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Quadrupole Transverse Beam Optics Chris Rogers 2 June 05

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Plan 1.Equations of Motion 2.Transfer Matrices 3.Beam Transport in FoDo Lattices 4.Bunch Envelopes 5.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!

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

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

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

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

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

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

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

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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?

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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

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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

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

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

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Multiple Components We can use the matrix formalism to deal with multiple components in a neat manner Say we have transport matrices M 10 from z 0 to z 1 and M 21 from z 1 to z 2 so What is transfer matrix from z 0 to z 2 ? and

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Multiple Components We can use the matrix formalism to deal with multiple components in a neat manner Say we have transport matrices M 10 from z 0 to z 1 and M 21 from z 1 to z 2 so Then –with and

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

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Stability Criterion What are the requirements for beam containment - is FoDo really stable? Transfer Matrix for n FoDos in series: For stability require that M tot 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

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

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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 Then state the transfer matrix for n FoDos

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

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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

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

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Transfer Matrices for Bunches We can transport this ellipse in a straight- forward manner –We have –What will V 1, the matrix at z 1, look like?

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

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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

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Emittance Conservation Claim: Emittance is constant at constant momentum – 0 = 1 if |V 1 | = |V 0 | –Use |A B| = |A| |B| –Then requirement becomes |M 10 |=1 Consider as an example –State principle that to 1st order |M|=1 for all “linear” optics

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

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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

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