at SwissFEL Pavle Juranić  ; Eduard Prat  $\\textcircled{1}$ DCheck for updates Articles You May Be Interested In Perspective: Opportunities for ultrafast science at SwissFEL Struct Dyn (January 2018) Physical optics simulations with Pඐඉඛඍ for SwissFEL beamlines AIP Conference Proceedings (July 2016) Optical design of the ARAMIS-beamlines at SwissFEL AIP Conference Proceedings (July 2016) Testing high-resolution transverse profile monitors by measuring the dependence of the electron beam size on the beam energy at SwissFEL Cite as: Rev. Sci. Instrum. 94, 073301 (2023); doi: 10.1063/5.0155444 Submitted: 20 April 2023 $\\cdot \\cdot$ Accepted: 17 June 2023 • Published Online: 5 July 2023 Pavle Jurani´ca) and Eduard Prata) AFFILIATIONS Paul Scherrer Institut, Forschungsstrasse 111, 5232 Villigen, Switzerland a)Authors to whom correspondence should be addressed: pavle.juranic@psi.ch and eduard.prat@psi.ch ABSTRACT Transverse profile monitors are essential devices to characterize particle beams in accelerators. Here, we present an improved design of beam profile monitors at SwissFEL that combines the use of high-quality filters and dynamic focusing. We reconstruct the profile monitor resolution in a gentle way by measuring the electron beam size for different energies. The results show a significant improvement of the new design compared to the previous version, from 20 to $1 4 \\mu \\mathrm m$ .
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What are the filter settings for the DSCR screens?
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The screens use two filters with 10% and 1% attenuation, that give attenuations of 10%, 1%, and 0.1%.
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[ScreenUpgrade]RevSciInst_94_073301(2023).pdf
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V. CONCLUSION The method presented in this article shows the ability to improve the screen resolution by using high-quality filters and dynamic focusing. The profile monitor resolution is reconstructed in a gentle way by measuring the electron beam sizes for different beam energies. Our results show a significant improvement of the beam size resolution (from 20 to $1 4 \\mu \\mathrm m \\dot { }$ ) with the new optical design. These improvements allow for regular evaluations of the resolution and the ability to set the focusing to different values to compensate for the change in optical properties of the setup, enabling a more reliable and consistent performance of the diagnostic for transverse beam profile measurements. ACKNOWLEDGMENTS The authors would like to thank and acknowledge the work of the technical groups that maintain and operate SwissFEL. Of those, special thanks go to Didier Voulot who was instrumental in setting up a script for automatic beam energy scaling in the SwissFEL accelerator that preserved the electron beam optics. Further thanks go to Thomas Schietinger and Rasmus Ischebeck for proofreading the manuscript. AUTHOR DECLARATIONS Conflict of Interest The authors have no conflicts to disclose. Author Contributions Pavle Jurani´c: Conceptualization (equal); Data curation (supporting); Formal analysis (supporting); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (lead); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Eduard Prat: Conceptualization (equal); Data curation (lead); Formal analysis (lead);
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What are the filter settings for the DSCR screens?
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The screens use two filters with 10% and 1% attenuation, that give attenuations of 10%, 1%, and 0.1%.
