1 Introduction

Now, it is well-established that plane interface between the media of permittivities with different signs can sustain the surface electromagnetic TM-waves. The research on these waves began with pioneering works [1, 2] and continue till nowadays. The most studied is the sustaining the surface electromagnetic waves at the interface between plasma (or metal) and vacuum [3]. Not so long ago it was discovered that the surface electromagnetic waves of TM- and TE-polarisations can propagate at the interface between vacuum and the so-called left-handed material [4]. This left-handed material possesses both negative permittivity and negative permeability. This weird term “left-handed material” was introduced due to the fact that in such unbounded medium, the three vectors of the electric, magnetic fields, and wave vectors of a plane waves form the left triple. Afterwards, a number of studies of the surface electromagnetic waves on a plane interface between media with various combinations of signs \(\varepsilon\) and \(\mu\) were carried out (see, for example [5,6,7]).

We want to clarify with emphasis a more basic issue the surface waves propagation at the interface of vacuum with a medium with negative permeability. Most likely, such metamaterials are easier to manufacture than double negative metamaterials.

2 Problem statement and results

We want to study the so far the unconsidered basic electrodynamics case of surface electromagnetic waves propagating at the interface between an artificial medium with a vacuum. Let’s consider the simple case of the plane single interface \((x=0)\) between a vacuum \((x<0)\) \(\epsilon _{1}=1, \mu _{1}=1\) and the ideal (lossless) homogeneous and isotropic frequency dependent mu-negative medium (for example, [8]) \((x>0)\) \(\epsilon _{2}=1\), \(\mu _{2}(f)=1-af^{2}/(f^{2}-f_{0}^{2})<0, a=0.56, f_{0}= 4\) GHz. We want to search the possibility of propagation of surface electromagnetic waves of frequency f and wavelength \(\lambda\) which amplitudes decay exponentially inside both media. In what follows we assume the anzats

$$\begin{aligned} A_{j}=A_{0j}\exp (-\kappa _{j}|x|)\exp (i(\beta z -\omega t)), j=1,2 \end{aligned}$$
(1)

where \(\kappa _{1}=\sqrt{\beta ^{2}-\epsilon _{1}\mu _{1}k^{2}},\kappa _{2}=\sqrt{\beta ^{2}-\epsilon _{2}\mu _{2}k^{2}},\beta =2\pi /\lambda ,\omega =2\pi f,k=\omega /c, c -\) speed of light in vacuum. From splitting of Maxwell’s equations, we have two subsystems of equations,one of which describes the wave disturbances (1) of TE-polarization, e.g. with an electric \(\mathbf {E}=\{0;E_{y};0\}\) and magnetic fields \(\mathbf {H}=\{H_{x};0;H_{z}\}\). These components are interconnected by such relationships:

$$\begin{aligned} \frac{d^{2}E_{y}}{dx^{2}}= & {} \kappa _{j}^{2}E_{y} \end{aligned}$$
(2)
$$\begin{aligned} H_{x}= & {} \left( \frac{\beta }{k\mu _{j}}\right) E_{y}; H_{z}=\left( \frac{1}{k\mu _{j}}\right) \frac{dE_{y}}{dx} \end{aligned}$$
(3)

Satisfying the proper boundary conditions give rise to the dispersion equation for the surface electromagnetic wave of TE-polarisation

$$\begin{aligned} \frac{\kappa _{1}}{\mu _{1}}+\frac{\kappa _{2}}{\mu _{2}}=0 \end{aligned}$$
(4)

We will find the solution of the dispersion equation (4) inside frequency interval in which the effective permeability \(\mu _{2}(f)<0.\) It should be noted that there are only the solutions for the dispersion equation for TE-polarization in this frequency interval but not for TM-polarization. There is a dependence on wavelength versus frequency for this wave mode in the Fig. 1. In the Fig. 2 we present the results of calculations of the normalised by speed of light both phase and group velocities versus wave frequency. As we can see, this wave is forward because the directions of the phase velocity and the group velocity coincide. The phase velocity decrease monotonically from c to 0.3c with increasing frequency. In similar manner the group velocity change from 0.35c to 0.05c in the frequency band under consideration. As demonstrated in Fig. 3, the ratio of the penetration depths \(L_{j}=1/\kappa _{j}\) into the mu-negative medium and into vacuum are depending on the frequency differently. The values of these two penetration depths are very close at final frequencies \(f\le 4.7\) GHz of the considered interval, so that \(\mu _{2}(f)\le -1.\)

With decreasing frequency, the penetration depth into the mu-negative medium slightly varies from 0.1 to 0.5 cm, but the penetration depth into vacuum increases up to 10 cm. The amplitudes of wave fields have accordingly similar dependencies versus distance from the boundary at high frequencies ( \(f\le 4.7\) GHz).

Fig. 1
figure 1

The wavelength versus wave frequency for TE surface mode

Full size image
Fig. 2
figure 2

The normalised phase and group velocities versus wave frequency

Full size image
Fig. 3
figure 3

The wave fields penetration depth into the mu-negative medium and into vacuum versus wave frequency

Full size image
Fig. 4
figure 4

The distribution of the amplitude of transverse electric wave field \(E_{y}\) in vacuum \((x<0)\) and metamaterial \((x>0)\) at the 4.672 GHz

Full size image
Fig. 5
figure 5

The distribution of the amplitude of transverse electric wave field \(E_{y}\) in vacuum \((x<0)\) and metamaterial \((x>0)\) at the 4.126 GHz

Full size image

This gives rise symmetric (with respect to the boundary) distribution of the amplitude of wave fields in the metamaterial and in vacuum (for the wave frequency 4.672 GHz see Fig. 4). But at lower frequency (4.126 GHz) distribution of the amplitude of wave fields is highly asymmetric: in vacuum the wave amplitude decreases weakly with distance from the boundary (Fig. 5). This variability will allow you to apply this wave mode in different situations, depending on the goal.

3 Conclusions

In this paper, we have found that the interface between mu-negative media and vacuum (or air) can sustain the surface electromagnetic waves. This eigenmode has TE-polarisation, the phase and group velocities that are unidirectional with absolute values less then speed of light. The robust features of this wave mode inspire its practical usage. Possible applications of this mode are wide: control of the movement of charged particles, signal processing and transmission, surface diagnostics, and many others.