Summary

Abstract

In this final chapter, we give a brief summary of the dissertation, and present some useful results for practitioners.

In this final chapter, we give a brief summary of the dissertation, and present some useful results for practitioners.

6.1 Summary of the Dissertation

The contribution of this dissertation consists of three parts: in Chaps. 2 and 3, we answer the question of how to realize SSMB; in Chap. 4, we investigate what radiation characteristics can be obtained from the formed SSMB; and in Chap. 5, we experimentally demonstrate the working mechanism of SSMB in a real machine for the first time.

In Chap. 2, to account for the impact of local phase slippage factors on beam dynamics in a quasi-isochronous electron storage ring, we have developed and applied the Courant-Snyder formalism in longitudinal dimension to derive new formulae of bunch length, energy spread and longitudinal emittance beyond the classical scaling laws. The method of optimizing the global and local phase slippages simultaneously to minimize the longitudinal $\beta$ function at the bending magnets has been proposed based on the analysis, to generate an ultrashort bunch length and ultrasmall longitudinal emittance, as required by SSMB. Further, we have derived the scaling law of the theoretical minimum bunch length and longitudinal emittance with respect to the bending radius and angle of the bending magnet. The use of transverse gradient bends for minimizing the longitudinal emittance has also been investigated. The application of multiple RF cavities, or laser modulators in an SSMB storage ring, for longitudinal strong focusing has been discussed using the same formalism with important observations made. Considering the momentum compaction of a laser modulator, its thick-lens linear and nonlinear maps have been derived and simulated for a more accurate modeling of beam dynamics in it. We have also studied the application of the higher-order terms of phase slippage for high-harmonic bunching and longitudinal dynamic aperture optimization. Based on the investigations in this chapter, we have presented in Table 6.1 an example parameters set of a longitudinal weak focusing SSMB storage ring for high-power infrared radiation generation.

In Chap. 3, we have presented a concise analysis of the bending magnet-induced passive bunch lengthening from transverse emittance of the particle beam. After that, we have generalized the analysis and proved three theorems on the active applications of transverse-longitudinal coupling (TLC) for efficient harmonic generation or bunch length compression. These theorems dictate the relation between the modulation lick strength and the lattice optical functions at the modulator and radiator, respectively. 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 in applying these TLC schemes in a storage ring. These 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. The relation between our TLC analysis and the transverse-longitudinal emittance exchange is also briefly discussed. In addition to the investigation on linear TLC dynamics, we have also reported the first experimental validation of particle energy widening and distortion by a nonlinear TLC effect in a quasi-isochronous ring, which originates from an average path-length dependence on the betatron oscillation amplitudes. The result could be important for quasi-isochronous rings, SSMB, nonscaling fixed-field alternate gradient accelerators, etc., where very small phase slippage factor or large chromaticity is required. Based on the investigations in this chapter, we have presented in Table 6.2 an example parameters set of a transverse-longitudinal coupling SSMB storage ring for high-power EUV and soft X-ray radiation generation.

In Chap. 4, we have presented theoretical and numerical studies of the average and statistical property of the coherent radiation from SSMB. Our results show that kW-level average power of 13.5 nm-wavelength EUV radiation can be obtained from an SSMB ring, provided that an average current of 1 A and bunch length of 3 nm microbunch train can be formed at the radiator. Such a high-power EUV source is a promising candidate to fulfill the urgent need of semiconductor industry for EUV lithography. Together with its narrowband feature, the EUV photon flux can reach $10^{15}\;\sim \;10^{16}$ phs/s within a 0.1 meV energy bandwidth, which is appealing for fundamental condensed matter physics research. In the theoretical investigation, we have generalized the definition and derivation of the transverse form factor of an electron beam which can quantify the impact of its transverse size on the coherent radiation. In particular, we have shown that the narrowband feature of SSMB radiation is strongly correlated with the finite transverse electron beam size. Considering the pointlike nature of electrons and quantum nature of radiation, the coherent radiation fluctuates from microbunch to microbunch, or for a single microbunch from turn to turn. Some important results concerning the statistical property of SSMB radiation have been presented, with a brief discussion on its potential applications for example the beam diagnostics. The presented work is of value for the development of SSMB and better serve the potential synchrotron radiation users. In addition, it also sheds light on understanding the radiation characteristics of free-electron lasers, coherent harmonic generation, etc.

