Up to now, we discussed EEX with thin-lens approximation of the TDC. However, the thickness of TDC perturbs the beam energy depending on the longitudinal position and generates non-zero diagonal 2× 2 block matrices (also called thick-cavity effect). Below is the EEX beamline transfer matrix when Eq.
(2.8) was considered in Eq. (2.11),
REEX,thick=
0 L6c κ(L+D+L3c) η+κ ξ(L+D+L3c)
0 0 κ κ ξ
κ ξ η+κ ξ(L+D+L3c) L6cκ2ξ L6cκ2ξ2 κ κ(L+D+L3c) L6cκ2 L6cκ2ξ
. (2.22)
TheR65of Eq. (2.8) inducedR12,R55,R56,R65, andR66of Eq. (2.22). Since these elements are not zero, the final phase spaces are coupled with their initial phase spaces. This thick-cavity effect is harmful to our LPS manipulation: increasing the bunch length, degrading the time–energy correlation control.
Several methods were devised to mitigate or compensate the thick-cavity effect [23, 24, 38, 39]. First method is controlling the incoming chirp before the EEX beamline. This method cannot eliminate the all thick-cavity effects, but can minimize it. The accelerating cavities before the EEX beamline enable to impart longitudinal chirp depending on the rf phase. The transfer matrix of the chirp control + EEX
beamline can be approximated as,
0 L6c κ(L+D+L3c) η+κ ξ(L+D+L3c)
0 0 κ κ ξ
κ ξ η+κ ξ(L+D+L3c) L6cκ2ξ L6cκ2ξ2 κ κ(L+D+L3c) L6cκ2 L6cκ2ξ
·
1 0 0 0
0 1 0 0
0 0 1 0
0 0 h 1
(2.23)
=
0 L6c κ(L+D+L3c)(1+hξ) +hη η+κ ξ(L+D+L3c)
0 0 κ(1+hξ) κ ξ
κ ξ η+κ ξ(L+D+L3c) L6cκ2ξ(1+hξ) L6cκ2ξ2 κ κ(L+D+L3c) L6cκ2(1+hξ) L6cκ2ξ
, (2.24)
wherehis a linear chirp from the accelerating cavities. Left matrix of Eq. (2.23) is a chirp control matrix and the initial beam is assumed to have the chirp of zero. If h=−1
ξ, thenR55 andR65 of Eq. (2.24) become zero.
Second method is inserting fundamental mode cavity (FMC) beside the TDC. This FMC enables to eliminate theR65of Eq. (2.8) using acceleration/deceleration rf field. Because the source of the pertur- bation was vanished, the diagonal 2×2 block matrices become zero. Below is the transfer matrix of FMC and TDC,
RFMC·RT DC,thick=
1 Lf 0 0
0 1 0 0
0 0 1 0
0 0 −L6fκ2 1
·
1 Lc L2cκ 0
0 1 κ 0
0 0 1 0
κ L2cκ L6cκ2 1
(2.25)
=
1 Lc+Lf L2cκ 0
0 1 0 0
0 0 1 0
κ L2cκ 0 1
, (2.26)
whereLf is the length of FMC. Since the FMC will cancel out the acceleration/deceleration from the TDC, the rf voltage of the FMC is set to getR65=−L6fκ2.
The single EEX scheme has some limitations in practical scenarios [40, 41]. Typical photoinjectors produce larger longitudinal emittance electron beams than its transverse one. Hence, a single EEX in- creases the final horizontal emittance, which eventually limits the horizontal focusing, brightness, and spatial resolutions of applications. Therefore, a second EEX that can exchange the phase spaces once again was introduced to maintain the transverse emittance low [38].
Another drawback is that both timing and energy jitters before the EEX beamline become horizontal jitters after the exchange. The horizontal jitters are fatal to the magnet alignments, wakefield effects,
and FEL lasing. The double emittance exchange (DEEX) technique, that will be explained in the next chapter, can alleviate this limitation because the second EEX brings these jitters back to the LPS [40].
Chapter 3
Theoretical background: Double EEX for LPS manipulations
The two limiting factors (emittance, jitters) necessitate additional EEX to return them to original direc- tions. In case of double emittance exchange (DEEX), transverse manipulations at the first EEX entrance are not able to perform the LPS manipulation due to the following second EEX. DEEX has transverse manipulation section in middle of two EEXs. That allows transversely manipulated beam is exchanged into final LPS phase space at the second EEX.
The middle section could contain transverse manipulation elements (such as magnets and mask) to control final LPS. Linear elements (e.g., quadrupole) enable to perform the bunch compression and the linear longitudinal chirp control. Since the bunch compression does not rely on the incoming chirp to the DEEX beamline, all the accelerating cavities could be operated on-crest to maximize the energy gain. The longitudinal chirp control after the DEEX beamline has the same advantage. In addition, higher-order magnets, such as a sextupole and an octupole, in the middle section nonlinearly control the final LPS. This may not require the X-band linearizer because the sextupole can mitigate quadratic curvatures from RF accelerating cavities. The octupole magnets also can cure the double-horn feature in the longitudinal profile, which decreases accelerator performance and damages equipment, such as undulators [20, 21, 42, 43].
