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**ILC School Chicago, Oct. 21, 2008 T. Higo, KEK**

LINAC-I ILC School Chicago, Oct. 21, 2008 T. Higo, KEK

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**LC School, Chicago, 2008, LINAC-I**

Contents of LINAC-I Acceleration for energy Requirement on acceleration to high energy Maxwell’s eq. to describe microwave transmission Cavity as a unit base of acceleration Coupled cavity system in a SW regime Pillbox cavity as a simple base to represent practical cavities Circuit modeling of cavity SW versus TW TW linac Summary of LINAC-I T. Higo LC School, Chicago, 2008, LINAC-I

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**LC School, Chicago, 2008, LINAC-I**

Energy and luminosity T. Higo LC School, Chicago, 2008, LINAC-I

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**Acceleration for energy**

T. Higo LC School, Chicago, 2008, LINAC-I

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**Requirements for high energy machine**

High gradient High efficiency Emittance preservation Stable operation Low cost Construction, operation, Electric consumption, cooling, T. Higo LC School, Chicago, 2008, LINAC-I

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**Evolution of gradient and energy**

Type Beam Energy Acceleration scheme Cockcroft Walton 1MeV 1MV/m DC Rectify Van de Graaff 10MeV 10MV/m DC Charging Cyclotron 100MeV 1MV/Dee Cyclic Synchrotron 100GeV 100MV/ring Cyclic Linear accelerator 1TeV 100MV/m Periodic Plasma accelerator 10TeV? >10GV/m Plasma T. Higo LC School, Chicago, 2008, LINAC-I

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**LC School, Chicago, 2008, LINAC-I**

Early accelerator developments after pioneering experiment with electrons or protons 1896 Thomson vacuum tube with thermal electrons Cathode ray tube 1911 Rutherford Radioactive material Alpha particle scattering by nucleus 1928 Wideröe Linear accelerator idea 1931 Sloan & Laurence Linear accelerator experiment 1932 Cockcroft & Walton ~1MeV High voltage by rectifier 1933 Van de Graaff ~a few MeV High voltage by carrying charge by belt T. Higo LC School, Chicago, 2008, LINAC-I

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**Linear accelerators Alvarez Sloan Lawrence Wideröe**

Independent driven electrodes Wideröe Driven electrodes Alvarez Resonant cavity with drift tube without acceleration/deceleration Sobenin text p18 加速器 p200 L. W. Alvarez, Rev. Sci. Instr. 26, p111, 1955 T. Higo LC School, Chicago, 2008, LINAC-I

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**Development for higher energy**

1931 Lawrence Several to higher than 10MeV, proton Cyclotron acceleration 1945 McMillan and Veksler higher and higher energy Synchrotron acceleration for proton 1952 Courant, Snyder: Strong focus AGS (alternating gradient synchrotron) 1950~1960’s Stanford electron linear accelerators GeV electron ~1955: Ginzton, Hansen, Chodorow: Mark-II~III Microwave technology the legacy of world war II 1967: Panofsky: 2-mile accelerator 20GeV ~10MV/m Late 1980’s Richter: Stanford Linear Collider Energy doubler by pulse compression technique T. Higo LC School, Chicago, 2008, LINAC-I

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**For higher energy machine**

TW DLS for electron high energy machine for years Disk Loaded Structure, 6MeV Stanford Univ. 1947 LANL SCS: Side coupled structure SW Side-coupled cavity for proton high energy machine SLAC: Targeting highest energy with electron Picture of Ginzton et al., from SLAC Beam Line ?? T. Higo LC School, Chicago, 2008, LINAC-I

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**Storage ring, collider to linear collider**

Storage ring in e+e- colliding mode PEP / PETRA ~17.5X2 GeV TRISTAN 30X2 GeV LEP X2 GeV Linear collider plans 1980’s VLEPP 14GHz, 100MV/m 1990’s TESLA GHz, 23.4MV/m 500Gev higher? 1990’s GLC/NLC 11.4GHz, 50MV/m 1TeV ILC 500GeV CLIC 3TeV Further higher energy machine plasma, laser, etc…….. T. Higo LC School, Chicago, 2008, LINAC-I

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**For efficiency toward high energy LEP and for stability toward high current KEKB**

LEP cavity with storage cavity for efficiency improvement ARES for KEKB with storage cavity for beam stability Wilson and H Henke, “The LEP main ring accelerator structure”, CERN 89-09 Y. Yamazaki and T. Kageyama, Part. Accel (1994). T. Higo LC School, Chicago, 2008, LINAC-I

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**Higher and higher energy for lepton linear collider**

14GHz DLS VLEPP single-bunch, high rep-rate 3GHz DLS S-band DESY multi-bunch 1.3GHz 9-cell SCC cavity TESLA DESY 11.4GHz DLS SLAC/KEK DDS (weakly damped & detuned) 30GHz DLS CLIC (Heavily damped) 1.3GHz 9-cell SCC cavity ILC developing 12GHz DLS in study Only two types; TW DLS and SW 9-cell shaped cavity. Simple and low cost. T. Higo LC School, Chicago, 2008, LINAC-I

