Broadband one-way propagation and rainbow trapping of terahertz radiations

Surface magnetoplasmon (SMP) supported at an interface between magnetized plasmonic and dielectric materials has been widely explored; however, it suffers with narrow bandwidth for one-way propagation. Here we propose a novel metal-semiconductor-dielectricmetal (MSDM) structure showing the large bandwidth for the complete one-way propagation (COWP). Because of the compression of the zone for two-way propagating modes in the semiconductor layer by reducing semiconductor thickness, the bandwidth is significantly increased by several times. More importantly, in such MSDM structure, the SMP dispersion can be engineered by controlling the semiconductor thickness, and based on this, slowing wave and trapping rainbow can be realized in a tapered system at terahertz frequencies. © 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement


Introduction
One-way propagating electromagnetic (EM) modes, which are similar to the chiral edge states found in the quantum-Hall effect [1], have received strong interest recently [2][3][4][5].One-way EM modes are such modes that are allowed to propagate in only one direction, and because of the absence of backpropagating mode in the waveguide system, they are immune to backscattering at imperfections or bends.One-way modes provide a basic mechanism for realizing new classes of photonic devices that would be impossible using conventional reciprocal modes [6].It was reported that one-way EM modes can be sustained by the edges of two-dimensional photonic crystals made of magnetic-optical (MO) materials, where time-reversal symmetry is broken by applying dc magnetic field [7][8][9][10].The existence of such modes has been experimentally verified by using (gyromagnetic) yttrium-iron-garnet (YIG) in the microwave regime [11].One-way EM mode of different type, which is based on surface plasmons (SPs) [12], has also been proposed [13].It seems to be more attractive owing to its robustness in mechanism and simpleness in configuration [14][15][16].
Surface plasmons (SPs) sustained by the interface of plasmonic material and dielectric become nonreciprocal under an external dc magnetic field [17,18].When the cyclotron frequency of electron (ω c ) in the plasmonic material is comparable with the plasma frequency, the SP asymptotic frequency differs remarkably in the forward and backward directions, making SP propagate unidirectionally within the interval between the two different asymptotic frequencies [13,15].SPs in magnetized plasmonic material is generally referred to as surface magnetoplasmons (SMPs) [17].Generally a semiconductor possesses a plasma frequency in terahertz regime and the electron effective mass is far less than the electron mass, thus the electron cyclotron frequency is often comparable with the plasma frequency under normal magnetic field intensity.Therefore, one-way SMPs is realizable at terahertz frequencies by using semiconductors [14].In a semiconductor-dielectric-metal (SDM) layered structure, it was shown that one-way SMPs can be immune to backscattering, thus exhibiting the complete one-way propagation (COWP) [15,19].
It is reported more recently that by using robust one-way SMPs, extraordinary cavities that breaks time-bandwidth limit in physics can be constructed [6].However, in the previous SDM structure, the COWP bandwidth of SMPs is very limited, e.g., the maximal value is only 0.03ω p (where ω p is the plasma frequency) in the case if the dielectric is air.In this paper, we will investigate theoretically SMPs in a metal-semiconductor-dielectric-metal (MSDM) layered structure.It will be shown that the semiconductor thickness in the MSDM structure can greatly affect the COWP band.Compared with the previous SDM structure, the dependence of the COWP bandwidth on ω c is significantly improved, thus broadband SMPs become available in the MSDM structure.It will also be shown that in the MSDM structure, the COWP bandwidth can be largely increased by optimizing the semiconductor thickness.Moreover, interesting phenomenon based on the effect of the semiconductor thickness is numerically demonstrated.
Slowing EM waves is believed to be an attractive technique for enhanced nonlinear optics [20], light harvesting [21], and optical signal processing [22,23].Recently, the concept of a "trapped rainbow" has generated considerable interest for potential use in optical data storage and processing [23].It aims to trap different frequency components of the wave packet at different positions in space permanently.More recently, it was reported that trapped rainbow can be realized by using SMPs [24].In our proposed MSDM structure, it will be shown that the SMP dispersion can be engineered by controlling the semiconductor thickness.Based on that mechanism, slowing wave and trapping rainbow can be achieved in a tapered MSDM system.Different from Ref. 21, where a tapered external magnetic field is required, only the semiconductor thickness is tapered in our system for trapped rainbow, so our approach seems to be more feasible in practice.The phenomena of trapped rainbow in a tapered MSDM system will be numerically demonstrated.

