SSMB Radiation

Abstract

Having discussed the methods to form and preserve microbunching in the last two chapters, now we present the theoretical and numerical study of the average and statistical property of coherent radiation from SSMB. Our results show that a kW-level average power quasi-continuous-wave 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 which is assumed to be an undulator. 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 three orders of magnitude higher than that in a conventional synchrotron source, allowing sub-meV resolution in angle-resolved photoemission spectroscopy (ARPES) and providing new opportunities for fundamental 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 coherent radiation.

Having discussed the methods to form and preserve microbunching in the last two chapters, now we present the theoretical and numerical study of the average and statistical property of coherent radiation from SSMB. Our results show that a kW-level average power quasi-continuous-wave 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 which is assumed to be an undulator. 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 three orders of magnitude higher than that in a conventional synchrotron source, allowing sub-meV resolution in angle-resolved photoemission spectroscopy (ARPES) and providing new opportunities for fundamental 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 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 are 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 (FELs), coherent harmonic generation (CHG), etc. Parts of the work presented in this chapter have been published in Ref. [1].

4.1 Formulation of Radiation from a 3D Rigid Beam

For simplicity, as the first step we consider only the impacts of particle position x, y and z, but ignore the particle angular divergence $x'$, $y'$ and energy deviation $\delta$, on the radiation. Under this approximation, concise and useful analytical formulas of the coherent radiation can be obtained. This approximation is accurate when the transverse and longitudinal beam size do not change much inside the radiator, i.e., $\beta _{x,y}\gtrsim L_{r}$ and $\beta _{z}\gtrsim R_{56,r}$, where $\beta _{x,y,z}$ are the Courant-Snyder functions of the beam in the horizontal, vertical and longitudinal dimensions [2], $L_{r}$ and $R_{56,r}$ are the length and momentum compaction of the radiator, respectively. Here in this dissertation, we call this approximation the three-dimensional (3D) rigid beam approximation, as the beam sizes do not change much during radiation. We will see later in Sect. 4.6 that the conditions of rigid beam approximation is generally satisfied in the envisioned EUV SSMB. In addition, we will briefly discuss the impact of beam divergence and energy spread on coherent radiation in Sect. 4.4.

Fig. 4.1

Coordinate system used to calculate the undulator radiation spectrum. The magnetic field is in the y-direction, and the electron wiggles in the x-z plane

Assuming that the vector potential of radiation from the reference particle at the observation location is $\textbf{A}_{\text {point}}(\theta ,\varphi ,t)$, with $\theta$ and $\varphi$ being the polar and azimuthal angles in a spherical coordinate system, respectively, as shown in Fig. 4.1. Under far-field approximation, the vector potential of radiation from a 3D rigid electron beam containing $N_{e}$ electrons is then

$$\begin{aligned} \begin{aligned} \textbf{A}_{\text {beam}}(\theta ,\varphi ,t)&=N_{e}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\\&\textbf{A}_{\text {point}}\left( \theta ,\varphi ,t+\frac{x\sin \theta \cos \varphi +y\sin \theta \sin \varphi }{c}+\frac{z}{\beta c}\right) \rho (x,y,z)dxdydz,\\ \end{aligned} \end{aligned}$$

(4.1)

in which $\beta$ is the particle velocity normalized by the speed of light in vacuum c, and $\rho (x,y,z)$ is the normalized charge density satisfying $\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\rho (x,y,z)dxdydz=1.$ Note that we have assumed that the particle motion pattern, therefore also the radiation pattern of a single electron, does not depend on x, y and z of the particle. In other words, x, y, z of a particle influences only the arrival time of the radiation at the observation. This is the reason why their impacts can be treated within a single framework. The impacts of $x'$, $y'$ and $\delta$ are different. Generally, their impacts are two-fold. First, they affect the radiation of the single particle itself, i.e., the radiation pattern. Second, they affect the electron beam distribution, therefore the coherence of different particles, during the radiation process.

According to the convolution theorem, for a 3D rigid beam, we now have

$$\begin{aligned} \textbf{A}_{\text {beam}}(\theta ,\varphi ,\omega )=N_{e}\textbf{A}_{\text {point}}(\theta ,\varphi ,\omega )b(\theta ,\varphi ,\omega ), \end{aligned}$$

(4.2)

where

$$\begin{aligned} \begin{aligned} b(\theta ,\varphi ,\omega )&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\rho (x,y,z)e^{-i\omega \left( \frac{x\sin \theta \cos \varphi +y\sin \theta \sin \varphi }{c}+\frac{z}{\beta c}\right) }dxdydz. \end{aligned} \end{aligned}$$

(4.3)

Since $\textbf{A}(\theta ,\varphi ,t)$ is real, then $\textbf{A}(\theta ,\varphi ,-\omega )=\textbf{A}^{*}(\theta ,\varphi ,\omega )$. The energy radiated per unit solid angle per unit frequency interval is [3]

$$\begin{aligned} \frac{d^{2}W}{d\omega d\Omega }(\theta ,\varphi ,\omega )=2|\textbf{A}(\theta ,\varphi ,\omega )|^{2}. \end{aligned}$$

(4.4)

Therefore, we have

$$\begin{aligned} \frac{d^{2}W}{d\omega d\Omega }(\theta ,\varphi ,\omega )\Bigg |_{\text {beam}}=N_{e}^{2}{|b(\theta ,\varphi ,\omega )|^{2}\frac{d^{2}W}{d\omega d\Omega }}(\theta ,\varphi ,\omega )\Bigg |_{\text {point}}. \end{aligned}$$

(4.5)

The total radiation energy spectrum of a beam can be obtained by the integration with respect to the solid angle

$$\begin{aligned} \frac{dW}{d\omega }\Bigg |_{\text {beam}}=\int _{0}^{\pi }\sin \theta d\theta \int _{0}^{2\pi }d\varphi \frac{d^{2}W}{d\omega d\Omega }(\theta ,\varphi ,\omega )\Bigg |_{\text {beam}}, \end{aligned}$$

(4.6)

and the total radiation energy of the beam can be calculated by further integration with respect to the frequency

$$\begin{aligned} W_{\text {beam}}=\int _{0}^{+\infty }d\omega \frac{dW}{d\omega }\Bigg |_{\text {beam}}. \end{aligned}$$

(4.7)

The reason why the lower limit in the above integration is 0, instead of $-\infty$, is that there is a factor of 2 in the right hand side of Eq. (4.4). The above formulas can be used to numerically calculate the characteristics of radiation from an electron beam, once its 3D distribution is given. Note that in the relativistic case, we only need to account for $\theta$ several times of $\frac{1}{\gamma }$, as the radiation is very collimated in the forward direction.

4.2 Form Factors

When the longitudinal and transverse dimensions of the electron beam are decoupled, we can factorize $b(\theta ,\varphi ,\omega )$ as

$$\begin{aligned} \begin{aligned} b(\theta ,\varphi ,\omega )&=b_{\bot }(\theta ,\varphi ,\omega )\times b_{z}(\omega ), \end{aligned} \end{aligned}$$

(4.8)

where

$$\begin{aligned} \begin{aligned} b_{\bot }(\theta ,\varphi ,\omega )&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\rho (x,y)e^{-i\omega \left( \frac{x\sin \theta \cos \varphi +y\sin \theta \sin \varphi }{c}\right) }dxdy, \end{aligned} \end{aligned}$$

(4.9)

and

$$\begin{aligned} b_{z}(\omega )=\int _{-\infty }^{\infty }\rho (z)e^{-i\omega \frac{z}{\beta c}}dz. \end{aligned}$$

(4.10)

Note that $\rho (x,y)$ and $\rho (z)$ are then the projected charge density. $b_{z}(\omega )$ is the usual bunching factor found in literature and is independent of the observation angle. This however is not true for $b_{\bot }(\theta ,\varphi ,\omega )$. For example, in the case of a 3D Gaussian x-y-z decoupled beam,

$$\begin{aligned} \begin{aligned} |b_{\bot }(\theta ,\varphi ,\omega )|^{2}&=\text {exp}\left\{ -\left( \frac{\omega }{c}\right) ^{2}\left[ \left( \sigma _{x}\sin \theta \cos \varphi \right) ^{2}+\left( \sigma _{y}\sin \theta \sin \varphi \right) ^{2}\right] \right\} ,\\ |b_{z}(\omega )|^{2}&=\text {exp}\left[ -\left( \frac{\omega }{\beta c}\right) ^{2}\sigma _{z}^{2}\right] , \end{aligned} \end{aligned}$$

(4.11)

where $\sigma _{x,y,z}$ are the root-mean-square (RMS) size of the beam in the horizontal, vertical and longitudinal dimension, respectively.

In order to efficiently quantify the impact of the transverse and longitudinal distributions of an electron beam on the overall radiation energy spectrum, here we define the transverse and longitudinal form factors of an electron beam as

$$\begin{aligned} \begin{aligned} FF_{\bot }(\omega )&=\frac{\int _{0}^{\pi }\sin \theta d\theta \int _{0}^{2\pi }d\varphi |b_{\bot }(\theta ,\varphi ,\omega )|^{2}\frac{d^{2}W}{d\omega d\Omega }(\theta ,\varphi ,\omega )\bigg |_{\text {point}}}{\int _{0}^{\pi }\sin \theta d\theta \int _{0}^{2\pi }d\varphi \frac{d^{2}W}{d\omega d\Omega }(\theta ,\varphi ,\omega )\bigg |_{\text {point}}}, \end{aligned} \end{aligned}$$

(4.12)

and

$$\begin{aligned} FF_{z}(\omega )=|b_{z}(\omega )|^{2}, \end{aligned}$$

(4.13)

respectively. The overall form factor is then

$$\begin{aligned} FF(\omega )=FF_{\bot }(\omega )FF_{z}(\omega ). \end{aligned}$$

(4.14)

The total radiation energy spectrum of a beam is related to that of a single electron by

$$\begin{aligned} \frac{dW}{d\omega }\Bigg |_{\text {beam}}=N_{e}^{2}FF(\omega )\frac{dW}{d\omega }\Bigg |_{\text {point}}. \end{aligned}$$

(4.15)

4.2.1 Longitudinal Form Factor

The longitudinal form factor is the usual bunching factor squared, and have been discussed extensively in literature. For example, the longitudinal form factor for a single Gaussian microbunch is given Eq. (4.11). When there are multi microbunches separated with each other a distance of the modulation laser wavelength $\lambda _{L}$ like that in SSMB, the longitudinal form factor is that of the single bunch multiplied with a macro form factor,

$$\begin{aligned} FF_{z\text {MB}}(\omega )=FF_{z\text {SB}}(\omega )\left( \frac{\sin \left( N_{b}\frac{\omega }{c}\frac{\lambda _{L}}{2}\right) }{N_{b}\sin \left( \frac{\omega }{c}\frac{\lambda _{L}}{2}\right) }\right) ^{2}, \end{aligned}$$

(4.16)

where the subscripts $_{\text {MB}}$ and $_{\text {SB}}$ mean multi bunch and single bunch, respectively, and $N_{b}$ is the number of microbunches. This macro form factor of multi bunches is a periodic function of the radiation frequency, with a period of the modulation laser frequency. The full width at half maximum (FWHM) linewidth around each laser harmonics is $\Delta \omega _{\text {FWHM}}=\frac{\omega _{L}}{N_{b}}$. When $N_{b}$ goes to infinity, this macro form factor will become the periodic delta function. Figure 4.2 presents an example plot of the macro form factor for three different $N_{b}$.