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**Higher electron energy being developed**

With care against wake field in various manners. Continue with TW DLS for higher-energy electron linear machine Super-conducting cavity for higher-energy electron machine SW SCC cavity TESLA TDR 9-cell cavity with HOM damping port T. Shintake, KEK-Priprint CLIC Quadrant-type DLS DLS: Medium-damped detuned structure T. Higo LC School, Chicago, 2008, LINAC-I

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**LC School, Chicago, 2008, LINAC-I**

LHC ILC/CLIC Livingston plot Original figure taken from SLAC Beam Line ?? 2010 2020 2030 T. Higo LC School, Chicago, 2008, LINAC-I

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**LC School, Chicago, 2008, LINAC-I**

Acceleration scheme Electric field to accelerate Voltage across electrodes DC: once RF in ring: many times n/freq=circumference/v RF in line: once Synchronization along a line It is important to focus electric field along beam axis to effectively accelerate beam. T. Higo LC School, Chicago, 2008, LINAC-I

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**How to reach higher energy**

Energy gradient X length DC: Van de Graaf, CockCroft Walton RF based Sloan Wideroe Alvarez SW and TW Independent RF source or Two beam acceleration Plasma accelerator T. Higo LC School, Chicago, 2008, LINAC-I

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**Limiting factors against ultimate gradient**

Peak power available Breakdown in structure Quench Mechanical stability He cooling Thermal / mechanical Dark current loading Phase coherency along a long line T. Higo LC School, Chicago, 2008, LINAC-I

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**LC School, Chicago, 2008, LINAC-I**

In this lecture RF acceleration is the only technology with which we can reach TeV range accelerator in very near future We focus here, as the examples, on the RF acceleration at microwave range L-band (1.3GHz) superconducting cavity X-band (11-12GHz) normal conducting cavity T. Higo LC School, Chicago, 2008, LINAC-I

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**Wave length / Frequency / Band**

60cm 20cm 10cm 2.6cm 1cm 3mm Wave length 1.3GHz Frequency 11GHz Band From Microwave Tubes by A. S. Gilmour, Jr. T. Higo LC School, Chicago, 2008, LINAC-I

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**Two types of linear accelerators**

I try to overview two types of acceleration scheme as an introduction of linear accelerator. Super-conducting / Normal conducting Lower frequency / Higher frequency Standing wave / Travelling wave These happen to be two candidates for linear collider which I believe we can explore in near future. ILC / CLIC T. Higo LC School, Chicago, 2008, LINAC-I

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**Linac example parameters ILC and CLIC**

units ILC(RDR) CLIC(500) Injection / final linac energy ELinac GeV 25 / 250 / 250 Acceleration gradient Ea MV/m 31.5 80 Beam current Ib A 0.009 2.2 Peak RF power / cavity Pin MW 0.294 74 Initial / final horizontal emittance ex mm 8.4 / 9.4 2 / 3 Initial / final vertical emittance ey nm 24 / 34 10 / 40 RF pulse width Tp ms 1565 242 Repetition rate Frep Hz 5 50 Number of particles in a bunch N 109 20 6.8 Number of bunches / train Nb 2625 354 Bunch spacing Tb ns 360 0.5 Bunch spacing per RF cycle Tb/ TRF 468 6 CLIC500 Alexej Grudief, 3rd. ACE, CLIC Advisory Committee, CERN, Sep. 2008, T. Higo LC School, Chicago, 2008, LINAC-I

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**Linac example parameters ILC and CLIC**

units ILC(RDR) CLIC(500) RF frequency F GHz 1.3 12 Beam phase w.r.t. RF degrees 5 15 EM mode in cavity SW TW Number of cells / cavity Nc 9 19 Cavity beam aperture a/l 0.152 0.145 Bunch length sz mm 0.3 0.044 ILC parameters are taken from Reference Design Report of ILC for 500GeV. CLIC500 parameters are taken from the talk by A. Grudief, 3rd. ACE, CLIC Advisory Committee, CERN, Sep. 2008, T. Higo LC School, Chicago, 2008, LINAC-I

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**Maxwell’s eq. to describe microwave transmission**

T. Higo LC School, Chicago, 2008, LINAC-I

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**Maxwell’s equation and wave propagation**

Then, Maxwell’s eq. becomes; This has a solution of a propagation along the z-direction; If all quantities vary time harmonically; T. Higo LC School, Chicago, 2008, LINAC-I

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**LC School, Chicago, 2008, LINAC-I**

to wave equation a0 attenuation and b0 wave number along the propagation direction Using Z0 and g0 to rewrite Maxwell’s eq.; In vacuum, s=0 velocity In a plane wave, Ex and Hy From these, the wave equation becomes; Wave impedance becomes; T. Higo LC School, Chicago, 2008, LINAC-I

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**Wave propagation in uniform medium**

