# Electron beam polarimeter and energy spectrometer

N. Yu. Muchnoi11footnotetext: Corresponding author.

###### Abstract

The backscattering of laser radiation by a relativistic electron beam is a
widely used method for measuring the spin polarization of electrons. We
consider again the properties of the scattering process paying special
attention to recoil electrons. Based on this consideration we propose the
concept of the Compton polarimeter in which, in addition to all the
polarization components, it becomes possible to accurately measure the energy
and other parameters of the electron beam. To demonstrate the capabilities of
the method we conduct a Monte Carlo simulations of the polarimeter developed
for the FCC-ee project.

##  1 Inverse Compton scattering

Shortly after the invention of lasers it was pointed out in refs. [1, 2] that
Inverse Compton Scattering (ICS) of laser light by relativistic electron beam
would produce useful yields of high energy polarized photons. In 1969 such a
beam had been realized at SLAC for a study of $\gamma p$ interactions [3]. The
effect of self-polarization of relativistic electrons due to emission of the
spin-flip synchrotron radiation was predicted in ref. [4] and observed at the
first ever $e^{+}e^{-}$ collider ACO [5] due to polarization effects in intra-
beam scattering. Soon the existence of this phenomenon was confirmed on the
storage rings VEPP-2 and SPEAR as described in review [6]. Theoretical and
experimental studies of spin dynamics in storage rings made it possible to
develop a method of resonant depolarization described e. g. in ref. [7] and
references therein. For the first time it was used for the beam energy
calibration in experiments on precise determination of the $\Phi$ and $K$
mesons masses [8]. At higher beam energies in addition to other drawbacks the
intra-beam scattering ceases to be a good tool for polarization measurement.
ICS was proposed in ref. [9] as one of the processes allowing to measure the
transverse spin polarization of high-energy electron beam. At the spin
polarimeters of the VEPP-4 [10] and VEPP-2M [11] colliders the synchrotron or
undulator radiation was scattered in the $e^{+}e^{-}$ collision area. ICS of
laser light for transverse beam polarization measurement was first used at
SPEAR [12] and then at circular colliders PETRA [13], DORIS-II [14], LEP [15,
16], HERA [17] and VEPP-4M [18]. The ICS for longitudinal spin polarimetry was
realized at SLAC linear collider [19], AmPS ring at NIKHEF [20], HERA collider
at DESY [21] and Jlab [22]. The ICS polarimeters are also planned for future
collider experiments ILC [23, 24], FCC-ee [25], CEPC [26, 27] and EIC [28,
29].

The practical realization of the ICS looks like it is shown in figure 1. Laser
radiation is inserted into the accelerator vacuum chamber and focused to the
interaction point. The plane of the figure is the plane of machine, the
$z$-axis of the curvilinear coordinate system is always directed along the
electron beam momentum. The $x$-axis is shown in figure 1 while the $y$-axis
is perpendicular to the plane of machine. The dipole magnet separates the
scattered photons from the electron beam. The beam is bent by the angle
$\theta_{0}$ and the recoil electrons are deflected by a larger angle
$\theta_{0}+\Delta\theta$.

Figure 1: Layout of ICS experiment.

Angle $\theta_{0}$ depends on the ratio of the bending field integral to the
electron energy $\varepsilon_{0}=\gamma mc^{2}$:

$\theta_{0}=\displaystyle\frac{\int\\!\\!B_{\perp}dl}{\gamma(mc/e)},\text{~{}~{}where~{}~{}}\frac{mc}{e}=1.704509024\cdot
10^{-3}[\text{T~{}m}].$ (1.1)

Consider the scattering of a low energy photon with wavelength $\lambda_{0}$
and energy $\hbar\omega_{0}$ by an electron with energy $\varepsilon_{0}$.
Denoting the energies of the photon and electron after scattering as
$\hbar\omega$ and $\varepsilon$, their ratio is the dimensionless parameter
$u=\hbar\omega/\varepsilon$ as in refs. [30, 31].

When $\gamma\gg 1$ one can neglect
$\hbar\omega_{0}\ll\varepsilon_{0},\varepsilon,\hbar\omega$ in energy
conservation equation and thus

$u\equiv\displaystyle\frac{\hbar\omega}{\varepsilon}=\displaystyle\frac{\hbar\omega}{\varepsilon_{0}-\hbar\omega}=\displaystyle\frac{\varepsilon_{0}-\varepsilon}{\varepsilon}=\displaystyle\frac{\theta_{e}}{\theta_{\gamma}},$
(1.2)

where $\theta_{\gamma}$ and $\theta_{e}$ are the scattering angles of photon
and electron with respect to the initial electron momentum. The equality
$\varepsilon\theta_{e}=\hbar\omega\theta_{\gamma}$ follows from the law of
conservation of the transverse momentum of the system.

Parameter $u$ lies in the range $u\in\left[0,\kappa\right]$, where $\kappa$ is
twice the initial energy of the photon in the rest frame of the electron,
expressed in units of the electron rest energy:

$\kappa=2\cdot\frac{2\gamma\hbar\omega_{0}}{mc^{2}}=\frac{4\hbar\omega_{0}\varepsilon_{0}}{(mc^{2})^{2}}.$
(1.3)

If $\alpha\neq\pi$ but $\tan(\alpha/2)\gg 1/\gamma$,
$\kappa\sin^{2}\\!\left(\alpha/2\right)$ should be used instead of $\kappa$.
Table 1 shows how the parameters of scattered photon and recoil electron are
derived from $u$ and $\kappa$ variables.

Table 1: Parameters of scattered particles. $\boxed{\begin{aligned}
&&\text{scattered photon:}&&\text{recoil electron:}\\\
\text{Energy:}&&\hbar\omega=\frac{\varepsilon_{0}u}{1+u}&,&\varepsilon=\frac{\varepsilon_{0}}{1+u}.\\\
\text{Edge
energy:}&&\hbar\omega_{max}=\frac{\varepsilon_{0}\kappa}{1+\kappa}&,&\varepsilon_{min}=\frac{\varepsilon_{0}}{1+\kappa}.\\\
\text{Scattering
angle:}&&\theta_{\gamma}=\frac{1}{\gamma}\sqrt{\kappa/u-1}&,&\theta_{e}=\frac{u}{\gamma}\sqrt{\kappa/u-1}.\\\
\text{Angle from bend:}&&\theta_{0}&,&u\theta_{0}.\\\ \text{Maximum
bend:}&&\theta_{0}&,&\Delta\theta=\kappa\theta_{0}.\\\ \text{Azimuthal
angle:}&&\varphi&,&\varphi+\pi.\\\ \text{Horizontal
angle:}&&\eta_{x}=\theta_{\gamma}\cos\varphi-\theta_{0}&,&\theta_{x}=u\theta_{0}-\theta_{e}\cos\varphi.\\\
\text{Vertical
angle:}&&\eta_{y}=\theta_{\gamma}\sin\varphi\;\;\;\;\;\;\;\;&,&\theta_{y}=\;\;\;\;\;\;-\;\theta_{e}\sin\varphi.\\\
\end{aligned}}$

