Backscattering-immune one-way surface magnetoplasmons at terahertz frequencies

Linfang Shen; Yun You; Zhuoyuan Wang; Xiaohua Deng

doi:10.1364/OE.23.000950

1. Introduction

One-way propagating electromagnetic (EM) modes, which were proposed by Raghu and Haldan [1, 2] as analogues of quantum Hall edge states [3] in photonic crystals (PhCs) [4], are not only interesting for fundamental science but also important for practical applications. One-way EM modes are such that they are allowed to propagate in only one direction, and because of the absence of a back-propagating mode in the waveguide system, they are immune to backscattering at imperfections or bends [2, 5], which is of particular interest for slow light systems. It was predicted that one-way EM modes can be confined at the edges of certain two-dimensional (2D) PhCs made of magnetic-optical (MO) materials, where time-reversal symmetry is broken by applying a dc magnetic field [1, 2, 5, 6]. The existence of such modes was first experimentally demonstrated by Wang and his colleagues using MO PhCs in the microwave regime [7], and since then one-way EM modes have received increasing attention. Different schemes for realizing EM one-way propagation have also been proposed [8–13]; among them, the one based on surface plasmons seems to be most attractive owing to its robust mechanism and simple configuration [8,14–17].

Surface plasmons (SPs) sustained by the interface between a dielectric and a metal will become nonreciprocal under an external dc magnetic field [8]. It has been shown that, when the electron cyclotron frequency is comparable to the plasma frequency of the metal, the SP asymptotic frequency will differ significantly in the forward and backward directions, causing SPs to propagate in only one direction within the interval between the two different asymptotic frequencies. To make one-way SPs immune to backscattering, Yu et al. introduced a PhC structure into the dielectric to remove the coupling between SP and bulk modes, where the latter is completely suppressed by the PhC’s band gap [8]. Though such one-way SPs seem unrealistic because of the required magnetic field strength (B₀ ∼ 10⁴ T), the scheme can be extended to the terahertz regime and they may become realizable with the use of semiconductors. A semiconductor generally has a plasma frequency in the terahertz regime, and meanwhile its electron effective mass is far less than the electron mass. As a result, the electron cyclotron frequency is often comparable to the plasma frequency for a semiconductor under normal magnetic field intensities. It is interesting to note that the study of SPs in magnetized semiconductors, referred to as surface magnetoplasmons (SMPs), began forty years ago [18–21]. We refer the reader to the work of Brion et al. for unidirectional propagation of SMPs in certain frequency regions. However, the SMPs studied in [18] are not robust against imperfections on the semiconductor surface, as the dielectric cladding is semi-infinite and bulk modes are allowed to propagate within it. Certainly, the bulk modes can be suppressed by introducing a PhC structure into the dielectric as proposed in [8]. For the terahertz regime, because metal resembles a perfect electric conductor, bulk mode suppression can be achieved by simpler means, i.e., by terminating the dielectric with a metal slab. One-way SMPs in such a basic physical model have recently been studied by Hu et al., and some interesting results were reported [15].

It is well known that SMP dispersion is characterized by its asymptotic frequencies, at which the propagation constant approaches infinity while the depth of field penetration into the dielectric approaches zero. Accordingly, in the basic model for the terahertz regime, the influence of the metal slab on the SMPs vanishes at the asymptotic frequencies. However, the SMP dispersion behaviour near one asymptotic frequency in Hu’s work [15] is found to be quite different from that in Brion’s [18]. Moreover, the dielectric layer sandwiched by a metal and a semiconductor (with a band gap) in the basic SMP model may support regular modes, which are guided based on total internal reflection. Evidently, regular modes are generally allowed to propagate in both the forward and backward directions, and their existence may make SMPs lose their robustness of one-way propagation. Unfortunately, attention has yet to be paid to the regular modes in the basic model. Furthermore, the loss of the semiconductor cannot be neglected generally for terahertz frequencies. Therefore, for basic one-way SMPs, it is necessary to examine whether the introduction of the metal slab will significantly increase the EM energy fraction in the semiconductor and cause serious propagation loss. In this paper, we will carefully investigate the characteristics of SMPs in the basic one-way propagation model and then physically clarify all the questions mentioned above.

