Electrostatic theory of rectangular waveguides filled with anisotropic media

The electrostatic (or, in a better word, quasi-electrostatic) theory of waves propagation in a long, rectangular waveguide having perfect electric conductor walls that filled with an anisotropic medium (here, a medium of nanowire-based hyperbolic metamaterials) is presented. Some data on characteristics of these waves are prepared. The presented results include electrostatic field configurations (modes) that can be supported by such structures and their corresponding cutoff frequencies, group velocities, power flows and storage energies.

are the radius of a nanowire and the separation between the nanowires, respectively. To simplify the analysis of the structure, in this section we reduce the problem to a 2D one (its length in the y direction is infinite) so that ∂/∂y = 0 . Although in practice the dimensions of the structure are finite, the 2D approximation not only simplifies the structure but also sheds insight into the characteristics of the structure. In general case, the cross section of the slab is a rectangular with finite height and width.
We now investigate the behaviors of electrostatic bulk modes of a planar slab of nanowire-based HMMs, based on the field analysis. Let us assume that the waves are traveling in the x direction, and the structure is infinite in that direction as illustrated in panel (a) of Fig. 1. We shall obtain the general expression for dispersion relation of the electrostatic waves. For the present problem, the relative dielectric tensor of the anisotropic medium is where ε ⊥ and ε that are the effective transversal and longitudinal permittivities, respectively, may be expressed by 19 where ε m = ε ∞ − ω 2 p /ω(ω + iγ ) shows the relative permittivity of a nanowire (for example a narrow-gap semiconductor nanowire such as InSb 27,28 ), ε ∞ is the high-frequency bulk permittivity, ω p is the electron plasma angular frequency, γ is the damping constant, f = πr 2 N is the filling factor of nanowires in the xy section of medium and ε d is the relative permittivity of insulator host.
Under the electrostatic approximation, the electric field can be represented by the gradient of an electric potential . The Maxwell's equation ∇ · D = 0 , gives the wave equation for the electrostatic potential of an unbounded medium of nanowire-based HMMs Side view of a planar slab of nanowire-based HMMs. The array of nanowires has areal density N with axes parallel to z-axis. Also, r is the radius of a nanowire and d = 1/ √ N is the separation between the nanowires. The PEC interfaces separating the slab ( −a ≤ z ≤ a ). (b) Isometric view of a rectangular waveguide filled with a medium of nanowire-based HMMs with its appropriate dimensions and PEC boundaries. www.nature.com/scientificreports/ where all the field quantities are assumed to have the harmonic time dependence of the form exp(−iωt) . Note that ω is angular frequency of an electrostatic wave in the system. In deriving the electrostatic mode spectrum of a planar slab of nanowire-based HMMs, we use the following general separated-variable solution for traveling waves in the +x direction Substituting this into Eq. (4), we get Equation (6) shows that a homogeneous bulk electrostatic plane wave corresponding to real k and κ is possible only when ε ⊥ /ε � is negative. Field existing within this waveguide must be characterized by zero tangential components of electric field at the PEC walls. However, the boundary conditions at the two PEC walls can be written as Now, by applying the mentioned boundary conditions we can find the relationship between the constants A and B, as To satisfy Eq. (8) we consider separately two categories: even and odd modes. For an even mode A = 0 and κa = (n + 1/2)π with n = 0, 1, 2, . . . Therefore κa = (n + 1/2)π is the dispersion relation for electrostatic waves with symmetric potential functions. For an odd mode B = 0 and we have κa = nπ that is the dispersion relation for electrostatic waves with anti-symmetric potential functions. The even and odd mode dispersion relations can be combined into a single equation. The result is where this equation corresponds to an even mode for n odd, and to an odd mode for n even. In the special case, for ε ∞ = 1 = ε d and γ = 0 , from Eq. (9), we find If k → 0 , then ω + → 1 − f ω p / √ 2 and ω − → f ω p . If k → ∞ , then ω + → 1 + f ω p / √ 2 and ω − → 0 . This means that this system represents a bulk electrostatic band filter with the frequency bands Note that the mentioned effective medium approximation that we use in the present study is valid when the radius of a nanowire is a lot smaller than the wavelength of the waves under consideration. Also the nanowires must be well separated. In particular, the conductor walls must behave as PEC walls and thus they must have constant potential. Therefore, the angular frequency is assumed to be restricted to the range 0 < ω − /ω p ≤ f with the consequence that we can choose the appropriate frequency of plane electrostatic wave in the microwave regime and appropriate radius of a nanowire in the present HMM medium. For instance, for InSb wires the electron plasma frequency ω p /2π = 4.9 THz 27,28 is approximately 61 µ m in wavelength. If one uses the angular frequency region of ω − ≈ f ω p , the wavelength under consideration is 61/ f µ m. Thus, for r = 5 nm and d = 1 µ m, we find f ≈ 8.8 × 10 −3 and therefore ≫ r, d which indicates the mentioned effective medium approximation is valid in the present study. Furthermore, we find ω − /2π ≤ 43.4GHz and therefore one may conclude the condition that conductor walls must behave as PEC walls is satisfied. Note that for the angular frequency range 0 < ω − /ω p ≤ f , one can find that ε ⊥ is positive and ε is negative, thus the present system for our purpose is a HMM of type I.
The dispersion curves of modes of a planar slab of nanowire-based HMMs with r = 5nm, d = 1 µ m for various values of n, using Eq. (10) are depicted in Fig. 2. An infinite number of bulk electrostatic waves with different values of n can exist in the slab, which depend on the conditions of excitation. It is clear from Fig. 2 that the frequency of an electrostatic mode decreases monotonically with wavenumber throughout the allowed frequency range 0 < ω − ≤ f ω p . Therefore, these electrostatic modes are backward waves. Furthermore, all backward modes have equal cutoff frequency ω − = f ω p . Hence, there is no frequency band where only a single mode propagates. Note that using the simple formula v g = dω/dk the group velocity of bulk modes of the system can be simply computed in an analytical way.