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[ScreenUpgrade]RevSciInst_94_073301(2023).pdf
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$^ { 8 } { \\mathrm { C } } .$ Wiebers, M. Hoz, G. Kube, D. Noelle, G. Priebe, and H.-C. Schroeder, in Proceedings of the 2nd International Beam Instrumentation Conference (IBIC 2013) (JACOW, Oxford, UK, 16-19 September 2013), p. 807. ${ ^ \\circ _ { \\mathrm { H } } } .$ D. T. ChoiKim, M. Chae, J. Hong, S.-J. Park, and C. Kim, in Proceedings of the 4th International Particle Accelerator Conference (IPAC 2013) (JACOW, Shanghai, China, 12-17 May 2013), p. 610. $^ { 1 0 } \\mathrm { R } .$ Ischebeck, E. Prat, V. Thominet, and C. O. Loch, Phys. Rev. Spec. Top.–Accel. Beams 18, 082802 (2015). ${ } ^ { 1 1 } \\mathrm { H }$ . Loos, in Proceedings of the 3rd International Beam Instrumentation Conference (IBIC 2014) (JACOW, Monterey, CA, 14-18 September 2014), p. 475. $^ { 1 2 } \\mathrm { Y }$ . Otake, H. Maesaka, S. Matsubara, S. Inoue, K. Yanagida, H. Ego, C. Kondo, T. Sakurai, T. Matsumoto, and H. Tomizawa, Phys. Rev. Spec. Top.–Accel. Beams 16, 042802 (2013). $^ { 1 3 } \\mathrm { B }$ . Walasek-Hohne, C. Andre, P. Forck, E. Gutlich, G. Kube, P. Lecoq, and A. Reiter, IEEE Trans. Nucl. Sci. 59(5), 2307–2312 (2012).
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What are the filter settings for the DSCR screens?
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The screens use two filters with 10% and 1% attenuation, that give attenuations of 10%, 1%, and 0.1%.
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Fact
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[ScreenUpgrade]RevSciInst_94_073301(2023).pdf
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$^ { 1 4 } \\mathrm { M }$ . Castellano and V. A. Verzilov, Phys. Rev. Spec. Top.–Accel. Beams 1, 062801 (1998). $^ { 1 5 } \\mathrm { E }$ . Prat, P. Craievich, P. Dijkstal, S. Di Mitri, E. Ferrari, T. G. Lucas, A. Malyzhenkov, G. Perosa, S. Reiche, and T. Schietinger, Phys. Rev. Accel. Beams 25, 104401 (2022). $^ { 1 6 } \\mathrm { E }$ . Prat, P. Dijkstal, M. Aiba, S. Bettoni, P. Craievich, E. Ferrari, R. Ischebeck, F. Löhl, A. Malyzhenkov, G. L. Orlandi, S. Reiche, and T. Schietinger, Phys. Rev. Lett. 123, 234801 (2019). 17E. Prat, P. Dijkstal, E. Ferrari, A. Malyzhenkov, and S. Reiche, Phys. Rev. Accel. Beams 23, 090701 (2020). $^ { 1 8 } \\mathrm { J }$ . Rossbach and P. Schmuser, in CAS—CERN Accelerator School: 5th General Accelerator Physics Course, edited by S. Turner (CERN, 1994), pp. 17–88. ${ } ^ { 1 9 } \\mathrm { H } .$ . J. Qian, M. Krasilnikov, A. Lueangaramwong, X. K. Li, O. Lishilin, Z. Aboulbanine, G. Adhikari, N. Aftab, P. Boonpornprasert, G. Georgiev, J. Good, M. Gross, C. Koschitzki, R. Niemczyk, A. Oppelt, G. Shu, F. Stephan, G. Vashchenko, and T. Weilbach, Phys. Rev. Accel. Beams 25, 083401 (2022). $^ { 2 0 } \\mathrm { S } .$ . Tomin, I. Zagorodnov, W. Decking, N. Golubeva, and M. Scholz, Phys. Rev. Accel. Beams 24, 064201 (2021).
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Expert
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What central wavelength of THz radiation was measured experimentally?