2. Basic physical model of one-way SMPs

The basic physical model of SMPs consists of a semi-infinite semiconductor (x ≤ 0) with dielectric cladding (x > 0) terminated by a metal slab (see the upper panel of Fig. 1). The metal slab is used to suppress bulk modes in the dielectric with relative permittivity ε_r, and the thickness of the dielectric layer is denoted by d. The semiconductor is uniformly magnetized by an external dc magnetic field B₀ in the −y direction, and SMPs propagate along the z direction. Gyroelectric anisotropy is induced in the semiconductor, with the permittivity tensor taking the form [18]

{\overset{\leftrightarrow}{ε}}_{s} = ε_{0} [\begin{matrix} ε_{1} & 0 & i ε_{2} \\ 0 & ε_{3} & 0 \\ - i ε_{2} & 0 & ε_{1} \end{matrix}],

with

ε_{1} = ε_{\infty} (1 - \frac{ω_{p}^{2}}{ω^{2} - ω_{c}^{2}}),

ε_{2} = ε_{\infty} \frac{ω_{c} ω_{p}^{2}}{ω (ω^{2} - ω_{c}^{2})},

ε_{3} = ε_{\infty} (1 - \frac{ω_{p}^{2}}{ω^{2}}),

where ω is the angular frequency, ω_p is the plasma frequency of the semiconductor, ω_c = eB₀/m^* (where e and m^* are, respectively, the charge and effective mass of the electron) is the electron cyclotron frequency, and ε_∞ is the high-frequency (relative) permittivity of the semiconductor. The semiconductor is assumed to be lossless herein, and its loss will be later taken into account when we consider the SMP propagation loss. Moreover, we assume the metal slab to be a perfect electric conductor, which is a good approximation for the terahertz regime. It should be noted that the metal slab in this system also has a mirror effect, making the system equivalent to a more complicated system, as shown in the lower panel of Fig. 1, where the latter requires the application of two oppositely directed magnetic fields.

Fig. 1 Schematic of the basic physical model (upper panel) of terahertz one-way SMPs. Owing to the mirror effect of the metal slab, this waveguide system is equivalent to that in the lower panel, where two opposite magnetic fields are applied.

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SMPs are transverse-electric-polarized in the (2D) basic model, and the nonzero component of the magnetic field (H) can be written as

H_{y} (x, z) = [A_{1} \exp (- α_{d} x) + A_{2} \exp (α_{d} x)] \exp [i (k z - ω t)]

in the dielectric layer for 0 < x ≤ d and

H_{y} (x, z) = B \exp (α x) \exp [i (k z - ω t)]

in the semiconductor for x ≤ 0, where k is the propagation constant,

α_{d} = \sqrt{k^{2} - ε_{r} k_{0}^{2}}

with k₀ = ω/c (where c is the light speed in vacuum), and

α = \sqrt{k^{2} - ε_{v} k_{0}^{2}}

with

ε_{v} = ε_{1} - ε_{2}^{2} / ε_{1}

being the Voigt permittivity. The nonzero components (E_x and E_z) of the electric field (E) can be obtained straightforwardly from H_y. The tangential component E_z must vanish at the metal boundary x = d, yielding A₂ = A₁ exp(−2α_dd). The boundary conditions further require the field components E_x and H_y to be continuous at the interface x = 0; based on this we can obtain

α + \frac{ε_{2}}{ε_{1}} k + \frac{ε_{v}}{ε_{r}} α_{d} \tanh (α_{d} d) = 0,

which is the SMP dispersion relation. The second term in the dispersion equation is linear with respect to k, meaning that the SMP dispersion relation is asymmetric about k = 0. It should be pointed out that the SMP dispersion relation in the guiding system as shown in the lower panel of Fig. 1 is completely identical to Eq. (4).

3. SMP dispersion characteristics

SMP dispersion is characterized by its asymptotic frequencies, at which k→ ±∞. Owing to the nonreciprocity of SMPs, we analyze their dispersion behaviour for the cases of k > 0 and k < 0 separately. We first consider the case of k > 0. As k → +∞, α ≈ k → +∞ and α_d ≈ k→ +∞, and from Eq. (4) two solutions can be found for ω:

ω_{a} = \frac{1}{2} (\sqrt{ω_{c}^{2} + 4 ω_{p}^{2}} - ω_{c}),

ω_{s p}^{+} = \frac{1}{2} (\sqrt{ω_{c}^{2} + 4 ω_{p}^{2} \frac{ε_{\infty}}{ε_{\infty} + ε_{r}}} + ω_{c}) .

