A method for the asymmetric focusing of electron bunches, based on the active plasma lensing technique, is proposed. This method takes advantage of the strong inhomogeneous magnetic field generated inside the capillary discharge plasma to focus on the ultrarelativistic electrons. The plasma and magnetic field parameters inside the capillary discharge are described theoretically and modeled with dissipative magnetohydrodynamic computer simulations enabling analysis of the capillaries of rectangle cross-sections. Large aspect ratio rectangular capillaries might be used to transport electron beams with high emittance asymmetries, as well as assist in forming spatially flat electron bunches for final focusing before the interaction point.
I. INTRODUCTION
The laser acceleration of charged particles provides an approach toward the development of compact electron accelerators with high energies and low emittance needed for coherent light sources and linear colliders.1,2 Although there has been a significant progress made recently in the understanding and production of high energy and quality electron beams via laser plasma interactions,1,3,4 the application of these beams to coherent light sources and linear colliders has yet to be achieved. Linear electron-positron colliders need asymmetric emittances and polarized electrons, with the smaller vertical emittance required to be of the order of 0.01 mm mrad. The asymmetric emittance, i.e., flat beams (see Ref. 5 and references therein), are needed to reduce the beam-induced synchrotron radiation (beam-strahlung6–9) at the interaction point. Since one of the laser plasma accelerator advantages is the short acceleration length (the 4.25 GeV electron beams were produced using a 9 cm long capillary3), compared to the conventional ones, it is desirable and important to keep the size of the transport and focusing sections of such accelerators in the same order.
Within the framework of the Laser Wake Field Accelerator concept,10 Kando et al.5 suggested a method for the electron injection via the transverse wake wave breaking,11 when the electron trajectory self-intersection leads to the formation of an electron bunch elongated in the transverse direction, thus implying the flat electron beam generation.
Here, we propose a plasma-based method for flat electron bunch formation, which is based on the active lensing concept,12–14 using a strong inhomogeneous magnetic field generated in the capillary discharge plasma. The capillary discharges have applications in laser electron accelerators as waveguides,1,3,15–20 and for electron beam focusing.12,13,18 The azimuthally symmetric transverse magnetic field generated in the circular cross-section capillaries vanishes at the axis and has a radially increasing strength. This solenoidal field provides the electron beam focusing in both transverse planes in contrast to the permanent-magnet quadruples. Moreover, the magnetic field gradient of the permanent-magnet quadruples approximately equals a few hundreds of T/m (see Ref. 21), whereas the capillary-discharge field gradient can be ten times higher, reaching several kT/m.14 We note that an active plasma lensing was demonstrated for ion beams in z-pinch plasma discharges22–25 and in non-pinching capillary discharges.26
Although a majority of experiments use circular cross-section capillaries, a square cross-section is more suitable for several plasma diagnostic techniques.19,27 We also note that in contrast to the circular cross-section capillaries, in a square cross-section capillary, the magnetic field is not azimuthally symmetric.28 It is plausible to assume that in the case of rectangle cross-section capillaries with a large aspect ratio (which is the ratio of the long to short side length of the capillary in the transverse plane), the magnetic field near the axis is almost one-dimensional. Its gradient along the short capillary side is substantially larger than that along the long capillary side. It would result in different focus lengths of the electrons (or ions) in different directions, leading to the electron (ion) beam being first compressed in the direction corresponding to the larger magnetic field gradient. We show in this work that the current distribution inside the capillary supports this idea. One application of such a magnetic system is the asymmetric focusing of charged particle beams.
In this paper, rectangle cross-sections capillaries and charged particle focusing by the magnetic field inside such capillaries are investigated. The paper is organized as follows: Section II describes the configuration and parameters of the magnetohydrodynamic (MHD) simulations. Section III contains the results of the dissipative MHD simulations of the rectangle, large-aspect-ratio, cross-section capillary. In Sec. IV, we present an analytical description of the plasma discharge density, temperature, and magnetic field distributions during the quasi-stationary phase of the capillary discharge. The electron (ion) beam focusing by the capillary magnetic field is considered in Sec. V. The scalings of the key capillary discharge parameters are given in Sec. VI. Conclusions and discussions are presented in Sec. VII.
