1. Introduction

We consider surface waves of wave number $k_z$ and frequency $\omega$, generated by a plasma beam travelling in a duct interfaced with vacuum by using the Vlasov equation. Surface waves are given rise to by satisfying the kinetic and electromagnetic boundary conditions on the interface between the plasma and the vacuum surrounding it. The electric and magnetic fields in the plasma are connected with the vacuum side fields by the appropriate connection formula, and the latter can be deciphered from the basic equations themselves that we adopt. The connection formula can be easily worked out mathematically if the density gradient across the plasma and the other side is very steep. We talk about ‘a sharp interface’ if the density gradient is theoretically infinite. In this case, the connection formula can be obtained by ‘infinitesimal integration’ across the interface, which is the operation performed on a certain relevant equation in the manner $\int _{-\epsilon }^\epsilon (\cdots )\,{{\rm d} x}$. If the quantity $(\cdots )$ is a perfect differential, this operation yields a non-vanishing surface term that contributes to the connection formula. Usually, the surface term is the surface charge or surface current, In this way, the well-known electromagnetic and dynamic boundary conditions on the boundary can be derived (Lee & Cho Reference Lee and Cho1997). In a gross picture, the surface wave and the vacuum side wave are two different manifestations of ‘the same wave’ given rise to in an extreme inhomogeneous plasma.

A cold drifting plasma has a characteristic boundary condition (Lee & Cho Reference Lee and Cho1997,  Reference Lee and Cho1999), which can be expressed in the form

(1.1)\begin{equation} [B_y]=\frac{u}{c}[E_x] \end{equation}

or equivalently,

(1.2)\begin{equation} [D_x]=\frac{ck_z}{\omega}[B_y] \end{equation}

where $[\cdots ]$ signifies the jump across the interface, $D_x$ is the normal to the interface component of ${\boldsymbol D}$, the electric displacement, $u$ is the drift velocity in the $z$-direction, $B_y$ is the tangential component of the magnetic field and $c$ is the speed of light, and the other symbols have the usual meaning. The casual use of $[D_x]=0$ or $[B_y]=0$ in a drifting plasma leads to erroneous results as, discussed in earlier works (Lee & Cho Reference Lee and Cho1997; Lee Reference Lee2005). The physical origin of the boundary relation in (1.1) is due to the surface current formed in a cold drifting plasma, as is evident in the equation.

In a bounded Vlasov plasma, the kinematic boundary condition that is usually referred to as the specular reflection condition is assumed to be satisfied on a sharp boundary (Landau Reference Landau1946). The kinetic theory of surface waves in semi-infinite plasmas is well known (Barr & Boyd Reference Barr and Boyd1972; Alexandrov, Bogdankevich & Rukhadze Reference Alexandrov, Bogdankevich and Rukhadze1984). The kinetic dispersion relation of a surface wave in a slab plasma was worked out earlier (Lee & Lim Reference Lee and Lim2007). In this work, we investigate surface waves of a moving Vlasov plasma in a duct. We consider an infinite duct formed by the intersections of four planes: $x=0,a$ and $y=0,b$, with $-\infty < z<\infty$. Thus the specular reflection condition on the $x=a$ plane, for example, requires for the distribution function $f({\boldsymbol r},{\boldsymbol v},t)$ to satisfy $f(a,y,z, v_x,v_y,v_z,t)=f(a,y,z, -v_x,v_y,v_z,t)$ or, on the $y=b$ plane, $f(x,b,z, v_x,v_y,v_z,t)=f(x,b,z, v_x,-v_y,v_z,t)$, with similar equations for $x=0$ and $y=0$. In our duct-bounded plasma, the kinematic conditions on the four planes are satisfied by introducing an extended electric field in the fashion

(1.3a,b)\begin{gather} E_x({-}x,y,z)={-}E_x(x,y,z),\quad E_x(2a-x,y,z)={-}E_x(x,y,z), \end{gather}

(1.4a,b)\begin{gather} E_y(x,-y,z)={-}E_y(x,y,z),\quad E_y(x,2b-y,z)={-}E_y(x,y,z). \end{gather}

This scheme is workable if $f_0({\boldsymbol v})$, the zero-order distribution function, is invariant with respect to the reflections $v_x\rightarrow -v_x$ and $v_y\rightarrow -v_y$, and clearly this reflectional property is satisfied by the moving Maxwellian, to be introduced later.