The second solution, Eq. (6), is surely the SMP asymptotic frequency for k > 0, as it reduces to $ω_{s p}^{+} = \sqrt{ε_{\infty} / (ε_{\infty} + ε_{r})} ω_{p} for ω_{c} = 0$ , which is just the result for SPs without magnetization. The first solution, Eq. (5), is found to correspond to a zero point of ε_v. Thus, for ω = ω_a, α = k since ε_v = 0, and the dispersion equation (4) reduces to (1 +ε₂/ε₁)k = 0. This equation seems to hold for any (positive) k value, since the factor (1 +ε₂/ε₁) vanishes in this case. To obtain a physical solution for ω = ω_a, Eq. (4) should be solved in a proper way. Let us consider the limit case of ω → ω_a, for which ε_v → 0 and $α \approx k - ε_{v} k_{0}^{2} (2 k)$ . In this situation, by deleting the factor (1 +ε₂/ε₁), Eq. (4) can be rewritten as

k + \frac{ε_{1} - ε_{2}}{ε_{r}} [α_{d} \tanh (α_{d} d) - \frac{ε_{r} k_{0}^{2}}{2 k}] = 0,

which is the actual equation that can be used to directly determine k for the special case of ω = ω_a. Obviously, k is generally finite at ω = ω_a; thus, ω_a is not an asymptotic frequency.

Because SMPs are a nonradiative mode, their dispersion curve lies outside the zone of bulk modes in the semiconductor, whose boundaries are described by $ω = c | k | / \sqrt{ε_{v}}$ . For a magnetized semiconductor, there exist two bulk-mode zones where ε_v ≥ 0. The lower zone is located in the frequency region of ω_a ≤ ω ≤ ω_s, where $ω_{s} = \sqrt{ω_{c}^{2} + ω_{p}^{2}}$ , and ε_v increases from 0 at ω = ω_a to +∞ at ω = ω_s. The upper zone is located in the region of ω ≥ ω_b, where ω_b is the second zero point of ε_v and is given by

ω_{b} = \frac{1}{2} (\sqrt{ω_{c}^{2} + 4 ω_{p}^{2}} + ω_{c}) .

Correspondingly, there exist two band gaps for the semiconductor: one for ω <ω_a and the other for ω_s < ω < ω_b. By using the bulk-mode zones as the position reference, the SMP dispersion diagram can be classified into three types for k > 0, depending on the applied magnetic field intensity, and we refer to them as dispersion diagrams I, II, and III, for which $ω_{s p}^{+}$ is, respectively, located in the regions ω < ω_a, ω_a ≤ ω ≤ ω_s, and ω > ω_s. Dispersion diagram I with $ω_{s p}^{+} < ω_{a}$ pertains when $ω_{c} < ω_{c}^{(1)}$ , where the critical value $ω_{c}^{(1)}$ is given by

ω_{c}^{(1)} = \frac{ε_{r}}{\sqrt{2 (ε_{\infty} + ε_{r}) (2 ε_{\infty} + ε_{r})}} ω_{p} .

In this case, the SMP dispersion relation has only a single band and lies completely within the lower semiconductor band gap. The dispersion band starts from the origin, rises to the right of the light line in the dielectric, which is described by $ω = c | k | / \sqrt{ε_{r}}$ , and finally approaches $ω_{s p}^{+}$ as k → +∞. Dispersion diagram II with $ω_{a} \leq ω_{s p}^{+} \leq ω_{r}$ pertains when $ω_{c}^{(1)} \leq ω_{c} \leq ω_{c}^{(2)}$ , where $ω_{c}^{(2)}$ is given by

ω_{c}^{(2)} = \frac{ε_{r}}{\sqrt{ε_{\infty}^{2} - ε_{r}^{2}}} ω_{p .}

Similar to dispersion diagram I, the SMP dispersion relation in this case has only a single dispersion band, but the dispersion band extends outside the lower semiconductor band gap.

If the applied magnetic field is so large that $ω_{c} > ω_{c}^{(2)}$ , the SMPs follow dispersion diagram III with $ω_{s p}^{+} > ω_{s}$ . In this case, the dispersion relation is split into two branches by the lower bulk-mode zone of the semiconductor. The lower branch is just terminated by the boundary of the bulk-mode zone. The upper branch starts from ω_s at $k = k_{c f}^{+}$ and approaches $ω_{s p}^{(+)}$ as k → +∞. As ω → ω_s, ε₁ → 0 and ε_v → ∞, and from Eq. (4) we derive

{(k_{c f}^{+})}^{2} = \frac{\tanh^{2} (α_{d} d)}{\tanh^{2} (α_{d} d) - ε_{r}^{2} ω_{s}^{2} / ε_{\infty}^{2} ω_{c}^{2}} k_{s}^{2},

where

k_{s} = \sqrt{ε_{r}} ω_{s} / c

and

α_{d} = \sqrt{{(k_{c f}^{+})}^{2} - k_{s}^{2}}

. If the upper branch starts from the light line of the dielectric, it can be derived from Eq. (4) that

