Stopping terahertz radiation without backscattering over a broad band

Linfang Shen; Xiaodong Zheng; Xiaohua Deng

doi:10.1364/OE.23.011790

1. Introduction

Completely stopping light or electromagnetic (EM) radiation in all-optical systems is not only of great interest in fundamental science, but also has a variety of prospective applications relying on various manipulation means. It was predicted that surface plasmon polaritons (SPPs) can be asymptotically stopped in a tapered plasmonic waveguide when they tend to the tip, where the group velocity of SPPs reduces to zero [1]. This adiabatic stopping of SPPs causes accumulation of EM energy and giant local fields at the stopping point, thus it has multiple possible applications in nanooptics and nanoprobing. By using metamaterials, other schemes to stop light or EM radiation have also been reported [2–4], i.e., so-called ’trapped rainbow’ or similar phenomena, in which incident light or EM waves with different frequencies stop at correspondingly different locations in linearly tapered waveguides. However, some researchers later claimed that instead of being trapped, incident EM fields in these tapered waveguides were totally reflected, as a result of the coupling between the forward and backward modes through the tapering effect [5]. Based on dynamic processes, in which the propagation of light is dynamically altered by modulating the refractive index of the system, several approaches of different type for stopping light have also been proposed [6–8], e.g., coupled resonator optical waveguide system [6], which is constructed from the analog of classical electromagnetic induced transparency. These methods to coherently store light in a tunable manner have profound implications for classical and quantum information processing.

Recently, one-way EM edge modes [9, 10], which were first proposed as analogues of quantum Hall edge states [11] in photonic crystals [12], have received much intention. Such modes can propagate in only one direction, and, because of the absence of a back-propagating mode in the system, they are immune to backscattering at imperfections or bends [13–15]. There also exists an alternative form of one-way edge modes, namely, one-way surface magnetoplasmons (SMPs) [16–19], which are sustained by the interface between a dielectric and a plasmonic material under an applied static magnetic field. Compared to one-way edge modes in the photonic crystal form, one-way SMPs are more simple in geometry and more robust in mechanism. Evidently, if one-way edge modes can be blocked, then their (EM) fields will be completely trapped, since there is no back-propagating mode in the guiding system. Stopping light or EM radiation in such a way should be easily realized in experiment, because one-way edge modes are robust against surface imperfections or roughness. Moreover, this stopping process happens in a forcible manner, thus it can be accomplished even in the cases with serious material losses, which is quite different from the situation of adiabatic stopping approaches. In this paper, by using a gyroelectric semiconductor, we will show that one-way SMPs at terahertz frequencies can be stopped at the interface of different cladding structures on the semiconductor, thus terahertz radiation is completely trapped and consequently a hotspot with extremely enhanced fields is generated, which has promising applications in terahertz sensing, surface-enhanced Raman spectroscopy, and enhanced nonlinearity.

2. Physical model

We first analyze the dispersion properties of SMPs in a semiconductor-dielectric-metal (SDM) structure under an applied static magnetic field (B₀), as shown in Fig. 1(a). To make one-way SMPs immune to backscattering, the cladding dielectric in this structure is terminated by the metal plate, thus radiation modes within the dielectric are suppressed. The magnetic field B₀ is assumed to be applied in the −y direction, and the gyroelectric semiconductor in this SDM structure has the permittivity tensor [18]

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

with

\begin{array}{l} κ_{1} = 1 - \frac{(ω + i ν) ω_{p}^{2}}{ω [{(ω + i ν)}^{2} - ω_{c}^{2}]}, \\ κ_{2} = \frac{ω_{c} ω_{p}^{2}}{ω [{(ω + i ν)}^{2} - ω_{c}^{2}]}, \\ κ_{3} = 1 - \frac{ω_{p}^{2}}{ω (ω + i ν)}, \end{array}

where ω is the angular frequency, ω_p is the (effective) plasma frequency, and ν is the scattering frequency; ω_c = eB₀/m^* (where e and m^* are, respectively, the charge and effective mass of electrons), being the electron cyclotron frequency. The fields of SMPs in the SDM structure are transverse-magnetic (TM) polarized, 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,