The function $E_x(x)$, as defined in (1.3a,b), is a periodic function of a piecewise continuous function of period $'a'$ extending over the range $-\infty < x<\infty$ with discontinuity at $x=\pm 2na$ with a jump of $A_1$ (say) and with discontinuity at $x=\pm (2n-1)a$ with a jump of $A_2$ (say), where $n$ is an integer. The profile of the piecewise function $E_x(x)$ is plotted in Lee (Reference Lee2019). The algebra involved in carrying out the Fourier transform of the piecewise discontinuous functions with the aforementioned discontinuous jumps is quite taxing (Lee & Lim Reference Lee and Lim2007). However, it turns out that, after all the algebraic hard work, the discontinuities that are present in the extended field components $E_x(x,y)$ and $E_y(x,y)$ in (1.3a,b) and (1.4a,b) at the locations $x=\pm 2na$ and $x=\pm (2n-1)a$ and $y=\pm 2nb$ and $y=\pm (2n-1)b$ are mathematically (as well as physically) tantamount to placing fictitious surface charges at the corresponding jump locations in the form

(1.5)\begin{gather} S(x,y,z,t)= A_1\sum_{n=0,1,2,\ldots} \delta(x\pm2na)+A_2\sum_{n=1,2,\ldots} \delta(x\pm(2n-1)a)\nonumber\\ +B_1\sum_{n=0,1,2,\ldots} \delta(y\pm2nb)+B_2\sum_{n=1,2,\ldots} \delta(y\pm(2n-1)b). \end{gather}

This is the vital point of improvement in this work as compared with the earlier work (Lee & Lim Reference Lee and Lim2007). The surface charges are associated with the surface currents by satisfying the charge conservation equation

(1.6)\begin{equation} \frac{\partial S}{\partial t}+\hat{{\boldsymbol z}}\boldsymbol{\cdot} {\boldsymbol J}_s=0. \end{equation}

Therefore, we can assume the presence of the fictitious surface currents

(1.7)\begin{equation} {\boldsymbol J}_s({\boldsymbol r},\omega)=\hat{{\boldsymbol z}}\,{\rm i}\omega S(x,y,z,\omega). \end{equation}

The surface charges in (1.5) and the surface currents in (1.7) should be included in the Maxwell equations for our duct plasma wave analysis.

The basic equations are the linearized Vlasov equation and the Maxwell equations for electrons. Ions are assumed to be stationary and only form the neutralizing background

(1.8)\begin{gather} \frac{\partial}{\partial t} f({\boldsymbol r}, {\boldsymbol v},t)+{\boldsymbol v}\boldsymbol{\cdot}\frac{\partial f}{\partial {\boldsymbol r}}- \frac{e}{m}\left({\boldsymbol E}({\boldsymbol r},t)+\frac{1}{c}{\boldsymbol v}\times{\boldsymbol B}\right)\boldsymbol{\cdot} \frac{\partial f_0(v)}{\partial{\boldsymbol v}}=0, \end{gather}

(1.9)\begin{gather} \boldsymbol{\nabla}\times{\boldsymbol E}={-}\frac{1}{c}\frac{\partial{\boldsymbol B}}{\partial t}, \end{gather}

(1.10)\begin{gather} \boldsymbol{\nabla}\times{\boldsymbol B}=\frac{4{\rm \pi}}{c}{\boldsymbol J}+\frac{1}{c}\frac{\partial{\boldsymbol E}}{\partial t}+\hat{{\boldsymbol z}}J_s, \end{gather}

where

(1.11)\begin{gather} J_s= A_1'\sum_{n=0,1,2,\ldots} \delta(x\pm2na)+A_2'\sum_{n=1,2,\ldots} \delta(x\pm(2n-1)a)\nonumber\\ +B_1'\sum_{n=0,1,2,\ldots} \delta(y\pm2nb)+B_2'\sum_{n=1,2,\ldots} \delta(y\pm(2n-1)b), \end{gather}

(1.12)\begin{gather} {\boldsymbol J}({\boldsymbol r},t)={-}e\int {\boldsymbol v} f({\boldsymbol r}, {\boldsymbol v},t)\,{\rm d}^3v, \end{gather}

(1.13)\begin{gather} \boldsymbol{\nabla}\boldsymbol{\cdot}{\boldsymbol E}={-}4{\rm \pi} e\int f \,{\rm d}^3v+S(x,y,z), \end{gather}

