Life Hacks for Solenoids and Magnetized Beams David Douglas

Disclaimers, Caveats, Warnings There’s nothing new here. It’s all in Kwang-Je’s 2003 PRST-AB paper. This is about coupled beams in coupled lattices. BEAMS AND LATTICES ARE DIFFERENT. Descriptions of coupling have been rediscovered or reinvented about every 5 years since early descriptions by Gluckstern et. al (PAC’69) and Teng (FN-0229, 1971) appeared during the Nixon administration… I don’t understand any of them.

Motivation Design activities for the JLEIC cooler ERL & DarkLight encountered interesting ambiguities, leading to strange questions… – How does the symmetry inherent in a magnetized beam constrain behavior… Despite “full” coupling… In either uncoupled or coupled beam lines? – Which emittance is which?? – WHY ISN’T THAT MATRIX SYMPLECTIC???? Need to establish consensus on what “emittance” means, and how to evaluate it… – Phase space volume? Moment of beam distribution? – "emittance", "canonical emittance", "drift emittance", "cyclotron emittance", "Larmor emittance", "invariant (Courant-Snyder) emittance", "projected emittance", "coupled emittance", "eigenmode emittance", … – Choice of variables… (x,x’,y,y’…), (x,p x, y, p y,…) (x, p x -qA x, y, p y -qA y,…) – Solenoids are atypical beamline elements: transverse components in vector potential

Motivation Design activities for the JLEIC cooler ERL & DarkLight encountered interesting ambiguities, leading to strange questions… – How does the symmetry inherent in a magnetized beam constrain behavior… Despite “full” coupling… In either uncoupled or coupled beam lines? – Which emittance is which?? – WHY ISN’T THAT MATRIX SYMPLECTIC???? Need to establish consensus on what “emittance” means, and how to evaluate it… – Phase space volume? Moment of beam distribution? – "emittance", "canonical emittance", "drift emittance", "cyclotron emittance", "Larmor emittance", "invariant (Courant-Snyder) emittance", "projected emittance", "coupled emittance", "eigenmode emittance", … – Choice of variables… (x,x’,y,y’…), (x,p x, y, p y,…) (x, p x -qA x, y, p y -qA y,…) – Solenoids are atypical beamline elements: transverse components in vector potential

Motivation Design activities for the JLEIC cooler ERL & DarkLight encountered interesting ambiguities, leading to strange questions… – How does the symmetry inherent in a magnetized beam constrain behavior… Despite “full” coupling… In either uncoupled or coupled beam lines? – Which emittance is which?? – WHY ISN’T THAT MATRIX SYMPLECTIC???? Need to establish consensus on what “emittance” means, and how to evaluate it… – Phase space volume? Moment of beam distribution? – "emittance", "canonical emittance", "drift emittance", "cyclotron emittance", "Larmor emittance", "invariant (Courant-Snyder) emittance", "projected emittance", "coupled emittance", "eigenmode emittance", … – Choice of variables… (x,x’,y,y’…), (x,p x, y, p y,…) (x, p x -qA x, y, p y -qA y,…) – Solenoids are atypical beamline elements: transverse components in vector potential

What We’re After We want to know – How to make and simulate a magnetized beam using existing tools – How to look at the output of the simulations wherever we want Like inside a solenoid Why we care – Standard tools lump solenoid fringe fields and body dynamics together. This can make it hard to sort out what is going on and to be sure matching has been successful – ERLs are nonequilibrium systems, so Beam and lattice are distinct - and must thus be described independently, but… Need to reconcile the descriptions & use modeling tools in a manner consistent with their capabilities

First Hack: Making a Magnetized Beam Y. Derbenev saw the simplest way: start with a flat beam and push it through a flat beam transform (FBT) There are several descriptions of FBTs. Note – Burov/Danilov (Fermilab-TM-2043) and Brinkmann (EPAC’02) give a recipe that has negative angular momentum, and thus requires negative solenoid strength – We’ll follow the recipe of JLAB-TN (which entirely inadvertently produces the opposite sign of angular momentum, and thus would use a positive solenoid…) If we want the beam to be matched to a solenoid, we just use an FBT with  equal to the matched  of the solenoid (2  Larmor ) and  of zero – Flat beam emittance is chosen to get the “right” beam size in the solenoid

Aside: “Matched” FBT [  establishes connection of “beam” and “lattice”]

Hacking a Matched Magnetized Beam

Second Hack: The Solenoid

Third Hack: Solenoid Fringes NOT SYMPLECTIC!!!

