2 Time-resolved diagnostic with TDS
2.2 Diagnostics with TDS
2.2.1 Longitudinal current profile
2 Time-resolved diagnostic with TDS
2.2 Diagnostics with TDS
imaging screen atsis given by Eq.1.3
us= Rs0→s·RthinTDS·us
0, (2.16)
whereRs0→s is the transfer matrix from the centre of the TDSs0to the location of the screen ats.
The horizontal position of the particle on the screen is of interest and can be found by xs= (R11,s0→sxs0 +R12,s0→sx′s0)
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
x0s
±R12,s0→sKz, (2.17)
where the result from Eq.2.14has been used. The first part of this equationxs0represents the hor-izontal displacement in the case of the TDS being switched off (V0 =0 → K = 0), and the second part the deflecting effect of the TDS. Substituting the definition of the matrix element R12,s0→s =
√βx(s)βx(s0)sin(∆µx)from Eq.1.9into Eq.2.17leads to
xs=xs0+S±z, S±= ±√
βx(s)βx(s0)· sin(∆µx)K, (2.18) whereβx(s)andβx(s0)are the horizontal beta-functions (see Eq.1.6) at the screen and the centre of the TDS, respectively, and ∆µxdenotes the horizontal phase advance (see Eq.1.8) between these two points. The parameterS±is commonly calledstreak parameterand describes the strength of the linear deflection of the TDS.
Without initial correlation
The simplest case is first considered for a bunch of electrons: the electrons display no correlation in(x,z)and(x′,z) at the entrance of the TDS, and thusx0s is not correlated to the longitudinal coordinatezeither, i.e.⟨xs0z⟩ =03. The second moment of the horizontal positions of the electrons at the screen, i.e. the squared rms beam size of the bunchσx2is given as
σx2= ⟨(xs0)2⟩ +S2±⟨z2⟩ = (σx0)2+S2±σz2, (2.19) where it is presumed that the first moments of the electrons in a nominal bunch are⟨x0s⟩ = 0 and
⟨z⟩ =0. By measuring the intrinsic beam sizeσx0(by switching off the TDS), the streaked beam size σxatS+(orS−), and the corresponding streak parameterS+(orS−, see Eq.2.36), the bunch length σzcan be determined to be
σz=
¿Á Á
Àσx2− (σx0)2
S2± . (2.20)
It is worth noting that the sought-after bunch length does not depend on the sign of the streak param-eter (i.e. the value of the zero-crossing phaseψ0) with this simplifying assumption of an uncorrelated initial transverse-longitudinal phase space.
3The bracket⟨. . .⟩is the operator for the statistical mean of the variables inside the bracket.
2 Time-resolved diagnostic with TDS
The longitudinal resolutionRz of the bunch length measurement is determined by the intrinsic beam size, which is the smallest measurable transverse beam size on the screen, divided by the streak parameter:
Rz= σx0
∣S±∣ =
√βx(s)εN/γ
√βx(s)βx(s0)· sin(∆µx)K =
√εN/γ
√βx(s0)· sin(∆µx)K, (2.21) withεN the normalized emittance andγthe Lorentz-factor of the bunch (see Eq.1.15). In order to optimize the longitudinal resolution, a large horizontal beta-functionβx(s0)at the centre of the TDS and a horizontal phase advance of ∆µx=π/2+nπ,n∈N0from the TDS to the screen are preferred.
The beta-functionβx(s)at the screen does not influence the longitudinal resolution, but should be chosen to give a beam size on the screen that is measurable taking into account the spatial resolution of the imaging system.
It is common to describe the longitudinal parameters using the time axist =z/cinstead of the longitudinal axisz, which leads to the expressions of the bunch lengthσtand longitudinal resolution Rtin the time domain:
σt=σz/c, Rt=Rz/c. (2.22)
Influence of initial correlations
The assumption of no correlation in the transverse-longitudinal plane was an idealized case. A real electron bunch usually displays correlations in(x,z)or (x′,z), or both of them. This initial cor-relation leads to a corcor-relation between the intrinsic offset (when the TDS is switched off) and the longitudinal coordinate, which modifies Eq.2.18to givexs = x0s(z) +S±z. In this case, Eq.2.19 be-comes invalid for the determination of the bunch length due to⟨x0sz⟩ ≠0, and the longitudinal profile cannot be directly obtained. One simple method based on Refs. [Ban90,Loo] to reconstruct the lon-gitudinal profile will be described in the following paragraphs, with one example of the application of the reconstruction method presented in Figs.2.2and2.3.
Since the motion inyis not affected by TDS, it is convenient to consider only thexandzplanes.
The initial phase space densityρ(x,z)of a bunch is assumed to be
ρ(x,z) =λ(z)δ(x− f(z)), z∈ [z0−,z0+] (2.23) withλ(z)being a line density function andz0−,z0+ (z0− < z0+) the start and end position of the bunch, respectively. Two approximations are made here: (i) each longitudinal slice inzhas an in-finitesimal width inx(expressed by theδ-function in Eq.2.23) and an infinitesimal divergence inx′, (ii) the horizontal-longitudinal correlation of each slice is given byx = f(z)andx′ = g(z), where f(z)andg(z)are two arbitrary correlation functions. The longitudinal density function equals to λ(z)due to the relation
ρz(z) = ∫ ρ(x,z)dx= ∫ λ(z)δ(x−f(z))dx=λ(z). (2.24) When the TDS is switched off, only the transverse plane(x,x′)is transformed, while the longitudinal
2.2 Diagnostics with TDS
density functionρz(z)remains unchanged. Therefore the intrinsic density distribution of the bunch on the screenρ0(x,z)is given as
ρ0(x,z) =λ(z)δ(x−x0(z)), (2.25)
withx0(z)being a linear combination off(x)andg(x)describing the horizontal-longitudinal cor-relation of the bunch at the position of the screen when the TDS is switched off.
