Abstract
After dedicated efforts devoted to the longitudinal dynamics to realize an ultrasmall longitudinal emittance and ultrashort bunch length for coherent radiation generation, we need to make sure that the coupling arising from transverse dynamics does not degrade or even destroy the longitudinal fine structures. Such an argument is based on the observation that the transverse beam size in an SSMB ring can be orders of magnitude larger than the desired microbunch length. This is the basic motivation for us to investigate the transverse-longitudinal coupling (TLC) dynamics. In this chapter, we start from the linear TLC and then investigate the nonlinear TLC dynamics. For the linear dynamics, first we analyze the passive bunch lengthening induced by bending magnets. We then emphasize the fact that TLC can actually be actively applied for efficient bunch compression and high harmonic generation when the transverse emittance is small. We present three theorems on the application of such TLC schemes, with their implications discussed. Further, we have analyzed the contribution of modulators to the vertical emittance from quantum excitation, to obtain a self-consistent evaluation of the required modulation laser power when applying these coupling schemes in a storage ring. The theorems and related analysis provide the theoretical basis for the application of TLC in SSMB to lower the requirement on the modulation laser power, by taking advantage of the fact that the vertical emittance in a planar ring is rather small. Based on the investigations, we have presented example parameters sets for the envisioned SSMB storage ring to generate high-power EUV and soft X-ray radiation at the end of this dissertation.
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After dedicated efforts devoted to the longitudinal dynamics to realize an ultrasmall longitudinal emittance and ultrashort bunch length for coherent radiation generation, we need to make sure that the coupling arising from transverse dynamics does not degrade or even destroy the longitudinal fine structures. Such an argument is based on the observation that the transverse beam size in an SSMB ring can be orders of magnitude larger than the desired microbunch length. This is the basic motivation for us to investigate the transverse-longitudinal coupling (TLC) dynamics. In this chapter, we start from the linear TLC and then investigate the nonlinear TLC dynamics. For the linear dynamics, first we analyze the passive bunch lengthening induced by bending magnets. We then emphasize the fact that TLC can actually be actively applied for efficient bunch compression and high harmonic generation when the transverse emittance is small. We present three theorems on the application of such TLC schemes, with their implications discussed. Further, we have analyzed the contribution of modulators to the vertical emittance from quantum excitation, to obtain a self-consistent evaluation of the required modulation laser power when applying these coupling schemes in a storage ring. The theorems and related analysis provide the theoretical basis for the application of TLC in SSMB to lower the requirement on the modulation laser power, by taking advantage of the fact that the vertical emittance in a planar ring is rather small. Based on the investigations, we have presented example parameters sets for the envisioned SSMB storage ring to generate high-power EUV and soft X-ray radiation at the end of this dissertation. The relation between our TLC analysis and the transverse-longitudinal emittance exchange is also briefly discussed. For the nonlinear dynamics, we present the analysis and the first experiment proof of a second-order TLC effect on the equilibrium beam parameters, which can help to improve the stable beam current and coherent radiation power of a ring working in quasi-isochronous regime. Parts of the work presented in this chapter have been published in Refs. [1,2,3,4].
3.1 Linear Transverse-Longitudinal Coupling Dynamics
3.1.1 Passive Bunch Lengthening
In a linear transport line without bending magnets, the transverse and longitudinal motions are decoupled in a first-order approximation. However, the situation changes when there are bending magnets. Particles with different horizontal (vertical) positions and angles will pass through the horizontal (vertical) bending magnets along different paths, resulting in differences in the longitudinal coordinate. The transverse motion can thus be coupled to the longitudinal dimension. When traversing the bending magnets, particles with different energies will also pass along different paths and exit with different horizontal (vertical) positions and angles. The longitudinal motion can thus also be coupled to the transverse dimension. The physical pictures of the linear TLC introduced by the bending magnets are shown in Fig. 3.1. Although this passive TLC is a well-understood effect [5,6,7,8,9], here we present a concise analysis of this effect with an emphasis on its vital role in microbunching formation and transportation for both the transient and steady-state cases.
Linear transverse-longitudinal coupling induced by a bending magnet. Particles with different horizontal positions (a) and angles (b) pass the horizontal bending magnet along different paths, resulting in longitudinal coordinate differences. Particles with different energies (c) also pass the horizontal bending magnet along different paths, resulting in horizontal position and angle differences
We start with a planar x-y uncoupled lattice and assume that the RF cavities are placed at dispersion-free locations. We temporarily ignoring the vertical dimension, and use the state vector \({\textbf {X}}=(x,x',z,\delta )^{T}\). The subscripts \(_{5,\ 6}\) are used for \(z,\ \delta \) for consistency with literature. Hereafter, the subscript \(_x\) in this section is omitted unless necessary. As introduced in Sect. 2.1.1, the betatron coordinate, defined by \({\textbf {X}}_{\beta }={\textbf {B}}{} {\textbf {X}}\), is first used to parametrize the transport matrix in a diagonal form. The transport matrix of \({\textbf {X}}_{\beta }\) from \(s_{1}\) to \(s_{2}\) is then
Following Courant and Snyder [10], we write \({\textbf {M}}_{x_\beta }(s_{2},s_{1})\) as
with
and
where
is the betatron phase advance from \(s_{1}\) to \(s_{2}\). For \({\textbf {M}}_{z_\beta }(s_{2},s_{1})\), the expression is similar to \({\textbf {M}}_{x_\beta }(s_{2},s_{1})\), but note that if we want to calculate the synchrotron phase advance similar to Eq. (3.5), the distance s should be replaced by the effective longitudinal drift space, i.e., \(F=-\tilde{\eta }(s_{2},s_{1})C_{0}\) defined in Eq. (2.50). If there is no RF cavity between \(s_{1}\) and \(s_{2}\), we have
The transport matrix of \({\textbf {X}}\) from \(s_{1}\) to \(s_{2}\) is then
After some straightforward algebra, \({\textbf {M}}(s_{2},s_{1})\) can be expressed as
This matrix can then be used to analyze both the transient and steady-state cases of TLC. Note that for a given lattice, F is a function of the initial dispersion and dispersion angle \(({D}_{1},{D}_{1}')\) at \(s_{1}\), although the transfer map \({\textbf {M}}(s_{2},s_{1})\) for the state vector \({\textbf {X}}\) is not, as the transport matrix is fixed once the lattice is given. This dependence is a result of the matrix parametrization.
We consider first the influence of the betatron oscillation on the longitudinal coordinate. With the help of the Courant-Snyder parametrization, the betatron oscillation position and angle at the starting point \(s_{1}\) can be expressed as
where \( J=\frac{1}{2}\left( \gamma x^{2}+2\alpha xx'+\beta x'^{2}\right) \) is the betatron invariant or action of the particle. The longitudinal coordinate displacement relative to the ideal particle due to the betatron oscillation from \(s_{1}\) to \(s_{2}\) is then
where to obtain the final concise result, D and \({D}'\) have been expressed in terms of the chromatic \(\mathcal {H}\)-function and the chromatic phase \(\chi \), defined as
where \(\mathcal {H}=\gamma {D}^{2}+2\alpha {D}{D}'+\beta {D}'^{2}\). If there is no dipole kick between point 1 and point 2, \(\mathcal {H}\) stays constant and \( \chi _{2}-\chi _{1}=\psi _{2}-\psi _{1}, \) which means \(\Delta z=0\). Physically, this means the contribution of transverse emittance to the bunch length does not change during drifting or experiencing quadrupole kicks, as these manipulations only affect the beam distribution in the transverse phase space. This argument can also be clearly observed in Fig. 2.3c.
From Eq. (3.10), the root-mean-square (RMS) value of the transient bunch lengthening of a longitudinal slice from \(s_{1}\) to \(s_{2}\) caused by this linear TLC can be calculated to be
The RMS bunch lengthening of an electron beam longitudinal slice after n complete revolutions in a ring, due to betatron oscillation, is then
The above equations can be used to explain the dependence of the coherent synchrotron radiation (CSR) repetition rate on the betatron tune in the bunch slicing experiment reported in Ref. [8]. A similar approach can be applied to analyze microbunching preservation with beam deflection, for example in FEL multiplexing. These equations are also useful for evaluating the influence of the coupling effect in the SSMB proof-of-principle experiment [11, 12], which is to be presented in Chap. 5.