α + k_{c f}^{+} ε_{2} / ε_{1} = 0

. This equation cannot hold for positive

k_{c f}^{+}

, since both ε₁ and ε₂ are positive for ω > ω_s, so the upper branch cannot start from the light line. In the case when ω_c is close to

ω_{c}^{(2)}

(but ω_{c} > ω_{c}^{(2)})

,

k_{c f}^{+}

should depart much farther from the light line, i.e.,

k_{c f}^{+} ≫ k_{s}

, so tanh(α_dd) ≈ 1 and Eq. (11) can be rewritten as

k_{c f}^{+} = \sqrt{\frac{1}{[1 - {(ω_{c}^{(2)} / ω_{c})}^{2}] (1 - ε_{r}^{2} / ε_{\infty}^{2})}} k_{s} .

Clearly, the value of $k_{c f}^{+}$ approaches +∞ as $ω_{c} \to ω_{c}^{(2)}$ (as expected), and it decreases when ω_c increases from $ω_{c}^{(2)}$ .

We now consider the case of k < 0. Similar to the case of k > 0, as k→ −∞, two solutions are found from Eq. (4). One solution is ω = ω_b and the other is $ω = ω_{s p}^{-}$ with

ω_{s p}^{-} = \frac{1}{2} (\sqrt{ω_{c}^{2} + 4 ω_{p}^{2} \frac{ε_{\infty}}{ε_{\infty} + ε_{r}}} - ω_{c}) .

Evidently, $ω_{s p}^{-}$ is really the SMP asymptotic frequency for k < 0. The first solution ω_b is just the second zero of ε_v, at which Eq. (4) holds for any negative k value, similar to the case of k > 0 at ω = ω_a. For ω = ω_b, we can also similarly derive

- k + \frac{ε_{1} + ε_{2}}{ε_{r}} [α_{d} \tanh (α_{d} d) + \frac{ε_{r} k_{0}^{2}}{2 k}] = 0,

which is the actual equation that can be used to directly determine k for this special frequency. For the case of k < 0,

ω_{s p}^{-}

is smaller than ω_a for any ω_c, so there always exists a complete SP dispersion band, which starts from the origin, rises, and approaches

ω_{s p}^{-}

as k→ −∞. However, for k < 0, an additional dispersion branch can always be found in the region where ω > ω_s. This branch starts from the light line of the dielectric and ends at the boundary of the upper bulk-mode zone of the semiconductor. At the starting point of this additional branch, α_d = 0; thus, from Eq. (4) we have α +kε₂/ε₁ = 0 with

k = \sqrt{ε_{r}} k_{0}

, from which we find that ε₁ is equal to ε_r and the cutoff frequency is

ω_{c f}^{-} = \sqrt{ω_{c}^{2} + \frac{ε_{\infty}}{ε_{\infty} - ε_{r}} ω_{p}^{2}}

for the additional branch. Evidently, the existence of the additional branch only requires ε_∞ >ε_r, and this condition is generally met by transparent dielectrics in the terahertz regime.

To validate our analysis above, we numerically calculate the SMP dispersion relation with Eq. (4) for various ω_c values. As an example, the basic parameters of the system are taken as follows: ε_∞ = 15.6 (InSb) [22], ε_r = 2.28 (polymer) [23], and d = 0.16λ_p (where λ_p is the vacuum wavelength for the plasma frequency ω_p). From now on, the semiconductor is assumed to be InSb in all numerical calculations. For the present system, we find that $ω_{c}^{(1)} = 0.066 ω_{p}$ and $ω_{c}^{(2)} = 0.148 ω_{p}$ . Figures 2(a)–2(c) display the SMP dispersion relations for the cases of ω_c = 0.05ω_p, 0.1ω_p, and 0.2ω_p, respectively. It is obvious that, as expected, they correspond to dispersion diagrams I, II and III, respectively. As can be seen from Figs. 2(a)–2(c), SMPs always follow two dispersion branches for k < 0, and the upper one (i.e., the additional branch) occurs in a narrow range compared with the lower one. Figure 2(d) shows a magnified view of the upper branch (with k < 0) in Fig. 2(a); it clearly indicates that this branch starts from the light line and ends at the boundary of the bulk-mode zone. For SMPs with k > 0, the second dispersion branch is observed only in Fig. 2(c). Our numerical analysis also shows that the second dispersion branch can be found only when ω_c > 0.148ω_p. All the numerical results agree well with our analytical analysis.