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Approximately 881 ?m
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hermann-et-al-2022-inverse-designed-narrowband-thz-radiator-for-ultrarelativistic-electrons.pdf
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$$ The design obtained from the gradient-based technique of adaptive moment estimation $( \\mathrm { A d a m } ) ^ { 2 . 5 }$ is depicted in Figure 1b. The structure features two rows of pillars, shifted by half a period with respect to each other. The rows of pillars are followed by three slabs on each side, which can be easily identified as distributed Bragg reflectors forming a microresonator around the electron channel. The channel width is $2 7 2 \\ \\mu \\mathrm { m } ,$ , even larger than the initially defined clearance of 150 $\\mu \\mathrm { m }$ . These slabs exhibit grooves, which perhaps act as a grating as well as a reflector. We note that these features are good examples of the superiority of inverse design over intuitionbased designs. To fabricate the geometry obtained with inverse design, we used an additive manufacturing process for poly(methyl methacrylate) (PMMA). A stereolithography device, featuring a resolution of $1 4 0 \\ \\mu \\mathrm { m } ,$ , is capable of reproducing the structure with subwavelength accuracy. The so-obtained structure is 6 mm high and $4 5 \\ \\mathrm { m m }$ long (Figure 1d). The holder of the structure was manufactured together with the structure, and filaments connect the pillars and slabs on top of the structure for increased mechanical stability. We selected the Formlabs High Temperature Resin as a material for this study due to its excellent vacuum compatibility after curing in a heated vacuum chamber.24 Afterward, the fabricated Smith‚àíPurcell radiator was inserted into the ACHIP experimental chamber26 at SwissFEL27 (Figure 2a). The photoemitted electron bunch is accelerated to an energy of $3 . 2 ~ \\mathrm { \\ G e V }$ with the normalconducting radio frequency accelerator at SwissFEL. A twostage compression scheme using magnetic chicanes is employed to achieve an electron bunch length of approximately 30 fs at the interaction point. At this location, the transverse beam size was measured to be around $3 0 \\ \\mu \\mathrm { m }$ in the horizontal and $4 0 \\ \\mu \\mathrm { { m } }$ in the vertical direction.
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What central wavelength of THz radiation was measured experimentally?
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Approximately 881 ?m
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hermann-et-al-2022-inverse-designed-narrowband-thz-radiator-for-ultrarelativistic-electrons.pdf
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$$ \\lambda = \\frac { a } { m } \\Biggl ( \\frac { 1 } { \\beta } - \\cos { \\theta } \\Biggr ) $$ where $\\beta$ is the normalized velocity of the electrons, a is the periodicity of the structure, and $m$ is the mode order. Smith‚àí Purcell emission from regular metallic grating surfaces has been observed in numerous experiments, first using $3 0 0 ~ \\mathrm { k e V }$ electrons11 and later also using ultrarelativistic electrons.12,13 If electron pulses shorter than the emitted wavelength are used, the fields from individual electrons add coherently, and the radiated energy scales quadratically with the bunch charge.14 The typically used single-sided gratings emit a broadband spectrum,15 which is dispersed by the Smith‚àíPurcell relation (eq 1). To enhance emission at single frequencies, a concept called orotron uses a metallic mirror above the grating to form a resonator.16,17 Dielectrics can sustain fields 1‚àí2 orders of magnitude larger than metals18 and are therefore an attractive material for strong Smith‚àíPurcell interactions. Inverse design is a computational technique that has been successfully employed to advance integrated photonics.19 Algorithms to discover optical structures fulfilling desired functional characteristics are creating a plethora of novel subwavelength geometries: applications include wavelengthdependent beam splitters19,20 and couplers,21 as well as dielectric laser accelerators.22
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What central wavelength of THz radiation was measured experimentally?
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Approximately 881 ?m
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hermann-et-al-2022-inverse-designed-narrowband-thz-radiator-for-ultrarelativistic-electrons.pdf
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The geometric acceptance angle of the Michelson interferometer $\\Delta \\theta$ in the plane of the electron beam and the THz radiation defines the accepted bandwidth of the setup. According to the Smith‚àíPurcell relation (eq 1), it is given by $$ \\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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What central wavelength of THz radiation was measured experimentally?
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Approximately 881 ?m
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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 central wavelength of THz radiation was measured experimentally?
