We introduce the technique of perturbation of boundary condition to the problem of terahertz two-wire metallic waveguides with different radii. Based on the quasi-TEM analytical mode fields derived by use of Möbius transformation, a concise expression for the complex effective index is obtained analytically. The expression is in good agreement with the simulation result. Further, the dispersion and attenuation are obtained from the expression. In addition, we find a zero value point of the group velocity dispersion around 1.268 THz. The results show that the technique of perturbation of boundary condition is helpful in the analysis and design of terahertz metal waveguide.
© 2015 Optical Society of America
Terahertz (THz) waves, typically defined as electromagnetic radiation in the frequency range of 0.1–10 THz (or from 30 μm to 3 mm in wavelength), has attracted a lot of interest owing to its potential applications in spectroscopy, imaging, communications, sensing, security and defense [1–5]. In particular, THz waveguide, one important branch of terahertz technology, has become an active research area. Different types of THz waveguides have been proposed including highly-doped silicon plasmonic waveguides [6,7], porous dielectric waveguides [8,9], and metallic waveguides [10–36]. In 2004, Wang and Mittleman reported that a simple single metal wire can effectively guide THz wave . Since then, many interesting works on the metal wire waveguides have been carried out [11–26]. The THz waveguide effect of the single metal wire is referred to as azimuthally polarized surface plasmons , cylindrical surface plasmons [12,13], or Sommerfeld wave . Surface plasmons confine to and travel along flat metal-dielectric interfaces have been known for decades , which can be excited by periodic structures like metal gratings . Unlike surface plasmons at flat interface, azimuthally polarized surface plasmons have radially polarized electric fields and are difficult to be coupled from the commonly used linearly polarized THz sources by direct end-fire input coupling . Thus, efforts have been made to enhance the coupling efficiency, for example by use of radially symmetric antennas , and by a segmented half-wave-plate mode converter with a polarization-controlled beam .
The two-wire metallic waveguides with identical radii combines low loss and efficient coupling properties [27–29]. The approximately linearly polarized mode supported by this type of waveguides is very similar to the field emitted from the commonly used linearly polarized source, resulting in efficient coupling of the electromagnetic energy from the THz source into the mode . Two-wire metallic waveguide have been used to connect two optoelectronic chips with prototype 250 GHz bandwidth , and to enhance the throughput by directly generating a THz field within the waveguide . Also a tapered dual elliptical plasmon waveguide  has been suggested as a high efficient coupler from an approximate plate waveguide to a two-wire waveguide for THz waves. Low loss foams and porous dielectric fibers are proposed to support the two-wire waveguides [33,34].
In order to analyze the dispersion and attenuation of the THz waveguide, complex effective index neff, whose definition can be found in , needs to be obtained. The real part Re(neff) of the complex effective index is directly related to the dispersion and the imaginary part Im(neff) to the attenuation. For two-wire metallic waveguide, even with identical radii, it is a big challenge to obtain neff. However, owing to the fact that the absolute values of the relative permittivities of metals in the THz region are large, the two-wire metallic surface plasmon has quasi-TEM mode field. The field of the surface plasmon outside the metals can be well approximated by the TEM mode field just like the metals are perfect conductors, which is a great simplification by involving only the 2D Laplace equation . Based on the mode field distribution, one can approximately obtain the attenuation, by use of energy conservation [28,35]. The method is direct and has intuitive appeal, but fails to yield the value of Re(neff), thus unable to estimate the dispersion analytically. Therefore, an explicit expression for the complex effective index is welcome.