When the radiation wavelength is a high harmonic of the modulation laser wavelength, corresponding to these delta function lines in the longitudinal form factor or radiation energy spectrum, there will be interference rings in the spatial distribution of the coherent radiation from different microbunches. The polar angles of these rings, corresponding to the delta function lines in energy spectrum, are determined by the off-axis resonant condition. Note that when we use the above form factor $FF_{z\text {MB}}(\omega )$ to calculate the radiation energy or spectrum from $N_{b}$ microbunches, the number of electrons used should be $N_{b}N_{e\text {SB}}$, with $N_{e\text {SB}}$ the number of electrons per microbunch. We remind the readers that the electron beam energy spread and angular divergence will make the linewidth of the delta function lines analyzed above become non-zero. For example, the relative bandwidth of the radiation caused by an energy spread of $\sigma _{\delta }$ is $2\sigma _{\delta }$.

Fig. 4.2

Macro form factor of multi bunches, as a function of the number of bunches

To make our results more useful, here we also present some analysis applies for FEL and CHG. As analyzed in Sect. 2.2.1, the bunching factor for a coasting beam-based CHG is

$$\begin{aligned} \begin{aligned} b_{z,\text {coasting}}(\omega )&=\sum _{n=0}^{\infty }\delta \left( \frac{\omega }{c}-nk_{L}\right) J_{n}\left[ -\frac{\omega }{c}R_{56}A\right] \text {exp}\left[ -\frac{1}{2}\left( \frac{\omega }{c}R_{56}\sigma _{\delta }\right) ^{2}\right] , \end{aligned} \end{aligned}$$

(4.17)

where

$$\begin{aligned} \delta (x)={\left\{ \begin{array}{ll} 1,&{} x=0,\\ 0,&{} \text {else}. \end{array}\right. } \end{aligned}$$

(4.18)

Let us now consider the more-often confronted case of a finite bunch length. We assume that the initial beam current before microbunching is Gaussian with an RMS bunch length of $\sigma _{z}$. According to the convolution theorem, then

$$\begin{aligned} \begin{aligned} b_{z,\text {bunched}}(\omega )&=b_{z,\text {coasting}}(\omega )\otimes b_{z,\text {Gaussian}}(\omega ), \end{aligned} \end{aligned}$$

(4.19)

where $\otimes$ means convolution and $b_{z,\text {Gaussian}}(\omega ) =\text {exp}\left[ -\frac{1}{2}\left( \frac{\omega }{c}\sigma _{z}\right) ^{2}\right] .$ Therefore, the longitudinal form factor of the beam is now $FF_{z,\text {bunched}}(\omega )=|b_{z,\text {bunched}}(\omega )|^2.$ The convolution with $b_{z,\text {Gaussian}}(\omega )$ results in a non-zero bandwidth of each laser harmonic line in the longitudinal form factor spectrum

$$\begin{aligned} \left( {\Delta \omega }\right) _{\text {FWHM}}=2\sqrt{2\ln 2}\frac{c/\sigma _{z}}{\sqrt{2}}=\frac{4\sqrt{2}\ln {2}}{\left( \Delta t\right) _{\text {FWHM}}}, \end{aligned}$$

(4.20)

where $\Delta t$ is the electron bunch length in unit of time. Then the relative bandwidth of the longitudinal form factor at the H-th laser harmonic can be expressed as

$$\begin{aligned} \begin{aligned} \left( \frac{{\Delta \omega }}{\omega }\right) _{\text {FWHM}}&=\frac{2\sqrt{2}\ln {2}}{\pi }\frac{1}{(c\Delta t)_{\text {FWHM}}/\lambda } =\frac{2\sqrt{2}\ln {2}}{\pi }\frac{\lambda _{L}}{H(c\Delta t)_{\text {FWHM}}}. \end{aligned} \end{aligned}$$

(4.21)

Note that the coherent radiation pulse length is $\frac{1}{\sqrt{2}}$ of the electron bunch length due to the scaling of $P_{\text {coh}}\propto N_{e}^{2}$, and the above formula means that the coherent radiation is Fourier-transform limited. Note also that the absolute width $({\Delta \omega })_{\text {FWHM}}$ is independent of H, while the relative bandwidth $\left( \frac{{\Delta \omega }}{\omega }\right) _{\text {FWHM}}\propto \frac{1}{H}$.

4.2.2 Transverse From Factor

Now let us investigate the transverse form factor. Since the transverse form factor depends on the radiation process, there is not a universal formula involving only the beam distribution. Here for our interest, we focus on the case of undulator radiation. We use a planar undulator as an example. The formulation for a helical undulator is similar.

As well established in literature, the planar undulator radiation of a point charge in the H-th harmonic is [4]

$$\begin{aligned} \begin{aligned}&\frac{d^{2}W_{H}}{d\omega d\Omega }(\theta ,\varphi ,\omega )\Bigg |_{\text {point}}=\frac{2e^2\gamma ^2}{\pi \epsilon _{0} c}G(\theta ,\varphi )F(\epsilon ),\\&F(\epsilon )=\left( \frac{\sin (\pi N_{u}\epsilon )}{\pi \epsilon }\right) ^{2},\ \epsilon =\frac{\omega }{\omega _{r}(\theta )}-H=\frac{\omega }{2c k_{u}\gamma ^{2}}\left( 1+K^{2}/2+\gamma ^{2}\theta ^{2}\right) -H,\\&G(\theta ,\varphi )=G_{\sigma }(\theta ,\varphi )+G_{\pi }(\theta ,\varphi ),\\&G_{\sigma }(\theta ,\varphi )=\left[ \frac{H\left( \frac{K}{\sqrt{2}}\mathcal {D}_{1}+\frac{\gamma \theta }{\sqrt{2}}\mathcal {D}_{2}\cos \varphi \right) }{1+K^{2}/2+\gamma ^{2}\theta ^{2}}\right] ^{2},\ G_{\pi }(\theta ,\varphi )=\frac{1}{2}\left( \frac{H\gamma \theta \mathcal {D}_{2}\sin \varphi }{1+K^{2}/2+\gamma ^{2}\theta ^{2}}\right) ^{2},\\&\mathcal {D}_{1}=-\frac{1}{2}\sum _{m=-\infty }^{\infty }J_{H+2m-1}(H\alpha )\left[ J_{m}(H\zeta )+J_{m-1}(H\zeta )\right] ,\\&\mathcal {D}_{2}=\sum _{m=-\infty }^{\infty }J_{H+2m}(H\alpha )J_{m}(H\zeta ),\\&\alpha =\frac{2K\gamma \theta \cos \varphi }{1+K^{2}/2+\gamma ^{2}\theta ^{2}},\ \zeta =\frac{K^{2}/4}{1+K^{2}/2+\gamma ^{2}\theta ^{2}}, \end{aligned} \end{aligned}$$

(4.22)

in which e is the elementary charge, $\gamma$ is the Lorentz factor, $\epsilon _{0}$ is the permittivity of free space, $\omega _{r}(\theta )$ is the fundamental resonant angular frequency at the observation with a polar angle of $\theta$, $k_{u}=\lambda _{u}/2\pi$ is the wavenumber of the undulator, $K=\frac{eB_{0}}{m_{e}ck_{u}}=0.934\cdot B_{0}[\text {T}]\cdot \lambda _{u}[\text {cm}]$ is the dimensionless undulator parameter, with $B_{0}$ being the peak magnetic flux density of the undulator and $m_{e}$ being the mass of an electron, $J_{m}$ means the m-th order Bessel function of the first kind.

Now we try to get some analytical results for the transverse form factor. Instead of a general discussion, here we consider only the simplest case of a transverse round Gaussian beam, i.e,

$$\begin{aligned} \begin{aligned} |b_{\bot }(\theta ,\varphi ,\omega )|^{2}&=\text {exp}\left[ -\left( \frac{\omega }{c}\sigma _{\bot }\sin \theta \right) ^{2}\right] , \end{aligned} \end{aligned}$$

(4.23)

where $\sigma _{\bot }$ is the RMS transverse size of the electron beam. As the radiation is dominantly in the forward direction in relativistic case, and $e^{-\left( \frac{\omega }{c}\sigma _{\bot }\sin \theta \right) ^2}$ approaches zero with the increase of $\theta$, therefore in Eq. (4.12), the upper limit of $\theta$ in the integration can be effectively replaced by infinity, and $\sin \theta$ can be replaced by $\theta$ in $e^{-\left( \frac{\omega }{c}\sigma _{\bot }\sin \theta \right) ^2}$. Then

$$\begin{aligned} \begin{aligned}&\int _{0}^{2\pi }d\varphi \int _{0}^{\pi }\sin \theta d\theta e^{-\left( \frac{\omega }{c}\sigma _{\bot }\sin \theta \right) ^2}\frac{d^2W_{H}}{d\omega d\Omega }(\theta ,\varphi ,\omega )\Bigg |_{\text {point}}\\&\approx \int _{0}^{2\pi }d\varphi \int _{0}^{\infty }\theta d\theta e^{-\left( \frac{\omega }{c}\sigma _{\bot }\theta \right) ^2}\frac{d^2W_{H}}{d\omega d\Omega }(\theta ,\varphi ,\omega )\Bigg |_{\text {point}}\\&\approx \frac{e^{2}}{\pi \epsilon _{0}c}\int _{0}^{2\pi }d\varphi G(\theta _{1},\varphi )\int _{0}^{\infty }d(\gamma \theta )^{2}e^{-\left( \frac{\omega }{c}\sigma _{\bot }\theta \right) ^2}F(\epsilon ), \end{aligned} \end{aligned}$$

(4.24)

where $\theta _{1}=\frac{1}{\gamma }\sqrt{\left( 1+K^{2}/2\right) \left( \frac{H\omega _{0}}{\omega }-1\right) },\ \omega _{0}=\omega _{r}(\theta =0)=\frac{2\gamma ^{2}}{1+K^{2}/2}\omega _{u}.$ Here we have made use of the fact that there is only one value of $\theta$, i.e., $\theta _{1}$, that contributes significantly to the integration over the solid angle $\Omega$ due to the sharpness of $F(\epsilon )$ when the undulator period number $N_{u}\gg 1$, as the spectral width of $F(\epsilon )$ is $1/N_{u}$.

