Theoretical model of current propagation in a helical coil with varying geometry and screen tube

C. L. C. Lacoste; A. Hirsch; E. d’Humières; V. T. Tikhonchuk; P. Antici; M. Bardon

doi:10.1063/5.0221820

I. INTRODUCTION

Traveling-wave tube amplifiers are devices used to amplify radio-frequency (RF) signals in the microwave range. The concept of the traveling wave tube (TWT) was conceived in 1947,1 and this was followed by the development of analytical models in the 1950s.2–5 The TWT is a wave–particle interaction system that involves an electron beam and an RF signal propagating in a conducting helix. The waveguide is engineered to create a phase velocity smaller than the speed of light, enabling wave–particle interactions when the wave phase velocity nearly matches the particle velocities. Electrons are slowed down to amplify the signal, which is transmitted to an antenna. The growing interest in these systems is mainly related to high-power and high-frequency microwave devices such as gyrotrons,6,7 relativistic TWTs,8 and free-electron lasers.9 Applications of TWTs include satellite communication systems, electronic countermeasures, and radar systems10,11 combining a large bandwidth, high gain, and light weight.

Laser-driven particle acceleration is achieved by the interaction of a high-intensity short pulse (picosecond or lower) laser pulse with a micrometer-thick solid target or gas jet.14,15 This is a promising acceleration scheme because of its compactness and simplicity,16–18 a high number of particles per bunch $(> 1 0^{11})$ , high ion energies (up to 100 MeV19), low emittance,20 and short duration.21 Applications of laser-accelerated ions include isochoric heating,22 isotope23 or neutron24 production, plasma radiography,25,26 nuclear fusion in fast ignition schemes,27,28 testing of materials,29 diagnosis of materials of interest for cultural heritage,30 and synthesis of microcrystals and microstructures.31 A simple and routinely available scheme for ion acceleration with lasers is target normal sheath acceleration (TNSA).32,33 Here, a high-intensity laser beam interacts with a solid target and accelerates electrons from the front side. Some electrons escape the target and leave positively charged ions in the bulk, but a large fraction are retained by the positive charge, generating a sheath electric field of the order of several TV/m at the rear side of the target. This field accelerates ions from the rear side to tens of MeV energies in the direction normal to the surface within an angle of about ±20°. However, for the applications mentioned above, the angular divergence of TNSA-accelerated ions is too high, and the energy distribution of protons is difficult to control.

To overcome these limitations, it was suggested to propagate the ions through a helical coil (HC) electrically connected to the target as shown in Fig. 1(a).34 During the laser–plasma interaction, the escape of electrons causes the target to become positively charged, which drives a discharge current pulse through the helix, generating an electromagnetic pulse (EMP) that post-accelerates and focuses the TNSA proton beam. The physics of current propagation in the helix is analogous to what occurs in coupled transmission systems.2,3 However, optimizations are needed for an efficient proton post-acceleration. Dispersion of the current pulse limits the length along which the wave–particle interaction is efficient35 and the use of a longer HC does not produce higher ion energies. Similar issues have been observed in coupled transmission systems.4,5 It has been suggested that the helix dispersion be controlled by adding a conducting tube around it. The current dispersion has been found to depend on the distance between the helix and the tube.

Figure 1.(a) Schematic of a single helical coil. (b) A helical coil with a tube. In the model, the helix is replaced by a thin cylinder that is symmetric under rotation around the z axis, as in Fig. 2. Reproduced with permission from Hirsch-Passicos et al., “Helical coil design with controlled dispersion for bunching enhancement of protons generated by the target normal sheath acceleration,” Phys. Rev. E 109, 025211 (2024). Copyright 2024 American Physical Society.¹²

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A schematic of a helical coil with a screening tube (HCT) is shown in Fig. 1(b). Similarly to the conventional TWT shown in Fig. 2, a conducting tube is set around the helical coil.12 In contrast to the low-current TWT, however, one cannot use ties to support the helical coil, owing to the high current, typically of a few kA, which might generate electrical breakdown between the HC and the tube. The HCT configuration reduces current dispersion, increases the ion maximum energy, and produces beam bunching. The choice of HC radius and pitch defines the propagation velocity of the electromagnetic fields, which can be synchronized with the ion velocity. The electromagnetic fields accelerate ions within a specific energy range depending on the field amplitude and resonance condition. Thus, the ion acceleration modifies the particle distribution, but the current dispersion limits the HCT effectiveness.

Figure 2.(a) View of TWT along the symmetry axis. (b) Schematic of loaded TWT.¹³ The helical coil (orange) is confined in the tube (gray) and supported by nonconductive ties (green). Reproduced with permission from Jiao et al., “A key design and experiment of a broadband high-power pulsed helix TWT,” in 2019 International Vacuum Electronics Conference (IVEC) (IEEE, 2019), pp. 1–2. Copyright 2019 IEEE.