Good conductor Insulator / vacuum Poor conductor Lossy propagation E and H at 45 degrees Velocity in free space T. Higo LC School, Chicago, 2008, LINAC-I

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**Reflection from good conductor and surface resistance**

This means phase lag between E and H. Better conductor makes E smaller and smaller. x z e, m, s e0, m0 Assume plane wave incident in z-direction, Wave equation for the transmitted wave into material becomes Treatment on reflection from conducting medium R. E. Collin, “Foundations for Microave Engineering,” McGraw-Hill, Kogakusha, Ltd. p47 Exponential decaying filed into material. Skin depth = e-folding depth. ~ 0.6micron in copper at 12GHz. T. Higo LC School, Chicago, 2008, LINAC-I

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**LC School, Chicago, 2008, LINAC-I**

Surface resistance Corresponding magnetic field in medium is Then wave impedance in the medium (Cu case) is From boundary condition at the surface, reflection coefficient becomes Almost full reflection Surface current Surface current Magnetic field is terminated by the surface current Js within the thickness ds. Loss occurs in the volume current with equivalent surface resistance T. Higo LC School, Chicago, 2008, LINAC-I

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**EM wave along a uniform guide**

In H field, also the same story. This equation can be solved with proper cutoff propagation constant to satisfy boundary condition; In non conducting case, Then, no propagation at low frequency; T. Higo LC School, Chicago, 2008, LINAC-I

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**Propagation field along a uniform guide**

Where Et, Ht are transverse component vector, while Ez, Hz are both scalar and Them, Maxwell’s equation gives A function Ez, Hz, which satisfy the wave equation, make the transverse component. T. Higo LC School, Chicago, 2008, LINAC-I

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**LC School, Chicago, 2008, LINAC-I**

TE (H) wave / TM (E) wave We can choose either Ez=0 or Hz=0, making These are classified into two modes; a pure TE (no longitudinal E field) or a pure TM (no longitudinal H field) If the waveguide has a modulation along z direction, pure TE nor pure TM can exist. This is the reality and we call it HEM, hybrid mode. T. Higo LC School, Chicago, 2008, LINAC-I

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**Transverse field pattern in rectangular waveguide**

Solving wave equation with satisfying boundary condition a b x y (x,y) T. Higo LC School, Chicago, 2008, LINAC-I

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**Typical field patterns**

E. Marcuvitz ed., Microwave Handbook E. Marcuvitz ed., Microwave Handbook, p71 Fig. 2.6, McCGRAW-HILL BOOK Company, Inc., 1951. E-mode / TM-mode H-mode / TE-mode T. Higo LC School, Chicago, 2008, LINAC-I

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**Transverse field pattern in cylindrical waveguide**

Solving wave equation with satisfying boundary condition TM mode case (r,q) r=a 野口メモ p3 11式 T. Higo LC School, Chicago, 2008, LINAC-I

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**Typical field patterns**

E. Marcuvitz ed., Microwave Handbook E. Marcuvitz ed., Microwave handbook, p71 Fig. 2.6, McCGRAW-HILL BOOK Company, Inc., 1951. E01 for acceleration E11 and H11 (or TM11 and TE11 and HEM11) for dipole mode, transverse kick E-mode / TM-mode H-mode / TE-mode T. Higo LC School, Chicago, 2008, LINAC-I

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**LC School, Chicago, 2008, LINAC-I**

Dispersion relation In TM case with m=0, Then the propagation of the form No acceleration in z-direction with TE mode because Ez=0. No acceleration in z-direction with TM mode because phase velocity is not c, making phase slip. Therefore, it cannot accelerate electrons in a long distance. freq betaz Linear Accelerator P43 eq. (3) Ez, Er, Htheta 野口表式との無矛盾かくにんせよ It is not posible to be in phase with beam for a long distance. T. Higo LC School, Chicago, 2008, LINAC-I

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**Reducing phase velocity to meet with beam velocity**

Add periodical perturbation with its period=d. If d=half wavelength, then reflection from each obstacle add coherently, making large reflection, resulting in a stop band. Then wave component with harmonics bz=2p/d suffer from significant reflections, making a stop band. ｄ To reduce and meet the phase velocity to beam T. Higo LC School, Chicago, 2008, LINAC-I

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**Expansion to space harmonics**

p/d 2p/d -p/d -2p/d b w This is equivalent to the Floquet’s theorem. Now it can be tuned to have a phase velocity of light. This is required for high energy linac structure. The accelerating field contains infinite number of space harmonics, driven at frequency w. There are stop bands. No propagation mode exists. Linear Accelerator p44 eq (4) T. Higo LC School, Chicago, 2008, LINAC-I

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**Uniform waveguide to isolated cavity**

ｄ d If the perturbation becomes large, reflection from the obstacle is so large that each cell becomes almost isolated cavity. Power propagation only through a very small aperture. In this extreme, the system can better be analyzed by a weakly coupled cavity chain model. Now let us start from isolated cavities. If the perturbation becomes large, reflection from the obstacle is so large that each cell becomes almost isolated cavity. Power propagation only through a very small aperture. In this extreme, the system can better be analyzed by a weakly coupled cavity chain model. T. Higo LC School, Chicago, 2008, LINAC-I

