An electromagnetic or acoustic beam splitter can divide an incident wave into two or more outgoing beams [1]. It was initially intended for use with optical waves, and is mainly used in interferometers and laser output-power trackers [2,3]. However, beam splitters for low-frequency electromagnetic/acoustic waves have many other applications, such as in a microwave/acoustic signal demultiplexer/channelizer [4–6], a spatial power divider [7–9], and a microwave/acoustic interferometer for measurement/imaging [10–13]. In recent years, various electromagnetic/acoustic beam splitters for low-frequency waves have been proposed, e.g., a microwave beam splitter based on metamaterials/metasurfaces [14–16], acoustic splitters based on metasurfaces [17,18], or acoustic crystals [19,20]. However, existing beam splitters can only produce beam-splitting effects for electromagnetic waves alone or for acoustic waves alone; there is still no structure that can create a simultaneous beam-splitting effect for both acoustic and electromagnetic waves [21]. In addition, the splitting ratio of the two split beams is often fixed once the beam splitter has been fabricated. Although the splitting ratio may be tuned by coding [18] and dynamic modulation [14], these splitters are active devices, so these methods require additional energy consumption. Also, many existing electromagnetic/acoustic splitters perform one-way beam splitting, in which reflection and transmission are used to generate the two split beams (i.e., reflected and transmitted waves), meaning that the incident light and one of the split beams are in the same optical path. Therefore, there is still a lack of an effective passive electromagnetic-acoustic beam splitter that can have a simultaneous beam-splitting effect on both electromagnetic and acoustic waves, leading to two transmitted split beams (i.e., the reflectivity is zero), and can achieve a tunable splitting ratio without additional energy consumption.

In this study, a novel electromagnetic-acoustic splitter (EAS) is proposed that uses an arc-shaped array of copper plates with a subwavelength separation Δ to simultaneously split an electromagnetic beam and acoustic beam. As shown in Fig. 1, the EAS can be realized by cutting a fan-shaped region with a shear angle α from a 1/4 arc structure composed of copper plates with radii ranging from R₁ and R₂. When an electromagnetic wave or an acoustic wave is incident on the input surface of the EAS, two split beams can be created at the output surface of the EAS, and their energy transmittances can be tuned by simply changing the incident angle β.

 

Fig. 1. Schematic diagram of the electromagnetic-acoustic splitter. An arc-shaped array of copper plates (colored yellow) of thickness Δ are separated by a subwavelength distance Δ, where the inner and outer radii of the array are R₁ and R₂, respectively. The regions between the copper plates are filled with air. When electromagnetic or acoustic waves are incident on the input surface of the splitter, two split beams (i.e., output beams 1 and 2) are created at the output surface of the splitter. The energy ratio of two split beams can be tuned by changing the incident angle β or the shear angle α.

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To quantitatively describe the performance of the beam splitting effect, the energy transmittances of the split beams are defined as

 

Fig. 2. 3D numerical simulation results. (a) Normalized amplitude of the magnetic field’s z component and (b) the acoustic pressure when the TM-polarized electromagnetic wave [in (a)] and acoustic wave [in (b)] are normally incident (β = 0°) on the input surface of the designed EAS with α = 30°. The parameters of the EAS are chosen as R₁= 15λ₀, R₂= 35λ₀, Δ = λ₀/4, and the working wavelength is λ₀= 0.04 m. Panels (c) and (d) show that the exit angles and transmittances, respectively, of the two output beams can be fine-tuned within a certain range by changing the shear angle α when the incident angle is fixed at β = 0° (see Visualization 1). Panels (e) and (f) show that the exit angles and transmittances, respectively, of the two output beams can be tuned by changing the incident angle β when the shear angle is fixed at α = 30° (see Visualization 2). If the wavelength deviates from the designed wavelength, the EAS can still act as a beam splitter (see Visualization 3).

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All of the numerical simulations performed in this study are conducted by COMSOL Multiphysics 5.6 with the license number 9406999. All simulations are 2D cases where the wave optics module and acoustic module with steady-state solver are selected to simulate electromagnetic waves and acoustic waves, respectively. Free tetrahedral meshing is used, where the maximum grid is one-tenth of the working wavelength (λ₀/10). As the proposed EAS can produce exactly the same function for electromagnetic and acoustic waves., i.e., the distribution of acoustic pressure distribution is exactly the same as the magnetic field distribution, subsequent calculations of exit angles and energy transmittances are based on numerical simulations of electromagnetic waves.

