2. Principle

Above all, two approximates can ensure the accuracy of this diagnosis. Firstly, the electric field is very similar to electrostatic field in the plasma. Thence, for the process from infinity to infinity, all the incident and exit potential of probe beams are zero, and there is no energy decrease or increase when the electron beam has penetrated the system. Secondly, for the scattered electron beams, the E/B field area is filled with very small amount of materials to intervene the electron scattering angle, just like the ultrashort intense laser-foil interaction process [Reference Chen, Li and Chen19].

The whole diagnosis process is divided into two parts to describe the interaction of beam line with plasma and the electrons transport in the point-to-point HEER beam line.

For the first subprocess, high-energy electron beams will be deflected by a magnetic field when penetrating the system. The deflection can be described by the following formula:

$\begin{array}{}\text{(1)}& \theta =\int \omega dt=\int \frac{e{B}_{\perp }dt}{\gamma m_0}=\int \frac{e{B}_{\perp }}{\gamma m_{0}c}dz,\end{array}$

where

$\begin{array}{}\text{(2)}& \gamma =\frac{1}{\sqrt{1-\left({\text{v}}^2/{\text{c}}^2\right)}},\end{array}$

where θ is deflection of electron beam, ω is angular velocity of electrons in magnetic field, v is velocity of electrons, m ₀ is the electron rest mass, c is speed of light in vacuum. $B_{\perp }$ is the weight of magnetic vector at x-y plane, and z is the position of magnetic area.

Based on the above description, if the incident electrons have a specific angle distribution with high symmetry in polar coordinate at x-y plane, there is one-to-one correspondence between transmittance and deflection.

The angle distribution of monoenergtetic electron interacting with matter can be described by the following function:

$\begin{array}{}\text{(3)}& f\left(\varphi \right)=\frac{N\left(t,\varphi \right)}{N_0}=\frac{1}{{\varphi }_0\left(t\right)\sqrt{2\pi }}{e}^{-\left(1/2\right){\left(\varphi /{\varphi }_0\left(t\right)\right)}^2},\end{array}$

where

$\begin{array}{}\text{(4)}& {\varphi }_0\left(t\right)=\frac{13.6\text{MeV}}{\beta cp}\sqrt{t}\left(1+0.038\ln \left(t\right)\right),\end{array}$

where ϕ is scattered angle of electrons, $\text{t}=X/X_0$ , X is the thickness of target, and X ₀ is the radio length of specimen materials.

From formula (3), after interactions between the incident electron and target, the electron beams will have a specific angle distribution. And the distribution should have high symmetry in polar coordinate. In other words, the angle distribution of incident electrons can be controlled by scattering target so as to meet diagnostic demand.

The next step is deflection measured by HEER beam line. For monoenergetic electron beams, the transport in beam line can be explained by the following matrix:

$\begin{array}{}\text{(5)}& \left(\begin{array}{c}{x}_i\\ x_i^{\prime }\\ y_i\\ y_i^{\prime }\end{array}\right)=\left(\begin{array}{cccc}{R}_{11}& R_{12}& 0& 0\\ R_{21}& R_{22}& 0& 0\\ 0& 0& R_{33}& R_{34}\\ 0& 0& R_{43}& R_{44}\end{array}\right)\left(\begin{array}{c}{x}_0\\ x_0^{\prime }\\ y_0\\ y_0^{\prime }\end{array}\right),\end{array}$

where x_i, y_i and $x_i^{\prime }$ , $y_i^{\prime }$ are the position and angle of electrons at the observation plane, and R_ij are the conversion factor of the beam line. The x ₀, y ₀ and $x_0^{\prime }$ , $y_0^{\prime }$ are the position and angle of electrons at the emission plane [Reference Brown27].

At the image plane, transport matrix parameters R ₁₂ = R ₃₄ = 0. It means that the electron position at the image plane is irrelevant with the angle of electrons at the object plane. This beam line design is called point-to-point radiography. Between the image plane and the object plane, by optimizing, there will be a plane named Fourier plane which matches the transport matrix parameters R ₁₁ = R ₃₃ = 0. It means that the electron angle information at object plane has been translated to position information at this plane. Thence, an aperture at the Fourier plane can achieve angle selection for transmitted electrons. In the past years, the application of aperture is to get better spatial and areal density resolutions. Now, the magnetic strength diagnosis depends on the characteristics of the aperture.