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

in the semiconductor for x ≤ 0, and

H_{y} (x, z) = C \exp (- α_{m} x) \exp [i (k z - ω t)]

in the metal for x ≥ d, where k is the propagation constant,

α_{s} = \sqrt{k^{2} - ε_{v} k_{0}^{2}}

with k₀ = ω/c (the vacuum wavenumber) and

ε_{v} = ε_{\infty} (κ_{1} - κ_{2}^{2} / κ_{1})

(the Voigt permittivity);

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

, ε_r and d are the relative permittivity and thickness of the dielectric layer, respectively; and

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

, and ε_m is the relative permittivity of the metal. The nonzero components (E_x and E_z) of the electric field (E) can be obtained straightforwardly from H_y. The boundary conditions require the field components E_x and H_y to be continuous at the interfaces x = 0 and d; based on this we can obtain

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

which is the dispersion relation of SMPs. In the terahertz regime, |ε_m|≫1 and |ε_rα_m/ε_mα_r|~|ε_m|⁻¹^/²≪1, so Eq. (5) can be approximated as

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

Fig. 1 (a), (b) Schematics of the SDM and SM structures under an applied static magnetic field. (c) Dispersion relations for SMPs in the SDM and SM structures. The upper two shaded areas represent the zones of bulk modes in the semiconductor, and the lowest one represents the COWP region for the SDM structure. The parameters are ε_∞ = 15.6 and ω_c = 0.25ω_p for both structures and ε_r = 11.68 and d = 0.08λ_p for the SDM structure.

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When we analyze the dispersion properties of SMPs, semiconductor and metal loses are neglected for clarity in physics. The dispersion feature of SMPs is characterized by their asymptotic frequencies, at which |k| → ∞. From Eqs. (5) or (6), the asymptotic frequencies are found to be

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

where the + and − signs denote the forward and backward directions, respectively. Equation (7) indicates that SMPs are only allowed to propagate forward in the frequency range

[ω_{s p}^{-}, ω_{s p}^{+}]

. If

ω_{s p}^{+}

is below the lower frequency limit (ω_a) of bulk modes in the semiconductor, which is given by

ω_{a} = (\sqrt{ω_{c}^{2} + ω_{p}^{2}} - ω_{c}) / 2

, one-way SMPs are then immune to backscattering, because their couplings with the bulk modes are prohibited, and we refer to this phenomenon as complete one-way propagation (COWP). Clearly, when

ω_{s p}^{+} < ω_{a}

, the COWP region of SMPs ranges from

ω_{s p}^{-}

to

ω_{s p}^{+}

; but, when

ω_{s p}^{+} < ω_{a}

, it ranges from

ω_{s p}^{-}

to ω_a.

We next considered another SMP waveguide, which is a semiconductor-metal (SM) structure, as shown in Fig. 1(b). Evidently, the SMP dispersion relation for the SM structure can be obtained from Eq. (5) by setting d = 0, which gives

α_{s} + \frac{κ_{2}}{κ_{1}} k + \frac{ε_{v}}{ε_{m}} α_{m} = 0.

The dispersion of SMPs in the SM structure was found to be completely different from that in the SDM structure. For the region ω < ω_a, ε_v < 0 and ε_m < 0, and κ₂/κ₁ < 0, so k in Eq. (8) must be positive, meaning that SMPs are only allowed to propagate forward. In the case of k → ∞, we found that the SMP asymptotic frequency is ω_sp = ω_c. In fact, for the terahertz regime and ω ≤ ω_a, where |ε_vα_m/ε_mα_s|≪1, Eq. (8) may be approximated as $k = \sqrt{ε_{1}} k_{0}$ , where ε₁ = ε_∞κ₁.

