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Theory and Practice of Free-Electron Lasers

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Theory and Practice of Free-Electron Lasers Theory and Practice of Free-Electron Lasers Particle Accelerator School Day 3 Dinh Nguyen, Steven Russell & Nathan Moody Los Alamos National Laboratory US Particle AcceAcceleratorlerator School 2009 LA-UR 09-03171 University of New Mexico - Albuquerque NM 3-1 Course Content Chapter 1. Introduction to Free-Electron Lasers Chapter 2. Basics of Relativistic Dynamics Day 1 Chapter 3. One-dimensional Theory of FEL Chapter 4. Optical Architectures Day 2 Chapter 5. Wigglers Day 3 Chapter 6. RF Linear Accelerators Chapter 7. Electron Injectors Day 4 US Particle AcceAcceleratorlerator School 2009 LA-UR 09-03171 University of New Mexico - Albuquerque NM 3-2 Chapter 5 Wigglers US Particle AcceAcceleratorlerator School 2009 LA-UR 09-03171 University of New Mexico - Albuquerque NM 3-3 Wigglers • Maxwell Equations • Wiggler Designs – Pure Permanent Magnet –Hybrid • Wiggler Natural Focusing • Two-plane Focusing – Weak Focusing – Strong Focusing • Tapered Wigglers US Particle AcceAcceleratorlerator School 2009 LA-UR 09-03171 University of New Mexico - Albuquerque NM 3-4 Maxwell Equations Gauss’ law for electricity Maxwell Equations in simple forms E D 0E 0 ⋅D = r Gauss’ law for magnetism B 0 ⋅B = 0 Faraday’s law B ¶B E ´E = - t ¶t Ampere’s law 1 ¶E ¶D ´BJ =m + BH= m0 ´HJ = + 0 ct2 ¶ ¶t US Particle AcceAcceleratorlerator School 2009 LA-UR 09-03171 University of New Mexico - Albuquerque NM 3-5 Physical Constants Speed of light c 2.9979 x 108 m/s 1 c 00 Permeability of vacuum 4x 10-7 H/m 0 1 -12 0 2 Permittivity of vacuum 0 8.8542 x 10 F/m 0c 1 Free space impedance 0 376.73 0 Zc00 00c Electronic charge e 1.6022 x 10-19 C -31 Electron mass m0 9.1094 x 10 kg Electron rest energy m c2 0.511 MeV 0 1 e2 r0 = 2 -15 4pe mc Classical electron radius r0 2.81794 x 10 m 0 ec 1 mc3 Alfven current I 1.7 x 104 A 0 0 I0 == re004pe US Particle AcceAcceleratorlerator School 2009 LA-UR 09-03171 University of New Mexico - Albuquerque NM 3-6 Wiggler Designs • Pure Permanent Magnet (PPM) – No power required – Highest magnetic field at very short wiggler periods – Can accommodate two-plane quadrupole focusing magnets – Susceptible to demagnetization • Hybrid (Permanent Magnet + Iron or Vanadium Permendur) – No power required – Higher field if the gap-to-period ratio is less than 0.4 – Magnetic field non-uniformity is reduced if ferromagnetic is saturated – Uses more magnet materials than PPM – Susceptible to demagnetization • Electromagnets – No risk of demagnetization – Lower magnetic field at very short wiggler periods – Require external power sources and cooling US Particle AcceAcceleratorlerator School 2009 LA-UR 09-03171 University of New Mexico - Albuquerque NM 3-7 Magnet Material Properties • Rare-earth Magnets –SmCo5 –Sm2Co17 –Nd2Fe14B • Ferromagnetic Materials –Iron – Vanadium Permendur • Room-temperature Electromagnets We will not consider • Superconducting Electromagnets electromagnet wigglers here –NbTi –NbSn – YBCO US Particle AcceAcceleratorlerator School 2009 LA-UR 09-03171 University of New Mexico - Albuquerque NM 3-8 