E. Todesco, Milano Bicocca January-February 2016 Appendix A: A digression on mathematical methods in beam optics Ezio Todesco European Organization for Nuclear Research (CERN)

E. Todesco, Milano Bicocca January-February 2016 CONTENTS The optics of particle accelerators makes use of interesting mathematical methods Very similar to formalisms developed in 20 th century physics, in particular Matrix and tensor formalism (general relativity, spin) Discrete formalism (Dirac delta function) Propagator theory Usually we study differential equations and we have little experience with matrix, discrete formalisms and propagators Matrix is the prototype of operator Here we will make a digression on way of solving Hills equations with two particular applications The motion in a detector (propagation in free space) The FODO cell (delta-like functions – quadrupoles – plus propagation in free space) The advantage is to have an example of classical physics to get used to these formalisms Appendix A - 2

E. Todesco, Milano Bicocca January-February 2016 CONTENTS Hills equations Propagation in free space The FODO cell Appendix A - 3

E. Todesco, Milano Bicocca January-February 2016 HILLS EQUATIONS We recall Hills equation in one degree of freedom It is a harmonic oscillator with both frequency and amplitude depending on time, plus a periodic condition on time (that is the longitudinal coordinate of the accelerator) The function K gives the gradient of the quadrupole, and it is zero in the dipoles We write the solution in the following form The advantage of the square root is that there is a simple relation between  and  Appendix A - 4

E. Todesco, Milano Bicocca January-February 2016 HILLS EQUATIONS We differentiate the Ansatz Replacing in and setting to zero the cos term Appendix A - 5

E. Todesco, Milano Bicocca January-February 2016 HILLS EQUATIONS So we obtain Therefore the first derivative can be written as And one can write the solution in terms of a matrix Appendix A - 6

E. Todesco, Milano Bicocca January-February 2016 HILLS EQUATIONS One can write the solution in terms of a matrix Here we are going to a Hamiltonian-like formalism: two coupled first order differential equations, with two variables x and x’ It is x and p in the Hamiltonian formalism Instead of the Newton formalism: one second order differential equation Appendix A - 7

E. Todesco, Milano Bicocca January-February 2016 HILLS EQUATIONS The inverse of the matrix is We can define the normalized coordinates in this space the motion is simply a rotation So we can write the solution from s 0 to s 1 as Appendix A - 8

E. Todesco, Milano Bicocca January-February 2016 HILLS EQUATIONS When the gradient K 1 is constant, the Hill equation can be solved and one can compute the transfer matricies Case 1: a drift Case 2: a focusing quadrupole Appendix A - 9

E. Todesco, Milano Bicocca January-February 2016 HILLS EQUATIONS Case 2: a focusing quadrupole It is interesting to consider the approximation of thin lens Where the quadrupole effect is No change in position x Kick in the angle x’ This is a delta like operator inducing a discontinuity in the derivative It is a very good approximation since in the lattice the quadrupole length is <10% of the quadrupole spacing (in the LHC, l q =3 m, L=50 m) Appendix A - 10

E. Todesco, Milano Bicocca January-February 2016 CONTENTS Hills equations Propagation in free space The FODO cell Appendix A - 11

E. Todesco, Milano Bicocca January-February 2016 FREE SPACE PROPAGATION We now consider the case of an empty space of length 2l *, with a symmetry condition in the middle This is the case of the beta function in the detector We first try with classical methods: apparently, very easy equation Using the relation between beta function and phase Appendix A - 12

E. Todesco, Milano Bicocca January-February 2016 FREE SPACE PROPAGATION So the equation for the beta function in free space is Not easy to solve Assuming a second order polynomial, by symmetry first order term must be zero And replacing in the previous one has And finally Appendix A - 13

E. Todesco, Milano Bicocca January-February 2016 FREE SPACE PROPAGATION Now we try the formalism based on matrix plus propagation The general form is We apply to the propagation from the centre of the IP, where  =  and  ’  =0, to a position at a distance l This gives Appendix A - 14

E. Todesco, Milano Bicocca January-February 2016 FREE SPACE PROPAGATION We know that the solution is a straight line Therefore in matrix formalism And comparing to the previous result One has And finally Appendix A - 15

E. Todesco, Milano Bicocca January-February 2016 CONTENTS Hills equations Propagation in free space The FODO cell Appendix A - 16

E. Todesco, Milano Bicocca January-February 2016 THE FODO CELL If we use the standard approach of differential equations we can reduce Hill equations to Hard to find an analytical solution … it is a nonlinear second order differential equation with discontinuous K … Appendix A - 17

E. Todesco, Milano Bicocca January-February 2016 FODO CELL For the FODO cell we have to consider a periodic condition, so the matrix in s 0 and s 1 are the same We can compute the transfer matrix And we find Appendix A - 18

E. Todesco, Milano Bicocca January-February 2016 FODO CELL We now compose the matrix of a quadrupole, a drift, another quadrupole with opposite gradient, and another drift The aim is to compute the matrix elements, and recover explicit expressions for the beta function From we find Appendix A - 19

E. Todesco, Milano Bicocca January-February 2016 FODO CELL And from we find For 90 degrees phase advance  =  /2 and Taking the DOFO case we can compute the beta function in F: Appendix A - 20

E. Todesco, Milano Bicocca January-February 2016 SUMMARY The mathematical methods used in beam optics are extremely useful to understand, in a classical problem, formalisms used aso in quantum mechanics and relativity The usual methods of differential equations fail In particular, one makes use of Matrix formalism Propagation of solution in a free space, with delta-like interactions that give non differentiable solutions Reminds Feynmann … Appendix A - 21