Abstract
In this chapter, we study the single-particle longitudinal dynamics of SSMB. The motivation is to answer the question: how to realize the short bunch length and small longitudinal emittance in an electron storage ring, as required by SSMB? Note that the curvilinear (Frenet-Serret) coordinate system and the state vector \({\textbf {X}}=(x,x',y,'y',z,\delta )^{T}\), with \(^{T}\) representing the transpose, are used throughout this dissertation. For the longitudinal dynamics without coupling from the transverse dimension, what we can play are the momentum compaction and RF systems, for SSMB the laser modulators.
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In this chapter, we study the single-particle longitudinal dynamics of SSMB. The motivation is to answer the question: how to realize the short bunch length and small longitudinal emittance in an electron storage ring, as required by SSMB? Note that the curvilinear (Frenet-Serret) coordinate system and the state vector \({\textbf {X}}=(x,x',y,'y',z,\delta )^{T}\), with \(^{T}\) representing the transpose, are used throughout this dissertation. For the longitudinal dynamics without coupling from the transverse dimension, what we can play are the momentum compaction and RF systems, for SSMB the laser modulators. The momentum compaction is a measure of particle energy dependence of the recirculation path length
where \(C_{0}\) is the ring circumference, \(E_{0}\) is the particle energy, \({D}_{x}\) is the horizontal dispersion which is a measure of the energy dependence of particle horizontal position, \(\rho \) is the bending radius. Considering the energy-dependent velocity, the particle energy dependence of the revolution time can be quantified by a parameter named phase slippage factor
with \(\gamma \) the Lorentz factor. For linear dynamics, the phase slippage to longitudinal dimension is like the drift space to transverse dimension, while the RF kick in linear approximation to longitudinal dimension is like the quadrupole to transverse dimension. The difference is that the sign of phase slippage can either be positive or negative, while the drift space can only have a positive physical length. To account for the impact of local or partial phase slippage on the evolution of longitudinal optics around the ring, Courant-Snyder analysis can be invoked for linear dynamics study beyond adiabatic approximation. Such new derivations are necessary to accurately describe the dynamics of the SSMB mechanism. Usually there is only one RF cavity in a storage ring, the longitudinal optics can be manipulated with more freedom with multiple RFs. For example, the strong focusing principle can be implemented in the longitudinal dimension to realize ultrashort bunch length, not unlike its transverse counterpart. For nonlinear dynamics, both the nonlinearity of the phase slippage and the sinusoidal modulation waveform can lead to subtle and rich beam dynamics. In the following we will investigate along this brief review. Parts of the work presented in this chapter have been published in Refs. [1,2,3,4].
2.1 Linear Longitudinal Dynamics
2.1.1 Longitudinal Courant-Snyder Formalism
SSMB means an ultrashort electron bunch in an equilibrium state. One successful method of realizing short bunches in an electron storage ring is the implementation of a quasi-isochronous lattice, which means particles with different energies complete one revolution using almost the same time. The reason behind is the well-known \(\sigma _{z}\propto \sqrt{|\eta |}\) scaling law of the “zero-current” bunch length given by Sands [5], in which \(\eta =\alpha -\frac{1}{\gamma ^{2}}\) is the global phase slippage factor of the ring as introduced just now. However, from single-particle dynamics perspective, there is a fundamental effect limiting the lowest bunch length realizable in an electron storage ring originating from the stochasticity of photon emission time or location. This stochasticity results in a diffusion of the electron longitudinal coordinate z even if the global phase slippage of the ring is zero as we cannot make all the local or partial phase slippages zero simultaneously. The partial phase slippage factor from \(s_{1}\) to \(s_{2}\) is defined as
The physical picture of the partial phase slippage and quantum excitation in both the particle energy and longitudinal coordinate, therefore the longitudinal emittance, is shown in Fig. 2.1.
Physical picture of the partial phase slippage factors and quantum excitation. Particles undergo diffusion in both the particle energy and the longitudinal coordinate in each turn, giving rise to longitudinal emittance growth
Due to this quantum diffusion, there exists a lower bunch length limit and the energy spread diverges when the bunch length is pushed close to the limit. This effect is of vital importance for SSMB and other ideas invoking ultrashort electron bunches or ultrasmall longitudinal emittance in storage rings. It is first theoretically investigated by Shoji et al. [6, 7], and recently more accurately analyzed by us using the longitudinal Courant-Snyder formalism [2, 3, 8]. The key to understanding the effect is to change from the global viewpoint to a local one, i.e., the quantum excitation at different places around the ring actually contribute to the longitudinal emittance with different strengths, just like its transverse counterpart.
For an accurate analysis of this effect, here we invoke Chao’s solution by linear matrices (SLIM) formalism [9]. SLIM is an early effort to generalize the classical Courant-Snyder theory [10] from 1D (2D phase space) to higher dimensions. It invokes \(6\times 6\) transport matrices and applies to 3D (6D phase space) general coupled lattice without the assumption of a small synchrotron tune. Concerning the evaluation of equilibrium beam parameters in an electron storage ring, SLIM can be viewed as a method of solving linear Fokker-Planck equation [11, 12] without adopting the adiabatic approximation. Therefore, it can account for the variation of one-turn map around the ring. In other words, the contribution of diffusion and damping, for example the quantum excitation and radiation damping, to the eigen emittances depends on the local one-turn map.
The three eigen emittances \(\epsilon _{k}\) of a particle beam, with \(k=I,II,III\), are defined as the positive eigenvalues of \(i\boldsymbol{\Sigma }{} {\textbf {S}}\), where i is the imaginary unit, \(\boldsymbol{\Sigma }=\langle {\textbf {X}}{} {\textbf {X}}^{T}\rangle \) are the second moments of the beam and
The eigen emittances are invariants with respect to a linear symplectic transportation \({\textbf {T}}\), as
in which the last step has invoked the symplecticity of \({\textbf {T}}\), i.e., \({\textbf {T}}^{T}{} {\textbf {S}}{} {\textbf {T}}={\textbf {S}}\). Therefore, \(i\Sigma _{\text {new}}{} {\textbf {S}}\) is related to \(i\Sigma _{\text {old}}{} {\textbf {S}}\) by a similarity transform, thus having the same eigenvalues.
In an electron storage ring, the equilibrium state is a balance between quantum excitation and radiation damping. According to SLIM [9], the equilibrium eigen emittances are given by
and the second moments of the beam are
where \(\alpha _{k}\) are the damping constants of the three eigen modes, \(C_{L}={55r_{e}\hbar }/{(48\sqrt{3}m_{e})}\), with \(\hbar \) the reduced Planck’s constant, \(r_{e}\) the electron classical radius, \(\text {Re}[]\) means taking the real part of a complex number, \(^{*}\) means complex conjugate, and \({\textbf {E}}_{k}\) are eigenvectors of the \(6\times 6\) symplectic one-turn map, satisfying the following normalization condition
and \({\textbf {E}}_{k}^{\dagger }{} {\textbf {S}}{} {\textbf {E}}_{j}=0\) for \(k\ne j\), in which \(^{\dagger }\) means complex conjugate transpose. \({\textbf {E}}_{ki}\) is the i-th component of the eigenvector \({\textbf {E}}_{k}\).
To simplify the discussion, here we only consider the horizontal and longitudinal dimensions and use the state vector \({\textbf {X}}=\left( x,x',z,\ \delta \right) ^{T} \). Under the assumptions that the ring is planar x-y uncoupled and the RFs are placed at dispersion-free locations, which is the typical setup for present synchrotron radiation sources, the betatron coordinate \({\textbf {X}}_{\beta }={\textbf {B}}{} {\textbf {X}}\) can be introduced to parametrize the transfer matrix in a diagonal form, with the dispersion matrix given by
The one-turn map \({\textbf {M}}\) of \({\textbf {X}}\) is related to the one-turn map \({\textbf {M}}_{\beta }\) of \({\textbf {X}}_{\beta }\) by
with
in which \(\Phi _{x}=2\pi \nu _{x}\) and \(\Phi _{z}=2\pi \nu _{s}\) are the betatron and synchrotron phase advance per turn. The eigenvectors of \({\textbf {M}}_{\beta }\) can be expressed using the Courant-Snyder functions as
where \(\Phi _{I}\) and \(\Phi _{III}\) are phase factors which do not affect the normalization of eigenvector and the calculation of physical quantities. Therefore, the eigenvectors of \({\textbf {M}}\) are
According to SLIM, the equilibrium horizontal and longitudinal emittance are then
in which
are the horizontal and longitudinal action of a particle, and \(\langle \rangle \) here means particle ensemble average, \(\alpha _{H}\) and \(\alpha _{L}\) are the horizontal and longitudinal damping constants,
where \(U_{0}\) is the radiation energy loss of a particle per turn, \(\mathcal {D}=\frac{\oint \frac{(1-2n)D_{x}}{\rho ^{3}}ds}{\oint \frac{1}{\rho ^{2}}ds}\), with \(n=-\frac{\rho }{B}\frac{\partial B}{\partial \rho }\) the field gradient index, \(\alpha _{F}=\frac{1}{137}\) is the fine-structure constant, \(\bar{\lambda }_{e}={\lambda _{e}}/{2\pi }=386\ \text {fm}\) is the reduced Compton wavelength of electron and \(\mathcal {H}_{x}=\gamma _{x}{D}_{x}^{2}+2\alpha _{x}{D_{x}}{D_{x}'}+\beta _{x}{D_{x}'}^{2}=\frac{D_{x}^{2}+\left( \alpha _{x}D_{x}+\beta _{x}D_{x}'\right) ^{2}}{\beta _{x}}\) is the horizontal chromatic function. Therefore, it is the longitudinal beta function \(\beta _{z}\) at the bending magnets that matters in determining the contribution of quantum excitation to the longitudinal emittance \(\epsilon _{z}\). A physical picture is given in Fig. 2.2 to help better understand this argument.