If the particle starts with a relative energy deviation of \(\delta \), then
Note that in Eq. (3.14), the betatron invariant and phase should be calculated according to
For a periodic system, if we observe the particle at the same place n periods later, then
where \(2\pi \nu _{x}\) is the horizontal betatron phase advance per period.
We can also obtain the equilibrium second moments in a storage ring by following the Courant-Snyder parametrization one step further. The result is the same with Eq. (2.18) obtained by SLIM. As can be seen from Eqs. (2.18) and (2.19), if there is only passive TLC introduced by bending magnet, the transverse emittance can lengthen the bunch at places where \(\mathcal {H}_{x}\ne 0\),
Similarly the energy spread can broaden the beam width at places where \(D\ne 0\),
To give the readers a more concrete feeling about the bunch lengthening from this passive TLC, we have presented in Fig. 2.3c some calculations based on the MLS lattice. As can be seen, indeed that the coupling from horizontal emittance can contribute significantly, or even dominant the bunch length at places where \(\mathcal {H}_{x}\) is large. This observation will especially be true in an SSMB ring, where the transverse size is much larger than the microbunch length. Therefore, the dispersion and dispersion angle should be controlled in precision at places where ultrashort bunch is desired, for example at the radiator. The bunch lengthening from the transverse emittance will make the current distribution in an SSMB storage ring less sharp and more like a coasting beam as places where \(\mathcal {H}_{x}\ne 0\), as shown in Fig. 3.2. Here we make a remark that this coupling effect may be helpful for suppressing unwanted CSR and may mitigate the intrabeam scattering (IBS) in SSMB or other applications, as extremely short bunches emerge only at dispersion-free locations.
Beam current distributions at places with different \(\mathcal {H}_{x}\). Bunch length in an SSMB ring can easily be dominated by the transverse emittance in places where \(\mathcal {H}_{x}\ne 0\)
3.1.2 Coupling for Harmonic Generation and Bunch Compression
The analysis in the above section may lead us to conclude that TLC always lengthens the bunch and degrades the microbunching. This, however, is not true, as the above analysis is based on the assumption of a planar x-y uncoupled lattice with only the passive coupling induced by the bending magnets. In addition to this passive coupling, an RF cavity (laser modulator in SSMB) placed at a dispersive location, a transverse deflecting RF cavity, etc., are other sources of coupling that can be used for subtle manipulation of particle beam in 6D phase space. In fact, one can take advantage of TLC for efficient harmonic generation or bunch compression when the transverse emittance is small. The reason is that there is some flexibility in tailoring the projection of the three eigen emittances of a beam into different physical dimensions, although their values cannot be changed in a linear symplectic lattice. Here in this and the following sections we investigate the active application of TLC for harmonic generation and bunch compression by taking advantage of the fact that the vertical emittance of an electron beam in a planar x-y uncoupled ring is rather small.
A schematic layout of applying y-z coupling for bunch compression
3.1.2.1 Problem Definition
Let us first define the problem we are trying to solve. We use y-z coupling as an example for the analysis, since we aim to exploit the small vertical emittance. The case of x-z coupling is similar. Suppose the beam at the entrance of the bunch compression section is y-z decoupled, i.e., its second moments matrix is
where \(\alpha \), \(\beta \) and \(\gamma \) are the Courant-Snyder functions [10], the subscript \(_{i}\) means initial, and \(\epsilon _{y}\) and \(\epsilon _{z}\) are the eigen emittances of the beam corresponding to the vertical and longitudinal mode, respectively. For the application of TLC for bunch compression, it means that the final bunch length at the radiator depends only on the vertical emittance \(\epsilon _{y}\) and not on the longitudinal one \(\epsilon _{z}\). The magnet lattices are all planar and x-y decoupled.
The schematic layout of a TLC-based bunch compression section is shown in Fig. 3.3. We divide such a section into three parts, with their transfer matrices given by
with \({\textbf {M}}_{1}\) representing “from entrance to modulator”, \({\textbf {M}}_{2}\) representing “modulation kick” and \({\textbf {M}}_{3}\) representing “modulator to radiator”. Note that \({\textbf {M}}_{1}\) and \({\textbf {M}}_{3}\) are in their general thick-lens form as analyzed in last section. The transfer matrix from the entrance to the radiator is then
From the problem definition, for \(\sigma _{z}(\text {Rad})\) to be independent of \(\epsilon _{z}\), we need
3.1.2.2 Three Theorems on Transverse-Longitudinal Coupling
Given the above problem definition, we have three theorems which dictate the relation between the modulator kick strength with the optical functions at the modulator and radiator, respectively.
Theorem one: If
which corresponds to the case of a normal RF or a TEM00 mode laser modulator, then
Theorem two: If
which corresponds to the case of a transverse deflecting RF or a TEM01 mode laser modulator or other schemes for angular modulation, then
Theorem three: If
whose physical correspondence is not as straightforward as the previous two cases, then
3.1.2.3 Proof
Here we present the details for the proof of Theorem one. The proof of the other two is just similar. From the problem definition, for \(\sigma _{z}(\text {Rad})\) to be independent of \(\epsilon _{z}\), we need
Note that the harmonic generation schemes in FEL like phase-merging enhanced harmonic generation (PEHG) [13, 14] and angular dispersion-induced microbunching (ADM) [15] can be viewed as specific examples of the above general relations [2]. Under the conditions of Eq. (3.29), we have
with
The bunch length squared at the modulator and the radiator are
According to Cauchy-Schwarz inequality, we have
The equality holds when \(\frac{-\left( \beta _{yi}r_{53}-\alpha _{yi}r_{54}\right) }{T_{54}}=\frac{r_{54}}{\left( \beta _{yi}T_{53}-\alpha _{yi}T_{54}\right) }.\) The symplecticity of \({\textbf {T}}\) requires that \({\textbf {T}}{} {\textbf {S}}{} {\textbf {T}}^{T}={\textbf {S}}\), where \({\textbf {S}}=\left( \begin{array}{cc} {\textbf {J}}&{}0\\ 0&{}{\textbf {J}} \end{array} \right) \) and \( {\textbf {J}}=\left( \begin{array}{cc} 0&{}1\\ -1&{}0 \end{array} \right) ,\) so we have
As shown in Eq. (3.31), \({\textbf {E}}=\left( \begin{matrix} 0&{}0\\ h&{}r_{56}h+1 \end{matrix} \right) ,\) then \({\textbf {E}}{} {\textbf {J}}{} {\textbf {E}}^{T}=\left( \begin{matrix} 0&{}0\\ 0&{}0 \end{matrix} \right) \). Therefore,
which means \({\textbf {C}}\) is also a symplectic matrix. So we have
where \(\text {det}({\textbf {C}})\) means the determinant of \({\textbf {C}}\). The theorem is thus proven.
3.1.2.4 Dragt’s Minimum Emittance Theorem
Theorem one in Eq. (3.24) can also be expressed as
Note that in the above formula, \(\sigma _{zy}(\text {Mod})\) means the bunch length at the modulator contributed from the vertical emittance \(\epsilon _{y}\). So given a fixed \(\epsilon _{y}\) and desired \(\sigma _{z}(\text {Rad})\), a smaller \(h(\text {Mod})\), i.e., a smaller RF gradient or modulation laser power (\(P_{\text {laser}}\propto |h(\text {Mod})|^2\)), means a larger \(\mathcal {H}_{y}(\text {Mod})\), thus a longer \(\sigma _{zy}(\text {Mod})\), is needed. As \(|h(\text {Mod})|\sigma _{z}(\text {Mod})\) quantifies the energy spread introduced by the modulation kick, we thus also have
Similarly for Theorem two and three, we have
and
respectively, and also Eq. (3.38). Note that in the above formulas, the vertical beam size or divergence at the modulator contains only the betatron part, i.e., that from the vertical emittance \(\epsilon _{y}\).
Equation (3.38) is actually a manifestation of the classical uncertainty principle [16], which states that
in which \(\epsilon _{\text {min}}\) is the minimum one among the three eigen emittances \(\epsilon _{I,II,III}\). In our bunch compression case, we assume that \(\epsilon _{y}\) is the smaller one compared to \(\epsilon _{z}\). Actually there is a stronger inequality compared to the classical uncertainty principle, i.e., the minimum emittance theorem [16], which states that the projected emittance cannot be smaller than the minimum one among the three eigen emittances,
3.1.3 Normal RF or TEM00 Mode Laser for Coupling
Now we investigate in more detail about the application of TLC in SSMB for bunch compression, using a TEM00 mode laser modulator for the modulation kick. This belongs to the category of Theorem one.