Maximum scattering angle of a recoil electron
$\max(\theta_{e})=2\hbar\omega_{0}/mc^{2}$ does not depend on the initial
electron energy $\varepsilon_{0}$ and happens when $u=\kappa/2$. Since the
energy of recoil electron is $\varepsilon=\varepsilon_{0}/(1+u)$, due to eq.
(1.1) such an electron will be bent by the magnet to the angle
$\theta_{0}(1+u)$. According to figure 1, the first term $\theta_{0}$ defines
the coordinate system rotation and is denoted in table 1 as the “angle from
bend” and “maximum bend” for scattered photon. The second term $u\theta_{0}$
defines the bend of the recoil electron due to its energy loss. This is the
“angle from bend” of recoil electron defined in table 1. The “maximum bend” of
recoil electron $\Delta\theta=\kappa\theta_{0}$ happens when $u=\kappa$ and
$\theta_{\gamma}=\theta_{e}=0$. So far as $\Delta\theta/\theta_{0}=\kappa$ the
measurement of the ratio of angles $\Delta\theta$ to $\theta_{0}$ provides the
possibility to determine the average energy of electrons in the beam according
to the definition of $\kappa$ given in eq. (1.3):

$\varepsilon_{0}=\frac{(mc^{2})^{2}}{4\hbar\omega_{0}}\frac{\Delta\theta}{\theta_{0}}.$
(1.4)

This approach was proposed in ref. [32]. As it follows from eq. (1.1),
$\Delta\theta$ does not depend on $\varepsilon_{0}$:

$\Delta\theta=\frac{4\hbar\omega_{0}}{mc^{2}}\frac{\int\\!\\!B_{\perp}dl}{mc/e}.$
(1.5)

Since $u$ is just a fractional part of $\kappa$, the initial energy of an
electron $\varepsilon_{0}$ does not matter for the “angle from bend”
$u\theta_{0}$ of recoil electron with whatever momentum transfer. In a sense,
eq. (1.5) is similar to the formula (1) from ref. [24] which describes the
angle of spin $\theta_{s}$ with respect to the direction of motion of a
relativistic electron:

$\theta_{s}=\frac{\mu^{\prime}}{\mu_{0}}\frac{\int\\!\\!B_{\perp}dl}{mc/e},$
(1.6)

where $\mu^{\prime}$ and $\mu_{0}$ are the anomalous and normal parts of the
electron magnetic moment.

It is worth noting that there is the established method for the beam energy
determination based on direct measurement of the maximal photon energy
$\hbar\omega_{max}$ from table 1, the details could be found in review [33]
and references therein.

###  1.1 Cross section

ICS cross section depends on polarization states of all initial and final
particles. In case of averaging over the polarizations of the final states the
cross section depends solely from the initial photon and electron
polarizations. Stokes parameters $\xi_{1},\xi_{2},\xi_{3}$ describe the
polarization of laser light as it is explained in Appendix A. The electron
beam polarization has three components $\zeta_{x}$, $\zeta_{y}$, $\zeta_{z}$,
the total degree of polarization
$\sqrt{\zeta^{2}_{x}+\zeta^{2}_{y}+\zeta^{2}_{z}}\in[0:1]$. We take
differential cross section from ref. [34] and after Lorentz transformations it
is represented in $u$ and $\varphi$ variables by the sum of the six terms:

$\displaystyle\frac{1}{r_{e}^{2}}\frac{d\sigma_{0}}{du\,d\varphi}$
$\displaystyle=\frac{1}{\kappa(1+u)^{3}}\left[1+(1+u)^{2}-4\frac{u}{\kappa}\left(1-\frac{u}{\kappa}\right)(1+u)\right],$
(1.7) $\displaystyle\frac{1}{r_{e}^{2}}\frac{d\sigma_{\xi_{1}}}{du\,d\varphi}$
$\displaystyle=\frac{4\,\xi_{1}}{\kappa(1+u)^{2}}\frac{u}{\kappa}\left(1-\frac{u}{\kappa}\right)\cos(2\varphi),$
$\displaystyle\frac{1}{r_{e}^{2}}\frac{d\sigma_{\xi_{2}}}{du\,d\varphi}$
$\displaystyle=\frac{4\,\xi_{2}}{\kappa(1+u)^{2}}\frac{u}{\kappa}\left(1-\frac{u}{\kappa}\right)\sin(2\varphi),$
$\displaystyle\frac{1}{r_{e}^{2}}\frac{d\sigma_{x}}{du\,d\varphi}$
$\displaystyle=\frac{-2\,\xi_{3}\,\zeta_{x}}{(1+u)^{3}}\;\frac{u}{\kappa}\sqrt{\frac{u}{\kappa}\left(1-\frac{u}{\kappa}\right)}\cos(\varphi),$
$\displaystyle\frac{1}{r_{e}^{2}}\frac{d\sigma_{y}}{du\,d\varphi}$
$\displaystyle=\frac{-2\,\xi_{3}\,\zeta_{y}}{(1+u)^{3}}\;\frac{u}{\kappa}\sqrt{\frac{u}{\kappa}\left(1-\frac{u}{\kappa}\right)}\sin(\varphi),$
$\displaystyle\frac{1}{r_{e}^{2}}\frac{d\sigma_{z}}{du\,d\varphi}$
$\displaystyle=\frac{\xi_{3}\,\zeta_{z}}{(1+u)^{3}}\;\frac{u}{\kappa}\left(u+2\right)\left(1-2\frac{u}{\kappa}\right).$

In (1.7) $r_{e}$ is the classical electron radius and $d\sigma_{0}$ represents
the cross section averaged over all polarization terms. Integration of
$d\sigma_{0}$ gives the total unpolarized cross section:

$\sigma_{0}=\frac{2\pi
r_{e}^{2}}{\kappa}\left[\left(1-\frac{4}{\kappa}-\frac{8}{\kappa^{2}}\right)\log(1+\kappa)\,+\frac{1}{2}\left(1-\frac{1}{(1+\kappa)^{2}}\right)+\frac{8}{\kappa}\right].$
(1.8)

If $\kappa\ll 1$ (1.8) tends to Thomson cross section
$\sigma_{0}=\frac{8}{3}\pi r_{e}^{2}\left(1-\kappa\right)$. The last term in
(1.7) could modify the cross section given by (1.8) in case
$\xi_{3}\,\zeta_{z}\neq 0$. Its integration gives the result:

$\sigma_{z}=\xi_{3}\,\zeta_{z}\frac{2\pi
r_{e}^{2}}{\kappa}\left[\left(1+\frac{2}{\kappa}\right)\log(1+\kappa)-\frac{1}{2}\left(4+\left(\frac{\kappa}{1+\kappa}\right)^{2}\right)\right].$
(1.9)

###  1.2 Photons

The cross section with respect to the photon scattering angles $\eta_{x}$ and
$\eta_{y}$ from table 1 could be derived from eqs. (1.7) using the following
transformations:

$u=\frac{\kappa}{1+\gamma^{2}\eta_{x}^{2}+\gamma^{2}\eta_{y}^{2}},\;\;\;\;dud\varphi=\frac{2u^{2}\gamma^{2}}{\kappa}d\eta_{x}d\eta_{y},$
(1.10)

where we take $\theta_{0}=0$ for brevity. The parts of the cross section are:

$\displaystyle\frac{1}{r_{e}^{2}}\frac{d\sigma_{0}}{d\eta_{x}d\eta_{y}}$
$\displaystyle=\frac{2}{(1+u)^{3}}\left(\frac{\gamma
u}{\kappa}\right)^{2}\left[1+(1+u)^{2}-4\frac{u}{\kappa}\left(1-\frac{u}{\kappa}\right)(1+u)\right],$
(1.11)
$\displaystyle\frac{1}{r_{e}^{2}}\frac{d\sigma_{\xi_{1}}}{d\eta_{x}d\eta_{y}}$
$\displaystyle=\frac{8\,\xi_{1}}{(1+u)^{2}}\left(\frac{\gamma
u}{\kappa}\right)^{4}\left(\eta_{x}^{2}-\eta_{y}^{2}\right),$
$\displaystyle\frac{1}{r_{e}^{2}}\frac{d\sigma_{\xi_{2}}}{d\eta_{x}d\eta_{y}}$
$\displaystyle=\frac{16\,\xi_{2}}{(1+u)^{2}}\left(\frac{\gamma
u}{\kappa}\right)^{4}\eta_{x}\,\eta_{y},$
$\displaystyle\frac{1}{r_{e}^{2}}\frac{d\sigma_{x}}{d\eta_{x}d\eta_{y}}$
$\displaystyle=\frac{-4\,\xi_{3}\,\zeta_{x}}{(1+u)^{3}}\left(\frac{\gamma
u}{\kappa}\right)^{3}u\,\eta_{x},$
$\displaystyle\frac{1}{r_{e}^{2}}\frac{d\sigma_{y}}{d\eta_{x}d\eta_{y}}$
$\displaystyle=\frac{-4\,\xi_{3}\,\zeta_{y}}{(1+u)^{3}}\left(\frac{\gamma
u}{\kappa}\right)^{3}u\,\eta_{y},$
$\displaystyle\frac{1}{r_{e}^{2}}\frac{d\sigma_{z}}{d\eta_{x}d\eta_{y}}$
$\displaystyle=\frac{2\xi_{3}\,\zeta_{z}}{\gamma(1+u)^{3}}\left(\frac{\gamma
u}{\kappa}\right)^{3}(u+2)(\kappa-2u).$

Equations (1.11) are applicable for describing the spatial distribution of
scattered photons measured e. g. by pixel detector with rectangular geometry.

###  1.3 Electrons

To make the formulae in this section more compact and easy to use we introduce
modified variables

$\vartheta_{x}=\gamma\theta_{x},\;\vartheta_{y}=\gamma\theta_{y},\;\vartheta_{0}=\gamma\theta_{0}.$
(1.12)

From definitions given in table 1 we obtain the equation:

$(\vartheta_{x}-u\vartheta_{0})^{2}+\vartheta_{y}^{2}=u(\kappa-u),$ (1.13)

whose roots are

$u_{\pm}=\frac{\kappa/2+\vartheta_{0}\vartheta_{x}\pm\sqrt{\kappa^{2}/4-\vartheta_{x}^{2}-\vartheta_{y}^{2}(1+\vartheta_{0}^{2})+\kappa\vartheta_{0}\vartheta_{x}}}{1+\vartheta_{0}^{2}}.$
(1.14)

In the $\vartheta_{x},\vartheta_{y}$ plane all recoil electrons are enclosed
within the ellipse described by the discriminant of eq. (1.14). The center of
this ellipse is located at the point
$[\vartheta_{x}=\kappa\vartheta_{0}/2;\vartheta_{y}=0]$, its vertical semi-
minor axis equals $\kappa/2$ while horizontal semi-major axis is
$\sqrt{1+\vartheta_{0}^{2}}$ times longer. The transformation from $u,\varphi$
to $\vartheta_{x},\vartheta_{y}$ variables is specified by the Jacobian
matrix:

$\begin{bmatrix}\displaystyle\frac{\partial\vartheta_{x}}{\partial
u}&\displaystyle\frac{\partial\vartheta_{x}}{\partial\varphi}\\\\[10.00002pt]
\displaystyle\frac{\partial\vartheta_{y}}{\partial
u}&\displaystyle\frac{\partial\vartheta_{y}}{\partial\varphi}\end{bmatrix}=\begin{bmatrix}[r]\displaystyle\vartheta_{0}-\frac{\kappa/2-u}{\sqrt{u(\kappa-u)}}\cos\varphi&&\displaystyle\sqrt{u(\kappa-u)}\sin\varphi\\\\[10.00002pt]
\displaystyle-\frac{\kappa/2-u}{\sqrt{u(\kappa-u)}}\sin\varphi&&\displaystyle-\sqrt{u(\kappa-u)}\cos\varphi\end{bmatrix}.$
(1.15)

The Jacobian determinant:

$\displaystyle\det(J(u,\varphi))$ $\displaystyle=$
$\displaystyle\kappa/2-u-\vartheta_{0}\sqrt{u(\kappa-u)}\cos\varphi=$ (1.16)
$\displaystyle=\det(J(\vartheta_{x},\vartheta_{y}))$ $\displaystyle=$
$\displaystyle\pm\sqrt{\kappa^{2}/4-\vartheta_{x}^{2}-\vartheta_{y}^{2}(1+\vartheta_{0}^{2})+\kappa\vartheta_{0}\vartheta_{x}}.$

$D=|\det(J(\vartheta_{x},\vartheta_{y}))|$ equals the absolute value of the
square root of discriminant in eq. (1.14). The following expressions follow
from the definitions given in table 1:

$\displaystyle\cos(\varphi)$
$\displaystyle=\frac{u\vartheta_{0}-\vartheta_{x}}{\sqrt{u(\kappa-u)}},$
$\displaystyle\sin(\varphi)$ $\displaystyle=$
$\displaystyle\frac{-\vartheta_{y}}{\sqrt{u(\kappa-u)}},$ (1.17)
$\displaystyle\cos(2\varphi)$
$\displaystyle=\frac{(u\vartheta_{0}-\vartheta_{x})^{2}-\vartheta_{y}^{2}}{u(\kappa-u)},$
$\displaystyle\sin(2\varphi)$ $\displaystyle=$
$\displaystyle\frac{-2(u\vartheta_{0}-\vartheta_{x})\vartheta_{y}}{u(\kappa-u)}.$

Lets apply the variables transformation to eqs. (1.7):

$\displaystyle\frac{1}{r_{e}^{2}}\frac{d\sigma_{0}}{d\vartheta_{x}d\vartheta_{y}}$
$\displaystyle=\frac{1}{D\kappa(1+u)^{3}}\left[1+(1+u)^{2}-4\frac{u(\kappa-u)}{\kappa^{2}}(1+u)\right],$
(1.18)
$\displaystyle\frac{1}{r_{e}^{2}}\frac{d\sigma_{\xi_{1}}}{d\vartheta_{x}d\vartheta_{y}}$
$\displaystyle=\frac{4\xi_{1}}{D\kappa^{3}(1+u)^{2}}\left((u\vartheta_{0}-\vartheta_{x})^{2}-\vartheta_{y}^{2}\right),$
$\displaystyle\frac{1}{r_{e}^{2}}\frac{d\sigma_{\xi_{2}}}{d\vartheta_{x}d\vartheta_{y}}$
$\displaystyle=\frac{-8\xi_{2}}{D\kappa^{3}(1+u)^{2}}(u\vartheta_{0}-\vartheta_{x})\vartheta_{y},$
$\displaystyle\frac{1}{r_{e}^{2}}\frac{d\sigma_{x}}{d\vartheta_{x}d\vartheta_{y}}$
$\displaystyle=\frac{-2\xi_{3}\,\zeta_{x}}{D\kappa^{2}(1+u)^{3}}u(u\vartheta_{0}-\vartheta_{x}),$
$\displaystyle\frac{1}{r_{e}^{2}}\frac{d\sigma_{y}}{d\vartheta_{x}d\vartheta_{y}}$
$\displaystyle=\frac{2\xi_{3}\,\zeta_{y}}{D\kappa^{2}(1+u)^{3}}u\vartheta_{y},$
$\displaystyle\frac{1}{r_{e}^{2}}\frac{d\sigma_{z}}{d\vartheta_{x}d\vartheta_{y}}$
$\displaystyle=\frac{\xi_{3}\,\zeta_{z}}{D\kappa^{2}(1+u)^{3}}\;u(u+2)(\kappa-2u).$

The total cross section is defined by the sum of these terms, each of which is
calculated twice (for $u_{+}$ and for $u_{-}$). With the following linear
transformation instead of the ellipse we obtain the circle of unit radius with
the center located at the origin of new coordinate system:

$\displaystyle x=A\frac{\vartheta_{x}}{\kappa/2}-B;\hskip
14.22636pty=\frac{\vartheta_{y}}{\kappa/2},\text{ where}$ (1.19)
$\displaystyle A=\frac{1}{\sqrt{1+\vartheta_{0}^{2}}}\text{ and
}B=\frac{\vartheta_{0}}{\sqrt{1+\vartheta_{0}^{2}}}.$

The condition $\vartheta_{0}\gg 1$ obviously provides better separation of
recoil electrons from the beam and in this case $A\simeq 1/\vartheta_{0}$ and
$B\simeq 1$. Exact calculations give the following:

$\displaystyle\frac{d\vartheta_{x}d\vartheta_{y}}{D}$
$\displaystyle=\frac{\kappa\,dx\,dy}{2\sqrt{1-x^{2}-y^{2}}},$ (1.20)
$\displaystyle u_{\pm}$ $\displaystyle=\frac{\kappa}{2}\left(1+Bx\pm
A\sqrt{1-x^{2}-y^{2}}\right)=\frac{\kappa}{2}\left(1+\Delta_{\pm}\right),$
$\displaystyle(u\vartheta_{0}-\vartheta_{x})_{\pm}$
$\displaystyle=\frac{\kappa}{2}\left(\;\;-\,Ax\pm
B\sqrt{1-x^{2}-y^{2}}\right)=\frac{\kappa}{2}\delta_{\pm}.$

Parameters $\Delta^{\pm}$ and $\delta^{\pm}$ in eqs. (1.20) are the
shorthand’s for corresponding parenthesized expressions. We see that
$d\vartheta_{x}d\vartheta_{y}/D$ provides the cross section “amplification”
near the border of the circle, while the difference $u_{+}-u_{-}$ is at
maximum in the center of the circle. Finally we obtain:

$\displaystyle\left.\frac{1}{r_{e}^{2}}\frac{d\sigma_{0}}{dxdy}\right|_{\pm}$
$\displaystyle=$
$\displaystyle\frac{1+(1+u_{\pm})^{2}-(1+u_{\pm})(1-\Delta_{\pm}^{2})}{2(1+u_{\pm})^{3}\sqrt{1-x^{2}-y^{2}}},$
(1.21)
$\displaystyle\left.\frac{1}{r_{e}^{2}}\frac{d\sigma_{\xi_{1}}}{dxdy}\right|_{\pm}$
$\displaystyle=$ $\displaystyle\xi_{1}$
$\displaystyle\frac{\delta_{\pm}^{2}-y^{2}}{2(1+u_{\pm})^{2}\sqrt{1-x^{2}-y^{2}}},$
$\displaystyle\left.\frac{1}{r_{e}^{2}}\frac{d\sigma_{\xi_{2}}}{dxdy}\right|_{\pm}$
$\displaystyle=$ $\displaystyle\xi_{2}$
$\displaystyle\frac{-\delta_{\pm}y}{(1+u_{\pm})^{2}\sqrt{1-x^{2}-y^{2}}},$
$\displaystyle\left.\frac{1}{r_{e}^{2}}\frac{d\sigma_{x}}{dxdy}\right|_{\pm}$
$\displaystyle=$ $\displaystyle\xi_{3}\zeta_{x}$
$\displaystyle\frac{-u_{\pm}\delta_{\pm}}{2(1+u_{\pm})^{3}\sqrt{1-x^{2}-y^{2}}},$
$\displaystyle\left.\frac{1}{r_{e}^{2}}\frac{d\sigma_{y}}{dxdy}\right|_{\pm}$
$\displaystyle=$ $\displaystyle\xi_{3}\zeta_{y}$
$\displaystyle\frac{u_{\pm}y}{2(1+u_{\pm})^{3}\sqrt{1-x^{2}-y^{2}}},$
$\displaystyle\left.\frac{1}{r_{e}^{2}}\frac{d\sigma_{z}}{dxdy}\right|_{\pm}$
$\displaystyle=$ $\displaystyle\xi_{3}\zeta_{z}$
$\displaystyle\frac{-u_{\pm}(u_{\pm}+2)\Delta_{\pm}}{2(1+u_{\pm})^{3}\sqrt{1-x^{2}-y^{2}}}.$