A physical picture to explain why a larger longitudinal beta function \(\beta _{z}\) means a larger contribution to longitudinal emittance \(\epsilon _{z}\), with a given strength of quantum excitation
After getting the equilibrium eigen emittances, we can obtain the second moments of the beam according to Eqs. (2.7) and (2.13), more specifically,
and
where
The distribution of a Gaussian beam is related to the second moments matrix of the beam according to
2.1.2 Classical \(\sigma _{z}\propto \sqrt{|\eta |}\) Scaling
Now we first reproduce the classical \(\sigma _{z}\propto \sqrt{|\eta |}\) scaling using this longitudinal Courant-Snyder parameterization. To simplify the discussion further, in this section and the following we focus on the longitudinal dimension only and the state vector \({\textbf {X}}=\left( z, \delta \right) ^{T} \) is used. We treat first the case where there is only one RF placed at a dispersion-free location. In this case, the linear longitudinal one-turn map observed in the middle of the RF cavity is
with \( h={eV_{\text {RF}}k_{\text {RF}}\cos \phi _{\text {s}}}/{E_{0}} \) quantifying the RF acceleration gradient, where e is the elementary charge, \(V_{\text {RF}}\) is the RF voltage, \(k_{\text {RF}}=2\pi /\lambda _{\text {RF}}\) is the RF wavenumber, \(\phi _{\text {s}}\) is the synchronous phase and \(E_{0}=\gamma m_{e}c^2\) is the electron energy. The \(R_{56}=-\eta C_{0}\), a measure for the dependence of z on \(\delta \), of the ring and the RF kick h can be viewed as the longitudinal drift space and quadrupole, in correspondence to their transverse counterparts, respectively. Note however that as mentioned before, the \(R_{56}\) can be either positive or negative, while the physical length of a drift space is always positive.
The linear stability requires that
Actually if the sinousidual modulation waveform is taken into account, the longitudinal dynamics is more accurately modeled by a standard kick map [13]. We then need \(h\eta C_{0}\) be small enough to avoid strong chaotic dynamics. An empirical safe criterion is that \(0<h\eta C_{0}\lesssim 0.1\). For rings working in the longitudinal weak focusing regime, \(|\nu _{s}|\ll 1\), we then have
Therefore the longitudinal beta function \(\beta _{z}\) at the RF center is
In this dissertation, we use the subscript \(_{\text {S}}\) to denote results which are the same with that obtained in Sands’ classical analysis [5], although the method used here to get these results is different from that of Sands. As \(|\nu _{s}|\ll 1\), therefore
We will see later in Sect. 2.1.6 that in a longitudinal strong focusing ring, \(|\nu _{s}|\) can be close to or even larger than 1, and \(\beta _{z}\) can then be the same level of or smaller than \(|-\eta C_{0}|\).
Using this \(\beta _{z\text {S}}\) to represent \(\beta _{z}\) of the whole ring, we then get the longitudinal emittance obtained in Sands’ analysis
For a ring consisting of isomagnets, \(\rho \) is a positive constant and
with \(J_{s}=2+\mathcal {D}\) the longitudinal damping partition number [5] and nominally \(J_{s}\approx 2\), we have
where \(C_{q}=\frac{55{\bar{\lambda }}_{e}}{32\sqrt{3}}=3.8319\times 10^{-13}\) m. Therefore, to generate short bunches in an electron storage ring, we need to implement a quasi-isochronous lattice, i.e., a small \(\eta \), and a high RF acceleration gradient, i.e., a large h. We also note that the energy spread of an electron beam in the classical analysis is dominantly determined by the beam energy and bending radius of the bending magnets, and has little dependence on the bunch length or global phase slippage of the ring.
2.1.3 Beyond the Classical \(\sigma _{z}\propto \sqrt{|\eta |}\) Scaling
2.1.3.1 Analysis
Using a single \(\beta _{z\text {S}}\) to represent that of the whole ring is valid in usual rings where the relative variation of \(\beta _{z}\) is negligible and the electron distribution in the longitudinal phase space is always upright. But when the global phase slippage is small, the partial phase slippage can be significantly larger than the global one and the variation of \(\beta _{z}\) and beam orientation in the longitudinal phase space around the ring can be significant, thus the classical \(\sigma _{z}\propto \sqrt{|\eta |}\) scaling fails. Now we present an accurate analysis of this effect using the longitudinal Courant-Snyder formalism.
If there is only a single RF cavity placed at a dispersion-free location in the ring, then at a specific position \(s_{j}\), the ring can be divided into three parts, with their longitudinal transfer matrices given by
where \(\tilde{\eta }(s_{\text {RF}},s_{j})+\tilde{\eta }(s_{j},s_{\text {RF}})=\eta \), In the analysis, the RF cavity is assumed to be a zero-length one. The one-turn map at \(s_{j}\) is then
Therefore,
Note that \(\beta _{z}\) is always positive, and
which is different from the conventional relation \(\frac{d\beta _{x,y}}{ds}=-2\alpha _{x,y}\) in transverse dimensions [4].
The first term in the numerator of Eq. (2.31) is the conventional global phase slippage. The second term reflects the impact of the partial phase slippage on \(\beta _{z}\). In usual rings, the second term is much smaller than the first term, therefore \(\beta _{z}\) is almost a constant value around the ring. As mentioned, the classical formulas of bunch length \(\sigma _{z\text {S}}\), energy spread \(\sigma _{\delta \text {S}}\), and longitudinal emittance \(\epsilon _{z\text {S}}\) in last section are actually obtained with such approximation. Now with both terms in the numerator of Eq. (2.31) considered, the more accurate formula of the longitudinal emittance is then
Note that \(\langle \rangle _{\rho }\) here means the radiation-weighted average around the ring, defined as
i.e., the average is actually conducted at places with nonzero bending field. After getting the longitudinal emittance and Courant-Snyder functions, the bunch length and energy spread at a specific location \(s_{i}\) are then
We remind the readers that the energy spread and \(\gamma _{z}\) are unchanged outside the RF cavity. In addition, if the contribution of \(\frac{1}{\gamma ^{2}}\) is negligible in the definition of \(\eta \), then \(\alpha _{z}\) and \(\beta _{z}\) will vary notably only inside the bending magnets. Actually, the chromatic \(\mathcal {H}_{x}\) function, a parameter quantifying the coupling of horizontal emittance to bunch length as can be seen from Eq. (2.19), also changes only inside the bending magnets. Both arguments reveal the fact that in ultrarelativistic cases, bunch length changes only inside the bending magnets. We will see this clearly in Fig. 2.3.
By investigating the bunch length at the RF cavity
we observe that there exists a lower bunch length limit when \(\eta \) approaches zero
This limit is the main consequence of the unavoidable quantum diffusion of longitudinal coordinate in a storage ring. It has little dependence on the global phase slippage and RF voltage, once the beam energy and dispersion function pattern around the ring is given. Since \(\sigma _{z\text {S}}\propto \sqrt{|\eta |}\), the above bunch length limit means \(\frac{\epsilon _{z}}{\epsilon _{z\text {S}}}\) will diverge as \(\eta \) approaches zero. The energy spread will thus diverge in this process.
While the bunch length at the RF cavity will saturate at the limit given by Eq. (2.37) with the decrease of \(\eta \), the bunch length at other places, from which the partial phase slippage to the RF cavity is large, may first decrease and then increase. The reason is that the increased energy spread will lead to bunch lengthening through the partial phase slippage from the RF cavity to the specific location. In other words, the longitudinal beta function ratio between that at the RF cavity and that at the specific location may increase with the lowering of \(\eta \).
2.1.3.2 Experimental Verification
The above analysis has been confirmed by numerical simulation as presented in Ref. [2]. Now we introduce our experimental work on this quantum diffusion effect. The experiment was conducted at the Metrology Light Source (MLS) [14,15,16] of the Physikalisch-Technische Bundesanstalt in Berlin. For usual rings, the bunch length limit given by Eq. (2.37) is a couple of 10 fs to about 100 fs, while the typical bunch length in operation is in 10 ps level. So this effect is negligible in almost all existing rings. However, with the accelerator physics and technologies continue to advance, more ambitious goals of bunch length can be envisioned and realized in the future to benefit more from the electron beam. For example, in an SSMB storage ring, the desired bunch length is sub-micron or even nanometer, which corresponds to sub-fs in unit of time. The quantum diffusion investigated here then becomes the first fundamental issue that needs to be resolved. With such motivation to develop an SSMB light source, and considering that it is a fundamental physical effect by itself, we believe it is important to experimentally verify this effect.
To observe the influence of this effect, we need the second term in the bracket of Eq. (2.33) to be comparable or larger than 1, which is non-trivial for many of the existing storage rings. Other collective and single-particle effects stand in the way before arriving at such a small value of \(\eta \). However, due to the dedicated quasi-isochronous lattice design and the individually independent magnet power supplies of the MLS storage ring, there is great flexibility in tailoring the lattice optics to obtain a locally large and globally small phase slippage simultaneously, thus opening the possibility to see this effect in an existing machine. Another characteristic making the MLS an ideal test bed of single-particle beam dynamical effects is that it can operate with a beam current ranging from 1 pA (a single electron) to 200 mA.
