1. Introduction

Surface waves (SWs), such as surface plasmon polaritons (SPPs) and spoof SPPs (SSPPs), are electromagnetic modes that are tightly confined at specific interfaces between different media [1–3]. The "two-dimensional" nature of the SWs offers significant flexibility in engineering photonic integrated circuits for optical communications and optical computing [4–8]. Effective manipulation of SWs plays a critical role in realizing these applications and remains a primary objective of frontier research. Traditional optical elements, like micro prisms and lenses, manipulate SWs by modulating the mode index of the SWs that pass through them [9,10]. However, these devices tend to be bulky in size, which makes them unsuitable for integration into optics applications. Bragg devices, made of subwavelength grooves or slit arrays on plasmonic metals, also have been proposed as an alternative means of controlling the in-plane propagation of SWs on plasmonic surfaces. Non-periodic nanoarray configurations in Bragg devices can reconstruct SW beams with complex wavefronts, leading to fascinating effects such as SW deflection [11], Airy beams [12], and beam focusing [13,14]. However, these devices face limitations in controlling wavelength-scale, generating multiple modes, and achieving quasi wavevector matching due to coherent interferences of scattered waves by periodically or quasi-periodically arranged nanostructures. Recently, metasurfaces [15–18], consisting of arrays of artificially designed sub-wavelength compositions, provide a powerful tool to control SWs. By accurately engineering its subwavelength inclusions to modulate the local wavefront, many fascinating phenomena and applications have been demonstrated, including SW excitations [17,19], beam focusing [20–22], Bessel beams [23,24], and so on [25,26]. However, most metasurfaces are driven by free-space waves, making their on-chip integration challenging.

Despite the widespread applications of traditional SWs, their susceptibility to backscattering by disorders, defects, and structural imperfections on reciprocal plasmonic platforms limits their potential for application in optical devices. Developing nonreciprocal plasmonic platforms [27] that enable unidirectional SW propagation is, therefore, of great importance. Such unidirectional SWs occur in topologically nontrivial systems made of nonreciprocal materials, where the magnetic field breaks time-reversal symmetry, and they are known as unidirectional surface magnetoplasmons (USMPs) [28–31]. These USMPs are immune to backscattering at disorders, making them ideal for realizing new classes of optical devices that are impossible using conventional reciprocal modes, such as optical cavities that overcome the time-bandwidth limit [32] and so forth [33,34]. However, so far, the coupling of USMPs and traditional SWs based on reciprocal plasmonic platforms, especially, controlling SWs with the unique propriety of USMPs, has not been reported.

In this paper, with carefully designed USMP waveguide arrays, we propose a new strategy to directly manipulate the wavefronts of SWs (Fig. 1). As the dispersion of USMPs is closely related to the structural details, hence, different phase accumulations can be achieved by controlling the structural parameters. Arranging the USMP arrays possessing designed phase accumulations distribution with traditional reciprocal plasmonic platforms, ultimately leading to the coupling from USMPs to traditional SWs, and especially to wavefront dressed SWs. Due to the unique and robust unidirectionality of USMPs, the power output from the unidirectional waveguide is independent of the waveguide length, if the material loss of the waveguide can be neglected. On the other hand, the output phase of the unidirectional waveguide can be directly determined by the propagation constant of the USMP and the waveguide length, making the design of the desired output SW wavefront simple and precise. As a proof-of-concept, we designed two metal-air-YIG unidirectional waveguide arrays working in the microwave regime, numerically demonstrated SW manipulation effects based on these devices, including Bessel beam and focusing. Furthermore, we extended the concept to the terahertz regime, where a metal-air-semiconductor unidirectional waveguide array was designed to achieve beam bending of SWs. Our research may enrich the application scenarios of SPP near-field manipulation and facilitate on-chip integration of SWs.

 

Fig. 1. Working principle of the proposed USMP waveguide array. The dispersion, and thus the phase accumulation of USMPs inside the unidirectional waveguide can be tuned individually, an array of subwavelength-spaced waveguides work collaboratively to form USMPs with certain phase distribution. Then, the USMPs are "pushed" to traditional reciprocal SWs modes on the plasmonic metal with certain wavefront and fulfill diverse functions, such as beam focusing.