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The resonant coupling of the ion beam with the current can be improved by varying the helical geometry along the propagation direction (z axis) to accelerate the current pulse to maintain synchronization with the accelerated protons. Here, we consider two geometries shown in Fig. 3. The pitch of the helix and/or its radius vary, thus providing an extended length of synchronization. We refer to a helix with varying geometry and with a tube as a varying helical coil with tube (VHCT).

Figure 3.Examples of helical coil geometries with (a) varying radius and (b) varying pitch along the z axis. The tube is symmetric under rotation around the symmetry axis as in Fig. 2.

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Pitch-tapering is a common feature of TWTs, but the codes currently used for TWT simulation, such as CST,36 KARAT,37 MAFIADIMOHA,38 DIMOHA,39 MVTRAD,40 CHRISTINE,41 and BWIS42 do not effectively account for it.

The main objectives of the analytical model developed in this paper are to provide an understanding of the physics of current dispersion and the optimization of VHCT geometry for an efficient ion acceleration and bunching. Compared with direct particle-in-cell (PIC) simulations, the analytical model allows us to consider a large parameter space and save computation time. Using this model, one may increase the cutoff ion energy, optimize beam bunching, and increase the number of high-energy ions. The model is applied to the optimization of a laser-driven ion beamline such as those that can be obtained on commercially available multi-hundred TW laser systems, for example, the advanced laser light source (ALLS) laser facility at INRS EMT.17 In this facility, the ion beamline is driven by a Ti:sapphire laser system generating pulses with an on-target energy of 3.3 J, a pulse duration of 22 fs, a repetition rate of 2.5 Hz, a central wavelength of 800 nm, and a focal spot of 5 μm. The peak intensity on target is ∼1.3 × 10²⁰ W/cm².

This work is based on the theoretical model of a helical coil developed in Refs. 2–5, and further applied to an HC in Refs. 12 and 35. Expressions for the electromagnetic fields are derived from Maxwell’s equations in vacuum, with the boundary conditions linking fields to the surface of the helix and the tube. The helix parameters, namely, its pitch and radius, are assumed to vary slowly along the z axis, and so the fields are calculated using the Wentzel–Kramers–Brillouin approximation (WKB). The dispersion relations are obtained in Fourier space, and then expressions for the fields in real space are analyzed.

The remainder of the paper is organized as follows. Section II presents the theoretical background: Maxwell’s equations, the current continuity equation, and the boundary conditions for VHC and VHCT geometries. These equations are compared with known analytical models and PIC simulations in Sec. III. Expressions for the electric and magnetic fields are implemented in the DoPPLIGHT code.43 Our conclusions are summarized in Sec. IV. Four appendixes present expressions for all components of the electromagnetic field in real space and discuss some results related to the consistency of our model, its limits, and approximations.

II. COUPLING THE CURRENT TO ELECTROMAGNETIC FIELDS

A general schematic of the model development is presented in Fig. 4. The first step consists in the definition of the current propagating in the helix, which is considered to be an infinitely thin perfect conductor. The inner and external electromagnetic fields are then defined through the boundary conditions. The electromagnetic fields are computed from the differential equations for the electrostatic and vector potentials. The equation for current propagation in the helix is given in Sec. II A. Following Ref. 2, the helix is modeled as a continuous cylinder with current propagating in an azimuthal direction (see Fig. 5). Equations for the electromagnetic fields are given in Sec. II B in the Fourier representation. The boundary conditions for the VHC case (without tube) are given in Sec. II C. They determine a connection between the current and the electromagnetic fields, as well as the dispersion relation. Expressions for the electromagnetic field for the VHC in real space are obtained in Sec. II D. The same process is then repeated for the VHCT geometry—a helix screened with a conducting tube. The boundary conditions and the dispersion equation are given in Sec. II E, and the electromagnetic fields in real space are given in Sec. II F. The analytical model is compared with PIC simulations in Sec. III.

Figure 4.Schematic of model construction.

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Figure 5.Diagram showing the direction of current propagation at the surface of a cylinder at an angle Ψ to the azimuthal direction.

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A. Current pulse propagation

Following the original sheath helix model,2 we assume the current is propagating along a thin cylindrical surface of radius a and length L in a direction defined by the angle Ψ with respect to the azimuthal axis θ. This angle defines a relation between the helix radius a and pitch h: $h = 2 π a \tan Ψ,$ (1)as shown in Fig. 5.