In this paper, we are able to solve the above problem by use of the technique called perturbation of boundary conditions, for not only the THz two-wire metallic waveguides with identical radii, but also their general form: THz two-wire metallic waveguides with different radii. The paper is organized as follows. In Section 2, we shall introduce the perturbation of boundary conditions and provide an integral solution of the neff for THz two-wire metallic waveguide with different radii. The whole waveguide is embedded in one dielectric cladding. In Section 3, we shall further transform the integral solution to a simple approximate analytical one for the case of THz two-wire metallic waveguides. It comes from the fact that the mode fields of the waveguide can be approximated by the TEM mode fields derived by the Möbius transformation. In Section 4, the analytical results (both mode fields and complex effective indices neff) for the waveguides with or without a dielectric cladding are compared with the simulation ones. In Section 5, we calculate the group velocity dispersion (GVD) and the attenuation. In Section 6, we discuss the validity range of our theoretical model and then we conclude this paper in Section 7. In this paper, we consider the single mode propagation of monochromatic harmonic waves. The time factor is e-iωt and can be omitted by replacing ∂/∂t by –iω, and the gradient along z axis ∂/∂z can be replaced by ikz, where parameter i is the imaginary unit, ω is the angular frequency, and kz = k0neff is the complex propagation wavenumber. The wavenumber in the free space k0 = ω/c = 2π/λ, where c is the speed of light and λ is the wavelength in the free space. For simplicity, we only discuss the nonmagnetic case, namely the relative permeabilities of the metal and the cladding are always 1.
2. Perturbation of boundary conditions
We now introduce the perturbation of boundary conditions. The method of perturbation of boundary conditions  is capable of, at least in principle, obtaining answers to any desired degree of accuracy, although we apply it only to the lowest order. The basic idea is as follows: First, one consider the (unperturbed) scalar potential ψ0 for case of perfectly conducting wall. The solution ψ0 is usually solvable. Second, one should then apply the perturbed boundary condition by substituting the scalar potential ψ and ∂ψ/∂n at the real metal wall with the initial unperturbed solutions ψ0 and ∂ψ0/∂n. Finally, one could derive the complex effective index neff. For the TEM mode in the two-wire metallic waveguide, we set a modified scalar potential Φh to be ψ and set the scalar potential Φ0 to be ψ0.
Let us consider the problem of THz two-wire metallic waveguiding. In order to make it more general, we assume that two wires with different radii are embedded in a dielectric cladding, as shown in Fig. 1. The THz wave propagates along the z axis, whose unit vector is z. The vector n and τ are the unit normal from the metal and the unit tangent along the curve C in the cross section, respectively. The vectors z, n, and τ satisfy the right-handed rule, namely z = n × τ. The permittivity of the metal wires εm is in the second quadrant for the time factor e-iωt, that is Re(εm)<0, Im(εm)>0. εd is the relative permittivity of the cladding and the permeability μd is 1. Then the index of the cladding nd satisfies = εd. The radii of the wires, and the distance between the two wires are much larger than the skin depth. They are not smaller than 10 μm in our case. The above selection of the parameters is to guarantee the validity of the impedance boundary condition (see Appendix A).
The tangential electric field, E|| = Eτ|Cτ + Ez|Cz, on the metal wall is exactly zero when the metal is perfect conductor. The subscript z denotes the longitudinal part of the field, and τ denotes the tangential part of the field on the curve C in the cross section. The subscript C denotes the field on the curve. However, for real metal, E|| is not zero, although it is still much smaller than the normal one on the wall just outside the metal. The small tangential electric field E|| is the key to the problem of the two-wire metallic waveguiding, and is difficult to be carried out by directly solving the Helmholtz equation. However, for the metal with very large absolute permittivity |εm|, there is a simple approximate relation between the tangential electric field E|| and the tangential magnetic field, H|| = Hτ|Cτ + Hz|Cz, on the metal wall (see Appendix A):Equation (1) is usually referred to as the impedance boundary condition (IBC) [39,40]. However, we choose permittivity εm rather than conductivity σ to describe the metal for convenience. It is more convenient to do that, because εm of the metal can be estimated from the fitted Drude model in the THz region . With the right-handed rule, z = n × τ, one can easily get:Eqs. (2) and (3), although |Ez|C is already quite small compared with |En|C. That is |En|C>>|Ez|C >>|Eτ|C.