Fig. 2 SMP dispersion diagrams for (a) ω_c = 0.05ω_p, (b) ω_c = 0.1ω_p, and (c) ω_c = 0.2ω_p. Solid lines are the SMP dispersion curves, and dashed lines are the light line in the dielectric. The upper two shaded areas represent the zones of bulk modes in the semiconductor. (d) Magnified view of the upper branch in (a). The other parameters are ε_∞ = 15.6, ε_r = 2.28, and d = 0.16λ_p for all cases.

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4. Complete SMP one-way propagation

It is evident that, for each kind of dispersion diagram, there always exist certain frequency ranges in which SMPs are allowed to propagate unidirectionally. For dispersion diagrams I and II, as shown in Figs. 2(a) and 2(b), SMPs only propagate forward in the frequency range from $ω_{s p}^{-}$ to $ω_{s p}^{+}$ , while its upper branch with $ω \geq ω_{c f}^{-}$ allow only backward propagation. In dispersion diagram III, there may exist three unidirectional-propagation ranges, as shown in Fig. 2(c); two correspond to the forward direction and the third to the backward direction. However, only the portion of the unidirectional-propagation range located within the semiconductor band gap is of interest, and we refer to it as the complete one-way region (or range). In the complete one-way region, SMPs are immune to backscattering resulting from imperfections in the system, as their coupling with bulk modes in the semiconductor is completely eliminated. Because a larger complete one-way range is expected for a normal (external) magnetic field, in what follows, we restrict ourselves to the complete one-way region within the lower semiconductor band gap (ω < ω_a). This complete one-way region ranges from $ω_{s p}^{-}$ to $ω_{s p}^{+}$ for dispersion diagram I $(ω_{c} \leq ω_{c}^{(1)})$ , and it ranges from $ω_{s p}^{-}$ to ω_a for dispersion diagrams II and III $(ω_{c} \leq ω_{c}^{(1)})$ . Correspondingly, the complete one-way bandwidth (Δω_ow) has the form

Δ ω_{o w} = ω_{c}

for

ω_{c} \leq ω_{c}^{(1)}

, or

Δ ω_{o w} = \frac{2 ω_{p}^{2} ε_{r} / (ε_{\infty} + ε_{r})}{\sqrt{ω_{c}^{2} + 4 ω_{p}^{2}} + \sqrt{ω_{c}^{2} + 4 ω_{p}^{2} ε_{\infty} / (ε_{\infty} + ε_{r})}}

for

ω_{c} > ω_{c}^{(1)}

. Obviously, Δω_ow first increases linearly with ω_c, then reaches a maximum of

ω_{c}^{(1)}

at

ω_{c} = ω_{c}^{(1)}

, and finally decreases as ω_c grows further.

The dependence of Δω_ow on ω_c is plotted in Fig. 3(a). Three different dielectrics are analyzed: air, polymer (ε_r = 2.28), and silica (ε_r = 11.68), all of which are transparent media commonly used in the terahertz regime [23]. As seen from Fig. 3(a), Δω_ow is independent of ε_r when ω_c < 0.031ω_p, which just corresponds to the value of $ω_{c}^{(1)}$ for air. Therefore, under a small magnetic field, it is impossible to increase Δω_ow by choosing the dielectric. However, the maximal value of Δω_ow, which is equal to the value of $ω_{c}^{(1)}$ and occurs at $ω_{c} = ω_{c}^{(1)}$ , closely depends on ε_r. The larger ε_r is, the larger the maximal Δω_ow is. It is found that the maximal values of Δω_ow are 0.031ω_p, 0.066ω_p, and 0.24ω_p for air, polymer, and silica, respectively. These values also represent the peak positions $ω_{c}^{(1)}$ for the corresponding dielectrics. Figure 3(b) shows the complete one-way range for the case of air. The upper limit of the one-way range increases with ω_c when ω_c < 0.031ω_p, and it decreases with ω_c when ω_c > 0.031ω_p, while the lower limit always decreases with ω_c. Finally, we should note that the semiconductor is inherently lossy as it is dispersive; consequently, SMPs propagating in it are lossy. In general, the SMP loss will become very serious when $k ≫ \sqrt{ε_{r}} k_{0}$ . To reduce the one-way SMP loss to an acceptable level, ω_c should be chosen to be a little larger than $ω_{c}^{(1)}$ . By doing so, the SMP dispersion curve in the complete one-way region does not deviate too far from the light line, while the one-way bandwidth is still large.

Fig. 3 (a) Complete one-way bandwidth of SMPs as a function of ω_c. Solid, dashed, and dotted lines correspond to the dielectrics of air, polymer (ε_r = 2.28), and silica (ε_r = 11.68), respectively. (b) Complete one-way range versus ω_c for the case of air. The other parameters are the same as in Fig. 2.