Physical model
The physical model for sustaining broadband one-way SMPs at terahertz frequencies is illustrated in Fig. 1(a).This waveguide system is a metal-semiconductor-dielectric-metal (MSDM) layered structure, with an external dc magnetic field applied to break the time-reversal symmetry.The metal is assumed to be perfectly electric conductor, which is a good approximation for the terahertz regime.The dielectric layer with the relative permittivity ε r has a thickness of d 1 , and the semiconductor layer has a thickness of d 2 .The external magnetic field introduces gyroelectric anisotropy in the semiconductor, thus its relative permittivity has the form [15,17] with where ω is the angular frequency, ω p is the plasma frequency, ω c = eB 0 /m * (where e and m * are, respectively, the charge and effective mass of a electron) is the electron cyclotron frequency, and ε ∞ is the high-frequency (relative) permittivity.Compared with the previous system for one-way SMPs, which is a semiconductor-dielectric-metal (SDM) layered structure as shown in Fig. 1(b), the semiconductor is terminated by a metal slab in the present system.Therefore, the EM modes in the semiconductor become bound modes in the present system, which must satisfy transverse resonance condition, and their modal properties should be closely dependent on the semiconductor thickness [25].These bound modes are guided by means of zigzag reflections at the metal surface and semiconductor-dielectric interface, and they are usually allowed to propagate both forward and backward.We refer to these two-way (propagating) modes as normal modes in order to distinguish them with SMPs.We first investigate SMPs in the system under our consideration.The EM fields of SMPs are H-polarized, and the nonzero component of the magnetic field can be written as in the dielectric layer for 0 < x ≤ d 1 and (3) where k 0 is the wavenumber in vacuum), and /ε 1 being the Voigt permittivity.The nonzero components (E x and E z ) of the electric field can be obtained straightforwardly from H y .The tangential component E z must vanish at the metal boundaries . The boundary conditions further require the field components E z and H y to be continuous at the interface x = 0, and based on which we can obtain (4) which is the dispersion relation for SMPs.The linear term in Eq. ( 4) with respect to k indicates that SMPs are nonreciprocal in the system.From Eq. ( 4) we find for k → +∞, and for k → −∞.Evidently, ω (1)  sp and ω (2)  sp represent the two asymptotic frequencies of SMPs (in the opposite directions) which are supported by the interface of the semiconductor and dielectric, and they are identical to those in the previous SDM system [15].The dispersion relation of SMPs in the SDM system reads which can be obtained from Eq. ( 4) by letting d 2 → ∞. ω (3)  sp in Eq. ( 7) corresponds to the asymptotic frequency of another SMP branch, which is supported by the interface of the semiconductor and metal in the MSDM system.In this paper, the semiconductor is assumed to be InSb with ∞ = 15.6 and ω p = 4π × 10 12 rad/s [26].The dielectric layer is air with ε r = 1, and its thickness is set to be d 1 = 0.1λ p (where λ p = 2πc/ω p with c being the light speed in vacuum), for which it can sustain no mode which is guided by a means of zigzag reflections at the surfaces of the metal and semiconductor [15].
Figure 2(a) shows the dispersion relation of SMPs in the MSDM system.As an example, the parameters are taken as follows: d 2 = 0.2λ p and ω c = 0.25ω p (i.e., B 0 = 0.25T).For comparison, the dispersion relation of SMPs in the corresponding SDM system is plotted in Fig. 2(b).Obviously, for SMPs supported by the semiconductor-dielectric interface (solid lines), the dispersion curves are similar for k < 0 in the two systems, but they are rather different for k > 0 because of difference between the normal-mode and bulk-mode zones.Note that the dispersion branch of SMPs with k > 0 are actually cut into two segments by the normal-mode zone or bulk-mode zone.In the SDM system, bulk modes in the semiconductor are space waves, which are described by k 2 x + k 2 z = √ ε v k 0 .The bulk-mode zones in the SDM system are represented by the shaded areas in Fig. 2(b), and there are two zones where ε v > 0. The lower bulk-mode zone has a low-frequency cutoff given by which determines the upper limit of the COWP region.For ω c = 0.25ω p , we find that ω c f ,b = 0.8828ω p .In the present MSDM system, normal modes guided by the semiconductor layer are somewhat similar to bulk modes, whose transverse components (k x ) of wavevectors are real-valued in the semiconductor, and the dispersion relation for them can be derived from Eq. ( 4) by letting k = k z and α s = ik x (where k x and k z are both real-valued), and it has the form where which is allowed to be real or imaginary number).Note that k x and k z must also satisfy the dispersion relation for the semiconductor, i.e., k 2 x + k 2 z = √ ε v k 0 .Therefore, for a given ω, there exist only discrete solutions for k z or k x , which correspond to the transverse resonances of different orders in the MSDM system.With Eq. ( 10), the normal mode with the lowest order, which possesses the lowest cutoff, is calculated and plotted in Fig. 2(a) as dotted line.For consistency, this cutoff is also denoted by ω c f ,b , and it is found to be ω c f ,b = 0.933ω p , which is obviously larger than the value for the SDM system.