For single mode propagation, in the dielectric cladding, the transverse components of the electric and magnetic fields can be determined by their longitudinal parts :Eq. (4). With Eq. (3), one can obtain a relation among ∂Ez/∂τ, ∂Hz/∂n, and Hz on C:Eqs. (4) and (5), the gradient of Ez can be expressed asEq. (8), one can find that and where Φ0 is the scalar potential and satisfies the 2D scalar Laplace equation:
For the real metal walls with huge |εm|, Ez and are not zero, and the 2D gradient of Ez [Eq. (8)] is quite unlike . However the absolute value of is very small compared with that of neffEt and -z × (Z0Ht). That is to say, although the distributions of Et and -z × (Z0Ht) are very similar to , they are not exactly the same. In fact, there is a relation between Et and Ht in the dielectric cladding (see Appendix B):Eq. (9). The first term in left hand side of Eq. (11) is zero. Then by substituting with Eq. (8) and use of Eq. (10), We can getEq. (2), then Eq. (12) becomesEq. (14), Laplace equation of Φ0 [Eq. (9)], and Green’s first identity [Eq. (11)], Eq. (13) can be reduced to
Since the electric and magnetic fields of the quasi-TEM mode of the two-wire real metal waveguide are very similar to that of the TEM mode of the two-wire perfect metal waveguide , we can apply the method of perturbation of boundary condition to the lowest order. That is to say, the modified scalar potential Φh can be well approximated by the unperturbed field Φ0. Φh can be expressed asEq. (15) becomesEq. (17) contains only the field of the TEM mode, which concerns only the Laplace equation, simplifies the problem dramatically.
3. Analytical expressions of the effective indices
We now focus on the potential Φ0 of the TEM mode. The conformal mapping as an important mathematical method is very useful for solving these problems [28,36]. By choosing an appropriate mapping, one can transform the inconvenient geometry into a much more convenient one. As one of the conformal mapping, the Möbius transformation  can directly maps the cross section of the two-wire waveguide to a simpler geometry of two concentric circles, similar to the cross section of a coaxial cable, as shown in Fig. 2.
The complex expression is as follows:
The scalar potential Φ0 in the gap region between the two concentric circles is given byEq. (21) from Eq. (18), one can get the potential Φ0 in the cross section of the two-wire waveguide, and thus the transverse electric field can be derived from the gradient of the potential:Eqs. (21) and (22), and doing the line integral of the right part of Eq. (17) in the w space, the analytical expression of neff can be given (see Appendix C) by
The first product factor in the right hand side of Eq. (23) indicates the contribution of the metal, cladding material and the wavelength. The second factor indicates that neff-nd is in inverse proportion to the geometry size. It means that smaller geometry size leads to higher loss, and vice versa. Thus, one needs to balance between geometry size and losses. The third factor is dimensionless and indicates the contribution of the geometry shape. For the two-wire perfect metal waveguide in the free space, |εm| is infinity and neff = nd = 1. The exact TEM mode without any loss and dispersion can exist in the waveguide. For the case with real metal, |εm| is finite but still huge, |εm| of copper for example is ~106 in the THz region, and it causes the low loss and dispersion of the mode in the waveguide. Generally speaking, the larger the value |εm|, the better waveguide with lower loss. In addition, for the two-wire waveguide in the free space (nd = 1), one can also see from Eq. (23) that if the permittivity of metal εm is a negative number, Im(neff) is zero. Then, the wave in the waveguide has no loss. The last two factors in the right hand side of Eq. (23) do not affect the zero result of Im(neff) when the imaginary part of permittivity is zero. This zero result of Im(neff) is irrelevant to the geometric size, which means one can achieve small mode field size with no loss by scaling down the geometric size.
Equation (23) can be extended to other frequencies such as RF frequencies. In the RF frequencies, the permittivities of the metal εm are almost pure imaginary. Then for the waveguide in the free space (nd = 1), Eq. (23) becomesEq. (24) means that the real part of is simply equal to the imaginary one. From the second factor, one can evaluate the losses by the ratio of skin depth to the geometry size. The third factor indicates the contribution of the geometry shape to the loss as stated above.
