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Why I wanted to learn more about coupling:
“Coupling mismatch” What is the associated emittance blow-up???
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Example of harmonic motion: the pendulum
m·g
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Coupled pendulums: Normal modes:
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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”.
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Driving terms: skew quadrupoles and solenoids
“position coupling” from skew quadrupole field, “velocity coupling” from solenoid field
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Equations of motion: Ansatz: Assume slowly varying amplitude functions. Retain only terms of lowest order in perturbation. Assume frequencies approximately equal.
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Take derivative: Eliminate Y: Solution: Re-define coefficient:
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Re-written solution: Note behaviour when coupling goes to zero! Inserted in the Ansatz:
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Result with smooth machine model:
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y y x x Normal modes are inclined due to skew quadrupole term... y y ...and elliptic due to solenoid term. x x
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Amplitude function: Bound motion:
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Assume initial displacement in x only:
Then:
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Amplitude interchange:
|X2| Modulation depth: Emax Emin Interchange wavelength: s |Y2| s
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Coupling coefficients:
Remember: Modulation depth gives: where T is interchange period
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Difference in frequency between normal modes
In terms of tunes (multiply by R): Uncoupled tunes |C| Standard method to measure magnitude of coupling
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“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”
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Assume only skew quads:
Then:
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Sum resonances in real machine:
-Non-zero driving term requires non-uniform distribution (P0) of errors in the machine. -Unstable motion e.g. Also more difference resonances:
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Transfer matrix for coupled motion:
44=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.
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Teng/Edwards parameterisation:
(normal modes in rotated, “de-coupled”, phase space) 4 new parameters: rotation angle and three free parameters of the matrix D
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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
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Definitions of emittance:
two sensible definitions -Invariants of normal modes (conserved) ...of theoretical interest -Projected emittances (not conserved) ...of practical interest
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Projected emittances along a transfer line:
yields Projected emittances:
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-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.
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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:
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The equations de-couple when:
or The new equations are then:
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Emittance invariants:
Parameterisation of initial distribution:
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Initial distribution in expressions for normal mode emittances:
For the sum of emittances, using explicit dependences of the parameters on ks: ...remember that
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Emittance blow-up in simple smooth approximation
…is quite small! What happens in a “real” machine??? (probably larger, due to form factors)
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