To clearly illustrate SMPs in the SDM and SM structures, their dispersion relations were numerically calculated and the results are displayed in Fig. 1(c). In this paper, the semiconductor was assumed to be InSb with ε_∞ = 15.6 and ω_p = 4π×10¹² rad/s (f_p = 2 THz) [20], and the metal was silver (Ag) with $ε_{m} = 1 - ω_{p e}^{2} / [ω (ω + i ω_{τ})]$ , where ω_pe = 1.367 × 10¹⁶ rad/s and ω_τ = 2.733 × 10¹³ rad/s [21]. For the SDM structure, the dielectric layer was silicon (Si) with ε_r = 11.68 [22], and its thickness was d = 0.08λ_p (where λ_p = 2πc/ω_p). For both types of waveguide, ω_c was taken to be 0.25ω_p (corresponding to B₀ = 0.25 T). As seen in Fig. 1(c), the COWP region for the SDM structure lies between 0.64ω_p and 0.88ω_p, and it is located above the COWP region for the SM structure, which is below 0.25ω_p. Let us consider a system composed of SDM and SM structures, as shown in Fig. 2(a); in such a system, robust one-way SMPs traveling forward in the SDM section (guiding part) must be stopped by the SM section (stopping part) without any backscattering, as SMPs are forbidden in the latter. Evidently, stopping SMPs can be achieved over the whole COWP band of the guiding part, provided that $ω_{s p}^{-}$ is larger than ω_c, which requires that

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

The critical value for ω_c described in Eq. (9) was found to be 0.53ω_p for Si.

Fig. 2 (a) Simulated E amplitudes in the system consisting of the SDM and SM structures. The magnetic current line source to excite SMPs is located at x = 6 μm and z = −200 μm. (b), (c) Distributions of the normalized E amplitude along the lines x = 0 and z = 0 in (a). The frequency is f = 1.5 THz, and the other parameters are the same as in Fig. 1(c).

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3. Numerical simulation and analysis

Using the finite element method (COMSOL Multiphysics), we simulated wave transmission in the proposed system for stopping terahertz radiation. In the simulation, both InSb and Ag losses were taken into account, and as an example, we took ν = 0.001ω_p for InSb. A magnetic current line source, which was located 6 μm above the semiconductor surface (x = 0) at a distance of 200 μm from the interface (z = 0) of the SDM and SM sections, was used to excite SMPs in the system. The operation frequency was set at f = 1.5 THz (ω = 0.75ω_p). The simulated electric field (E) amplitudes are plotted in Fig. 2(a). It is observed that excited SMPs in the SDM section only travel forward and are finally stopped by the SM section without any backscattering. A trapped spot, which is a hotspot with extremely enhanced field, occurs near the interface z = 0. Our numerical calculation showed that the maximal E amplitude lies at x = 0 and z = −0.01 μm. The distribution of the E amplitude along the semiconductor surface is plotted in Fig. 2(b), where the field amplitude is normalized to a reference value, which corresponds to the (maximal) E amplitude of SMPs at the same propagation distance as the trapped SMPs (i.e., 200 μm) in the simulation for a single SDM structure with sufficient length. The maximum of the normalized E amplitude was found to be nearly 520, and this field-enhancement value is independent of the distance between the source and the SDM-SM interface. Figure 2(c) shows the distribution of the normalized E amplitude along the interface of the SDM and SM sections. At this interface, the maximal normalized E amplitude is nearly 143. Figure 3(a) clearly displays the field pattern of the trapped spot. It should be physically reasonable that the sizes (δ_t, δ_l) of the trapped spot are defined as the transverse or longitudinal separation between the locations where the E amplitude has fallen to 1/e of its maximal value. Thus, the sizes of the trapped spot were found to be δ_t = 0.13 μm and δ_l = 5.2 μm.

Evidently, the field enhancement achieved by stopping one-way SMPs closely depends on the loss level of the semiconductor. To clarify this, we also performed the simulation of wave transmission for a system with ν = 0.01ω_p. The distribution of the normalized E amplitude along the semiconductor surface for this case (the line with squares) is plotted in Fig. 4, where the corresponding results for the case ν = 0.001ω_p (the line with solid circles) are included for comparison. It is seen that, when ν increases from 0.001ω_p to 0.01ω_p, the maximum of normalized E amplitude (i.e., the field-enhancement value) decreases from 520 to 333, which is far larger than expected. The field pattern of the trapped spot for ν = 0.01ω_p is shown in Fig. 3(b). The sizes of this hotspot were found to be δ_t = 0.14 μm and δ_l = 0.85 μm, the latter one being much smaller than that for ν = 0.001ω_p, which may explain why the trapped spot still has large field amplitudes for ν = 0.01ω_p. The influence of the metal loss on the trapped spot was also examined. The wave transmission in a system, in which ν = 0.001ω_p but the metal was approximated as a perfect electric conductor (PEC), was simulated, and the results are plotted with open circles in Fig. 4, which are almost the same as those for the Ag case. Compared to the Ag case, the maximal E amplitude of the trapped spot increases by only 1.2% in the PEC case. Therefore, we conclude that metal loss has a negligible influence on the trapped spot.