We have prepared two quasi-isochronous lattice optics at the MLS, named lattice A and B, respectively. Lattice A is the standard quasi-isochronous lattice, while lattice B is developed and dedicated for this experiment. The optical functions of the two lattices are shown in Fig. 2.3. Other related parameters of the two lattices are given in Table 2.1. The key difference of these two lattices is that lattice B has a much larger partial phase slippage and average value of \(\langle \beta _{z}\rangle _{\rho }\). Therefore, the bunch length limit in lattice B (469 fs at 630 MeV) due to this quantum diffusion is larger than that in lattice A (115 fs at 630 MeV). Note that with the given parameters set, \(\beta _{z}\) in lattice A is almost a constant value around the ring, while \(\beta _{z}\) in lattice B varies significantly and at many places is much larger than that in lattice A.
Two lattices used in the experiment. Evolution of a \(\beta _{x,y}\), b \(D_{x}\) and \(\beta _{z}\), c \(\sigma _{z}\), d \(\sigma _{\delta }\), around the ring. In this plot, the RF cavity is placed at \(s_{\text {RF}}=0\) m and \(V_{\text {RF}}=600\) kV is applied, the global phase slippage used is \(\eta =1\times 10^{-5}\). The dipole magnets are shown at the top as blue rectangles. Each dipole has a length of 1.2 m and bends the electron trajectory for an angle of \(\pi /4\). \(\beta _{x,y}\) and \(D_{x}\) are obtained by fitting a model to the BPM-corrector response matrix (LOCO) [17]. The bunch length and energy spread evolution are calculated based on the longitudinal Courant-Snyder formalism and SLIM formalism
As can be seen in Fig. 2.3, the magnitudes of horizontal dispersion function \(D_{x}\) of lattice B are large at some of the bending magnets, which according to Eq. (2.3) means the local phase slippage increases or decreases sharply within them, leading to a large variation of local phase slippage \(\tilde{\eta }\) and \(\beta _{z}\). The small global phase slippage \(\eta \) is realized by canceling the contribution of positive and negative \(D_{x}\) at different bending magnets. We remind the readers that this lattice can also be used for the delayed alpha-buckets study in which the momentum differences of particles in different alpha-buckets can be translated into large arrival time differences through the large partial phase slippage [16], which might be useful for some user experiments.
To evaluate the possibility of verifying this effect experimentally, the bunch length and energy spread evolution around the ring in these two lattices have also been presented in Fig. 2.3. Note that the bunch length formula in Eq. (2.35) contains only the contribution from longitudinal emittance. Considering the bunch lengthening by horizontal emittance at dispersive locations, according to Eq. (2.19), the more accurate formula of bunch length is [1, 18]
Strictly speaking, Courant-Snyder and dispersion functions are only well-defined in a planar uncoupled lattice and only when the RF cavity is placed at a dispersion-free location. For a general coupled lattice, the more accurate SLIM formalism should be referred, i.e.,
On the other hand, although the RF cavity is placed at a dispersive location in lattice B, we have confirmed that the Courant-Snyder parametrization for beam dynamics analysis in this case is still largely valid, since the difference of result between that given by the longitudinal Courant-Snyder formalism and the more accurate SLIM formalism is very small.
As can be seen in Fig. 2.3, in which the global phase slippage \(\eta \) is lowered to be \(1\times 10^{-5}\), which corresponds to a synchrotron frequency of \(f_{s}=2.2\) kHz with \(V_{\text {RF}}=600\) kV, the energy spread grows to be \(\sigma _{\delta }=7.9\times 10^{-4}\), while the classical energy spread is \(\sigma _{\delta \text {S}}=4.4\times 10^{-4}\). Such an amount of energy spread growth is detectable by measuring the spectra of Compton-backscattered (CBS) photons from the head-on collision between a CO\(_{2}\) laser with the electron beam at the MLS [14, 19]. In addition, the bunch length difference in these two lattices are large enough to be observable by evaluating the spectra and power of coherent THz radiation, and invoking streak camera measurement.
To exclude the influence of collective effects, the beam current is lowered to around \(6\ \upmu \)A/bunch in a multi-bunch filling mode in the experiment. There is no indication of microwave or other collective instabilities. The beam is stable (no fluctuation of radiation source point observed) and its width and energy spread are independent of the beam current when the single-bunch current is as low as the value applied in the experiment. The horizontal chromaticity has been carefully corrected close to zero (about 0.05) to minimize the beam energy widening arising from the betatron motion of particles as will be reported in Sect. 3.2. The longitudinal chromaticity has also been corrected to a small value to mitigate longitudinal nonlinear dynamics. We note that a large quantum diffusion of longitudinal coordinate (a root-mean-square value of 0.54 \(\upmu \)m or 1.8 fs per turn in lattice B at 630 MeV) actually helps suppress collective beam instability of ultrahigh frequency, as it will disperse any fine time structure in an electron beam like density modulation and energy modulation [7].
Experiment measurement results and comparison with theory. a Theoretical bunch length and measured coherent THz radiation power observed at \(s=38.775\) m, in lattice A and B, respectively. b Theoretical, measured raw data (shifted 8 ps downwards) and fitted bunch length at \(s=24\) m, in lattice A and B, respectively. c Theoretical and measured electron beam energy spreads \(\sigma _{\delta }\) normalized by the classical energy spread \(\sigma _{\delta \text {S}}\) versus the synchrotron frequency \(f_{s}\), in lattice A and B, respectively
To get an idea about the bunch length in the two lattices, first we measure the coherent THz radiation spectra and power as a function of the synchrotron tune in the two lattices. The shorter the electron bunch, the higher frequency range the coherent THz radiation spectra extends and the larger radiation power we can obtain. In the experiment, the synchrotron frequency \(f_{s}\), thus the global phase slippage \(\eta \) (\(f_{s}\propto \sqrt{|\eta |}\)), is controlled by slightly changing the quadrupole currents while keeping the dispersion function pattern unchanged. The THz beamline has its source point at \(\frac{\pi }{16}\) bending angle (\(s=38.775\) m) at the 7-th dipole, counted from \(s=0\) m in Fig. 2.3 which is where the RF cavity is placed. To get the coherent synchrotron radiation emission spectra in the THz spectral range, a commercial, Michelson-type FTIR spectrometer (Vertex 80v) in combination with a 4K liquid helium cooled composite silicon bolometer was used for measuring interferograms. After fast Fourier transform of the data, the emitted spectrum can directly be accessed. For this experiment a series of 128 interferograms have been acquired and the average Fourier transformed.
The measured coherent THz radiation power, integrated with wavenumber from 1 to 20 cm\(^{-1}\), together with the theoretical bunch length at the THz observation calculated using Eq. (2.38), are shown in Fig. 2.4a. The measurement results agree with our expectation reasonably well. In particular, we notice that in lattice B, the THz power first increases and then decreases, with the lowering of the synchrotron tune, while the radiation power in lattice A monotonically decreases and then saturates in this process. This observation agrees well with our theoretical prediction of the bunch length evolution in these two lattices. Not presented here, we also notice that the frequency range of the spectra evolves consistently with the integrated power, i.e., a larger THz power corresponds to a higher frequency range coverage. To be more rigorous, we remind the readers that the bunch length in lattice A at the THz radiation observation point in principle will also diverge, as explained in last section, if we push the phase slippage factor of the ring even closer to zero, which in practice is a demanding work.
During the measurement of coherent THz radiation, we at the same time employed a streak camera to measure the electron bunch length directly. The streak camera at the MLS is installed at the undulator beamline (opposite the RF cavity, \(s=24\) m in Fig. 2.3). For the experiment, the undulator was closed from the “open” gap of 180 mm to 45.7 mm to have the fundamental-mode undulator radiation at a visible wavelength available for the streak camera. The measurement results of bunch length and the comparison with theory is presented in Fig. 2.4b. Note that we have shifted the measured raw data downwards by 8 ps in the plot. The errorbars in the plot are the standard deviation of the fitted results for each single column of the recorded streak camera image. Again we observe the significant difference in the two lattices concerning the bunch length evolution as a function of the synchrotron tune, which agrees qualitatively with theory. However, quantitatively the measured raw data of bunch length deviates notably from the theoretic prediction.
Realizing that there will be unavoidable systematic errors concerning the streak camera measurement because we are close to its resolution limit, we try to use the model below to fit the data with the theory,
where \(\Delta z\) means the bunch length. Note that here we use the full width at half maximum (FWHM), instead of the root mean square, to quantify the bunch length, since in the real case, the bunch profile is unavoidable non-Gaussian to some extent, especially when the phase slippage factor of the ring is small. The noise in the above equation is used to model the square sum-type error, while the offset accounts for the systematic shift concerning the measurement results. The fitted data (\(\text { noise}=7\) ps and \(\text { offset}=3\) ps applied) agrees well with the theoretical curve as shown in Fig. 2.4b. We remind the readers that all the data points in the plot are modeled with the same \(\text { noise}\) and \(\text { offset}\).
Further, we have measured the electron beam energy spread in the two lattices, using the head-on CBS between a CO\(_{2}\) laser with the electron beam. Note that the RF voltage applied in the above bunch length measurements is 500 kV, while now it is 600 kV when doing the energy spread measurement. The measurement of CBS photon spectra and the evaluation of electron beam energy spread based on it is a well-established method implemented at the MLS, and is used in this experiment to confirm the energy widening as we push the bunch length close to the limit, by lowering the global phase slippage \(\eta \). More details about this CBS method can be found in Refs. [14, 19]. Quantitative analysis revealing the energy spreads \(\sigma _{\delta }\) normalized by the classical energy spread \(\sigma _{\delta \text {S}}\), and its comparison with the theoretical prediction from Eq. (2.35) for the two different lattices are shown in Fig. 2.4c. The error bars in Fig. 2.4c are the root-mean-square uncertainties of the measurements and are due to calibration errors and counting statistics. The data acquisition time of a photon spectrum is 15 min. It can be seen from Fig. 2.4 that in lattice B the energy spread grows significantly with the decrease of \(\eta \), in the figure synchrotron frequency \(f_{s}\), to the level of \(1\times 10^{-5}\), while the energy spread stays almost constant in lattice A. Again the measurement agrees qualitatively with the theory.
