1. Electromagnetic Devices and Optics - PHY743 - Devices are based on the electrodynamics' character of moving charged particles in presence of electro- magnetic fields – Especially magnetic field Basic principle is originated from Lorentz force F = q[E + (v  B)/c]  Electric force in the direction of E – Acceleration  Magnetic force normal to both v and B – Circular motion The characters can be described by “Optics” The characters defined the variety of “Elements”

2. Circular motion of charged particle in uniform B field Magnetic Dipole Uniform B Field Pointing Out of Paper Circular Motion:  = 33.356  p B  – Radius in meter P – Momentum in GeV/c B – Field in Tesla (kGauss)  is a function of momentum p p...  

3. Momentum Dispersion by Magnetic Dipole Function of Magnetic Dipole: ◦ Change charged particles’ trajectory orientation ◦ Disperse trajectory orientation according to momentum Magnetic Dipole – Cont. Optics: Prism Wavelength Dispersion................... Magnetic Dipole p  p p+  p p -  p p Momentum Dispersion

4. Basic Structure of a Dipole Magnetic Dipole – Cont. “H” Dipole “C” Dipole York Iron (High  ) Electric Coil Magneti c Flux Uniform Field Region Large uniform field area Suitable for large particle trajectory profile - Spectrometer Small uniform field area but small size Suitable for small particle trajectory profile – Beam Line Element or Special application

5. Effective Field Boundary (EFB) Magnetic Dipole – Cont. B y /B 0 Z/G B y – Normal field B 0 – Total field strength Z – Trajectory distance G – Gap of the opening 1.0 Uniform Field Region None-uniform field - “Fringe Field Distribution” F.F.D. I =  B y dZ B y  Z =  B y dZ EFB Actual Pole Iron Boundary Boundary shaping outlined by EFB line and detailed F.F.D. are important parameters for design and optical description of a dipole B x and B z are mot zero in fringe field region

6. Important Optical Parameters for a Dipole ◦ B 0 and L (path length) ◦ and  These are first order parameters ◦ and  ◦ Shaping of EFB’s ◦ Fringe field description These are second and higher order parameters Magnetic Dipole – Cont.   B0B0  

7. Basic Structure of a Quadrupole ◦ York iron with 4 inner circular symmetric poles ◦ Four sets of connected coils ◦ Field flux flows oppositely: Up-Down and Left-Right ◦ B = 0 at r = 0, B max at r = R Magnetic Quadrupole R R B max (Tip Field) 0 B(r) (For Illustration only)

8. Profile of Charged Particles w/ the Same Momentum It works just like an optical lens Quadrupole magnet – Magnetic Lens Magnetic Quadrupole – Cont. Point Source Point Image Optical Lens Symbol of Focusing Lens Horizontal (or Vertical) Plane Vertical (or Horizontal) Plane Quadrupole focuses the charged particles. Multipoles and quadrupoles are needed to focus the particles in full phase space

9. Magnetic Multipoles have the same concept as Quadrupole except number of poles They are: ◦ Hexapole (Axial Symmetry – 2 nd order in optics) ◦ Octapole (Point Symmetry – 3 rd order in optics) ◦ Decapole (Axial Symmetry – 4 th order in optics) ◦ Dodecapole (Point Symmetry – 5 th order in optics) Hardware Hexapole Magnetic Multipoles York Iron Coils Pole Face Imperfect Quadrupole produces Multipole fields Reference to perfect Quadrupole Axial asymmetry of pole spacing Point asymmetry of pole spacing Others defects: Combined asymmetries, imperfect individual pole location and rotation, and imperfect pole face curvatures. These are unavoidable.

10. Quadrupoles are used for beam line and spectrometer to confine or focus the beam profile since Dipole changes the profile size due to incident angle and momentum spreads Hexapoles are used commonly in beam line to control the beam profile at hardware level Multipole Fields from spectrometer Quadrupoles are commonly described or corrected in the “Optics” description Optical Parameters for Quadrupole and Multipoles: ◦ Tip field strength – B max, radius R, and effective length L (1 st order) ◦ Strength of Multipole field contents (2 nd and higher orders) ◦ Fringe field distribution description (2 nd and higher orders) Magnetic Multipoles – Cont.