Since the direction of current propagation is defined, it is sufficient to consider only the z component of the current, j_z(t, z). This is related to the charge density per unit length q by the continuity equation: $\partial_{z} j_{z} = - \partial_{t} q + 2 j_{0} (t) δ (z),$ where the second term on the right-hand side is the source term j₀ prescribed at the origin z = 0, and δ(z) is the Dirac function. The helix is represented as a lossless transmission line with conductance C and inductance per unit length L. The charge and current are related by the telegraph equation: $- C^{- 1} \partial_{z} q = L \partial_{t} j_{z} .$ Taking the derivative of the continuity equation on z, inserting it into the propagation equation, and performing a Fourier transform in time, we obtain the following equation for the current: $\partial_{z}^{2} j_{z} (ω, z) + k^{2} (ω, z) j_{z} (ω, z) = 2 j_{0} (ω) δ^{'} (z),$ (2)where k² = LCω² and ω is the frequency. The value of LC depends on the transmission line structure and may depend on the frequency and coordinate.

To solve this equation, we assume that k(ω, z) is a slow function of the coordinate and represents the current wavenumber in the WKB approximation, j_z = Ae, where A(ω, z) is the slowly varying amplitude and ϕ(ω, z) is the phase. Inserting this expression into Eq. (2), we neglect the second derivative of A as being of third order. Conversely, the phase ϕ varies strongly with z, and its derivative ϕ′ is of first order, the same as k. Consequently, we obtain two equations for the amplitude and phase: ${(ϕ^{'})}^{2} = k^{2}, 2 i A^{'} ϕ^{'} e^{i ϕ} + i ϕ^{″} A e^{i ϕ} = 2 j_{0} δ^{'} (z) .$ (3)Solving the first equation as $ϕ = \int_{0}^{z} k (ω, z_{η}) d z_{η}$ , we obtain the following expression for the current: $j_{z} = j_{0} κ (ω) g (z),$ (4)where k₀ = k(ω, z = 0), $κ (ω) = 1 - i k_{0}^{'} / 2 k_{0}^{2}$ , and the function g describes the current variation along the axis of the helix: $g (z) = \sqrt{\frac{k_{0}}{k}} \exp [i \int_{0}^{z} k (ω, z_{η}) d z_{η}] .$ (5)

B. Propagation of electromagnetic fields

Starting from Maxwell’s equations, we obtain differential equations for the axial components of electric and magnetic fields. They are solved by performing separation of variables in the radius and z axis: E_z = f(r)g(z) and B_z = h(r)g(z), where the function g(z) [Eq. (5)] describes the field distribution along the propagation direction in the WKB approximation. A general solution for the radial dependence of the field components in the axisymmetric geometry can be represented as a linear combination of two modified Bessel functions: f(r) = B₁I₀(αr) + B₂K₀(αr) and h(r) = B₃I₀(αr) + B₄K₀(αr), where B_i are constants defined by the boundary conditions and $α (ω, z) = \sqrt{k^{2} - ω^{2} / c^{2}}$ (6)is the radial propagation constant. Using this notation, expressions for all the field components can be written as follows: $E_{z} = g (z) [B_{1} I_{0} (α r) + B_{2} K_{0} (α r)],$ (7) $E_{r} = \frac{i k}{α} Λ g (z) [- B_{1} I_{1} (α r) + B_{2} K_{1} (α r)],$ (8) $B_{θ} = - \frac{i ω}{α c^{2}} ℵ g (z) [B_{1} I_{1} (α r) - B_{2} K_{1} (α r)],$ (9) $B_{z} = g (z) [B_{3} I_{0} (α r) + B_{4} K_{0} (α r)],$ (10) $B_{r} = \frac{i k}{α} Λ g (z) [- B_{3} I_{1} (α r) + B_{4} K_{1} (α r)],$ (11) $E_{θ} = \frac{i ω}{α} g (z) [B_{3} I_{1} (α r) - B_{4} K_{1} (α r)],$ (12)where $Λ (ω, z) = 1 + \frac{i k^{'}}{2 k^{2}}, ℵ (ω, z) = 1 + \frac{k^{″}}{2 k ω^{2}} - \frac{3 {(k^{'})}^{2}}{4 k^{2} ω^{2}} .$

C. Dispersion relation without tube

The constants B_i are defined by the boundary conditions. Since the current sheet is localized at r = a, there are two distinct zones, r < a and r > a, inside and outside the helix where the constants take different values. These are denoted with superscripts − and +, respectively.