Application of TLC for bunch compression and harmonic generation, using a TEM00 mode laser modulator. Parameters used in this example plot: \(\lambda _{L}=1064\) nm, \(\sigma _{zi}=30\) nm and \(\sigma _{zf}=3\) nm for the case of a pre-microbunched beam, \(\sigma _{\delta i}=3\times 10^{-4}\), \(\sigma _{yi}=2\ \upmu \)m, \(\sigma _{y'i}=1\ \upmu \)rad. The figures show the beam distribution evolution in the longitudinal phase space. Depending on the specific lattice scheme, the different colors in the plot correspond to different particle vertical positions, angles, or combination of them. The modulation waveforms are shown in the figure as the red curves
3.1.3.1 Physical Picture
According to Theorem one, given a vertical emittance \(\epsilon _{y}\) and modulation kick strength h, in principle we can realize as short \(\sigma _{z}(\text {Rad})\) as we want by lengthening \(\sigma _{zy}(\text {Mod})\). In other words, we can lower \(\mathcal {H}_{y}(\text {Rad})\) by increasing \(\mathcal {H}_{y}(\text {Mod})\). However, such great flexibility of a TLC coupling scheme is not obtained without sacrifice. For a premicrobunched beam, and considering that the modulation waveform is actually a nonlinear sinusoidal, a bunch lengthening at the modulator will result in bunching factor degradation at the radiator. Another key point is that the modulator itself will contribute to the vertical emittance through quantum excitation since it is placed at a place where \(\mathcal {H}_{y}\ne 0\). We will elaborate these points more in this section.
To give the readers a better picture before going into the mathematical details, here we summarize in Fig. 3.4 the main information to be presented in this section: (i) Compared to bunch compression or harmonic generation scheme in longitudinal dimension alone like high-gain harmonic generation (HGHG), TLC schemes like PEHG or ADM can reduce the required energy chirp strength, to realize the same bunch length compression ratio or harmonic generation number, when the transverse emittance is small. (ii) This lowering of energy chirp strength is realized through the bunch lengthening from transverse emittance at the modulator, which can degrade the bunching factor at the radiator for a pre-microbunched beam due to the nonlinear nature of the sine modulation. (iii) Addition of the RF or laser harmonics is an effective way to mitigate this bunching factor degradation by broadening the linear zone of the modulation waveform.
3.1.3.2 Bunching Factor
Now we derive the bunching factor degradation at the radiator due to the bunch lengthening at the modulator, using ADM as an example. PEHG has a similar result as we have proven in the last section the general theorem of bunch lengthening in this kind of TLC schemes. Thin-lens kick maps in the last section are again used for the analysis, but now with the fact that the modulation waveform is a nonlinear sine taken into account.
Putting in the optimized bunch compression conditions for ADM, namely \( hR_{56}+1=0\) and \( -d' D+R_{56}=0 \), and using the mathematical identity \( e^{ia\sin (b)}=\sum _{m=-\infty }^{\infty }e^{imb}J_{m}[a], \) the final bunching factor at the n-th laser harmonic in ADM is
For a coasting beam, \(\left\langle e^{-ink_{L}\left[ \left( 1-\frac{m}{n}\right) \left( d' y_{i}+z_{i}\right) \right] }\right\rangle \) will be non-zero only if \(m=n\), where the bracket \(\langle \cdot \cdot \cdot \rangle \) means the average over all the particles. Therefore,
Note that \(\sigma _{z}(\text {Rad})=|D|\sigma _{y'i}=\sqrt{\epsilon _{y}\mathcal {H}_{y}(\text {Rad})}\) in this section follows the definition in the linear matrix analysis of the previous section, and does not represent the real bunch length at the radiator considering the nonlinear modulation waveform. Note also that considering the nonlinear sine modulation waveform, the optimal microbunching condition for a specific harmonic is slightly different from our simplified linear analysis, and \(J_{n}(n)\) in Eq. (3.44) should be replaced by \(J_{n}(-nR_{56}h)\). In the following discussions, we will use the simplified optimal bunch compression conditions, as the main physics is the same.
For a pre-microbunched beam, \(\left\langle e^{-ink_{L}\left[ \left( 1-\frac{m}{n}\right) \left( d' y_{i}+z_{i}\right) \right] }\right\rangle \) will be non-zero for all m, thus
with \(\sigma _{z}(\text {Mod})=\langle d' y_{i}+z_{i}\rangle =\sqrt{\epsilon _{y}\mathcal {H}_{y}(\text {Mod})+\epsilon _{z}\beta _{z}(\text {Mod})}\). Note that the bunch length \(\sigma _{z}(\text {Mod})\) here contains contribution from both \(\epsilon _{y}\) and \(\epsilon _{z}\).
Now we first investigate two limiting cases. If \(\sigma _{z}(\text {Mod})=0\), then we have
This result is the same as that assuming the modulation waveform is linear. Second, if \(\sigma _{z}(\text {Mod})\) is much longer than the modulation laser wavelength, i.e., \(k_{L}\sigma _{z}(\text {Mod})\gg 1\), then the summation terms in Eq. (3.45) will be nonzero only for \(m=n\) and we arrive at the same result as the coasting beam case Eq. (3.44) as expected.
Now we conduct a bit more general discussion. Compared to the linear modulation case, the reduction factor of the bunching factor Eq. (3.45) is
Figure 3.5 shows the flat contour plot for the bunching factor reduction factor \(|R_{n}|\) of Eq. (3.47) as a function of the harmonic number n and the modulation wavelength-normalized bunch length at the modulator \(k_{L}\sigma _{z}(\text {Mod})\). As can be seen from the figure, the bunch lengthening at the modulator indeed degrades the bunching factor at the radiator, due to the nonlinearity nature of sine modulation. The longer this bunch lengthening, the more degradation of the bunching factor. The higher the harmonic number, the more significant the impact is. The limit of \(R_{n}\) with an infinite long \(\sigma _{z}(\text {Mod})\) is \(J_{n}(n)\). Equation (3.47) and Fig. 3.5 is the general result of this bunching factor degradation analysis. We emphasize the fact that the discussion of bunching factor degradation is more relevant for a pre-microbunhed beam, like that in some SSMB scenarios, and is generally not an issue for a coasting beam where the bunch duration is much longer than the modulation wavelength, like that in an FEL.
Flat contour plot for the bunching factor reduction factor \(|R_{n}|\) of Eq. (3.47) as a function of the harmonic number n and the modulation wavelength-normalized bunch length at the modulator \(k_{L}\sigma _{z}(\text {Mod})\)
Left: the bunching factor reduction factor \(|R_{n}|\) of Eq. (3.50) as a function of the harmonic number n for \(k_{L}\sigma _{z}(\text {Mod})=1\), with \(h_{3}=0\) (red) and \(h_{3}=-0.15h_{1}\) (blue), respectively. Right: the bunching factor reduction factor \(|R_{n}|\) of Eq. (3.50) as a function of the modulation wavelength-normalized bunch length at the modulator \(k_{L}\sigma _{z}(\text {Mod})\) for \(n=79\), with \(h_{3}=0\) (red) and \(h_{3}=-0.15h_{1}\) (blue), respectively. \(n=79\) corresponds to the case for example a modulation wavelength of \(\lambda _{L}=1064\) nm and a radiation wavelength of \(\lambda _{R}=\lambda _{L}/79=13.5\) nm
As the decrease of bunching factor originates from the nonlinearity of the sine modulation, we expect that this reduction will be less if we make the modulation waveform more like linear, for example by adding a third-harmonic RF or laser to broaden the linear zone of the modulation waveform, as also suggested before in Refs. [17, 18]. The energy modulation then becomes \( \delta =\delta +\frac{h_{1}}{k_{L}}\sin (k_{L}z)+\frac{h_{3}}{3k_{L}}\sin (3k_{L}z). \) The optimized bunch compression conditions for ADM are now \( (h_{1}+h_{3})R_{56}+1=0 \) and \(-d' D+R_{56}=0.\) The n-th laser harmonic bunching factor at the radiator is then
for a coasting beam, and
for a pre-microbunched beam. Therefore, the reduction factor of the bunching factor Eq. (3.49), compared to the linear modulation case, is now
The limit of \(R_{n}\) with an infinite long \(\sigma _{z}(\text {Mod})\) is \(\sum _{m_{1}+3m_{3}=n}J_{m_{1}}\left( \frac{h_{1}}{h_{1}+h_{3}}n\right) J_{m_{3}}\left( \frac{h_{3}}{h_{1}+h_{3}}\frac{n}{3}\right) \). It is straightforward to generalize the above derivation and result to the case of adding more laser harmonics.
