2 Time-resolved diagnostic with TDS
2.1 Principle of TDS
The idea of transverse deflecting structures was originally proposed for the separation and identifica-tion of charged high energy particles in particle physics [Phi61]. Based on the theorem from Panofsky and Wenzel [PW56], suitable RF separator imposing transverse deflecting field to separate particles was developed and investigated at SLAC [ALL64] in the 1960’s. Figure2.1shows a schematic drawing in cutaway view of the LOLA-type1TDS invented at SLAC, which is an iris-loaded RF travelling wave waveguide.
With the invention of FELs, challenging requirements were put on the methods for measuring the short bunch length of relativistic electron bunches. The physics of the RF separators was re-viewed in the 1990’s, highlighting TDS as a promising tool for the time-resolved diagnostics of elec-tron bunches [EFK00]. Successful applications of the existing LOLA-type structures incorporated at LCLS and FLASH have confirmed the reliable performance of the TDS for longitudinal electron beam diagnostics [Röh08]. Recent success in using the TDS to reconstruct the temporal profile of the FEL pulse has further raised its potentials [BDD+14].
Figure 2.1:Schematic drawing in cutaway view of the LOLA-type TDS invented at SLAC. It is anS-band RF travelling wave structure operated at 2.856 GHz. The inner side cavity and the iris have a radius of 5.895 cm and 2.032 cm, respectively. To avoid possible rotation of the deflecting field, two suppressor holes with a diameter ofρ=1.905 cm are added symmetrically aside each iris at a distance ofC=3.620 cm from the centre of the iris to the centre of the suppressor hole. The deflection direction of the illustrated structure is in the vertical plane. Figure adapted from Ref. [ALL64].
The LOLA-type TDS has proven to be a reliable design, and will be adopted at the European XFEL. In the following section, the beam dynamics of a relativistic electron passing through an iris-loaded RF travelling wave TDS is considered. The transformation of the transverse and longitudinal phase space of the particle is derived.
1Named after the inventors of the structures: O. H. Altenmueller, R. R. Larsen, and G. A. Loew [ALL64].
2.1 Principle of TDS
2.1.1 Beam dynamics within a TDS
A TDS deflecting in the horizontal directionx with a length ofLis studied in the following. The electric and magnetic field distribution inside the structure are discussed and given in Ref. [ALL64].
After transformation from cylindrical to cartesian coordinate(x,y,z), the Lorentz forceF experi-enced by an electron with a charge ofecan be calculated:
Fx=eE0sin(ψ), (2.1)
Fy =0, (2.2)
Fz=eE0cos(ψ)kx. (2.3)
ψis the RF phase relative to the phase with the maximum gradient of the transverse electric field component (i.e. RF zero-crossing). E0describes the amplitude of a travelling waveE0ei(kz−ωt)and kis the wave number of the structure. The transverse force vanishes in theydirection, and remains constant inxindependent of the positions in yandz, resulting in an aberration-free deflection in purexdirection. The longitudinal forceFz reduces to zero in the centre of the structure atx = 0.
However, it increases linearly with the off-axis positionx.
Due to the vanishing force in the vertical directiony, only the horizontal motion(x,x′)of the particle and its momentum are affected by the TDS, and will be derived in the following. Now a relativistic electron (∣v∣ ≈c), travelling in a bunch of electrons, with an initial status of(x0,x0′,z,p)T before injection into the TDS is treated. Here x0 is its distance from the design trajectory, x0′ = dx/dsthe horizontal slope with respect to the design trajectorys,zthe longitudinal distance to the bunch centre and p = ∣p∣the momentum. When the energy stays constant, the slopex′describes the transverse momentum via px ≈ px′as well. The RF phase will be replaced withψ = kz+ψ0, whereψ0gives the RF phase between the centre of the bunch and the RF zero-crossing. The term kz is negligible, for example: for the LOLA-type TDS at FLASH with f =2.856 GHz and a typical uncompressed bunch length of 300µm,kzamounts to 2π f/c·z ≈18 · 10−3. As a result, the cosine and sine of the phaseψcan be approximated by their Taylor series aroundkz=0 up to the first order:
sin(kz+ψ0) ≈kzcos(ψ0) +sin(ψ0)and cos(kz+ψ0) ≈cos(ψ0) −kzsin(ψ0).
At a positionsinside the TDS (s<L), the particle will undergo an accumulated deflection of x′(s) =x0′+ ∫0px(s)dpx
p =x0′+ ∫0s 1 pFxds
c =x0′+Fx
pcs
≈x0′+eE0(kzcos(ψ0) +sin(ψ0))
pc s. (2.4)
For the purpose of beam diagnostics, a deflection with maximum linear dependence on the longitu-dinal positionzis highly desirable and can be achieved by operating the TDS at the zero-crossings of the RF, which means settingψ0to 0 orπ. Therefore, Eq.2.4becomes
x′(s) =x0′±eE0kz
pc s, (2.5)
2 Time-resolved diagnostic with TDS
with the plus and minus sign corresponding toψ0=0 andπ, respectively. In the following, only the situations of operating the TDS atψ0=0,πare considered.