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2. Coupling from USMPs to SWs

In order to investigate the proposed USMP waveguide array, we first analyze the dispersion property of the YIG-air-metal structure which can support USMPs at microwave frequencies, as illustrated in Fig. 2(a). The dispersion characteristics of this structure have been extensively studied [35–38]. We briefly summarize as follows: surface magnetoplasmons (SMPs) in this system are transverse-electric (TE) polarized, the nonzero component of electric field is $E_{y}$. The dispersion relation of SMPs in this YIG-air-metal system is

 

Fig. 2. Designs and performance of SWs coupling from unidirectional waveguide to plasmonic metal. (a) Schematic diagram of waveguides supporting USMPs at microwave frequencies, and (b) the corresponding dispersion relations of USMPs for various thicknesses of the air layer. The shaded rectangular area indicates the one-way region for the waveguide, and the other shaded areas indicate the zones of bulk modes in the gyromagnetic materials ($\varepsilon _{m}=15$ ). (c) Sample picture of the artificial plasmonic metal supporting TE polarized SWs, composed of square metallic patches and a metallic ground plane separated by a $2.4$ mm thick dielectric substrate ($\varepsilon _{r}=3.9+0.002i$, with periodicity $p=8.5$ mm). (d) FEM-simulated dispersion relation of TE polarized SW mode for plasmonic metal shown in (c). (e) Sample picture of the coupling device composed of unidirectional waveguide depicted in (a) and artificial plasmonic metal depicted in (c), and the FEM simulated results of (f) $E_y$, (g) $H_z$ and (h) $H_x$ field distribution.

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The linear term with respect to $k$ in Eq. (1) implies that the waveguide is intrinsically non-reciprocal, allowing for the unidirectional SMP propagation. It is worth highlighting that the air layer thickness $d$ plays a crucial role in managing the dispersion property. To vividly depict the relationship between the dispersion property and the air layer thickness $d$. The dispersion Eq. (1) for the waveguide was numerically solved, yielding results displayed in Fig. 2(b), where three values of $d$ were analyzed. Herein, the basic parameters of the system are taken as follows: $\omega _m=2\pi f_m=10\pi \times 10^9$ rad/s ($f_m=5$ GHz), and $\omega _0$ is set at $\omega _m$, which corresponds to $H_0=1785$ Gs. As can be seen from Fig. 2(b), the dispersion relations for SMPs in the waveguide exhibit a single branch $[1.5\omega _m,2\omega _m]$ (i.e. $[7.5,10]$ GHz here) that extends across the entire light cone in air. The positive slop of the dispersion curve unambiguously establishes the USMP nature of sustained SMPs. The bulk modes (the uppermost and lowest shaded areas) of the magnetized YIG are given by $k^2<\varepsilon _m\mu _vk^2_0$, which limit the frequency scope of USMPs. Moreover, as $d$ increases, the dispersion curve descends, enabling the tailoring of the phase constant $k$ by adjusting the thickness of the air layer. For instance, we chose 8.5 GHz as the working frequency, at $f=8.5$ GHz, $k$ increases from $-2.22k_0$ to 1.92$k_0$ as $d$ grows from $1$ mm to $3$ mm. When the loss of gyromagnetic material ($\nu =10^{-3}$) is taken into account, the propagation constant of USMP become a complex, i.e., $k=k_{r} +ik_{i}$, and the propagation length of USMP is determined by $L_{p} =1/\left ( 2k_{i} \right )$. Our calculations show that as $d$ varies from 1 to 3 mm, the propagation length $L_{p}$ increases from $4.6\lambda$ to $11.3\lambda$. Therefore, different phase accumulations can be achieved by controlling the air layer thickness $d$.