Spectroscopy of electrostatic modes of a rectangular waveguides filled with a medium of nanowire-based HMMs
Now, consider a long, rectangular waveguide of nanowire-based HMM having PEC walls of dimensions w and h in the y and z directions respectively. Geometry of the problem is presented in panel (b) of Fig. 1. It is our purpose to determine the various electrostatic modes that can exist inside this rectangular electrostatic waveguide.
In particular, the yz cross section electric field and their corresponding cross sections of equipotential surfaces for the first nine electrostatic modes of a square waveguide are plotted in Fig. 4. Field configurations of a rectangular waveguide are very similar with Fig. 4 (not shown here). By comparing the results in Fig. 4 of 8 with Fig. 4 in the present work, one can conclude that the electric field pattern of electrostatic mode (m, n) is similar with electric field pattern of TM mn modes for a hollow waveguide. Furthermore, in Fig. 5 we show the yz cross section of bulk charge density of a square waveguide corresponding to the Fig. 4. The white color corresponds to positive charge and black color to negative charge. The electrostatic waves that are created and propagating inside the present waveguide have power associated with them. To find the power flowing down the waveguide, it is first necessary to find the cycle-averaged of power density directed along the axis of the waveguide. The power flowing along the waveguide can then be obtained by integrating the axial directed power density over the cross section of the waveguide.
For the waveguide geometry of Fig. 1, the x-directed power density can be written as 29 in the complex-number representation, where * denotes complex conjugation, and Re denotes taking the real part. Use of Eq. (18) into Eq. (25), allows the x-directed power density of Eq. (25) for the electrostatic modes can be written as S x = − ε 0 ωk 2 ε ⊥ |A mn | 2 sin 2 mπy w sin 2 nπz h . www.nature.com/scientificreports/ Note that ε ⊥ and ε are real since we neglected damping in the present study. The associated power is obtained by integrating Eq. (26) over the cross section A = hw of the guide, as Also, the cycle-averaged of energy density distribution associated with the waves can be written as 29 After substitution Eq. (18) into Eq. (28), we obtain The associated storage energy is obtained by integrating Eq. (29) over the cross section A of the waveguide, as As mentioned before, using the simple formula v g = dω/dk and Eq. (22), the group velocity of electrostatic modes of the system can be simply computed. However, in the absence of the damping effects, the group velocity of the waves is also equal with the ratio of the power flow to the storage energy, such as Since all investigations in this work is based on electrostatic theory, the equality of above formula with the result obtained using v g = dω/dk is a manifestation of self-consistency and general validity of presented results in the electrostatic theory.

Conclusion
We have presented the theory of electrostatic bulk waves propagation in a long, rectangular PEC waveguide containing an anisotropic medium (here, a medium of nanowire-based HMMs). We have derived general expression for dispersion equation of the waves and then presented the plots the electrostatic field distributions in such electrostatic waveguides. The results show that the electric field pattern of electrostatic modes are similar with electric field pattern of TM mn modes in the rectangular hollow waveguides 8 . Also, we have found that all modes with m = n have equal cutoff frequencies and mode (m = 3, n = 1) is the dominant mode for electrostatic modes (m, n) (with n = 1, 2, 3 and m = 1, 2, 3 ) in the electrostatic waveguides. However, in general the dominate mode of the system cannot be found. Finally, we have verified the obtained results by showing that group velocity of the waves is the same as energy velocity (i.e, the ratio of the power flow to the storage energy). Because of the possibility of electrostatic waves propagation in a rectangular waveguides filled with anisotropic media, they may be used in the development of new waveguides using guided electrostatic waves.