The cross section in $x,y$ plane is the sum of “+” and “-” solutions. It’s
time to explain why the variables $x$ and $y$ were introduced. To build a
function that describes the number of events in a pixel of the detector of
recoil electrons, we need the integral of the cross section over a rectangular
area. The antiderivative of $f(x,y)=1/\sqrt{1-x^{2}-y^{2}}$ function is an
analytical expression

$F(x,y)=x\operatorname{atan2}2(y,\rho)+y\operatorname{atan2}2(x,\rho)-\operatorname{atan2}2(xy,\rho),$
(1.22)

where $\rho=\sqrt{1-x^{2}-y^{2}}$. The integral over the pixel with the
corners $x_{0},y_{0}$ and $x_{1},y_{1}$ is

$I(x_{0},y_{0},x_{1},y_{1})=F(x_{0},y_{0})+F(x_{1},y_{1})-F(x_{0},y_{1})-F(x_{1},y_{0}).$
(1.23)

##  2 Monte Carlo experiment

###  2.1 General considerations

Consider again the diagram shown in figure 1. In the Monte-Carlo experiment
scattered photons and electrons hit the notional detectors located far enough
from the interaction area and the bending magnet. These detectors are
installed along the $x$-axis on the right-side of figure 1. It is also worth
paying attention that there are no magnetic elements between the magnet and
the detectors.

If $R_{0}$ is the bending radius of a beam electron, then the radius of the
recoil electron with minimal energy equals $R_{\text{min}}=R_{0}/(1+\kappa)$.
Consider the bending dipole of length $L$ with rectangular shape of the
magnetic poles. All the electrons enter the dipole along the same line – the
beam orbit. The equality $\Delta\theta=\kappa\theta_{0}$ in table 1 was
obtained under the assumption that the field integral does not depend on the
electron energy. The accuracy of this assumption depends on the parameters
combination $\Lambda=\kappa L/2R_{0}$. At the end of the dipole an electron
with minimal energy $\varepsilon_{min}$ will have transverse horizontal
distance from the beam orbit $\Delta x\simeq\Lambda L$, which sets the
requirement for a transverse size of the uniform field region inside the
magnet. Relative difference in the arcs lengths inside the magnet for
$\varepsilon_{0}$ and $\varepsilon_{min}$ electrons is estimated as $\Delta
L/L\simeq\Lambda^{2}/2$. With $\Lambda\simeq 10^{-3}$ the relative difference
in the magnetic field integrals $\Delta L/L\simeq 5\cdot 10^{-7}$. Small
enough values of $\Lambda$ are accessible with small bending angles
$L/R_{0}\ll 1$ when $\kappa\geq 1$. At low electron beam energies, when
$\kappa\ll 1$, the bending angle could be increased.

Use of two-dimensional pixel detector for recoil electrons was proposed for
transverse beam polarization measurement in ref. [35]. Figure 2 illustrates
the scattered photons, recoil electrons and corresponding detectors in $x,y$
plane.

Figure 2: Detection plane. 2D pixel detectors for photons and electrons are
represented by dotted rectangles.

It is assumed that the detectors are precisely positioned relative to each
other. Each detector has a certain geometry, number of pixels and pixel size
in both dimensions. In the numerical experiment the detectors are represented
by the two-dimensional ($x,y$) histograms where each bin corresponds to one
pixel. The following parameters are used in further Monte-Carlo simulations:

  * •

$\gamma$ is the mean value of the beam electron Lorentz factor.

  * •

$\theta_{0}$ is the equilibrium orbit bending angle.

  * •

$\lambda_{0}$ and $\hbar\omega_{0}=hc/\lambda_{0}$ are the wavelength and
energy of a laser photon.

  * •

$\kappa=4\gamma\hbar\omega_{0}/mc^{2}$ is the mean value of the Compton
scattering parameter $\kappa$.

  * •

Stokes vector $[\xi_{1},\xi_{2},\xi_{3}]$ describes the laser beam
polarization.

  * •

Vector $[\zeta_{x},\zeta_{y},\zeta_{z}]$ describes the electron beam
polarization.

  * •

$u$ and $\varphi$ parameters are generated according to the sum of eqs. (1.7).

  * •

Lorentz factor of a beam electron is $\gamma^{\prime}=\gamma(1+\Delta)$ where
$\Delta$ is extracted from the normal distribution with zero mean value and
standard deviation described by the relative energy spread of the electron
beam $\sigma_{\gamma}/\gamma$.

  * •

The transverse horizontal coordinate of the electron at the moment of
scattering due to synchrotron motion is $D_{x}\Delta$ where $D_{x}$ is the
dispersion function.

  * •

The transverse coordinates $x,y$ of the electron due to betatron motion are
generated by the normal distributions with zero mean value and
$\sigma_{x}=\sqrt{\epsilon_{x}\beta_{x}}$,
$\sigma_{y}=\sqrt{\epsilon_{y}\beta_{y}}$ described by corresponding emittance
and $\beta$-function.

  * •

The transverse angles $x^{\prime},y^{\prime}$ of the electron due to betatron
motion are generated by the normal distributions with zero mean value and
$\sigma_{x}^{\prime}=\sqrt{\epsilon_{x}/\beta_{x}}$,
$\sigma_{y}^{\prime}=\sqrt{\epsilon_{y}/\beta_{y}}$.

  * •

The length of the interaction region of laser radiation with electron beam is
assumed to be very short ($\ll L_{1}$) and thus it is not taken into account.