COWP bandwidth of SMPs
As in the SDM system, there exists a frequency range in the MSDM system, where SMPs with k > 0 exhibit complete one-way propagation (COWP).The COWP band must lie within the band gap of bulk or normal modes in the semiconductor.In both Figs.2(a) and 2(b), the COWP band ranges from ω (2)  sp to ω c f ,b , and its bandwidth is ∆ω cow p = ω c f ,b − ω (2)  sp .The COWP region is indicated by the shaded rectangle.We find that ∆ω cow p = 0.0806ω p in Fig. 2(a) and ∆ω cow p = 0.0304ω p in Fig. 2(b).The COWP bandwidth in the MSDM system is 2.65 times larger than that in the SDM system.But note that ∆ω cow p = ω c for both the systems if ω (1)  sp ≤ ω c f ,b .Both ω (1)  sp and ω c f ,b are closely dependent on ω c , and we denote the critical ω c value by ω c,1 , at which ω (1)  sp = ω c f ,b .For the SDM system, it is found that and it has ω c,1 = 0.0306ω p (B 0 = 0.0306T) for ε r = 1.For the MSDM system, however, our numerical calculation shows that ω c,1 = 0.04956ω p (B 0 = 0.04956T) when d 2 = 0.2λ p . Figure 3(a) shows the dependence of ∆ω cow p on ω c for the both systems, where the parameters are the same as in Fig. 2 (except for ω c ).For the MSDM system, ∆ω cow p = ω c when ω c ≤ ω c,1 , where ω c,1 = 0.04956ω p , and interestingly, ∆ω cow p grows with ω c even when ω c > ω c,1 , which is quite different from the situation for the SDM system.For the SDM system, ∆ω cow p decreases with ω c when ω c > ω c,1 , and it has a maximum of 0.0306ω p at ω c = ω c,1 , where ω c,1 = 0.0306ω p .For the MSDM system with d 2 = 0.2λ p , when ω c grows from ω c,1 to 0.5ω p (B 0 = 0.5T), ∆ω cow p increases from 0.04956ω p to 0.1ω p , as seen in Fig. 3(a).So large COWP bandwidth can be achieved for the MSDM system by increasing external magnetic field.However, as seen in Fig. 3(a), when ω c further increases and is larger than a certain value, which we denote by ω c,2 , ∆ω cow p begins to quickly decrease with ω c .This critical value for ω c is found to be ( 12) at which ω (2) sp = ω (3)  sp .We find that ω c,2 = 0.6855ω p for ε r = 1.Note that for the MSDM system ω c,1 depends on d 2 but ω c,2 does not.In the case when ω c > 0.6855ω p , ω (3)  sp > ω (2)  sp , thus the lower COWP limit is determined by ω (3)  sp instead of ω (2)  sp .To show this, the dispersion diagram for the MSDM system with ω c = 0.75ω p (B 0 = 0.75T) is displayed in Fig. 3(b).
As the cutoff frequency ω c f ,b shifts up in the MSDM system when d 2 decreases, it is naturally desired that the COWP bandwidth can be effectively increased by reducing d 2 .To clarify this, the MSDM systems with different d 2 values are analyzed.The other parameters of the MSDM systems are kept to be ω c = 0.5ω p (B 0 = 0.5T), d 1 = 0.1λ p , and ε r = 1.Figures 4(a)-4(d) show the dispersion diagrams for the cases of d 2 = 0.2λ p , 0.1λ p , 0.05λ p , and 0.03λ p , respectively.Obviously, the cutoff of the lowest-order normal mode, which determines the upper COWP limit (ω u ), shifts up when d 2 is reduced.We find that ω c f ,b = 0.7511λ p for d 2 = 0.2λ p , ω c f ,b = 0.7669λ p for d 2 = 0.1λ p , ω c f ,b = 0.8824λ p for d 2 = 0.05λ p , and ω c f ,b = 0.9864λ p for d 2 = 0.03λ p .However, in the later three cases, the SMP dispersion curve with k < 0 is out of shape, and a peak appears at a finite |k | value.In this situation, the lower limit (ω l ) of the COWP band is determined by the top of that peak instead of ω − sp .The dependences of ω u and ω l on d 2 are shown in Fig. 5(a).