The transverse form factor corresponding to the H-th harmonic can thus be defined as

$$\begin{aligned} \begin{aligned} FF_{\bot }(H,\omega )&=\frac{\int _{0}^{\infty }d(\gamma \theta )^{2}e^{-\left( \frac{\omega }{c}\sigma _{\bot }\theta \right) ^2}\text {sinc}^{2}(N_{u}\pi \epsilon )}{\int _{0}^{\infty }d(\gamma \theta )^{2}\text {sinc}^{2}(N_{u}\pi \epsilon )}. \end{aligned} \end{aligned}$$

(4.25)

The radiation spectrum of the H-th harmonic is then

$$\begin{aligned} \frac{dW_{H}}{d\omega }\Bigg |_{\text {beam}}=N_{e}^{2}FF_{\bot }(H,\omega )FF_{z}(\omega )\frac{dW_{H}}{d\omega }\Bigg |_{\text {point}}, \end{aligned}$$

(4.26)

and the total radiation spectrum of an electron beam is

$$\begin{aligned} \frac{dW}{d\omega }\Bigg |_{\text {beam}}=\sum _{H=1}^{\infty }\frac{dW_{H}}{d\omega }\Bigg |_{\text {beam}}. \end{aligned}$$

(4.27)

Denote

$$\begin{aligned} \begin{aligned} \kappa _{1}&\equiv N_{u}\pi \left( \frac{\omega }{\omega _{0}}-H\right) ,\ \kappa _{2}\equiv N_{u}\pi \frac{\omega }{\omega _{0}}\frac{1}{1+K^{2}/2},\ \kappa _{3}\equiv \left( \frac{\omega \sigma _{\bot }}{ c\gamma }\right) ^2, \end{aligned} \end{aligned}$$

(4.28)

then the denominator in Eq. (4.25) is

$$\begin{aligned} \begin{aligned} \mathcal {D}(H,\omega )&=\int _{0}^{\infty }dx\text {sinc}^{2}\left( \kappa _{1}+\kappa _{2}x\right) =\frac{\frac{\pi }{2}-\text {Si}\left( 2 \kappa _1\right) +\frac{\sin ^2\left( \kappa _1\right) }{\kappa _1}}{\kappa _2}, \end{aligned} \end{aligned}$$

(4.29)

where $\text {Si}(x)=\int _{0}^{x}\frac{\sin (t)}{t}dt$ is the sine integral, and the numerator in Eq. (4.25) is

$$\begin{aligned} \begin{aligned} \mathcal {N}(H,\omega )&=\int _{0}^{\infty }dxe^{-\kappa _{3}x}\text {sinc}^{2}\left( \kappa _{1}+\kappa _{2}x\right) \\&=\frac{e^{\frac{\kappa _1 \kappa _3}{\kappa _2}}}{4 \kappa _1 \kappa _2^2}\left\{ -4\kappa _2\sin ^{2} (\kappa _{1})e^{-\frac{\kappa _1\kappa _3}{\kappa _2}}-2 \kappa _1 \kappa _2 i\left[ \text {Ei} \left( 2\kappa _1 i-\frac{\kappa _1\kappa _3}{\kappa _2}\right) -\text {Ei} \left( -2\kappa _1 i-\frac{\kappa _1\kappa _3}{\kappa _2}\right) \right] \right. \\&\left. \ \ \ + \kappa _1 \kappa _3\left[ \text {Ei} \left( 2\kappa _1 i-\frac{\kappa _1\kappa _3}{\kappa _2}\right) - 2\text {Ei}\left( -\frac{\kappa _1 \kappa _3}{\kappa _2}\right) +\text {Ei} \left( -2\kappa _1 i-\frac{\kappa _1\kappa _3}{\kappa _2}\right) \right] \right\} ,\\ \end{aligned} \end{aligned}$$

(4.30)

where $\text {Ei}(x)=\int _{-\infty }^{x}\frac{e^{t}}{t}dt$ is the exponential integral. The transverse form factor is then

$$\begin{aligned} FF_{\bot }(H,\omega )=\frac{\mathcal {N}(H,\omega )}{\mathcal {D}(H,\omega )}. \end{aligned}$$

(4.31)

When $\omega = H\omega _{0}$, then $\kappa _{1}=0$, the transverse form factor has a simpler form,

$$\begin{aligned} \begin{aligned} FF_{\bot }(S)&\equiv FF_{\bot }(H,H\omega _{0})=\frac{\int _{0}^{\infty }dxe^{-\kappa _{3}x}\text {sinc}^{2}\left( \kappa _{2}x\right) }{\int _{0}^{\infty }dx\text {sinc}^{2}\left( \kappa _{2}x\right) }\\&=\frac{2}{\pi }\left[ \tan ^{-1}\left( \frac{1}{2S}\right) +S\ln \left( \frac{(2S)^{2}}{(2S)^{2}+1}\right) \right] , \end{aligned} \end{aligned}$$

(4.32)

where

$$\begin{aligned} S\left( \sigma _{\bot },L_{u},\omega \right) =\frac{\kappa _{3}}{4\kappa _{2}}=\frac{\sigma _{\bot }^2k_{u}\frac{\omega }{c}}{ 2N_{u}\pi }=\frac{\sigma _{\bot }^2\frac{\omega }{c}}{ L_{u}} \end{aligned}$$

(4.33)

is the diffraction parameter, with $L_{u}=N_{u}\lambda _{u}$ being the length of the undulator. This form factor $FF_{\bot }(S)$ is a universal function and has been obtained before in Ref.  [5]. Here we have reproduced the result following the general definition of the transverse form factor. The variable S is a parameter used to classify the diffraction limit of the beam,

$$\begin{aligned} FF_{\bot }(S)={\left\{ \begin{array}{ll} &{}1,\ S\ll 1,\ \text {below diffraction limit,}\\ &{}\frac{1}{2\pi S},\ S\gg 1,\ \text {above diffraction limit.} \end{array}\right. } \end{aligned}$$

(4.34)

This function along with its asymptotic result above diffraction limit are shown in Fig. 4.3.

Note that the decrease of $FF_{\bot }(S)$ with the increase of $\sigma _{\bot }$ ($S\propto \sigma _{\bot }^{2}$) means that the coherent radiation at the frequency $\omega =H\omega _{0}$ becomes less when the transverse electron beam size becomes larger. This reflects the fact that for a given radiation frequency $\omega$, there is a range of polar angle $\theta$ that can contribute. For $\omega =H\omega _{0}$, not only $\theta =0$, but also $\theta$ very close to 0 contribute. With the increase of $\sigma _{\bot }$, the effective bunching factor $b(\theta ,\varphi ,\omega )$ at $\omega =H\omega _{0}$ drops for these non-zero $\theta$ due to the projected bunch lengthening, therefore the coherent radiation becomes less. Another way to appreciate the drop of $FF_{\bot }(S)$ with the increase of $\sigma _{\bot }$ is that there is a transverse coherence area whose radius is proportional to $\sqrt{L_{u}\lambda _{0}/H}$ with $\lambda _{0}=2\pi \frac{c}{\omega _{0}}$, and less particles can cohere with each other when the transverse size of the electron beam increases.

Fig. 4.3

The universal function $FF_{\bot }(S)$ and its asymptotic value above diffraction limit. The solid line comes from Eq. (4.32), the dashed line from the asymptotic relation above diffraction limit Eq. (4.34)

Note that our definition Eq. (4.12) and derivation of the transverse form factor Eq. (4.31) is more general than that given in Ref. [5], as it covers other frequencies in addition to a single frequency $\omega _{0}$. Therefore, it contains more information than Eq. (4.32) as will be presented soon. However, Eq. (4.31) is still not simple enough for efficient analytical evaluation to provide physical insight. A further approximation is thus introduced,

$$\begin{aligned} \begin{aligned} FF_{\bot }(H,\omega )&=\frac{\int _{0}^{\infty }dxe^{-\kappa _{3}x}\text {sinc}^{2}\left( \kappa _{1}+\kappa _{2}x\right) }{\int _{0}^{\infty }dx\text {sinc}^{2}\left( \kappa _{1}+\kappa _{2}x\right) }\\&=e^{\frac{\kappa _1 \kappa _3}{\kappa _2}}\frac{\int _{\kappa _{1}}^{\infty }dye^{-\frac{\kappa _{3}}{\kappa _{2}}y}\text {sinc}^{2}(y)}{\int _{\kappa _{1}}^{\infty }dy\text {sinc}^{2}(y)}\\&\approx e^{\frac{\kappa _1 \kappa _3}{\kappa _2}}\frac{\int _{0}^{\infty }dye^{-\frac{\kappa _{3}}{\kappa _{2}}y}\text {sinc}^{2}(y)}{\int _{0}^{\infty }dy\text {sinc}^{2}(y)}\\&=e^{-4N_{u}\pi S\left( H-\frac{\omega }{\omega _{0}}\right) }FF_{\bot }(S). \end{aligned} \end{aligned}$$

(4.35)

The condition of applying such simplification is $\frac{\kappa _{3}}{\kappa _{2}}\left( \omega =H\omega _{0}\right) \ll 1\ \text {or}\ S\left( \omega =H\omega _{0}\right) \ll 1,$ i.e., the beam is below diffraction limit for the on-axis radiation $\omega =H\omega _{0}$. Therefore, the conditions of applying Eq. (4.35) are

$$\begin{aligned} \begin{aligned}&N_{u}\gg 1\ \text {and}\ \sigma _{\bot }\ll \frac{1}{\sqrt{ H}}\sqrt{\frac{L_{u}\lambda _{0}}{2\pi }}. \end{aligned} \end{aligned}$$

(4.36)

If the second condition is not satisfied, the more accurate result Eq. (4.31) should be referred.

Fig. 4.4

The comparison of the transverse form factor, between that calculated from our simplified analytical formula Eq. (4.35) and that from the direct numerical integration of Eq. (4.12) for the case of $H=1$, with $N_{u}=10$ (left) and $N_{u}=100$ (right), respectively. Other related parameters used in the calculation: $E_{0}=400$ MeV, $\lambda _{0}=13.5$ nm, $\lambda _{u}=1$ cm, $K=1.14$, $\sigma _{\bot }=5\ \upmu$m

As a benchmark of our derivation, here we conduct some calculations of the transverse form factor based on direct numerical integration of Eq. (4.12), and compare them with our simplified analytical formula Eq. (4.35). The parameters used are for the envisioned EUV SSMB to be presented in Sect. 4.6, and are given in the figure caption. As can be seen in Fig. 4.4, their agreement when $N_{u}=100$ is remarkably well. Even in the case of $N_{u}=10$, the agreement is still satisfactory. There are two reasons why the agreement is better in the case of a large $N_{u}$. The first is that in the derivation we have made use of the sharpness of $F(\epsilon )$, whose width is $1/N_{u}$. The second is that $S\propto \frac{1}{N_{u}}$ with a given transverse beam size and undulator period length, and our simplified analytical formula Eq. (4.35) applies when $S\left( \omega =H\omega _{0}\right) \ll 1$.