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5. Regular mode guided by the dielectric layer

In the basic one-way SMP model, the dielectric layer can still support (transverse-electricpolarised) regular modes, which are guided by a means of zigzag reflections at the surfaces of the metal and the semiconductor (with a band gap). As regular modes are generally allowed to propagate in both forward and backward directions, their existence causes SMPs to lose their robustness of one-way propagation against imperfections. The dispersion equation for regular modes can be obtained from Eq. (4) by replacing α_d with −ip, where $p = \sqrt{ε_{r} k_{0}^{2} - k^{2}}$ is real-valued, and it has the form

α + \frac{ε_{2}}{ε_{1}} k - \frac{ε_{V}}{ε_{r}} p \tan (p d) = 0.

Obviously, the dispersion relation for regular modes is also asymmetric about k = 0. Because regular modes have a tangential electric field component (E_z) that vanishes at the metal surface, they should exhibit transverse resonances and related cutoff frequencies. These cutoffs increase as d decreases; hence, it is possible to remove the regular modes from the complete one-way region by reducing the dielectric layer thickness.

Owing to the nonreciprocity, the cutoff point of a regular mode may deviate from k = 0. However, this deviation should be very small under a normal magnetic field, as ω_c is generally much smaller than ω_p (e.g., for InSb at room temperature, ω_c is equal to 0.1ω_p for B₀ = 0.1 T). If we let k ≈ 0, then $p \approx \sqrt{ε_{r}} k_{0}$ , and from Eq. (16) we have $α = ε_{v} k_{0} \tan (\sqrt{ε_{r}} k_{0} d) / \sqrt{ε_{r}}$ . In the region below ω_a, ε_v is always negative, so Eq. (17) has no solution if $\sqrt{ε_{r}} k_{0} d < π / 2$ . This means that even the fundamental regular mode with the lowest cutoff will vanish at this frequency. By setting k₀ = ω_a/c, the critical thickness (d_c) of the dielectric layer is found to be

d_{c} = \frac{1}{8 \sqrt{ε_{r}}} (\sqrt{4 + ω_{c}^{2} / ω_{p}^{2}} + ω_{c} / ω_{p}) λ_{p,}

below which no regular mode appears in the region for ω < ω_a; i.e., the complete one-way region is preserved for SMPs. Obviously, d_c is inversely proportional to

\sqrt{ε_{r}}

. For ω_c = 0.1ω_p, we find that d_c = 0.26λ_p for ε_r = 1, d_c = 0.17λ_p for ε_r = 2.28, and d_c = 0.077λ_p for ε_r = 11.68.

To illustrate the regular mode guided by the dielectric layer, we numerically calculate its dispersion relations using Eq. (16) for various d values, and the results are plotted in Fig. 4, where ε_r = 2.28 and ω_c = 0.1ω_p (corresponding to d_c = 0.17λ_p) for all cases. Figures 4(a)–4(d) show the dispersion relations (lines with filled circles) of the (fundamental) regular mode for d = d_c, 0.2λ_p, 0.25λ_p, and 0.29λ_p, respectively. As expected, in the case of d = d_c, the fundamental regular mode has a cutoff of ω_a, and the complete one-way SMP region is preserved. When d increases from d_c, the cutoff of the regular mode shifts downward. As a result, the complete one-way region is compressed, and its upper limit is just determined by the cutoff frequency. Figure 4(d) indicates that, when d = 0.29λ_p, the cutoff drops to $ω_{s p}^{-}$ and consequently the complete one-way region vanishes completely.

Fig. 4 Dispersion relations of all modes in the guiding system. Solid lines with and without filled circles correspond to the dispersion curves for regular and SMP modes, respectively. Dashed lines represent the light line in the dielectric. (a) d = d_c = 0.17λ_p, (b) d = 0.2λ_p, (c) d = 0.25λ_p, and (d) d = 0.29λ_p. In (a)–(c), the lower shaded area represents the complete one-way region, and the upper one represents the zone of bulk modes in the semiconductor. The other parameters are ε_∞ = 15.6, ε_r = 2.28, and ω_c = 0.1ω_p for all cases.