Now we can use the above formula of \(R_{n}\) to do comparison between the cases with and without the third-harmonic laser. If \(h_{3}=0\), then Eq. (3.50) reduces to Eq. (3.47). As can be seen in Fig. 3.6, indeed addition of a third-harmonic laser is effective in mitigating the bunching factor degradation arising from the bunch lengthening at the modulator.
3.1.3.3 Contribution of Modulators to Vertical Emittance and Scaling of Required Modulation Laser Power
We have stated that the main motivation of applying TLC scheme for bunch compression in SSMB is to lower the requirement on the modulation laser power \(P_{L}\). This is based on the fact that the vertical emittance \(\epsilon _{y}\) in a planar x-y uncoupled ring is rather small. However, since the modulator in this TLC scheme is placed at a dispersive location, i.e., \(\mathcal {H}_{y}(\text {Mod})\ne 0\), therefore quantum excitation at the modulator will also contribute to \(\epsilon _{y}\). With this consideration taken into account, below we try to give a self-consistent analysis of the required modulation laser power \(P_{L}\) in these TLC schemes.
To make sure that the TLC-based bunch compression can repeat turn-by-tun in a ring, usually two laser modulators are placed upstream and downstream of the radiator, respectively, to form a pair. The lattice scheme between these two modulators can either be a symmetric one, or a reversible seeding one. In both cases, the chromatic function \(\mathcal {H}_{y}\) at two modulators are identical. Assuming the modulator undulator is planar, the contribution of these two modulators to \(\epsilon _{y}\) is then
with the vertical damping constant
where \(C_{\gamma }=8.85\times 10^{-5}\frac{\text {m}}{\text {GeV}^{3}}\), \(\rho _{\text {ring}}\) is the bending radius of dipoles in the ring, and \(\rho _{0\text {Mod}}\) is the minimum being radius corresponding to the peak magnetic flux density \(B_{0\text {Mod}}\) of the modulator. Note that in the above analysis we have ignored the contribution of the dispersive lattice sections upstream and downstream of the modulators to the vertical emittance, as in principle we can minimize their contribution by choosing weak bending magnets in them. On the other hand, we cannot choose as weak modulator as we want since it will also affect the energy modulation efficiency. This is the reason why the contribution of modulators to the vertical emittance is of more fundamental importance.
The resonant condition of the laser-electron interaction inside a planar undulator is
with \(K=\frac{eB_{0}\lambda _{u}}{2\pi m_{e}c}=0.934\cdot B_{0}[\text {T}]\cdot \lambda _{u}[\text {cm}]\) the dimensionless undulator parameter. The effective modulation voltage of a laser modulator using a planar undulator is related to the laser and undulator parameters according to [19]
in which \([JJ]=J_{0}(\chi )-J_{1}(\chi )\) and \(\chi =\frac{K^{2}}{4+2K^{2}}\), \(P_{L}\) is the modulation laser power, \(Z_{0}=376.73\ \Omega \) is the impedance of free space, \(Z_{R}\) is the Rayleigh length of the modulation laser, \(L_{u}\) is the undulator length. The linear energy chirp strength around zero-crossing phase is therefore
where \(k_{L}=2\pi /\lambda _{L}\) is the wavenumber of the modulation laser.
For simplicity, we set \(\epsilon _{y}=\Delta \epsilon _{y}(\text {Mod, QE})\), i.e., the vertical emittance is purely from the contribution of these two modulators, and assuming that equality holds in Theorem one, then the required modulation laser power is
Now we try to derive more useful scaling laws to offer guidance in our parameters choice for a TLC SSMB ring. To maximize the energy modulation, we need \(\frac{Z_{R}}{L_{u}}=0.359\approx \frac{1}{3}\). When \(K>\sqrt{2}\), we approximate the resonance condition as \( \lambda _{L}\approx \frac{K^{2}}{4\gamma ^{2}}\lambda _{u}, \) and \([JJ]\approx 0.7\). Then we have
The corresponding modulator length scaling is
Putting in the numbers for the constants, we obtain the quantitative expressions of the above scalings for practical use
The above scaling laws are accurate when \(K>\sqrt{2}\).
Note that \(\epsilon _{y}\) does not appear explicitly in the scaling of the required laser power. It however affects the bunch length at the modulator and therefore the bunching factor at the radiator as we have explained. Also it affects the required modulator length. In other words, to obtain a desired bunching factor, the smaller \(\epsilon _{y}\) is, the larger \(\sigma _{z}(\text {Rad})\) we can use, thus a lower modulation laser power. Generally a shorter modulation laser wavelength and lower beam energy is preferred in lowering the required laser power. But we need to keep in mind that when the beam energy is too low, intrabeam scattering (IBS) could blow up \(\epsilon _{y}\) [20, 21]. From the scaling, a weaker \(B_{0\text {Mod}}\) means a smaller modulation laser power will be needed. But we should be aware that the corresponding length of modulator \(L_{u}\propto \frac{1}{B^{3}_{0\text {Mod}}}\). In Table 6.2 of the final chapter, we have presented an example parameters set of a TLC SSMB storage ring for high-power EUV and soft X-ray radiation generation, based on the investigations presented here.
3.1.4 Transverse Deflecting RF or TEM01 Mode Laser for Coupling
Now we investigate in more detail about the application of TLC in SSMB for bunch compression, using a TEM01 mode laser modulator for the modulation kick. This belongs to the category of Theorem two.
3.1.4.1 TEM01 Mode Laser Modulator for Bunch Compression
A laser modulator implementing a TEM01 mode laser is like a transverse deflecting RF cavity in the optical wavelength range. The electric field of a Hermite-Gaussian TEM01 mode laser polarized in the vertical direction is [19]
with \(Q=\frac{i}{Z_{R}\left( 1+i\frac{z}{Z_{R}}\right) }\). The relation between \(E_{y0}\) and the laser peak power for a TEM01 mode laser is given by
Note there is a factor of two difference in the above laser power formula compared to the case of a TEM00 mode laser. The electron wiggles in a vertical planar undulator according to
and the laser-electron exchanges energy according to
Assuming that the laser beam waist is in the middle of the undulator, and when \(x,y\ll w(z)\), which is typically the case in SSMB, we drop \(\text {exp}\left[ i\frac{k_{\text {L}}Q}{2}(x^2+y^2)\right] \) in the laser electric field. Since usually \(Z_{R}\gg \lambda _{u}\), we can also drop the contribution from \(E_{z}\) on energy modulation. Then the integrated modulation voltage induced by the laser in the undulator is
The linear energy chirp with respect to y introduced is then
Note that the symplecticity of the dynamics requires that the vertical angle of the particle after modulation will depend on its initial longitudinal location. This observation is also supported by the Panofsky-Wenzel theorem [22]
where \(\Delta y'\) and \(\Delta \gamma \) are the electron angular kick and energy change in the laser modulator. It is interesting to note that the modulation kick strength depends on the ratio between \(Z_{R}\) and \(L_{u}\), instead their absolute values, and the maximal modulation is realized when \(Z_{R}=\frac{L_{u}}{2}\).
Now we can do some evaluation based on the formulas. For an example choice of parameters, \(E_{0}=400\) MeV, \(\lambda _{L}=270\) nm, \(K[JJ]=5\), \(P_{L}=1\) MW, \(Z_{R}=\frac{L_{u}}{2}\), we have \(t=4\ \text {m}^{-1}\). If \(\epsilon _{y}=1\) pm and the desired bunch length is 3 nm at the radiator, i.e., \(\sigma _{z}(\text {Rad})=3\) nm, then according to Theorem two we have \( \beta _{y}(\text {Mod})\ge \frac{\epsilon _{y}}{t^{2}(\text {Mod})\sigma _{z}^{2}(\text {Rad})}=6.9\times 10^{3}\ \text {m}. \) The application of TLC with a TEM01 mode laser to compress the bunch length therefore faces the issue of a too large \(\beta _{y}\) at the modulator, if the desired bunch length is at nm level, which is needed for example in 13.5 nm coherent EUV radiation generation. However, if our target wavelength region is \(\lambda _{R}\gtrsim 100\) nm, then the idea looks appealing as the required \(\beta _{y}\) is within the reasonable reach then. We remind the readers that there could be other more effective angular modulation scheme invented such that the issue of large \(\beta _{y}\) at the modulator can be solved, even if our target radiation wavelength is still in EUV.