11. Used to separate particles w/ the same momentum, i.e. purify the secondary beam content Basic Structure: Location and size of the slit selects the particles Optical Parameters: Effective path length – L and E x (first order) Gap and width of electrodes and fringe field (Higher orders) Electric Separator – Velocity Separator Vacuum Chamber Window HV (+250kV) HV (-250kV) Uniform E field: F = qE = ma Mass Slit (Move Up-Down) Heavier Particle Lighter Particle

12. Commonly used for collision physics or large acceptance reactional or decay physics Basics structure (Assuming for reactional or decay physics): Optical parameters:  Length of solenoid  Diameter of solenoid  Asymptotic magnetic field of solenoid, i.e. B = 0.4  IN/LSolenoids x x x x x x x x x x x x x x x x x................... Cylindrical York IronEnd Cap York Electric Cylindrical Coil  Uniform Axial B Field Detector Region Target Momentum of Detected Particle Transverse momentum is measured by the r Longitudinal momentum is measured by TOF

13. Example: The Hadron Hall at J-PARC Put All The Elements Together for Hadronic Beam Lines 50 GeV/c proton beam to primary production target Secondary lines for  +, K +, or p beam Secondary lines for  -, K -, or p beam Beam line dipoles Dipole spectrometer Quadru-/multi-poles Separators/Mass Slits

14. Example: Continuous Electron Beam Accelerator Facility (CEBAF) A B CCH North Linac +400MeV South Linac +400MeV Injector West Arc East Arc

15. Example: Continuous Electron Beam Accelerator Facility (CEBAF) 499 MHz,  = 120  Optical fiber-based, RF-pulsed drive lasers ARC’s and Hall A/C lines require a series of beam line dipoles to separate passes and reorient the beam direction Many quadrupoles and multipoles are required to confine the beam profile, remittance, achromatic in momentum at target

16. Example: Hall C at Jlab (CEBAF) SOS HMS Quadra-Poles and Dipoles They form specialized magnetic optical instruments that analyze the momentum of the scattered charged particles from the experimental target

17. Coordinate Matrices ◦ At target:X t = (x t, x’ t, y t, y’ t, 0,  p), x t = y t = 0 for point “target” ◦ At focal plane:X fp = (x fp, x’ fp, y fp, y’ fp, L,  p), measured at focal plane ◦ x’ and y’ are the angles in dispersion and non-dispersion planes ◦ p is momentum in % with respect to the central momentum Transportation Matrices – Representing the Optical Character of the Spectrometer System ◦ M – Forward optical matrix from target to focal plane ◦ M -1 – Backward optical matrix from focal plane to target Matrix Representation of Optical Transportation and Reconstruction ◦ Forward: X fp = M X t Backward: X t = M -1 X fp ◦ p can be found when M (or M -1 ) and the rest elements in X t, and X fp matrices are known, i.e. Matrix Representation of Magnetic Optics Using Spectrometer at CEBAF as Example  p = F(known coordinates) where F is also written in matrix At CEBAF: x’ t = F’(known coordinates and  p); y’ t = F”(known coordinates and  p) Reconstruction matrices, F, F’, and F”, are all derived from M or M -1

18. By Polynomial expansion, M is written in series of orders in which the 1 st order matrix represents the basic optical nature of a specifically designed spectrometer. 1 st order matrix M(6x6): Using 1,2,3,4,5,6 for x, x’, y, y’, L,  p Each element represents an “Amplification” or a “Correlation” from individual X t to X fp coordinates Matrix Representation of Magnetic Optics – Cont. M1M1M1M1 R11, R12, R13, R14, R15, R16 R21, R22, R23, R24, R25, R26 R31, R32, R33, R34, R35, R36 R41, R42, R43, R44, R45, R46 R51, R52, R53, R54, R55, R56 R61, R62, R63, R64, R65, R66 XtXt X fp

19. Example: ◦ R11 and R33 are x fp /x t and y fp /y t ratios, i.e. image (or spot size) “Amplifications” Matrix Representation of Magnetic Optics – Cont. Object size (x t ) Illustrated by a simple single lens optics f Image size (x fp )