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Using the finite element method (FEM), we performed a simulation to demonstrate the regular mode in the basic SMP model. For this purpose, the parameters of the system are taken to be the same as in Fig. 4(c). In the simulation, a magnetic current line source is used to excite SMPs and two operation frequencies of f_A = 0.905 f_p and f_B = 0.94 f_p (where f_p = ω_p/2π) are considered, as marked with points A and B in Fig. 4(c). The plasma frequency is set to be f_p = 2 THz, so f_A = 1.81 THz and f_B = 1.88 THz. To examine the robustness of one-way propagating SMPs, we consider both cases with and without a metal rod obstacle. The metal rod with a radius of 8 μm is inserted at the polymer–semiconductor interface at a horizontal distance of 150 μm from the source. All simulated results are displayed in Fig. 5. The left panels show the SMP transmission at f_A = 1.81 THz, where the source is placed at the polymer–semiconductor interface. It can be clearly observed from the upper left panel that SMPs propagate only in the forward direction. Furthermore, SMPs at this frequency can go around the obstacle without any power loss, as no backward-propagating wave emanates from the left side of the source (see the lower left panel). For f_B = 1.88 THz, to avoid strong direct excitation of the regular mode, the source is placed inside the semiconductor at 30 μm below the surface. Nevertheless, a weak backward-propagating wave, which corresponds to the regular mode and is directly excited by the source, is still observed in the upper right panel of Fig. 5. As shown in the lower right panel, the backward-propagating wave is enhanced in the presence of the obstacle. Evidently, in this case, coupling between SMPs and the backward-propagating regular mode occurs via scattering at the obstacle. The results shown in the right panels indicate that SMPs at f_B = 1.88 THz are no longer immune to backscattering owing to the existence of the regular mode in the dielectric layer. The simulated results agree well with our previous analysis.

Fig. 5 Simulated electric field amplitudes. For f = 1.81 THz, the source is placed at the polymer–semiconductor interface; for f = 1.88 THz, the source is placed inside the semiconductor, 30 μm below the surface. In the lower panels, a metal rod with a radius of 8 μm is inserted at the polymer–semiconductor interface, which lies at a horizontal distance of 150 μm from the source. The other parameters are the same as in Fig. 4(c).

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6. Mirror effect of the metal slab

As a dispersive medium, the semiconductor is inherently lossy, and, consequently, SMPs suffer a propagation loss in it. For a lossy semiconductor, the expressions for ε₁ and ε₂ in Eq. (1) become

\begin{array}{l} ε_{1} = ε_{\infty} {1 - \frac{(ω + i ν) ω_{p}^{2}}{ω [{(ω + i ν)}^{2} - ω_{c}^{2}]}}, \\ ε_{2} = ε_{\infty} \frac{ω_{c} ω_{p}^{2}}{ω [{(ω + i v)}^{2} - ω_{c}^{2}]}, \end{array}

where ν is the electron scattering frequency. In the lossy case, the propagation constant k in Eq. (4) becomes complex, i.e., k = k_r + ik_i (where both k_r and k_i are real-valued), and the SMP propagation length is determined by L = 1/(2k_i). Evidently, in the case of ν≪ω_p, the SMP loss is proportional to the fraction of EM energy in the semiconductor. Correspondingly, the larger the EM energy fraction of SMPs is distributed within the dielectric layer, the larger the propagation length is. In the basic one-way SMP model, the metal slab has two opposing effects on the SMP energy distribution: 1. As the metal slab compresses the dielectric region, the energy fraction in the dielectric layer is decreased, and 2. the metal slab introduces the mirror effect, which will enhance the EM energy density and thus increase that energy fraction. Owing to the mirror effect, a SMP in the present model may be viewed as a symmetric combination of a SMP associated with a single (dielectric–semiconductor) interface and its image produced by the metal slab; the field in the dielectric layer is therefore enhanced compared with the case of a single interface. However, the mirror effect is effective only when d ≤ 1/α_d, which is the field penetration depth into the dielectric. Thus, for robust one-way SMPs, an interesting question arises as to whether the propagation loss is aggravated or lessened (compared with SMPs at a single interface) by the metal slab or whether the mirror effect of the metal slab is the dominant effect on the SMP energy distribution.

To clarify this question, we numerically calculate the propagation length L using Eq. (4) for a lossy semiconductor. As a numerical example, we take ν = 0.0025ω_p for the semiconductor, and other parameters of the system are as follows: ε_∞ = 15.6, ε_r = 2.28, and ω_c = 0.1ω_p. Figure 6(a) shows the calculated L value as a function of d for the frequency ω = 0.918ω_p. This frequency lies at the centre of the complete one-way region for d ≥ d_c, i.e., $ω = (ω_{s p}^{-} + ω_{a}) / 2$ . With the above parameters, the critical thickness is d_c = 0.17λp. For comparison, we also calculate the propagation length (L₀) for SMPs at a single interface, and we find that L₀ = 7.7λ (λ = 2πc/ω) for ω = 0.918ω_p, which is indicated in Fig. 6(a) by the horizontal dashed line. From Fig. 6(a), it is seen that L < L₀ for small d, meaning that the metal slab aggravates the SMP loss owing to the compression of the dielectric region. L rapidly increases with d, becoming larger than L₀ at d = d_c, meaning that the mirror effect becomes the dominant effect on the energy distribution. Our calculation gives L = 8.3λ for d = d_c (L is inversely proportional to ν, and for this case L = 20.75λ if ν = 0.001 ω_p). L reaches a maximal value of 8.4λ at d = 0.20λ_p, but the cutoff of the fundamental regular mode in this case is found to be below ω. The inset of Fig. 6(a) shows the distribution of L in the complete one-way region for d = d_c, and the corresponding results for L₀ are also plotted with a dashed line for comparison. Clearly, owing to the mirror effect of the metal slab, SMPs in the present system can possess a larger propagation length than SMPs at a single interface for the entire one-way region.

