Electromagnetic Devices and Electromagnetic Devices and OpticsOptics- PHY743 -- 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”

Circular motion of charged particle in uniform B field

Magnetic DipoleMagnetic Dipole

Uniform B Field Pointing Out of

Paper

Circular Motion:

=

33.356 pB

– Radius in meterP – Momentum in GeV/cB – Field in Tesla (kGauss)

is a function of momentum

p

p

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Momentum Dispersion by Magnetic Dipole

Function of Magnetic Dipole:◦ Change charged particles’ trajectory orientation◦ Disperse trajectory orientation according to

momentum

Magnetic Dipole – Magnetic Dipole – Cont.Cont.

Optics: Prism

Wavelength Dispersion

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Magnetic Dipole

pp

p+p

p -pp

Momentum Dispersion

Basic Structure of a Dipole

Magnetic Dipole – Magnetic Dipole – Cont.Cont.

“H” Dipole

“C” DipoleYork Iron

(High )

Electric Coil

Magnetic Flux

Magnetic

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

Effective Field Boundary (EFB)

Magnetic Dipole – Magnetic Dipole – Cont.Cont.

By/B0

Z/G

By – Normal fieldB0 – Total field strengthZ – Trajectory distanceG – Gap of the opening

1.0

Uniform Field

Region

None-uniform field - “Fringe Field

Distribution” F.F.D.

I = BydZ

ByZ = BydZ

EFBActual 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

Bx and Bz are mot zero in fringe field

region

Important Optical Parameters for a Dipole

◦B0 and L (path length)

◦ and These are first order parameters

◦ and ◦Shaping of EFB’s◦Fringe field descriptionThese are second and higher order parameters

Magnetic Dipole – Magnetic Dipole – Cont.Cont.

B0

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, Bmax at r = R

Magnetic QuadrupoleMagnetic Quadrupole

R

R

Bmax (Tip

Field)

0

B(r)(For Illustration

only)

Profile of Charged Particles w/ the Same Momentum

It works just like an optical lens

Quadrupole magnet – Magnetic Lens

Magnetic Quadrupole – Magnetic Quadrupole – Cont.Cont.

PointSource

PointImage

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

Magnetic Multipoles have the same concept as Quadrupole except number of poles

They are:◦ Hexapole (Axial Symmetry – 2nd order in optics)◦ Octapole (Point Symmetry – 3rd order in optics)◦ Decapole (Axial Symmetry – 4th order in optics)◦ Dodecapole (Point Symmetry – 5th order in optics)

Hardware Hexapole

Magnetic MultipolesMagnetic Multipoles

York Iron

Coils

Pole Face

Imperfect Quadrupole produces Multipole fieldsReference to perfect

QuadrupoleAxial asymmetry of pole spacingPoint asymmetry of pole spacing

Others defects: Combined asymmetries, imperfect individual pole location and rotation, and imperfect pole face curvatures. These are unavoidable.

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 – Bmax, radius R, and effective length L (1st order)

◦ Strength of Multipole field contents (2nd and higher orders)◦ Fringe field distribution description (2nd and higher orders)

Magnetic Multipoles – Magnetic Multipoles – Cont.Cont.

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 Ex (first order)

Gap and width of electrodes and fringe field (Higher orders)

Electric Separator – Velocity Electric Separator – Velocity SeparatorSeparator

Vacuum Chamber

Window

HV (+250kV)

HV (-250kV)

Uniform E field: F = qE = ma Mass Slit

(Move Up-Down)

Heavier

Particle

Lighter Particl

e

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 = 0.4IN/L

SolenoidsSolenoids

x x x x x x x x x x x x x x x x x

. . . . . . . . . . . . .

. . . . . .

Cylindrical York Iron

End 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

Example: The Hadron Hall at J-Example: The Hadron Hall at J-PARCPARC

Put All The Elements Together for Hadronic Beam LinesPut 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

Example: Continuous Electron Beam Accelerator Facility (CEBAF)

AB

CCH

NorthLinac

+400MeV

SouthLinac

+400MeV

InjectorWest Arc

East Arc

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

Example: Hall C at Jlab (CEBAF)Example: Hall C at Jlab (CEBAF)

SOS

HMS

Quadra-Poles

and DipolesThey form specialized magnetic optical instruments

that analyze the momentum of the scattered charged particles from the experimental target

Coordinate Matrices◦ At target: Xt = (xt, x’t, yt, y’t, 0, p), xt = yt = 0 for point

“target”

◦ At focal plane: Xfp = (xfp, x’fp, yfp, 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: Xfp = M Xt Backward: Xt = M-1 Xfp

◦ p can be found when M (or M-1) and the rest elements in Xt, and Xfp matrices are known, i.e.

Matrix Representation of Magnetic Matrix Representation of Magnetic OpticsOptics

Using Spectrometer at CEBAF as ExampleUsing Spectrometer at CEBAF as Example

p = F(known coordinates) where F is also written in matrixAt 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

By Polynomial expansion, M is written in series of orders in which the 1st order matrix represents the basic optical nature of a specifically designed spectrometer.

1st 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 Xt to Xfp coordinates

Matrix Representation of Magnetic Optics Matrix Representation of Magnetic Optics – – Cont.Cont.

MM

11

R11, R12, R13, R14, R15, R16R21, R22, R23, R24, R25, R26R31, R32, R33, R34, R35, R36R41, R42, R43, R44, R45, R46R51, R52, R53, R54, R55, R56R61, R62, R63, R64, R65, R66

Xt

Xfp

Example: ◦ R11 and R33 are xfp/xt and yfp/yt ratios, i.e. image

(or spot size) “Amplifications”

Matrix Representation of Magnetic Optics Matrix Representation of Magnetic Optics – – Cont.Cont.

Object size (xt)

Illustrated by a simple single lens optics

f

Image size (xfp)

Matrix Representation of Magnetic Optics Matrix Representation of Magnetic Optics – – Cont.Cont.

Example – Cont.: ◦ R12 and R34 are xfp/x’t and yfp/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 xfp and xfp- x’t correlation

Matrix Representation of Magnetic Optics Matrix Representation of Magnetic Optics – – Cont.Cont.

Example – Cont.: ◦ Element R16 (p/xt) 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

1st order focal plane

Image size due to “Wavelength” or

Momentum acceptance

FP

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 1st order focusing.

Matrix Representation of Magnetic Optics Matrix Representation of Magnetic Optics – – Cont.Cont.

p = 0

p = +

p = -

p = +

p = -

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 Matrix Representation of Magnetic Optics – – Cont.Cont.

x

z

R12 0

yR44 0

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 OpticsHigher Order of Electro-magnetic Optics

Higher order matrix elements:◦ 2nd order: Ri|jk, i, j, k = 1 – 6, e.g. Rx’|x’y’=R2|24

Total of ~63/2 elements◦ 3rd order: Ri|jkl, i, j, k, l = 1 – 6, e.g. Rx’|x’yy’=

R2|234 Total of ~64/22 elements

◦ 4th order: Total of ~65/23 elements◦ …

Number of orders needed: 6 – 10 for accuracyNumber of elements: Often more than

thousand

Higher Order of Electro-magnetic Optics Higher Order of Electro-magnetic Optics – – Cont.Cont.

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 OpticsSummary of Magnetic Optics