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Abstract
Simultaneous and independent modulation of the amplitude and phase of surface waves (SWs) is critical in photonics and plasmonics. Here, we propose a method for flexible complex-amplitude modulation of SWs based on a metasurface coupler. Benefiting from the full range complex-amplitude modulation ability of the meta-atoms over the transmitted field, the coupler can convert the incident wave into a driven surface wave (DSW) with an arbitrary combination of amplitude and initial phase. By placing a dielectric waveguide that supports guided SWs below the coupler, the DSWs can resonantly couple to SWs while preserving complex-amplitude modulation. The proposed scheme provides a practical way for freely tailoring the phase and amplitude profiles of SWs wavefronts. As verification, meta-devices for normal and deflected SW Airy beam generation and SW dual focusing are designed and characterized in the microwave regime. Our findings may stimulate various advanced surface optical meta-devices.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Surface waves (SWs) are special electromagnetic (EM) modes, which are bounded at the interface of two distinct media [1]. Due to the peculiar features of subwavelength field confinement and field enhancement, SWs provide numerous exotic applications, such as biochemical sensing [2], enhanced nonlinear optics [3], super-resolution imaging [4], and on-chip photonic circuits [5,6]. Flexible and efficient control of the SWs wavefront, namely arbitrary modulation of the amplitude and phase of SWs, is crucial to these applications. Various approaches have been proposed to achieve this goal, including specially designed photonic crystals [7,8], and nanostructures arranged in specific patterns [9–15], such as nanoaperture arrays and paired nanoaperture. In these approaches, by appropriately choosing the orientation angles, sizes, and positions of the nanostructures, intriguing functionalities such as near-field focusing [9,11], plasmonic Airy beams [10,15], arbitrary bending [13], and holograms [14] are realized. However, due to the underlying scattering or propagation phase accumulation mechanism, these methods may suffer from bulky configuration, and limited functionality and efficiency.
Recently, metasurfaces have shown unprecedented abilities to manipulate the propagating waves (PWs) and SWs via a proper arrangement of elaborately designed ultrathin and subwavelength meta-atoms. Various novel effects, such as cloaks [16], optical vortex [17], holography [18], and anomalous deflection [19], have been realized for PWs. As for SWs, phase gradient metasurfaces (PGMs) which can provide artificial parallel wave vectors greater than those in the free space, were used for the high-efficiency coupling of PWs to SWs [20–22]. Furthermore, by controlling the initial phase of PGM [23,24], introducing polarization-sensitive nanostructures [25–29], and other advanced strategies [30,31], a variety of SW phase and/or amplitude modulation methods and exotic SW wavefront manipulation applications were proposed. However, most of these methods can only provide either phase-only modulation, amplitude modulation with binary phase, or modulation with severe polarization dependence. A flexible and universal full-range SW complex-amplitude modulation scheme is still highly desirable.
In this paper, we propose a full range SWs complex-amplitude modulation scheme based on a metasurface coupler, which is built with meta-atoms with the capability of full complex-amplitude control over the transmitted spin waves. Benefitting from the phase gradient and amplitude control of the transmitted wave, circularly polarized waves normally incident to the coupler can be converted to driven surface waves (DSWs) with an arbitrary combination of the initial phase and amplitude, which can be further coupled to SW modes supported by a dielectric waveguide while reserving the complex-amplitude profile. As a demonstration of the proposed scheme, three microwave regime metasurface couplers are designed for the SW Airy beam generation, Airy beam deflection, and dual focusing, respectively, which reveal the full range complex-amplitude modulation ability of the SW wavefront. The calculated and measured results validate our proposal, which may pave the way to various advanced surface optical meta-devices.