First, we apply the condition that fields should decrease to zero far from the helix, at r → ∞. Taking the expressions (7) and (10) for E_z and B_z, we find $B_{1}^{+} = 0, B_{3}^{+} = 0 .$ (13)Another condition is that the fields must have finite values at the helix axis, r = 0. Again taking the expressions (7) and (10) for E_z and B_z, we find $B_{2}^{-} = 0, B_{4}^{-} = 0,$ (14)Two more conditions apply to the electric field at the position of the current sheet, r = a. The component of the electric field tangential to the current should be continuous and perpendicular to the current. By applying the continuity condition to the components E_z and E_θ and accounting for Eqs. (13) and (14), we find $B_{2}^{+} = B_{1}^{-} \frac{I_{0} (α a)}{K_{0} (α a)}, B_{3}^{-} = - B_{4}^{+} \frac{K_{1} (α a)}{I_{1} (α a)} .$ (15)The component of the electric field parallel to the current reads $E_{‖} = E_{θ} \cos Ψ + E_{z} \sin Ψ .$ (16)Setting this field to zero at r = a, we find $B_{4}^{+} = - i B_{2}^{+} \frac{α}{ω} \frac{K_{0} (α a)}{K_{1} (α a)} \tan Ψ .$ (17)Thus, by applying these boundary conditions, we have expressed all eight constants $B_{i}^{\pm}$ in terms of one, $B_{1}^{-}$ , which is proportional to the current amplitude j₀. However, we have one more continuity condition that applies to the component of the magnetic field parallel to the current: $B_{‖}^{+} (r = a) = B_{‖}^{-} (r = a)$ . Taking the expressions (9) and (10) for B_z and B_θ along with Eqs. (13) and (14), we obtain a relationship between the frequency ω and wavenumber k, which is the dispersion relation $ω^{2} = k^{2} c^{2} \frac{1 + [\frac{3 {(k^{'})}^{2}}{4 k^{4}} - \frac{k^{″}}{2 k^{3}}] \cot^{2} Ψ \frac{I_{1} (α a) K_{1} (α a)}{I_{0} (α a) K_{0} (α a)}}{1 + \cot^{2} Ψ \frac{I_{1} (α a) K_{1} (α a)}{I_{0} (α a) K_{0} (α a)}} .$ (18)

This dispersion relation is plotted in red in Fig. 6 for a helix angle Ψ = 6.36°. At large wavenumbers ka ≫ 1, dispersion is negligible and ω ≈ kc sin Ψ, where c sin Ψ is the velocity of electric current propagation along the z axis. By contrast, dispersion effects are important at wavelengths comparable to the helix radius, and the phase velocity ω/k approaches the speed of light. Moreover, the second term in the numerator in Eq. (18) is assumed to be smaller than 1 in the domain of validity of the WKB approximation. The limits of validity are discussed further in Appendix A. From these limits, we neglect the first and second derivatives of k in the equations.

Figure 6.Dependence of frequency ω on wavenumber k for an HC of radius a = 0.5 mm and linearly varying pitch (on a length of l_c = 40 mm) of h = 0.35–0.7 mm, without tube (red), with tubes of radii b/a = 1.4 (blue) and 1.8 (green), and in a perfect case without any dispersion (dashed black).

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To find a relation between the remaining constant $B_{1}^{-}$ and the current j₀, we apply the condition of discontinuity of the tangential component of the magnetic field perpendicular to the direction of electric current at r = a. According to Ampère’s theorem, $B_{⊥}^{+} (r = a) - B_{⊥}^{-} (r = a) = \frac{μ_{0}}{h} j_{0} \cot Ψ,$ (19)where h/cos Ψ is the length of the current path over one period, j₀/sin Ψ is the full current propagating along the wire, and μ₀ is the vacuum magnetic permittivity. Considering the expressions (9) and (10) for B_z and B_θ, we find $B_{1}^{-} = - \frac{i ω μ_{0} j_{0} κ}{2 π a α} \frac{I_{1} (α a) K_{1} (α a)}{I_{0} (α a) J (α a)} \cot^{3} Ψ \cos Ψ,$ (20)where J(αa) = K₁(αa)I₀(αa) + K₀(αa)I₁(αa). Knowing all the constants and the dispersion relation, we obtain expressions for the electric and magnetic fields as functions of time.

D. Expressions for fields without tube

Taking the inverse Fourier transforms of Eqs. (7)–(12) with appropriate values of the constants, we obtain expressions for the electric and magnetic fields as functions of r, z, and t. Here, we give the expressions for E_z and B_z for r < a: $\begin{align} E_{z}^{-} (r, z, t) = & - \frac{i μ_{0}}{2 π^{2} a} \int_{0}^{\infty} d ω \frac{j_{0} ω κ}{α} \sqrt{\frac{k_{0}}{k}} \frac{I_{1} (α a) K_{1} (α a)}{I_{0} (α a) J (α a)} \\ \times \cot^{3} Ψ \cos Ψ I_{0} (α r) \exp [i \int_{0}^{z} k (ω, z_{η}) d z_{η} - i ω t], \end{align}$ (21) $\begin{align} B_{z}^{-} (r, z, t) = & \frac{μ_{0}}{2 π^{2} a} \int_{0}^{\infty} d ω j_{0} κ \sqrt{\frac{k_{0}}{k}} \frac{K_{1} (α a)}{J (α a)} \\ \times \cot^{2} Ψ \cos Ψ I_{0} (α r) \exp [i \int_{0}^{z} k (ω, z_{η}) d z_{η} - i ω t] . \end{align}$ (22)Expressions for the other components of the electric and magnetic fields are given in Appendix B.