There is still some deviation of the measured energy widening and the theoretical prediction for lattice B. Candidate explanations are: first, there is some uncertainty in the determination of synchrotron frequency \(f_{s}\), especially when \(f_{s}\) is lowered to \(2\sim 3\) kHz, considering the fact that the peak of the synchrotron frequency spectrum then can be as wide as 0.5 kHz; second, there could be some remaining higher-order phase slippages which may contribute to the energy spread growth when \(\eta \) is small due to its impact on the longitudinal phase space bucket, while the theory assumes a linear phase slippage.
The above presented measurements of bunch length and energy spread are very demanding and are moving on the edge of the experimentally accessible parameter space. Nevertheless, we see a nice qualitative agreement with the theory presented in this section, proofing important experimental evidence to support the theoretical analysis. As far as we know, this type of investigation can actually not be performed at any other operating storage ring. However, we recognize that the deviation of the quantitative numbers between the measurements and theory concerning both the bunch length and energy spread emphasizes the need for an even more improved model. Summarizing we state that our experimental work supports the existence of the analyzed quantum diffusion effect, and the argument that the quantum excitation on longitudinal emittance at a given location depends on the longitudinal beta function there. The evidence however is not strong enough to claim this is a fully consistent proof of the effect.
2.1.4 Campbell’s Theorem
We point out that quantifying the impact of variation of \(\beta _{z}\) around the ring on \(\epsilon _{z}\) using partial phase slippage variance \(\langle \tilde{\eta }^{2}\rangle _{\rho }-\langle \tilde{\eta }\rangle _{\rho }^2\), as that done in Ref. [6] and also our previous publication Ref. [1], is not generally correct. The reason is that while the photon emission process is stochastic, the evolution of partial phase slippage around the ring is deterministic. So the diffusion of z each turn \(d_{z}^{2}\) due to quantum excitation is
instead of
as that given in Ref. [6], where \(\tilde{\eta }\) is the partial alpha slippage calculated using the final observation location as the ending point, \(\langle \mathcal {N}\rangle \) is the expected number of emitted photons, u is the photon energy, \(\langle u^2 \rangle \) and later also \(\langle u \rangle \) mean the average is taken with respect to the photon energy spectrum.
This result can be understood with the help of Campbell’s theorem [20]. From this theorem some expectation result for the Poisson point process follows. For example, for the application in synchrotron radiation, we have \(\delta =-\sum _{i} \frac{u_{i}}{E_{0}}\), where the subscript \(_{i}\) means the i-th photon emission. Then according to Campbell’s theorem we have
where \(\dot{\mathcal {N}}\) is the number of photons emitted per unit time in the dipoles and \(T_{\text {dipole}}\) is the total time within dipoles. Equation (2.43) is why \(\dot{\mathcal {N}}\langle u^{2}\rangle \) appears so often in the calculation of energy spread, emittance, etc., in electron storage ring physics. Note that the relation in Eq. (2.43) holds as long as the radiation is a Poisson point process. It is independent of whether \(\langle u\rangle =0\) or not, and is also independent of the detailed spectrum of the photon energy. In other words, the key of a Poisson point process is the randomness in whether there is a kick or not, i.e, the kick number, and not in the randomness of the size of the kicks. The importance of this theorem for electron dynamics was first pointed out by Sands [21]. A proof can be found in the article of Rice [22] and a less rigorous but simpler one in the lecture note of Jowett [23].
Now we can understand Eq. (2.41) as follows. Suppose that the RF is our observation point. We divide the ring into many sections, and in each section \(\tilde{\eta }(s_{\text {RF}},s_{j})\) does not change much. Then the change of electron longitudinal coordinate in one turn is \(z=\sum _{j}z_{j}\), with \(z_{j}=\sum _{i}C_{0}\tilde{\eta }(s_{\text {RF}},s_{ji})\frac{u_{ji}}{E_{0}}\) the contribution due to photon emissions within the section j. According to Campbell’s theorem, the variance of \(z_{j}\) is
where \(t_j\) is the time within the dipoles in section j. As the photon emissions in different sections are uncorrelated, then the variance of z is the sum of variance of \(z_{j}\)
in which \(T_{\text {total}}=\sum _{j}t_{j}\) is the total time within the dipoles. So now we have obtained Eq. (2.41) following Campbell’s theorem.
We can also view the above argument from another way. Given the same dispersion function pattern, which means the same \(\langle \tilde{\eta }^{2}\rangle _{\rho }-\langle \tilde{\eta }\rangle _{\rho }^2\) as it is independent of the observation point, a different longitudinal beta function pattern can be generated if the RF is placed at a different location, therefore resulting in a different longitudinal emittance according to Eq. (2.14).
Changing the RF location means shifting \(\tilde{\eta }(s_{j},s_{\text {RF}})\) up or down as a whole. According to Eq. (2.33), the equilibrium emittance is a parabolic function of the shifted value. When the RF is placed at a location such that \(\langle \tilde{\eta }(s_{j},s_{\text {RF}}\rangle _{\rho }=\frac{\eta }{2}\), we arrive at the minimum longitudinal emittance
The maximum longitudinal emittance is realized when the RF is placed at a place such that \(\left| \langle \tilde{\eta }(s_{j},s_{\text {RF}}\rangle _{\rho }-\frac{\eta }{2}\right| \) reaches the maximum possible value. When the minimum longitudinal emittance is reached, the bunch length at the RF is
In the case of ultrasmall \(\eta \), we have
and
The equality holds when \( \beta _{z\text {S}}=\sqrt{\frac{\eta C_{0}}{h}}=C_{0}\sqrt{\langle \tilde{\eta }^{2}\rangle _{\rho }-\langle \tilde{\eta }\rangle _{\rho }^2}. \) Therefore, the variance of partial phase slippage can be viewed as a parameter to quantify the lowest possible contribution of this effect to the equilibrium bunch length at the RF and the longitudinal emittance with a dispersion function pattern given, if we can choose the location of the RF as we want. However, in a real machine, the RF location is fixed, and Eqs. (2.33) and (2.35) should be referred. This is why we state that using \(\langle \tilde{\eta }^{2}\rangle _{\rho }-\langle \tilde{\eta }\rangle _{\rho }^2\) to quantify the impact of this effect is not generally correct.
2.1.5 Minimizing Longitudinal Emittance
It is clear that the quantum diffusion of z needs to be carefully treated for the realization and long-term maintenance of ultrashort bunch or small longitudinal emittance in either a multi-pass device or a single-pass transport line with bending magnets and large dispersion. A lower operating energy is preferred for suppressing the strength of quantum excitation. Note that the energy scaling laws of this effect are different in the one-turn or single-pass and steady-state cases; for the single-pass case, i.e., Eq. (2.41), the root-mean-square diffusion of longitudinal coordinate \(d_{z}\propto \gamma ^{2.5}\), while for the steady-state case, i.e., Eq. (2.48), \(\sigma _{z,\text {min}}\propto \gamma \), because the radiation damping time also depends on \(\gamma \).
As can seen in Eq. (2.14), the longitudinal beta function \(\beta _{z}\) with respect to the longitudinal dimension plays a role similar to that of the chromatic function \(\mathcal {H}_{x}\) in the transverse dimension. As both the longitudinal and transverse emittances originate from quantum excitation, for a ring consisting of identical isochronous cells, the same scaling law of the theoretical minimum emittance (TME), i.e., \(\epsilon _{x,z,\text {TME}}\propto \gamma ^{2}\theta ^{3}\), concerning the beam energy and bending angle of the dipole can be expected. Note that the TME is independent of the bending radius. But we will show soon that the bunch length limit does depend on the bending radius. According to the scaling, a ring consisting of a larger number of isochronous cells, each with a smaller bending angle, can better minimize the emittance than a ring consisting of fewer cells with larger bending. Generally, it is easier to realize small emittance in a larger ring.
Equation (2.48) gives the lower bunch length limit by optimizing the location of RF cavity, with a given dispersion function pattern. To make this limit as small as possible, in addition to ensuring a small global phase slippage, the variation in the partial phase slippage should also be well confined by means of dedicated lattice design. More specifically, the strategy is to tailor the horizontal dispersion function, thus to minimize the longitudinal beta function at the bending magnets. At the MLS, the small global phase slippage is achieved by means of an overall integration cancellation between the large positive and large negative horizontal dispersions at different dipoles [15, 16]. Therefore, the partial phase slippage varies sharply within the dipoles, leading to a large partial phase slippage variation and significant quantum diffusion of z. To obtain small global and partial phase slippages simultaneously, such cancellation should be done as locally as possible, and the magnitudes of the dispersion at the dipoles should also be minimized, thus making the partial phase slippage vary as gently as possible. In other words, each partial component of the ring should be made as isochronous as possible. In this sense, the dispersion function pattern in Fig. 2.5 is the most locally isochronous bending magnet [24], i.e., each half of the bending magnet is isochornous.