The DEEX beamline and its functions that are demonstrated herein would provide various new opportunities in a similar way as the initial development of bunch compression methods enabled modern accelerator facilities.
3.1 Transfer matrix of DEEX beamline
3.1.1 Ideal DEEX beamline
For simplicity of calculations, we assume a TDC thickness of zero. Among the two EEXs, the first EEX beamline transfer matrix can be given using Eq. (2.11),
REEX1=
0 0 κ(L+D1) η+κ ξ(L+D1)
0 0 κ κ ξ
κ ξ η+κ ξ(L+D1) 0 0
κ κ(L+D1) 0 0
, (3.1)
where D1 is the distance between the dogleg and TDC. The second EEX beamline has doglegs that are similar to those of the first one, except for the bending directions. This means that the matrix for the second EEX can be obtained using η → −η together with setting κ → −κ (to ensure the EEX condition). Furthermore, we need to switchD1→D2because the distance between the two doglegs at the second EEX beamline can differ from the first one. Detailed derivations are provided in Ref. [36].
The middle section consists of n quadrupoles and n+1 drift spaces between them. The higher-order magnets were not considered for simplicity. Here, we express the transfer matrix of the middle section in terms of parametersC,S,C′, andS′for simplicity.
Rmid=Rdrift,n+1·
1
∏
i=n
Rquad,i·Rdrift,i
=
C S 0 0
C′ S′ 0 0
0 0 1 0
0 0 0 1
. (3.2)
The DEEX beamline is composed of two EEX beamlines and a middle section between them. Hence, the transfer matrix of the entire DEEX beamline can be expressed as
R=REEX2·Rmid·REEX1
=
R11 R12 0 0 R21 R22 0 0 0 0 R55 R56 0 0 R65 R66
, (3.3)
where
R11=1−2κ2ξ(D2+L), (3.4)
R12= (D1+D2+2L)−2κ2ξ(D1+L)(D2+L), (3.5)
R21=−2κ2ξ, (3.6)
R22=1−2κ2ξ(D1+L), (3.7)
R55=−κ
(D1+L)
κ ξC+ (η+κ ξ(D2+L))C′ +
κ ξS+ (η+κ ξ(D2+L))S′ , (3.8) R56=−(η+κ ξ(D1+L))
κ ξC+ (η+κ ξ(D2+L))C′ −κ ξ
κ ξS+ (η+κ ξ(D2+L))S′ , (3.9) R65=−κ2
(D1+L)
C+ (D2+L)C′ +
S+ (D2+L)S′ , (3.10)
R66=−κ
(η+κ ξ(D1+L))
C+κ(D2+L)C′ +κ ξ
S+κ(D2+L)S′ . (3.11)
From Eq. (3.3), we note that if the initial beam is decoupled before the EEX, the final beam after the EEX remains decoupled in a linear order. Similarly, the rms emittances in each plane remain constant.
In particular, we emphasize that the matrix elementsR55,R56,R65, and R66, which determine the final longitudinal beam phase space, contain the transverse beam optics parameters, such asC,S,C′, andS′. Hence, the LPS can be effectively manipulated using transverse optics, which is considerably simpler and more flexible.
3.1.2 Thick-cavity effects in DEEX beamline
As mentioned in Sec. 2.3, the thick-cavity effect induced the non-zero block matrices of the EEX transfer matrix. There are two solutions to mitigate or eliminate the effect. Our experiment used the chirp control to minimize the thick-cavity effect from each EEX beamline. Through the accelerating cavity, the beam chirp can be controlled as−1/ξ which only suppresses the thick-cavity effect from the first EEX. The second EEX beamline is also required the incoming chirp of −1/ξ. This chirp can be generated via transverse manipulation at the first EEX beamline entrance. The bunch length (σz,3) and the chirp (C3) after the first EEX beamline can be expressed as,
σz,32 =σx,02 (R51+R52sx,0)2+R52εx,02
σx,02 +σz,02 (R55+R56C0)2+R56εz,02
σz,02 , (3.12)
σzδ,3=R51R61σx,02 +R52R62σx2′,0+ (R51R62+R52R61)σxx′,0 (3.13) +R55R65σz,02 +R56R66σδ,02 + (R55R66+R56R65)σzδ,0,
C3=σzδ,3
σz,32 . (3.14)
The initial bunch length and the chirp are fixed at the first EEX beamline entrance while the transverse properties are controllable using quadrupoles. Hence, we can set the appropriateσz,3andC3of−1/ξ to minimize the thick-cavity effect from the second EEX beamline.