E. Dispersion relation with tube

Here, we calculate the constants B_i for a helix with a screening tube (VHCT). Considering the tube as an ideal conducting cylinder of radius b, the boundary conditions correspond to zero electric and magnetic fields at r = b. Taking the expressions (7) and (10) for E_z and B_z, we find $B_{2}^{+} = - B_{1}^{+} \frac{I_{0} (α b)}{K_{0} (α b)}, B_{4}^{+} = B_{3}^{+} \frac{I_{1} (α b)}{K_{1} (α b)} .$ (23)The condition on the fields at the helix axis r = 0 is unchanged and given by Eq. (14). The conditions on the electric field at the position of the current sheet r = a are also the same as in the case without a tube, Eqs. (15) and (17). The condition (17) remains unchanged, and, accounting for the conditions (23) at the tube, we can represent Eq. (15) as follows: $B_{1}^{+} = B_{1}^{-} \frac{K_{0} (α b) I_{0} (α a)}{K_{0} (α b) I_{0} (α a) - I_{0} (α b) K_{0} (α a)},$ (24) $B_{3}^{+} = B_{3}^{-} \frac{K_{1} (α b) I_{1} (α a)}{K_{1} (α b) I_{1} (α a) - I_{1} (α b) K_{1} (α a)} .$ (25)Thus, by applying these boundary conditions, we have expressed all eight constants $B_{i}^{\pm}$ in terms of one, $B_{1}^{-}$ , which is proportional to the current amplitude j₀. We then use the continuity condition for the component of the magnetic field parallel to the current: $B_{‖}^{+} (r = a) = B_{‖}^{-} (r = a)$ . Taking the expressions (9) and (10) for B_z and B_θ along with Eq. (23), we obtain the dispersion relation for the helix with tube: $ω^{2} = k^{2} c^{2} \frac{1 + [\frac{3 {(k^{'})}^{2}}{4 k^{4}} - \frac{k^{″}}{2 k^{3}}] \cot^{2} Ψ \frac{I_{1} (α a) I_{0} (α b) J (α a) N (α a)}{I_{0} (α a) F (α a)}}{1 + \cot^{2} Ψ \frac{I_{1} (α a) I_{0} (α b) J (α a) N (α a)}{I_{0} (α a) F (α a)}},$ (26)where $F (α a) = K_{0} (α b) I_{0} (α a) - I_{0} (α b) K_{0} (α a),$ $N (α a) = \frac{K_{1} (α b) I_{1} (α a) - I_{1} (α b) K_{1} (α a)}{I_{1} (α b) J (α a)} .$ This dispersion relation is plotted in Fig. 6 for b/a = 1.4 (blue curve) and b/a = 1.8 (green curve). As can be seen, the mode frequency decreases as the tube approaches the coil, i.e., the ratio b/a decreases.12

To find a relation between the remaining constant $B_{1}^{-}$ and the current j₀, we apply the condition of discontinuity of the tangential component of the magnetic field perpendicular to the direction of electric current at r = a given by Eq. (19). Considering the expressions (9) and (10) for B_z and B_θ, we find $B_{1}^{-} = \frac{i ω μ_{0} j_{0} κ}{2 π a α} \frac{I_{1} (α a) N (α a)}{I_{0} (α a)} \cot^{3} Ψ \cos Ψ .$ (27)

F. Expressions for the fields with tube

Taking the inverse Fourier transforms of Eqs. (7) and (10) with the corresponding expressions for the constants B_i, we obtain the electric and magnetic fields in real space. As an example, we give here the expressions for $E_{z}^{-}$ and $B_{z}^{-}$ : $\begin{align} E_{z}^{-} (r, z, t) = & \frac{i μ_{0}}{2 π^{2} a} \int_{0}^{\infty} d ω \frac{j_{0} ω κ}{α} \sqrt{\frac{k_{0}}{k}} \frac{I_{1} (α a) N (α a)}{I_{0} (α a)} \\ \times \cot^{3} Ψ \cos Ψ I_{0} (α r) \exp [i \int_{0}^{z} k (ω, z_{η}) d z_{η} - i ω t], \end{align}$ (28) $\begin{align} B_{z}^{-} (r, z, t) = & - \frac{μ_{0}}{2 π^{2} a} \int_{0}^{\infty} d ω j_{0} κ \sqrt{\frac{k_{0}}{k}} N (α a) \\ \times \cot^{2} Ψ \cos Ψ I_{0} (α r) \exp [i \int_{0}^{z} k (ω, z_{η}) d z_{η} - i ω t] . \end{align}$ (29)The other components of the electric and magnetic fields are given in Appendix B. The electric and magnetic fields in the coil given by Eqs. (21), (22), (28), and (29) are calculated with the DoPPLIGHT code described in Sec. III

III. MODEL VALIDATION: COMPARISON WITH PIC SIMULATIONS

The analytical model describing the current propagation along the helix and electromagnetic fields, is implemented in the DoPPLIGHT code, introduced in Sec. III B, and compared with the results obtained with a PIC code, SOPHIE, presented in Sec. III A. The validation of the model consists of a comparison of expressions for electromagnetic fields and dispersion equations with known formulas for specific conditions3,5 and numerical simulations presented in Sec. III C. Furthermore, the proton energy distributions obtained with DoPPLIGHT and from PIC simulations are compared in Sec. III D.