II. SIMULATION SET-UP
To simulate the plasma dynamics and magnetic field evolution in the capillary discharge plasma, we chose an oblong rectangular capillary with the aspect ratio of 10:1 and the size of in the (x, y)-plane, where m and . The capillary is pre-filled with pure hydrogen gas, which is initially (at t = 0) distributed homogeneously, and its density is equal to , corresponding to the electron density when the gas is fully ionized.
Initially, there is no electric current inside the channel. The discharge is initiated by a pulse of current driven by an external electric circuit. The total electric current through the discharge, I(t), is considered to be a given function of time. The current profile with the peak of 777 A at 160 ns is shown in Fig. 1 and its form is similar to the experimental one in Ref. 3. In order to initiate the discharge in the simulations, hydrogen is assumed to be slightly ionized at t = 0.
For the sake of simulation simplicity, we calculate the evolution of the corresponding magnetic field implementing an extended computational domain with an insulator around the capillary similar to the case analyzed in Ref. 28. Total transverse size of the domain is mm. The following MHD equations are solved in this extended domain:
where ρ, v, pe, and pi are the plasma density, plasma velocity, and electron and ion pressures, respectively. In Eq. (2), an artificial viscosity α is subtracted from the full pressure in order to preserve the monotony of the numerical solution. The electric and magnetic fields, E and B, obey Maxwell's equations (3), (5), and (6) within the framework of the quasi-static approximation (see Ref. 29). In Eqs. (4) and (7), σ is the electric conductivity. The electron and ion specific energies are (including the cost of ionization) and , respectively. The electron thermal conductivity coefficient is equal to κe. The function Qie describes the energy exchange between the electron and ion components. The equation of state and dissipation coefficients take into account the degree of the gas ionization similar to the method used in Ref. 17. Here, κe, σ and Qie are given by the expressions
respectively. The functions and depend on the parameters , where is the electron Larmor frequency, characterizing whether or not the electrons are magnetized, and the Lorentz parameter w, showing the relative role of the electron–electron and electron–ion collisions (for details see Ref. 30). In the case under consideration, the functions and are of the order of unity. The electron-ion collision frequency νei equals to
where z is the mean ion charge and Λei is the Coulomb logarithm for electron-ion collisions, which in the case under consideration is equal to
It is necessary to note that at low temperatures, as long as the mean ion charge is considerably less than unity, there is a noticeable fraction of neutral particles. Both fully ionized plasma and partially ionized plasma are treated as a single two-temperature fluid. We assume that the conditions for the local thermodynamic equilibrium are satisfied separately for the electrons and ions. The equilibrium state of ionization, z, is determined by the generalized to the case when Saha equation for a plasma composed of atoms and singly-ionized ions. To take into account the partial ionization of hydrogen plasma, we re-normalize the electron–ion collision frequency νei by taking into account the contribution of neutral particles to the electron scattering.
The insulator medium is treated as immobile (v = 0) and with zero electric conductivity, σ = 0, hence the electric current density vanishes, j = 0. The coupling between the external electric circuit and the simulated discharge is determined by the boundary condition for the time-dependent azimuthal component of magnetic field at the boundary of simulated domain . The simulation time is within the range, . The MHD code MARPLE (Magnetically Accelerated Radiative PLasma Explorer)28,31,32 is used to perform the simulations.
III. SIMULATION RESULTS
The behavior of the discharge plasma in y-direction is quite similar to that of the circular and square capillary plasmas17,28,31 as can be seen from simulations. After a relatively long period of plasma ionization (∼100 ns), the plasma is heated by the Joule effect; however, it remains relatively cold in a small layer running along the capillary walls, which becomes more pronounced near the shorter walls. This leads to the formation of a plasma temperature maximum at the mid line of the slot and to the redistribution of the plasma density in the y-direction as shown in Fig. 2. A quasi steady state of the discharge plasma is reached after approximately 100 ns. At the steady state, the plasma is at the mechanical and thermal equilibrium, as well as magnetic field diffusion processes across the plasma are at the equilibrium.17 This steady state evolves slowly due to the total electric current changing in time and due to minor plasma dynamics in the x-direction. We see from Fig. 3 that a quasi-steady state is also reached after about 200 ns in the x-direction. The parameters of such quasi steady state discharge are weakly dependent on the x coordinate as can be seen in Fig. 4. At the quasi-equilibrium stage, the magnetic field pattern corresponds to the asymmetric focusing of electron bunches (see Fig. 5).