λ(z) q0
λ(z)
z0− z0 0 z0+
∆x(q0)
x0(z)
µ+(z)
µ−(z)
z
x
ν+(x)
q0 ν−(x)
q0
density functionν(x)
0 q0 1
q+(x)
q−(x)
∆x(q0)
particle fractionq(x)
Figure 2.2:Illustration for the principle of the profile reconstruction method using measurements at both TDS RF zero-crossings. The example is shown for a Guassian distributed longitudinal density functionλ(z).
An arbitrary correlation functionx0(z)is assumed. The reconstructed longitudinal density functionq′(z) is compared with the referenceλ(z)in Fig.2.3.
The TDS introduces extra correlation in the horizontal-longitudinal plane, resulting in
µ±(z) =x0(z) +S±z. (2.26)
It is assumed thatµ±(z)is a monotonic function4due to the large value of the streak parameterS±
(see Fig.2.2left). Replacing the correlation functionx0(z)in Eq.2.25withµ±(z)leads to the phase space density function of the streaked bunch
ρ±(x,z) =λ(z)δ(x−µ±(z)). (2.27)
The measured horizontal density distribution on the screen is defined as the integral over the
longi-4for allz∈ [z0−,z0+],µ′+(z) =x0′(z) +S+>0 andµ′−(z) =x0′(z) +S−<0.
2 Time-resolved diagnostic with TDS
tudinal coordinate (see Fig.2.2middle):
ν±(x) = ∫ ρ±(x,z)dz. (2.28)
From the measuredν±(x), the particle fractionq±can be calculated to be
q+(x) = ∫−∞x ν+(x)dx, (2.29)
q−(x) = ∫x∞ν−(x)dx, (2.30)
with the different integral limits indicating that they both give the particle fraction contained in the bunch part starting from the trailing end of the bunch (i.e. from the small longitudinal coordinate z0−). The particle fraction functionsq±(x)are invertible and their inverse functions are denoted as x±(q). For a given particle fraction q = q0 = ∫zz0−0 λ(z)dzin the range ofz0− toz0, the positions x+(q0)andx−(q0)refer to the transformation of the particles from the same longitudinal position z0(see Fig.2.2right):
x+(q0) =µ+(z0) =x0(z0) +S+z0, (2.31) x−(q0) =µ−(z0) =x0(z0) +S−z0. (2.32) Taking the difference of the above two equations, the following relation is obtained
∆x(q0) =x+(q0) −x−(q0) = (S+−S−)z0. (2.33) This equation indicates that the longitudinal positionz0can be solved for a given particle fraction q0by measurements at both TDS streak parameters S+andS−. By replacingq0with an arbitrary particle fractionq, the longitudinal positionzcan be expressed as a function ofq
z(q) = ∆x(q)
S+−S−. (2.34)
The inverse function ofz(q)isq(z), whose derivative gives the longitudinal line density function of the bunch (see Fig.2.3)
λ(z) =q′(z). (2.35)
This method retrieves the longitudinal distributionλ(z)of a bunch with arbitrary initial transverse-longitudinal correlation using measurements at both TDS RF zero-crossings. The finite slice beam width and slice divergence is neglected.
Calibration of the streak parameter S
In both cases described above (with and without initial correlations), the reconstruction of the lon-gitudinal profile using TDS requires the knowledge of the streak parameterS±. The value ofS±can be calculated directly according to Eq.2.18when the values of the beta-functions and phase advance
2.2 Diagnostics with TDS
z0− z0 0 z0+
0 q0
1
q(z)
z
q(z)
z0− z0 0 z0+
z
densitydistribution q′(z) λ(z)
Figure 2.3:Illustration for the principle of the profile reconstruction method using measurements at both TDS RF zero-crossings. The longitudinal density functionq′(z)is reconstructed for the example shown in Fig.2.2.
are known. The latter is usually not the case during real measurements, and therefore the streak parameter has to be determined experimentally.
From Eq.2.18the position of the centre of the bunch on the screen is given by⟨xs⟩ = ⟨x0s⟩+S±⟨z⟩, using the fact that the expectation is a linear operation. This relation is valid for both cases with and without initial transverse-longitudinal correlation. The change of the centre position of the bunch on the screen ∆⟨xs⟩depends linearly on the variation of the longitudinal centre position ∆⟨z⟩. According to ∆⟨z⟩ ≈c∆t=c∆ϕ/2π f with f being the RF frequency of the TDS and ∆ϕa slight change of the TDS RF phase around the zero-crossing phase, the longitudinal centre position can be varied by changing the TDS RF phase as well:
∆⟨xs⟩ =S± c
2π f∆ϕ. (2.36)
By measuring the change of the horizontal position of the bunch centre ∆⟨xs⟩and the corresponding change of the TDS RF phase ∆ϕ, the streak parameterS±can be derived. One example of simulating the calibration procedure is shown in Fig.3.36.