  * •

The $X_{\gamma},Y_{\gamma},X_{e},Y_{e}$ coordinates of scattered photon and
electron at the detection plane are obtained by the following equations:

$\displaystyle X_{\gamma}$
$\displaystyle=\frac{1}{\cos\\!\theta_{0}}\left(D_{x}\Delta+x+L_{1}\tan\\!\left[x^{\prime}+\frac{\cos\\!\varphi\sqrt{\kappa/u-1}}{\gamma(1+\Delta)}\right]\right)-L_{2}\tan\\!\theta_{0},$
(2.1) $\displaystyle Y_{\gamma}$ $\displaystyle=\hskip
31.29802pty+L_{1}\tan\left[y^{\prime}+\frac{\sin\\!\varphi\sqrt{\kappa/u-1}}{\gamma(1+\Delta)}\right],$
$\displaystyle X_{e}$
$\displaystyle=\frac{1}{\cos\\!\theta_{0}}\left(D_{x}\Delta+x+L_{1}\tan\\!\left[x^{\prime}-\frac{\cos\\!\varphi\sqrt{u(\kappa-u)}}{\gamma}\right]\right)+L_{2}\tan\\!\theta_{0}\left[u-\frac{\Delta}{1+\Delta}\right],$
$\displaystyle Y_{e}$ $\displaystyle=\hskip
31.29802pty+L_{1}\tan\left[y^{\prime}-\frac{\sin\\!\varphi\sqrt{u(\kappa-u)}}{\gamma}\right].$

If the photon $X_{\gamma},Y_{\gamma}$ or electron $X_{e},Y_{e}$ coordinates
hits the pixel of the photon/electron detector, the content of the
corresponding histogram bin is increased by one. Note that the eqs. (2.1)
contain the lattice functions $\beta_{x},\beta_{y},D_{x}$ but not their
derivatives. Each beam electron at the area of interaction with laser
radiation has some deviations from the equilibrium parameters due to betatron
and synchrotron motion. The transverse coordinates of this electron at the
plane of the detectors installation could be obtained from the full set of its
initial parameters by tracking through the spectrometer. According to eq.
(1.5) bending of the recoil electrons differs from bending of the beam
electrons by an additional angle which does not depend on the initial electron
parameters. It means that the (gaussian) blur of the two-dimensional
distribution of recoil electrons described by eqs. (1.21) is equal to the
transverse beam size ($\sigma_{x},\sigma_{y}$) at the plane of detectors.

It is worth noting that in this simulation it is implicitly assumed that the
scattering probability does not depend on the transverse coordinates of the
electron. This condition is satisfied if the size of the laser spot
significantly exceeds the size of the electron beam. Such a configuration of
the interaction area is necessary in any case when we are going to measure the
true average parameters of the electron beam.

###  2.2 FCC-ee beam polarimeter

FCC-ee is a lepton collider with centre-of-mass energies between 90 and 350
GeV [36]. It is considered as a potential intermediate step towards the
realization of the hadron facility. Beam energy calibration by resonant
depolarization is the basis for the precise measurements of the $Z$ mass and
width with a precision of <100 keV, and of the $W$ mass and width with a
precision of the order of 500 keV. The conceptual design of the Compton
polarimeter may be found in [36, 37] and the same setup is considered here.

Table 2: Simulation parameters. All designations have been defined in the text above. $\varepsilon_{0}=\;45.6$ GeV |  $\gamma=$ 89237 | $\epsilon_{x}=270$ | pm rad |  $\beta_{x}=100$ m |  $L_{1}=117$ m
---|---|---|---|---|---
$\lambda_{0}=\;532$ nm | $\kappa=1.6279$ | $\epsilon_{y}=1$ | pm rad |  $\beta_{y}=\;\,20$ m |  $L_{2}=100$ m
$\omega_{0}=2.331$ eV |  $\vartheta_{0}=$ 190.44 | $\theta_{0}=2.1341$ | mrad |  $D_{x}=\,25$ mm |  $\sigma_{\gamma}/\gamma=$ 0.001

Table 2 contains the list of simulation parameters which will be identical in
all numerical experiments below. Bunch revolution frequency at FCC-ee is
$3\cdot 10^{3}\;\text{s}^{-1}$ while the rate of Compton scattering events is
estimated as $2\cdot 10^{6}\;\text{s}^{-1}$. The parameters of the pixel
detectors for photons and electrons are presented in Table 3.

Table 3: Detectors: geometry, number of pixels, size of pixels. Detector | Size ($X\times Y$) |  $N$pix ($X\times Y$) | Pixel size ($X\times Y$)
---|---|---|---
Photons |  $10\times 10$ mm | $100\times 100$ |  $100\times 100\;\mu$m
Electrons |  $400\times 4$ mm | $1600\times 80$ |  $250\times 50\;\mu$m

####  2.2.1 Experiment I

The goal of the first numerical experiment is to verify the above formulae for
the cross sections and check the fitting procedure. The subject of this study
is the trivial case when the Stokes vector of laser polarization is
$[\xi_{1},\xi_{2},\xi_{3}]=[0,0,1]$ and the electron beam is unpolarized
$\zeta_{x},\zeta_{y},\zeta_{z}=[0,0,0]$.

Figure 3: Photon spot at the detector. Number of bins is reduced for better
visualization.

Figure 3 shows the distribution of scattered photons at the detector obtained
with $10^{7}$ Monte-Carlo events. In FCC-ee conditions [37] this amount is
available within 5 s measurement time with approximately $1.5\cdot 10^{4}$
collisions between laser pulse and electron bunch.

The fit function for the scattered photons distribution is based on eqs.
(1.11) with nine fit parameters $X_{0}$, $Y_{0}$, $\sigma_{x}$, $\sigma_{y}$,
$\xi_{1}$, $\xi_{2}$, $\xi_{3}\zeta_{x}$, $\xi_{3}\zeta_{y}$,
$\xi_{3}\zeta_{z}$ representing the spot center position, blur effect from
electron beam emittance and five independent polarization components. The
tenth parameter is the normalization. The distance from the interaction area
to the detector $L_{1}$ and the scattering parameter $\kappa$ are fixed
according to the setup. The results of the fit are presented in Table 4 (NDF
is the acronym for Number of Degrees of Freedom).

Table 4: Photon spot fit results. $X_{0}=-213.538\pm 0.001$ mm |  $Y_{0}=-0.002\pm 0.001\;$mm
---|---
$\sigma_{x}=255\pm 3\;\mu$m |  $\sigma_{y}=30\pm 18\;\mu$m
$\xi_{1}=0.000\pm 0.002$ | $\xi_{2}=-0.001\pm 0.001$
$\xi_{3}\zeta_{x}=0.004\pm 0.006$ | $\xi_{3}\zeta_{y}=-0.008\pm 0.006$
$\xi_{3}\zeta_{z}=0.000\pm 0.002$ | $\chi^{2}/\text{NDF}=9796.9/9990$

It can be seen that the position of the spot is established with accuracy
about 1 $\mu$m. The $\sigma_{x}$ and $\sigma_{y}$ values carry the information
about the combination of beam parameters at the interaction area. The accuracy
of $\sigma_{y}$ is poor because it is three times smaller than the pixel size.
It would be useful to recall that in order to detect gamma rays, they will be
converted into $e^{+}e^{-}$ pairs, which will lead to additional blurring and
reduce the number of registered events. However all polarization parameters
correspond to their zero set values with absolute accuracy of better than one
percent.

