An asymmetric system based on gyrotropic material allows manipulation of electromagnetic (EM) wave scattering, transmission, and propagation in a nonreciprocal and tunable manner ^[1–5]. The nonreciprocal property of the light–matter interaction has been intensely studied with the development of nontrivial physical phenomena (e.g., breaking the time-bandwidth limit) ^[6] and nonreciprocal photonic devices (e.g., optical diode and circulator) ^[7,8]. More importantly, the time-reversal symmetry broken by the gyrotropic material in the external static magnetic field has vast potential for the realization of photonic topological insulators ^[9,10], which is a promising material and a current research hotspot.

Nonreciprocal behaviors of Cherenkov radiation and dipole emission in a half-plane gyrotropic material have been investigated in recent work ^[11,12]. In these systems, the surface plasmon polaritons (SPPs) are excited directionally around the associated plasmon resonances. Therefore, the directional propagation of the EM modes is stringently dependent on the operation wavelength ^[13–17]. Due to the single boundary between the gyrotropic material and the adjacent dielectric, only one surface mode can be supported in each propagation direction. This inevitably restricts the freedoms of the nonreciprocal devices owing to the limited working channels. In this work, we find that the working channels can be multiplied by adopting the gyrotropic-based heterostructure. A plasmonic material (e.g., metals) is added on the top of the dielectric loaded with a gyrotropic material. The EM modes induced on the surfaces of the plasmonic material and gyrotropic material can be strongly coupled to each other, and thus generate nonreciprocal supermodes when their coupling strength is larger than the dissipation of the system ^[18,19]. These nonreciprocal supermodes can extend the waveband channels for additional nonreciprocal operations.

Moreover, it is a hot research topic currently to enhance and separate the coherent high-order harmonic produced during the nonlinear process with strategies such as backward phase matching in negative-index materials ^[20–26]. But the conditions of strategies such as backward phase matching are rather strict, requiring a unique dispersion relation of the materials and a specific wavevector of the incidence (e.g., n(ω)=−n(2ω), 2k(ω)=k(2ω), where n is the refractive index and k is the wavevector ^[23]). Thus, the perturbation of the incidence angle can drastically deteriorate nonlinear conversion efficiency. In this paper, we show that the extraction of nonlinear signal by adopting the nonreciprocal system can overcome the previous problem. The propagation direction of the EM mode at a given wavelength is intrinsically determined by the nonreciprocity of the asymmetric system, but unrelated to the wavevector of the source. Thus, the high-order harmonic and the fundamental signal can be separated by tuning the fundamental frequency to an appropriate one. Furthermore, we show that when the transparent dielectric is replaced by a second-order nonlinear material in our multi-channel nonreciprocal system, the second harmonic can be emitted and propagate antiparallelly to the fundamental signal in two different working wavebands, enhancing the freedom of the photonic dichroism.

The nonreciprocal waveguide is constructed with a transparent dielectric slab with a thickness of 2d, loaded between a plasmonic material and a gyrotropic material, as shown in Fig. 1(a). The dielectric function of the gyrotropic material in the static magnetic field B is expressed as ^[2,6,25] where ϵt=1−ωp12(ω+iγ1)ω(ω+iγ1)2−ωωc2, ϵg=1ωωcωp12ωc2−(ω+iγ1)2, and ϵa=1−1ωωp12ω+iγ1. Particularly, ω is the operation frequency of the incident light, ωc is the cyclotron frequency and proportional to the magnetic field of the incident light, ωp1=2×1015rad/s is the plasmon frequency of the gyrotropic material, γ1=0.01ωp1 is the corresponding collision frequency, and ϵ∞1=1 is high-frequency permittivity. Meanwhile, the polarization of magnetic field B is along the z direction. The dielectric function of the plasmonic material is given by the Drude model as ^[26] Here, we assume that the plasmon frequency ωp2=1.1ωp1, collision frequency γ2=0.01ωp2, high-frequency permittivity ϵ∞2=1, and relative permittivity of the dielectric medium is ϵd=5. The thickness of the dielectric is set as 2d=50nm to make the coupling strength of the EM modes larger than the loss rate of the system ^[18,19]. In the end, the strong coupling between the EM modes on the adjacent interfaces leads to the formation of supermodes [i.e., the even mode and odd mode shown in Fig. 1(b)].