In Chap. 5, we have reported the first demonstration of the mechanism of SSMB at the Metrology Light Source in Berlin. We have shown that electron bunches stored in a quasi-isochronous ring can yield sub-micrometre microbunching and narrowband coherent radiation, one complete revolution after energy modulation induced by a 1064 nm wavelength laser, and this microbunching can preserve for multiple turns. These results verify that the optical phases, i.e, the longitudinal coordinates, of electrons can be correlated turn by turn in a storage ring at a precision of sub-laser wavelengths. On the basis of this phase correlation, we expect that SSMB will be realized by applying a phase-locked laser that interacts with the electrons turn by turn. This demonstration represents the first milestone towards the implementation of an SSMB-based high-power, high-repetition photon source.

6.2 Useful Formulas and Example Parameters for SSMB Storage Rings

To make our investigations more useful for practitioners, especially concerning the parameters choice for an SSMB storage ring, here we present some important formulas. Generally we group our formulas into two categories, i.e., a longitudinal weak focusing storage ring for a desired radiation wavelength $\lambda _{R}\gtrsim 100$ nm, and a transverse-longitudinal coupling storage ring for a desired radiation wavelength $1~\text {nm}\lesssim \lambda _{R}\lesssim 100$ nm. In each category, we have presented an example parameters set for the corresponding SSMB storage ring.

6.2.1 Longitudinal Weak Focusing SSMB

The relation of bending radius $\rho$ and magnetic flux density B of the bending magnet is

$$\begin{aligned} \frac{1}{\rho }=0.2998\frac{B[\text {T}]}{E_{0}[\text {GeV}]}, \end{aligned}$$

(6.1)

with $E_{0}$ the electron energy.

Assuming that the storage ring consists of isomagnets, then the radiation loss of an electron per turn is

$$\begin{aligned} U_{0}=C_{\gamma }\frac{E_{0}^{4}}{\rho _{\text {ring}}}, \end{aligned}$$

(6.2)

with $C_{\gamma }=8.85\times 10^{-5}\frac{\text {m}}{\text {GeV}^{3}}$, $\rho _{\text {ring}}$ the bending radius of bending magnets in the ring.

The horizontal, vertical and longitudinal radiation damping constants for a planar uncoupled ring are

$$\begin{aligned} \begin{aligned} \alpha _{H}&=\frac{U_{0}}{2E_{0}}(1-\mathcal {D}),\\ \alpha _{V}&=\frac{U_{0}}{2E_{0}},\\ \alpha _{L}&=\frac{U_{0}}{2E_{0}}(2+\mathcal {D}), \end{aligned} \end{aligned}$$

(6.3)

where $\mathcal {D}=\frac{\oint \frac{(1-2n)D_{x}}{\rho ^{3}}ds}{\oint \frac{1}{\rho ^{2}}ds}$, with $n=-\frac{\rho }{B}\frac{\partial B}{\partial \rho }$ the field gradient index and $D_{x}$ is the horizontal dispersion. Nominally for a planar uncoupled ring using bending magnets with no transverse gradient, we have $\mathcal {D}\ll 1$.

The horizontal, vertical and longitudinal radiation damping times are

$$\begin{aligned} \tau _{H,V,L}=\frac{C_{0}/c}{\alpha _{H,V,L}}, \end{aligned}$$

(6.4)

with $C_{0}$ the ring circumference and c the speed of light in vacuum.

The natural energy spread of electron beam in a longitudinal weak focusing ring is

$$\begin{aligned} \sigma _{\delta \text {S}}=\sqrt{\frac{C_{q}}{J_{s}}\frac{\gamma ^{2}}{\rho }}, \end{aligned}$$

(6.5)

with $C_{q}=\frac{55\bar{\lambda }_{e}}{32\sqrt{3}}=3.8319\times 10^{-13}$ m, ${\bar{\lambda }}_{e}=\frac{\lambda _{e}}{2\pi }=386$ fm is the reduced Compton wavelength of electron, $J_{s}=2+\mathcal {D}$ is the longitudinal damping partition number, $\gamma$ is the Lorentz factor.