A symmetric dispersion function pattern which makes each half of the bending magnet isochronous, which is desired in minimizing the theoretical minimum bunch length and longitudinal emittance when there is only a single RF or laser modulator in the ring. The lattice design for the realization of the dispersion function pattern can be found in Ref. [25]
2.1.5.1 Constant Bending Radius
Now we present some quantitative analysis of the minimization of longitudinal emittance. In this section we use the partial \(R_{56}\), defined as
for the analysis. As can be seen Eqs. (2.48) and (2.49), it is \(\sqrt{\langle F^{2}\rangle _{\rho }-\langle F\rangle _{\rho }^2}\) that determines the theoretical minimum bunch length and longitudinal emittance. Here we present an analysis of the above scaling by evaluating \(\sqrt{\langle F^{2}\rangle _{\rho }-\langle F\rangle _{\rho }^2}\) of a single isochronous bending magnet with constant bending-radius, and whose dispersion function is symmetric with respect to the dipole middle point as shown in Fig. 2.5. To simplify the analysis, we start from the middle point of the dipole where the dispersion angle is zero \(D'_{\text {m}}=0\), then the dispersion as a function of angle \(\phi \) is
where \(D_{\text {m}}\) is the dispersion at the middle of the dipole. As we are mainly interested in the relativistic case, ignoring the contribution of \(\frac{1}{\gamma ^{2}}\) on F, the condition of isochronicity of each half of the dipole is
with \(\theta \) the bending angle of each dipole. Substituting Eq. (2.51) into Eq. (2.52), we get
where \(D_{\text {e}}\) is the dispersion at the entrance and exit of the dipole. Then we have
Note that the F in this section is defined with the middle point of the bending magnet as the starting point. Therefore, for a ring consisting of such isochronous isomagnets (note that the global phase slippage of the ring is non-zero for a stable beam motion), we have
and
We can also derive the above emittance scaling using directly the longitudinal beta function evolution in the dipole
where \(\alpha _{z\text {m}}\), \(\beta _{z\text {m}}\), \(\gamma _{z\text {m}}\) are the longitudinal Courant-Snyder functions at the middle of the dipole. Then
The minimum average of \(\beta _{z}\), \(\langle \beta _{z}\rangle _{\rho ,\text {min}}\), thus the minimum longitudinal emittance, \(\epsilon _{z,\text {min}}\), is realize when
The corresponding minimum longitudinal emittance is
which is the same as that given in Eq. (2.56). If the longitudinal damping partition number \(J_{s}=2\), then for practical use we have
This is the main result of our analysis of theoretical minimum bunch length and longitudinal emittance in a longitudinal weak focusing ring. A comprehensive study of minimizing longitudinal emittance can also be found in Ref. [4].
Actually if we relax the condition of isochronousity for each dipole, and make \(D_{\text {m}}\) and \(D_{\text {m}}'\) also variables that can be optimized, then we have
When \(D_{\text {m}}=-\frac{\rho \theta ^{2}}{40}\), \(D_{\text {m}}'=0\), and \(\beta _{\text {zm}}=\frac{\sqrt{7}}{840}\rho \theta ^{3}\), we reach the minimal of \(\langle \beta _{z}\rangle _{\rho ,\text {min}}=\frac{\sqrt{7}}{420}\rho \theta ^{3}\) and longitudinal emittance
This is also the result Eq. (29) in Ref. [4]. The emittance given in Eq. (2.63) is smaller than that given in Eq. (2.60) or equivalently Eq. (2.61). However, we should note that when isochronousity of each dipole is broken, to make the longitudinal beta function in different dipoles identical, we need RF kick between each two neighboring dipoles to adjust \(\alpha _{z}\) there, or we need a very long drift space between them if we consider the contribution of \(\frac{1}{\gamma ^2}\) on \(R_{56}\). This means at least N RFs or laser modulators are needed if there are N dipoles in the ring. This is not very feasible in reality. In addition, placing RFs at dispersive locations will make the dynamics becomes transverse-longitudinal coupled. In the more-confronted or practical case of a single RF in the ring, Eq. (2.61) is a more self-consistent evaluation of the theoretical minimum bunch length and longitudinal emittance.
Since high-power EUV radiation is of particular interest for EUV lithography [26], let us now do some evaluation based on our investigations to see if we can realize high-power EUV radiation in a longitudinal weak focusing SSMB storage ring. For coherent 13.5 nm EUV radiation generation, we need an electron bunch length around 3 nm or shorter. The lower limit of bunch length \(\sigma _{z,\text {min}}\) should be smaller than this desired bunch length to avoid significant energy widening. Here we assume \(\sigma _{z,\text {min}}\le 2\) nm. If \(E_{0}=400\) MeV and \(\rho =4\) m (\(B=0.334\) T), then \(\sigma _{\delta \text {S}}=1.7\times 10^{-4}\). To realize \(\sigma _{z,\text {min}}\le 2\) nm, according to Eq. (2.61), we need \( \theta \le 0.0797\ \text {rad}\approx \frac{2\pi \ \text {rad}}{79}, \) which means 79 bending magnets are required in the ring. If the length of each isochronous cell with a single bending magnet can be designed to be around 2 m, then the circumference of the ring can be about 180 m, considering the sections of laser modulation, radiation generation and the energy supply system, etc.
Applying the lowest phase slippage factor realizable in practice at present, which is about \(\eta =1\times 10^{-6}\), to realize a 3 nm bunch length in such a ring, the required energy chirp strength is then \( h=\eta C_{0}\left( \frac{\sigma _{\delta {\text {S}}}}{\sigma _{z\text {S}}}\right) ^{2}=5.88\times 10^{5}\ \text {m}^{-1}. \) The effective modulation voltage of a laser modulator using a planar undulator is related to the laser and undulator parameters as [27]
in which \([JJ]=J_{0}(\chi )-J_{1}(\chi )\) with \(J_{n}\) the n-th order Bessel function of the first kind and \(\chi =\frac{K^{2}}{4+2K^{2}}\), \(K=\frac{eB_{0}}{m_{e}ck_{u}}=0.934\cdot B_{0}[\text {T}]\cdot \lambda _{u}[\text {cm}]\) is the dimensionless undulator parameter, determined by the undulator period and magnetic flux density, \(P_{L}\) is the modulation laser power, \(Z_{0}=376.73\ \Omega \) is the impedance of free space, \(Z_{R}\) is the Rayleigh length of the 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. If \(\lambda _{L}=270\) nm, \(\lambda _{u}=4\) cm, \(B_{0}=1.02\) T, \(L_{u}=1.6\) m, \(Z_{R}=\frac{L_{u}}{3}\), then the required modulation laser power to get \( h=5.88\times 10^{5}\ \text {m}^{-1} \) is \( P_{L}=2.75\ \text {GW}. \) At present, 1 MW stored average laser power is the state-of-art value we can realize in practice for an optical enhancement cavity. So we can only operate the cavity in pulsed mode, which means the average radiation power will be limited. To put it another way, if a CW optical cavity and a practical global phase slippage is applied, a longitudinal weak focusing SSMB storage ring can only realize bunch length as low as tens of nanometer with a beam energy of several hundred MeV. Such an SSMB ring can provide high-power radiation with wavelength \(\lambda _{R}\gtrsim 100\) nm. We have presented in Table 6.1 of the final chapter an example parameters set of SSMB ring for high-power infrared radiation. We remind the readers again that considering the nonlinear modulation waveform, we actually need \(0<h\eta C_{0}\lesssim 0.1\) to avoid strong chaotic dynamics in a longitudinal weak focusing ring.
The above limitation of longitudinal weak focusing scheme is the motivation for us to develop the longitudinal strong focusing SSMB and transverse-longitudinal coupling SSMB, or generalized longitudinal strong focusing SSMB, to compress the bunch length further for coherent EUV and soft X-ray radiation generation. We will present the details of these advanced scenarios in the following part of this dissertation.
2.1.5.2 Transverse Gradient Bends
The above analysis of theoretical minimum bunch length and emittance is for a constant bending radius. To minimize the longitudinal emittance further, transverse and longitudinal gradient bending magnets (TGB and LGB) can be invoked. Below we conduct some calculations based on the similar dispersion configuration as shown in Fig. 2.5, but this time using a TGB. The Hill’s equation for the dispersion is
For simplicity, here we only investigate the case of a constant bending radius \(\rho (s)=\rho \) and a constant transverse gradient \(k(s)=k\). To simplify the writing, we denote
If \(g>0\), then the solution of Eq. (2.66) is
where \(D_{\text {i}}\) and \(D'_{\text {i}}\) are the initial dispersion and dispersion angle at the origin \(s=0\) m, respectively. If \(g<0\), then the solution of Eq. (2.66) is
Below, we present the derivations for the case of \(g>0\) and the results are similar when \(g<0\). Like our previous calculations, we set the origin at the middle of the dipole where \(D'_{\text {m}}=0\), the dispersion as a function of angle \(\phi \) is then
Substitute Eq. (2.70) into the isochronicity condition Eq. (2.52), we get
The \(\sqrt{\langle F^{2}\rangle _{\rho }-\langle F\rangle _{\rho }^2}\) in this case can be calculated as follows
For example, to reduce \(\sqrt{\langle F^{2}\rangle _{\rho }-\langle F\rangle _{\rho }^{2}}\) by a factor of two compared to the case of no transverse gradient, we need
For the example shown in last section, \(\rho =4\) m, \(\theta =\frac{2\pi }{79}\), then \(k=63\ \text {m}^{-2}\), which is a practical gradient.