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Approximately 881 ?m
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hermann-et-al-2022-inverse-designed-narrowband-thz-radiator-for-ultrarelativistic-electrons.pdf
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File Name:hermann-et-al-2022-inverse-designed-narrowband-thz-radiator-for-ultrarelativistic-electrons.pdf Inverse-Designed Narrowband THz Radiator for Ultrarelativistic Electrons Benedikt Hermann,# Urs Haeusler,# Gyanendra Yadav, Adrian Kirchner, Thomas Feurer, Carsten Welsch, Peter Hommelhoff, and Rasmus Ischebeck\\* Cite This: ACS Photonics 2022, 9, 1143−1149 ACCESS 山 Metrics & More 国 Article Recommendations ABSTRACT: THz radiation finds various applications in science and technology. Pump−probe experiments at free-electron lasers typically rely on THz radiation generated by optical rectification of ultrafast laser pulses in electro-optic crystals. A compact and cost-efficient alternative is offered by the Smith−Purcell effect: a charged particle beam passes a periodic structure and generates synchronous radiation. Here, we employ the technique of photonic inverse design to optimize a structure for Smith− Purcell radiation at a single wavelength from ultrarelativistic electrons. The resulting design is highly resonant and emits narrowbandly. Experiments with a 3D-printed model for a wavelength of $9 0 0 \\mu \\mathrm { m }$ show coherent enhancement. The versatility of inverse design offers a simple adaption of the structure to other electron energies or radiation wavelengths. This approach could advance beam-based THz generation for a wide range of applications. KEYWORDS: THz generation, Smith−Purcell radiation, inverse design, light−matter interaction, free-electron light sources $\\mathbf { C }$ aopuprlciecs iof ,TiHnzc rdaidniagtiwoinr aerses ofmimntuenriecsat fonr ,n2uelmeecrtrouns acceleration,3−5 and biomedical and material science.6,7 Freeelectron laser (FEL) facilities demand versatile THz sources for pump−probe experiments.8 Intense, broadband THz pulses up to sub-mJ pulse energy have been demonstrated using optical rectification of high-power femtosecond lasers in lithium niobate crystals.9,10 The Smith−Purcell effect11 offers a compact and cost-efficient alternative for the generation of beam-synchronous THz radiation at electron accelerators. This effect describes the emission of electromagnetic waves from a periodic metallic or dielectric structure excited by electrons moving parallel to its surface. The wavelength of Smith− Purcell radiation at an angle $\\theta$ with respect to the electron beam follows:11
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What central wavelength of THz radiation was measured experimentally?
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Approximately 881 ?m
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hermann-et-al-2022-inverse-designed-narrowband-thz-radiator-for-ultrarelativistic-electrons.pdf
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RESULTS The goal of our inverse design optimization was a narrowband dielectric Smith‚àíPurcell radiator for ultrarelativistic electrons $\\mathit { \\check { E } } = 3 . 2 \\ \\mathrm { G e V }$ , $\\gamma \\approx 6 0 0 0 ,$ ). To simplify the collection of the THz radiation, a periodicity of $a = \\lambda$ was chosen, resulting in an emission perpendicular to the electron propagation direction, $\\theta \\ : = \\ : 9 0 ^ { \\circ }$ . The optimization was based on a 2D finite-difference frequency-domain (FDFD) simulation of a single unit cell of the grating (Figure 1a). Periodic boundaries in direction of the electron propagation ensure the desired periodicity, and perfectly matched layers in the transverse xdirection imitate free space. The design region extends $4 . 5 \\mathrm { m m }$ to each side of a $1 5 0 \\ \\mu \\mathrm { m }$ wide vacuum channel, large enough to facilitate the full transmission of the electron beam with a width of $\\sigma _ { x } = 3 0 \\ \\mu \\mathrm { m }$ (RMS). The electric current spectral density $\\scriptstyle \\mathbf { J } ( x , y , \\omega )$ of a single electron bunch acts here as the source term of our simulation and is given by
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Expert
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What central wavelength of THz radiation was measured experimentally?