The natural bunch length at the laser modulator is

$$\begin{aligned} \sigma _{z\text {S}}=\sigma _{\delta \text {S}}\beta _{z\text {S}}, \end{aligned}$$

(6.6)

where $\beta _{z\text {S}}$ is the longitudinal beta function at the laser modulator to be given soon.

The effective modulation voltage of a laser modulator using a planar undulator is [1]

$$\begin{aligned} V_{L}=\frac{[JJ] K}{\gamma }\sqrt{\frac{4P_{L}Z_{0}Z_{R}}{\lambda _{L}}}\tan ^{-1}\left( \frac{L_{u}}{2Z_{R}}\right) . \end{aligned}$$

(6.7)

in which $[JJ]=J_{0}(\chi )-J_{1}(\chi )$ and $\chi =\frac{K^{2}}{4+2K^{2}}$, $J_{n}$ is the n-th order Bessel function of the first kind, $K=\frac{eB_{0}}{m_{e}ck_{u}}=0.934\cdot B_{0}[\text {T}]\cdot \lambda _{u}[\text {cm}]$ is the undulator parameter, determined by the peak magnetic flux density $B_{0}$ and period $\lambda _{u}$ of the undulator, $P_{L}$ 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 related to the laser and modulator undulator parameters according to

$$\begin{aligned} h=\frac{eV_{L}}{E_{0}}k_{L}=\frac{e[JJ] K}{\gamma ^{2}mc^{2}}\sqrt{\frac{4P_{L}Z_{0}Z_{R}}{\lambda _{L}}}\tan ^{-1}\left( \frac{L_{u}}{2Z_{R}}\right) k_{L}, \end{aligned}$$

(6.8)

where $k_{L}=2\pi /\lambda _{L}$ is the wavenumber of the modulation laser.

Linear stability of the longitudinal motion requires

$$\begin{aligned} 0<h\eta C_{0}<4, \end{aligned}$$

(6.9)

where $\eta$ is the phase slippage factor of the ring.

Considering the fact that the modulation waveform is sinusoidal and the longitudinal dynamics is more accurately modeled by a “standard kick map”, to avoid strong chaotic dynamics, an empirical criterion is

$$\begin{aligned} 0<h\eta C_{0}\lesssim 0.1. \end{aligned}$$

(6.10)

In a longitudinal weak focusing ring $(\nu _{s}\ll 1)$, the synchrotron tune is

$$\begin{aligned} \nu _{s}\approx \frac{\eta }{|\eta |}\frac{\sqrt{h\eta C_{0}}}{2\pi }. \end{aligned}$$

(6.11)

In a longitudinal weak focusing ring, the longitudinal beta function at the laser modulator is

$$\begin{aligned} \beta _{z\text {S}}\approx \sqrt{\frac{\eta C_{0}}{h}}. \end{aligned}$$

(6.12)

The micro-bucket half-height is

$$\begin{aligned} \hat{\delta }_{\frac{1}{2}}=\frac{2}{\beta _{z\text {S}}k_{L}}. \end{aligned}$$

(6.13)

If there is a single RF or laser modulator in the ring, and $J_{s}=2$, then the theoretical minimum bunch length and longitudinal emittance in a longitudinal weak focusing ring with respect to the bending radius $\rho$ and angle $\theta$ of each bending magnet are

$$\begin{aligned} \begin{aligned} \sigma _{z,\text {min}}[\mu \text {m}]&\approx 4.93\rho ^{\frac{1}{2}}[\text {m}]E_{0}[\text {GeV}]\theta ^3[\text {rad}],\\ \epsilon _{z,\text {min}}[\text {nm}]&\approx 8.44E^{2}_{0}[\text {GeV}]\theta ^3[\text {rad}]. \end{aligned} \end{aligned}$$

(6.14)

Scaling law of the horizontal emittance in an SSMB storage ring is

$$\begin{aligned} \epsilon _{x}[\text {nm}]\approx -366.5E^{2}_{0}[\text {GeV}]\theta ^3[\text {rad}]\left[ \frac{1}{9} \tan \left( \frac{\Phi _x}{2}\right) +\frac{1}{10} \cot \left( \frac{\Phi _x}{2}\right) \right] , \end{aligned}$$

(6.15)

with $\Phi _{x}$ the horizontal betatron phase advance per cell which usually lies in $(\pi ,2\pi )$. The above scaling is derived by considering only the contribution of main cells, and ignoring that from the matching section.