Besides the influence on \(\sqrt{\langle F^{2}\rangle _{\rho }-\langle F\rangle _{\rho }^{2}}\), the transverse gradient may also affect the damping partition and hence has an impact on the bunch length and longitudinal emittance. For the specific case of a constant bending radius with a constant transverse gradient we are treating, we have
where \(I_{2}\) and \(I_{4}\) are the radiation integrals [5]. Then \(J_{s}=2+\frac{I_{4}}{I_{2}}=2\). So a dipole with a constant bending radius and a constant transverse gradient is not very flexible in controlling the damping partition number, due to the constraint of isochronous condition. A varying transverse gradient may be helpful to minimize the longitudinal emittance, and optimization of the transverse gradient profile based on numerical method can be invoked. The application of TGB can also be analyzed following the same formalism, which we do not detail in this dissertation.
2.1.5.3 Transverse Emittance Scaling
For completeness, now we present the horizontal emittance scaling in a longitudinal weak focusing SSMB storage ring. To realize the dispersion function pattern shown in Fig. 2.5, in thin-lens approximation, the horizontal optical functions at the dipole middle point are correspondinglyFootnote 1
with \(\Phi _{x}\) the betatron phase advance per isochronous cell, which usually lies in \((\pi ,2\pi )\). We have assumed there is only a single dipole each isochronous cell. The normalized eigenvector corresponding to the horizontal plane at the dipole middle point is
The transfer matrix of a sector dipole with no transverse gradient is
Then
Note that \( \mathcal {H}_{x}(-\alpha )=\mathcal {H}_{x}(\alpha ). \) Putting in \( \alpha _{H}\approx \frac{\alpha _{L}}{2}\approx \frac{U_{0}}{2E_{0}}=\frac{1}{2}C_{\gamma }\frac{E_{0}^{3}}{\rho }, \) then according to Eq. (2.14), the equilibrium horizontal emittance is
Putting in the numbers, we have
Note that the horizontal emittance diverges as \(\Phi _{x}\) approaches \(\pi \) or \(2\pi \).
2.1.6 Longitudinal Strong Focusing
2.1.6.1 Analysis
The analysis in the above sections considers the case with only a single RF. When there are multiple RFs, for the longitudinal dynamics, it is similar to implement multiple quadrupoles in the transverse dimension, and the beam dynamics can have more possibilities. Longitudinal strong focusing scheme for example can be invoked [8, 28], not unlike its transverse counterpart which is the foundation of modern high-energy accelerators [29, 30]. The linear beam dynamics with multiple RFs can be treated using SLIM the same way as that with a single RF. When all the RFs are placed at dispersion-free locations, the Courant-Snyder parametrization can be applied as analyzed in previous sections. Here we use a setup with two RFs as an example to show the scheme of manipulating \(\beta _{z}\) around the ring. The schematic layout of the ring is shown in Fig. 2.6. The treatment of cases with more RFs is similar.
A schematic layout of a storage ring using two RF systems for longitudinal strong focusing and an example beam distribution evolution in the longitudinal phase space. Note that the tilted angles of the beam distribution and bunch length ratios at different places do not strictly correspond to the parameters in Table 2.2, but only to present the qualitative characteristics
We divide the ring into five sections, i.e., three longitudinal drifts (\(R_{56}\)) and two longitudinal quadrupole kicks (h), with the transfer matrices given by
Then the one-turn map at the radiator center is
Linear stability requires that \(|\text {Tr}\left( {\textbf {M}}_{\text {R}}\right) |<2\), where Tr() means the trace of. For the generation of coherent radiation, we usually want the bunch length to reach its minimum at the radiator, then we need \(\alpha _{z}=0\) for \({\textbf {M}}_{\text {R}}\).
With the primary purpose to present the principle, instead of a detailed design, here for simplicity we only discuss one special case: \(R_{56}^{(1)}=R_{56}^{(3)}\), \(h_{1}=h_{2}=h\). The treatment of more general cases with different signs and magnitudes of \(R_{56}^{(1)}\) and \(R_{56}^{(3)}\) and \(h_{1}\) and \(h_{2}\) is similar, but the same-signed \(R_{56}^{(1)}\) and \(R_{56}^{(3)}\) might be easier for a real lattice to fulfill. For example if \(R_{56}^{(1)},R_{56}^{(3)}>0\), a possible realization of them are chicanes.
For the special case of \(R_{56}^{(1)}=R_{56}^{(3)}\), \(h_{1}=h_{2}=h\) and denote \( \zeta _{1}\equiv 1+R_{56}^{(1)}h,\ \zeta _{2}\equiv 2+R_{56}^{(2)}h, \) we then have
The linear stability requires \( |\zeta _{1}\zeta _{2}-1|<1, \) and the synchrotron tune is
Here we give one example parameter set with a stable linear motion as shown in Table 2.2. According to the longitudinal Courant-Snyder functions given in Table 2.2 (note the values of \(\beta _{z}\) and the signs of \(\alpha _{z}\)), the evolution of electron distribution in the longitudinal phase space around the ring (note the bunch lengths and orientations) is qualitatively shown in Fig. 2.6. If we can realize \(\epsilon _{z}\lesssim 5\) pm in such a strong focusing ring, then we have \(\sigma _{z}(s_{\text {rad}})=\sqrt{\epsilon _{z}\beta _{z}(s_{\text {rad}})}\lesssim 3\) nm. We remind the readers that the contribution of modulators to longitudinal emittance should be carefully counted in a longitudinal strong focusing SSMB storage ring. Note that the energy chirp strength needed here is one order of magnitude smaller than the example of using a single RF or laser modulator to realize 3 nm as discussed just now. This benefit originates from a much compressed \(\beta _{z}\) at the radiator in a strong focusing ring. However, we recognize that the laser power needed (20 MW level if 270 nm-wavelength laser is applied) is still demanding, and here our primary goal is to present the principle based on which the interested readers can choose and optimize the parameters for their target applications. We will discuss in Chap. 3 the application of transverse-longitudinal coupling scheme to lower the requirement on the modulation laser power further, to make the optical cavity can be operated in CW mode, thus to improve the filling factor of electron beam in the ring and the average output radiation power.
2.1.6.2 Discussions
Here we make several observations from the above analysis and numerical example, which we believe are important. First, \(\beta _{z}\) in a longitudinal strong focusing ring can be at the same level of or even smaller than the ring \(|R_{56}=-\eta C_{0} |\), while in a longitudinal weak focusing ring \(\beta _{z}\gg |-\eta C_{0}|\). Therefore, the bunch length can thus be much smaller than that in a longitudinal weak focusing ring. This is the reason behind the application of longitudinal strong focusing in SSMB to realize extreme short bunches [28, 31]. We remind the readers that the longitudinal emittance of electron beam in a longitudinal strong focusing ring still cannot be smaller than that given in Eq. (2.63), due to the intrinsic partial phase slippage, thus the evolution of longitudinal beta function, in a dipole.
Second, \(\beta _{z}\) changes significantly around the ring in the longitudinal strong focusing regime. Therefore, the bunch length and beam orientation in the longitudinal phase space varies greatly around the ring, as shown qualitatively in Fig. 2.6. This means the adiabatic approximation cannot be applied for the longitudinal dimension anymore. Actually the adiabatic approximation also breaks down in the case corresponds to Fig. 2.3, where the change of \(\beta _{z}\) around the ring is significant although the total synchrotron phase advance per turn is small. Therefore, the global synchrotron tune is not a general criterion in the classification of whether the adiabatic approximation fails. The evolution of \(\beta _{z}\) is more relevant. The argument is based on the fact that \(R_{56}\) can be either positive or negative, therefore the local synchrotron phase advance can be either positive or negative. While in the transverse dimension, the drift length and betatron phase advance are always positive.
The breakdown of adiabatic approximation may have crucial impacts on the study of both the single-particle and collective effects. For linear single-particle dynamics, the longitudinal and transverse dimensions should be treated the same way on equal footing and SLIM formalism can be invoked. The treatment of nonlinear single-particle dynamics is more subtle as the longitudinal dynamics now is strongly chaotic. For the collective effects, many classical treatments should be re-evaluated and some new formalism needs to be developed. For example, the Haissinski equation [32] for calculating the equilibrium beam distortion cannot be applied directly then. Also, to our knowledge, there is no discussion on coherent synchrotron radiation (CSR)-induced microwave instability in a longitudinal strong focusing ring. The scaling law obtained in the longitudinal weak focusing [33] cannot be applied directly. 3D CSR effects and also the impact of bunch lengthening from transverse emittance on CSR needs more in-depth study. This is especially true for an SSMB ring, considering the fact that the beam width there is much larger than the microbunch length, while the contrary is true in a conventional ring. The contribution from horizontal emittance can easily dominate the bunch length at many places in an SSMB ring. This on the other hand, will be helpful to suppress unwanted CSR and may also be helpful in mitigating intrabeam scattering (IBS) [34, 35], as extreme short bunches occur only at limited locations. The IBS in a longitudinal strong focusing ring, and a general coupled lattice, also deserves special attention. To our knowledge, the IBS formalism of presented in Refs. [36, 37] can be applied for such purposes, as they are based on \(6\times 6\) general transport matrices. An IBS formalism can also be developed based on SLIM formalism [9], in which eigen analysis has been invoked and applies to 3D general coupled lattice with longitudinal strong focusing.
2.1.7 Thick-Lens Maps of a Laser Modulator
In the previous discussions, we have approximated the function of a laser modulator by a thin-lens RF-like kick. This means that we have ignored the phase slippage or \(R_{56}\) of the laser modulator itself. We need to know if this approximation is valid or under what circumstance we can use this approximation.