Time evolutions of electron (a) density and (b) temperature along x = 0 (solid black lines in Fig. 4) for the range .
Time evolutions of electron (a) density and (b) temperature along x = 0 (solid black lines in Fig. 4) for the range .
Time evolutions of electron (a) density and (b) temperature along y = 0 (dashed black lines in Fig. 4) for the range mm.
Time evolutions of electron (a) density and (b) temperature along y = 0 (dashed black lines in Fig. 4) for the range mm.
Distribution of the electron (a) density and (b) temperature at t = 240 ns in the (x, y)-plane. (c) electron density distributions along the line y = 0 (dashed red line) and along the x = 0 (solid red line) and electron temperature vs x along the y = 0 line (dashed blue line) and vs the coordinate y along the x = 0 (solid blue line).
Distribution of the electron (a) density and (b) temperature at t = 240 ns in the (x, y)-plane. (c) electron density distributions along the line y = 0 (dashed red line) and along the x = 0 (solid red line) and electron temperature vs x along the y = 0 line (dashed blue line) and vs the coordinate y along the x = 0 (solid blue line).
Distribution of the magnetic field at t = 240 ns inside the capillary. (a) Magnetic field lines in the (x, y)-plane. (b) Bx vs y along the line x = 0 (solid black line) and By vs x along the line y = 0 (dashed black line).
Distribution of the magnetic field at t = 240 ns inside the capillary. (a) Magnetic field lines in the (x, y)-plane. (b) Bx vs y along the line x = 0 (solid black line) and By vs x along the line y = 0 (dashed black line).
In Fig. 4, we present the electron density and temperature distributions at t = 240 ns after the discharge was started. At this moment, the variation in these parameters along the long side of the capillary (i.e., along the dashed lines on the figure, y = 0) reaches its minimum.
We note that inside the capillary at some distance from its short sides, e.g., for mm, the distributions of electron density and temperature along the line y = 0 are almost homogeneous. The variation in the electron density ne is 7% and the electron temperature Te variation is 3% in the defined region, i.e., the density and temperature depend mostly on the y-coordinate. As can be seen from Fig. 3, the almost homogeneous distribution of the plasma parameters is reached after approximately 200 ns of the discharge and keeps constant during the remaining 300 ns of the simulation. At the same time, as we see in Fig. 2, the quasi steady state along the y-coordinate is reached after approximately 150 ns and and is maintained till the end of the simulation.
We estimate the size of that region, where plasma parameters depend mostly on y-coordinate, as equal to . It equals the length of the capillary long side 20 L without two narrow regions of the width approximately equal to the length of the capillary short side.
We assumed above that the magnetic field in the central part of an oblong capillary is almost one-dimensional, i.e., its gradient along the short side is substantially larger than that along the long side. In Fig. 5, we see that the magnetic field dependence on the coordinates corresponds to this assumption.
The simulation gives the following dependence of the electron density ne and of the Bx component of the magnetic field on y along the line x = 0 in the quasi-stationary regime
Here, the electron density at the capillary center equals . The magnetic field gradients along the x and y directions at t = 240 ns, when I = 524 A, are
As we see, the ratio is of the order of the capillary aspect ratio, which is equal to 10. We also note that according to the simulation results, this ratio changes during the initial stage of the discharge from zero to the quasi-stationary value, which may provide a tunable magnetic element for achieving focusing and/or transport of electron beams with different degrees of asymmetry. This property of the rectangular capillary discharges will be investigated elsewhere. The relationships (14), (15), and (17) obtained in simulations can be compared with Eqs. (36), (42), and (64), respectively. The latter ones are obtained in the frame of a simple analytic theory of the discharge presented in Sec. IV. This comparison will be presented in Sec. VI A.