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Approximately 881 ?m
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hermann-et-al-2022-inverse-designed-narrowband-thz-radiator-for-ultrarelativistic-electrons.pdf
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$$ { \\bf J } ( x , y , \\omega ) = \\frac { q } { 2 \\pi } { \\cdot } ( 2 \\pi \\sigma _ { x } ^ { 2 } ) ^ { - 1 / 2 } { \\cdot } e ^ { - x ^ { 2 } / 2 \\sigma _ { x } ^ { 2 } } { \\cdot } e ^ { - i k _ { \\mathrm { y } } y } \\hat { \\bf y } $$ with the electron wavevector $k _ { y } = 2 \\pi / \\beta \\lambda$ and the line charge density $q$ . The absolute value of $q$ is irrelevant for the optimization, but rough agreement with 3D simulations is found by choosing $q \\sim Q / { \\dot { d } } ,$ where $Q$ is the bunch charge and $d$ the charge-structure distance.23 The optimization problem was to find a design (parametrized by the variable $\\phi$ ) that maximizes the radiation to both sides of the grating. Exploiting the full symmetry of the double-sided, perpendicular emission process, we enforced mirror and point symmetry with respect to the center of a unit cell of the grating. The design is defined by its relative permittivity $\\bar { \\varepsilon ( x , y , \\phi ) }$ and can only take the two values of vacuum, $\\varepsilon ~ = ~ 1$ , or the structure material, $\\varepsilon \\ = \\ 2 . 7 9$ . For simplicity, we neglected the small imaginary part $\\varepsilon ^ { \\prime \\prime } = 0 . 0 8$ of the material.24
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What central wavelength of THz radiation was measured experimentally?
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Approximately 881 ?m
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hermann-et-al-2022-inverse-designed-narrowband-thz-radiator-for-ultrarelativistic-electrons.pdf
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The objective function $G$ , quantifying the performance of a design $\\phi ,$ is given by the line integral of the Poynting vector $\\begin{array} { r } { { \\bf S } ( x , y ) = \\mathrm { R e } \\left\\{ \\frac { 1 } { 2 } { \\bf E } \\times { \\bf H } ^ { * } \\right\\} } \\end{array}$ in the $x$ -direction along the length of one period, evaluated at a point $x _ { S }$ outside the design region: $$ 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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Expert
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What central wavelength of THz radiation was measured experimentally?
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Approximately 881 ?m
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Fact
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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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augmentation
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Expert
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What central wavelength of THz radiation was measured experimentally?
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Approximately 881 ?m
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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 central wavelength of THz radiation was measured experimentally?
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Approximately 881 ?m
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Fact
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hermann-et-al-2022-inverse-designed-narrowband-thz-radiator-for-ultrarelativistic-electrons.pdf
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Our work naturally extends to the field of subrelativistic electrons. Here, simultaneous arrival of THz radiation and electron bunches is readily achieved by compensating for the higher velocity of the radiation with a longer path length (Figure 5b). Besides its application for pump‚àíprobe experiments, our structure can be more generally applied as a radiation source at wavelengths that are otherwise difficult to generate. An advantage lies in the tunability that arises from changing the periodicity, either by replacing the entire structure or using a design with variable periodicity (Figure 5c), or from tuning the electron velocity. For the visible to UV regime, the idea of a compact device with the electron source integrated on a nanofabricated chip has recently sparked interest.30,31 METHODS Structure Parametrization. Our inverse design process was carried out with an open-source Python package32 suitable for 2D-FDFD gradient-based optimizations25 of the chosen objective function $G ( \\phi )$ with respect to the design parameter $\\phi$ . A key step lies in the parametrization of the structure $\\varepsilon ( \\phi )$ through the variable $\\phi$ in a way that ensures robust convergence of the algorithm and fabricability of the final design. In the most rudimentary case, $\\varepsilon ( x , y ) \\stackrel { \\cdot } { = } \\phi ( x , y )$ is a two-dimensional array with entries $\\in [ 1 , 2 . 7 9 ]$ for each pixel of the design area. Instead of setting bounds on the values of $\\phi ,$ we leave $\\phi$ unbounded and apply a sigmoid function of the shape
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Expert
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What central wavelength of THz radiation was measured experimentally?