Okay, So, I Lied… I do kinda sorta understand one of the descriptions of coupled motion… the one where you evaluate the eigenmodes of the motion, transform everything to that basis (establishing coupling amplitudes and angles in the process), and using a decoupled Twiss parameterization of the motion in the eigenbasis. Not that I can do it, but I sort of understand it. Why do I admit this? – Confession is good for the soul, and – Solenoid basis vectors are low-hanging fruit - that relate to that FBT/emittance thingy that’s motivating this rant…

Hack 5: Basis Vectors Set v 1 =(1,0,0,K), v 2 =(0,-K,1,0),v 3 =(1,0,0,-K), and v 4 =(0,K,1,0) Note any (x,x’,y,y’)=av 1 +bv 2 +cv 3 +dv 4 with a=(x+y’/K)/2, b=(y-x’/K)/2, c=(x-y’/K)/2, and d=(y+x’/K)/2 Any trajectory can be decomposed into a superposition of these four modes; – what do each of them do inside the solenoid? – what does the full solenoid do to each of them?

Effect of Fringe & Body Fields The basis vectors transform as follows under the front end fringe map F: Fv 1 =(1,0,0,0) Fv 2 =(0,0,1,0) Fv 1 =(1,0,0,-2K) Fv 1 =(0,2K,1,0) Pushing them through the solenoid body map gives further indication of what each basis element means; after using  Twiss =2  Larmor =1/K, we get: L(Fv 1 )=(1,0,0,0) L(Fv 2 )=(0,0,1,0) L(Fv 3 )=(cos(  Larmor ), -sin(  Larmor )/  Larmor, -sin(  Larmor ), -cos(  Larmor )/  Larmor ) L(Fv 4 )=(-sin(  Larmor ), -cos(  Larmor )/  Larmor,-cos(  Larmor ), sin(  Larmor )/  Larmor ) The first two describe the “quiet” – i.e., magnetized – components of the beam The second pair describes Larmor motion about the field lines Applying the back-end fringe map B will then give the effect of the complete solenoid

Effect of Full Solenoid on Basis

Connection to Flat Beams and Emittances The magnetized components v 1 and v 2 come from the horizontal plane upstream of the initial FBT – These correspond to the drift emittance and define the beam size during acceleration and transport The Larmor components v 3 and v 4 come from the vertical components in the original (nearly) flat beam – These correspond to the Larmor/cyclotron emittance that provides (or impedes) cooling Does this suggest we can get both emittances by simply applying the inverse FBT? – This would give a really simple way to evaluate beam quality and assess the level of degradation… – Possible spoiler: collective-effect-induced coupling; The foregoing analysis assumes an initially uncoupled flat beam. Collective interactions/nonlinearities at an inopportune point (e.g. solenoid fringe?) might open up a channel to generate coupling that could then result in a coupled beam after the inverse FBT…

Semi-Final Hack: General FBT on General Decoupled (But Matched) Beam

Beam sizes set by notionally familiar Twiss beta and effective – or drift – emittance Can relate lattice and beam parameters via the FBT – Define ,  with incoming beam parameters – Match lattice (FBT) acceptance to beam – Output has “same” beam parameters, can be put into a lattice with appropriate acceptance

Final Hack: FBT of Round Beam

Exegesis Any trajectory can be described by the basis we’ve evaluated: (x,x’,y,y’)=av 1 +bv 2 +cv 3 +dv 4 with a=(x+y’/K)/2, b=(y-x’/K)/2, c=(x- y’/K)/2, and d=(y+x’/K)/2 An inverse FBT segregates the basis : the magnetized modes go to one plane, the Larmor modes go to the other This provides a diagnostic: apply inverse FBT to particle tracking data – magnetized modes  drift emittance  “x plane” emittance after FBT Defines notional “beam size” in cooling channel – Larmor modes  cyclotron emittance  “y plane” emittance after FBT Notionally controls “cooling rate” – If back-transform of simulated reality doesn’t look “right” – e.g. it’s not decoupled, the emittances grow, etc. – can quantify degradation via Emittance dilution Coupling in beam matrix