3.1.4.2 Contribution of Modulators to Vertical Emittance
As said the advantage of TLC for bunch compression is based on a small vertical emittance. Like before let us now investigate the contribution of modulators to \(\epsilon _{y}\). We remind the readers that for bunch compression using a TEM01 mode laser, in principle, we can place the modulator at a dispersion-free location. Note however, if we aim at a complete y-z emittance exchange, the modulator needs to be placed at a dispersive location as will be discussed in next section. Here to minimize the contribution of modulators to \(\epsilon _{y}\), we choose to place the modulator at dispersion-free location, which means \(d=0\) and \(d'=0\), then the bunch compression condition is
Although we have placed the modulator at a dispersion-free location, there is still some residual contribution to \(\epsilon _{y}\) since the transfer matrix of a TEM01 mode laser modulator is intrinsically transverse-longitudinal coupled, and the physical length \(L_{u}\) and \(r_{56}=2N_{u}\lambda _{L}\) of the modulator are nonzero. The thick-lens transfer matrix of the laser modulator can be obtained by slicing the laser modulator to tiny slices and use the thin-lens kick and drift method to get the total map. If we consider only terms to first order of t, \(r_{56}\) and \(L_{u}\), then the thick-lens matrix of the TEM01 laser modulator is
Note that \({\textbf {M}}_{2}\) is symplectic to first order of t. The transfer matrix of the state vector from the modulator entrance to a distance of s in it is
Here for simplicity we have assumed that the laser is a plane wave such that the induced angular modulation strength is proportional to the distance traveled inside the modulator.
Assuming that the one-turn map observed at the entrance of modulator is
in which \(\Phi _{y}=2\pi \nu _{y}\) and \(\Phi _{z}=2\pi \nu _{s}\). The eigenvector of the one-turn map corresponding to the vertical mode at the position s inside the modulator is then
Note that to ensure the TLC-based bunch compression can repeat turn-by-tun in a ring, usually two laser modulators are placed upstream and downstream of the radiator, respectively, to form a pair. According to Chao’s SLIM formalism [23], we can calculate the contribution of the two modulators to \(\epsilon _{y}\)
When \(N_{u}\gg 1\), due to the fast oscillating behaviour of \(\sin \left[ 2N_{u}\pi \left( \frac{s}{L_{u}}\right) \right] \), we can adopt the approximation
For simplicity, we assume \(\alpha _{y}=0\) at the modulator entrance, and usually \(\beta _{y}(\text {Mod})\gg L_{u}\), then
Now let us put in some numbers to get a more concrete feeling. For example, if \(E_{0}=400\) MeV, \(\rho _{\text {ring}}=1\) m (\(B_{\text {ring}}=1.33\) T), \(\lambda _{L}=270\) nm, \(\lambda _{u}=4\) cm, \(K=3.8\), \(B_{0}=1.02\) T, \(N_{u}=10\), \(r_{56}=2N_{u}\lambda _{L}=5.4\ \upmu \)m, \(L_{u}=0.4\) m, \(\epsilon _{y}=1\) pm, \(\beta _{y}(\text {Mod})=100\) m, \(\sigma _{y}(\text {Mod})=\sqrt{\epsilon _{y}\beta _y(\text {Mod})}=10\ \upmu \)m, \(\sigma _{z}(\text {Rad})=2\) nm, \(t=\frac{1}{\sqrt{\beta _{y}(\text {Mod})\mathcal {H}_{y}(\text {Rad})}}=\frac{\epsilon _{y}}{\sigma _{y}(\text {Mod})\sigma _{z}(\text {Rad})}=50\ \text {m}^{-1}\), then the contribution of the two modulators to \(\epsilon _{y}\) is \( \Delta \epsilon _{y}(\text {Mod, QE}) \approx 2.06\ \text {fm}. \) So generally, the contribution of the two modulators to \(\epsilon _{y}\) is a small value, if the modulators are placed at dispersion-free locations.
3.1.5 Emittance Exchange
3.1.5.1 Lattice Condition
For completeness of the investigation, it might also be helpful to make a short discussion on the relation between our TLC analysis and the transverse-longitudinal emittance exchange (EEX). For a complete EEX, we need the transfer matrix of the form
Therefore, EEX is a special case in the context of our problem definition of TLC-based bunch compression, i.e., in EEX the final beam is also y-z decoupled. As can be seen from Eq. (3.31), the application of a normal RF or TEM00 mode laser modulator cannot accomplish a complete EEX, as \(T_{65}=h\ne 0\). PEHG and ADM can thus be viewed as partial EEXs. In contrast, a transverse deflecting RF or TEM01 mode laser modulator can be used to obtain a complete EEX. All we need is to add another condition to Eq. (3.22), i.e.,
After some straightforward algebra, the relations in Eqs. (3.22) and (3.76) can be summarized in an elegant form as follows [24]
Note that the above relations mean that the lattices upstream and downstream the transverse deflecting RF are not mirror symmetric with respect to each other [25]. Under the conditions given in Eq. (3.77), we have
3.1.5.2 Two EEXs as an Insertion
In order to apply EEX to generate short bunch in a storage ring on a turn-by-turn basis, another EEX might be needed following the radiator to swap back the \(\epsilon _{y}\) and \(\epsilon _{z}\) to maintain the ultrasmall vertical emittance \(\epsilon _{y}\). If there is only one transverse-longitudinal EEX, the ring will then be a transverse-longitudinal Möbius accelerator [26], which is also an interesting topic we are not going into in this dissertation.
Application of two transverse-longitudinal emittance exchangers to manipulate the bunch length in a storage ring
Now we consider the application of two y-z EEXs for bunch length manipulation in a storage ring as shown in Fig. 3.7. The motivation is still to make use of the fact that the vertical emittance \(\epsilon _{y}\) is rather small in a planar x-y uncoupled ring. The first natural idea is to add an inverse EEX unit following the EEX,
then the total insertion will be an identity matrix and be transparent to the ring. The issue of this approach, however, is that we need to design the downstream beamline with an \(R_{56}\) having opposite sign to the upstream beamline, which might be a challenging task if we aim at a compact lattice.
The second natural idea is to implement the mirror symmetry of the upstream beamline as the downstream beamline, which is straightforward for the lattice design. The transfer matrix of the mirror image is related to that of the original beamline according to [27, 28]
Therefore,
Note, however, the transfer matrix of the total insertion in this case is generally not an identity matrix,
with
A special case of EEX is the phase space exchange (PSX), i.e., the exchange happens in the phase space variables apart from a magnification factor. In this case, a PSX followed by its mirror can form an identity or a negative identity matrix.
Case one:
Case two:
According to Eq. (3.78), for case one, we need
and
For case two, we need
and
3.2 Nonlinear Transverse-Longitudinal Coupling Dynamics
3.2.1 Average Path Length Dependence on Betatron Amplitudes
After investigating the linear TLC, we will now examine nonlinear coupling. However, here we consider only the second-order path lengthening or shortening from betatron oscillations, and its impact on equilibrium beam parameters. A general discussion of the nonlinear dynamics is beyond the scope of this dissertation. The second-order TLC considered here originates from a dependence of the average path length on the betatron oscillation amplitudes, which can be expressed by a concise formula
where \(\Delta C\) is the average path-length deviation relative to the ideal particle, and \(\xi _{x,y}=\frac{d\nu _{x,y}}{d\delta }\) and \(J_{x,y}\) are the horizontal (vertical) chromaticity and betatron invariant, respectively. This simple relation is a result of the symplecticity of the Hamiltonian dynamics [29,30,31]. It is called a second-order coupling because the betatron invariant is a second-order term with respect to the transverse position and angle. Note that Eq. (3.90) is accurate only for the cases of multiple passes or multiple betatron oscillation periods as it is a betatron-phase-averaged result. For the case of a single pass with only a few betatron oscillations, there will be an extra term, depending on the betatron phase advance, in the path length formula.