Fig. 4.5

Flat contour plot of the transverse form factor $FF_{\bot }(H,\omega )$ for $H=1$, as a function of the radiation frequency $\omega$ and transverse electron beam size $\sigma _{\bot }$, calculated using our simplified analytical formula Eq. (4.35). Parameters used in the calculation: $E_{0}=400$ MeV, $\lambda _{0}=13.5$ nm, $\lambda _{u}=1$ cm, $K=1.14$, $N_{u}=2\times 79$

To appreciate the implication of the generalized transverse form factor further, an example flat contour plot of the transverse form factor as a function of the radiation frequency $\omega$ and transverse electron beam size $\sigma _{\bot }$ is shown in Fig. 4.5. As can be seen, a large transverse electron beam size will suppress the off-axis red-shifted coherent radiation due to the projected bunch lengthening from the transverse size, thus the effective bunching factor degradation, when observed off-axis. Therefore, a large transverse electron beam size will make the coherent radiation more collimated in the forward direction, and more narrowbanded around the harmonic lines. But note that not only the red-shifted radiation is suppressed, the radiation strength of each harmonic line $\omega =H\omega _{0}$ actually also decreases with the increase of the transverse electron beam size, the reason of which we have just explained.

Now we evaluate the bandwidth and opening angle of the radiation at different harmonics due to the transverse form factor. In particular, we are interested in the case where the off-axis red-shifted radiation is significantly suppressed by the transverse size of the electron beam, which requires that

$$\begin{aligned} N_{u}\pi S\left( \omega =H\omega _{0}\right) \gg 1, \end{aligned}$$

(4.37)

i.e., $\sigma _{\bot }\gg \sqrt{\frac{H}{2}}\frac{\sqrt{\lambda _{u}\lambda _{0}}}{2\pi }$. Note that to apply Eq. (4.35), we still need the conditions in Eq. (4.36). For example, to apply the analytical estimation for the example EUV SSMB calculation to be presented in Sect. 4.6, in which $\lambda _{u}=1$ cm, $\lambda _{0}=13.5$ nm and $N_{u}=2\times 79$, we need $1.3\ \upmu \text {m}\ll \sigma _{\bot }\ll 58\ \upmu \text {m}$. The typical transverse electron beam size in an EUV SSMB ring is in this range.

With these conditions satisfied, the value of the exponential factor in Eq. (4.35) is more sensitive to the change of $\omega$, compared to the universal function $FF_{\bot }(S)$. Therefore, here we consider only the exponential term when $\omega$ is close to $H\omega _{0}$. We want to know the $\omega$ at which the exponential term gives $e^{-4N_{u}\pi S\left( H-\frac{\omega }{\omega _{0}}\right) }=e^{-1}.$ Putting in the definition of $S=\frac{\sigma _{\bot }^2k_{u}\frac{\omega }{c}}{ 2N_{u}\pi }$, we have $\omega _{e^{-1}}=\frac{1+\sqrt{1-\frac{2}{H^{2}\sigma _{\bot }^{2}k_{u}k_{0}}}}{2}H\omega _{0}$. Then

$$\begin{aligned} \begin{aligned} \Delta \omega _{e^{-1}}\bigg |_{\bot }&=H\omega _{0} - \omega _{e^{-1}}=\frac{1-\sqrt{1-\frac{2}{H^{2}\sigma _{\bot }^{2}k_{u}k_{0}}}}{2}H\omega _{0}. \end{aligned} \end{aligned}$$

(4.38)

As $\sigma _{\bot }\gg \sqrt{\frac{H}{2}}\frac{\sqrt{\lambda _{u}\lambda _{0}}}{2\pi }$, therefore $\frac{2}{H^{2}\sigma _{\bot }^{2}k_{u}k_{0}}\ll 1$, we have

$$\begin{aligned} \frac{\Delta \omega _{e^{-1}}}{H\omega _{0}}\bigg |_{\bot }\approx \frac{1}{2H^{2}\sigma _{\bot }^{2}k_{u}k_{0}}. \end{aligned}$$

(4.39)

Correspondingly, the opening angle of the coherent radiation due to the transverse form factor is

$$\begin{aligned} \begin{aligned}&\frac{\gamma ^{2}\theta _{e^{-1}}^{2}}{1+K^{2}/2}\approx \frac{\Delta \omega _{e^{-1}}}{H\omega _{0}}\bigg |_{\bot }\Rightarrow \theta _{e^{-1}}\bigg |_{\bot } \approx \frac{\sqrt{2+K^{2}}}{2H\gamma \sigma _{\bot }\sqrt{k_{u}k_{0}}}. \end{aligned} \end{aligned}$$

(4.40)

It is interesting to note that

$$\begin{aligned} \frac{\Delta \omega _{e^{-1}}}{H\omega _{0}}\bigg |_{\bot }\propto \frac{1}{H^{2}}. \end{aligned}$$

(4.41)

As a comparison, the relative bandwidth at the harmonics due to the longitudinal form factor is

$$\begin{aligned} \frac{\Delta \omega }{H\omega _{0}}\bigg |_{z}\propto \frac{1}{H}. \end{aligned}$$

(4.42)

Note also that $\frac{\Delta \omega _{e^{-1}}}{H\omega _{0}}\bigg |_{\bot }$ and $\theta _{e^{-1}}\bigg |_{\bot }$ are independent of $N_{u}$, although the approximations adopted in the derivation actually involve conditions related to $N_{u}$.

4.3 Radiation Power and Spectral Flux

In many cases, the microbunching is formed based on an electron bunch much longer than the radiation wavelength, for example in an FEL or CHG. In these cases, the linewidth of the longitudinal form factor at the harmonics are usually even narrower than that given by the transverse form factor. This also means that the coherent radiation of a long continuous electron bunch-based microbunching will be dominantly in the forward direction, as the bunching factor of the off-axis red-shifted frequency is suppressed very fast compared to the on-axis resonant ones. For a more practical application, here we derive the coherent radiation power and spectral flux at the undulator radiation harmonics in these cases. As we will see, the results can be viewed as useful references for SSMB radiation.

We assume that the long electron bunch, before microbunching, is Gaussain. The transverse form factors around the harmonics do not change much, i.e., we assume $e^{-4N_{u}\pi S\left( H-\frac{\omega }{\omega _{0}}\right) }\approx 1$ when $\omega$ is close to $H\omega _{0}$. Therefore, we only need to take into account the Gaussian shape of the longitudinal form factors at the harmonics. The RMS bandwidth of the longitudinal form factor for a Gaussian bunch with a length of $\sigma _{z}$ is $\Delta \omega |_{z}=\frac{c}{\sigma _{z}/\sqrt{2}}$. Therefore, the coherent radiation energy at the H-th harmonic is

$$\begin{aligned} \begin{aligned} W_{H}&=\left[ N_{e}^{2}FF_{\bot }(H,\omega )FF_{z}(\omega )\frac{dW_{H}}{d\omega }\Bigg |_{\text {point}}\right] \left( \omega =H\omega _{0}\right) \times \int _{-\infty }^{\infty }\text {exp}\left( -\frac{(\omega -H\omega _{0})^2}{(c/\sigma _{z})^{2}}\right) d\omega \\&= N_{e}^{2}FF_{\bot }(S)|b_{z,H}|^{2}\frac{2e^{2}}{\epsilon _{0}c}{G}(\theta =0)\frac{1+K^2/2}{H}\frac{N_{u}}{2}\times \sqrt{\pi }c/\sigma _{z}. \end{aligned} \end{aligned}$$

(4.43)

For a planar undulator, the $\sigma$-mode radiation dominates and from Eq. (4.22) we have

$$\begin{aligned} G_{\sigma }(\theta =0)=\left[ \frac{HK/\sqrt{2}}{2(1+K^{2}/2)}\right] ^{2}[JJ]_{H}^{2}, \end{aligned}$$

(4.44)

in which the denotation $[JJ]_{H}^{2}=\left[ J_{\frac{H-1}{2}}\left( H\chi \right) -J_{\frac{H+1}{2}}\left( H\chi \right) \right] ^{2}$, with $\chi =\frac{K^{2}}{4+2K^{2}}$, is used. Note however, the above expression is meaningful only for an odd H, as the on-axis even harmonic radiation is rather weak. The peak power of the odd-H-th harmonic coherent radiation is then

$$\begin{aligned} \begin{aligned} P_{H,\text {peak}}&=\frac{W_{H}}{\sqrt{2\pi }\frac{\sigma _{z}/c}{\sqrt{2}}}=\frac{\pi }{\epsilon _{0}c}N_{u}H\chi [JJ]_{H}^{2}FF_{\bot }(S)|b_{z,H}|^{2}I_{P}^{2}, \end{aligned} \end{aligned}$$

(4.45)

where $I_{P}=\frac{N_{e}e}{\sqrt{2\pi }\sigma _{z}/c}$ is the peak current of the Gaussian bunch before microbunching. For a more practical application of the derived formula, we put in the numerical value of the constants, and arrive at

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

(4.46)

Note that the above formula applies when the radiation slippage length $N_{u}\lambda _{0}$ is smaller than the bunch length $\sigma _{z}$. If not, the above formula will overestimate the coherent radiation peak power, as the RMS radiation pulse length is then longer than $\frac{c/\sigma _{z}}{\sqrt{2}}$. Note also that given the same bunch charge and form factors, $P_{H,\text {peak}}\propto I_{P}^{2}\propto \frac{1}{\sigma _{z}^{2}}$ while $W_{H}\propto \frac{1}{\sigma _{z}}$. The reason a shorter bunch radiates more total energy is because that more particles are within the coherence length.

At a first glance of Eq. (4.45), the coherent radiation power $P_{\text {coh}}$ seems to be proportional to $N_{u}$, while an intuitive picture of the longitudinal coherence length $l_{\text {coh}}\propto N_{u}$ says that the scaling should be $P_{\text {coh}}\propto N_{u}^{2}$, as the electron number within the coherence length is proportional to $N_{u}$. This is actually because that $FF_{\bot }(S)$ is also a function of $N_{u}$. It is interesting to note that

$$\begin{aligned} P_{\text {coh}}={\left\{ \begin{array}{ll} &{}\propto N_{u},\ \text {below diffraction limit,}\\ &{}\propto N_{u}^{2},\ \text {above diffraction limit,} \end{array}\right. } \end{aligned}$$

(4.47)

which can be obtained from the asymptotic expressions of $FF_{\bot }(S)$ as shown in Eq. (4.34). So for a given transverse beam size, $P_{\text {coh}}\propto N_{u}^{2}$ at first when $N_{u}$ is small. When $N_{u}$ is large enough such that the electron beam is below diffraction limit, then $P_{\text {coh}}\propto N_{u}$. Physically this is because with the increase of $N_{u}$, the diffraction of the radiation will prevent the radiation from one particle so effectively affect the particles far in front of it, as the on-axis field from this particle becomes weaker with the increase of the radiation slippage length.