In a source-free system, the tangent field components in the frequency domain are of the form ^[11] Here, BTM± and CTM± represent reflection coefficients and transmission coefficients of the transverse magnetic field, and their superscripts + and − denote the upper interface and lower interface, respectively. Meanwhile, Sd=kydεd, Sm=kymεm, Sg=εtkyg−(iεg)kxεt2+(iεg)2, kx is the wavevector along x direction, kyd=ϵdk02−kx2, kym=ϵmk02−kx2, and kyg=εeffk02−kx2(k0 is the wavevector of the incident light in free space). The effective permittivity of the gyrotropic material can be obtained as εeff=εt2+(iεg)2εt. The continuity of the tangent field components (i.e., Hzf and Ezf) cancels out all the reflection and transmission coefficients, so that the eigenmode equation can be derived as Then, we assume that a linearly polarized emitter with x-oriented dipole momentum of pe is placed on the center of the waveguide. The Hz component and Ex component of the dipole source within the dielectric slab (−d<y<d) can be expressed as ^[11] Finally, components of the total field are combined as respectively. In this case, the reflection coefficients BTM± and the transmission coefficients CTM± are determined by matching the boundary conditions of the total field (i.e., the continuity of Ext and Hzt) as Here, A=−iωpekx4πkyd is the amplitude of the dipole source.

As shown in Fig. 1(b), without the static magnetic field (the cyclotron frequency ωc=0), the dispersion relations of the guiding modes are symmetric with two branches. They are, respectively, odd modes (blue line) and even modes (red line) induced by the strong coupling between the SPPs at adjacent interfaces ^[27]. We notice that the odd modes have a positive equivalent refractive index (neff>0), and the corresponding energy flux propagates parallel to the wavevector of the SPPs, whereas the energy flux and the wavevector are antiparallel to each other for the even modes whose equivalent refractive index is negative.

By introducing a non-zero static magnetic field in the gyrotropic material, the symmetric dispersion relations shown in Fig. 2(a) are broken, which leads to a unidirectional emission of the dipole. In fact, the directionality of the EM modes is determined mainly by the dominant mode confined on the structure and the sign of the corresponding equivalent refractive index neff. Accordingly, we classify the operation wavebands as channel 1, channel 2, channel 3, and channel 4, respectively, showing different nonreciprocal behaviors. The corresponding emission field is given in Fig. 2(b), calculated by Eqs. (8)–(13). In channel 1, the dipole source emits both forward (+x direction) and backward (−x direction) EM modes. However, when kx>0, the dispersion curve is closer to the light line. Therefore, the EM field with a positive wavevector is weakly confined in the waveguide, resulting in a fast decay in free space. Owing to the direction of the energy flux coinciding with that of the wavevector (neff>0), the backward emission is dominant [see Fig. 2(biv)]. Although the EM field in channel 4 is confined more effectively in the waveguide when kx>0, the negative neff implies that the EM modes of the positive wavevector are mainly backward propagating [see Fig. 2(bi)]. It is also seen that channel 2 supports only the EM mode of the positive wavevector, while the EM modes of the negative wavevector are supported only in channel 3. Due to the opposite sign of the neff in these two situations, the emission of the dipole is routed forward in channels 2 and 3 [see Figs. 2(bii) and 2(biii)].

The directionalities of the nonreciprocal waveguide in different waveband channels are evaluated and shown in Fig. 3. The forward and backward directionalities are determined by Eqs. (14) and (15), respectively: in which PL(ω) [P−L(ω)] denotes the power density harvested at the distance of L (−L) away from the dipole source. It is shown that the directionality is around 95% in channel 1 and over 70% in channel 4. However, the directionalities of these two channels are lower than those in channels 2 and 3 (the directionalities in channels 2 and 3 are 100%) because in channels 1 and 4, the EM emissions are weakly unidirectional, while in channels 2 and 3, the EM emissions are truly unidirectional.