ω u decreases with d 2 , and this is also the situation for ω l in the case when d 2 < 0.14λ p .But ω l becomes a constant of ω (2)  sp when d 2 ≥ 0.14λ p .In this case, the peak of the dispersion curve with k < 0 disappears, and its highest frequency is the asymptotic frequency ω (2)  sp , by which ω l is determined.From the above analysis, it is clear that the COWP band is greatly affected by the semiconductor thickness.To vividly illustrate this, by using the finite-element method (FEM), we performed the simulation of wave transmission in the MSDM system, and three cases were considered: d 2 = 0.2λ p , d 2 = 0.1λ p , and d 2 = 0.03λ p .The other parameters are the same as in Fig. 4. In the simulation, a linear magnetic current, which lay at d 1 /2 (i.e., 7.5 µm) above the semiconductor surface, was used to excite waves, and its frequency was set to be f = 1.8 THz, i.e., ω = 0.9ω p .To obtain a steady-state solution in the frequency domain, the semiconductor loss was taken into account.For lossy semiconductor, the expressions for ε 1 and ε 2 in Eq. (1) become where ν is the electron scattering frequency.We took ν = 0.01ω p in the simulation.Figures ], exhibiting the one-way propagation behaviour.However, in the case when d 2 = 0.03λ p , the situation [see Fig. 6(c)] seems to revert to the first case of d 2 = 0.2λ p .These phenomena are in good agreement with the dispersion properties of the MSDM system in Fig. 4, where the frequency ω = 0.9ω p is indicated by a horizontal dashed line.
To verify the robustness of the one-way propagation in the MSDM system, we further introduced an obstacle into the MSDM system with d 2 = 0.1λ p .The obstacle consists of a air and semiconductor columns, which are respectively located below and above the semiconductor surface at a distance of 320 µm from the source [see Fig. 6(d)].Both columns have the same rectangular cross-section with length 10 µm and width 5 µm.The simulated results are plotted in Fig. 6(d).Evidently, when propagating-forward wave (excited by the source) reaches the obstacle, it goes around the obstacle.The backscattering of wave by the obstacle is completely suppressed, because there exists no backward-propagating mode in this MSDM system.Compared with Fig. 6(b), the field of SMPs in Fig. 6(d) is only modified locally around the obstacle.In Fig. 6(d), the fields in front of and behind the obstacle are almost identical.This is more clearly displayed in Fig. 6(e), where the magnetic field amplitudes along the semiconductor surface (x = 0) are plotted for the cases in Figs.6(b) and 6(d).
Finally, we examine the influence of semiconductor loss on SMPs and make a related comparison between the MSDM and SDM systems.The parameters of the systems are taken as follows: ω c = 0.5ω p (B 0 = 0.5 T), d 1 = 0.1λ p , d 2 = 0.2λ p and ν = 0.01ω p .In the lossy case, the propagation constant of SMPs becomes complex, i.e., k = k r + ik i (where both k r and k i are real-valued), and it can be accurately solved from Eq. ( 4).The propagation length of SMPs can be defined by L p = 1/(2k i ).For the frequency f = 1.52 THz (i.e., ω = 0.76ω p ), we find that L p = 3024 µm for the MSDM system and L p = 2403 µm for the SDM system.Compared with the SDM system, the portion of the EM energy of SMPs in the semiconductor should be substantially reduced for the MSDM system, therefore SMPs has larger propagation length there.7. SMP bands (with the asymptotic frequency ω (2)  sp ) for different d 2 values.From top to bottom: d 2 = 0.02λ p , 0.032λ p , 0.045λ p , 0.06λ p , and 0.09λ p .ω c = −0.5ωp , and the other parameters are the same as in Fig. 2.