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**Cavity as a unit for acceleration**

T. Higo LC School, Chicago, 2008, LINAC-I

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**The extreme of isolated cavities**

Should be synchronized with beam With external reference With phasing among cavities External control Need many input circuits along linac Space factor is not high. Coupling between cavities The only way to practically apply for very long linac as linear collider Two ways, TW and SW. T. Higo LC School, Chicago, 2008, LINAC-I

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**LC School, Chicago, 2008, LINAC-I**

Inside a cavity Frequency Field phase should be synchronous to beam Acceleration field on axis Focusing Ez to beam axis R/Q=V2/2wU focusing the field within stored energy R shunt impedance R=V2/Pwall keep field by feeding power Loss factor beam cavity interaction kL= V2/4wU beam loading energy kq2 beam loading voltage -kqCosq T. Higo LC School, Chicago, 2008, LINAC-I

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**Most basic parameter: Frequency**

SCC cavity tuning Blue nominal freq Freq up green Freq down red Resonant frequency in electric circuit Cavity frequency can be tuned by changing L and/or C by perturbing magnetic field and/or electric field. Slater’s perturbation theory states; Four dimple tuning per cell in NCC. Slater perturbation theory p81 eq. (7.1) in J. C. Slater, “Microwave Electronics”, D. Van Nostrand Company, Inc., 1950. Actual cavity tuning can be done by deforming cell shape, local dimple tuning, inserting rod, etc. T. Higo LC School, Chicago, 2008, LINAC-I

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**Acceleration related parameters**

Basic acceleration-related parameters. In a cavity or in a unit length. Geometrical factor due to geometry. Surface resistance due to surface loss mechanism. Wall loss by surface integral Stored energy by volume integral T. Higo LC School, Chicago, 2008, LINAC-I

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**Efficient acceleration R/Q**

How to concentrate the Ez field on axis to make an efficient acceleration? Increase R/Q. For higher R/Q Smaller beam aperture smaller cell-to-cell coupling. Nose cone same as above need other coupling mechanism ILC super-conducting cavity smooth, polish with liquid, high pressure rinse, etc. with circle-ellipsoid smooth connection, nose cone is difficult less effort on higher R/Q, simply decreasing beam hole aperture because storing large energy with longer period is possible Choke mode cavity needs field at choke area to establish imaginary short sacrifice several % loss in R/Q Shaped disk-loaded structure only change R/Q by beam hole aperture T. Higo LC School, Chicago, 2008, LINAC-I

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**LC School, Chicago, 2008, LINAC-I**

Loss factor Loss factor KL described later The energy left after a bunch, with change q, passes a cavity is Larger R/Q makes bigger energy left in the cavity. It may cause various problems; Phase rotation of accelerating mode Transverse kick field Heating beam pipe In a ring application, such as storage ring and DR, sometimes R/Q should be reduced. In the linac application, it usually tuned to be maximized to get a better acceleration efficiency. T. Higo LC School, Chicago, 2008, LINAC-I

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**Acceleration: Transit time factor**

Assume TM010 mode in a pillbox of length L In p mode cavity Maximum acceleration occurs if the electric field is maximum when the beam passes the center of the cavity. Then transit time factor becomes In case of thin cavity, where L<< c / f, The acceleration felt by the beam decays as time, Voltage acquired by beam is then x=p Transit time factor: T. Higo LC School, Chicago, 2008, LINAC-I

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**LC School, Chicago, 2008, LINAC-I**

Surface loss and Q0 Super conductor, Nb case: Normal conductor: Equivalent surface current in thin skin depth ds with surface resistance Rs. Rs depend on mostly choice of material. at 1.3GHz, T=2K << 9K rBCS=11nW Higher freq larger BCS loss. Possible to increase geometrical factor, G by shaping. It reduces cryogenic power consumption. Actually, Rs = RBCS + Rresidual Need to keep smaller Rs by making proper material surface. Suppressing multipacting and field emission loading. sCu=5.8X107(1/W) Rs~28mW Higher Rs makes larger pulse surface heating during short pulse. T. Higo LC School, Chicago, 2008, LINAC-I

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**LC School, Chicago, 2008, LINAC-I**

How to increase Q0 ILC SCC TESLA to LL shape expanding cell outer area reduce H field against quench eventually increase Q0 by larger G Heavily damped cavity Loss in choke mode cavity, to establish imaginary short by storing power at choke Loss in heavily damped cavity with damping waveguide opening toward damping waveguide, magnetic field gets higher For disk-loaded structure good to have higher Q0 to reduce wall power, higher transfer efficiency near round cell shape, make it close to sphere T. Higo LC School, Chicago, 2008, LINAC-I

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**Suppression of local field enhancement**