Our derivation of the coherent radiation power above is for a Gaussian bunch-based microbunching. For a coasting or DC beam, we just need to replace $I_{P}$ in Eq. (4.45) by the average current $I_{\text {A}}$, and the peak power is then the average power. For a helical undulator, we need to replace $K_{\text {planar}}/\sqrt{2}$ with $K_{\text {helical}}$, and $[JJ]_{1}^{2}$ with 1, in the evaluation of the radiation power at fundamental frequency.

We remind the readers that Eq. (4.45) is for the case of a long continuous bunch-based microbunching, for example in FELs and CHG. In some of the SSMB scenarios, the microbunches are cleanly separated from each other according to the modulation laser wavelength as will be shown in Fig. 4.9, and usually the radiation wavelength is at a high harmonic of the modulation laser. Therefore, there actually could be significant red-shifted radiation generated in SSMB as we will see in the example calculation in Sect. 4.6. If we put the average current of SSMB in Eq. (4.45), what it evaluates is the radiation power whose frequency content is close to the on-axis harmonic and will underestimate the overall radiation power.

After investigating the radiation power, let us now have a look at the spectral flux, which is the number of photons per unit time in a given small bandwidth. The spectral flux of coherent radiation at the odd-H-th harmonic can be calculated according to $\frac{dW_{H}}{d\omega }$ as

$$\begin{aligned} \begin{aligned} \mathcal {F}(\omega =H\omega _{0})&=\left[ N_{e}^{2}FF_{\bot }(H,\omega )FF_{z}(\omega )\frac{dW_{H}}{d\omega }\Bigg |_{\text {point}}\times \frac{\Delta \omega }{\hbar \omega }\right] \left( \omega =H\omega _{0}\right) \\&=\frac{1}{1000}\frac{e^{2}}{2\epsilon _{0}c\hbar }N_{u}H\chi [JJ]_{H}^{2}FF_{\bot }(S)|b_{z,H}|^{2}N_{e}^{2}\ \text {(phs/pass/0.1}\% \text {b.w.)}, \end{aligned} \end{aligned}$$

(4.48)

where $\hbar$ is the reduced Planck’s constant. Again we put in the numerical value of the constants, and arrive at

$$\begin{aligned} \begin{aligned} \mathcal {F}(\omega =H\omega _{0})&=4.573\times 10^{-5}N_{u}H\chi [JJ]_{H}^{2}FF_{\bot }(S)|b_{z,H}|^{2}N_{e}^{2}\ \text {(phs/pass/0.1}\% \text {b.w.)}. \end{aligned} \end{aligned}$$

(4.49)

Note that the above spectral flux is for a single pass of the microbunched electron beam through the radiator undulator. For the evaluation of the average spectral flux in an SSMB storage ring, we need to multiply it with the number of microbunches passing a fixed location in one second, namely $F\frac{\bar{v}_{z}}{\lambda _{L}}$, with F being the filling factor of microbunches in the ring, $\bar{v}_{z}$ being the average longitudinal speed of electron wiggling in the modulator undulator, and $\lambda _{L}$ being the modulation laser wavelength. We remind the readers that the above statement means we do not account for the radiation overlapping between different microbunches if the radiation slippage length is larger than $\frac{c}{\bar{v}_{z}}\lambda _{L}$. If there is such radiation overlapping, the flux will be boosted further since the electrons in neighboring microbunches can now cohere with each other. To give the readers a more concrete feeling about the high spectral flux in SSMB, we just need to multiply the spectral flux of the usual incoherent undulator radiation with a factor of $N_{e}FF_{\bot }(S)|b_{z,H}|^{2}$, with $N_{e}$ being the number of electrons per microbunch. For example, in the envisioned EUV SSMB to be presented in Sect. 4.6, $N_{e}=2.2\times 10^{4}$, and $FF_{\bot }(S)|b_{z,H}|^{2}$ can be as large as 0.1. Therefore, the EUV spectral flux in an SSMB storage ring can thus be three orders of magnitude higher than that in a conventional synchrotron source.

4.4 Impact of Electron Beam Divergence and Energy Spread

Now we take into account the impact of beam divergence and energy spread on the coherent radiation. With an aim to obtain some efficient evaluation, here we simplify the analysis by considering only the impact of particle’s $x'$, $y'$ and $\delta$ on the arrival time of the radiation, not on the radiation pattern. This approximation is valid when the beam divergence and energy spread are small enough, such that $\sigma _{x',y'}<\frac{1}{\gamma }$ and $\sigma _\delta <\frac{1}{N_{u}}$. Basically we want to get a formula of the effective transverse and longitudinal form factors considering the beam size evolution during radiation.

As an example, here we assume that the beam is a 6D Gaussian one, and round in the transverse dimension. Further we assume the beam reaches its minimal in all three dimensions at the radiator undulator center, which is desired to get high-power radiation, then the effective transverse and longitudinal form factors are

$$\begin{aligned} \begin{aligned} FF_{\bot }(\omega )&=\frac{1}{L_{u}}\int _{-\frac{L_{u}}{2}}^{\frac{L_{u}}{2}}FF_{\bot }\left( \frac{\left( \sigma _{\bot }^2+(\sigma _{\theta _\bot }s)^{2}\right) \frac{\omega }{c}}{ L_{u}}\right) ds,\\ FF_{z}(\omega )&=\frac{1}{L_{u}}\int _{-\frac{L_{u}}{2}}^{\frac{L_{u}}{2}}e^{-\left( \frac{\omega }{c}\right) ^2 \left[ \sigma _z^2+\left( \sigma _{\delta }\frac{s}{L_{u}2N_{u}\lambda _{0}}\right) ^{2}\right] }ds\\&=e^{-\left( \frac{\omega }{ c}\right) ^2 \sigma _z^2}\frac{\sqrt{\pi } }{2} \frac{\text {erf}\left( \frac{\omega }{c}\sigma _{\delta } N_u \lambda _0\right) }{ \frac{\omega }{c} \sigma _{\delta } N_u \lambda _0},\\ \end{aligned} \end{aligned}$$

(4.50)

where $\sigma _{\bot }$, $\sigma _{\theta _\bot }$, $\sigma _{z}$ and $\sigma _{\delta }$ are the transverse beam size, divergence, bunch length and energy spread at the undulator center, with $\sigma _{\bot }\sigma _{\theta _\bot }=\epsilon _{\bot }$ and $\sigma _{z}\sigma _{\delta }=\epsilon _{z}$, and $\text {erf}(x)=\frac{2}{\sqrt{\pi }}\int _{0}^{x}e^{-t^{2}}dt$ is the error function. Note that $\frac{\sqrt{\pi } }{2}\frac{\text {erf}(x)}{x}<1$ for $x\ne 0$.

Figure 4.6 is an example plot of the effective form factors based on the above formulas, and the comparison with the case for a 3D rigid beam. It can be seen that given a transverse and longitudinal emittance, there is an optimal transverse beam size and bunch length at the radiator center considering the impact of beam divergence and energy spread. This is expected, since the beam size or bunch length of an over-focused beam grows very fast.

Fig. 4.6

An example plot to show the optimization of beam sizes in the middle of the radiator, considering the beam divergence and energy spread. Parameters used for calculation: $\epsilon _{\bot }=1$ nm, $\epsilon _{z}=4$ pm, $\lambda _{0}=13.5$ nm, $N_{u}=2\times 79$, $L_{u}=1.58$ m

4.5 Statistical Property of Radiation

In the previous sections, we have ignored the quantum discrete nature of radiation. Besides, we have derived the coherent radiation property using a smooth distributed charge, i.e., we have treated the charge as a continuum fluid. The number of photons radiated from a charged particle beam actually fluctuates from turn to turn or bunch to bunch if the quantum nature of radiation and the pointlike nature of electrons are taken into account. The first mechanism exists even if there is only one electron, and the second mechanism is related to the interference of fields radiated by different electrons [6]. Using the classical language, the second fluctuation mechanism is from the fluctuation of the bunching factor or form factor of electron beam.

There have been studies on the statistical property of the radiation in FELs [7] and also the storage ring-based synchrotron radiation sources [6, 8]. Rich information about the electron beam is embedded in the radiation fluctuations, or more generally the statistical property of the radiation. For example, the turn-by-turn fluctuation of the incoherent undulator radiation can be used to measure the transverse emittance of electron beam [8]. The previous treatment, however, usually cares about the cases where the bunch length is much longer than the radiation wavelength, i.e., the radiation is temporally incoherent (in SASE FEL, incoherent at the beginning). In SSMB, the bunch length is comparable or shorter than the desired radiation wavelength, and the dominant radiation is temporally coherent. Although numerical calculation is doable following the general theoretical formulation, an analytical formula for the fluctuation in this temporally coherent radiation dominant regime is of value for a better understanding of the physics and investigation of its potential applications.

4.5.1 Pointlike Nature of Electron

Here for SSMB we consider first the second mechanism of fluctuation, i.e., the radiation power fluctuation arising from the pointlike nature of the radiating electron. In this section, to simplify writing, we use the vector notation

$$\begin{aligned} \begin{aligned} \textbf{k}&=\frac{\omega }{c}\left( \sin \theta \cos \varphi ,\sin \theta \sin \varphi ,1\right) ,\\ \textbf{r}&=(x,y,z). \end{aligned} \end{aligned}$$

(4.51)

Then the bunching factor with the pointlike nature of electrons taken into account is

$$\begin{aligned} \begin{aligned} {b}(\textbf{k})={b}(\theta ,\varphi ,\omega )=\frac{1}{N_{e}}\sum _{n=1}^{N_{e}}e^{-i\textbf{k}\cdot \mathbf {r_{n}}}. \end{aligned} \end{aligned}$$

(4.52)

First we want to evaluate the coherent radiation power fluctuation at a specific frequency and observation angle. As the radiation power is proportional to $N_{e}^{2}|b(\textbf{k})|^{2}$, therefore we need to know the fluctuation of $|b(\textbf{k})|^{2}$. Since

$$\begin{aligned} \begin{aligned} |b(\textbf{k})|^{2}&=\frac{1}{N_{e}^{2}}\sum _{n=1}^{N_{e}}\sum _{m=1}^{N_{e}}e^{-i\textbf{k}\cdot (\textbf{r}_{n}-\textbf{r}_{m})}=\frac{1}{N_{e}^{2}}\left[ N_{e}+\sum _{m\ne n} e^{-i\textbf{k}\cdot (\textbf{r}_{n}-\textbf{r}_{m})}\right] , \end{aligned} \end{aligned}$$

(4.53)

we have

$$\begin{aligned} \begin{aligned} \left\langle |b(\textbf{k})|^{2}\right\rangle&=\frac{1}{N_{e}}+\left( 1-\frac{1}{N_{e}}\right) |\overline{b}(\textbf{k})|^{2}, \end{aligned} \end{aligned}$$

(4.54)

with

$$\begin{aligned} \begin{aligned} \overline{b}(\textbf{k})=\overline{b}(\theta ,\varphi ,\omega )=\int \rho (\textbf{r})e^{-i\textbf{k}\cdot \textbf{r}}d\textbf{r} \end{aligned} \end{aligned}$$

(4.55)

being the bunching factor we calculated before using a continuum fluid charge distribution in Eq. (4.3). As can be seen from Eq. (4.54), when $N_{e}=1$, which corresponds to the case of a single point charge, then $\left\langle |b(\textbf{k})|^{2}\right\rangle =1$. When $N_{e}\gg 1$ and $N_{e}|\overline{b}(\textbf{k})|^{2}\ll 1$, which corresponds to the case of incoherent radiation dominance, then $\left\langle |b(\textbf{k})|^{2}\right\rangle =\frac{1}{N_{e}}$. When $N_{e}\gg 1$ and $N_{e}|\overline{b}(\textbf{k})|^{2}\gg 1$, which corresponds to the case of coherent radiation dominance, then $\left\langle |b(\textbf{k})|^{2}\right\rangle =|\overline{b}(\textbf{k})|^{2}$. These results are as expected.