The essence of the proposed SWs complex-amplitude modulation approach is imparting arbitrary independent amplitude and initial phase to the DSWs, which is obtained with a metasurface coupler that is capable of full-range complex-amplitude modulation over the transmitted field. By introducing lossy components [32] or exploiting multiple degrees of freedom of meta-atom such as size, position, orientation, etc., various schemes have been proposed to achieve full complex-amplitude modulation over propagating EM fields [33–37]. By tuning the orientation angles of meta-atom composed of cascaded bilayer anisotropic structures, a straightforward complex-amplitude modulation strategy for spin waves is proposed in our previous works [38,39]. The amplitude of the transmitted circular polarization conversion component is determined by the difference between two orientation angles, whereas the phase is determined by their summation. We adopt this strategy for simplifying the meta-device design. Noting that, inspired by this work, other complex-amplitude modulation strategies may also be explored to realize the metasurface coupler and to implement the proposed SWs complex-amplitude modulation under non-circularly polarized incident waves. For example, the C-shape split-ring resonators [33,40] have the capability of arbitrary amplitude and phase control of the transmitted cross-polarized wave under linearly polarized incidence. Therefore, it can also be used to implement the proposed SWs complex-amplitude modulation under linearly polarized incident waves.
2. Concept and metasurface coupler design
The proposed SWs complex-amplitude modulation scheme is conceptually shown in Fig. 1. The metasurface coupler which controls the amplitude of transmitted waves and provides the large parallel phase gradient is used to convert the normally incident circularly polarized waves to the DSWs with designed complex-amplitude. A dielectric waveguide (dielectric slab with the metallic ground) supporting guided SWs is placed below the metasurface coupler for resonantly coupling the DSWs to the SWs while endowing the designed amplitude and initial phase. Specifically speaking, as conceptually shown in Fig. 1, by suitably arranging the orientation angles of the meta-atoms in a supercell, an arbitrary combination of the amplitude and initial phase of the generated DSW can be obtained, and consequently, the complex amplitude of SW is also decided.
Fig. 1. Schematic illustration of the SWs complex-amplitude modulation meta-device. The metasurface coupler consists of replicating supercells in the + x direction, and orientation angles (α and β) of each row are preset to determine the amplitude A and phase φ of SWs converted by different rows.
To implement the proposed SWs complex-amplitude modulation, a bilayer complementary split ring resonator (CSRR) structure is adopted as a meta-atom to build the metasurface coupler. As shown in Fig. 2(a) and (b), two layers of metallic structures are separated by a layer of F4B dielectric substrate (εr = 2.65, tanδ = 0.0017) with a thickness of d = 2 mm. The CSRR structures are fabricated with copper (the electric conductivity is 5.8 × 107 S/m) film with a thickness of 0.018 mm. The orientation angles of CSRRs are defined as the angle between their symmetric axes and the x-axis and labeled as α and γ. Except for the orientation angles, the top and bottom CSRR structures have identical geometric parameters, which are the radii R1 = 2.7 mm, R2 = 2.3 mm, R3 = 2.1 mm, R4 = 1.9 mm, R5 = 1.7 mm, the widths of two gaps W1 = 1 mm, W2 = 0.9 mm, and the lattice constant p = 6 mm. Figure 2(c) shows the transmission spectra on a circular basis for meta-atom with orientation angles α and γ both being zero. The calculations are performed with the commercial software Eastwave [41], the background material is set to be vacuum, the circular polarization plane wave is set to propagate along the + z axis as the incident wave, and transmitted circular polarized electric fields are recorded to obtain desired transmission coefficients. For these calculations, periodic boundary conditions are applied at the x and y boundaries, and perfect matched layers at the z boundaries. Clearly, the transmission coefficients of circular polarization conversion components reach a maximum of 0.9 at 12.13 GHz. In addition, as proposed in our previous works [38,39], full-range complex-amplitude control for the circular polarization conversion component can be achieved by tuning the two orientation angles. As verification, the transmission spectra of right-handed circular polarization (RCP) to left-handed circular polarization (LCP) conversion components are calculated for meta-atoms with other orientation angles. The normalized amplitude and phase shift compared to the (α, γ) = (0°, 0°) state at 12.13 GHz are respectively shown in Fig. 2(d) and (e). Obviously, a free combination of phase shift in the range of [0°, 360°] and normalized amplitude in the range of [0,1] can be obtained with the proposed meta-atom. Specifically speaking, the normalized amplitude is mainly decided by the orientation angle difference β = γ - α and decreases continuously from unity to zero as β varying from 0° to 90°. Meanwhile, for any value of β, the full range of phase shift can be obtained by exploiting the Pancharatnam-Berry (PB) geometric phase, i.e., tuning orientation angle α in the range of [0°, 180°).