Here we first derive the phase slippage factor of the undulator and then get the thick-lens transfer matrix of a laser modulator. The path length of an electron with a relative energy deviation of \(\delta \) wiggling in a planar undulator is
in which \(k_{u}=2\pi /\lambda _{u}\) is the undulator wavenumber, \(\gamma _{r}\) is the Lorentz factor corresponding to the resonant energy. The \(R_{56}\) of an undulator is then
where \(N_{u}\) is the number of undulator periods, \(\lambda _{0}=\frac{1+K^{2}/2}{2\gamma _{r}^{2}}\lambda _{u}\) is the central wavelength of the on-axis fundamental spontaneous radiation. As can be seen from Eq. (2.86), the undulator \(R_{56}\) is twice the slippage length of the undulator radiation.
As mentioned, the RF or laser modulator kick in linear approximation is like a longitudinal quadrupole and the \(R_{56}\) of the laser modulator is like the longitudinal drift space length of this longitudinal quadrupole. Assuming that the energy modulation is uniform along the undulator, then similar to the thick-lens quadrupole in the transverse dimension, we have the thick-lens transfer matrix of a laser modulator
A laser modulator in linear approximation is therefore like a thick-lens quadrupole in the longitudinal dimension. A thin-lens approximation is applicable when \(|R_{56}h|\ll 1\).
After discussing the linear map, now we take into account the fact that the modulation waveform of a laser is actually sinusoidal. In principle, we can get an approximate analytical nonlinear thick-lens transfer map of the laser modulator using the techniques of drift and kick and Lie algebra [38, 39], by slicing the interaction into several smaller pieces and concatenate the maps of thin-lens kickes and drift spaces. Here we use a more straightforward method, i.e., to implement a symplectic kick map as below in a numerical code, to give the readers a picture. The kick map implemented in the code is as follows
In other words, we have split the undulator into \(N_{u}\) small “\(\frac{1}{2}\)dispersion + modulation + \(\frac{1}{2}\)dispersion”. For a plane wave \( A_{i}=\frac{A}{N_{u}} \) with A being the total energy modulation strength, while for a Gaussian laser beam \(A_{i}\) is a function of i.
Impact of modulator undulator period number \(N_{u}\) (thus \(R_{56}\)) on the single-pass modulation process in a laser modulator. The injected beam is “\(\delta =0\) and \(z\in [0,\lambda _{0}]\)”. (x, y) in the figures from left to right: (\(z_{\text {entrance}}\), \(\delta _{\text {exit}}\)); (\(z_{\text {entrance}}\), \(z_{\text {exit}}\)); (\(z_{\text {exit}}\), \(\delta _{\text {exit}}\)). Parameters used: \(\lambda _{L}=1\ \upmu {m},\ A=1\times 10^{-3}\)
To simplify the analysis, in the example numerical simulation, we consider the case of a plane wave, i.e., \( A_{i}=\frac{A}{N_{u}}\). First, we want to see how the single-pass modulation waveform is like based on Eq. (2.88). We choose parameters \(\lambda _{L}=\lambda _{0}=1\ \upmu {m},\ A=1\times 10^{-3}\). The single-pass modulation of a line beam, with “\(\delta =0\) and \(z\in [0,\lambda _{0}]\)”, as a function of \(N_{u}\), namely \(R_{56}\), of the undulator is shown in Fig. 2.7. When changing \(N_{u}\), we keep A unchanged. As can be seen, the waveform deviates from sine wave when \(N_{u}\) increases. The beam distribution in phase space at the undulator exit (right sub-figure of Fig. 2.7) is similar to the beam de-coherence in an RF bucket.
Now we consider the multi-pass cases, i.e., we consider the impact of modulator \(R_{56}\) on the phase space bucket. But here we do simulation only for the longitudinal weak focusing with a single laser modulator, as we only aim to give the readers a picture about such impact. We use parameters of \( \lambda _{L}=\lambda _{0}=1\ \upmu {m},\ A=1\times 10^{-3},\ C_{0}=100\ \text {m},\ \eta =5\times 10^{-7} \) in the simulation, and choose to observe the beam opposite the modulator center where \(\alpha _{z}=0\) with \(N_{u}=0,40,320\), respectively. Note that \(\eta \) is the phase slippage factor of the whole ring, including the modulator. When changing the undulator priod number \(N_{u}\), we keep \(\eta \) a constant. The results are shown in Fig. 2.8. It can be seen that the modulator \(R_{56}\) only distorts the bucket slightly when \(N_{u}\) is 40. But when \(N_{u}\) is as large as 320, it will have a profound effect on the longitudinal phase space topology. Its impact on longitudinal strong focusing is more subtle as the particle motion in a longitudinal strong focusing ring is strongly chaotic if the nonlinear modulation waveform is taken into account. The study of such effect can refer more straightforwardly to numerical simulations. Besides, the undulator \(R_{56}\) could also have an impact on the coherent radiation induced collective instability in the laser modulator [40, 41].
Impact of modulator undulator period number \(N_{u}\) (thus \(R_{56}\)) on the longitudinal phase space bucket in longitudinal weak focusing regime. Parameters used: \( \lambda _{L}=1\ \upmu {m},\ A=1\times 10^{-3},\ C_{0}=100\ \text {m},\ \eta =5\times 10^{-7} \)
2.2 Nonlinear Longitudinal Dynamics
After resolving the issue of quantum diffusion of z by means of dedicated lattice design, we can then apply the quasi-isochronous or low-phase slippage method to realize short bunches in SSMB. However, the phase slippage is actually a function of the particle energy
When \(\eta _{0}\) is sufficiently small, the higher-order terms in Eq. (2.89) may become relevant or even dominant, and the beam dynamics can be significantly different from those in a linear-phase slippage state. Proper application of dedicated sextupoles and octupoles may be needed to control these higher-order terms.
The beam dynamics of the quasi-isochronous rings have been studied by many authors [15, 16, 42]. Here, we wish to emphasize two points that have not been well investigated before and might be important, for example, in the SSMB proof-of-principle experiment to be introduced in Chap. 5 and the longitudinal dynamic aperture optimization in SSMB.
2.2.1 For High-Harmonic Bunching
For seeding techniques such as coherent harmonic generation (CHG) [43] and high-gain harmonic generation (HGHG) [44, 45], it seems that to date, linear phase slippage or \(R_{56}\) has been applied for microbunching formation. Here, we wish to point out that one can actually take advantage of the nonlinearity of the phase slippage for high harmonic generation. Intuitively, this is because a sinusoidal energy modulation followed by a nonlinear phase slippage can lead to a distorted current distribution, which, in some cases, can lead to large bunching at a specific harmonic number. Figure 2.9 shows an example simulation of using \(\eta (\delta )=\eta _{1}\delta \) for microbunching. It can be seen that there is significant bunching in the second and fourth harmonics, while no bunching is produced in the fundamental and third harmonics. The reason can be found from the following derivation of the bunching factor.
The beam evolution in longitudinal phase space, final current distribution and bunching factor, when \(\eta (\delta )=\eta _{1}\delta \) is used for microbunching in CHG or HGHG, as modeled by Eq. (2.90)
The microbunching process in the case of a single energy modulation followed by a dispersion section, as that in CHG and HGHG, can be modeled as
where \(k_{L}=2\pi /\lambda _{L}\) is the wavenumber of the modulation laser, A is the electron energy modulation strength induced by the laser. The bunching factor at the wavenumber of k is defined as
where \(\rho (z)\) is the normalized longitudinal density distribution of the electron beam satisfying \(\int _{-\infty }^{\infty }dz\rho (z)=1\). According to Liouville’s theorem, we have \(dzd\delta =dz'd\delta '\). Therefore, the bunching factor can be calculated in accordance with the initial distribution of the particles \(\rho _{0}(z,\delta )\) as
Here we consider the simple case of \(\eta (\delta )\equiv \eta _{0}+\eta _{1}\delta \), then
Adopting the notation \(Y\equiv k\left( \eta _{0}C_{0}A+2\eta _{1}C_{0}\delta A\right) ,\ Z\equiv -k\eta _{1}C_{0}A^{2}/{2}\), and using the mathematical identity \( e^{ix\sin (k_{L}z)}=\sum _{n=-\infty }^{\infty }e^{ink_{L}z}J_{n}[x], \) we have
If the initial beam is much longer than the laser wavelength, and considering that \( \frac{1}{2\pi }\int _{-\infty }^{\infty }e^{-i\omega t}d\omega =\delta (t), \) where \(\delta (t)\) is the Dirac delta function, the bunching factor will not vanish only if \( k=(p-2q)k_{L} \). The bunching factor at the n-th harmonic of the modulation laser is then
Here we consider the simple case of an initial Gaussian energy distribution \(\rho _{0}(\delta )=\frac{1}{\sqrt{2\pi }\sigma _{\delta }}\text {exp}\left( -\frac{\delta ^{2}}{2\sigma _{\delta }^{2}}\right) \), where \(\sigma _{\delta }\) is the initial RMS energy spread. If \(\eta _{1}=0\), then \(Y=nk_{L}\eta _{0}C_{0}A\), \(Z=0\), and \(\sum _{m=-\infty }^{\infty }J_{n+2m}[Y]J_{m}[Z]=J_{n}[nk_{L}\eta _{0}C_{0}A]\), and we have
which is a familiar result for HGHG [44] if we adopt the notation \(R_{56}=-\eta _{0}C_{0}\). If \(\eta _{0}=0\), then \(Y=2nk_{L}\eta _{1}C_{0}\delta A\), \(Z=-nk_{L}\eta _{1}C_{0}A^{2}/{2}\), meaning that we have
The two exponential terms in the integral are even functions of \(\delta \), while \(J_{n+2m}[Y]J_{m}[Z]\) is an odd function of \(\delta \) when n is odd; thus, \(b_{n}\) is nonzero only for an even n. This is why bunching occurs only in the second and fourth harmonics but not in the fundamental and third harmonics when we use \(\eta (\delta )=\eta _{1}\delta \) for microbunching, as shown in Fig. 2.9.