Coherent undulator radiation power at the odd-H-th harmonic from a transversely-round electron beam is

$$\begin{aligned} P_{H,\text {peak}}[\text {kW}]=1.183N_{u}H\chi [JJ]_{H}^{2}FF_{\bot }(S)|b_{z,H}|^{2}I_{P}^{2}[\text {A}], \end{aligned}$$

(6.16)

where $N_{u}$ is the number of undulator periods, $[JJ]_{H}^{2}=\left[ J_{\frac{H-1}{2}}\left( H\chi \right) -J_{\frac{H+1}{2}}\left( H\chi \right) \right] ^{2}$, with $\chi =\frac{K^{2}}{4+2K^{2}}$, and the transverse form factor is

$$\begin{aligned} FF_{\bot }(S)=\frac{2}{\pi }\left[ \tan ^{-1}\left( \frac{1}{2S}\right) +S\ln \left( \frac{(2S)^{2}}{(2S)^{2}+1}\right) \right] , \end{aligned}$$

(6.17)

with $S=\frac{\sigma ^{2}_{\bot }\frac{\omega }{c}}{L_{u}}$ and $\sigma _{\bot }$ the RMS transverse electron beam size, $b_{z,H}$ is the bunching factor at the H-th harmonic, and $I_{P}$ is the peak current.

The relative fluctuation of coherent radiation power considering the pointlike nature of electrons is

$$\begin{aligned} \begin{aligned} \frac{\text {Var}\left[ |b(\textbf{k})|^{2}\right] }{\langle |b(\textbf{k})|^{2}\rangle ^{2}}&=\frac{2}{N_{e}}\left( \frac{|\overline{b}(\textbf{k})|^{2}+\text {Re}\left[ \overline{b}(2\textbf{k})\overline{b}^{2}(-\textbf{k})\right] }{|\overline{b}(\textbf{k})|^{4}}-2\right) +\mathcal {O}\left( \frac{1}{N_{e}^{2}}\right) , \end{aligned} \end{aligned}$$

(6.18)

where $b(\textbf{k})$ is the bunching factor at the wavevector $\textbf{k}$, and $N_{e}$ is the number of electrons.

Table 6.1 Example parameters set of a longitudinal weak focusing SSMB storage ring for infrared radiation generation

Based on the above formulas, here we present an example parameters set in Table 6.1 of a longitudinal weak focusing SSMB storage ring, aimed for high-power infrared radiation generation. As can be seen, such a compact SSMB storage ring can be used for power amplification of the injected seed laser. The requirement on the stored laser power is easy to realize in practice. All the other parameters are also within practical range. A sharp reader may notice that the microbucket half-height is only twice the natural energy spread of the electron beam. Therefore, in addition to these shallow microbuckets, we need a larger bucket, for example a barrier bucket formed by an induction linac, to constrain the particles in the ring to ensure a large enough beam lifetime.

6.2.2 Transverse-Longitudinal Coupling SSMB

For a transverse-longitudinal coupling (TLC) based SSMB, or a generalized longitudinal strong focusing SSMB [2], using TEM00 mode laser modulator for energy modulation, we have the following important formulas.

Relation between energy chirp strength and optical functions at the modulator and radiator

$$\begin{aligned} h^2(\text {Mod})\mathcal {H}_{y}(\text {Mod})\mathcal {H}_{y}(\text {Rad})\ge 1, \end{aligned}$$

(6.19)

where $\mathcal {H}_{y}$ is a chromatic function quantifying the contribution of vertical emittance to bunch length.