20. Matrix Representation of Magnetic Optics – Cont. Example – Cont.: ◦ R12 and R34 are x fp /x’ t and y fp /y’ t, i.e. “Correlation” dependence of image or spot size at FP to the incident angular acceptance x’ t and y’ t. Illustrated by a simple single lens optics f Point Object Two different angles: x’ t ~ 0 and x’ t at maximum Crossing z over a  z Causing an enlarged and smeared image size x fp and x fp - x’ t correlation

21. Matrix Representation of Magnetic Optics – Cont. Example – Cont.: ◦ Element R16 (  p/x t ) represents the enlarged image size due to momentum accaptance or “bite”. ◦ D/M = R11/R16 defines an important character for a spectrometer: Momentum Dispersion in unit of cm/%. In principle, the larger D/M the better momentum resolution for a spectrometer. Illustrated by a simple single lens optics f Point Object Rays with different “Wavelength”, i.e. “Momentum” Rays with lowest momentum Rays with highest momentum 1 st order focal plane Image size due to “Wavelength” or Momentum acceptance FP

22. General considerations of a specific optical system ◦ Optimize all first order parameters, including all drift spaces to achieve specific optical features for a system ◦ D/M for required momentum resolution of a spectrometer ◦ To achieve Point-to-point focusing, minimize R12 and R34, i.e. no angular and size correlations: Better momentum resolution. ◦ To achieve Point to Parallel focusing, minimize R22 and R44, i.e. no angular and angular correlations: Better angular acceptance but poor 1 st order focusing. Matrix Representation of Magnetic Optics – Cont.  p = 0  p = +  p = -  p = +  p = -

23. General considerations of a specific optical system – Cont. ◦ Mixed: Point-to-Point in x but Point-to-Parallel in y. Enhance resolution by good D/M and x focusing but increase angular acceptance from y’. ◦ Achromatic optics for beam line: R16  0 (or D/M  0) To minimize the beam size and dispersion to connect optical systems or send beam on experimental target. ◦ Issues to be considered for a spectrometer:  Momentum resolution  Momentum and angular acceptances  Total path length  Focal plane size  Total spin precession for polarized particle Matrix Representation of Magnetic Optics – Cont. x z R12  0 y R44  0

24. First order optics defines the intrinsic and general features of an optical system (a spectrometer or a sub-section of beam line). It is an ideal approximation that analogs to the small lens approximation of optics. Higher order optics come from non-ideal features of a system, thus represent the “realities”. Inclusion of higher order matrices in M is to reproduce the “Real Optics” of a “Realistic” system. Therefore, it is extremely important and crucial to evaluate and obtain the realistic higher order optics in order for the system to work or achieve the design goal. The sources contributed to higher order optics: ◦ Fringe field effect from each electro-magnetic element ◦ Dipole EFB boundary shape and non-parallel of dipoles ◦ Asymmetries from symmetric elements ◦ Alignment errors and relative rotations between elements ◦ Precision of field setting ◦ Field interference between elements Higher Order of Electro-magnetic Optics

25. Higher order matrix elements: ◦ 2 nd order: Ri|jk, i, j, k = 1 – 6, e.g. Rx’|x’y’=R2|24 Total of ~6 3 /2 elements ◦ 3 rd order: Ri|jkl, i, j, k, l = 1 – 6, e.g. Rx’|x’yy’= R2|234 Total of ~6 4 /2 2 elements ◦ 4 th order: Total of ~6 5 /2 3 elements ◦…◦… Number of orders needed: 6 – 10 for accuracy Number of elements: Often more than thousand Higher Order of Electro-magnetic Optics – Cont.

26. Magnetic devices and systems are similar as optical components and systems, such as Quadra-poles  Lens and Dipole  Prism, … Magnetic devices and systems can be designed and used based on magnetic optics. Commonly used optics software are:  Transport – Up to third orders, used for basic design, obtain matrix  Turtle – Use matrix to evaluate profiles to optimize acceptance  Raytrace – Describe field up to fifth orders, use field map to evaluate “realistic” optics  COSY – Combined all above, include higher orders and obtain matrix Accurate optical matrix is essential for designing and using the magnetic systems – beam line and spectrometer Summary of Magnetic Optics