To demonstrate the separation of high-order harmonic from the fundamental signal in our proposed multi-channel nonreciprocal waveguide, we replace the transparent dielectric by a second-order nonlinear material with the nonlinear susceptibility χ(2)=2pm/V. We implement the nonlinear model by COMSOL Multiphysics 5.3a. The method of undepleted pump approximation is used to simplify the problem, assuming that the nonlinear polarization is much smaller than the linear polarization. Under this assumption, we obtain the second-harmonic field ESHG generated by the defined external current as Jext=i2ωχ(2)EFF2, where EFF is the fundamental electric field ^[28,29]. When the fundamental signal is propagated mainly along the −x direction (backward) with the frequency ω=0.2ωp1 in channel 1, the generated second harmonic with frequency 2ω=0.4ωp1 is totally propagated along the +x direction (forward) in channel 2 [see Fig. 4(a)]. In contrast, when the fundamental signal with the frequency ω=0.4ωp1 in channel 2 is propagated along the +x direction (forward), the induced second harmonic with frequency 2ω=0.8ωp1 is emitted mainly in the −x direction (backward) in channel 4 [see Fig. 4(b)].

Figure 5 shows the power spectra of the fundamental signal (blue line) and the induced second harmonic (red line) corresponding to the above two schemes. For the first scheme, the power of the second harmonic reaches a maximum value of 8.8×10−5W/m when the fundamental frequency is ω=0.177ωp1. But it vanishes when the operation frequency is beyond the cutoff frequency of channel 1. In the second scheme, the induced second harmonic has a peak power value of 1.65×10−5W/m when the fundamental frequency is ω=0.42ωp1. However, the operation bandwidth of the second harmonic is limited by the narrow operation waveband of channel 4. Moreover, we unveil that the second harmonic generated by the first scheme is more effective than that induced by the second scheme. This is attributed mainly to the weaker directionality of channel 4 than that of channel 1.

Finally, we present a possibility to construct such a multi-channel nonreciprocal waveguide using doped indium antimonide (InSb) and strained silicon (Si), shown as the inset in Fig. 6(a). The InSb in the top layer behaves as a plasmonic material in the terahertz frequency band ^[30]. Under the static magnetic field, the InSb in the bottom layer exhibits the features of gyrotropic material. The critical optical parameters of the InSb are obtained from Ref. ^[6] (i.e., ϵ∞=15.6, and ωp=4π×1012rad/s). The cyclotron frequency of the InSb is determined by ωc=eBm*, where the effective mass of the InSb m*=0.0126m0 (e is the unit of the electron charge, and m0 is the remaining mass of the electron) ^[31]. Moreover, the thickness of the Si slab in the center is 2d=50nm, relative permittivity of the Si is ϵd=11.68, and second-order susceptibility of the Si is selected as χ(2)=2pm/V, which can be tuned with different strain levels ^[32]. With the static magnetic field enhanced from 0 to 0.54 T, the waveguide system shows increasing nonreciprocity in the terahertz frequency band, and the wavebands of working channels are changed accordingly [see Fig. 6(a)]. This allows the design of photonic dichroism in the terahertz frequency band flexibly. For instance, if B=0.54T, the quantum dot (QD) with the frequency of 1.13 THz can emit light only in the forward direction, while its second harmonic is routed backward [see Fig. 6(b)].

In conclusion, we proposed a four-channel nonreciprocal waveguide composed of a plasmonic material layer, a transparent dielectric layer, and a gyrotropic material layer. We have shown rigorously that the emission of the dipole source can be unidirectionally routed at will by controlling the operation frequency of the proposed waveguide. Notably, truly unidirectional emission is realized in channels 2 and 3, whereas weakly unidirectional emission occurs in other channels. We anticipate that additional unidirectional coupling could be observed in similar waveguide structures if the linear dipole is replaced by a spinning dipole ^[33,34]. The associated spin-orbit interaction can also bring intriguing effects, such as spin-Hall effect of light ^[35,36], which is a promising way to precisely locate nanoparticles or probe local optical fields. In addition, our numerical results unambiguously reveal that the high-order harmonic from the fundamental signal is separated during the nonlinear process by applying the multi-channel nonreciprocal waveguide. Our proposed multi-channel nonreciprocal waveguide offers significant guidance for the design of novel photonic dichroism with a tunable waveband, high integration, and multiple channels.