We next design an appropriate "plasmonic metal" that can support SSPPs with TE polarization to match the TE-mode USMPs at the target frequency. We design an SW eigenmode plate made of isotropic eigen-units to support TE-mode SW, as shown in Fig. 2(c). The isotropic eigen-unit consists of a metallic patch with parameters $a=7$ mm on the top of metallic plate separated by a $h=2.4$ mm thick dielectric spacer ($\epsilon _r=3.9+0.002i$, with periodicity $p=8.5$ mm). Figure 2(d) depicts the Finite element method (FEM) simulated dispersion relation of the SW mode supported by such a system, which exhibits an eigen wave-vector $k_{SW}=1.45k_0$ with $k_0$ being the free-space wave-vector at the frequency 8.5 GHz. We then connect the unidirectional waveguide with the designed plasmonic metal as schematically illustrated in Fig. 2(e). Figures 2(f)–2(h) display the FEM simulated nonzero components of the electric field ($E_y$) and the magnetic field ($H_z,H_x$), respectively. In the simulation, the USMP was excited by a line current source located at the left side of the air layer along the $y$ direction. As can be seen from Figs. 2(f)-2(h), the TE-mode USMPs sustained in the unidirectional waveguide were coupled to TE-mode SWs supported by plasmonic metal. The coupling efficiency is found as 67%, which is calculated as the ratio between the energy of output SWs supported by plasmonic metal and the input USMPs sustained by the unidirectional waveguide. Evidently, the coupling efficiency is relatively high, and the reduced efficiency primarily stems from modes mismatch between the SWs and USMPs. For a real metal at microwave frequencies, its absorption power ($P_{ab}$) can be calculated by $P_{ab} =\frac {1}{2} \sqrt {\pi \mu _{0}f/\sigma } \int _{S}^{} \left | H_{t} \right | ^{2}ds$ [40], where $H_{t}$ (obtained in the PEC approximation) is the tangential component of the magnetic field on the metal surface, $\sigma$ is the metal conductivity, $f$ is the wave frequency, and S represents the metal surface. Using this formula, we calculate the absorption power ($P_{a}$, caused by the metal) per unit length in the USMP waveguide, and then evaluate the propagation length of USMP, which is given by $L=1/\left (2\alpha \right )$, where $\alpha =P_{a} /\left ( 2P_{0} \right )$ and $P_{0}$ is the mode power. The metal is assumed to be copper with $\sigma =5.8\times 10^{7}$Sm$^{-1}$. Our calculations show that as $d$ varies from 1 to 3 mm, the propagation length of USMP increases from $7.2\lambda$ to $15.7\lambda$. For SSPPs, we calculate the absorption power ($P_{b}$, caused by the metal) in the unit cell, and then evaluate its the propagation length by $L=P_{0} p/P_{b}$, where $P_{0}$ is the SSPP power and $p$ is the period. The propagation length of SSPP is found to be $L_{SSPP} =56.5\lambda$. To sum up, for USMPs, the losses caused by the absorption of gyromagnetic materials and metallic structures are of the same magnitude. However, for SSPPs, the loss is only the absorption of the metal. As metal blocks in our periodic structure are very thin, the SSPP field cannot be tightly confined to the metal, and the field energy is mainly distributed in the air, resulting in relatively lower metal absorption loss. It is also worth noting that the USMP waveguides in this configuration may appear substitutable with conventional reciprocal waveguides. However, if the USMP waveguides are replaced with conventional reciprocal waveguides, both the output power and phase of the waveguide have complex dependence on its length, as there exist waves propagating in both forward and backward directions within the waveguide due to partial reflections at its two waveguide ends. This association between output phase and power makes the design of the output SW wavefront complicated and difficult. In contrast, unidirectional waveguides, with their robust unidirectionality, eliminate wave reflection at the output end, highlighting the indispensable role of USMP waveguides in this setup.

3. Design of USMP waveguide arrays for generating SWs with preset wavefronts

We now introduce how to build a USMP waveguide array to realize arbitrary wavefront manipulation of SWs. We first study the aforementioned waveguide slit shown in the inset of Fig. 3(a), here, the unidirectional waveguide has a length of 100 mm. FEM is utilized to simulate the coupling efficiency from USMPs to SWs in the waveguide slit and the phase accumulation of USMPs at the terminal of unidirectional waveguide with varied air thickness $d$, as shown in Fig. 3(a). Figure 3(a) shows that the phase accumulation of transmitted USMPs can cover the full range of $2\pi$ with the thickness $d$ varying from 2.8 mm to 4.2 mm at the working frequency of 8.5 GHz. Meanwhile, the coupling efficiency can also keep a high value (larger than 0.6) in the whole spectrum. With the waveguide slit in hand, we can configure the appropriate phase function $\phi _ {0} \left ( y \right )$ to achieve various fascinating wavefront manipulation phenomena of SWs.