Fig. 3 E amplitudes of trapped spots for ν = 0.001ω_p (a) and 0.01ω_p (b). The other parameters are the same as in Fig. 2.

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Fig. 4 Influence of the semiconductor or metal losses on the distribution of normalized E amplitude along the semiconductor surface. The other parameters are the same as in Fig. 2.

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It is interesting to examine whether the trapped spot is just the evanescent mode of SMPs whose fields decay in the backward direction. To clarify this, we calculated the dispersion relation for backward propagating and evanescent SMPs in the region (0,ω_a). In the lossless case, where ν = 0 and the metal is a PEC, the propagation constant k of SMPs was found to be a real number in the interval (0, $ω_{s p}^{-}$ ), and |k| → ∞ as $ω \to ω_{s p}^{-}$ . In the interval ( $ω_{s p}^{-}$ ,ω_a), k is a complex number with nonzero real (k_r) and imaginary (k_i) parts, as shown in Fig. 5 (see solid lines), and this means that SMPs are an evanescent mode. In the lossy case, where ν ≠ 0 and the metal is Ag, k becomes a complex number in the whole region (0,ω_a). The real part k_r is almost the same as that for the lossless case, except in the vicinity of $ω_{s p}^{-}$ , where k_r now becomes finite, as shown in Fig. 5(a) (see dashed line with circles). On the other hand, in the interval ( $ω_{s p}^{-}$ ,ω_a), the k_i values of the evanescent SMPs in the lossy (dashed line with circles) and lossless (solid line) cases almost overlap, as shown in Fig. 5(b). Our numerical calculations showed that at ω = 0.75ω_p, k_i = 2.766k_p for the lossless case, and k_i = 2.762k_p and 2.726k_p for the lossy cases with ν = 0.001ω_p and 0.01ω_p, respectively. The decaying lengths for the two lossy cases are 8.64 and 8.76 μm, which are remarkably larger than the longitudinal sizes of the trapped spots, especially in the latter case with ν = 0.01ω_p. Therefore, the trapped spot is not simply an evanescent mode of SMPs, and its fields should be formed by the summation over the evanescent mode of SMPs and evanescent modes sustained by the dielectric layer, which correspond to transverse resonances of different orders and have real transverse components of wave vector in the dielectric layer.

Fig. 5 Dispersion relation for backward propagating and evanescent modes of SMPs in the SDM structure. (a) Real part, (b), imaginary part. Solid lines correspond to the lossless case, and dashed lines with circles correspond to the lossy case where ν = 0.01ω_p and the metal is Ag. The other parameters are the same as in Fig. 1(c).

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Because the proposed system can stop one-way SMPs over a broad band [0.64ω_p,0.88ω_p], it can also completely trap terahertz pulses. To verify this, the finite-difference time-domain (FDTD) method was employed to simulate the transmission of a terahertz pulse in this system. The FDTD simulation equations in the gyroelectric semiconductor can be written as

\begin{array}{l} \frac{\partial E_{y}}{\partial x} - \frac{\partial E_{x}}{\partial y} = - μ_{0} \frac{\partial H_{z}}{\partial t}, \\ \frac{\partial H_{z}}{\partial y} = ε_{0} ε_{\infty} \frac{\partial E_{x}}{\partial t} + (J_{x}^{-} + J_{x}^{+}) + i (J_{y}^{-} - J_{y}^{+}), \\ - \frac{\partial H_{z}}{\partial x} = ε_{0} ε_{\infty} \frac{\partial E_{y}}{\partial t} - i (J_{x}^{-} - J_{x}^{+}) + i (J_{y}^{-} + J_{y}^{+}), \\ \frac{\partial J_{x, y}^{\pm}}{\partial t} + (ν \mp i ω_{c}) J_{x, y}^{\pm} = \frac{1}{2} ε_{0} ω_{p}^{2} E_{x, y} . \end{array}

