Approximate periodically focused solutions to the nonlinear Vlasov-Maxwell equations for intense beam propagation through an alternating-gradient field configuration

This paper considers an intense non-neutral ion beam propagating in the $z$ direction through a periodic-focusing quadrupole or solenoidal field with transverse focusing force, $-\left[{\kappa }_x\right(s)x{\hat{\mathbf{e}}}_x+{\kappa }_y(s\left)y{\hat{\mathbf{e}}}_y\right]$, on the beam ions. Here, $s={\beta }_b\mathrm{ct}$ is the axial coordinate, $({\gamma }_b-1\mathrm{})m_b{c}^2$ is the directed axial kinetic energy of the beam ions, and the (oscillatory) lattice coefficients satisfy ${\kappa }_x(s+S)={\kappa }_x\left(s\right)$ and ${\kappa }_y(s+S)={\kappa }_y\left(s\right)$, where $S=\mathrm{const}$ is the periodicity length of the focusing field. The theoretical model employs the Vlasov-Maxwell equations to describe the nonlinear evolution of the distribution function $f_b({x,y,x}^{\prime }{,y}^{\prime },s)$ and the (normalized) self-field potential $\psi (x,y,s)$ in the transverse laboratory-frame phase space $\left({x,y,x}^{\prime }{,y}^{\prime }\right)$. Here, $\hat{H}({x,y,x}^{\prime },y^{\prime },s)=\left(1/2\right)\left(x^{\prime 2}{+y}^{\prime 2}\right)+\left(1/2\right)\left[{\kappa }_x\right(s)x^2+{\kappa }_y(s\left)y^2\right]+\psi (x,y,s)$ is the (dimensionless) Hamiltonian for particle motion in the applied field plus self-field configurations, where $(x,y)$ and $\left(x^{\prime }{,y}^{\prime }\right)$ are the transverse displacement and velocity components, respectively, and $\psi (x,y,s)$ is the self-field potential. The Hamiltonian is formally assumed to be of order $\epsilon$, a small dimensionless parameter proportional to the characteristic strength of the focusing field as measured by the lattice coefficients ${\kappa }_x\left(s\right)$ and ${\kappa }_y\left(s\right)$. Using a third-order Hamiltonian averaging technique developed by P. J. Channell [Phys. Plasmas 6, 982 (1999)], a canonical transformation is employed that utilizes an expanded generating function that transforms away the rapidly oscillating terms. This leads to a Hamiltonian, $H(\tilde{X},\tilde{Y},{\tilde{X}}^{\prime },{\tilde{Y}}^{\prime },s)=\left(1/2\right)({\tilde{X}}^{\prime 2}+{\tilde{Y}}^{\prime 2})+\left(1/2\right){\kappa }_f({\tilde{X}}^2+{\tilde{Y}}^2)+\psi (\tilde{X},\tilde{Y},s)$, correct to order ${\epsilon }^3$ in the “slow” transformed variables $(\tilde{X},\tilde{Y},{\tilde{X}}^{\prime },{\tilde{Y}}^{\prime })$. Here, the transverse focusing coefficient in the transformed variables satisfies ${\kappa }_f=\mathrm{const}$, and the asymptotic expansion procedure is expected to be valid for a sufficiently small phase advance ($\sigma <\pi /3=60°$, say). Properties of axisymmetric beam equilibrium distribution functions, $F_b^0\left(H^0\right)$, with $\partial /\partial s=0=\partial /\partial \Theta$, are calculated in the transformed variables, and the results are transformed back to the laboratory frame. Corresponding properties of the periodically focused distribution function $f_b({x,y,x}^{\prime }{,y}^{\prime },s)$ are calculated correct to order ${\epsilon }^3$ in the laboratory frame, including statistical averages such as the mean-square beam dimensions, $〈x^2〉\left(s\right)$ and $〈y^2〉\left(s\right)$, the unnormalized transverse beam emittances, ${\epsilon }_x\left(s\right)$ and ${\epsilon }_y\left(s\right)$, the self-field potential, $\psi (x,y,s)$, the number density of beam particles, $n_b(x,y,s)$, and the transverse flow velocity, ${\mathbf{V}}_b(x,y,s)$. As expected, the beam cross section in the laboratory frame is a pulsating ellipse for the case of a periodic-focusing quadrupole field or a pulsating circular cross section for the case of a periodic-focusing solenoidal field.