Fig. 2. (a) Top and (b) perspective views of the proposed meta-atom. (c) Calculated transmission spectra in circular basis for unit cell with α = 0°, γ = 0°, where the first and the second subscripts in the legend refer to the polarization state of the transmitted and incident waves, respectively. The calculated (d) normalized amplitude and (e) phase shift of RCP-to-LCP conversion component compared to the (α, γ) = (0°, 0°) orientated meta-atom as functions of α and β at 12.13 GHz.
With the aforementioned meta-atom, we can now design the metasurface coupler for SW complex-amplitude modulation. Since natural SPPs do not exist in the microwave regime, we adopt a dielectric waveguide that supports guided SWs, which is a 1.5 mm thick dielectric slab (F4B, εr = 2.65, tanδ = 0.0017) with a metallic ground plane. The dispersion relation of guided SW modes is well defined [42–44], and the calculated result is shown in Fig. 3(b), which indicates the wave vector of SW mode is kSW = 1.0303k0 at 12.13 GHz with k0 being the wave vector in free space. To generate DSWs that can best match the SW wave vector, identical meta-atoms (same orientation angle β) are arranged in a periodic array with their orientation angles α successively rotated by a constant step of Δφ = -45.02° along the x direction. Thus, we realize the required phase gradient of ξ = 2Δφ/p = 1.0303k0, given that the phase modulation of the transmitted wave is based on the well-defined PB phase mechanism. As shown in Fig. 3(a), one supercell of metasurface coupler is designed with 12 sequentially rotated meta-atoms along the x direction, and placed at an optimized distance of h = 8 mm above the dielectric waveguide. Note that such a supercell is designed to convert normal incident RCP wave to the DSW and then coupled to the SW propagating in the dielectric waveguide along the + x direction. As for the LCP incidence case, the rotation step should be Δφ = 45.02° to generate SW propagating along the + x direction. Figure 3(c) shows the calculated Re(Ez) distribution as the metasurface coupler is illuminated by a normally incident RCP wave at 12.13 GHz. In these calculations, the boundaries of x and -z are the perfect matched layers, the –z boundary is set to be the electric boundary, and the periodic boundary conditions are applied at the y boundaries. RCP incident plane wave is set to be along the + z direction. For the supercell forming the metasurface coupler, the orientation angles of the leftmost meta-atom are (α0, β) = (0°, 0°). All the other 11 meta-atoms share the same orientation angle of β = 0° to ensure uniform transmission, while the orientation angles α successively rotate to obtain a proper phase gradient to generate desired DSW. The incident RCP wave is coupled to the SWs as expected. In addition, based on the calculated wavelength λSW = 23.97 mm, we can further determine the wave vector of SW to be 2π/λSW ≈ 1.0303k0, which is in good agreement with the theoretical result at 12.13 GHz. Note that the supercell has only two degrees of freedom, that is, two orientation angles of the leftmost meta-atom (α0, β), which are further used to implement the full complex-amplitude modulation of SW.
Fig. 3. (a) Schematic illustration of the meta-device. (b) The calculated dispersion curve of guided SW in the dielectric waveguide. (c) Calculated Re(Ez) distribution for meta-device with parameters (α0, β) = (0°, 0°) under RCP normally incident plane wave at 12.13 GHz. The origin (x = 0 position) is set to be by the coupler boundary. The black arrow indicates the position of the sampling point. (d) Normalized amplitude and (e) phase shift of SWs compared to the (α0, β) = (0°, 0°) meta-device as a function of α0 and β at 12.13 GHz.