Magnetization Curves Br Remanence Second quadrant First quadrant Hc Coercivity Third quadrant Fourth quadrant Hysteresis curve starts from the origin and follows the dashed line to saturation (a). After the magnetizing force is removed, the curve moves to (b) and the magnets retains remanent field Br. Reversing the magnetizing force causes the magnets to be demagnetized beginning at (c) in the 3rd quadrant (danger zone). US Particle AcceAcceleratorlerator School 2009 LA-UR 09-03171 University of New Mexico - Albuquerque NM 3-9 Permanent Magnet Materials Working point Load line PM (BH)max Remanence Br Temp Coefficient Hc Coercivity HcB Temp Coefficient Radiation Material (kJ/m3) (mT) (% / K) (kA/m) (% / K) Hardness SmCo5 170 800-1000 -0.042 2400 -0.25 High Sm2Co17 220 1000-1100 -0.032 2000 -0.19 Medium NdFeB 300 1100-1400 -0.10 1400 -0.60 Low - Medium US Particle AcceAcceleratorlerator School 2009 LA-UR 09-03171 University of New Mexico - Albuquerque NM 3-10 Halbach PPM Wiggler Design Magnet arrangement lM h g w w y Magnetic field lines z x Halbach PPM design uses 4 magnets per period for each jaw (8 on both jaws) to produce a sinusoidal vertical field (By) along the z axis. US Particle AcceAcceleratorlerator School 2009 LA-UR 09-03171 University of New Mexico - Albuquerque NM 3-11 Peak On-axis Magnetic Field l M Peak on-axis magnetic field g BBffe 2 w h 0 rMH g 2h lM sin w f H 1 e w w f M g: Full gap between magnet jaws M Br: Remanence (a material property) fM: Magnet filling factor lM: Length of individual magnet (along z) Optimized Halbach design g M: Number of magnets per period (one jaw) w fH: Magnet height factor B0 1.78Ber h: Height of individual magnet (along y) US Particle AcceAcceleratorlerator School 2009 LA-UR 09-03171 University of New Mexico - Albuquerque NM 3-12 Hybrid Wiggler Designs Straight poles Permanent magnet Electron beam Iron or Vanadium Permendur w Wedged poles Electron beam US Particle AcceAcceleratorlerator School 2009 LA-UR 09-03171 University of New Mexico - Albuquerque NM 3-13 On-axis Field vs Gap-Period Ratio Empirical formula for peak on-axis field gg 10 bc ww Bae0 abc Hybrid() x Hybrid 1 PM() x SmCo 3.33T 5.47 1.8 PPM NdFeB 3.69T 5.07 1.52 On-axis magnetic field 0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 x g/w Hybrid wigglers are generally useful for maximizing on-axis field at small gap-to-period ratios. Hybrid wigglers also produce good field uniformity if the ferromagnetic material (iron or Vanadium Permendur) is saturated. US Particle AcceAcceleratorlerator School 2009 LA-UR 09-03171 University of New Mexico - Albuquerque NM 3-14 Planar Wiggler Magnetic Field Planar wiggler with infinite x dimension Bx = 0 ByBkykzyww= ˆ 0 cosh( )cos( ) BzBkykzzww=-ˆ 0 sinh( )sin( ) Lorentz force (written as second derivative with respect to z) æöe ÷ yBB¢¢ =-ç ÷()uu ç 2 ÷ x zzx èøgmc0 Magnetic field in the y-z plane Consider only the first term (Bx = 0). æöe yxB¢¢=ç ÷ ¢ çgmc÷ z èø0 Equation of motion in the y direction Using small k y approximation w æökeB ¢¢ç w 0 ÷ ¢ B Bk ysin k z yxkzy=-ç ÷ sin()w zww0 ç ÷ èøgmc0 US Particle AcceAcceleratorlerator School 2009 LA-UR 09-03171 University of New Mexico - Albuquerque NM 3-15 Wiggler Natural Focusing Expand B