A. Description of SOPHIE code

The PIC code SOPHIE,44 developed at CEA-CESTA, solves Maxwell’s equations for the electric and magnetic fields in 3D geometry in vacuum with boundary conditions defined at surfaces of arbitrary shape made of conducting, dielectric, or magnetic materials. Maxwell’s equations are solved on the Yee mesh with centered electric and magnetic fields using the finite-difference time-domain (FDTD) method.45 The particles are moved according to the relativistic dynamical equations using a Boris and Shanny solver.46 Self-consistency is achieved using a Buneman currents collector.

SOPHIE is not designed to simulate laser–plasma interaction, which requires resolution of the laser wavelength. It is complementary to laser–plasma interaction PIC codes such as CALDER47 and Smilei.48 It is used to model current propagation in vacuum on large spatial scales (of the order of several centimeters) and long time scales (of the order of a few nanoseconds). The code is benchmarked on test cases and compared with other similar codes.

B. Description of DoPPLIGHT code

DoPPLIGHT43 is a numerical realization of a model describing charged-particle motion in a coil with prescribed electric and magnetic fields varying in space and time. It is used to run fast computations for a large number of cases. The scheme of this code is presented in Fig. 7. It includes the following steps:•The helix geometry and the current are defined at the beginning of the helix in the time domain.•At the same time, the particles’ source terms are defined in terms of particle density as a function of time, along with the angular and energy distributions.•DoPPLIGHT then performs a Fourier transform of the current and uses it to calculate the fields according to the equations given in Sec. II.•The code then calculates the space-charge fields,^43,49 assuming a Gaussian-shaped nonrelativistic proton beam.•Finally, the protons and electrons are injected into the coil, and their trajectories are calculated using the Boris pusher⁵⁰ using the fields interpolated at every time step on each particle’s position.

Figure 7.Diagram of DoPPLIGHT code. The coil parameters correspond to the radius a, the pitch h and the length L. The source term provides the distribution of particles in energy, angle, and time. The electric and magnetic fields in the helix are calculated according to Eqs. (28) and (29). The space-charge fields account for Coulomb repulsion. Particle displacement is calculated with the Boris pusher.

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This numerical model operates in a 2D axisymmetric geometry. It is time-resolved and, in contrast to a PIC code, not self-consistent. Its simplicity enables the use of many diagnostics, such as the proton spectrum at the HC exit or values of the electromagnetic fields at the particle trajectory at every time step. In Sec. III C, the results obtained with this code are compared with those of the PIC simulations using SOPHIE.

C. Comparisons

1. Coil with constant pitch and radius

By considering k to be independent of the coordinate, our results can be compared with known analytical expressions. The dispersion Eq. (18) for a helix without tube reads $ω^{2} = k^{2} c^{2} {[1 + \cot^{2} Ψ \frac{I_{1} (α a) K_{1} (α a)}{I_{0} (α a) K_{0} (α a)}]}^{- 1} .$ This agrees with the dispersion equation given in Refs. 2 and 3. Similarly, for a coil with tube, Eq. (26) agrees with Refs. 4 and 5.

A simple expression can be obtained for the magnetic field on the coil axis in the case of a DC current j(z, t) = J₀. Inserting this expression into Eq. (22) we find $B_{z} = \frac{μ_{0}}{h} J_{0} \cos Ψ .$ In particular, in the case of small pitch (Ψ → 0), this expression reduces to the well-known formula for the magnetic field of a solenoid, B_z = μ₀NJ₀, where N = 1/h is the number of turns per unit length.

Figure 8(a) presents a comparison of the z component of the electric field on the axis calculated with DoPPLIGHT and SOPHIE for an input current of $j_{0} = J_{0} \exp [- {(t - t_{0})}^{2} / 2 t_{p}^{2}]$ , where J₀ = 800 A, t₀ = 16 ps is the delay time between proton emission and current emission, and t_p = 3 ps is the pulse duration FWHM. The temporal distribution of the current is responsible for the different frequencies of the pulse, which itself governs the current dispersion, as explained in Secs. II C and II E. Our model estimates that the electric field component is 300 MV/m for the radial and axial components and 20 MV/m for the azimuthal component. Concerning the magnetic field, the estimates given by DoPPLIGHT are of the order of 5 T for the axial component and 150 mT for the radial and azimuthal components. These estimates indicate that the radial and axial electric fields dominate the particle motion in the helix. These parameters correspond to the coil used in an experiment on proton acceleration on the ALLS facility.12 Good agreement between the PIC simulation and the reduced model is evident. A similar comparison for a helix with a tube of radius b/a = 1.4 is presented in Fig. 8(b). Agreement is also good, thus confirming the consistency of our model.