Let us take advantage of the opportunities provided by the numerical
experiment and place the electron detector so that it detects all recoil
electrons, including those that propagate very close to the beam. Such a
distribution of scattered electrons is presented in figure 4.

Figure 4: Recoil electrons at the detector. Number of $x$-bins is reduced to
100 for better visualization. Electrons detector covers the beam at the
coordinates $X=0,Y=0$, which is impossible to do in practice.

The fit procedure is done by Pearson’s method, the expected bin error is
estimated as $\sqrt{1+\mathcal{F}}$, where the fit function $\mathcal{F}(X,Y)$
is calculated in three steps. First, geometry parameters $X_{1}$, $X_{2}$,
$Y_{1}$, $Y_{2}$ shown in figure 2 specify ellipse of scattered electrons
defined in eqs. (1.14) and (1.16). With linear transformations, based on the
known detector geometry, these parameters are converted to dimensionless
variables $x$ and $y$ defined in eqs. (1.19). The integrals from (1.23) are
calculated as well as the corresponding cross section terms from eqs. (1.21)
at the center of each pixel. Scattering parameter $\kappa$ is fixed as it was
done with photons while neither $L_{1}$ nor $L_{2}$ are not used. Finally the
distribution, obtained by pixel by pixel multiplication of the integrals and
the cross sections, is convoluted with two-dimensional Gaussian to account for
transverse beam sizes at the detection plane. Totally the fit function has
twelve parameters including normalization, while only non-empty bins are
accounted in NDF. The results of the fit are presented in table 5.

Table 5: Scattered electrons ellipse fit results. $X_{1}=0.0035\pm 0.0016$ mm |  $X_{2}=347.635\pm 0.003$ mm
---|---
$Y_{1}=-1.0682\pm 0.0001$ mm |  $Y_{2}=1.0682\pm 0.0001$ mm
$\sigma_{x}=320.7\pm 1.5\;\mu$m |  $\sigma_{y}=27.06\pm 0.03\;\mu$m
$\xi_{1}=0.001\pm 0.001$ | $\xi_{2}=0.432\pm 0.198$
$\xi_{3}\zeta_{x}=1.000\pm 0.195$ | $\xi_{3}\zeta_{y}=-0.001\pm 0.002$
$\xi_{3}\zeta_{z}=0.000\pm 0.001$ | $\chi^{2}/\text{NDF}=51568.9/52270$
$\varepsilon_{0}=45.5997\pm 0.0007$ GeV

It can be seen that the accuracy of the left ellipse border position $X_{1}$
is 0.16 $\mu$m which is approximately twice better than the accuracy of the
right border position $X_{2}$. This is explained by larger amount of recoil
electrons near $X_{1}$ (the beam position). The difference $Y_{2}-Y_{1}$ is
defined with relative precision of $5\cdot 10^{-5}$, which allows to measure
(or just check) $L_{1}$ with the same accuracy. The $X_{1}$ and
$(Y_{1}+Y_{2})/2$ define the electron beam transverse position relative to the
detector. Unlike the photons distribution, the $\sigma_{x}$ and $\sigma_{y}$
here are the parameters describing the transverse beam sizes at the detector
position and we see that both sizes could be measured with rather high
accuracy (compared to e. g. ref. [38]). As for polarization parameters,
$\xi_{2}$ and $\xi_{3}\zeta_{x}$ determination is wrong. This happens because
the sum of $u_{+}$ and $u_{-}$ solutions in corresponding terms of the cross
section (1.21) have opposite signs and almost cancel each other when
$\vartheta_{0}\gg 1$ (in our case $\vartheta_{0}=\gamma\theta_{0}\simeq 190$).
The other three polarization parameters $\xi_{1}$, $\xi_{3}\zeta_{y}$ and
$\xi_{3}\zeta_{z}$ however are determined even more precise than from the
photons distribution. The value of the beam energy $\varepsilon_{0}$ in the
last row of table 5 is obtained from $X_{0},X_{1},X_{2}$ fit parameters with
relative accuracy of about $1.5\cdot 10^{-5}$ by the formula which follows
from eq. (1.5):

$\varepsilon_{0}=\frac{(mc^{2})^{2}}{4\omega_{0}}\frac{X_{2}-X_{1}}{X_{1}-X_{0}}.$
(2.2)

####  2.2.2 Experiment II

In this experiment we study more realistic case when the recoil electron
detector is shifted 15 mm away from the electron beam. It is assumed that this
gap will provide a sufficient physical aperture for the electron beam. The
Stokes vector of laser polarization is chosen as
$[\xi_{1},\xi_{2},\xi_{3}]=[0.1,0.1,0.99]$ with small amount of vestigial
linear polarization which may exist due to imperfect polarization control. The
set of the electron beam polarization parameters is
$[\zeta_{x},\zeta_{y},\zeta_{z}]=[0.1,0.25,0.1]$. The expected polarization of
the pilot electron bunches at FCC-ee (averaged over thousands beam
revolutions) has $\zeta_{y}$ component only. The $\zeta_{x}$ and $\zeta_{z}$
components are added to the simulations in order to investigate the
possibility of measuring the electron beam polarization in general case.

We do not present the figure with the distribution of photons here since
visually it is very close to that shown in figure 3. The fitting procedure is
the same as in previous experiment and the results obtained from the fit are
presented in Table 6. From the table we can conclude that all polarization
parameters are determined correctly with absolute accuracies from 0.1% to
0.7%.

Table 6: Photon spot fit results. $X_{0}=-213.539\pm 0.002$ mm |  $Y_{0}=0.000\pm 0.001\;$mm
---|---
$\sigma_{x}=246\pm 4\;\mu$m |  $\sigma_{y}=13\pm 70\;\mu$m
$\xi_{1}=0.102\pm 0.002$ | $\xi_{2}=0.100\pm 0.001$
$\xi_{3}\zeta_{x}=0.095\pm 0.007$ | $\xi_{3}\zeta_{y}=0.247\pm 0.006$
$\xi_{3}\zeta_{z}=0.105\pm 0.002$ | $\chi^{2}/\text{NDF}=9935.8/9990$

The distribution of recoil electrons in figure 5 looks different mainly due to
the new location of the detector, the results of the fit are presented in
table 7.