Trapped rainbow in
It is interesting that one branch of SMPs with the asymptotic frequency ω (2)  sp in the MSDM system is humped in the negative k region when the semiconductor layer is thin enough.This humped SMP band can be transferred to the positive k region by reversing the external magnetic field, as shown in Fig. 7, where ω c = −0.5ωp .In such humped band, the group velocity (v g = dω/dk) vanishes at the top of the peak, i.e., at the highest frequency for SMPs propagating forward in the case of ω c < 0, and we denote this frequency by f 0 .Obviously, f 0 is larger than ω (2)  sp described in Eq. ( 6).Furthermore, f 0 is a function of d 2 , i.e., f 0 = f 0 (d 2 ), and it decreases when d 2 increases, as illustrated in Fig. 7, where the cases with various d 2 values are displayed.Suppose that f 0 decreases with the propagation distance in a tapered MSDM structure, where the semiconductor thickness is linearly increased along the propagation direction.In such tapered structure, an incident wave with a certain frequency f would be slowed down as it propagates forward.When it reaches a critical point, at which f 0 is equal to f , it would be stopped because its group velocity vanishes there.Behind the critical position, f 0 < f , so the wave is prohibited to propagate forward.Evidently, waves with different frequencies would be stopped at different positions along the propagation direction, and a phenomenon similar to the "trapped rainbow" would occur [23].
To verify the phenomenon of trapped rainbow, we performed the simulation of wave transmission in a properly designed system with the FEM.The system consists of two sections, one is a uniform MSDM structure with d 2 = 0.03λ p and the length 200 µm, the other is a tapered MSDM structure, where d 2 is linearly increased from 0.03λ p to 0.09λ p over its length 500 µm.In the simulation, we took ω c = −0.5ωp , and other parameters of the system were the same as in Fig. 6.The source (i.e., linear magnetic current) was placed at d 1 /2 above the semiconductor surface in the uniform section, and four different frequencies were considered: f = 1.6 THz, 1.7 THz, 1.8 THz, and 1.9 THz.The simulated electric-field amplitudes for the four frequencies are plotted in Figs.8(a)-8(d), respectively.As expected, waves at the four frequencies are all stopped in the tapered section, and the one with lower frequency is stopped at a larger distance from the source.At the position where a wave stops, a hotspot with significantly enhanced electric field is generated for each case.Figure 8(e) shows the electric field amplitudes along the semiconductor surface for the four cases.Obviously, the electric field is only increased by several times at the hotspot, compared with the fields in the uniform section.We think that the field enhancement is so limited because the hotspot has a large longitudinal size, as shown in Fig. 8.The hotspot occurring at larger d 2 has a larger field enhancement, which means that the related SMPs have a smaller modal size (in the transverse direction), and this agrees well with larger propagation constant at the peak of the corresponding dispersion band, as shown in Fig. 7.