Figure 8.Coordinate dependence of the axial electric field E_z at r = 0.2 mm for coils (a) without and (b) with tube. Dashed lines show the result of the PIC simulation (SOPHIE), and solid lines are obtained with the DoPPLIGHT code. The helix radius is 0.5 mm, the pitch 0.35 mm, and the tube radius 0.7 mm.

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It can be seen from Fig. 8(a) that the velocity of pulse propagation decreases with time. This is the dispersion effect. By contrast, the presence of the tube in Fig. 8(b) mitigates dispersion, and the pulse propagates faster. This has a positive effect on the proton acceleration and bunching, as shown in Sec. III C 2.

Small differences between the model and PIC simulation can be explained by the limits of the sheath helix model, which requires a low helix angle Ψ.2 However, these differences are smaller than the typical shot-to-shot variations in experiments,51,52 which are of the order of 25%. Furthermore, we estimate the current from the ejected electrons and protons measured during experiments.17 The average error of our model is within 20%, which is smaller than the measurement error.

A comparison between SOPHIE and DoPPLIGHT for the radial electric field and the current is presented in Appendix C. The good agreement found there confirms the validation of our model for the case of constant helix parameters.

2. Coil with a tube and varying pitch

Here, we consider a coil of length of l_c = 40 mm with pitch varying linearly from h₀ = 0.35 mm to h₁ = 0.70 mm, such that $h (z) = h_{0} + (h_{1} - h_{0}) z / l_{c} .$ (30)The coil and tube radii are a = 0.5 mm and b = 0.7 mm, respectively. The axial electric field E_z(0, z) is shown in Fig. 9 for coils without (VHC) and with tube (VHCT). Comparison of the dashed and solid lines reveals good agreement between the model and the PIC simulations for the axial component of the electric field. A similar level of agreement for the radial electric fields is demonstrated in Appendix C. The average error of 20% is smaller than the typical measurement errors.52 Small differences between the model and PIC simulation can be explained by the approximation made in the solution of Maxwell’s equations. The separation of variables in the r and z directions is not exact if the coil parameters vary along the axis. Also, the WKB approximation implies a slow variation of the coil parameters.

Figure 9.Coordinate dependence of the axial electric field E_z at r = 0.2 mm for coils (a) without and (b) with tube and pitch varying linearly with the coordinate [Eq. (30)]. Dashed lines show the results of PIC simulations (SOPHIE), and solid lines are obtained with the DoPPLIGHT code. Other parameters are given in the text.

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Other comparisons not shown here confirm the capacity of the model to describe the electromagnetic fields in the coil with sufficient accuracy, including the pulse propagation velocity and the field amplitude. DoPPLIGHT is able to compute the electric fields as functions of z and r in a few minutes, compared with some 10 000 central processing unit (CPU) hours for PIC simulations.

D. Energy distribution of protons accelerated in coil

In this subsection, we demonstate the capacity of our model to calculate the proton acceleration in the coil by comparing the results obtained with SOPHIE and DoPPLIGHT. The proton spectrum entering the coil is taken from experiments conducted at the ALLS facility.52,53 It is approximated by an exponential distribution $\frac{d N_{p}}{d ε} = \frac{N_{p}}{T_{p}} \exp (- \frac{ε}{T_{p}}),$ (31)with the total number of protons N_p = 7.5 × 10¹⁰, effective temperature T_p = 1 MeV, cutoff energy of 7 MeV, and angular divergence of ±20°. The protons are propagated through a coil of length l_c = 14 mm, radius a = 0.5 mm, and pitch varying linearly from h₀ = 0.35 mm to h₁ = 0.4725 mm. The spectra of protons exiting the coil are shown in Figs. 10(a) and 10(b) for cases respectively without and with a tube of radius b = 0.7 mm.

Figure 10.Energy distribution of the protons accelerated in the helical coil without tube (a) and with tube (b) calculated with SOPHIE code (red line) and DoPPLIGHT code (blue line) with a linearly varying pitch. Panel (c) compare the case of a linearly varying pitch and the constant pitch. Black line shows the input proton spectrum. The parameters of the coil and the current are given in the text.