Using Eq.2.5, the displacementx(s)of the electron at a positionscan be formulated as:
x(s) =x0+ ∫0sx′(s)ds=x0+x′0s± eE0kz
2pc s2. (2.6)
The non-vanishing longitudinal force at the off-axis positions inside the TDS induces extra change of the momentum of the electron. After travelling the total lengthLof the TDS, the particle gains an amount of momentum ∆pzin the longitudinal direction (using Eqs.2.3and2.6):
c∆pz= ∫0LFzds= ∫0LeE0k(x0+x0′s± eE0kz 2pc s2)ds
=eE0k[Lx0+1
2L2x0′±1 6eE0k
pc L3z]. (2.7)
The transverse deflecting force has an effect changing the electron momentum as well. Since the deflecting forceFxis constant over the aperture of the structure (see Eq.2.1), the resulting momentum change ∆pxof the particle in the transverse plane is given by
c∆px=Fx·x(s=L) =eEk(zx0+zLx0′±1 2eE0k
pc L2z2), (2.8)
where Eq.2.6is used. Compared to the longitudinal momentum gainc∆pz in Eq.2.7, each term in the expression for the transverse momentum gain is much smaller due to the multiplication with z/L(typicallyz <100µm). The total momentum gain ∆pinduced by the TDS is dominated by the change in the longitudinal direction and can be approximated byc∆p≈c∆pz. Using the substitution K= eVpc0k with the peak effective voltage defined asV0= E0L, the relative momentum deviation of the particle induced by the TDS can be obtained:
∆δ= c∆p
cp ≈ c∆pz
cp =Kx0+1
2KLx0′±1
6K2Lz. (2.9)
The first term describes a momentum gain due to the finite transverse beam size, the second term relates to the initial beam divergence and the last term is induced by the off-axis longitudinal force of the TDS.
Finally, the total displacement and deflection angle at the exit of TDS can be derived from Eqs.2.6 and2.5using the substitutions=L, so the final states of the electron are given by:
xfinal=x0+Lx0′±KL
2 z, (2.10)
xfinal′ =x′0±Kz, (2.11)
δfinal=δ+Kx0+1
2KLx′0±1
6K2Lz, (2.12)
2.1 Principle of TDS
whereδdescribes the initial momentum deviation of a particle with respect to a reference momen-tump0such that p=p0(1+δ). For a relativistic particle with∣v∣≈ c, the particle energyEcan be approximated byE ≈cp. Therefore, the relative momentum deviation describes the relative energy deviationδ≈ (E−E0)/E0with respect to the design energyE0as well.
2.1.2 Transfer matrix of TDS
Equations2.10,2.11and2.12reveal that the final state of the particle is a linear combination of the initial status. It is convenient to summarize the beam dynamics of the TDS in matrix formalism (to the first order), yielding
⎛⎜⎜
⎜⎜⎜
⎝ x x′ z δ
⎞⎟⎟
⎟⎟⎟
⎠final
=
⎛⎜⎜
⎜⎜⎜
⎝
1 L ±KL2 0
0 1 ±K 0
0 0 1 0
K KL2 ±K62L 1
⎞⎟⎟
⎟⎟⎟
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶⎠
RthickTDS
·
⎛⎜⎜
⎜⎜⎜
⎝ x0
x0′ z δ
⎞⎟⎟
⎟⎟⎟
⎠
. (2.13)
This transfer matrix of the TDSRthickTDS takes into account the finite length of the structure and is therefore referred to as “thick lens form”. An approximation forL→0 yields the simplified transfer matrix in “thin lens form”:
RthinTDS=
⎛⎜⎜
⎜⎜⎜
⎝
1 0 0 0
0 1 ±K 0
0 0 1 0
K 0 0 1
⎞⎟⎟
⎟⎟⎟
⎠
. (2.14)
The thin lens form is convenient to be used when only the transverse motion(x,x′)is of interest. In thin lens form, a TDS with a total length ofLis conceived to be composed of a drift space ofL/2, an instantaneous deflection ofKzat the centre of the structure, followed by another drift space ofL/2.
The expressions ofxandx′for such a thin lens TDS are exactly the same as that for a thick lens TDS as given in Eqs.2.10and2.11.
Since the other transverse planeyis not affected by the TDS, the matrices presented in Eqs.2.13 and2.14can be easily augmented to be
RthickTDS =
⎛⎜⎜
⎜⎜⎜⎜
⎜⎜⎜⎜
⎝
1 L 0 0 ±KL2 0
0 1 0 0 ±K 0
0 0 1 L 0 0
0 0 0 1 0 0
0 0 0 0 1 0
K KL2 0 0 ±K62L 1
⎞⎟⎟
⎟⎟⎟⎟
⎟⎟⎟⎟
⎠
and RthinTDS=
⎛⎜⎜
⎜⎜⎜⎜
⎜⎜⎜⎜
⎝
1 0 0 0 0 0
0 1 0 0 ±K 0
0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
K 0 0 0 0 1
⎞⎟⎟
⎟⎟⎟⎟
⎟⎟⎟⎟
⎠
. (2.15)
The 6-dimensional matrices describe the beam dynamics in both the transverse phase space and the longitudinal phase space, and can be applied to the phase space vector(x0,x′0,y0,y′0,z,δ)T of the electron. Expressions for a TDS deflecting in theydirection can be derived analogously.
2 Time-resolved diagnostic with TDS