High Brightness Electron Beam Diagnostics and their ... - CASA

60
0.25
40
20
emittances~"x;y,andrmsbeamsizesx;y)versustheoperatingacceleratingphaseofthelinac. Figure6.18:Evolutionofbeamparameters(bunchlengthz,rmsenergyspreadE,transverse
0
0
6
1
x
x
4
y
0.5
y
2
thelatterequationclearlyshowsthat,comparedtotheconstantenergyspreadequation,thereis whichcanbesolvedusingthestandardGreenfunctionperturbativetechnique[59]11Thesolution
0
0
anincrementinangleandpositionof: oftheaboveequationis: x(s)=cos(s=x)x(0)+xsin(s=x)x0(0)+x(1cos(s=x))(0)+Zs x0(s)=1=xsin(s=x)x(0)+cos(s=x)x0(0)+cos(s=x)(0)+Zs 0xsin(~s=x)(~s)d~s 0cos(~s=x)(~s)d~s(6.24)
−20 −15 −10 −5 0
−20 −15 −10 −5 0
∆φ (RF−Deg)
∆φ (RF−Deg)
achromaticbendingsystem,theachromaticcharacterisbrokenbecauseofEqn.(6.25).Another interestingpointisthatdependingonthebendingsystemdesign,onecanconceiveawayofmaking andhxx0i)andsubstitutethemintheeqn.(6.20).Itisinterestingtonotethatinthecaseofan Tocomputetheemittancegrowthweneedtocomputethesecondordermoments(hx2i,hx02i x(s)=Zs x0(s)=Zs 0xsin(~s=x)(~s)d~s
theaboveintegralverysmall(orideallyzero)sothatthenetemittancegrowthisnegligible.Such amethodhasbeendiscussedindetailinreferences[60]and[61]. 0cos(~s=x)(~s) (6.25)
principalsolutions(S(t)andC(t))accordinglyto: equationwritesx(t)=Rt 11Ifweconsidertherighthandsideofthepreviousequationasaperturbationtermp(t;s)thesolutionofthis 0p(~t)d~tG(t;~t)whereG(t;~t)isaGreen'sfunctionthatcanbeconstructedfromthetwo G(t;~t)=S(t)C(~t)C(t)S(~t)
σ z (mm)
ε x,y (mm−mrad)
0.5
∆E (keV)
σ x,y (mm)
100
80