Since the capability of the proposed coupler to couple circularly polarized incident waves to SWs is demonstrated, we can now introduce the scheme for SW complex-amplitude modulation. As we stated above, the incident circularly polarized wave is first converted to the DSW, and then coupled to the SW. Therefore, the amplitude and phase modulation on SW can be implemented by control of the amplitude and phase of converted DSW. The full range complex-amplitude modulation feature of meta-atom shown in Fig. 2 can be utilized to meet this end. For the amplitude control of SW, we can simply tune the orientation angles β of meta-atoms in the supercell. The amplitude of DSW, as well as the SW, continuously varies from the maximum to zero as β varies from 0° to 90°. Similar to the method in Ref. [23], the full range of SW phase modulation can be achieved by control of the initial phase of DSW, which can be simply implemented by tuning the orientation angle α0 of the leftmost meta-atom in a supercell from 0° to 180°. As verification, we have performed further calculations of the designed coupler with different parameters (α0, β) under RCP normal incidence at 12.13 GHz. The amplitude and phase of generated SWs are characterized by sampling the Ez filed component 1 mm above the dielectric waveguide and 315 mm away from the coupler boundary. The sampling point is denoted with a black arrow in Fig. 3(c). The results are compared to those for the coupler with parameters (α0, β) = (0°, 0°), and the normalized amplitude and phase shift are shown in Fig. 3(d) and (e). Clearly, an arbitrary combination of normalized amplitude from zero to unity and phase shift from 0° to 360° can be obtained for SW by tuning the parameters (α0, β). Having demonstrated the full complex-amplitude modulation capability of our meta-device, we now quantitatively evaluate the working efficiency of our scheme by numerical calculations. The calculations are performed with meta-device with parameters (α0, β) = (0°, 0°). The total powers carried by the excited SW and the input circularly polarized wave are numerically integrated and the ratio between them is defined as the working efficiency of our meta-device. The calculated efficiency is about 53.11% at 12.13 GHz, and it can be improved by optimization of the meta-atom and by designing artificial plasmonic metal that supports SWs of both TE and TM modes (see calculation details and further discussion in Supplement 1).
3. Meta-devices design and experiment
The proposed SW full complex-amplitude modulation scheme can be engineered for high-demanding arbitrary SW manipulation. As a demonstration, we first design and characterize a meta-device for generating SW Airy beam with unique self-bending, diffraction-free, and self-healing features [45,46], which enables exotic applications, such as particle manipulation [47], and curved plasma channel generation for energy routing [48].
For a one-dimensional (1-D) finite-energy Airy beam, the electric-field distribution on its initial plane can be described as
where A is the amplitude, Ai represents the Airy function, y0 is a reference coordinate for normalization, w is a scaling length, and α is the decay factor. The parameters are chosen to be y0 = 12 mm, w = 20.5 mm, α = 0.01, and the corresponding amplitude and phase distribution along the y direction are shown in Fig. 4(a) and (b). The SW Airy beam meta-device is designed with 51 aforementioned supercells (each consists of 12 meta-atoms along the x direction) along the y direction. The amplitude distribution of the Airy beam profile varies in the range of [0,1], and the phase presents the binary distribution of 0° and 180°. Then, the parameters (α0, β) of each supercell can be determined according to the relationship shown in Fig. 3(d) and 3(e). Namely, the orientation angle difference β is firstly derived through interpolation based on the SW amplitude values shown in Fig. 3(d), then the orientation angle α0 of the leftmost meta-atom can be derived through interpolation based on the phase values shown in Fig. 3(e).Fig. 4. The distribution of theoretical (red solid line) and sampled (blue dots) (a) normalized amplitude and (b) phase distributions of the SW Airy beam generation meta-device. (c) Photograph of experimental setup and sample. The insets show the detailed probe structure and meta-atoms for part of the coupler. (d) Schematic diagram of the near-field experimental setup. (e) Calculated and (f) measured Ez field intensity distributions on the transverse plane 1 mm above the dielectric waveguide at 12.13 GHz. The origin (x = 0 position) is set to be by the coupler boundary.