This path length effect has previously been theoretically analyzed by several authors in different contexts [29,30,31,32,33,34]. Due to this effect, particles with different betatron amplitudes lose synchronization with each other when traversing a lattice with nonzero chromaticity. This leads to a stringent requirement on the beam emittance for FELs in the X-ray regime (XFELs), as microbunching can be smeared out by this effect when the beam is traveling through the undulator [35]. This effect is also crucial in non-scaling fixed-field alternating-gradient (FFAG) accelerators for muon acceleration [31], as a muon beam typically has a large emittance. Furthermore, the natural chromaticities of a linear non-scaling FFAG accelerator are usually not corrected to achieve a large transverse acceptance. This effect may also have an impact on the momentum and dynamic aperture in a storage ring [36, 37], for example, due to the Touschek scattering-induced large betatron amplitude or the large natural chromaticity in a low-emittance lattice. In this dissertation, we will emphasize the importance of this nonlinear TLC effect in precision longitudinal dynamics in a storage ring, such as SSMB.
3.2.2 Energy Widening and Distortion
As mentioned, this second-order TLC effect can disperse microbunching in XFELs. Methods of overcoming this influence are referred to as “beam conditioning”. Several such methods have been proposed since the first publication of Ref. [35]. The basic idea of these proposals is to compensate the difference in path length through a difference in velocity by establishing a correlation between the betatron amplitude and the particle energy. In a storage ring, unlike in a single-pass device, the RF cavity will “condition” the beam automatically, causing all particles to synchronize with it in an average sense through phase stabilization (bunching). This is accomplished by introducing a betatron-amplitude-dependent energy shift to compensate for the path-length difference arising from the betatron oscillations,
where \(\alpha \) is the momentum compaction factor of the ring, defined in Eq. (2.1). This shift will result in the beam energy widening in a quasi-isochronous ring with nonzero chromaticity, because different particles have different betatron invariants [34]. This widening will become more significant with the decreasing of the momentum compaction.
Due to the energy shift, there will also be an amplitude-dependent shift in the betatron oscillation center at dispersive locations. The shift direction is determined by the signs of \(\alpha \), \(\xi _{x,y}\) and \(D_{x,y}\), and the magnitude of the shift is determined by the magnitudes of \(J_{x,y}\), \(\alpha \), \(\xi _{x,y}\) and \(D_{x,y}\). The physical pictures of the betatron center shift resulting from this effect are shown in Fig. 3.8.
Physical picture of the amplitude-dependent shift of betatron oscillation center in the case of a positive momentum compaction. Only a horizontal betatron oscillation is considered in this illustration
When quantum excitation is also taken into account, the total relative energy deviation of a particle with respect to the ideal particle is
where \(\delta _{\text {qe}}\) represents the quantum excitation contribution. Finding a general analytical formula for the steady-state distribution of the particles is a complex task and, at the same time, not very useful. Simpler expressions can be obtained based on reasonable approximations. Since the vertical emittance is usually much smaller than the horizontal emittance in a planar uncoupled ring, here, we consider only the contribution from the horizontal emittance. When the coupling is not very strong, the distributions of \(J_{x}\) and \(\delta _{\text {qe}}\) are still approximately exponential and Gaussian, respectively, and are independent of each other,
where \(\epsilon _{x0}\) and \(\sigma _{\delta 0}\) are the natural horizontal emittance and energy spread. The distribution of \(\delta \) is thus an exponentially modified Gaussian because it is the sum of an exponential and a normal random variable,
where \(\lambda =\frac{\alpha C_{0}}{2\pi \xi _{x}\epsilon _{x0}}\), sgn(x) is the sign function and erfc(x) is the complementary error function, defined as \( \text {erfc}(x)=1-\text {erf}(x)=\frac{2}{\sqrt{\pi }}\int _{x}^{\infty }e^{-t^{2}}dt. \) The direction of long non-Gaussian tail of the energy distribution is determined by the signs of \(\alpha \) and \(\xi _{x}\).
Because of the dispersion and dispersion angle, the non-Gaussian particle energy distribution can also be reflected in the transverse dimension. When this nonlinear coupling is considered, the horizontal position and angle of a particle in the storage ring are
It is assumed that the concept of the Courant-Snyder functions is still approximately valid in Eq. (3.95).
With these approximations, the variance of \(\delta \) is then
By assuming the MLS parameters shown in Table 3.1 and applying \(\alpha =1\times 10^{-4}\) and \(\xi _{x}=2\), one can find that the energy spread contributed by this effect can be as significant as its natural value.
As discussed in Ref. [34], a shift in the energy center corresponds to a shift in the synchronous RF phase \(\phi _{s}\),
where \(J_{s}\) is the longitudinal damping partition number and nominally \(J_{s}\approx 2\). Therefore, particles with different betatron amplitudes will oscillate around different fixed points in the longitudinal dimension, thus lengthening the bunch. The change in the synchronous RF phase in a unit of the longitudinal coordinate, \(\Delta z_{s}\), is related to the relative change in energy, \(\Delta \delta \), according to
where \(\nu _{s}\) is the synchrotron tune, \(f_{\text {rev}}\) is the particle revolution frequency in the ring, \(\sigma _{z0}\) and \(\sigma _{\delta 0}\) are the natural bunch length and energy spread, respectively.
Energy widening, bunch lengthening and distortion from a Gaussian distribution induced by a nonvanishing horizontal chromaticity. From up to bottom, the particle tracking results for distributions of x, \(x'\), z and \(\delta \) are shown at two dispersive locations in the MLS under three different horizontal chromaticities \(\xi _{x}\). The direction of the long non-Gaussian tail for \(\delta \) is related to the signs of \(\alpha \) and \(\xi _{x}\), while for x and \(x'\) they are also dependent on \(D_{x}\) and \(D_{x}'\), respectively. The simulation was conducted using the code ELEGANT [38] with a beam energy of 630 MeV, an RF voltage of 500 kV and the application of \(\alpha =1\times 10^{-4}\). In each simulation, eight particles were tracked for \(5\times 10^{6}\) turns, corresponding to approximately 73 longitudinal radiation damping times
The critical value of alpha, \(\alpha _{\text {c}}\), when the relative change of bunch length and energy spread are the same can be calculated to be
where \(\tau _{\delta }=1/\alpha _{L}=\frac{2E_{0}}{J_{s}U_{0}}T_{0}\) is the longitudinal radiation damping time. As an example, we use the MLS parameters given in Table 3.1 and consider the application of an RF voltage of 500 kV, which corresponds to a synchronous RF phase of \(\phi _{s}=0.018\) rad. The critical value of alpha is then \(|\alpha _{\text {c}}|\approx 2.1\times 10^{-9}\), which is about four orders of magnitude smaller than the alpha value reachable at the present MLS. Therefore, the relative bunch lengthening resulting from this effect is much less significant than the corresponding energy widening at the MLS.
Several particle tracking simulations were conducted using the MLS lattice with the parameters presented in Table 3.1 to confirm the analysis. Two dispersive locations, with different signs and magnitudes of \(D_{x}\), were selected as the observation points in the simulations. The simulation results are shown in Fig. 3.9. The energy widening and distortion from Gaussian behaviors are as expected and, indeed, are more significant than the bunch lengthening when \(\alpha =1\times 10^{-4}\). At the two observation points, widening and distortion of the particle energy distribution also manifest in the transverse dimension through \(D_{x}\) and \(D_{x}'\). The related optic functions at the two observation points are also shown in the profiles of x and \(x'\). Note that the directions of the long non-Gaussian tails of the profiles and their relations to the signs of \(\xi _{x}\), \(D_{x}\) and \(D_{x}'\). We conclude that the simulation results agree well with the analysis and physical pictures presented above.
3.2.3 Experimental Verification
Here we report the first experimental verification of the energy widening and particle distribution distortion from Gaussian due to this second-order TLC effect as analyzed above. At the MLS, the Compton-backscattering (CBS) method is applied to measure the electron energy [39]. Nevertheless, within certain limitations, the electron beam energy spread can also be evaluated from the CBS photon spectra [40]. The non-Gaussian momentum distribution makes the evaluation a bit more involved, but we can assume a Gaussian distribution with an equivalent mean energy spread. This is a good approximation as long as \(\xi _{x}\) is not too large.
