Lecture Note Part 2 001 Scope Beam-Beam effect Yue Hao Beam-beam effect happens when two charged beam collides Strong/Weak interaction ----Goal of the Collider Electric-Magnetic interaction ----parasitic, hard to avoid We are discussing the unwanted electro magnetic interaction of two colliding beam. Main limit from achieving high luminosity. Beam-beam and Space charge force Charged particle not only feel the force from the opposing beam, but also from the bunch which it resides. Beam-beam force, from the opposing beam Space-charge force, from the its own bunch. Determine the field Usually it is easier to calculate in the rest frame: From the charge distribution, both the scalar potential (Poisson equation) and electric field (Gauss Law):
Lecture Note Part 2 002 Transform to lab frame Beam-beam force The electric and magnetic field in the lab frame can be obtained: Lorentz force gives: In lab frame, the perpendicular field is much larger than the longitudinal component. Beam-beam effect is mainly transverse effect. In ultra-relativistic case, the reads: Simple Example Let s assume a 2D round Gaussian beam: Beam-beam kick What is more important is the transverse kick of the test particle when traveling through the opposing beam. The beam-beam force reads With classical radius:
Lecture Note Part 2 003 Beam-beam parameter For small amplitude test particle, the only effect is tune shift. Linear focusing in both transverse direction. Beam-beam parameter ξ is the linear tune shift. BEAM-BEAM EFFECT IN RING- RING COLLIDER Beam-beam parameter limitation LHC RHIC Tevatron B factory Type Hadron Hadron Hadron Lepton Energy 3500 250 980 3GeV*9GeV Beam Beam Parameter 0.035 0.01 0.03 0.16 Beam-beam effect is the major limitation factor from achieving higher luminosity. Reason is complicate Let s start from linear consideration Linear Model Linear beam beam force modifies the one turn map as Tune shift: Beta beat
Lecture Note Part 2 004 Linear model, cont d Linear stability condition Assuming the tune change is small compare with the one turn tune, The beam-beam parameter has to satisfy: Stability condition This wouldn t explain the observed beambeam parameter limitation of the existing machines. Nonlinear kick Multi-pole expansion Head on Collision Long Range Beam Beam
Lecture Note Part 2 005 Nonlinear tune shift Beam-beam simulation (0.685, 0.695) w/o beam-beam effect Every particle has it own tune shift as function of its transverse amplitude. Soft Gaussian Model Bunch always preserve Gaussian shape in transverse direction. Analytical formula exists even with non-round beam (Bassetti & Erskine) Incoherent effect. Nonlinear resonance induces the beambeam limit. Need sophisticate simulation efforts to understand and predict. Self consistent model, field is calculate from transverse distribution via Poisson Solver Symplectic Map To model the collision of two long bunches, the only known symplectic map, synchrobeam mapping can be used to keep symplecticity during long-term tracking. The effect is calculated for the slice of the opposing beam at position: Synchro-Beam Map With the redefined coordinate: The map:
Lecture Note Part 2 006 Beam-Beam compensation(wire) For long-range beam-beam force is proportional to 1/r Beam-Beam compensation (elens) To compensate head-on collision, no magnet can do the job. Only candidate is another charged bunch with same transverse distribution. Current-caring wire for long-range beambeam compensation. The current produces B φ to generate same kick amplitude with opposite sign. e-lens experiment @ RHIC for head-on beam-beam compensation e-lens, cont d < No e lens @ 2e11 p per bunch With e lens @2e11 p per bunch > e-lens, cont d Electron beam must have same Gaussian transverse distribution. When collide with ion beam, the the beam-beam force can compensate the one from the other ion beams. < With e lens @ 2.5e11 p per bunch With e lens @3e11 p per bunch > 0 ma (all bunches) 365 ma (last 2 b.) 780 ma (last 2 b.) [Simulation S. White]
Lecture Note Part 2 007 Disruption parameter A BRIEF INTRODUCTION TO BEAM-BEAM EFFECT IN LINACS The beam-beam limit in ring does not exist if the beam is used only once. Therefore beam-beam parameter can be much larger. Usually another parameter is important: Disruption parameter Disruption effect with large disruption parameter The strong nonlinear field will distort the transverse distribution. The beam will rotate in phase space while traveling through the opposing beam. Beamstrahlung In lepton colliders or lepton-ion colliders, the electron beam will radiate in the opposing beam. This is name Beamstrahlung, Beam+Bremsstrahlung. Beamstrahlung parameter: Emittance growth due to nonlinearity and mismatch. Much less then unity: classical SR Much larger than unity: Quantum effect