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Approximately 881 ?m
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Fact
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hermann-et-al-2022-inverse-designed-narrowband-thz-radiator-for-ultrarelativistic-electrons.pdf
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$$ \\epsilon _ { r } ( x , y ) = \\epsilon _ { \\mathrm { m i n } } + ( \\epsilon _ { \\mathrm { m a x } } - \\epsilon _ { \\mathrm { m i n } } ) { \\cdot } \\frac { 1 } { 2 } ( 1 + \\operatorname { t a n h } \\alpha \\phi ( x , y ) ) $$ where large values of $\\alpha$ yield a close-to-binary design with few values between $\\varepsilon _ { \\operatorname* { m i n } } = 1$ and $\\varepsilon _ { \\operatorname* { m a x } } = 2 . 7 9$ . To avoid small or sharp features in the final design, we convolved $\\phi ( x , y )$ with a uniform 2D circular kernel with radius $6 0 \\mu \\mathrm m$ before projection onto the sigmoid function tanh $( \\alpha \\tilde { \\phi } )$ with the convolved design parameter $\\tilde { \\phi }$ . By increasing $\\alpha$ from 20 to 1000 as the optimization progresses, we found improved convergence. We further accelerated convergence by applying mirror and point symmetry with respect to the center of a unit cell of the grating, which reduces the parameter space by a factor of 4. An exemplary design evolution is shown in Figure 6.
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Expert
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What central wavelength of THz radiation was measured experimentally?
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Approximately 881 ?m
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Fact
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hermann-et-al-2022-inverse-designed-narrowband-thz-radiator-for-ultrarelativistic-electrons.pdf
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Ultrarelativistic Optimization. The simulation of ultrarelativistic electrons poses challenges that have so far prevented inverse design in this regime.33 Here, we report on two main challenges. First, the electron velocity is close to the speed of light $( \\beta = 0 . 9 9 9 9 9 9 9 8 5$ for $E = 3 . 2 \\mathrm { G e V } ,$ ), which requires a high mesh resolution. If the numerical error is too large due to a low mesh resolution, the simulation may not be able to distinguish between $\\beta < 1$ and $\\beta > 1$ . In that case, the simulation could show Cherenkov radiation in vacuum instead of Smith‚àíPurcell radiation. Not only does a higher mesh resolution require more computational memory and time, but it may also hamper the inverse design optimization if the number of design parameters becomes too large. Hence, we parametrized our structures at a low resolution (mesh spacing $\\lambda / { 3 0 } \\mathrm { \\dot { } }$ ), which is still above the fabrication accuracy of $\\lambda / 5$ , and computed the fields at a high resolution (mesh spacing $\\lambda / { 1 5 0 } \\dot$ ).
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Expert
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What central wavelength of THz radiation was measured experimentally?
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Approximately 881 ?m
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Fact
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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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Expert
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What central wavelength of THz radiation was measured experimentally?
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Approximately 881 ?m
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Fact
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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 central wavelength of THz radiation was measured experimentally?
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Approximately 881 ?m
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Fact
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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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Yes
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Expert
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What central wavelength of THz radiation was measured experimentally?
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Approximately 881 ?m
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Fact
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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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Yes
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Expert
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What central wavelength of THz radiation was measured experimentally?
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Approximately 881 ?m
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Fact
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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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augmentation
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Yes
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Expert
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What central wavelength of THz radiation was measured experimentally?
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Approximately 881 ?m
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Fact
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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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augmentation
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Expert
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What central wavelength of THz radiation was measured experimentally?
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Approximately 881 ?m
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Fact
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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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