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The possibilities of the slow surface electromagnetic waves propagation along the flat boundary of a metasurface with a dissipative dielectric are studied. The metasurface is a thin flat slab of metamaterial with simultaneously negative permittivity and permeability with "amplification". All media were assumed to be isotropic. Dispersion dependences are obtained for the eigenmodes of such a waveguide structure. The possibility of full compensation of the energy losses of surface waves by the appropriate choice of the “gains” values is demonstrated.

The paper presents the results of the study of properties of fast surface electromagnetic waves that propagate along the flat interface between the active metamaterial and air (or vacuum). The case of homogeneous and isotropic metamaterial is considered. The dispersion properties, the wave spatial attenuation, the phase and group velocities, as well as the spatial distribution of the electromagnetic field of the eigen TE and TM modes of such a waveguide structure are studied in the frequency range where the metamaterial has a simultaneously negative permittivity and permeability. It is shown that fast surface electromagnetic waves can exist in this waveguide structure and their properties are studied. It is shown that the phase speed of TM mode is several times higher than the speed of light in vacuum, while the phase speed of TE mode is slightly higher than the speed of light in vacuum. The TM mode is a direct wave in which the phase and group velocities have the same direction. It is obtained that the group velocity of the TM mode varies from zero to the about half of speed of light in vacuum, and reaches a minimum at a certain value of wave frequency, which depends on the characteristics of the metamaterial. It is shown that the penetration depth of the TM mode into the metamaterial is much smaller than into the vacuum. The TE mode is a backward wave with opposite directed phase and group velocities. The absolute value of the group velocity of the TE mode is about six times less than the speed of light in vacuum. In contrast to the TM mode the penetration depth of the TE mode into the metamaterial is much greater than in vacuum. The obtained properties of the fast surface electromagnetic waves can be used for modeling and design of modern generation and amplification devices containing metamaterials.

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The slow surface electromagnetic waves that propagate along the planar waveguide structure that contains two identical isotropic left-handed material with the vacuum (or air) gap has been considered. The possibility of propagation of slow surface electromagnetic waves of TM and TE polarizations in the frequency range at which the dielectric permittivity and magnetic permeability are negative are shown. By selecting a frequency, one, two or three modes can be excited simultaneously. The group and phase velocities of the TE polarization waves are directed in opposite directions. The frequency range of the existence of the TE mode with an antisymmetric field distribution lies above the frequency range of the existence of the TE mode with a symmetric field distribution. The TM wave, depending on the wavelength, can be either forward or backward, or have zero group velocity. The wave of TM-polarization has a slightly varying group velocity in a sufficiently wide frequency range. A rich set of different properties of these waves make them promising in applications.