Lecture Note Part 2 008 Reference Lecture of Beam-beam effects, W. Herr Beam-beam effect study in ERL based erhic, Ph.D Thesis, Y. Hao, Indiana University Y. Luo, et.al. PRSTAB, 15, 051004, 2012
Lecture Note Part 2 009 Outline Laser Beam Manipulation and Interaction Describing the laser. Phase manipulation for coherent radiation Laser beam accelerator Compton Scattering Yue Hao Properties of Radiation Laser Basics Temporal Coherence Spatial Coherence A Gaussian shape is assumed: Monochromatic light,best temporal coherence Coherence time and length
Lecture Note Part 2 010 Free Electron Laser PHASE SPACE MANIPULATION Reach the wavelength that traditional laser cannot reach. (Hard/Soft X-ray source) Tuning ability. 1971, first FEL by Madey, theoretical and experimental 2009, first X-ray FEL, SLAC Coherent and Incoherent Radiation The radiation of N e -electrons Resonance condition for undulators FEL principles In phase Micro bunching and Instability u e Coherent radiation dominates when the bunch is much shorter than the wavelength. r u 2 2 (1 K 2 /2 2 2 ) da dz b a: normalized radiation field db dz ip b ei j : bunching factor dp dz a P j ei j : energy modulation
Lecture Note Part 2 011 Gain Length and FEL parameter Beam Quality Requirement The Power Gain length under 1-D Assumption: u LG 4 3 P P exp z/ L Given without proof: 0 Radiation energy at saturation Electron beam energy G The FEL parameter ρ A dimensionless parameter 1 10 2 2 2 ek[ JJ] ne 3 2 2 32 0 0mc ku 3 1/3 Saturation Efficiency Energy spread: Beam emittance: Beam size Beam current: Large enough to get reasonable FEL parameter SASE: Born from Noise Beam energy modulation We use laser + short undulator to modulate the beam. Short undulator, much shorter than the Rayleigh length. Beam average energy change is negligible. Plane EM wave assumption: One way to improve coherence is to generate the input signal ourselves, but no x-ray coherence seed!
Lecture Note Part 2 012 Chicane First thought, HGHG What is the time delay due to the chicane? HGHG, cont d Assume the beam initially has Gaussian energy distribution, HGHG cont d From Liouville's theorem: After the energy modulation, Drop the prime notation and calculate the bunch factor Then the dispersive section (chicane), the position:
Lecture Note Part 2 013 HGHG, cont d EEHG To maximize the short wavelength seed, we have to maximize the bunching factor. Bessel function has its maximum value when The exponential decay term then becomes The scaling to higher harmonic can be improved with Echo enabled harmonic generation, involving two modulator and two chicanes. At large n, it requires very small energy spread EEHG, cont d Follow the same method as in HGHG case, a EEHG, cont d c b d e f The distribution becomes: After 2 nd chicane the density modulation echoes back. a c b d Bn~ n-1/3 e f
Lecture Note Part 2 014 Cooled HGHG Cooled HGHG, cont d To eliminate the exponential decay term in HGHG bunching factor. In TGU Dispersive section Then after the modulator: Transverse Gradient Undulator Adjust the dispersion and transverse gradient to make vanish Laser Acceleration Nowadays, the peak power of laser pulse can reach PW, 10 15 W. A 100TW laser can be achieved with a reasonable small size, with laser spot ~10um. Energy flux density: ~10 20 W/cm 2. The electric field, therefore, can reach: 10 13 V/m=10 4 GV/m. Question: How to use the field effectively? Lawson-Woodward Theorem With two simple conditions: Laser in free-space Particle has constant speed and moves in a straight line. No net acceleration can be achieved! Proof: Calculate the work done on the charged particle due to the laser.
Lecture Note Part 2 015 Walk around, IFEL Energy gain in IFEL One way to walk around is to void the condition, the particle moves in a straight line Assuming a planar electric field: Velocity modulation by undulator: Energy Gain: Energy gain in IFEL, cont d Experiment @BNL ATF The average longitudinal velocity: Homework: find the optimum frequency of the drive laser and the possible energy gain.