 

Fig. 3. Demonstration of the SW manipulation concept based on USMP waveguide array. (a) FEM-simulated coupling efficiency from USMPs to SWs in the waveguide slit and the phase accumulation of USMPs at the terminal of unidirectional waveguide versus the air thickness $d$. (b) FEM-simulated near-field $E_y$ pattern on a plane above the YIG and plasmonic metal. Here, the $d$ distribution of the unidirectional waveguide arrays is designed to satisfy a phase distribution $\phi _0\left (y\right )=\xi \sin {\theta }\left |y\right |$, and $\xi =k_{SW} =1.45k_0$. In the simulation, the USMP propagate along $+x$ direction is excited by a linear electric current source. The working frequency is 8.5 GHz. (c) Normalized $\left |E_y\right |$ distribution along the line with $x=43$ mm marked as dashed red line in (b), and the comparison with theoretical formula for a zero-order Bessel functions $J_0\left (\alpha y\right )$. (d) $E_y$ distribution of Bessel beam scattered by a metallic sphere (diameter $D = \lambda /2$ ) placed at 43 mm away from the terminal of unidirectional waveguide.

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To preliminarily demonstrate the feasibility of the concept, we build a phase-gradient waveguide array to generate a SW Bessel beam as an illustration. The Bessel beam is a special type of non-diffraction beam exhibiting unique self-reconstruction properties [41,42]. The designed waveguide array is constructed by 21 waveguide slits which are terminated in the $y$ direction with subwavelength-separated ($p=8.5$ mm here) metal slabs (metal can be approximated as perfect electric conductors in the microwave regime) to prevent crosstalk. The air thickness $\left \{ d \right \}$ of the waveguide slits were determined from the database of Fig. 3(a) to make the accumulation phase satisfy distribution $\phi _{0}\left (y\right )=\xi \sin \theta \left | y \right |$, where $\xi =k_{SW} =1.45k_0$ and $\theta =45^{\circ }$ is the taper angle of Bessel beam. We performed FEM simulations to characterize the performance of the proposed waveguide array. In the simulation, the USMP was excited by a line current source along the $y$ direction. Figure 3(b) depicts the calculated $E_y$ field pattern generated on the system. It is clearly that the phase difference of USMPs between neighboring unidirectional waveguide slits increases as their propagation, and eventually the USMPs in the unidirectional waveguide array coupled to the plasmonic metal, generating SWs with a Bessel beam wavefront. Through energy flow integral, the efficiency of SW Bessel beam can be evaluated as about 67%. We also simulated the phase distribution of USMPs at the terminal of all unidirectional waveguide slits $\varphi \left (y\right )$ and plotted as black dots in Fig. 3(b). Evidently, the phase of USMPs satisfies the distribution $\phi _0\left (y\right )=\xi \sin \theta \left | y \right |$ (black solid line in Fig. 3(b)), further validating our approach. In Fig. 3(c), we compare the intensity distribution along the $y$ axis (at $x=1.2\lambda$), obtained by the FEM simulation, with the zero-order Bessel function $J_0 \left (\alpha y\right )$, where $\alpha =k_{SW} \sin \theta$. Good agreement between these results clearly demonstrate the high quality of the Bessel beam generated by our waveguide array. The full width at half maximum (FWHM) of this Bessel beam is calculated as 11.5 mm, which is close to its theoretical value, given by $\omega _{FWHM} = 2.25/\alpha =12.3$ mm. We now demonstrate the self-healing property of the generated SW Bessel beam, that is, its propagation behaviors are not affected by small obstacles. Placing a metallic sphere (with diameter $D = \lambda /2$) at 43 mm away from the waveguide array, we repeat the $E_y$ field simulation. Clearly shown in Fig. 3(d), while the presence of the metallic sphere does scatter the Bessel beam in the vicinity of the sphere, some waves can bypass the scatterer and reconstruct the desired Bessel beam field pattern in the far-field. The resulting far-field pattern is nearly identical with that of the original Bessel beam (Fig. 3(b)), unambiguously demonstrating the self-healing property.

