Beam Optics Manoel Couder University of Notre Dame

Good aim is sometime important

Sometime it is critical!

Not Talking About Accelerators Introduction to Accelerators: Evolution of Accelerators and Modern Day Applications Sarah Cousineau, Jeff Holmes, Yan Zhang USPAS January, 2011

The Plan: “Beamline” • Type of ion optics equipment? Don’t forget diagnostic!!

Outline • Optics • Charged particle optics • Beam – Collective description of particles • Spectrometers/Separators

Straight Light Rays • Z axis = Optics axis of a bundle of rays • Deviation of rays from bundle y x Will be using this a lot!!! Optics of Charged Particles By Hermann Wollnik

Straight Light Rays • Z axis = Optics axis of a bundle of rays • Deviation of rays from bundle • Angular dependence y Not really exiting … x Optics of Charged Particles By Hermann Wollnik

Straight Light Rays • Z axis = Optics axis of a bundle of rays • Deviation of rays from bundle • Angular dependence Transfer matrix from one profile plane to another In rotationally symmetric system Both matrix are identical…

What is a Focal Point? • https: //phet. colorado. edu/en/simulation/lega cy/geometric-optics

What is a Focal Point? f Rays that enter the system parallel to the optical axis are focused such that they pass through the “rear focal” point. Any ray that passes through it will emerge from the system parallel to the optical axis.

Thin Lens f • transfer from 2 to 3 Profile plane 2 AND 3 Optics of Charged Particles By Hermann Wollnik

Transport through… Optics of Charged Particles By Hermann Wollnik

How do we get a proper picture? Independence of final ray position on initial angle Under that condition there is a object-image relation between profile planes 1 and 4

Additional Definitions M is called the magnification Notation change to allow for generalization Can be a simple drift or a very complex system.

Examples What constraints on which matrix element needs to explain the plots? Optics of Charged Particles By Hermann Wollnik

Charged Particle Optics • Approximation

Quadrupole Lenses Focus only in one direction Electric The motion of charged particles in any of the element presented here is known. The transformation matrix is known as well. You can solve the equation of motion to find them. Magnetic

Combination of Quadrupole Lenses Point to parallel Point to point With appropriate diagnostic tune by slowly changing one field and correcting with the other Optics of Charged Particles By Hermann Wollnik

Symmetric Focusing A Single tuning parameter. Maximize transmission to appropriate diagnostic. Works at relatively low energies ~100 ke. V Max WHY?

Bending Charged Particles +V -V B

The Color Twist d could represent any variable They could be more Actually, most transport code Use at least 6 dimensions. Most of the matrix element are zero

Dispersion p+dp Dipole field B perpendicular to paper plane . Object (size x 0) x Radius r p Dispersion dx/dp used in magnetic analysis

Beam? Pedro Castro Introduction to Particle Accelerators DESY, July 2010

Beam? Same for the vertical plane Pedro Castro Introduction to Particle Accelerators DESY, July 2010

Concept of emittance Same for the vertical plane The emittance is conserved as long as the forces to which the beam is subjected to are conservative. Pedro Castro Introduction to Particle Accelerators DESY, July 2010

Concept of emittance

Concept of emittance The normalized emittance is conserved during acceleration Pedro Castro Introduction to Particle Accelerators DESY, July 2010

Emittance and Radioactive Beam Production Method • Depending on the production method • Emittance of RI Beam can be much worse than the one of stable beam – Equipment have to be designed to accommodate • Large energy and angular acceptance • Larger bore • Re. A 3(, 6, 12), Caribu 29

Equivalence of Transport of ONE Ray Û Ellipse Defining the s Matrix representing a Beam The 2 -dimensional case ( x, Q ) s = æs 11 s 21 ü ès 21 s 22 þ Real, pos. definite symmetric s Matrix s-1 = 1/e 2 æs 22 -s 21 ü è-s 21 s 11 þ

Equivalence of Transport of ONE Ray Û Ellipse Defining the s Matrix representing a Beam The 2 -dimensional case ( x, Q ) 2 -dim. Coord. vectors (point in phase space) X = æx ü èQþ X T = (x Q) Ellipse in Matrix notation: X T s-1 X = 1 Exercise: Show that Matrix notation is equivalent to known Ellipse equation: s 22 x 2 - 2 s 21 x Q + s 11 Q 2 = e 2

Beam Sigma Matrix and Transfer Matrix Ray X 0 from location 0 is transported by a 6 x 6 Matrix R to location 1 by: X 1 = RX 0 (1) Note: R maybe a matrix representing a complex system is : R = Rn Rn-1 … R 0 Ellipsoid in Matrix notation, generalized to e. g. 6 -dim. using s Matrix: X 0 T s 0 -1 X 0 = 1 (2) Inserting Unity Matrix I = RR-1 in equ. (2) it follows X 0 T (RTRT-1) s 0 -1 (R-1 R) X 0 = 1 from which we derive (RX 0)T (Rs 0 RT)-1 (RX 0) = 1 The equation of the new ellipsoid after transformation becomes X 1 T s 1 -1 X 1 = 1 where s 1 = Rs 0 RT (3) Conclusion: Knowing the TRANSPORT matrix R that transports one ray through an ion-optical system using (1) we can now also transport the phase space ellipse describing the initial beam using (3)

Beam/Ion Transport Calculation • Now we have a “framework” to calculate either individual particle trajectories or a full beam… – How do we use it? Do we have to calculate individual transfer matrix? – Code with various approach and level of details • • • TRANSPORT COSY INFINITY GIOS GICOSY … – SIMION, OPERA, …

Resolving Power Your favorite spectrometer/separator Half size of your target x 0

Resolving Power Half size of your target x 0 The resolving power is the inverse value of d 1 that still provide a separation x

Resolving Power The resolving power is the inverse value of d 1 that still provide a separation x 18 O(a, g)22 Ne @ 2. Me. V Recoil Beam

MDM @ Texas A&M

What Did we Ignore Truncation of equation of motion to allow for linear combination and usage of matrix formalism. Truncation is similar to selection of order of a Taylor expansion. Missing parts are called “Higher Order” Element “limitation”

Beam Space Charge Effects 1 n. A Space charge in the accelerating lens can cause broadening of the beam, which will affect all of the ions in the beam, independent of mass. 100 n. A 39

Higher Order