around y=0 y x B Bkykz1cos2 yww0 y > 0 y″ < 0 Velocity in x with respect to z z e 2 x Bkykz1sin 0 ww kmcw 0 x Focusing force in the y direction y < 0 y″ > 0 2 æöeB 2 ykykzy¢¢ =-ç 0 ÷ éù1sin +()2 () ç ÷ ëûêúww èøgmc0 Averaging rapid oscillations (twice the wiggler oscillation) yield the net focusing æö2 kaww é 2 ù ykyy¢¢ =-ç ÷ 1 +() èøç g ÷ ëê w ûú US Particle AcceAcceleratorlerator School 2009 LA-UR 09-03171 University of New Mexico - Albuquerque NM 3-16 Vertical and Horizontal Motions Betatron motion in y Wiggler motion in x Approximation for small kwy 2 xkx¢¢ +=0 yky¢¢ +=b 0 w eB0 kww a k mc0 2 Longitudinal velocity (relative to c) Transverse velocity (relative to c) 1 2 2 2 eB0 2 1 2 1 ky 22 w kmcw 0 Betatron motion is slow, large amplitude motion in the y direction over many wiggler periods due to gradient of By along the y direction. The field amplitude is proportional to the square of deviation from the center. US Particle AcceAcceleratorlerator School 2009 LA-UR 09-03171 University of New Mexico - Albuquerque NM 3-17 Vertical Envelope Equation • Vertical envelope equation with emittance 2 2 dRyny2 1 23kR y dz Ry dR2 • Find matched beam envelope radius by setting y to zero 2 2 dz R4 ny y0 ka Matched beam radius k ww k 2 2 ny Ry0 kaww If the focused electron beam is not the same as the matched beam radius, the y envelope will have betatron oscillations along the length of the wiggler. US Particle AcceAcceleratorlerator School 2009 LA-UR 09-03171 University of New Mexico - Albuquerque NM 3-18 Wiggler Focusing in a Planar Wiggler Conventional planar wigglers focus the electron beam only in the y direction. A pair of quadrupoles is used to focus the beam in x and y. y Wiggler x FD y x x-y beam profiles at different locations The electron beam is matched into the wiggler with external focusing (by a pair of quadrupole magnets) to form an ellipse (the long axis in x) at the entrance. The matched beam is round only at the center of the wiggler. US Particle AcceAcceleratorlerator School 2009 LA-UR 09-03171 University of New Mexico - Albuquerque NM 3-19 Two-plane Weak Focusing Parabolic Pole Face Dual Canted Pole Face Slotted Pole Face 2 2 Quadratic x dependence Approximate x dependence of By at the center 2 By() 1 0 10 1 xy US Particle AcceAcceleratorlerator School 2009 LA-UR 09-03171 University of New Mexico - Albuquerque NM 3-20 Two-plane Focusing Wigglers • Wiggler field in a two-plane focusing wiggler kx Bxxyw= xBˆ 0 sinh( k x )sinh( k y )cos(kz ) ky ByBkxkykzyxyw= ˆ 0 cosh( )cosh( )cos( ) kw Bzxyw=-zBˆ 0 cosh( k x )sinh( k y )sin(kz ) ky • For small values of x and y (near center) expand sinh and cosh terms 2 ByBkxykzxx ˆ 0 cos( w ) 22 22 kxx kyy ByBywˆ 0 1cos() kz 22 US Particle AcceAcceleratorlerator School 2009 LA-UR 09-03171 University of New Mexico - Albuquerque NM 3-21 Matched Beam Radius for Two-Plane Focusing Wigglers Equations for x and y envelope radii in two-plane focusing wigglers 2 2 dR2 2 dRxnx2 1 yny2 1 23kRxx 23kRyy dz Rx dz Ry where 22 2 kkkxy Equal two-plane focusing Matched beam radii in x and y k 2 kkxy 2 nx ny 2 R x 0 R y 0 kaww kaww Focusing in y is distributed to focusing in x in natural focusing wigglers.
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