Terahertz Range Surface-Wave Bragg Resonators with Optimized Ratio between Ohmic and Radiative Losses

Finite grating area at the surface of a metal plate can form a highly selective surface-wave Bragg resonator which can be used for sensing in the THz range, among other applications [1]. For 1D and 2D structures, we describe surface modes confined at a metallic plate. For a given material conductivity and grating dimensions, an optimum corrugation depth can be determined that provides the maximum value of the fundamental mode quality factor due to the interplay between radiative and Ohmic losses. Surface electromagnetic waves confined at the interface of metal and dielectric have been widely studied [2] since they possess a number of unique features making them useful for various applications, including antenna applications Raman scattering, etc. Since the evanescent waves are decelerated, Cherenkov devices based on interaction of electron beams with surface modes of the corrugated structures is of special interest in the short wavelength radiation bands, including the terahertz band [3]. These devices allow to solve the problem of longitudinal and transverse mode selection in oversized electrodynamical systems. In this paper, we consider a periodically corrugated section of a metallic plate as a surface-wave resonator (see Fig.1) in a dielectric media with a permittivity of  . We obtain the equations of wave coupling with Ohmic losses taken into account and derive an integral equation describing the eigenmodes. Based on this equation, we find the optimum values of the corrugation depth providing the maximum Q factor of the fundamental mode dependent on the  . We also discuss the 2D variant of a surface-wave resonator.

Finite grating area at the surface of a metal plate can form a highly selective surface-wave Bragg resonator which can be used for sensing in the THz range, among other applications [1]. For 1D and 2D structures, we describe surface modes confined at a metallic plate. For a given material conductivity and grating dimensions, an optimum corrugation depth can be determined that provides the maximum value of the fundamental mode quality factor due to the interplay between radiative and Ohmic losses.
Surface electromagnetic waves confined at the interface of metal and dielectric have been widely studied [2] since they possess a number of unique features making them useful for various applications, including antenna applications Raman scattering, etc. Since the evanescent waves are decelerated, Cherenkov devices based on interaction of electron beams with surface modes of the corrugated structures is of special interest in the short wavelength radiation bands, including the terahertz band [3]. These devices allow to solve the problem of longitudinal and transverse mode selection in oversized electrodynamical systems.
In this paper, we consider a periodically corrugated section of a metallic plate as a surface-wave resonator (see Fig.1) in a dielectric media with a permittivity of  . We obtain the equations of wave coupling with Ohmic losses taken into account and derive an integral equation describing the eigenmodes. Based on this equation, we find the optimum values of the corrugation depth providing the maximum Q factor of the fundamental mode dependent on the  . We also discuss the 2D variant of a surface-wave resonator. In the case when the depth of sinusoidal corrugation is small in the scale of wavelength and of the structure period d the electromagnetic field of the surface mode existing near the 1D corrugated metal plate can be presented as a sum of two quasi-optical wavebeams with slowly varying amplitudes A  which can be characterized by the magnetic field is the Bragg frequency. Propagation of electromagnetic field in the vicinity of the Bragg resonance can be described by a system of coupled parabolic wave-beam equations which take into account the diffraction in y direc- At the edges of the corrugation we set non-reflection boundary conditions: Here describes the Ohmic losses in the metal corresponding to the Leontovich boundary condition, skin d is the skin depth of the metal.
Equations (1) applied to a system infinite in z direction lead to the existence of a surface wave ~i z g y i t A e       with a dispersion law which in the lossless case can be put down as Dispersion characteristic corresponding to (2) is shown in the lower part of Fig. 2. In the case when the length z l of the structure is finite, a set of longitudinal modes can be found. Seeking modes of (1) at an eigenfrequency of  , and using the Fourier transform, one can obtain an integral equation equivalent to system [5] which possesses an infinite discrete eigenmode spectrum. Frequencies and spatial profiles of eigenmodes can be found numerically. Real parts of the eigenfrequencies of the first several modes are marked with dots close to the dispersion in lower part of Fig.2  surface wave resonator. Making use of the fact that according to the exact solution of (3), the longitudinal field structure of the mode tends to sine shape with increasing parameter 2 z hl  , an asymptotic formulas for the fundamental mode eigenfrequency and Q factor can be obtained in the following form: Total Q factor has a maximum in its  dependence for each z l . This maximum can be found from expressions (4) as 1   . According to (4), in this parameter area the small corrugation approximation is validated for structures of several dozen periods or longer. For instance, for a structure with a length of 100 periods ( 630 z hl  , see dotted line in Fig.5), a corrugation depth of 1/10 wavelength provides an optimal Q factor of about 1000 for copper at normal temperature. In a dielectric with 5   , Q value is much lower while the optimum corrugation depth remains almost the same.
Theory of 1D surface-wave resonators given above is applicable only at moderate (in the sense of Fresnel scales) transverse (x) sizes of the system, as it describes, in fact, a single transverse mode approximation. At larger x-dimensions of the system, various transverse modes with close Q factors would compete with each other. In [5], 2D surface-wave resonators were introduced to provide a means for synchronization of radiation from wide electron beams into the evanescent wave. For a 2D resonator (see Fig.1