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In spite of some differences related to a simplified description of the particle dynamics in the model, the proton spectra obtained with DoPPLIGHT and SOPHIE show similarities concerning the cutoff energy, position of peaks, and the overall shape for energies above 4 MeV. A similar agreement for the coils with a constant pitch can be seen in Appendix D. A comparison of Figs. 10(a) and 10(d) shows that the use of a tube allows an increase in the cutoff energy from 10 to 12 MeV. This is explained by the mitigation of the coil dispersion by the tube and a better coupling between the field and the particles. Moreover, the number of protons with energies above 3.5 MeV is increased by a factor of 20 compared with the input spectrum. This example shows the potential of coils with a tube and varying geometry to shape the proton spectrum according to the desired application. A comparison of the results for a helical coil with tube and a varying helical coil with tube in Fig. 10(c) shows a cutoff energy of up to 12 MeV for the VHCT, which is higher than the HCT cutoff. The total number of protons at the HC exit is only few percent of the input TNSA protons. However, because of the focusing effect of the radial electric field, we get a higher number of protons per solid angle unit.

A comparison between DoPPLIGHT and SOPHIE results concerning proton acceleration confirms the acceptable accuracy of our model. It enables the performance of a large number of simulations and facilitates the exploration of a broad range of coil geometries. Although SOPHIE and DoPPLIGHT give similar results for the HC geometries presented in this paper, there are three major differences between these two codes. The first comes from the particle retroaction. Actually, the effect on the electromagnetic field of particle acceleration/deceleration is not taken into account in DoPPLIGHT, in contrast to SOPHIE. The second difference comes from the space-charge modeling. This is done by solving Poisson’s equation in SOPHIE, whereas in DoPPLIGHT, the space charge is described as a field-created charged axisymmetric ellipsoid. The third difference comes from reflections during the pulse propagation along the helix. SOPHIE describes the real helical structure of the helix with a wire of a finite thickness. So, in addition to the fields induced by the current, it accounts for the secondary fields induced by reflections from conducting surfaces.

IV. CONCLUSION

In this work, we have developed an analytic description of current propagation in a helical coil with a screening tube and varying geometry. The tube mitigates the coil dispersion and maintains a quasi-constant pulse shape and propagation velocity. In addition, a gradual increase in the coil pitch enables the pulse to be accelerated along the propagation direction while maintaining resonance with the accelerated particles over a longer distance.

The model has been implemented numerically in the code DoPPLIGHT, and its accuracy has been demonstrated by comparing its results with those of PIC simulations. Good agreement has been obtained for coils with a constant pitch and radius with and without a tube. Furthermore, the comparison has been extended to the case of a linearly varying pitch.

We have demonstrated that coil screening and pitch stretching enables optimization of proton acceleration. An increase in the number of high-energy protons by an order of magnitude has been observed, together with an increase in the maximum proton energy by 20% and a focusing of the particles on the coil axis.

Thanks to this model, we can begin an optimization of helical coil geometries to increase the cutoff energy and the number of high-energy charges. The utilization of a more powerful laser beam could also be interesting because of the increased discharge current and thus better proton acceleration and focusing.

Further future improvements of the model could take account of current reflection due to inhomogeneity of the helix along the propagation direction. Currently, the model neglects coil resistance and the retro-action of the particles on the electric field. A perfect conductor is a good approximation because the resistance of a realistic coil is about 10 mΩ (accounting for the skin effect), and so the resistive electric field is much smaller than the inductive electric field. The model could be improved to take into account resistive losses (R ≠ 0) and the skin effect. The retro-action of the particles on the electric field in the helical coil could also be included. Moreover, the separation of variables for the electromagnetic fields introduces slight parasitic variations in z and r, particularly in the limit of small wavenumber, ka ≪ 1.

Nevertheless, the model is sufficiently accurate and allows the performance of a large number of simulations in a few minutes. It is an excellent tool for optimizing the helical coil geometry for proton post-acceleration, bunching, and focusing.

ACKNOWLEDGMENTS

Acknowledgment. This work is supported by the CEA/DAM Laser Plasma Experiments Validation Project and the CEA/DAM Basic Technical and Scientific Studies Project. It is also supported by the National Sciences and Engineering Research Council of Canada (NSERC) (Grant Nos. RGPIN-2023-05459 and ALLRP 556340-20), Compute Canada (Job pve-323-ac), and the Canada Foundation for Innovation (CFI). Access to the HPC resources of IRENE was granted under Allocation No. A0130512899 made by GENCI. We acknowledge financial support by the IdEx University of Bordeaux/Grand Research Program “GPR LIGHT” and the Graduate Program on Light Sciences and Technologies of the University of Bordeaux. The authors are grateful to O. Cessenat for his help and useful discussions.

Category:

Received: Jun. 3, 2024

Accepted: Jul. 23, 2024

Published Online: Jan. 8, 2025

The Author Email: C. L. C. Lacoste (clement.lacoste@inrs.ca)

DOI:10.1063/5.0221820