Conclusion
In conclusion, we have systemically studied dispersion properties of SMPs and normal modes in the MSDM structure.It has been shown that the lower cutoff of normal modes in the semiconductor layer shifts up when the semiconductor thickness reduces, thus the COWP band for SMPs can be largely broaden.Compared with the SDM structure, the COWP bandwidth of SMPs possesses essentially different relation with the external magnetic field in the MSDM structure, thus the previous limit to its growth is removed.For the example of the dielectric being air, the COWP bandwidth in the MSDM structure can be six times larger than the maximal value in the previous SDM structure.More importantly, it has been shown that the dispersion of SMPs in the MSDM structure can be engineered by controlling the semiconductor thickness, and the group velocity of SMPs may vanishes at a certain frequency and finite propagation constant.Based on that mechanism, "trapped rainbow" has been numerically demonstrated at terahertz frequencies in a MSDM system where the semiconductor thickness is tapered along the propagation distance.

Fig. 1 .
Fig. 1.Schematics of the metal-semiconductor-dielectric-metal (a) and related semiconductordielectric-metal (b) structures for sustaining one-way SMPs.An external dc magnetic field is applied along the −y direction in the two structures.

Fig. 2 .
Fig. 2. Dispersion relations of SMPs in the MSDM (a) and SDM (b) systems.Solid and dot-dashed lines correspond to SMPs supported by the dielectric-semiconductor and metal-semiconductor interfaces, respectively.Dotted lines in (a) represent the dispersion curves for normal modes with lowest order in the semiconductor layer, and shaded areas in (b) represent bulk-mode zones for the semiconductor itself.The shaded rectangle in (a) and (b) represents the COWP region and dashed lines the light line in air.The parameters of the two systems are as follows: ε r = 1, d 1 = 0.1λ p ; ω c = 0.25ω p , d 2 = 0.2λ p .

Fig. 3 .
Fig. 3. (a) COWP bandwidth (∆ω cow p ) as a function of ω c in the MSDM system.The corresponding COWP bandwidth (dashed line) for the SDM system is included for comparison.(b) Dispersion of SMPs in the MSDM system when ω c = 0.75ω p .Solid and dot-dashed lines correspond to SMPs supported by the dielectric-semiconductor and metal-semiconductor interfaces, respectively.Dotted line represents the dispersion curve for the normal mode with lowest order in the semiconductor layer, and shaded rectangle represents the COWP region.The other parameters are the same as in Fig.2.

Figure 5 (
b) shows the COWP bandwidth (∆ω cow p = ω u − ω l ) as a function of d 2 .It is found that ∆ω cow p reaches a maximum of 0.1956ω p at d 2 = 0.08λ p , which is 6.4 times larger than the maximal value for the SDM system.

Fig. 5 .
Fig. 5. (a) Upper (ω u ) and lower (ω l ) limit frequencies of the COWP band as functions of d 2 .(b) COWP bandwidth as a function of d 2 .ω c = 0.5ω p , and other parameter are the same as in Fig. 2.

Fig. 6 .
Fig. 6. (a)-(c) Simulated magnetic field amplitudes in the MSDM systems with different d 2 values.(a) d 2 = 0.2λ p , (b) d 2 = 0.1λ p , and (c) d 2 = 0.03λ p .(d) Simulated magnetic field amplitudes in the MSDM system of d 2 = 0.1λ p but with an obstacle.The obstacle consists of two square columns of air and semiconductor, which are respectively located below and above the semiconductor surface.The white dashed line in (a)-(d) indicates the semiconductor surface.(e) Magnetic field amplitudes along a horizontal line at the semiconductor surface in (b) (dashed line) and (d) (dotted line).For each case, the source with the frequency f = 1.8 THz lies at d 1 /2 above the semiconductor surface.The semiconductor is lossy with ν = 0.01ω p .ω c = 0.5ω p , and the other parameters are the same as in Fig. 2.

6
(a)-6(c) show the simulated magnetic field amplitudes.Obviously, in the case of d 2 = 0.2λ p , waves excited by the source are allowed to propagate in both the forward and backward directions [see Fig. 6(a)], and in the case when d 2 = 0.1λ p , waves are only allowed to propagate forward [see Fig. 6(b)

Fig. 8 .
Fig. 8. (a)-(d) Simulated electric field amplitudes for different frequencies in a system consisting of a uniform and tapered MSDM structures.(a) 1.6 THz, (b) 1.7 THz, (c) 1.8 THz,and (d)1.9THz.In the system, the semiconductor thickness is d 2 = 0.03λ p in the uniform section of the length 200 µm, and in the tapered section, it linearly increases from 0.03λ p to 0.09λ p over the length 500 µm.(e) Electric field amplitudes along the semiconductor surface in (a)-(d).The source lies at d 1 /2 above the semiconductor surface in the uniform section.ω c = −0.5ωp , ν = 0.01ω p , and the other parameters are the same as in Fig.2.