Fig. 6 (a) SMP propagation length L (solid line) as a function of d for ω = 0.918ω_p. The dashed line represents the L₀ value for SMPs sustained by the single interface (i.e., d → ∞). The inset shows the dependence of L on ω for d = d_c (solid) and d→∞ (dashed). The scattering frequency of the semiconductor is ν = 0.0025ω_p. (b) η_d versus d for ω = 0.918ω_p. The other parameters are ε_∞ = 15.6, ε_r = 2.28, and ω_c = 0.1ω_p.

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To gain a deeper insight into the mirror effect, we further calculate the SMP EM energy in the dielectric layer and semiconductor in the lossless case (ν = 0). The other parameters are kept the same as in Fig. 6(a). As the semiconductor is strongly dispersive, the EM energy density in it can be expressed as [24]

U_{s} = \frac{1}{4} E * \frac{d (ω {\overset{\leftrightarrow}{ε}}_{s})}{d ω} E + \frac{1}{4} μ_{0} | H |^{2} .

The SMP EM energy fraction in the dielectric layer is

η_{d} = \frac{\int_{0}^{d} U_{d} d x}{\int_{- \infty}^{0} U_{s} d x + \int_{0}^{d} U_{d} d x},

where U_d = (1/4)(ε₀ε_r|E|² +μ₀|H|²), which is the energy density in the dielectric layer. The calculated results of η_d for various d values are plotted in Fig. 6(b). As expected, the dependence of η_d on d is similar to that of L, as shown in Fig. 6(a). η_d first increases with d and then reaches a maximum at d ≈ 0.2λ_p. As d continues to grow, it decreases and finally approaches a limit, which corresponds to that for SMPs at a single dielectric–semiconductor interface. The behaviour of η_d clearly indicates that the mirror effect of the metal slab enhances the EM energy density in the dielectric layer and thus may increase the energy fraction.

7. Conclusions

In summary, we have theoretically studied terahertz SMPs in a basic physical model comprising a semi-infinite magnetized semiconductor with dielectric cladding terminated by a metal slab. It has been shown that, depending on the applied magnetic field intensity, such SMPs may follow three different kinds of dispersion diagrams. For each kind of dispersion diagram, there always exists a complete one-way SMP region, which lies within the lower band gap of the semiconductor. If the magnetic field is strong enough, another complete one-way band appears in the upper band gap of the semiconductor, but it generally has a bandwidth significantly smaller than that in the lower band gap. The bandwidth of the (complete) one-way propagation closely depends on the magnetic field intensity, and it reaches a maximum at a finite magnetic field. Regular modes guided by the dielectric layer have also been investigated. The regular modes are allowed to propagate in both the forward and backward directions, and when they are present, the complete one-way region is often compressed or even removed. However, regular modes can be suppressed by reducing the dielectric layer thickness. Moreover, the mirror effect of the metal slab in this basic one-way SMP model has been analyzed. It has been shown that, because the mirror effect can significantly increase the EM energy fraction in the dielectric layer, the SMPs immune to backscattering in the basic physical model can exhibit a larger propagation length than those at a single dielectric–semiconductor interface.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 61372005), the National Natural Science Foundation of China under a key project (Grant No. 41331070), the Science Foundation of Zhejiang Province (Grant No. LY14F030013), and the Natural Science Foundation of Ningbo (Grant No. 2013A610004).

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Backscattering-immune one-way surface magnetoplasmons at terahertz frequencies

Abstract

1. Introduction

2. Basic physical model of one-way SMPs

3. SMP dispersion characteristics

4. Complete SMP one-way propagation

5. Regular mode guided by the dielectric layer

6. Mirror effect of the metal slab

7. Conclusions

Acknowledgments

References and links

Cited By

Figures (6)

Equations (25)

Optics Express