Dielectric claddings are however needed to hold the two wire waveguide, yet they often have absorption losses in the THz region, which imply that the refractive index nd in Eq. (23) is complex valued.
4. Comparison of the analytical results with the simulations
The mode fields and the effective indices neff of the two-wire waveguides of different radii have also been calculated by full-wave simulation with a commercial software (COMSOL Multiphysics 4.3b, RF Module) as a comparison to the analytical results. The eigenvalue solver was used to find modes of the two-wire waveguide in the simulation. The frequency is chosen to be 0.5 THz. For the real metal, copper, the relative permittivity εm = –6.3 × 105 + 2.77 × 106i according to a fitted Drude model . In order to support the two wire waveguide, dielectric cladding is needed. As an example, we use the low loss material, polystyrene foam, as the cladding, whose complex index of refraction is nd = 1.0104 + 1.5059 × 10−4i at 0.5 THz according to the fitted formula and Fig. 8 in . The permeabilities of the materials are always 1.
In our study, we fix the radius of the right wire (R2) at 150 μm and vary the radius of the left wire (R1) in the range: 150-1500 μm, while keeping the width of the inter-wire gap (D-R1-R2) fixed at 200 μm.
Just as an example, we describe the field distribution and polarization in the gap region of the waveguide, whose radius of the left wire R1 = 300 μm, in the box with an area of 2 mm × 1.8 mm. Figure 3 shows the analytical normalized transverse electric field,[Fig. 3(a)], simulated normalized transverse electric field, Et [Fig. 3(b)], and the simulated normalized transverse magnetic field with a 90° rotation[Fig. 3(c)] for the two-wire waveguide in the free space. Figure 3(d) shows the simulated normalized transverse magnetic field also with a 90° rotation for the waveguide with the polystyrene foam. The electric field in Fig. 3(b) is perpendicular to the metal surfaces, and approximately linearly polarized in the gap region. The distributions of the fields shown in Fig. 3 are almost identical. Likewise, the polarizations of the fields shown in Fig. 3 are almost identical. These results imply the validity of approximation of Eqs. (14) and (16). Figs. 3(c) and 3(d) show the rotated magnetic field in order to comparewith One can imagine that the distributions of the real magnetic fields are exactly the same as shown in Figs. 3(c) and 3(d), but the polarization with a 90° rotation. That is the magnetic fields are parallel to the metal surfaces, not perpendicular to them. It is worth mentioning that, as we state in Section 2, although is very similar to Et, they are not exactly the same, even for the field distribution. However, the difference between and Et can be neglected compared with or Et.
In order to get an intuitive impression, we describe the field distribution in a quantitative way. The rotated magnetic field from simulation for the waveguide in the free space and the analytical expression of Eq. (22) are shown in Fig. 4. Figure 4(a) shows the normal components of and of along the left circle C1, and Fig. 4(b) shows the ones along the right circle C2. The absolute values of relative deviations between the normal components are less than 3%. The analytical results are in good agreements with the simulation results, which confirms the validity of the approximation, expression of Eqs. (14) and (16).
It is worth mentioning that the magnetic field distribution of the analytical result far away from the wires is quite different from that of the simulation result, shown in Fig. 4(c) [dashed line for and solid lines for along the midperpendicular of the line segment connecting the two opposite vertices, x = (D + R1-R2)/2]. The 90° rotated magnetic field of simulation in Fig. 4(c) decays exponentially with the distance far away from the two wires. This exponentially decaying field results from the none-zero value of , like the field far away from the single metal wire waveguide . However, the analytical field, decays by 1/r2. As most of the energy is restricted within the air gap, the field energy far away from the two wires can be neglected. Figure 4 only shows the fields of the waveguide in the free space. And for the waveguide with polystyrene foam cladding, there are similar behaviors.
