Why I wanted to learn more about coupling: “Coupling mismatch” What is the associated emittance blow-up???

Example of harmonic motion: the pendulum m·g

Coupled pendulums: Normal modes:

Hill’s equation: Smooth machine approximation: Either: Average focussing distributed evenly in machine. ...or: Observe the beam in one point of the machine, and use a “lumped machine model”.

Driving terms: skew quadrupoles and solenoids “position coupling” from skew quadrupole field, “velocity coupling” from solenoid field

Equations of motion: Ansatz: Assume slowly varying amplitude functions. Retain only terms of lowest order in perturbation. Assume frequencies approximately equal.

Take derivative: Eliminate Y: Solution: Re-define coefficient:

Re-written solution: Note behaviour when coupling goes to zero! Inserted in the Ansatz:

Result with smooth machine model:

y y x x Normal modes are inclined due to skew quadrupole term... y y ...and elliptic due to solenoid term. x x

Amplitude function: Bound motion:

Assume initial displacement in x only: Then:

Amplitude interchange: |X2| Modulation depth: Emax Emin  Interchange wavelength: s |Y2| s

Coupling coefficients: Remember: Modulation depth gives: where T is interchange period

Difference in frequency between normal modes In terms of tunes (multiply by R): Uncoupled tunes |C|  Standard method to measure magnitude of coupling

“Lumped” coupling coefficients in a “real” machine (a function of position): Where: (skew quads and solenoid end-field) (solenoid field) Solenoids and skew quads are “interchangeable”

Assume only skew quads: Then:

Sum resonances in real machine: -Non-zero driving term requires non-uniform distribution (P0) of errors in the machine. -Unstable motion e.g. Also more difference resonances:

Transfer matrix for coupled motion: 44=16 elements, but only 10 free parameters (symplectic constraints) How to parameterise the motion (and the transfer matrix)? “Teng/Edwards”: use four new parameters for transformation to de-coupled system and define new Twiss functions in this system “DESY”: Try to keep old Twiss functions and add four new (2 betas and 2 alphas) plus auxiliary parameters.

Teng/Edwards parameterisation: (normal modes in rotated, “de-coupled”, phase space) 4 new parameters: rotation angle and three free parameters of the matrix D

DESY parameterisation: (normal modes in x-y phase space) 4 new parameters: alphas and betas in “opposite planes” Redefined: “old” alphas (where longitudinal field) Auxiliary quantities: u’s and ’s Transfer matrix expressed in these new parameters

Definitions of emittance: two sensible definitions -Invariants of normal modes (conserved) ...of theoretical interest -Projected emittances (not conserved) ...of practical interest

Projected emittances along a transfer line: yields Projected emittances:

-The projected emittances are constant in between coupling elements. -The projected emittances are what is measured by a three-SEM-grid system (assuming no coupling elements in between the grids). -The projected emittances are the smallest achievable filamented emittances if the coupled beam is injected in an uncoupled machine. -The sum of projected emittances are always larger than or equal to the sum of normal mode emittances.

How much is the emittance increased when an uncoupled beam is injected into a coupled machine? Assume smooth machine with only “skew quad term”: Change to tilted co-ordinate system:

The equations de-couple when: or The new equations are then:

Emittance invariants: Parameterisation of initial distribution:

Initial distribution in expressions for normal mode emittances: For the sum of emittances, using explicit dependences of the parameters on ks: ...remember that

Emittance blow-up in simple smooth approximation …is quite small! What happens in a “real” machine??? (probably larger, due to form factors)