Electric field; Ep / Eacc Peak surface electric field Field emission source Breakdown in NCC Magnetic field; Hp / Eacc Surface temperature rise within a pulse in NCC Quenching of superconductor above magnetic field threshold How to decrease these ratios? Shaping global cell shape Make it smooth locally Need care on SCC EWB welding quality NCC remove burrs and sharp corners EBD: Electron beam welding T. Higo LC School, Chicago, 2008, LINAC-I

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**Cares on local field enhancement**

Smooth opening to reduce Hp Smooth shape to suppress Ep Enhancement at small edge on opening 2D calculation of cylinder with radial opening channel Care on the opening edge to damping waveguide. 0.2mm Bump over EBW line Bump above design contour Hs enhancement over a small sphere sitting on a spherical cavity. Height / radius < for dHs/Hs<a few % Care on EBW bead shape in SCC cavity. “Evaluation of Magnetic Field Enhancement Along a Boundary” Y. Iwashita and T. Higo TUP-95, LINAC2004, Lubeck, Germany, 2004. Presentation on T. Higo LC School, Chicago, 2008, LINAC-I

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**Shaping of accelerator cell profile examples**

SCC Typical cavity Damped cavity for storage ring With nose cone. SCC Reentrant cavity DAW cavity SW TM010 SCC p mode Smooth More Ez on axis Less Hs at outer Single cell Damped cavity HDDS for GLC/NLC TM020-like p/2 Floating washer Coupling mode in addition to accelerating mode. High Q, high R TW DLS TM010 5p/6-mode (figure shows p field) T. Higo LC School, Chicago, 2008, LINAC-I

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**Coupled cavity system in a SW regime**

T. Higo LC School, Chicago, 2008, LINAC-I

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**Cell-to-cell coupling**

Ep/Eacc increases, but easy to confine field to increase T if normalized in the field-existing area. Coupled cavity needs coupling between cells through some mechanism other than beam aperture. Weakly coupled-cell through beam aperture SCC cavity for ILC Transit time factor cannot be improved, similar to that of pillbox or less. T. Higo LC School, Chicago, 2008, LINAC-I

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**Weakly coupled resonators**

Each resonator has Internal freedom Eigen modes in the cavity in an almost closed surface Excited resonant modes couple to beam Acceleration, deceleration, transverse kick, etc. Total system Weak coupling usually to adjacent cavity through some apertures Total system is described as coupled resonator system Mathematically equivalent to Mechanically coupled oscillator model Electrically coupled resonant circuit model T. Higo LC School, Chicago, 2008, LINAC-I

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**Coupled resonator model to describe the total system**

Assume each cavity is represented by a resonant circuit. (described later) where p-mode p/9-mode Dispersion Refer Knapp paper D. E. Nagle, E. A. Knapp, B. C. Knapp, RSI 38, 1967. If Q>>1, T. Higo LC School, Chicago, 2008, LINAC-I

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**Perturbation analysis**

Equation: Solution: Frequency errors First order equation T. Higo LC School, Chicago, 2008, LINAC-I

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**Perturbation analysis (cont.)**

For p mode For p/2 mode Where is Fourier component of frequency perturbation of Both perturbation scales as 1/k. As for number of cells, p mode scales as N^2, while p /2 mode linearly on N. For longer structure, p mode becomes difficult. This is related to no energy exchange in p mode because of zero group velocity. For energy transfer, we need other mode than p mode, which destroys p mode itself. T. Higo LC School, Chicago, 2008, LINAC-I

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**LC School, Chicago, 2008, LINAC-I**

p-mode and p/2 mode Most basic but no net acceleration Stable cavity system but half acceleratin Good compromize in TW linac Most efficient but weak against perturbation Electrically p/2 but acceleration efficiency ~ p-mode 何故 LCではπ/２をとらないか？ 複雑な構造、 コスト、 Actual p-mode acceleration with p/2 coupling element outside accelerating cell. T. Higo LC School, Chicago, 2008, LINAC-I

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**SCC 9-cell cavity example**

Cell to cell coupling k~2% Dispersion curve f0/(1+ k Cosf)1/2 Band width BW ~ k f0 Small mode separation fp - f8p/9 ~ 0.06*BW Tuning of SCC 9-cell cavity see next page Shunt Impedance of Total system Rtotal = Rsingle X 9 if flat field and right frequency T. Higo LC School, Chicago, 2008, LINAC-I

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**Practical issues in SCC being analyzed with coupled resonator model**

SCC 9-cell cavity is basically expressed as a single chain of coupled resonators. Field flatness consideration and tuning Frequency of cells Lorentz force detuning EP deformation of cell shape Coupling between cells It makes coupling coefficient between resonators. It represent robustness of field flatness against perturbations It gives spacing to the nearest resonance, 8p/9 mode. Some other system such as super-structure Also can be described by a weekly coupled two 9-cell systems. T. Higo LC School, Chicago, 2008, LINAC-I

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**Frequency error and field distribution**

T. Higo LC School, Chicago, 2008, LINAC-I

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**Frequency error estimation from measured field**