The experiment is conducted with all 80 RF buckets equally filled. To exclude a severe impact from energy widening collective effects, the beam current is decayed till the horizontal beam size is not sensitively dependent on it. The average single-bunch current is below 12.5 \(\upmu \)A (1 electron/1 pA) while doing the CBS measurements. To mitigate the influence of a nonlinear momentum compaction, the longitudinal chromaticity has been corrected close to zero. The other parameters of the ring in the experiment are presented in Table 3.1.
Measurements of the CBS photons spectra at the 344.28 keV (a) and 778.90 keV (c) emission lines of \(^{152}\)Eu radionuclide to calibrate the channel numbers in terms of keV and the HPGe-detector resolution at the CBS cutoff edge (b) in the CBS method of measuring beam energy spread
To get the energy spread based on the CBS method with precision, the HPGe-detector used in the measurement should be calibrated in terms of the photon energy per channel. This is realized by recording the emission lines from a \(^{152}\)Eu radionuclide simultaneously during the measurement of the CBS photons. Moreover, the width of the fitted \(^{152}\)Eu lines that are close to the edge of the CBS photons have been used to determine the detector resolution \(\sigma _{\text {det}}\) at the photon energy of the CBS cutoff edge, \(E_{\text {edge}}\), in our case 707 keV. This is done by a linear interpolation of the width of the \(^{152}\)Eu lines at 344.28 and 778.90 keV. The detector resolution \(\sigma _{\text {det}}\) at 707 keV is thus determined to be 0.64(4) keV. The calibration scheme and result is shown in Fig. 3.10.
Figure 3.11a shows the typical CBS photon spectra close to the cutoff edge under the cases of different \(\xi _{x}\) and \(\alpha \). The adjustment of \(\xi _{x}\) is accomplished by the implementation of different chromatic sextupole strengths and the \(\alpha \) by slightly tuning quadrupole strengths. In the experiment the RF voltage is kept constant and the synchrotron frequency \(f_{s}\) is proportional to the square root of the magnitude of \(\alpha \). The edge in the figure is a convolution of a step function representing the CBS cutoff edge with a Gaussian function which attributes to the finite HPGe-detector energy resolution and the electron beam energy spread. The fitted line is basically an error function from which the energy width of the CBS photons at the edge \(\sigma _{\text {edge}}\), and therefore the electron beam energy spread \(\sigma _{\delta }\), can be deduced. It is assumed \(\sigma _{\text {edge}}\) is given by \(\sigma _{\text {edge}}=\sqrt{\sigma _{\text {det}}^2+(2E_{\text {edge}}\sigma _{\delta })^2}\). The second term in the square root is due to the electron beam energy spread and is based on the fact that the energy of the backscattered photon is proportional to the electron energy squared.
Measurement of the electron beam energy widening brought by the horizontal chromaticity using the CBS method. a The cutoff edges of CBS photon spectra under different \(\xi _{x}\) and \(f_{s}\) (therefore \(\alpha \)). b Quantitative evaluation of the cut-off edges revealing the energy spread and its comparison with theory Eq. (3.96). The error bars in both figures are the one sigma uncertainties of the measurements and are due to calibration errors and counting statistics. The data acquisition of each spectrum takes 15 min
It can be seen from Fig. 3.11a that the edge slope decreases with the magnitude increasing of \(\xi _{x}\) and lowering of \(\alpha \) when \(\xi _{x}\ne 0\), which indicates that there is an energy widening in the process. Quantitative evaluation of the edges revealing the energy spread and its comparison with theory of Eq. (3.96) are shown in Fig. 3.11b. The energy spread grows significantly with the magnitude decrease of \(\alpha \) when \(\xi _{x}=-3.14\) while it stays almost constant in the case of \(\xi _{x}=0\). The agreement between measurements and theory is quite satisfactory. This is the first direct experimental proof of the impact of this effect on the equilibrium beam parameters in a storage ring.
As analyzed before, the bunch lengthening due to this second-order coupling is much less notable and also due to the limited resolution of the present streak camera, we do not measure the bunch lengthening in the experiment. Nevertheless, a more comprehensive investigation of this effect can be conducted on the other beam characteristics like the transverse intensity distribution. As can be seen from Eq. (3.95), particles with different betatron amplitudes oscillate around different closed orbits, which is the amplitude dependent center shift [41]. Because of the dispersion, the non-Gaussian particle momentum distribution can also be reflected to the transverse dimension, which can be observed by the beam imaging systems installed at the MLS [42].
Figure 3.12 shows the typical transverse beam intensity distribution measured by the imaging systems at two dispersive locations, QPD0 and QPD1, with different values of \(\xi _{x}\) in both the negative and positive momentum compaction modes. The relevant optics functions, \(\beta _{x}\) and \(D_{x}\), at the two observation points are also shown in the figure. Note that \(D_{x}\) have different signs and magnitudes at QPD0 and QPD1. It can be seen that the horizontal beam distribution at these dispersive locations becomes asymmetric when \(\xi _{x}\ne 0\). The long tail direction and the magnitude of deviation from Gaussian are determined by the signs and magnitudes of \(\alpha \), \(D_{x}\), \(\xi _{x}\) and also the value of \(\epsilon _{x0}\) and \(\beta _{x}\), which fits with the expectations.
Transverse beam intensity distortion from Gaussian at dispersive locations due to a non-vanishing horizontal chromaticity measured by the imaging systems installed at QPD0 and QPD1. Three different \(\xi _{x}\) are applied in both the positive and negative momentum compaction modes with \(\alpha =\pm 8.4\times 10^{-5}\). There is some residual horizontal-vertical coupling in the positive momentum compaction case, which do not influence the principle observation of the non-Gaussian behavior
Horizontal beam profile distortion from Gaussian by horizontal chromaticty. a Typical horizontal beam profile at QPD1 with \(\alpha =-7\times 10^{-5}\) under three different \(\xi _{x}\). The closed orbit movements of the ideal particle due to the sextupole strengths changes when adjusting \(\xi _{x}\) have been compensated in the plot. Cross: beam imaging system measurement results. Dashed line: fit of the measurement data by an exponentially modified Gaussian function Eq. (3.100). Solid line: theoretical prediction. b Measured and theoretical asymmetry parameter d versus \(\xi _{x}\) at QPD0 and QPD1 with \(f_{s}=5\) kHz (\(\alpha =-7\times 10^{-5}\)); c Measured and theoretical asymmetry parameter d versus \(f_{s}\) at QPD0 and QPD1 with \(\xi _{x}=1.4\). All the theoretical curves are obtained based on Eqs. (3.93) and (3.95)
Figure 3.13a demonstrates the typical horizontal beam profiles measured at QPD1 in the negative momentum compaction mode under three different \(\xi _{x}\) and their good agreements with theory. It turns out that both the theoretical and experimental measured horizontal coordinate distribution \(\psi (x)\) can be excellently fitted by a skewed Gaussian function
The asymmetry parameter d in Eq. (3.100) is used to quantitatively describe the deviation from Gaussian and as a criterion to do comparison between measurements and theory. Figure 3.13b and c show the asymmetry parameter d versus \(\xi _{x}\) and \(f_{s}\), therefore \(\alpha \), from measurements and theory at QPD0 and QPD1. It can be seen that the larger the \(\xi _{x}\) and the smaller the \(\alpha \), the more asymmetric the distribution is. Also the asymmetry at QPD1 is more significant than that at QPD0 as the magnitude of \(D_{x}\) at QPD1 is larger while the \(\beta _{x}\) difference at two places is not much. The agreement between measurements and theory confirms that this effect distorts the beam from Gaussian in both the longitudinal and transverse dimensions.
While the energy widening and beam distortion could be a detrimental outcome for some applications, it may actually also be beneficial as it can help to stabilize collective instabilities. The bunch lengthening on the other hand is much less notable compared to the energy widening. So quasi-isochronous ring-based coherent radiation schemes, like some of the SSMB scenarios, may boost the stable coherent radiation power by taking advantage of this effect. For example, the stable single-bunch current at the MLS can grow for more than one order of magnitude by increasing the absolute value of the horizontal chromaticity from zero to a value larger than three, with the head-tail and the other collective effects like the longitudinal microwave instability properly suppressed. It has been proved at the MLS that the increase of THz power due to a higher stable beam current overcompensates the decrease due to the slight bunch lengthening of this effect. Therefore, this is now the standard low momentum compaction mode at the MLS for the application of Fourier Transform Spectroscopy.