IV. PLASMA EQUILIBRIUM AND MAGNETIC FIELD PROFILE
The capillary discharge plasma under the conditions, when the magnetic Reynolds number , where v is the velocity scale of the plasma motion, L is the space scale, and is the magnetic diffusivity, is small, , the pinch effect, is negligible, the axial electric field is uniform across the capillary, the magnetic field pressure is much less than the plasma pressure, and hence, the plasma pressure can be considered to be constant across the capillary, the electrons are unmagnetized, and the electron and ion temperatures are equal to each other, .17 In particular, these conditions are realized in the capillary schemes used for the laser pulse guiding, when the plasma is in the dynamic equilibrium i.e., when the plasma velocity vanishes. As a result, the plasma density dependence on the coordinate inside the capillary is determined by the balance between the Ohmic heating, jE with j and E being the electric current density and the electric field, respectively, and the cooling due to the electron heat transport. Here, we formulate a theoretical model of the plasma equilibrium inside the flat capillary. The model in this case is similar to that model, which has been formulated in Ref. 17 for the circular cross-section capillary.
In the case of the capillary of large aspect ratio cross section, i.e., with one side considerably larger than the other, we can assume that the parameters of the discharge at the equilibrium are almost independent of the x coordinate, provided we exclude regions of the size of the order of L in the vicinities of the shorter sides of the rectangle. Hence, all parameters of plasma and magnetic field depend only on the y coordinate. In this case, the dependence of the plasma temperature on the y coordinate is described by the heat transport equation
where we use the relationship between the electric current density and the electric field, , the coordinate y. Equation (18) is a nonlinear ordinary differential equation, which should be solved inside the interval with the boundary conditions
Here, Tw is the temperature of the capillary wall. For the sake of brevity, we simplify the problem under consideration as follows. Taking into account the weak dependence of the Coulomb logarithm, Λei, on the temperature and density, we assume that it is constant in Eqs. (9), (10), and (12) with the typical value approximately equal to 3, retaining in the expressions for the electron thermal conductivity (9) and for the electric conductivity (10) the dependence on the temperature only. Using this assumption, we rewrite Eqs. (9) and (10) in the form
Taking into account that the wall temperature is significantly lower than the plasma temperature at the capillary center, we assume that Tw = 0.
Introducing the dimensionless variable
and the function
with the parameter
we obtain from Eq. (18) the equation for the function
which should be solved for with the boundary conditions
which follow from Eq. (19). Here and below, a prime denotes differentiation with respect to the variable ξ.
Multiplying both sides of Eq. (24) by and integrating over ξ, we obtain the integral
where u0 is equal to the value of the function u at the capillary center, ξ = 0. The boundary conditions (25) have been used. Using the integral (26), we can write the solution of the boundary problem in quadratures
For the constant u0, we have
where is the Euler gamma function.33
The integral on the left-hand side of the Eq. (27) can be expressed via hypergeometric functions. As a result, the solution of the boundary problem (24) and (25) can be written in the implicit form
Figure 6 shows the dependence of the function on the coordinate ξ. Near the maximum, the function can be approximated by the parabola. It is easy to find that for
Near the boundaries at , the function depends on ξ as
Dependence of the function and of the normalized magnetic field on the coordinate ξ within the interval .
Dependence of the function and of the normalized magnetic field on the coordinate ξ within the interval .
The electron density distribution along the coordinate ξ can be found from the condition of constant pressure across the capillary. In the case of full ionization z = 1, we have
so that the electron density profile is given by
The electron density at the center is equal to
where
is a mean electron density.
Near the axis ξ = 0, it has the form
The magnetic field, , where ex is the unit vector in the x direction, should be found from the Eq. (3), which takes the form
The normalized gradient of the magnetic field near the capillary axis is given by
It follows from Eqs. (26) and (38) that the magnetic field dependence on the coordinate ξ has the following form:
This relationship can be written as (we note that is negative at )
Figure 6 presents the dependence of the normalized magnetic field on the coordinate ξ. As we see, the magnetic field vanishes at the axis, ξ = 0, where . Expanding the expression (40) near u = u0, i.e., for , we obtain the following expression for the magnetic field:
with . The magnetic field maximum is reached at the capillary boundary, ξ = 1, where , and is equal to
The total (per unit width) electric current in the capillary, , is
The electric current can also be derived from Eq. (24)
It follows from Eq. (26) that . Substituting U and u0 given by the Eqs. (23) and (28), respectively, into the right-hand side of Eq. (45), we obtain the expression (44) showing that the electric current I0 is proportional to the electric field in the power of 5/2 as in the case of circular capillary discharges.17
V. CHARGED PARTICLE FOCUSING
In what follows, we consider the motion of an electron in the magnetic field of the rectangular capillary discharge in the framework of a simplified one-dimensional model. We neglect any wakefield effects and assume electrons to be probe particles. The trajectory of a charged particle moving in a constant one-dimensional magnetic field can be found from the conservation of the particle energy and of the z-component of the generalized momentum
Here, is the relativistic Lorentz factor and Az is the z-component of the vector potential . The x component of the momentum, , is constant. These integrals of the particle motion follow from the independence of the magnetic field on time and z-coordinate. The normalizations of the momentum on and of the electromagnetic potential on are used. The magnetic field is equal to the curl of the vector potential, , i.e., .
