Quadrupole Transverse Beam Optics

1. Quadrupole Transverse Beam Optics Chris Rogers 2 June 05

2. Plan • Equations of Motion • Transfer Matrices • Beam Transport in FoDo Lattices • Bunch Envelopes • Emittance Invariant • I haven’t had time to do numerical techniques (e.g. calculating beta function) • I had hoped to introduce Hamiltonian dynamics but again no time. • This is quite a mathsy approach - not many pictures • I don’t claim expertise so apologies for mistakes!

3. Quadrupole Field Hyperbolic pole faces Field gradient • Quad Field “hyperbolic” • Transverse focusing & defocusing • Used for beam containment

4. Equations of Motion - in z • Lorentz force given by • Constant energy in B-field => relativistic g constant

5. Equations of Motion - in z • Lorentz force given by • Constant energy in B-field => relativistic g constant • Use chain rule so d/dt = vzd/dz and

6. Equations of Motion - in z • Lorentz force given by • Constant energy in B-field => relativistic g constant • Use chain rule so d/dt = vzd/dz • Substitute for B-field to get SHM (Hill’s equation) and and Signs! Focusing strength K (dependent on m/q)

7. Transfer Matrices 1 • Recall solution of SHM • Take e.g. K>0 solution with K>0 • Recall solution of SHM K=0 K<0

8. Transfer Matrices 1 K>0 • Recall solution of SHM • Take e.g. K>0 solution with • Use double angle formulae K=0 K<0

9. Transfer Matrices 1 K>0 • Recall solution of SHM • Take e.g. K>0 solution with • Use double angle formulae K=0 K<0

10. Transfer Matrices 2 • This is tidily expressed as a matrix • This is no coincidence • Actually, this is the first order solution of a perturbation series • Can be seen more clearly in a Hamiltonian treatment • What do the matrices for K=0, K<0 look like?

11. Transfer Matrices for other K • Quote: • Assumes K is constant between 0 and z • Introduce “effective length” l to deal with fringe fields K>0 K=0 K<0

12. FoDo Lattice - an example • It is possible to contain a beam transversely using alternate focusing and defocusing magnetic quadrupoles (FoDo lattice) • This is possible given certain constraints on the spacing and focusing strength of the quadrupoles • We can find these constraints using certain approximations

13. Thin Lens Approximation • In thin lens approximation, • Define focusing strength • Then

14. Thin Lens Transfer Matrices • Transfer matrices become • Should be recognisable from light optics

15. Multiple Components • We can use the matrix formalism to deal with multiple components in a neat manner • Say we have transport matrices M10 from z0 to z1 and M21 from z1 to z2 so • What is transfer matrix from z0 to z2? and

16. Multiple Components • We can use the matrix formalism to deal with multiple components in a neat manner • Say we have transport matrices M10 from z0 to z1 and M21 from z1 to z2 so • Then • with and

17. Transfer matrix for FoDo Lattice • Wrap it all together then we find that the transfer matrix for a FoDo lattice is

18. Stability Criterion • What are the requirements for beam containment - is FoDo really stable? • Transfer Matrix for n FoDos in series: • For stability re quire that Mtot is finite & real for large n. • Route is to solve the Eigenvalue equation (mathsy) • Then use this to get a condition on f and l

19. Eigenvalues of FoDo lattice • Standard way to solve matrix equation like this - take the determinant • Then we get a quadratic in l • Neat trick - define m such that • Giving eigenvalues • Try comparing with quadrupole transport M

20. Transfer Matrix ito eigenvalues • Then we recast the transfer matrix using eigenvalues, and remaining entirely general • Here I is the identity matrix and J is some matrix with parameters a, b, g • Then state the transfer matrix for n FoDos

21. Stability Criterion • For stability we require cos(nm), sin(nm) are finite for large n, i.e. • Recalling the transfer matrix for the FoDo lattice, this gives • or

22. Bunch Transport • We can also transport beam envelopes using the transfer matrices • Say we have a bunch with some elliptical distribution in phase space (x, x’ space) • i.e. density contours are elliptical in shape • Ellipse can be transported using these transfer matrices Density contour

23. Contour equation • General equation for an ellipse in (x,x’) given by • Or in matrix notation

24. Transfer Matrices for Bunches • We can transport this ellipse in a straight-forward manner • We have • What will V1, the matrix at z1, look like?

25. Transfer Matrices for Bunches • We can transport this ellipse in a straight-forward manner • We have • Define the new ellipse using • So that

26. Emittance • Define un-normalised emittance as the area enclosed by one of these ellipses in phase space • E.g. might be ellipse at 1 rms (so-called rms emittance) • Or ellipse that contains the entire/95%/whatever of the bunch • Area of the ellipse is given by • e.g. for rms emittance

27. Emittance Conservation • Claim: Emittance is constant at constant momentum • e0=e1 if |V1| = |V0| • Use |AB| = |A| |B| • Then requirement becomes |M10|=1 • Consider as an example • State principle that to 1st order |M|=1 for all “linear” optics

28. Normalised emittance • Apply some acceleration along z to all particles in the bunch • Px is constant • Pz increases • x’=Px/Pz decreases! • So the bunch emittance decreases • This is an example of something called Liouville’s Theorem • ~“Emittance is conserved in (x,Px) space” • Define normalised emittance

29. Summary • Quadrupole field => SHM • We can transport individual particles through linear magnetic lattices using transfer matrices • Multiple components can be strung together by simply multiplying the transfer matrices together • We can use this to contain a beam in a FoDo lattice • We can understand what the bunch envelope will look like • We can derive a conserved quantity emittance