As we can see from Fig. 5, the real and the imaginary parts of analytical neff agree well with the numerical simulations for the waveguides with the polystyrene foams claddings. We keep the gap (edge-to-edge distance) constants, D-R1-R2 = 200 μm, and set R2 = 150 μm. The radius ratio R1/R2 varies from 1 to 10. We can see that the values of Im(neff-nd) in Fig. 5 are very small, with a magnitude of about 10−4. This is attributed to the large value of |εm|, or the small ratio of the skin depth δ (~72.5 nm) to the geometric size x2-x1 (~0.5 mm) as stated below Eq. (24).
Here, it is necessary to clarify that one can either hold R2 or R1 constant while changing another. Due to the commutability of the geometry, we just show one case.
To see the dependency on the frequency, we calculate the analytical and the simulated values of neff for the two-wire waveguide in the free space (nd = 1). The metal is chosen to be copper, the two wires are identical, and R1 = R2 = 150 μm. The center-to-center distance D = 0.5 mm. The frequency-dependent εm of copper is obtained from the corresponding Drude model . The analytical and simulated in the range of 0.1-2 THz are shown in Fig. 6. The absolute values of relative deviations of and are less than 1.42% and 2.95%, respectively. Again, both the real and the imaginary parts of the analytical are in good agreements with the simulations.
The good agreements between the analytical and simulated results indicate that our simple approximate explicit formulas, Eqs. (22) and (23), are able to describe fields and the complex effective index neff of the THz two-wire metallic waveguides with different radii.
5. Group velocity dispersion and attenuation
With the expression of Eq. (23), one can describe the group velocity dispersion (GVD) and the attenuation of the two-wire metallic waveguide with nd = 1 . We adopt the same parameters in Fig. 6. The GVD can be obtained by d2β/dω2, where β = Re(k0neff). Figure 7 gives the GVD with respect to frequency from 0.1~2 THz, and the GVD is small in this range. Especially, in the range of 1.15~1.4 THz, the absolute value of GVD is below 1.0 × 10−3 ps2/m as shown in the insert of Fig. 7. We note that GVD has a zero value around 1.268 THz. One can deduce from Eq. (23) that this zero value point is only related to the permittivity of the metal and irrelevant to the geometric parameters.
The attenuation constant can be simply obtained by α = k0Im(neff), where Im(neff) is ~10−4. The attenuation constant is ~10−2 cm−1 in the frequency range of 0.1~2 THz.
The coupling efficiency is obtained by a 3D simulation with COMSOL, in which we directly couple a Gaussian beam from the free space into the two-wire waveguide. The waist of the incident Gaussian beam is one λ (λ = 600 μm), and the edge-to-edge distance of the two wires is set to be 200 μm. We obtain that the coupling efficiency is 17.3% for R1 = R2 = 150 μm, and 16.7% for R1 = 1500 μm, R2 = 150 μm. Although R1/R2 increases to 10, the coupling efficiency has a little decrease. Actually, the edge-to-edge distance plays a more important role in the coupling instead of R1/R2. Further, one could improve the coupling efficiency by selecting other coupling schemes [32,43].
As an exception, when the mode is decoupled, and the field is split to each wire, the hybrid mode becomes the angular independent TM mode, as shown in Fig. 8(a). The radii of the wires are identical, and R1 = R2 = 0.6 mm. The center-to-center distance D = 3.2 mm. The frequency is chosen to be 10 THz. The relative permittivity of the copper εm = –3.0457 × 104 + 6.684 × 103i according to the fitted Drude model . The simulated field in Fig. 8(a) is quite different from the analytical one [Eq. (22)] shown in Fig. 8(b). It means that the approximation [Eq. (16)] is not sufficient in this case, and Eqs. (17) and (23) are not able to describe the propagation in the waveguide. However, Eqs. (13)-(15) are still valid. We note that in this case, Hz is zero and is only the gradient of Ez according to Eq. (5). From the definition of Φh [Eq. (14)], one can see that Φh is now in proportion to Ez. And Ez satisfies the scalar Helmholtz equation and is also angular independent. The expression of Ez along the radial direction outside the metal can be seen in . Then Eq. (15) changes to11,17].