The aperture should be set at the Fourier plane where aperture can select the angle around the center axis when the electrons are not deflected. If the electron beam collective is deflected by the magnetic field, the angle selection will also be deflected from the center axis of angle distribution. Thence, for the incident electrons with specific angle distribution, the deflection of them has a negative correlation with its transmittance. Furthermore, it is necessary to make the ratio of the x-half-axis and y-half-axis equal R12F/R34F to ensure one-to-one correspondence between magnetic strength and transmittance. The footnote F means that the transport matrix parameters are from the object plane to Fourier plane. The principle is shown in Figure 1.

Figure 1: (color online). Schematic diagram of HEER for magnetic area diagnosis. (a) The angle distribution of incident electrons. (b) The electrons angle distribution after scattering target. (c) The electron position at Fourier plane; the green shadow area is the aperture position.

3. Simulations

3.1. General Setup

The sketch of integral simulation is shown in Figure 2. The incident electrons are scattered by an aluminum foil initially, and then, the scattered electrons penetrate the magnetic vacuum. Then, HEER magnetic lens are set to transport, select, and refocus the electrons behind target. Finally, the electron number density will be counted by the detector at image plane.

Figure 2: (color online). Schematic of simulation design. The magnetic area as the specimen is a quadrupole magnet (a). And the aperture is designed as an ellipse, where x-half-axis is 1 mm and y-half-axis is 0.5 mm in length (b).

3.2. Image Lens Optimization

The typical charged particle image beam line should satisfy that the transport matrix parameters are R ₁₂ = R ₃₄ = 0 at the image plane. But for the force field diagnosis HEER, as mentioned in Section 2, it is necessary to make the x-axis and y-axis Fourier plane coincident run along the z-axis. Under this requirement, the imaging lens system adapting 50 MeV kinetic energy electron beams is optimized by COSY INFINITY 9.1 [Reference Berz and Makino28]. It includes eight quadrupole magnets, as shown in Figure 2; the length of them is 10 cm, the inner diameter is 40 mm, and the MG of QA is 3.9097 T/m, and QB is -3.6295 T/m. The drift length L1 is 30 cm, and L2 is 10 cm. The total length of this beam line is 2.6 m. The optimization result is shown in Figure 3. And the relational transport matrix parameters of it are shown in Table 1.

Figure 3: Electron beam trace from object plane to image plane in the x-plane and y-plane.

Table 1: The transport matrix parameters of radiography beam line.

According to transport matrix parameters from the object plane to Fourier plane, the aperture in the simulation is designed as an ellipse shape as shown in Figure 2(b) and the corresponding selection angle range can be described by the following formula:

$\begin{array}{}\text{(6)}& {x_0^{\prime }}^2+{y_0^{\prime }}^2\le {\left(0.91\text{mrad}\right)}^2.\end{array}$

3.3. Specimen Design

The specimen is composed of scattering target and magnetic vacuum. In this work, we select a 5 μm thick aluminum foil as the scattering target. On the one hand, as a low Z element, it can ensure that the probe electrons have lower energy spread. On the other hand, aluminum is always selected as the target material in the research of intense laser-solid interaction or Z-pinch. Combined with the aperture selection, the transmittance corresponding to deflection is shown in Figure 4.

Figure 4: Angle distribution of scattered electrons.

For this Gaussian distribution, we delimit the transmittance larger than 0.01 as detectable. That means the detection range of deflection is from -3 mrad to 3 mrad. For 50 MeV electrons, the corresponding magnetic strength diagnostic range is from -0.51 T^∗mm to 0.51 T^∗mm.

Except for the lens design, there are two factors that might influence measuring accuracy in this design: deflection and magnetic gradient (MG) of the specimen. The deflection can be limited by the incident electron energy. To get microspatial resolution, we should control the deflection less than 10 mrad for 1 mm scale expansion size. To study the influence of MG on measurement accuracy, we select a quadrupole model as the magnetic vacuum in this simulation, and a group of MG is set as 1, 2, and 3 T/mm.