Table 4.1 The $N_{e}^{4}$ terms in the quadruple sum of Eq. (4.56) can be placed in 15 different classes, as shown in Ref. [9]

The calculation of $\left\langle |b(\mathbf {k_{1}})|^{2}|b(\mathbf {k_{2}})|^{2}\right\rangle$ is more involved. More specifically,

$$\begin{aligned} \begin{aligned} |b(\mathbf {k_{1}})|^{2}|b(\mathbf {k_{2}})|^{2}=\frac{1}{N_{e}^{4}}\sum _{n=1}^{N_{e}}\sum _{m=1}^{N_{e}}\sum _{p=1}^{N_{e}}\sum _{q=1}^{N_{e}}e^{-i\mathbf {k_{1}}\cdot (\textbf{r}_{n}-\textbf{r}_{m})-i\mathbf {k_{2}}\cdot (\textbf{r}_{p}-\textbf{r}_{q})}. \end{aligned} \end{aligned}$$

(4.56)

The $N_{e}^{4}$ terms in this summation can be placed in 15 different cases, as shown in Table 4.1. Corresponding to the 15 cases, we have

$$\begin{aligned} \begin{aligned} \left\langle |b(\mathbf {k_{1}})|^{2}|b(\mathbf {k_{2}})|^{2}\right\rangle&=\frac{1}{N_{e}^{4}}\left[ N_{e}\overline{b}(\mathbf {k_{1}}-\mathbf {k_{1}}+\mathbf {k_{2}}-\mathbf {k_{2}})\right. \\&\left. \ \ \ \ +N_{e}(N_{e}-1)\overline{b}(\mathbf {k_{1}}-\mathbf {k_{1}})\overline{b}(\mathbf {k_{2}}-\mathbf {k_{2}})\right. \\&\left. \ \ \ \ +N_{e}(N_{e}-1)(N_{e}-2)\overline{b}(\mathbf {k_{1}}-\mathbf {k_{1}})\overline{b}(\mathbf {k_{2}})\overline{b}(-\mathbf {k_{2}})\right. \\&\left. \ \ \ \ +N_{e}(N_{e}-1)\overline{b}(\mathbf {k_{1}}+\mathbf {k_{2}})\overline{b}(-\mathbf {k_{1}}-\mathbf {k_{2}})\right. \\&\left. \ \ \ \ +N_{e}(N_{e}-1)(N_{e}-2)\overline{b}(\mathbf {k_{1}}+\mathbf {k_{2}})\overline{b}(-\mathbf {k_{1}})\overline{b}(-\mathbf {k_{2}})\right. \\&\left. \ \ \ \ +N_{e}(N_{e}-1)\overline{b}(\mathbf {k_{1}}-\mathbf {k_{2}})\overline{b}(-\mathbf {k_{1}}+\mathbf {k_{2}})\right. \\&\left. \ \ \ \ +N_{e}(N_{e}-1)(N_{e}-2)\overline{b}(\mathbf {k_{1}}-\mathbf {k_{2}})\overline{b}(-\mathbf {k_{1}})\overline{b}(\mathbf {k_{2}})\right. \\&\left. \ \ \ \ +N_{e}(N_{e}-1)\overline{b}(\mathbf {k_{1}}-\mathbf {k_{1}}+\mathbf {k_{2}})\overline{b}(-\mathbf {k_{2}})\right. \\&\left. \ \ \ \ +N_{e}(N_{e}-1)\overline{b}(\mathbf {k_{1}}-\mathbf {k_{1}}-\mathbf {k_{2}})\overline{b}(\mathbf {k_{2}})\right. \\&\left. \ \ \ \ +N_{e}(N_{e}-1)\overline{b}(\mathbf {k_{1}}+\mathbf {k_{2}}-\mathbf {k_{2}})\overline{b}(-\mathbf {k_{1}})\right. \\&\left. \ \ \ \ +N_{e}(N_{e}-1)\overline{b}(-\mathbf {k_{1}}+\mathbf {k_{2}}-\mathbf {k_{2}})\overline{b}(\mathbf {k_{1}})\right. \\&\left. \ \ \ \ +N_{e}(N_{e}-1)(N_{e}-2)(N_{e}-3)\overline{b}(\mathbf {k_{1}})\overline{b}(-\mathbf {k_{1}})\overline{b}(\mathbf {k_{2}})\overline{b}(-\mathbf {k_{2}})\right. \\&\left. \ \ \ \ +N_{e}(N_{e}-1)(N_{e}-2)\overline{b}(\mathbf {k_{1}})\overline{b}(-\mathbf {k_{1}})\overline{b}(\mathbf {k_{2}}-\mathbf {k_{2}})\right. \\&\left. \ \ \ \ +N_{e}(N_{e}-1)(N_{e}-2)\overline{b}(\mathbf {k_{1}})\overline{b}(\mathbf {k_{2}})\overline{b}(-\mathbf {k_{1}}-\mathbf {k_{2}})\right. \\&\left. \ \ \ \ +N_{e}(N_{e}-1)(N_{e}-2)\overline{b}(\mathbf {k_{1}})\overline{b}(-\mathbf {k_{2}})\overline{b}(-\mathbf {k_{1}}+\mathbf {k_{2}})\right] . \end{aligned} \end{aligned}$$

(4.57)

The above result can be re-organized as

$$\begin{aligned} \begin{aligned} \left\langle |b(\mathbf {k_{1}})|^{2}|b(\mathbf {k_{2}})|^{2}\right\rangle&=\frac{1}{N_{e}^{4}}\left[ N_{e}^{2}+N_{e}^{2}(N_{e}-1)\left( |\overline{b}(\mathbf {k_{1}})|^{2}+|\overline{b}(\mathbf {k_{2}})|^{2}\right) \right. \\&\left. \ \ \ \ +N_{e}(N_{e}-1)(N_{e}-2)\left( \overline{b}(\mathbf {k_{1}}+\mathbf {k_{2}})\overline{b}(-\mathbf {k_{1}})\overline{b}(-\mathbf {k_{2}})+\text {c.c.}\right) \right. \\&\left. \ \ \ \ +N_{e}(N_{e}-1)(N_{e}-2)\left( \overline{b}(\mathbf {k_{1}}-\mathbf {k_{2}})\overline{b}(-\mathbf {k_{1}})\overline{b}(\mathbf {k_{2}})+\text {c.c.}\right) \right. \\&\left. \ \ \ \ +N_{e}(N_{e}-1)\left( |\overline{b}(\mathbf {k_{1}}+\mathbf {k_{2}})|^{2}+|\overline{b}(\mathbf {k_{1}}-\mathbf {k_{2}})|^{2}\right) \right. \\&\left. \ \ \ \ +N_{e}(N_{e}-1)(N_{e}-2)(N_{e}-3)|\overline{b}(\mathbf {k_{1}})|^{2}|\overline{b}(\mathbf {k_{2}})|^{2}\right] , \end{aligned} \end{aligned}$$

(4.58)

in which c.c. means complex conjugate.

If $\mathbf {k_{1}}=\mathbf {k_{2}}=\textbf{k}$, then

$$\begin{aligned} \begin{aligned} \left\langle |b(\textbf{k})|^{4}\right\rangle -\left\langle |b(\textbf{k})|^{2}\right\rangle ^{2}&=\frac{1}{N_{e}^{4}}\left[ N_{e}(N_{e}-1)+2N_{e}(N_{e}-1)(N_{e}-2)|\overline{b}(\textbf{k})|^{2}\right. \\&\left. \ \ \ \ +2N_{e}(N_{e}-1)(N_{e}-2)\text {Re}\left[ \overline{b}(2\textbf{k})\overline{b}^{2}(-\textbf{k})\right] \right. \\&\left. \ \ \ \ +N_{e}(N_{e}-1)|\overline{b}(2\textbf{k})|^{2}\right. \\&\left. \ \ \ \ -2N_{e}(N_{e}-1)(2N_{e}-3)|\overline{b}(\textbf{k})|^{4}\right] , \end{aligned} \end{aligned}$$

(4.59)

where Re[] means taking the real part of a complex number.

When $N_{e}\gg 1$ and $N_{e}|b(\textbf{k})|^{2}\ll 1$, which is the case for incoherent radiation dominance, we have $\left\langle |b(\textbf{k})|^{2}\right\rangle =\frac{1}{N_{e}}$ and

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

(4.60)

where Var[] means the variance of, and $\mathcal {O}(x^{n})$ means terms of order $x^{n}$ and higher. Therefore, the relative fluctuation of incoherent radiation is relatively large. This is also the reason why SASE-FEL radiation has a large shot-to-shot power fluctuation.

When $N_{e}\gg 1$ and $N_{e}|b(\textbf{k})|^{2}\gg 1$, which corresponds to the case of coherent radiation dominance like that in SSMB, we have

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

(4.61)

The above equation is the main result of our analysis of bunching factor fluctuation for the regime of coherent radiation dominance, and to our knowledge is new. The formula can be used to evaluate coherent radiation power fluctuation at a specific frequency and observation angle. If the transverse electron beam size is zero, or if we observe on-axis, then we can just replace $b(\textbf{k})$ with $b_{z}(\omega )$ in the above formula. We remind the readers that $b(\textbf{k})$ in general is a complex number.

Now we conduct some numerical simulations to confirm our analysis of coherent radiation fluctuation. As can be seen from Fig. 4.7, which correspond to the cases of a Gaussian and a rectangular distributed bunch, respectively, the simulation results agree well with our theoretical prediction.

