In this paper, we demonstrate that spoof surface magnon polaritons (SSMPs) can propagate along a corrugated perfect magnetic conductor (PMC) surface. From duality theorem, the existence of surface electromagnetic modes on corrugated PMC surfaces are manifest to be transverse electric (TE) mode compared with the transverse magnetic (TM) mode of spoof surface plasmon plaritons (SSPPs) excited on corrugated perfect electric conductor surfaces. Theoretical deduction through modal expansion method and simulation results clearly verify that SSMPs share the same dispersion relationship with the SSPPs. It is worth noting that this metamaterial will have more similar properties and potential applications as the SSPPs in large number of areas.
© 2014 Optical Society of America
Surface plasmon polaritons (SPPs), which are localized surface electromagnetic (EM) modes that propagate along the interface between a metal and a dielectric in optical frequencies and decay exponentially in the transverse direction [1,2], have attracted great attentions owing to their huge potential applications in large number of areas [3–5]. Thereafter, to produce highly confined surface EM waves at lower frequencies, a new plasmonic metamaterial called spoof surface plasmon polaritons (SSPPs) are proposed by Pendry et al.  in 2004 and has been verified experimentally in the microwave regimes by Hibbins et al.  in 2005. Owing to the advantages of this metamaterial, the cut-off frequency of the SSPPs can be adjusted at will by decorating the metal surfaces with one-dimensional (1D) arrays of subwavelength grooves , or 2D arrays of subwavelength holes [6–8], or 3D perfect electric conductor (PEC) wire with periodical array of radial grooves . And due to their ability to confine EM waves in subwavelength scale with high intensity, both SPPs and SSPPs have found lots of applications from optical to microwave frequency bands [10–19].
However, a number of works have shown that magnetic materials can excite surface magnon polaritons (SMPs) in recent years, i.e., strong coupling of photons to quantized magnetization waves due to collective spins . Ruppin showed that surface polariton can exist for both s- and p-polarized waves in left-handed metamaterials . Excitation of SMPs in gyromagnetic materials and anisotropic antiferromagnetic crystals have been discussed by Hartstein et al.  and Arakelian et al. , respectively. The presence of magnon polaritons was demonstrated in a YIG slab using the attenuated total reflectance technique . The dispersion relations for magnetic polaritons have been obtained for anisotropic systems of ordered ferromagnetic slabs  and uniaxial antiferromagnets [26,27]. It has been shown that s-polarized waves can propagate at the interface of media with negative refractive index and negative dielectric constant of low-dimensional nanowaveguides [28,29]. Then the underlying theoretical expressions of the tangential wave vectors and field distributions of s-and p-polarized waves at an interface of two media with arbitrary materials are proposed by Raymond Ooi et al. , making SMPs get more attentions nowdays. However, up to now, to our knowledge, the form and manner of existence and implementation of SMPs in the terahertz and microwave regimes are not realized.
So, borrowing the idea of the existence of the SSPPs, in this paper, we propose a corrugated perfect magnetic conductor (PMC) surface to support spoof surface magnon polaritons (SSMPs). In duality, the existence of surface EM modes on corrugated PMC surfaces are obvious to be TE mode compared with the TM mode of the SSPPs excited on corrugated PEC surfaces. Theoretical and simulation results clearly verify that the SSMPs share the same dispersion relationship with the SSPPs and we also show that, as long as the size and spacing of the grooves are much smaller than the wavelength, a perforated PMC surface behaves as an effective medium. It is also worth noting that this metamaterial has more similar properties and applications as the SSPPs in large number of areas.
2. Theoretical analysis
2.1. Modal expansion method
First of all, in Fig. 1(a), a corrugated PMC surface decorated by a periodic array of grooves with depth h, width a and period d is considered. According to the duality theorem in Maxwell equations, we can find the eigenmodes of the SSMPs supported by this surface with the modal expansion method . We are interested in s-polarized surface waves propagating along the x direction with the form of E = ŷEy and H = x̂Hx + ẑHz. In Region I above the surface (z > 0), the electric field component Ey, which is nonradiative and vanishes as z → ∞, can be written as
The magnetic field components Hx and Hz can be directly obtained from Ey through Maxwell’s equations thatEq. (5) over this interval, we can obtain
The magnetic field component Hx at the interface must be continuous over the whole period. Thus, for |x| ≤ a/2 we haveEq. (10) into Eq. (5), and neglecting the diffraction effects, we can obtain (9) in Ref. . Considering that no material parameters are involved in Eq. (11) and only the boundary condition is changed from PEC to PMC, the duality theorem holds for the case from SSPPs to SSMPs.
2.2. Effective medium approximation
Similarly, from the perspective of the effective medium , the same dispersion relation could be obtained if we replaced the array of grooves by a single homogeneous but anisotropic layer of thickness h on top of the PMC surface shown in Fig. 1(b). The homogeneous layer would have the following parameters:
As wave propagates in the grooves along the y and z directions with the velocity of light that
For the s-polarized plane wave impinging on the surface of a homogeneous layer of thickness h with μ and ε given by Eqs. (12) and (14), the specular reflection coefficient R can be written as Eq. (15) after some straightforward algebras,
This effective medium implies a bound surface state when there is a divergence in the reflection coefficient of the surface for large values to the case kx > k0 and looking at the zeros of the denominator of R, we can obtain the dispersion relation of the surface modes
3. Results and discussions
Note that the Eq. (16) coincides with Eq. (11) in the limit kxa ≪ 1. In Fig. 2, we plot the analytical dispersion relation with Eq. (16) and numerical results for the particular case a/d = 0.4, h/d = 0.8. Though the cutoff frequency of the analytical value is slightly higher than the numerical ones due to the reason that the diffraction effects are neglected, it is worth commenting that the dispersion relations between the SSMPs and the SSPPs are identical, and the cutoff frequency ω also approaches πc0/(2h) according to the dispersion relations and duality of Maxwell’s equations between the PEC and PMC cases. All numerical simulation results in this paper have been performed with the finite element method (FEM) . The corresponding eigenvalue problem is posed in a single unit cell where Bloch boundary conditions are used. The open space is mimicked with PEC or PMC boundary conditions, and the Bloch boundary conditions are employed to model the periodic arrays in the simulations shown in Fig. 3(a) and 3(b). The surface electric and magnetic field vectors on the xoz plane of the PEC and PMC cases are illustrated in Fig. 3(c) and 3(d) with exactly identical field distributions, which further verify the essential characteristics between the SSPPs and SSMPs.
In conclusion, we have shown that the existence of the SSMPs in corrugated PMC surfaces, and the dispersion relation is obtained through the duality of Maxwell’s equations combined with modal expansion method, and an effective medium analytical approach is used to depict the physical insight of the new metamaterial. Thanks to this new concept, a whole brunch of surface modes already known in optical region can be transformed to terahertz and microwave frequency bands. This kind of metamaterial may have a very significant impact on both fundamental and applied researches.
This work was supported by the National Natural Science Foundation of China for Young Scholars under Grant No. 61102033, the Fundamental Research Funds for the Central Universities under Grant NJ20140009, the Aeronautical Science Foundation of China under grant No. 20128052063 and the priority academic program development of Jiangsu Higher Education Institutions.
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