Lecture Note Part 2 016 What does IFEL provide us? What is not covered: Experiment demonstrates >50MeV Energy Gain ~100 MeV/m Gradient IFEL has advantages within GeV level More controllable (through undulator) Higher gradient Good beam quality. Laser Plasma acceleration Use the wake field that excited in the plasma by laser http://uspas.fnal.gov/materials/11odu/odu_lase rplasma.shtml Dielectric laser acceleration Nature 503, 91-94, 2013. Work lead by J. England Many other schemes Compton scattering Here we only interested in the incident angle of the photon as 180 degree In the rest frame Compton wavelength Doppler effect Transform from the rest frame to the lab frame. With the back scattering, i.e. the scattered angle is ~Pi, the wavelength relation in lab frame reads: The energy ratio: Quantum correction of the Thomson scattering.
Lecture Note Part 2 017 Cross Section The cross section of the Thomson scattering is: Example, a compact x-ray source The Klein-Nishina formula gives Where P is the ratio of the photon energy before and after collision. When the electron energy is much larger, the formula reduces to Thomson case. Proposed by W. Graves Reference E. Hemsing et al, Review of Modern Physics V.86 Manuscript, Lecture of Free Electron Laser, by L. H. Yu H.Deng and C.Feng, PRL 111, 084801, 2013 J. Duris et.al., PRSTAB 15, 061301, 2012
Lecture Note Part 2 018 Motivation Truncated Power Series Expension It is useful to find nonlinear map to very high orders. The map can be used to extract beam dynamics quantities. Complicate nonlinear lattice Very time-consuming to track A faster compromise? We want to establish a formula (maybe complicate) to describe such complex, timeconsuming. Usually no exact analytical solution. TPSA Truncated power series algorithm is suitable for such goal. It is a mathematical treatment, no physics inside. Express the map as polynomials: Convergence? Which order to stop?
Lecture Note Part 2 019 Determine the coefficient 1 D case Coefficient can be determined by the derivative with respect to the elements of initial vector. For N dimension problem, the coefficient for the ith element of order j gives Lets assume we need to construct 1D Taylor map: To get the value of all the derivatives at x 0, the simple and usual way is: How to find them effectively? The answer is TPSA, introduce by M. Berz in 1989 Does not guarantee accuracy. TPSA method Let s start with an example We know that With the following rule, we plug in Why TPSA works? TPSA may give the value of the derivatives without calculate the expression. In the 1 D, first derivative example, we substitute x with (a,1). Since (x,x ) x=a =(a,1) The rule also comes naturally,
Lecture Note Part 2 020 1D, higher orders Homework Substitute x with (a, 1, 0, 0,, 0), a Ω+1 dimension, we will get (y(a), y (a),, y (Ω) (a)) Multiplication rule: Assuming n>0, find the rules for For 2 nd order TPSA, find the rule for Binomial Coefficient! Special functions We have demonstrated the TPSA for arithmetic operatiors. How about special functions such as sin(x), exp(x), log(x), etc Answer: use taylor expansion: Note that Special function, cont d Therefore, for X=(x 0,x 1,x 2,,x n ), the arbitrary function f(x): Let s do exercise for Exp(X), Log(X) and sin(x) And special cases, exp(x=(x 0,1,0, )) and sin(x=(x 0,1,0, )) With m leading zeros
Lecture Note Part 2 021 Extension to multi dimension Application in accelerator Now consider the independent variables has more than one dimension, y=y(x 1,x 2,x 3,,x n ) One order has multiple terms How many terms for N dimension, M orders? Two variable example (x 1,x 2 ): Basic application Express 6D coordinate as its initial value. 6 Dimension with some preset orders Linear order gives the 6 by 6 matrix Derive linear/nonlinear properties, optics, nonlinear tune dependence, nonlinear driving terms, etc. Advanced application Include magnet parameter in to account, perfect for optimizing the machine. TPSA Library The initial library written in Fortran is widely used. A C++ version by L.Y. Yang is now used in PTC. A good library should Accept arbitrary order and dimension Optimize the memory usage. Optimize the calculation speed. Reference Alex Chao, Lecture Notes on Special Topics in Accelerator Physics.