Tuning: Mechanical deformation Frequency error Field uniformity Eq. circuit Field measurement Deviation from p mode T. Higo LC School, Chicago, 2008, LINAC-I

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**Pillbox cavity as a simple base to represent practical cavities**

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**SW Cavity example: pillbox**

d a In a cylindrical waveguide, two propagation modes exist; Forward wave and Backward wave For satisfying the boundary condition at both end plates, the solution with the superposition of these two counter-propagating modes in proper phase and amplitude becomes SW in a pillbox cavity. 野口メモ Pillbox ｐ６ TMに対する表式 T. Higo LC School, Chicago, 2008, LINAC-I

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**Bessel’s functional form representing pillbox field to satisfy boundary condition**

TM01 Ez and Hr TM11 Ez and Hr TM02 Ez and Hr Transverse kick Two polarizations Zero at center and linearly increase as r increases. Big kick at center H field. Acceleration Max at center and 0 at r=a Acceleration More energy storage for a given acceleration T. Higo LC School, Chicago, 2008, LINAC-I

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**Mode frequency in a pillbox cavity**

Modes are classified as TM and TE mode. Frequencies are determined to satisfy boundary condition at two end surface, TM mode frequencies. where rmnl is Bessel’s functions zero for TM or derivative of Bessel’s function becomes zero for TE. Modal density increases as higher frequency region. Accelerating mode is usually TM010 mode. If we want more stored energy with the same acceleration, TM011, TM020 or others can be used. T. Higo LC School, Chicago, 2008, LINAC-I

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**Q value of modes in pillbox cavity**

For TM modes; Volume integral; Noguchi note p8 T. Higo LC School, Chicago, 2008, LINAC-I

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**LC School, Chicago, 2008, LINAC-I**

Q (cont.) Finally we get, Surface integral; Therefore, for TM modes, Where ds is skin depth, Similarly we obtain for TE modes; T. Higo LC School, Chicago, 2008, LINAC-I

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**Acceleration in a single pillbox cavity**

For an even function, such as the case of TM010, the f=0 to maximize acceleration. This is the case of on-crest acceleration. T. Higo LC School, Chicago, 2008, LINAC-I

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**R/Q value of modes in pillbox cavity**

Normalization: Definition of field integral: Transit time: Impedance: T. Higo LC School, Chicago, 2008, LINAC-I

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**LC School, Chicago, 2008, LINAC-I**

R/Q for TM0nl field In a previously obtained field expression of pillbox mode, Then, to make the normalization, For TM010, Finally, T. Higo LC School, Chicago, 2008, LINAC-I

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**LC School, Chicago, 2008, LINAC-I**

For TM0nl mode For TM0nl, with T. Higo LC School, Chicago, 2008, LINAC-I

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**Circuit modeling of cavity**

T. Higo LC School, Chicago, 2008, LINAC-I

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**Cavity to equivalent circuit**

Ez Ez Hf Hf Magnetic field Inductive energy storage Current flow Power loss into wall Resistive loss Electric field Capacitive energy storage This system can intuitively be expressed with series resonant circuit in electric circuit The differential equiation is mathematically equivalent and the system can also be presented by parallel resonant circuit. PS PS T. Higo LC School, Chicago, 2008, LINAC-I

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**Circuit model of cavity**

Perry Wilson Lecture copy From P. Wilson Lecture Note. RF power source, transmission line and cavity can be described as an equivalent circuitry. T. Higo LC School, Chicago, 2008, LINAC-I

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**Cavity response and beam**

y: Cavity tuning angle y vbr vb vc -ib va f q vg vgr Phasor diagram If frequency changes to the order of Q band width, y becomes as big as 45deg! vg/vgr = 0.7 (reduced much) To save power, need smaller d <<1/QL Need vgr to keep f>0. Phasar diagram: Perry Lecture note p22, Fig. 3.13 T. Higo LC School, Chicago, 2008, LINAC-I

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**Transient field around cavity**

Emitted field from cavity stored field Emitted (out-going, reflected) field from cavity Cavity filled voltage βｃ 定義 Outgoing wave = sum of emitted field from cavity + reflected wave from input wave Energy conservarion ijnput power out going power + cavity wall loss + stored energy of cavity Reflaction coefficient ~ -1 Bc = Pout / Pc Tc = 2 QL / omega …… T. Higo LC School, Chicago, 2008, LINAC-I

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**Cavity transient response**

Reflected field Input power Cavity field bc=1 bc=0.5 bc=2 bc=5 Reflected power With square pulse input; Cavity Q value can be measured by free oscillation after power off. Reflacted power comes out from caivty at the onset and off of the power. T. Higo LC School, Chicago, 2008, LINAC-I

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**LC School, Chicago, 2008, LINAC-I**

SW cavity beam loading From energy conservation, With proper timing t=tr, we can make Then, we can make the beam loading compensation to keep the voltage gain the same within the bunch train. This is realized in a proper timing, feeding power and beam current to adjust to the transient behavior of the cavity. T. Higo LC School, Chicago, 2008, LINAC-I