Evidently, the concept of USMP waveguide array can be employed to generate SW beams with other arbitrary complicated wavefronts. Here, we show the generation of the SW focusing as an illustration. Through varying the thickness distribution $\left \{ d \right \}$ of the waveguide array, we can ensure that the accumulated phase of USMPs follows the parabolic distribution:

Then, the USMPs can be coupled to the plasmonic metal and generating SWs with focusing wavefront. We also employ the FEM simulations to calculate the electric field distribution of the designed device. Figure 4(a) depicts the simulated near-field $E_y$ distribution when USMP beam is launched in the waveguide array. Evidently, the USMPs is converted to SWs on the plasmonic metal and the SWs gradually converge to a focal point as expected. The efficiency of SW focusing is evaluated as 65%. The focal length is demonstrated as $2.33\lambda$, agrees well with the designed length $F=2.5\lambda$. The simulated phase distribution of USMPs at the terminal of unidirectional waveguide array $\varphi \left (y\right )$ in Fig. 3(a) also satisfies the targeted parabolic distribution, further validating the effect of SW focusing. Figure 4(b) shows the field intensity distribution of the focused SWs at the focal line marked as the red dashed line in Fig. 4(a). The full-width at the half of the maximum (FWHM) is about $0.48\lambda$, which shows the excellent focusing effect of this system.

 

Fig. 4. Demonstration of the SW focusing effect based on the model system. (a) FEM-simulated near-field $\left |E_y\right |^2$ pattern on a plane above the YIG and plasmonic metal. Here, the $d$ distribution of the USMP waveguide array is designed to satisfy a parabolic phase distribution $\phi _0\left (y\right )=\varphi _0+\xi \left (\sqrt {y^2+F^2} -F\right )$, and the focal length is demonstrated as $2.33\lambda$. In the simulation, the USMP propagate along $+x$ direction is excited by a linear electric current source. (b) The near-field $\left |E_y\right |^2/\left |E_{y0}\right |^2$ distribution of SWs at the focal length position versus the $y$ coordinate. Here, $E_{y0}$ denotes the field amplitude at the center of initial incidence USMP beam.

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4. Extensions to terahertz frequencies

This phenomena of USMPs closely relies on strong magnetic-optical materials. Currently, such materials only exist in microwave and terahertz regimes, and they are YIGs and semiconductors, respectively. Therefore, the scheme we discussed in the microwave regime can be easily extended to the terahertz frequencies. To demonstrate its effectiveness, we propose a specific design for enabling coupling from USMPs to SWs at terahertz frequencies. Moreover, we construct a USMP waveguide array and successfully achieve anomalous deflection of SWs beam at terahertz frequencies, showcasing the broader potential of our approach.

Fig. 5(a) shows the schematic of metal-air-InSb waveguide that support USMP transmission in terahertz region. Different from the gyromagnetic anisotropy of YIG at microwave frequencies, gyroelectric anisotropy is induced in the InSb under an external magnetic field, with the permittivity tensor taking the form [43]

 

Fig. 5. Basic physical model of SWs coupling from unidirectional waveguide to plasmonic metal in the terahertz regime. (a) SMPs dispersion diagram for $d=0.01 \lambda _p$, $0.05 \lambda _p$, $0.1 \lambda _p$ and $0.25 \lambda _p$. Solid lines are the SMPs dispersion curves, and dashed lines are the light line in the air. The two green shaded areas represent the zones of bulk modes in the InSb. Inset: Schematic diagram of waveguides supporting USMPs at terahertz frequencies. (b) Dispersion relation (red solid line) of TM polarized SW mode supported by plasmonic metal depicted in the inset. (c) Sample picture of the coupling device at terahertz frequencies, which is composed of unidirectional waveguide depicted in (a) and artificial plasmonic metal depicted in (b). The FEM simulated (d) $E_x$, (e) $E_z$ and (f) $H_y$ field distribution of the coupling device shown in (c).