Fig. 4.7

Fluctuation of the longitudinal form factor at 13.5 nm v.s. bunch length with $N_{e}=2.2\times 10^{4}$. The bunch distribution is assumed to be Gaussian in the left and rectangular in the right, and the theoretical fluctuation is calculated according to Eq. (4.61), omitting the term $\mathcal {O}\left( \frac{1}{N_{e}^{2}}\right)$. For each parameters choice, $1\times 10^{4}$ simulations have been conducted to obtain the fluctuation

After investigating the expectation and variance of $|b(\textbf{k})|^{2}$, one may be curious about its more detailed distribution. It can be shown that when $N_{e}|b(\textbf{k})|^{2}\gg 1$, the distribution of $|b(\textbf{k})|^{2}$ tends asymptotically toward Gaussian.

As explained before, for a fixed frequency $\omega$, there is a range of polar angle $\theta$ which can contribute. To evaluate the overall radiation power fluctuation at a specific frequency $\omega$, we then need to know the fluctuation of the form factor $FF(\omega )$ which involves calculation depending on the specific radiation process. For our interested undulator radiation, it seems not easy to get a concise closed-form analytical formula to evaluate the total radiation power fluctuation when the beam has a finite transverse beam size. So here we refer to numerical calculation to give the readers a more concrete feeling about the impact of transverse size on coherent radiation power fluctuation.

Fig. 4.8

Fluctuation of the transverse form factor at 13.5 nm v.s. transverse beam size with $N_{e}=2.2\times 10^{4}$. The bunch is assumed to have zero length and is round Gaussian in the transverse plane. For each parameters choice, $1\times 10^{3}$ simulations have been conducted to obtain the fluctuation. Parameters used for the calculation: $E_{0}=400$ MeV, $\lambda _{u}=1$ cm, $K=1.14$, $N_{u}=2\times 79$

For simplicity, we assume that the bunch length is zero and focus on the fluctuation of the transverse form factor. As can be seen from the simulation result in Fig. 4.8, the larger the transverse beam size, the larger relative fluctuation of the transverse form factor. We also observe that in a typical parameters set of the envisioned EUV SSMB, the fluctuation of 13.5 nm radiation power due to the finite transverse size is small. For example if $\sigma _{\bot }=16\ \upmu$m, then the relative fluctuation of the transverse form factor as shown in Fig. 4.8 is $0.14\%$. While the relative fluctuation of the longitudinal form factor at 13.5 nm when $\sigma _{z}=3$ nm according to Eq. (4.61) is about $2\%$. Assuming that the beam is transverse-longitudinal decoupled, then

$$\begin{aligned} \begin{aligned} \frac{\text {Var}[FF(\omega )]}{\langle FF(\omega )\rangle ^{2}}&=\frac{\text {Var}[FF_{\bot }]}{\langle FF_{\bot }\rangle ^{2}}+\frac{\text {Var}[FF_{z}]}{\langle FF_{z}\rangle ^{2}}+\frac{\text {Var}[FF_{\bot }]\text {Var}[FF_{z}]}{\langle FF_{\bot }\rangle ^{2}\langle FF_{z}\rangle ^{2}}. \end{aligned} \end{aligned}$$

(4.62)

Therefore, for the envisioned EUV SSMB, the fluctuation of the longitudinal form factor dominates.

After investigating the power fluctuation at a specific frequency $\omega$, now we look into the radiation energy fluctuation gathered within a finite frequency bandwidth and a finite angle acceptance. We use a filter function of $FT(\theta ,\varphi ,\omega )$ to account for the general case of frequency filter, angle acceptance, and detector efficiency. The expectation of the gathered photon energy and photon energy squared are

$$\begin{aligned} \begin{aligned} \langle W\rangle =&N_{e}^{2}\int _{0}^{\pi }\sin \theta d\theta \int _{0}^{2\pi }d\varphi \int _{0}^{\infty }d\omega FT(\theta ,\varphi ,\omega )\frac{d^{2}W}{d\omega d\Omega }(\theta ,\varphi ,\omega )\Bigg |_{\text {point}}\left\langle |b(\theta ,\varphi ,\omega )|^{2}\right\rangle \end{aligned} \end{aligned}$$

(4.63)

and

$$\begin{aligned} \begin{aligned} \langle W^{2}\rangle =&N_{e}^{4}\int _{0}^{\pi }\sin \theta d\theta \int _{0}^{2\pi }d\varphi \int _{0}^{\infty }d\omega \int _{0}^{\pi }\sin \theta ' d\theta ' \int _{0}^{2\pi }d\varphi '\int _{0}^{\infty }d\omega ' \\&FT(\theta ,\varphi ,\omega )FT(\theta ',\varphi ',\omega ')\frac{d^{2}W}{d\omega d\Omega }(\theta ,\varphi ,\omega )\Bigg |_{\text {point}}\frac{d^{2}W}{d\omega d\Omega }(\theta ',\varphi ',\omega ')\Bigg |_{\text {point}}\\&\left\langle |b(\theta ,\varphi ,\omega )|^{2}\right\rangle \left\langle |b(\theta ',\varphi ',\omega ')|^{2}\right\rangle g_{2}(\theta ,\theta ',\varphi ,\varphi ',\omega ,\omega '), \end{aligned} \end{aligned}$$

(4.64)

where

$$\begin{aligned} g_{2}(\theta ,\theta ',\varphi ,\varphi ',\omega ,\omega ')=\frac{\left\langle |b(\theta ,\varphi ,\omega )|^{2}|b(\theta ',\varphi ',\omega ')|^{2}\right\rangle }{\left\langle |b(\theta ,\varphi ,\omega )|^{2}\right\rangle \left\langle |b(\theta ',\varphi ',\omega ')|^{2}\right\rangle }, \end{aligned}$$

(4.65)

whose calculation can follow similar approach of calculating $\left\langle |b(\mathbf {k_{1}})|^{2}|b(\mathbf {k_{2}})|^{2}\right\rangle$ in Eq. (4.57). And the relative fluctuation of the gathered photon energy is

$$\begin{aligned} \sigma ^{2}_{W}=\frac{\langle W^{2}\rangle }{\langle W\rangle ^{2}}-1. \end{aligned}$$

(4.66)

4.5.2 Quantum Nature of Radiation

As mentioned, there is another source of fluctuation, i.e., the quantum discrete nature of radiation. As a result of the Campbell’s theorem [10], we know that for a Poisson photon statistics, the variance of photon number arising from this equals its expectation value. With both contribution from pointlike nature of electrons and quantum nature of radiation taken into account, the relative fluctuation of the radiation power or energy at a given frequency and a specific observation angle is

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

(4.67)

in which

$$\begin{aligned} \begin{aligned} \langle \mathcal {N}_{\text {ph}}(\omega )\rangle |_{\text {beam}}&=\left[ N_{e}+N_{e}(N_{e}-1)|\overline{b}(\omega )|^{2}\right] \langle \mathcal {N}_{\text {ph}}(\omega )\rangle |_{\text {point}}\\&\approx N_{e}^{2}|\overline{b}(\omega )|^{2}\langle \mathcal {N}_{\text {ph}}(\omega )\rangle |_{\text {point}} \end{aligned} \end{aligned}$$

(4.68)

is the expected radiated photon number from the electron beam, and $\langle \mathcal {N}_{\text {ph}}(\omega )\rangle |_{\text {point}}$ is the expected radiated photon number from a single electron. Note that to obtain a nonzero expected photon number $\langle \mathcal {N}_{\text {ph}}(\omega )\rangle |_{\text {beam}}$, a finite frequency bandwidth is needed. Therefore, Eq. (4.67) actually applies to a finite frequency bandwidth close to $\omega$ where $\overline{b}(\omega )$ does not change much.

From Eq. (4.67), it is interesting to note that with the narrowing of the energy bandwidth acceptance, i.e., the decrease of $\langle \mathcal {N}_{\text {ph}}(\omega )\rangle |_{\text {beam}}$, the contribution to the relative fluctuation from the quantum nature of radiation increases, while the contribution from the pointlike nature of electron does not change. This reflects the fact that one fluctuation is quantum, while the other is classical.

Note that in our interested case, $N_{e}\langle \mathcal {N}_{\text {ph}}(\omega )\rangle |_{\text {point}}$ is usually much larger than 1, then the second term in Eq. (4.67) dominants. In other words, the fluctuation due to the pointlike nature of electrons dominants. Only when $N_{e}\langle \mathcal {N}_{\text {ph}}(\omega )\rangle |_{\text {point}}$ is close to 1, will the first term become significant compared to the second term.

As the statistical property of the radiation embeds rich information about the electron beam, innovative beam diagnostics method can be envisioned by making use of this fact. One advantage of using radiation fluctuation in diagnostics is that it has a less stringent requirement on the calibration of detectors. Here we propose an experiment to measure the sub-ps bunch length accurately at a quasi-isochronous storage ring, for example the MLS, at a low beam current, by measuring and analyzing the fluctuation of the coherent THz radiation generated from the electron bunch. Equation (4.67) or some numerical code based on the analysis presented in this section will be the theoretical basis for the experimental proposal. In principle, we can also deduce the transverse distribution of electron beam by measuring the two-dimensional distribution of the radiation fluctuation. More novel beam diagnostics methods may be invented for SSMB and future light sources by making use of the statistical property of radiation.

Fig. 4.9

An example plot of the beam current and longitudinal form factor spectrum of the microbunch train at the radiator in the envisioned EUV SSMB. Up: beam current of the 3 nm microbunch train separated by the modulation laser wavelength $\lambda _{L}=1064$ nm. Bottom: longitudinal form factor $FF_{z}(\omega )$. The exponential decaying envelope corresponds to that of a single 3 nm Gaussian microbunch, and the green periodic delta function lines correspond to the periodic microbunch train in time domain. The desired radiation wavelength is $\lambda _{0}=\frac{\lambda _{L}}{79}=13.5$ nm

4.6 Example Calculations for Envisioned EUV SSMB

To summarize our investigations on the average and statistical property of SSMB radiation, here we present an example calculation for the envisioned EUV SSMB. Figure 4.9 is an example plot of the beam current and longitudinal form factor spectra of the envisioned EUV SSMB. In the envisioned example, the microbunch length is $\sigma _{z}\approx 3$ nm at the radiator where 13.5 nm coherent EUV radiation is generated, and these 3 nm microbunches are separated from each other with a distance of $\lambda _{L}=1064\ \text {nm}=79\times 13.5\ \text {nm}$, which is the modulation laser wavelength. The radiator is assumed to be an undulator. The beam at the radiator can be round or flat depending on the lattice scheme, and its transverse size can range from a couple of $\upmu$m to a couple of 10 $\upmu$m. Note that a Gaussian-distributed current at the radiator is assumed in the plot. This is the case corresponding the usual longitudinal strong focusing SSMB. We remind the readers that the current at the radiator in the TLC-based bunch compression scheme, i.e., generalized longitudinal strong focusing, is actually non-Gaussian considering the nonlinear modulation waveform, as shown in Fig. 3.4. As our goal is to give the readers a picture of the radiation characteristics, here for simplicity we consider the case of a round beam. We remind the readers that the parameters used in this example EUV SSMB radiation calculation is for an illustration and is not optimized.