Equations (40) and (26) yield the relationship between the z-component of the electromagnetic potential and the function
with
Using equations for the particle coordinates, and with ξ given by Eq. (21) and , where a dot stands for the time derivative, and Eq. (26), we can obtain the dependence of the particle normalized ζ-coordinate on the ξ-coordinate in quadrature
Since in the region near the z-axis for the magnetic field can be approximated by a linear function of the ξ-coordinate, the aberration in the focusing of high energy electrons can be made weak. This implies a smallness of two parameters
where a0 is approximately equal to the initial coordinate y0. The first small parameter shows that in the limit , the electron with a large enough momentum along the z-axis, or/and small initial position y0 moves inside the nonadiabatic region, where it is not magnetized. The electron oscillates in the y-direction around the position y = 0 and moves with approximately constant velocity along the z-axis. In addition, the smallness of the parameter δ1 indicates the role of the oscillation anharmonism due to the nonlinear dependence of the z-component of the electron momentum on the coordinate even in the case of the magnetic field linearly dependent on y. The second parameter δ3 should be small to make insignificant the effects of the nonlinear term in the magnetic field dependence on the coordinate (42).
Small but finite anharmonism results in the dependence of the oscillation frequency on the amplitude, which leads to the intersection of the particle trajectory with the y = 0 axis at different coordinates z. This leads to the aberration of the magnetic lens. The electrons being focused do not meet after the lens in one focal point. The further the trajectories initially are from the axis, the closer to the lens they intersect the axis. It is convenient to illustrate this by considering the electron motion near the axis in the magnetic field given by Eq. (42). Using the smallness of the electron displacement from the axis and retaining the terms of the order not higher than y3, the equations of electron motion can be written in the form of an anharmonic oscillator equation
with the oscillation frequency and the parameter of nonlinearity equal to
respectively. The nonlinearity parameter can also be written as with δ1 and δ3 given by Eq. (50).
We consider the electron focusing by two magnetic configurations, by a “long focusing system” and by a “short focusing system”. In the first case, the particles are focused at or near the capillary end. In the case of the short focusing system, the focus is located far from the capillary end.
In the long focusing system, the particle trajectory is given by the solution of Eq. (51), which to the third order of the small oscillation amplitude is34
It yields the relationship between the amplitude a0 and the initial coordinate y0, which reads
According to Eq. (53), the time at which the trajectory intersects the axis y = 0 is approximately equal to
It depends on the initial coordinate y0, which leads to the finite focus width equal to
where Y0 is the particle beam half-width. Figure 7(a) shows the typical trajectories of the electrons, moving in the magnetic field (42). The inset shows the close-up of the focus region. Initial momentum components are equal to and . The y and z coordinates at t = 0 are and . The oscillation frequency is , and the parameter κ approximately equals . The further the particles are from the axis at t = 0, at the larger distance along the z-direction they intersect the axis. This is an example of positive spherical aberration.
Electron trajectories in the magnetic field (42). (a) Long magnetic focusing system. The inset shows the close-up of the focus region. (b) Short magnetic focusing system. The inset shows the close-up of the focus region.
Electron trajectories in the magnetic field (42). (a) Long magnetic focusing system. The inset shows the close-up of the focus region. (b) Short magnetic focusing system. The inset shows the close-up of the focus region.
In the short focusing system case, the capillary length is substantially smaller than the focus length. As a result of the moving in the magnetic field, the particle acquires the transverse velocity
where Z is the capillary length and y0 is the particle coordinate at t = 0. Since the particle trajectory in the region outside of the capillary is given by , we can find the axis intersection coordinate, which is equal to
i.e., the focus width is
In Fig. 7(b), we present the trajectories of the electrons moving in the magnetic field (42), when the capillary is of a finite length equal to Z = 2.5. The inset shows the close-up of the focus region. Other parameters are the same as for the case shown in panel (a). Here, we also have an example of positive spherical aberration.