This nonlinear TLC may also be useful in some more applications. For example, it can be used for the real-time emittance evaluation in storage rings if the chromaticities, beta function and dispersion are known, which are usually easier to get than measuring the emittance directly. The amplitude dependent center shift can be applied to detect beam instabilities which blow up the transverse emittance [43]. A strongly asymmetric particle momentum distribution due to this effect cooperating with a large momentum compaction lattice can generate a strongly asymmetric distributed current, which is favored in some applications such as beam-driven wakefield acceleration [44].
References
Deng XJ, Chao AW, Feikes J, Huang WH, Ries M, Tang CX (2020) Single-particle dynamics of microbunching. Phys Rev Accel Beams 23:044002
Deng X, Huang W, Li Z, Tang C (2021) Harmonic generation and bunch compression based on transverse-longitudinal coupling. Nucl Instrum Methods Phys Res A 1019:165859
Li Z, Deng X, Pan Z, Tang C, Chao A (2023) A generalized longitudinal strong focusing storage ring, Generalized longitudinal strong focusing in a steady-state microbunching storage ring, submitted
Deng XJ, Klein R, Chao AW, Hoehl A, Huang WH, Li J, Lubeck J, Petenev Y, Ries M, Seiler I, Tang CX, Feikes J (2020) Widening and distortion of the particle energy distribution by chromaticity in quasi-isochronous rings. Phys Rev Accel Beams 23:044001
Shoji Y (2004) Bunch lengthening by a betatron motion in quasi-isochronous storage rings. Phys Rev ST Accel Beams 7:090703
Wüstefeld G (2005) Horizontal-longitudinal coupling in the fel beam line, Technical Report, BESSY TB 227/05
Huang X (2007) Matrix formalism of synchrobetatron coupling. Phys Rev ST Accel Beams 10:014002
Shimada M, Katoh M, Adachi M, Tanikawa T, Kimura S, Hosaka M, Yamamoto N, Takashima Y, Takahashi T (2009) Transverse-longitudinal coupling effect in laser bunch slicing. Phys Rev Lett 103:144802
Shoji Y (2011) Transient bunch lengthening by a betatron motion along bending sections. Nucl Instrum Methods Phys Res, Sect A 637(1):S120–S122
Courant ED, Snyder HS (1958) Theory of the alternating-gradient synchrotron. Ann Phys (NY) 3(1):1–48
Deng X, Chao A, Feikes J, Hoehl A, Huang W, Klein R, Kruschinski A, Li J, Matveenko A, Petenev Y et al (2021) Experimental demonstration of the mechanism of steady-state microbunching. Nature 590(7847):576–579
Kruschinski A, Feikes J, Li J, Ries M, Deng X, Hoehl A, Klein R (2023) Exploring the necessary conditions for steady-state microbunching at the metrology light source. In: Proceedings of the 14th international particle accelerator conference (IPAC’23), Vinece, Italy, 2023. JACoW, Geneva, p MOPA176
Deng H, Feng C (2013) Using off-resonance laser modulation for beam-energy-spread cooling in generation of short-wavelength radiation. Phys Rev Lett 111:084801
Feng C, Deng H, Wang D, Zhao Z (2014) Phase-merging enhanced harmonic generation free-electron laser. New J Phys 16(4):043021
Feng C, Zhao Z (2017) A storage ring based free-electron laser for generating ultrashort coherent euv and x-ray radiation. Sci Rep 7(1):4724
Dragt AJ (2021) Lie methods for nonlinear dynamics with applications to accelerator physics
Ratner D, Chao A (2011) Seeded radiation sources with sawtooth waveforms. In: Proceedings of the 33th international free electron laser conference (FEL’11), Shanghai, China, 2011, pp 53–56
Stupakov G, Zolotorev M (2011) Using laser harmonics to increase bunching factor in eehg. In: Proceedings of the 33th international free electron laser conference (FEL’11), Shanghai, China, 2011, pp 45–48
Chao A (2022) Focused laser, unpublished note
Piwinski A (1974) Intra-beam-scattering. In: Proceedings of the 9th international conference on high energy accelerators, Stanford, p 405
Bjorken JD, Mtingwa SK (1982) Intrabeam scattering. Part Accel 13(FERMILAB-PUB-82-47-THY):115–143
Panofsky W, Wenzel W (1956) Some considerations concerning the transverse deflection of charged particles in radio-frequency fields. Rev Sci Instrum 27(11):967–967
Chao AW (1979) Evaluation of beam distribution parameters in an electron storage ring. J Appl Phys 50(2):595–598
Fliller R (2006) General solution to a transverse to longitudinal emittance exchange beamline. Fermi Beams Document 2553–v2
Xiang D, Chao A (2011) Emittance and phase space exchange for advanced beam manipulation and diagnostics. Phys Rev ST Accel Beams 14:114001
Talman R (1995) A proposed möbius accelerator. Phys Rev Lett 74:1590–1593
Berz M, Makino K, Wan W (2015) An introduction to beam physics. CRC Press
Chao AW (2020) Lectures on accelerator physics. World Scientific
Forest É (1998) Beam dynamics: a new attitude and framework. Harwood Academic Publisher, Amsterdam
Chao AW (2022) Special topics in accelerator physics. World Scientific
Berg JS (2007) Amplitude dependence of time of flight and its connection to chromaticity. Nucl Instrum Methods Phys Res, Sect A 570(1):15–21
Artamonov A, Derbenev IS, Inozemtsev N (1989) The possibility of controlling the dispersion characteristics of particle motion in an undulator. Zh Tekh Fiz 59:214–216
Emery L (1993) Coupling of betatron motion to the longitudinal plane through path lengthening in low-\(\alpha _c\) storage rings. In: Rossbach J (ed.) Proceedings of the 15th international conference on high energy accelerators. World Scientific, Singapore, pp 1172–1174
Shoji Y (2005) Dependence of average path length betatron motion in a storage ring. Phys Rev ST Accel Beams 8:094001
Sessler AM, Whittum DH, Yu L-H (1992) Radio-frequency beam conditioner for fast-wave free-electron generators of coherent radiation. Phys Rev Lett 68:309–312
Takao M (2008) Impact of betatron motion on path lengthening and momentum aperture in a storage ring. In: Proceedings of the 11th European particle accelerator conference (EPAC’08), Genoa, Italy, 2008, pp 3152–3154
Hoummi L, Lopez JR, Welsch C, Loulergue A, Nagaoka R (2019) Beam dynamics in mba lattices with different chromaticity correction schemes. In: Proceedings of the 10th international particle accelerator conference (IPAC’19), Melbourne, Australia, 2019. JACoW, Geneva, pp 346–349
Borland M (2000) Elegant: a flexible sdds-compliant code for accelerator simulation. Technical report, Argonne National Lab
Klein R, Brandt G, Fliegauf R, Hoehl A, Müller R, Thornagel R, Ulm G, Abo-Bakr M, Feikes J, Hartrott MV, Holldack K, Wüstefeld G (2008) Operation of the metrology light source as a primary radiation source standard. Phys Rev ST Accel Beams 11:110701
Klein R, Mayer T, Kuske P, Thornagel R, Ulm G (1997) Beam diagnostics at the bessy i electron storage ring with compton backscattered laser photons: measurement of the electron energy and related quantities. Nucl Instrum Methods Phys Res A 384(2–3):293–298
Shoji Y, Takao M, Nakamura T (2014) Amplitude dependent shift of betatron oscillation center. Phys Rev ST Accel Beams 17:064001
Koschitzki C, Hoehl A, Klein R, Thornagel R, Feikes J, Hartrott M, Wüstefeld G (2010) Highly sensitive beam size monitor for pa currents at the mls electron storage ring. In: Proceedings of the 1st international particle accelerator conference (IPAC’10), Kyoto, Japan, 2010. JACoW, Geneva, pp 894–896
Kobayashi K, Schimizu J, Hara T, Seike T, Soutome K, Takao M, Nakamura T (2011) Amplitude dependent betatron oscillation center shift by non-linearity and beam instability interlock. Conf Proc 110904:2178–2180
Chen P, Su JJ, Dawson JM, Bane KLF, Wilson PB (1986) Energy transfer in the plasma wake-field accelerator. Phys Rev Lett 56:1252–1255
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Deng, X. (2024). SSMB Transverse-Longitudinal Coupling Dynamics. In: Theoretical and Experimental Studies on Steady-State Microbunching. Springer Theses. Springer, Singapore. https://doi.org/10.1007/978-981-99-5800-9_3
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