4.6.1 Average Property

First we present the result for the average property of the EUV radiation. The calculation is based on Eqs. (4.7), (4.11) and (4.22), and the result is shown in Fig. 4.10. The upper part of the figure shows the radiation energy spectrum. The lower part shows the spatial distributions of the radiation energy. The total radiation power is calculated according to

$$\begin{aligned} P=\frac{W}{\lambda _{L}/c}, \end{aligned}$$

(4.69)

where W is the total radiation energy loss of each microbunch. For the example radiator undulator parameters, corresponding to $\sigma _{\bot }=5,10,20\ \upmu$m, the total radiation power are 92 kW, 14 kW, 3.5 kW, respectively. As a reference, the radiation power calculated based on Eq. (4.45) for these three transverse beam sizes are 4 kW, 3.6 kW and 2.6 kW, respectively. The reason Eq. (4.45) gives a smaller value than the overall power as explained is that it does not take into account the red-shifted part of the radiation. Therefore, Eq. (4.45) can be used to evaluate the lower bound of the radiation power from SSMB, once the parameters set of electron beam and radiator undulator is given. It can be seen that generally, kW-level EUV radiation power can be straightforwardly anticipated from a 3 nm microbunch train with an average beam current of 1 A. Note that for simplicity, in this example calculation, the filling factor of microbunches in the ring is assumed to be 100%, i.e., one microbunch per modulation laser wavelength. Then 1 A average current corresponds to the number of electrons per microbunch $N_{e}=\frac{I_{\text {A}}\lambda _{L}/c}{e}=2.2\times 10^{4}$, if the modulation laser wavelength is $\lambda _{L}=1064$ nm.

Fig. 4.10

An example EUV SSMB radiation calculation with a microbunch length of $\sigma _{z}=3$ nm and different transverse sizes $\sigma _{\bot }$. The upper part shows the energy spectrum. Corresponding to $\sigma _{\bot }=5,10,20\ \upmu$m, the total radiation power are 92 kW, 14 kW, 3.5 kW, respectively. The shaded area corresponds to wavelength of $13.5\pm \frac{13.5}{100}$ nm. The bottom part shows spatial distribution of radiation energy. From left to right: $\sigma _{\bot }=5,10,20\ \upmu$m. Parameters used for the calculation: $E_{0}=400$ MeV, $I_{\text {avg}}=1$ A, $\lambda _{L}=1064$ nm, $\lambda _{\text {r}}=\frac{\lambda _{L}}{79}=13.5$ nm, $\lambda _{u}=1$ cm, $K=1.14$, $N_{u}=2\times 79$

Another important observation is that the spectral and spatial distribution of SSMB radiation depends strongly on the transverse size of the electron beam. A large transverse size results in a decrease of the overall radiation power, and also makes the radiation more narrowbanded and collimated in the forward direction. This is an important observation drawn from our investigation on the generalized transverse form factor. Using the example parameters, i.e., $E_{0}=400$ MeV, $\lambda _{L}=1064$ nm, $\lambda _{0}=\frac{\lambda _{L}}{79}=13.5$ nm, $\lambda _{u}=1$ cm, $N_{u}=2\times 79$, $K=1.14$, if $\sigma _{\bot }=10\ \upmu$m, then according to Eqs. (4.39) and (4.40), the relative bandwidth and opening angle due to the transverse form factor can be calculated to be

$$\begin{aligned} \begin{aligned} \frac{\Delta \omega _{e^{-1}}}{\omega _{0}}\bigg |_{\bot }&\approx 1.7\%,\\ \theta _{e^{-1}}\bigg |_{\bot }&\approx 0.21\ \text {mrad}. \end{aligned} \end{aligned}$$

(4.70)

which is in agreement with the result presented in Fig. 4.10.

Since $N_{u}=2\times 79$ is used in the calculation, which means the radiation slippage length is twice the modulation laser wavelength, the energy spectrum and spatial distributions presented in Fig. 4.10 is for that of two neighboring microbunches. The peaks in the energy spectrum, and the interference rings in the radiation spatial distribution as explained is due to the macro longitudinal form factor $\left( \frac{\sin \left( N_{b}\frac{\omega }{c}\frac{\lambda _{L}}{2}\right) }{N_{b}\sin \left( \frac{\omega }{c}\frac{\lambda _{L}}{2}\right) }\right) ^{2}$ of multiple microbunches, with $N_{b}$ the number of microbunches. There is a one-to-one correspondence between the peaks in the energy spectrum and the interference rings in radiation energy spatial distribution. The reason for the appearance of such peaks and rings is that our radiation wavelength is a high harmonic of the distance between the neighboring microbunches.

As a result of the high-power and narrowband feature of the SSMB radiation, a high EUV photon flux of $10^{15}\sim 10^{16}$ phs/s within a 0.1 meV bandwidth can be obtained, if we can realize an EUV power of $\gtrsim$1 kW per 1% b.w. as shown in Fig. 4.10. We remind the readers that the radiation waveform of SSMB is actually a CW or quasi-CW one, if induction linac is used as the energy compensation system and the microbunches occupy the ring with a large filling factor, as assumed in the example calculation. This kind of CW or quasi-CW narrowband photon source is favored in ARPES to minimize the space charge-induced energy shift, spectral broadening and distortion of photoelectrons in a pulsed photon source-based ARPES [11]. Therefore, the high photon flux within a narrow bandwith, together with its CW or quasi-CW waveform, makes SSMB a promising light source for ultrahigh-resolution ARPES. Such a powerful tool may have profound impact on fundamental physics research, for example to probe the energy gap distribution and electronic states of superconducting materials like the magic-angle graphene [12].

4.6.2 Statistical Property

Now we present the result for the statistical property of the radiation. For the case of a Gaussian bunch with $\sigma _{z}=3$ nm and $N_{e}=2.2\times 10^{4}$, from Eq. (4.61) we know that the relative fluctuation of the turn-by-turn or microbunch-by-microbunch on-axis 13.5 nm coherent radiation power will be around $2\%$.

Figure 4.11 gives an example plot for the longitudinal form factor spectrum of three possible realizations of such a Gaussian microbunch. As can be seen, the spectrum is noisy mainly in the high-frequency or short-wavelength range. Since our EUV radiation is mainly at the wavelength close to 13.5 nm, and the longitudinal form factor close to this frequency fluctuates together from turn to turn, or bunch to bunch. As shown in Fig. 4.8 and discussed before, for the envisioned EUV SSMB, the transverse form factor fluctuation is much smaller than that of the longitudinal form factor. So the overall radiation power fluctuation is also about 2% as analyzed above. This fluctuation is also the fluctuation of microbunch center motion induced by the coherent radiation. A small fluctuation as it is, its beam dynamics effects need further study.

Note that this 2% fluctuation of radiation power should have negligible impact for the application in EUV lithography, since the revolution frequency of the microbunch in the ring is rather high (MHz), let alone if we consider that there is actually a microbunch each modulation laser wavelength and the radiation waveform is CW or quasi-CW.

Fig. 4.11

The spectrum of the longitudinal form factor of three possible realizations of a Gaussian microbunch length of $\sigma _{z}=3$ nm and $N_{e}=2.2\times 10^{4}$. The shaded area corresponds to wavelength of $13.5\pm \frac{13.5}{100}$ nm

4.6.3 Discussions

To resolve possible concerns of readers on the validity of the short bunch length and high average current used in the example calculation, here we present a short discussion on the related single-particle and collective effects in SSMB. We recognize that realizing a steady-state bunch length as short as nanometer level in an electron storage ring is non-trivial. Both global and local momentum compaction should be minimized to confine the longitudinal beta function, therefore the longitudinal emittance, in an electron storage ring as analyzed in Sect. 2.1. By invoking this principle in the lattice design, a bunch length as short as tens of nanometer can be realized in a storage ring, with a momentum compaction factor of $1\times 10^{-6}$ and modulation laser power of 1 MW. 1 MW intra-cavity power is the state-of-art level of present optical enhancement cavity technology. Therefore, to realize nanometer bunch length at the radiator, we need to compress the bunch further. There are two scenarios being actively studied by us, namely the longitudinal strong focusing scheme and transverse-longitudinal coupling scheme. The longitudinal strong focusing scheme is similar to its transverse counterpart which is the basis of modern particle accelerators [13, 14]. In such a scheme, the longitudinal beta function and therefore the bunch length is strongly focused at the radiator, and the synchrotron tune of the beam in the ring can be at the level of 1, as analyzed in Sect. 2.1.6 and Ref. [15]. Although nanometer bunch length can be realized, this scheme requires a large modulation laser power (20 MW level if 270 nm laser is used), causing the optical cavity can work only in the pulsed laser mode and the average output radiation power is thus limited. To lower the modulation laser power, the transverse-longitudinal coupling scheme is thus applied in a clever way by taking advantage of the fact that the vertical emittance in a planar storage ring is rather small, as analyzed in Sect. 3.1. We refer to this turn-by-turn transverse-longitudinal coupling-based bunch compression scheme as the generalized longitudinal strong focusing [16], in which the phase space manipulation is 4D or 6D, in contrast to the conventional longitudinal strong focusing where the phase space manipulation is 2D. This generalized longitudinal strong focusing scheme can relax the modulation laser power, but its nonlinear dynamics optimization is a challenging task which we are trying to tackle.

Concerning the high average current, there are two collective effects of special importance, namely the intrabeam scattering (IBS) and coherent synchrotron radiation (CSR). IBS will affect the equilibrium emittance and thus can have an impact on the radiation power and also the modulation laser power in the generalized longitudinal strong focusing scheme. The IBS effect in an SSMB ring thus needs careful optimization and the operation beam energy is also mainly determined by IBS. CSR is the reason why SSMB can provide powerful radiation. On the other hand, CSR is also the effect which sets the upper limit of the stable beam current. In Ref. [17], there is some preliminary evaluation of the stable beam current for SSMB based on the 1D model of CSR-driven microwave instability. The investigation in this dissertation implies that the transverse dimension of the electron beam can have large impact on the coherent radiation in SSMB. In addition, the bunch lengthening from the transverse emittance in an SSMB storage ring can easily dominate the bunch length at many dispersive places of the ring, as the transverse size of microbunches is much larger than its longitudinal length. This bunch lengthening might be helpful in mitigating unwanted CSR. The 3D effect of the coherent radiation is expected to be also helpful in improving the stable beam current. With these beneficial arguments in mind, we realize that CSR in SSMB still deserves special attention. For example, the coherent radiation in the laser modulator could potentially drive single-pass and multi-pass collective instabilities in an SSMB storage ring [18, 19]. More in-depth study of collective effects in SSMB is ongoing.