The sample is fabricated by standard printed-circuit-board technique on F4B substrates (Taconic TLT-6). The experiment setup is shown in Fig. 4(c) and (d). The performance of the SW Airy meta-device is measured in an anechoic chamber with Keysight PNA-X 5242A Network Analyzer. An RCP horn antenna is adopted to generate normal incident waves and is placed 3 meters away to mimic a plane wave source. We adopt a small monopole antenna that is mounted on a translation stage as the probe, with which the Ez field component is measured in a step of 3 mm on the plane 1 mm above the dielectric waveguide. The calculated and measured results of Ez field intensity distribution on the sampling plane at 12.13 GHz are shown in Fig. 4(e) and (f). For simulations of the meta-devices, the boundaries of x, y and –z are perfect matched layers, the + z boundary is set to be electric boundary. The incident wave are also set to be RCP plane wave along the + z direction. The black dashed lines denote the theoretical trajectories of the main lobe, described by the theoretical relation:
yt represents the main lobe trajectory of the Airy beam versus the propagation distance x. The measurement result, which is in good agreement with the simulations ones, clearly indicates the generation of the SW Airy beam and shows evident self-bending features. The diffraction-free feature of the generated Airy beam is evaluated by the full-width half-maximum (FWHM) of the main lobe. Though the FWHM shows some oscillating features due to the interference of discrete supercells, the measured and calculated results are around the predesigned value of 33.6 mm, and are below 1.5 times the design value for a long propagation distance (detailed results are shown in Supplement 1). In addition, the self-healing feature of generated Airy beam is verified by placing a square hole on the propagation path of the main lobe (see calculation results in the Supplement 1).Note that the above-stated meta-device generates an Airy beam with a normal launch angle, only continuous amplitude modulation and two segments of 0° and 180° phase modulation are required. To demonstrate the full-range SW complex-amplitude modulation ability, we deflected the Airy beam, for which additional lateral phase gradient and full-range phase modulation are required. The modified phase distribution for generating the Airy beam with a deflection angle θ = 15° is depicted in Fig. 5(a), which is the combination of the normally launched Airy beam phase distribution and the additional phase gradient ξy = sinθ · kSW for deflection of the SWs. And the trajectory of the deflected Airy beam can be expressed with
The amplitude distribution is the same as that of the normally launched Airy beam shown in Fig. 4(a). The Airy beam deflection meta-device is designed with the same size as the normally launched one, and the calculated and measured results are shown in Fig. 5(b) and 5(c). Obviously, both results verified the deflection design acquired by the additional phase gradient, and the non-diffraction feature can also be clearly observed.
Fig. 5. (a)The phase distribution of theoretical (red solid line) and sampled (blue dots) of the SWs deflection Airy beam meta-device. (b) Calculated and (c) measured Ez field intensity distributions at 12.13 GHz.
For further validation of the proposed complex-amplitude modulation method, we design and characterize an SW lateral bifocal meta-device, which also demands continuous full-range control of amplitude and phase of the SW wavefront. Similar to a 1-D bifocal meta-lens for PWs, the required complex-amplitude distribution can be expressed as
Fig. 6. The distribution of theoretical (red solid line) and sampled (blue dots) (a) normalized amplitude and (b) phase distributions of the SWs bifocal meta-device. (c) Calculated and (d) measured Ez field intensity distributions on the dielectric waveguide at 12.13 GHz. (e) The calculated (red solid line) and measured (blue dots) Ez field intensity along the white dashed lines (x = 280 mm) are shown in Fig. 6(c) and (d). All intensities distributions are normalized to their global maximum.
4. Conclusion
In summary, we have presented a practical scheme for SWs complex-amplitude modulation. With a metasurface coupler that is capable of arbitrary complex-amplitude control of the transmitted field, a normally incident circularly polarized wave can be converted to DSW with an arbitrary combination of the amplitude and initial phase. Then, the guided SWs on a dielectric waveguide get stimulated by the DSW while reserving the full complex-amplitude modulation. As verification, three meta-devices are fabricated and characterized in a microwave regime. The proposed idea can be further extended to metasurfaces with polarization states beyond the circularly polarized ones mentioned in this work. Our findings can be generalized to other frequencies, such as terahertz, infrared, visible light, etc., and motivate high-performance plasmonic devices and applications.
Funding
National Natural Science Foundation of China (11774057, 11874286, 61205041); Fundamental Research Funds for the Central Universities (20153638, 22120190222).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request
Supplemental document
See Supplement 1 for supporting content.
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