Figure 5: Recoil electrons at the detector. Number of $x$-bins is reduced to 100 for better visualization. Electrons detector is shifted in $X$ by 15 mm away from the beam coordinates $X=0,Y=0$. Table 7: Scattered electrons ellipse fit results. $X_{1}=0.013\pm 0.007$ mm |  $X_{2}=347.632\pm 0.004$ mm
---|---
$Y_{1}=-1.0682\pm 0.0001$ mm |  $Y_{2}=1.0684\pm 0.0001$ mm
$\sigma_{x}=319.6\pm 4.3\;\mu$m |  $\sigma_{y}=27.15\pm 0.03\;\mu$m
$\xi_{1}=0.100\pm 0.001$ | $\xi_{2}=0.100$
$\xi_{3}\zeta_{x}=0.099$ | $\xi_{3}\zeta_{y}=0.246\pm 0.002$
$\xi_{3}\zeta_{z}=0.099\pm 0.001$ | $\chi^{2}/\text{NDF}=50152.7/51245$
$\varepsilon_{0}=45.5959\pm 0.0025$ GeV

The fitting was performed in the same way as in our previous experiment,
except that the parameters $\xi_{2}$ and $\xi_{3}\zeta_{x}$ are fixed
according to the results obtained from the photons distribution. We see that
the accuracy of determining parameters $X_{1}$ and $X_{2}$ is almost the same
now. The transverse beam sizes $\sigma_{x}$ and $\sigma_{y}$ are in perfect
agreement with the results of the first experiment. The polarization
parameters $\xi_{1}$, $\xi_{3}\zeta_{y}$ and $\xi_{3}\zeta_{z}$ are determined
with absolute accuracy of 0.1%, 0.2% and 0.1%. The beam energy
$\varepsilon_{0}$ is “measured” with relative accuracy of about $5.5\cdot
10^{-5}$ and differs from the true value by a little less than two standard
deviations. The accuracy is worse than in the first experiment cause the
electrons detector is shifted away from the electron beam by 15 mm and $X_{1}$
parameter is now determined by extrapolation of the shape of distribution.

##  3 Summary

Simultaneous measurement of the spatial distributions of gamma quanta and
recoil electrons makes it possible to significantly expand the capabilities of
the Compton polarimeter. In addition to all polarization components of the
laser and electron beams, another parameters can be measured with rather high
accuracy: the average energy of beam electrons, the integral of the field in
the spectrometer magnet, the transverse coordinates and dimensions of the
electron beam at the location of the detectors. All these parameters are
obtained by analysis of the two-dimensional spatial distribution of recoil
electrons. The recoil electrons have a sharp boundary in the scattering angle
which, after bending in the magnet, takes the form of an ellipse in the
detection plane. The sharp edges of the ellipse are blurred by the gaussian
beam sizes of the electron beam at the location of the detectors.

The considered approach is applicable when the vertical size of the ellipse of
recoil electrons is larger than the vertical beam size:

$L_{1}\frac{2\omega_{0}}{mc^{2}}>\sqrt{\varepsilon_{y}\beta_{y}}.$ (3.1)

Recoil electrons propagate to the inner side of the beam orbit and thus there
is no direct background from high energy synchrotron radiation. The angular
distribution of scattered photons partially overlaps with photons of
synchrotron radiation, this circumstance requires additional consideration if
we are dealing with high electron energies. Electrons are registered directly
by ionization losses while photons should be converted into pairs leading to a
decrease in the detection efficiency and spatial resolution. Despite the fact
that the fluxes of scattered photons and electrons are almost the same, the
flux density of electrons is much lower due to bending and corresponding
spatial separation by energies, making easier the simultaneous detection of
multiple recoil electrons.

The main technical difficulty for the practical implementation of the proposed
measurement scheme is the need to have a long spectrometer arm, in which the
recoil electrons are not exposed to magnetic fields. A high degree of field
uniformity in the bending magnet is required for accurate determination of the
beam energy. Additional simulations with realistic parameters of the magnetic
system should be carried out to estimate the systematic effects due to these
factors. If the realistic situation differs from the ideal one, the successful
fitting of the recoil electron distributions can be performed by introducing
additional fit parameters to the transformation from the detector coordinates
$X$ and $Y$ (figure 2) to the dimensionless coordinates $x$ and $y$ used in
eqs. (1.21).

The FCC-ee polarimeter was designed to provide fast and precise measurements
of the transverse polarization of the pilot bunches. This will allow to
continuously monitor the beam energy by resonant depolarization technique at
the beam energies up to 80 GeV approximately, when it becomes impossible to
keep the beam polarized. For these needs the analysis of the distribution of
recoil electrons will provide more accurate measurements at the same
acquisition time if compared with scattered photons. Meanwhile the energy
spectrometer option of the polarimeter could be cross-checked with the
resonant depolarization results and allow to use it at higher beam energies
and for interpolation purposes. Detection and analysis of distributions of
both photons and recoil electrons will reduce systematic errors in
measurements of all polarization components. Accurate determination of the
transverse sizes of the electron beam is, in a sense, a usefull additional
bonus.

##  Appendix A Stokes parameters

The Stokes parameters describe the polarization state of electromagnetic
radiation. Their definition is slightly different in different sources, so
below are the definition that we use here.

  * •

$\xi_{0}=E_{x}^{\,2}+E_{y}^{\,2}$ is the intensity of light.
With normalization $E_{x}^{\,2}+E_{y}^{\,2}=1$ for 100% polarized laser
radiation $\xi_{0}=\sqrt{\xi_{1}^{2}+\xi_{2}^{2}+\xi_{3}^{2}}=1$.

  * •

$\xi_{1}=E_{x}^{\,2}-E_{y}^{\,2}$.

    * $E_{x}=1,E_{y}=0,\xi_{1}=+1$: 100% linear polarization along x-axis.

    * $E_{x}=0,E_{y}=1,\xi_{1}=-1$: 100% linear polarization along y-axis.

  * •

$\xi_{2}=2E_{x}E_{y}\cos(\delta)$.

    * $E_{x}=E_{y},\delta=0,\xi_{2}=+1$: 100% linear polarization along $\varphi=+\pi/4.$

    * $E_{x}=E_{y},\delta=\pi,\;\xi_{2}=-1$: 100% linear polarization along $\varphi=-\pi/4.$

  * •

$\xi_{3}=2E_{x}E_{y}\sin(\delta)$.

    * $E_{x}=E_{y},\delta=+\pi/2,\;\xi_{3}=+1$: 100% right circular polarization.

    * $E_{x}=E_{y},\delta=-\pi/2,\;\xi_{3}=-1$: 100% left circular polarization.

## Acknowledgments

The author expresses his gratitude to Mikhail Achasov and Ivan Koop for useful
discussions, to Alain Blondel and Eugene Levichev for their interest and
motivation to continue these studies. The author would also like to thank
Katsunobu Oide who always found a solution to the difficult issue of
polarimeter integration to the FCC-ee lattice (otherwise these studies would
have been closed in the early stages of discussion) and Aurélian Martens for
his help in deriving the cross section formulae. This study is dedicated to
the upcoming centennial anniversary of the discovery of the Compton effect.

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