At the higher frequencies, typically higher than 2 THz, the analytical expression Eq. (23) starts to deviate from the simulated result as shown in Fig. 9. We keep R1 = R2 = 150 μm, D = 0.5 mm, and vary the frequency from 2 THz to 10 THz in Fig. 9. The frequency dependent relative permittivity of the copper is obtained from the fitted Drude model . It can be seen in Fig. 9 that the deviation becomes evident in the high frequency. The maximum absolute values of relative deviations of and are 15.6% and 24.7%, respectively. The higher the frequency, the greater the deviation. This is because in higher frequencies, the absolute values of relative permittivity |εm| gets smaller, the field of the single metal wire azimuthally polarized surface plasmons with a K1(.) distribution starts to affect the original quasi-electrostatic field near the metal surface. That is the distribution of starts to deviate from like the case shown in Fig. 8. However, we note that in most circumstances, especially in low frequencies (0.1-2 THz), the strong coupled field in the gap of two adjacent wires can be well approximated by the electrostatic field, as demonstrated in Section 4 and in . Therefore, our theoretical results, Eqs. (22) and (23), are able to give a simple, fast, and efficient description of the field and the complex effective index neff of THz wave in the two-wire metallic waveguide with different radii.
In addition, we need to state that the invalidation of the analytic expression at high frequency is not due to the simulation parameters. In the COMSOL simulations, a convergency analysis is implemented by varying the computational regions and the mesh sizes. The real and imaginary parts of neff vary by less than 1%.
We develop a theoretical procedure to analyze the field and propagation of THz wave in the two-wire metallic waveguide with different radii. Approximate explicit formulas of the field and complex effective index neff have been given for the waveguides. The analytical results are in good agreements with the simulations. Besides, we find a zero value point of the group velocity dispersion around 1.268 THz. It can be expected that these results are helpful for future design of two-wire metallic waveguides capable of low dispersion and loss propagation in THz, and lower frequencies. These results also indicate that the technique of perturbation of boundary condition could be helpful in the analysis and design of other THz metal waveguide.
Impedance boundary condition
The Maxwell curl equation in the nonmagnetic metal isEqs. (26a) and (26b) becomeEquation (29b) means that the magnetic field Hm inside the metal is parallel to the surface. We define the tangential H|| = Hτ|Cτ + Hz|Cz is the magnetic field right on the metal surface. The solution of Eq. (29a) is thenEq. (28a) and the solution Eq. (30), the electric field in the metal can be approximated:11]. Finally, from the continuity of the tangential E and H combined with Eq. (31), one can obtain the small tangential electric field on the metal surface in the dielectric cladding region as shown in Eq. (1).
One may find these steps in . However, we use εm rather than conductivity σ to describe the metal. It is more convenient, because εm of the metal can be estimated from the fitted Drude model in the THz region .
Conservation of charge
From Eqs. (4) and (5), one can easily getEquations (32) and (33) then giveEq. (34). We first consider the left circle C1 as an example. The line charge density along the longitudinal direction ρ1, and the total current in the longitudinal section I1 of the left wire can be expressed asEqs. (35)-(37) for the right circle C2, one may find that the right hand side of Eq. (34) is zero, and then Eq. (10) can be obtained.
Expression of Complex Effective Index for THz Two-Wire Waveguide with Different Radii
Although the field in the gap region of the two wires in Eq. (22) is given, it is difficult to solve Eq. (17) in the original two-wire ζ space. The solution is much simpler in the conformal coaxial space w. The absolute normal derivative of potential Φ0 on C1 can be expressed in the w space byEqs. (18) and (21), one can get:Eqs. (38)-(41) on C2 and combining Eq. (9), one can getEq. (23) by substituting Eqs. (40)-(43) to Eq. (17).
This work was financially supported by the National Natural Science Foundation of China with Grant Number 61275103.
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