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**LC School, Chicago, 2008, LINAC-I**

SW versus TW T. Higo LC School, Chicago, 2008, LINAC-I

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**SW and TW Superposition of Cos + j Sin Hf Example pillbox TM010 mode**

Ez Hf Superposition of Cos + j Sin Example pillbox TM010 mode Ez and Hf is 90 degrees out of phase Forward or backward wave F+B or F-B Er and Hf in phase to make Poynting vector T. Higo LC School, Chicago, 2008, LINAC-I

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**SW field TW real field**

Short b.c Open b.c. Cosine Sine T. Higo LC School, Chicago, 2008, LINAC-I

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**LC School, Chicago, 2008, LINAC-I**

SW TW R/QTW = R/QSW X 2 The space harmonics of SW propagating against beam cannot contribute to the net acceleration, while the reverse-direction power is needed to establish the SW field. SCC cavity operated in TW mode Resonant ring like: return the outgoing power from cavity to feed again into the upstream of the cavity. But need sophisticated system outside. Many NCC at high gradient use TW High field need high impedance. Power not used for beam acceleration nor wall loss need be absorbed by outside RF load. High gradient in SW cavity Probably resistive against damage from arcing due to easy detuning. Stability requires short cavity need more feeding points T. Higo LC School, Chicago, 2008, LINAC-I

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**LC School, Chicago, 2008, LINAC-I**

TW linac T. Higo LC School, Chicago, 2008, LINAC-I

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**TW basic idea different than standing wave**

Travelling wave = microwave power flow in one direction Beam to be coupled to the field associated to this power flow Power flow Acceleration field decreases along the structure Due to wall loss and energy transfer to beam Attenuation parameter is a key parameter Extracted power is absorbed by load or re-inserted into the structure Shaping of attenuation along a structure Accelerating mode CZ, CG Higher mode consideration Detuned, … Synchronization to beam Control of frequencies of cells to make the phase velocity right Input matching with mode conversion Matching to TM01-type mode in the periodic chain of cavities From TE10 in waveguide T. Higo LC School, Chicago, 2008, LINAC-I

88
**Acceleration field in TW linac**

m=0 m=1 m=-1 p/d 2p/d -p/d -2p/d b w P. B. Wilson High Energy Electron Linacs: Application to Storage Ring RF systems and Linear Colliders SLAC-PUB-2884, 1982 T. Higo LC School, Chicago, 2008, LINAC-I

89
**Structure parameters in a uniform structure (CZ)**

SW cavity TW linac Acceleration voltage in a cavity Acceleration gradient Stored energy in a cavity Stored energy in a unit length Power loss in a cavity Power attenuation in a unit length Group velocity and attenuation parameter Various cavity parameters are similar to those of SW cavity case. All parameters are re-read by the value per unit length. That is all. At the end of a structure, of length L T. Higo LC School, Chicago, 2008, LINAC-I

90
**DLS mode Cal. With SW field patterns**

p/2 mode 2p/3 mode 5p/6 mode p mode T. Higo LC School, Chicago, 2008, LINAC-I

91
**Shunt impedance vs phase advance**

Blue book or linear accelerators, p93, p94 n=number of disks per wavelength. Thinner disk makes larger Gross peak at 2p/3 mode Actual acceleration field on axis can be decomposed into space harmonics. Net acceleration for long distance is realized by only synchronous one. T. Higo LC School, Chicago, 2008, LINAC-I

92
**DLS basic design with changing (a,t)**

Simple geometry for DLS design “b” to tune frequency a t “vg” varies as (a,b,t) changes “Q” varies as (a,b,t) changes T. Higo LC School, Chicago, 2008, LINAC-I

93
**DLS Parameter variation with (a,t)**

“r” varies as (a,t) changes “Ep/Eacc” varies as (a,t) changes “r/Q” varies as (a,t) changes Hp/Eacc T. Higo LC School, Chicago, 2008, LINAC-I

94
**CG: Constant gradient structure**

vg vs a Assume CG case is realized. By neglecting weak variation of r, r vs a Group velocity should be varied linearly as Filling time becomes 2p/3 mode in X-band disk-loaded structure This is the same as CZ case T. Higo LC School, Chicago, 2008, LINAC-I

95
**ENL and ELD actual and CG characteristics**

Basically changing beam aperture “a” to make it roughly CG Roughly constant gradient by linearly tapered vg Reduce field near input coupler with initial taper T. Higo LC School, Chicago, 2008, LINAC-I

96
**Acceleration with/without beam**

Steady state case Field with beam is superposition of externally-driven field + beam-induced field Acceleration along a structure without beam Beam induced field (beam loading) in an empty structure Eb : Beam induced field I0 : DC current of beam P : Power flow of beam induced field Since , then This becomes simple when we consider CZ or CG structure, T. Higo LC School, Chicago, 2008, LINAC-I