If the fields of a pulse traveling in the system evolve into a stable state, the trapping effect will be vividly demonstrated. For this purpose, the system was first assumed to be lossless in our FDTD simulation, i.e., ν = 0 and the metal was a PEC. The length of the guiding part (i.e., the SDM structure) was taken to be L = 600 μm, and a magnetic current line source was placed 100 μm from its left end. The magnetic current line source varied with time as I_m = exp[−(t −t₀)²/τ²]exp(−i2π f₀t) (t ≥ 0), where τ = 4T_p, t₀ = 10T_p (T_p = 2π/ω_p), and f₀ = 1.5 THz. For this Gaussian pulse, the spectral half width, which is given by Δ f = 1/(πτ), was found to be 0.16 THz. The simulated E amplitudes at different evolution times are displayed in Fig. 6. Figure 6(a) shows the field pattern at t = 10T_p when the magnetic current source reaches its maximal amplitude, and Fig. 6(b) shows the results at t = 20T_p when the source finishes its excitation. At this time, the excited wave packet is centered at z = 217.5 μm, and it has a longitudinal size of δ_l = 195.5 μm and a transverse size of δ_t = 15 μm. The wave packet only travels forward and simultaneously broadens owing to the SMP dispersion effect, as shown in Fig. 6(c) for t = 26T_p, at which the wave packet is centered at z = 294 μm and has δ_l = 283.2 μm. When the wave packet begins to touch the interface of the SDM and SM structures, the field amplitudes nearby grow rapidly; meanwhile, it starts to be compressed, as shown in Fig. 6(d) for t = 32T_p. Figure 6(e) shows the field pattern for t = 48T_p, and at this time, the wave packet has almost been compressed to its half (longitudinal) size. The wave packet becomes a subwavelength hotspot when t = 66T_p, as shown in Fig. 6(f), and since then, it tends to be stable. In this situation, the trapped spot is centered at z = 596.75 μm, and its sizes are δ_l 6 μm and δ_t ≈ 0.8 μm. The maximal field amplitude of the trapped spot in Fig. 6(f) is nearly a factor of 17 larger than that in Fig. 6(b), where the initial wave packet was just formed. We also performed the FDTD simulation of the pulse transmission in a lossy system, in which ν = 0.001ω_p and the metal was Ag, and found that the field patterns at the various evolution times in Fig. 6 are almost identical to those in Fig. 6. Compared to the lossless case, the maximal field amplitude of the trapped spot drops by 13.6% at t = 66T_p, and when t = 100T_P, it drops by 22.3%. In the lossy case, the trapped spot has no stable state, and it finally vanishes when the evolution time is enough long.

Fig. 6 (a)–(f) FDTD simulated E amplitudes at different evolution times: (a) 10T_p (I_m = 1), (b) 20T_p (when the excitation finishes), (c) 26T_p, (d) 32T_p, (e) 48T_p, and (f) 66T_p. (g), (h) Distributions of E amplitude along the semiconductor surface for the various evolution times. The parameters of the Gaussian pulse are t₀ = 10T_p, τ = 4T_p, and f₀ = 1.5 THz. The lengths of the guiding and stopping sections are respectively 600 and 100 μm, and the magnetic current line source is placed at x = 6 μm and z = 100 μm. The other parameters are the same as in Fig. 2.

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4. Conclusions

In summary, we have demonstrated an approach for completely stopping terahertz radiation by using a gyroelectric semiconductor. It has been shown that one-way SMPs at terahertz frequencies can be completely stopped without any backscattering, and consequently a hotspot with strongly enhanced fields occurs on a subwavelength scale. As the proposed system for stopping one-way SMPs works over a broad band, terahertz pulses can also be completely trapped within it, and meanwhile, they are compressed to subwavelength scales. Compared to the previous adiabatic stopping approaches, the present one is a forcible method to trap EM radiation, thus it is available even for the cases with serious material losses. The proposed approach for stopping terahertz radiation has promising applications in various areas of terahertz technology, such as spectroscopy, sensing, enhanced nonlinearity, and enhanced absorption.

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), and, the Science Foundation of Zhejiang Province (Grant No. LY14F030013).

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Stopping terahertz radiation without backscattering over a broad band

Abstract

1. Introduction

2. Physical model

3. Numerical simulation and analysis

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (6)

Equations (11)

Optics Express