VI. SCALING
A. Capillary discharge parameters
From the theoretical model, formulated in Secs. IV and V, the scaling of the key capillary discharge parameters follows. The scaling provides the relationships between the discharge plasma parameters: the electric field E, resistance , magnetic field B, plasma density ne, temperature Te, and electric current I0, characterizing the external electric circuit, and the capillary size L. This yields for the electric field
The resistance per unit length of the discharge, , depends on I0 on and L as
For the electron temperature and density, we have
and
respectively. Here, the mean plasma density is defined by Eq. (35). The magnetic field gradient dependence on I0 on and L is given by
Here, we considered that, under the conditions considered in the present paper, the coefficients κ0 and σ0 are almost constant. They were calculated for the electron density equal to and for the electron temperature equal to the ion temperature eV. These coefficients are assumed to be constant within the framework of the theoretical model, presented earlier.
The correspondence between the predictions of the theoretical model and the simulation results is due the fact that the characteristic time for the capillary discharge to reach the equilibrium is substantially smaller than the timescale of the electric current in the electric circuit, which is approximately equal to 100–200 ns. Our estimates show that the characteristic time of establishing of the mechanical equilibrium τhy in the y-direction is of the order of 6 ns; the time for the thermal equilibration in the y-direction, τt, is approximately 22 ns; and the time for establishing the equilibrium distribution of electric over the y-direction τBy is about 1 ns. A characteristic time for establishing the constant electric field over the larger axis of the capillary slot is of the order of
which, for the parameters under consideration, is approximately equal to 35 ns. Here, is inductance of the discharge per its unit length, and is the resistance of the discharge per unit of its length. A mechanical equilibrium in the x-direction is established in about ns.
In Sec. IV, we presented the analytical model of the plasma equilibrium inside the rectangular capillaries. The model is based on the assumption that the plasma and magnetic field parameters depend only on the y-coordinate in the region inside the capillary far enough from its short sides. With this model, we obtained for the key parameters the scalings given by Eqs. (60)–(64). The results of the simulations with 2D MHD codes of plasma and magnetic field evolution in the capillary discharge show that the plasma reaches the equilibrium at ns. The coefficients in Eqs. (60)–(64), which were obtained from simulations, can be compared with the ones obtained from the analytical model in Sec. IV. The comparison is presented in Table I.
Comparison of the coefficients in the scaling given by Eqs. (60)–(64) with those obtained in the MHD simulation.
. | Simulation . | Theory . |
---|---|---|
E | 0.0694 | 0.0762 |
0.00694 | 0.00762 | |
Te | 7.10 | 7.58 |
ne | 0.867 | 0.798 |
7.28 | 7.87 |
. | Simulation . | Theory . |
---|---|---|
E | 0.0694 | 0.0762 |
0.00694 | 0.00762 | |
Te | 7.10 | 7.58 |
ne | 0.867 | 0.798 |
7.28 | 7.87 |
The relative accuracy of the analytical model prediction of the quasi-stationary stage parameters compared to the simulation results is of the order of 10%. Discrepancies of the same order take place in the coefficients describing spatial distribution of ne and Bx. They can be derived by comparing of Eqs. (14) and (15) with Eqs. (36) and (42), respectively. These minor discrepancies are caused by several reasons. The simulations take into account the weak non-stationarity of the discharge even in the quasi-stationary stage, the inhomogeneous distribution of the electric current, as well as weak plasma motion. Another source of minor inaccuracy of the analytic model of the same order is the replacement of the Coulomb logarithms by constant values.