(1.14)\begin{gather} \boldsymbol{\nabla}\boldsymbol{\cdot} {\boldsymbol B}=0. \end{gather}

It should be noted that the last term in (1.8) involving the magnetic field vanishes if $f_0$ is isotropic, but in a beam plasma, that term contributes to complexity. The zero-order distribution function is a moving Maxwellian

(1.15)\begin{equation} f_0({\boldsymbol v})=\left(\frac{m}{2{\rm \pi} T}\right)^{3/2}\exp\left[-\frac{m}{2T}({\boldsymbol v}-{\boldsymbol u})^2\right]. \end{equation}

We Fourier transform the above equations by performing $\int _{-\infty }^\infty \,{\rm d}^3r \exp ({{\rm i}{\boldsymbol k}\boldsymbol {\cdot }{\boldsymbol r}})(\cdots )$ and by assuming $\partial /\partial t={\rm i}\omega$. Then, the wave has a phasor $\exp ({{\rm i}k_zz-{\rm i}\omega t})$. The $(k_z,\omega )$ dependency in the Fourier amplitudes will be suppressed. Equation (1.8) is Fourier transformed to give

(1.16)\begin{equation} f({\boldsymbol k},{\boldsymbol v},\omega)=\frac{e}{m}\frac{{\rm i}}{\omega}\frac{E_j}{\omega-{\boldsymbol k}\boldsymbol{\cdot}{\boldsymbol v}} \left[(\omega-{\boldsymbol k}\boldsymbol{\cdot}{\boldsymbol v})\delta_{sj}+k_sv_j\right]\frac{\partial f_0}{\partial v_s},\end{equation}

where the repeated indexes are summed over. Here, and in the following, the Fourier-transformed variables are expressed by the argument; for example, $f(t)$ and $f(\omega )$ have different dimensions. If $f_0$ is isotropic, ${\partial f_0}/{\partial v_s}\sim f_0v_s$, and (1.16) reduces to the expression for the isotropic case.

Eliminating ${\boldsymbol B}$ between (1.9) and (1.10) gives

(1.17)\begin{equation} {\boldsymbol k}\times({\boldsymbol k}\times{\boldsymbol E})+\frac{\omega^2}{c^2}\varepsilon_{ij}\boldsymbol{\cdot}{\boldsymbol E}({\boldsymbol k},\omega)=\hat{{\boldsymbol z}}J_s({\boldsymbol k},\omega), \end{equation}

where $\varepsilon _{ij}$ is the dielectric tensor in a moving medium

(1.18)\begin{equation} \varepsilon_{ij}=\left(1-\frac{\omega_p^2}{\omega^2}\right) \delta_{ij}+\frac{\omega_p^2}{\omega^2}\int \,{\rm d}^3v\frac{v_iv_jk_s}{\omega-{\boldsymbol k}\boldsymbol{\cdot}{\boldsymbol v}} \frac{\partial f_0}{\partial v_s}, \end{equation}

with $\omega _p$ being the plasma frequency. In (1.17), $J_s({\boldsymbol k},\omega )$, the Fourier transform of the surface currents in (1.11), is

(1.19)\begin{gather} J_s({\boldsymbol k},\omega)=\delta(k_y)[A_1\varSigma_0\exp({\pm {\rm i}2nak_x})+A_2\varSigma_1 \exp({\pm {\rm i}(2n-1)ak_x})]\nonumber\\ +\delta(k_x)[B_1\varSigma_0\exp({\pm {\rm i}2nbk_y})+B_2\varSigma_1\exp({\pm {\rm i}(2n-1)bk_y})], \end{gather}

where the $A$ and $B$ values may be functions of $k_z$, the double signs are summed over and the notations $\varSigma _0$ and $\varSigma _1$ are the summations in (1.11).

The foregoing formulation can be applied for simpler boundaries: in a semi-infinite plasma, we can suppose a single surface charge sheet $S(x)=A\delta (x)$, and in a slab geometry $0< x< a$, the surface charges are assumed in the form of (1.5) with $B_1=B_2=0$. The dispersion relation of surface waves in a slab plasma was worked out earlier without introducing the fictitious surface charges by directly Fourier transforming the extended electric field in (1.3a,b) (Lee & Lim Reference Lee and Lim2007).