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Then, we design an artificial plasmonic structure supporting SWs in the terahertz regime, which is a metallic ground plane with a 9.8 $\mu$m-thick dielectric layer ($\varepsilon _r =15$) put on its top (see inset of Fig. 5(b)). Figure 5(b) depicts the FEM-simulated dispersion relation of the SW modes supported by such a system, which exhibits an eigen wave-vector $k_{SW} = 1.38 k_0$ with $k_0$ being the free-space phase constant at the frequency 1.78 THz. We then connect the unidirectional waveguide with the designed plasmonic metal as schematically illustrated in Fig. 5(c). Figures 5(d)-(f) display the FEM simulated nonzero components of the electric field ($E_x$, $E_z$) and the magnetic field ($H_y$) at $f=1.78$ THz, respectively. In the simulation, the USMP was excited by a line current source located at the center of the air layer along $y$ direction. Evidently, the electromagnetic wave can only propagate in $+x$ direction as expected, and the USMPs sustained in the TM-mode unidirectional waveguide coupled to TM-mode SWs supported by plasmonic metal, with the coupling efficiency found as 57.4%. Similarly, the coupling efficiency is relatively high, and the reduced efficiency primarily stems from modes mismatch between the SWs and USMPs.

We now move on to design a phase-gradient waveguide array to realize SW anomalous deflection at terahertz frequencies. The inset of Fig. 6(a) shows the simulation setup of a waveguide slit. FEM is utilized to simulate the phase accumulation and coupling efficiency as air layer depth varies, as displayed in Fig. 6(a). Figure 6(a) shows that the phase accumulation of transmitted USMPs can cover the full range of $2\pi$ with the thickness ($d$) varying from 11 to 25 $\mu$m at the working frequency of 1.78 THz. Meanwhile, the coupling efficiency can also keep a high value (larger than 0.5) in the whole spectrum. With the waveguide slit in hand, we can configure the appropriate phase function $\phi _{0}\left (y \right )$ to achieve various fascinating wavefront manipulation phenomena of SWs.

 

Fig. 6. Demonstration of the SW anomalous deflection at terahertz frequencies. (a) FEM-simulated coupling efficiency from USMPs to SWs, as well as the phase accumulation of the USMPs with respect to air thickness $d$. (b) FEM-simulated near-field $E_z$ pattern on a plane above the semiconductor and plasmonic metal, assuming that the USMP Gaussian beam propagate along $+x$ direction is excited by a linear magnetic current source in the simulation. Here, the $d$ distribution of the USMP waveguide array is designed to satisfy a phase gradient $\phi \left (y \right )=\varphi _{0} +\xi y$.

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Based on the relationship between the phase accumulation and $d$ plotted in Fig. 6(a), we build a phase-gradient waveguide array constructed by 61 waveguide slits with different $d$ that satisfies the phase accumulation distribution

5. Conclusions

In this work, we focused on implementing SW wavefront manipulation utilizing arrayed USMP waveguide platforms. We systematically investigated the phase accumulation of spacing-varied USMP waveguide arrays and their coupling with SWs sustained on reciprocal plasmonic platforms. The successful application of these platforms in SW wavefront shaping was demonstrated. As a proof of concept, we showcased precise SW Bessel beam generation and SW focusing in a series of spacing-varied YIG-air-metal waveguide arrays, with the efficiency reached up to 67% and 65% at 8.5 GHz. Furthermore, such a scheme can also be extended to higher frequencies. To illustrate this, we conducted full-wave simulations that showcased SW anomalous deflection at a frequency of 1.78 THz, exhibiting a deflection efficiency of 56%. In principle, this approach of utilizing the robust unidirectionality of USMPs to tailor the wavefront of SWs can be further extended to the optical domain and even other types of waves. These findings highlight the potential of spacing-varied USMP waveguide arrays as promising candidates for on-chip wavefront shaping. This research opens new possibilities for on-chip information processing and provides inspiration for the development of novel on-chip photonic devices.