B. Focusing of charged particle beams
In the homogeneous magnetic field, the charged particle rotates along the circle of the Larmor radius equal to , where p is the particle momentum. If the magnetic field is inhomogeneous and is vanishing at the surface y = 0 as in the case considered above, , the charged particle trajectory does not intersect the y = 0 plane, being localized in the region approximately equal to , provided the mean trajectory distance from the null plane y = 0 is substantially larger than . The magnetic field inhomogeneity results in the charged particle motion along the z-axis due to the gradient drift. In the limit , the particle Larmor radius formally tends to infinity; nonetheless, the trajectory is localized in the finite size region in the vicinity of the neutral plane y = 0. One can find the space scale of the particle trajectory localization in the vicinity of the y = 0 plane, , using the relationship
Here, we take into account that h1 and maximum magnetic field Bm given by Eq. (43) are related as with . The charged particle with the initial coordinate y0 larger than undergoes the so-called gradient drift directed perpendicularly to the magnetic field and to the magnetic field gradient, i.e., directed along the z axis. The beam in which the particle initial coordinates y0 are outside of the interval cannot be focused. Instead, if the charged particle initial coordinate y0 is significantly less than , the trajectory is localized within the region along the y axis of the size approximately equal to y0 intersecting the y = 0 plane at the distance . We notice that the initial coordinates of the trajectories presented in Fig. 7 are chosen to be inside the interval .
The scaling of the charged particle focusing characterized by the dependence of the focusing distance zf on the magnetic field parameters and charged particle energy can be considered in two limits.
In the ultra relativistic limit, assuming that the charged particle is an electron of the energy , we find
For example, for Bm = 10 kGs and L = 1 mm, that corresponds to the current line density of 16 kA/cm, and (i.e., for the electron with the energy of 1 MeV), it is approximately equal to 0.17 cm. If the electron energy is of 10 GeV (i.e., ) and Bm = 10 kGs and L = 1 mm, the focusing distance is equal to 17 cm.
In the case of the non-relativistic limit, assuming that it is an ion of the mass , where Ai is the ion atomic number and mp is the proton mass, and of the electric charge , the focusing distance can be written as
where is a normalized velocity of the ion with the energy . For the 100 MeV proton (Ai = 1 and Zi = 1) moving inside the L = 1 mm size capillary with the 1 kGs magnetic field, corresponding to current line density of 1.6 kA/cm, the focusing distance is equal to 10.8 cm. If the ion is carbon with atomic number Ai = 12, charge Zi = 6, and energy GeV moving in the L = 1 mm capillary magnetic field Bm = 10 kGs, the focus is at the distance cm.
As we see, with the capillary plasma generated magnetic field, the focusing length is substantially less than that in the case of the focusing systems used in the standard charged particle accelerator technology for both multi GeV electrons and multi hundred MeV ions.
VII. CONCLUSIONS
We suggest a novel method for the asymmetric focusing of electron beams. The scheme is based on the active lensing technique, which takes advantage of the strong inhomogeneous magnetic field generated inside the capillary discharge plasma. The plasma and magnetic field parameters inside a capillary discharge are described theoretically and modeled with dissipative magneto-hydrodynamic simulations to enable the analysis of capillaries of oblong rectangle cross-sections, implying that large aspect ratio rectangular capillaries can be used to form flat electron and/or ion bunches. In the case of flat electron bunches, the beamstrahlung in linear colliders or storage rings weakens. The effect of the capillary cross-section on the charged particle beam focusing properties is studied using the analytical methods showing the range of the capillary discharge parameters required for producing the high quality flat electron beams. Active lensing of charged particle beams with the inhomogeneous magnetic field generated in the capillary discharge plasma will enable the developing of compact laser plasma based systems for accelerating high energy charged particles and for manipulating the beam parameters.
ACKNOWLEDGMENTS
The work was supported in part by the Russian Foundation for Basic Research (Grant No. 15-01-06195), by the Competitiveness Program of National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), contract with the Ministry of Education and Science of the Russian Federation No. 02.A03.21.0005, 27.08.2013 and the basic research program of the Russian Academy of Sciences Mathematical Branch, Project 3-OMN RAS. The work at the Lawrence Berkeley National Laboratory was supported by U.S. DOE under Contract No. DE-AC02-05CH11231. The work at KPSI-QST was funded by the ImPACT Program of Council for Science, Technology and Innovation (Cabinet Oce, Government of Japan). At ELI-BL, it has been supported by the project ELI—Extreme Light Infrastructure—phase 2 (CZ.02.1.01/0.0/0.0/15 008/0000162) from the European Regional Development Fund, and by the Ministry of Education, Youth and Sports of the Czech Republic (Project No. LQ1606) and by the project High Field Initiative (CZ.02.1.01/0.0/0.0/15_003/0000449) from the European Regional Development Fund.