97
**LC School, Chicago, 2008, LINAC-I**

Beam loading voltage Beam loaded field along a structure is described as, Beam loading in CZ z By integrating along a structure z Beam loading in CG The net acceleration voltage then becomes, Where q is the off crest angle w.r.t. RF on-crest phase. T. Higo LC School, Chicago, 2008, LINAC-I

98
**Beam loaded field with beam**

ELD(0.8A) ELD(1.6A) ELD(2.4A) Estimation with CG case r=60 MW/m t=0.6 L=0.6 m E0=65 MV/m I= A Recursive calculation at Ib=0.9A with actual parameters (a,b,t) r, Q along the structure Field Beam loading voltage build up toward downstream end Z. Li reference?? T. Higo LC School, Chicago, 2008, LINAC-I

99
**LC School, Chicago, 2008, LINAC-I**

Full loading in CTF3 In a steady state regime, beam can fully absorb stored energy of the structure. This is fully loaded condition. Proc. LINAC06, Knoxville, P. Urschütz et al., “Efficient Long-Pulse Fullｙ Loaded CTF3 Linac Operation”, P. Urschütz et al., “Efficient Long-Pulse Fullｙ Loaded CTF3 Linac Operation”, LINAC06. T. Higo LC School, Chicago, 2008, LINAC-I

100
**Transient property of travelling wave propagation**

Wave propagation suffers from the dispersive effect of the periodic structure. p/2 mode f=0, only amplitude modulation Accelerated particle sees as function of time. Rising front of pulse transmission at a certain position q down the structure. (Linear Accelerators) Solid line: Field amplitude along the structure. Dotted line: Field a particle sees. (Linear Accelerators) Reference: Shintake Frontiers of Accelerator Technology, p435 Reference: Linear Accelerators p164 The same property is calculated straightforwardly based on the equivalent circuit model by T. Shintake. (Frontiers in Accelerator Technology, 1996) T. Higo LC School, Chicago, 2008, LINAC-Is

101
**TW accelerator structure in practice**

Design Cell fabrication Frequency control Bonding Tuning Matching Phase tuning Minimize small reflection HOM T. Higo LC School, Chicago, 2008, LINAC-I

102
**Typical TW structure made of stacked disks**

T. Higo LC School, Chicago, 2008, LINAC-I

103
**Small reflection and tuning**

Express wave propagation m From continuity at m-th cell, From coupled resonator model, Explicitly written as, D. Whittum: Frontiers of Accelerator Technology, US-CERN-Japan International School on Particle Accelerators, Hayama, 1996. Tuning condition, T. Higo LC School, Chicago, 2008, LINAC-I

104
**LC School, Chicago, 2008, LINAC-I**

Reflection (cont.) Finally reflection from cell m becomes, Assume; loss-less TW with 2p/3 mode, k~0.02, dw2/w2 ~ 10-4 If coherent error from 10 cells, those cohere so that R becomes as much as 0.1. Such systematic error is not allowed. In contrast, random error makes much smaller and such amount of error is allowed. T. Higo LC School, Chicago, 2008, LINAC-I

105
**LC School, Chicago, 2008, LINAC-I**

Phase error Dispersion relation; Group velocity formula; Frequency error to phase advance error At X-band, 1MHz gives phase advance error of 0.6degree/cell. In 20 cell structure such as CLIC, Systematic error of this order is not good but random error should be OK, as long as acceleration mode. T. Higo LC School, Chicago, 2008, LINAC-I

106
**Summary of LINAC-I in comparison of SCC and NCC**

How the linac design can be determined from gaining energy point of view? Energy mode SW or TW Confinement mechanism SCC or NCC Frequency choice 1GHz-10cm or 10GHz-1cm? T. Higo LC School, Chicago, 2008, LINAC-I

107
**Choice of material for EM field confinement**

SCC Nb G Geometrical factor Intrinsic limit Hs Mechanical strength Quench Cryogenic power NCC Cu R Shunt impedance Thermal Breakdown RF generation T. Higo LC School, Chicago, 2008, LINAC-I

108
**LC School, Chicago, 2008, LINAC-I**

Choice of frequency 1GHz 10cm Drawing or hydroforming Longer pulse ~1ms Longer power 10GHz 1cm High precision turning / milling Shorter pulse ~100ns High peak power T. Higo LC School, Chicago, 2008, LINAC-I

109
**Choice of EM mode & Efficiency**

SW SCC Power loss Cavity wall cryogenic Reflection from cavity TW NCC Power loss Cavity wall Transmitted to RF load T. Higo LC School, Chicago, 2008, LINAC-I

110
**Potential for higher energy**

Es/Ea Hs/Ea Multi pactor Field emission Lorentz detuning Quench Q0 cryogenic power Es/Ea Hs/Ea Surface temperature rise in a pulse Fatigue Breakdown R/Q * Q efficiency T. Higo LC School, Chicago, 2008, LINAC-I

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Lecture 2. Review lecture 1 Wavelength: Phase velocity: Characteristic impedance: Kerchhoff’s law Wave equations or Telegraphic equations L, R, C, G ?

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