Put the above relation in another way,

$$\begin{aligned} h\ge \frac{\epsilon _{y}}{\sigma _{zy}(\text {Mod})\sigma _{zy}(\text {Rad})}. \end{aligned}$$

(6.20)

Bunching factor at the n-th laser harmonic in TLC SSMB at the radiator

$$\begin{aligned} \begin{aligned} b_{n}=\ {}&\left( \sum _{m=-\infty }^{\infty }J_{m}\left( n\right) \text {exp}\left[ -\left( (n-m)k_{L}\sigma _{z}(\text {Mod})\right) ^2/2\right] \right) \text {exp}\left[ -\left( nk_{L}\sigma _{z}(\text {Rad})\right) ^2/2\right] , \end{aligned} \end{aligned}$$

(6.21)

where $\sigma _{z}(\text {Mod})=\sqrt{\epsilon _{z}\beta _{z}(\text {Mod})+\epsilon _{y}\mathcal {H}_{y}(\text {Mod})}$ and $\sigma _{z}(\text {Rad})=\sqrt{\epsilon _{y}\mathcal {H}_{y}(\text {Rad})}$ are the linear bunch length at the modulator and radiator, respectively.

Contribution of two modulators to $\epsilon _{y}$ from quantum excitation

$$\begin{aligned} \begin{aligned} \Delta \epsilon _{y}(\text {Mod})&=2\times \frac{55}{96\sqrt{3}}\frac{\alpha _{F}{\bar{\lambda }}_{e}^{2}\gamma ^{5}}{\alpha _{V}}\frac{\mathcal {H}_{y}(\text {Mod})}{\rho _{0\text {Mod}}^{3}}\frac{4}{3\pi }L_{u},\\ \end{aligned} \end{aligned}$$

(6.22)

where $\alpha _{F}=\frac{1}{137}$ is the fine-structure constant.

Assuming $\epsilon _{y}=\Delta \epsilon _{y}(\text {Mod})$, which means the vertical emittance is solely from the two modulators, then the required modulation laser power and modulator length scaling are

$$\begin{aligned} \begin{aligned} P_{L}[\text {kW}]&\approx 5.67\frac{\lambda _{L}^{\frac{7}{3}}[\text {nm}]E_{0}^{\frac{8}{3}}[\text {GeV}]B_{0\text {Mod}}^{\frac{7}{3}}[\text {T}]}{\sigma _{z}^{2}(\text {Rad})[\text {nm}]B_{\text {ring}}[\text {T}]},\\ L_{u}[\text {m}]&\approx 57\frac{B_{\text {ring}}[\text {T}]\epsilon _{y}[\text {pm}]}{\mathcal {H}_{y}(\text {Mod})[\mu \text {m}]B_{0\text {Mod}}^{3}[\text {T}]}, \end{aligned} \end{aligned}$$

(6.23)

where $B_{0\text {Mod}}$ is the peak magnetic flux density of the modulator undulator, $B_{\text {ring}}$ is the magnetic flux density of bending magnets in the ring. The above scaling laws are accurate when $K_{u}>\sqrt{2}$. For the more general case, refer to Eq. (3.56).

Table 6.2 Example parameters set of a transverse-longitudinal coupling SSMB storage ring for EUV and soft X-ray radiation generation

Based on the presented formulas, here we present an example parameters set in Table 6.2 of a TLC SSMB storage ring, aimed for high-power EUV and soft X-ray radiation. It can be seen that as long as we can realize a coasting beam of 1.5 A average current, and an optical cavity stored power of $\gtrsim$100 kW, we can realize 1 kW average power 13.5 nm EUV and 6.75 nm soft X-ray radiation. All the other parameters applied should be realizable, including the small $\epsilon _{y}$ considering IBS. Even if we can only realize an average beam current of 1 A or less, we can take advantage of the fact that $P_{\text {coh}}\propto I_{P}^{2}$ to realize an average radiation power of kW level, by decreasing the filling factor of electron beam in the ring but increasing the peak current as long as the value is below the collective instability threshold.¹ Since there is no requirement on the longitudinal emittance for a coasting beam, thus no requirement on the fine control of longitudinal $\beta$ function, the circumference of this ring has great flexibility, which means the ring can be very compact, for example a circumference of 100 m should be feasible. This compact high-power EUV radiation source is promising to fulfill the urgent need of EUV lithography for high volume manufacture, and also serve the future lithography like Blue-X which invokes 6.x nm-wavelength light source. Such an SSMB-based high-power soft X-ray photon source could be of great value for fundamental science like high-resolution angle-resolved photoemission spectroscopy and can also bridge the water window gap.