Following the derivations and according to the relation
it can be seen that the energy modulation at the fundamental frequency can be cast into \(\left[ i\times (n-2p)+j\times (n-2q)\right] \)-th harmonic bunching through the term \(\eta _{n-1}\delta ^{n}\) in the function of \(\eta (\delta )\). For an odd n, bunching at all harmonic numbers are possible, while for an even n, only bunching at the even harmonic numbers is possible. The optimal bunching condition for a specific harmonic requires the matching of \(\eta (\delta )\) with the energy modulation strength. However, the analytical formula for the bunching factor will become increasingly involved with more higher-order terms of the phase slippage considered. Thus, it would be better to refer to numerical code to calculate and optimize the bunching factor directly for a specific application case. For storage rings, another relevant point is that the distribution of the particle energy in the nonlinear phase slippage state may also have an impact on the high harmonic generation, and this phenomenon is also easier to be studied by means of numerical simulation.
The approach of applying a nonlinear phase slippage for high harmonic bunching can be considered to share some similarity with echo-enabled harmonic generation (EEHG) [46, 47], in which the sinusoidal energy modulation and dispersion in the first stage can be viewed as the source of the distorted current distribution in the second stage of modulation and dispersion for microbunching. We have also noticed the work on optimizing the nonlinearity of the dispersion to increase the bunching factor for EEHG [48]. Based on similar considerations, tricks can also be applied on the energy-modulation waveform using different harmonics of the modulation laser, for example, forming a sawtooth waveform to boost bunching, as will be discussed in Chap. 3.
2.2.2 For Longitudinal Dynamic Aperture
Similar to the transverse dimension, there is a region in the longitudinal phase space outside of which particle motion is not bounded and can be lost in a ring. We refer this stable region as the longitudinal dynamic aperture. Here in this section, we want to show that, by properly tailoring the nonlinear phase slippage, the longitudinal dynamic aperture can be enlarged significantly compared to the case of a pure linear phase slippage. Only symplectic dynamics is considered in this discussion.
2.2.2.1 Longitudinal Weak Focusing
The longitudinal dynamics of a particle in a ring with a single RF can be modeled by the kick map
where \(A\sin \phi _{\text {s}}=U_{0}/E_{0}\) where \(U_{0}\) is the radiation loss of a particle per turn. For the case of longitudinal weak focusing, the kicik map can be approximated by differentiation and Hamiltonian formalism can be invoked for the analysis. Denote \(\phi \equiv k_{\text {RF}}z\), then the equation of motion is
with \(T_{0}\) being the revolution period of the particle in the ring. For \(\eta (\delta )=\eta _{0}+\eta _{1}\delta +\eta _{2}\delta ^{2}\), the corresponding Hamiltonian is
In writing down the closed-form Hamiltonian, we have implicitly assumed that the motion is integrable, i.e., there is no chaos. But we need to keep in mind that the dynamics dictated by Eq. (2.99) is actually chaotic even with a linear phase slippage [13]. But here we ignore this subtle point as the chaotic layer is very thin in the longitudinal weak focusing regime. We remind the readers that the chaotic dynamics, for example the bucket bifurcation, can actually also be applied for ultrashort bunch generation [49].
To analyze the motion, we need to find the fixed points of the system
To determine whether a fixed point is stable or not, we need to check the trace of the Jacobian matrix around the fixed point. If \(\eta (\delta )=\eta _{0}\), there are two sets of fixed points:
in which SFP stands for stable fixed point while UFP stands for unstable fixed point. If \(\eta (\delta )=\eta _{0}+\eta _{1}\delta \), there are four sets of fixed points:
If \(\eta (\delta )=\eta _{0}+\eta _{1}\delta +\eta _{2}\delta ^{2}\), there are six sets of fixed points
The impact of \(\eta _{1}\) on the longitudinal phase space bucket in the longitudinal weak focusing regime. Simulation parameters: \(\lambda _{\text {RF}}=1\ \upmu \)m, \(A=1\times 10^{-3}\), \(C_{0}=100\) m, \(\eta _{0}=5\times 10^{-7}\)
The impact of \(\eta _{2}\) on the longitudinal phase space bucket in the longitudinal weak focusing regime. Simulation parameters: \(\lambda _{\text {RF}}=1\ \upmu \)m, \(A=1\times 10^{-3}\), \(C_{0}=100\) m, \(\eta _{0}=5\times 10^{-7}\)
To see the impact of \(\eta _{1}\) and \(\eta _{2}\) on the longitudinal phase space bucket, some numerical simulations are conducted. We choose the observation point at the middle of the RF, where \(\alpha _{z}=0\) and the beam distribution in the longitudinal phase space is upright. The results of the impact of \(\eta _{1}\) and \(\eta _{2}\) on longitudinal dynamical aperture are shown in Figs. 2.10 and 2.11, respectively. Note that in the plots, we have used the longitudinal coordinate z rather than the phase \(\phi \).
As we can see in Fig. 2.10, the emergence of \(\eta _{1}\) will make the bucket asymmetric in \(\delta \), which is as expected as the \(\eta =\eta _{0}+\eta _{1}\delta \) is asymmetric in \(\delta \). In both directions (positive or negative), the bucket size shrinks with the increase of \(\eta _{1}\) and the bucket becomes like an upright \(\alpha \)-shape, so they are usually referred to as \(\alpha \)-buckets. Note that we can also classify the bucket to be an RF-bucket or an \(\alpha \)-bucket according to whether \(\delta =0\) or \(\eta (\delta )=0\) at the bucket center, respectively. Such a classification is more reasonable from beam dynamics consideration. \(\alpha \)-bucket is also a method to generate short bunch and there are many interesting beam dynamics issues related to such buckets [16].
However, the impact of \(\eta _{2}\) is different. As can be seen from the simulation results presented in Fig. 2.11, the bucket is still symmetric in \(\delta \) as expected. Besides, when \(\frac{\eta _{2}}{\eta _{0}}<0\), the stable region of the bucket can be even larger than the case without \(\eta _{2}\). We will see later that in the case of longitudinal strong focusing, such observation can be even more notable. Therefore, we can tailor the phase slippage factor as a function of energy in the case of longitudinal strong focusing to enlarge the longitudinal dynamic aperture. This is very helpful as usually the longitudinal dynamic aperture in a longitudinal strong focusing ring is not a trivial issue and needs to be optimized to guarantee a sufficient quantum lifetime for example.
2.2.2.2 Longitudinal Strong Focusing
For a longitudinal strong focusing ring, the particle motion is strongly chaotic and not integrable, and we cannot get a closed-form Halmitonian for analysis anymore. Therefore, we use numerical simulations to study the dynamics directly. Instead of a comprehensive investigations, here in this section we aim to give some qualitative remarks on the role of the nonlinear phase slippage, i.e., a proper tailoring of the nonlinear phase slippage can enlarge the longitudinal dynamic aperture significantly.
The impact of \(\eta _{1}\) on the longitudinal phase space bucket in the longitudinal strong focusing regime. Simulation parameters: \(\lambda _{\text {RF}}=1\ \upmu \)m, \(h=-50000\ \text {m}^{-1}\), \(R_{56}^{(1)}=15\ \upmu \)m, \(C_{0}=100\) m, \(\eta _{0}=1\times 10^{-6}\)
Let us use the schematic layout shown in Fig. 2.6 and parameters choices given in Table 2.2 as an example for illustration. Note that the modulation wavelength used here is \(\lambda _{\text {RF}}=1\ \upmu \)m. We choose to observe the beam at the radiator center, and suppose the ring is symmetric with respect to the radiator. The one-turn kick map is then
As we aim to present the main physical picture, here we only consider the nonlinearity of the main ring first, i.e., \(R_{56}^{(2)}(\delta )=-C_{0}\left( \eta _{0}+\eta _{1}\delta +\eta _{2}\delta ^{2}\right) \). \(R_{56}^{(1)}\) and \(R_{56}^{(3)}\) in principle can also be a function of \(\delta \). The simulation results are shown in Figs. 2.12 and 2.13.
The impact of \(\eta _{2}\) on the longitudinal phase space bucket in the longitudinal strong focusing regime. Simulation parameters: \(\lambda _{\text {RF}}=1\ \upmu \)m, \(h=-50000\ \text {m}^{-1}\), \(R_{56}^{(1)}=15\ \upmu \)m, \(C_{0}=100\) m, \(\eta _{0}=1\times 10^{-6}\)
From Fig. 2.12, we know that, like that in the weak focusing case, \(\eta _{1}\) makes the bucket asymmetric in \(\delta \) and shrinks the bucket size whether \(\eta _{1}\) is positive or negative. From Fig. 2.13, we can see that when \(\frac{\eta _{2}}{\eta _{0}}<0\), a proper \(\eta _{2}\) can help to merge the island buckets with the main bucket and broaden the stable region of the phase space, i.e., the longitudinal dynamic aperture, significantly. A proper \(\eta _{2}\) makes the amplitude dependent tune shift favorable for the motion to be stable. Note that the fixed points of the island buckets may not have period-1 but period-n stability.
Notes
- 1.
Private communication with Zhilong Pan.
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Deng, X. (2024). SSMB Longitudinal Dynamics. In: Theoretical and Experimental Studies on Steady-State Microbunching. Springer Theses. Springer, Singapore. https://doi.org/10.1007/978-981-99-5800-9_2
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