The exit angles (θ₁ and θ₂) and energy transmittances of the two split beams can be changed by choosing different shear angles α (the incident angle is fixed at β = 0°), as shown in Figs. 2(c) and 2(d), respectively. Note that the directions of the exit angles are defined with respect to the incident wave direction (see the inset of Fig. 1). When the shear angle α changes from 26° to 34°, the exit angle θ₁ of output beam 1 approximately linearly increases from 96° to 137°, and the exit angle θ₂ of output beam 2 varies within the range between −6° and −19°. The convention that the exit angle is positive or negative is dependent on whether the rotation of the horizontal axis to the output beam is clockwise or counterclockwise, which is indicated in Fig. 1. As the shear angle α changes, both the exit angles in Fig. 2(c) and the energy transmittances (and also the splitting ratio) of the two output beams in Fig. 2(d) change correspondingly (see Supplementary Movie 1). When α = 26°, the transmitted energy is mainly carried by output electromagnetic/acoustic beam 1, i.e., the splitting ratio γ = 62.490. As α increases to 30.72°, the transmitted energy is divided equally between the output two beams, i.e., equal splitting of the light is achieved; γ = 1. When α reaches 34°, the transmitted energy is mainly carried by output beam 2, with a minimal splitting ratio γ = 0.0371. If the shear angle α is smaller than 26°, only transmitted beam 1 comes from the output surface (T₁= 1 and T₂= 0). If the shear angle α is larger than 34°, only transmitted beam 2 comes from the output surface (T₁= 0 and T₂= 1).

Actually, there is another conventional way to tune the energy transmittances and the splitting ratio of the two split beams for an EAS with a fixed shear angle α. As shown in Figs. 2(e) and 2(f), the exit angles (θ₁ and θ₂) and energy transmittances of the two split beams can be tuned by using different incident angles β (the shear angle of the EAS is fixed at α = 30°). When the incident angle β changes from −10° to 5°, exit angle θ₁ of output beam 1 varies within the range between 111° and 128° and exit angle θ₂ of output beam 2 varies within the range between −6° and −16°. The convention that the incident angle is positive or negative is dependent on whether the rotation of the horizontal axis to the incident light is clockwise or counterclockwise, which is indicated in Fig. 1. As the incident angle β changes, both the exit angles and the energy transmittances (and also the splitting ratio) of the two output beams in Fig. 2(f) change correspondingly (see Supplementary Movie 2), which can be utilized as a simple way to achieve a tunable splitting ratio without additional energy consumption.

Figures 3(a) and 3(b) show the energy transmittances of the two transmitted split beams as the incident angle β and shear angle α change. They show that the energy distribution of the two transmitted split beams can be tuned by changing the incident angle β of the incident electromagnetic/acoustic beam and the shear angle α. The energy reflectivity of the EAS (i.e., R = 1 − T₁ − T₂) can be designed to be zero (no reflected beam) by choosing α and β to be in the appropriate range [see Fig. 3(c)], which further verifies that the proposed EAS is an efficient transmissive splitter with a tunable splitting ratio.

 

Fig. 3. 3D simulated results: the energy transmittances of (a) output beam 1 and (b) output beam 2 as both the incident angle β and shear angle α change. (c) Energy reflectivity of the EAS.

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In our design, all electromagnetic-acoustic beams incident from the “complete plane” (i.e., the input surface) and exit from the “cut plane” (i.e., the output surface) to obtain the best total energy transmittances. If the EAS works in the ‘reverse state’ (i.e., the beams are incidents in the cut plane), we can still perform as a beam splitter for both electromagnetic and acoustic waves. However, the reflected beam will appear and the total energy transmittances of the two split transmitted beams will be reduced when the EAS works in the ‘reverse state.’

The EAS used in this study is realized by cutting a fan-shaped region with the shear angle from the output surface. If the fan-shaped region is cut from the input surface, the structure can still act as a beam splitter for both electromagnetic and acoustic waves. However, the total energy transmittance of the two split transmitted beams (T₁+ T₂) will decrease in this case. Therefore, the EAS can create the splitting effect with the best total energy transmittance when a fan-shaped region with the shear angle is cut from the output surface.

The proposed EAS can still perform as a beam splitter when the wavelength deviates from the designed wavelength (see Visualization 3), except that the total transmittance of the split beam changes as the wavelength changes. When the wavelength of the incident beam is larger than the designed wavelength (λ > λ₀), the transmittance of transmitted beam 2 remains almost constant (i.e., T₂ ∼ 50%), while the transmittance of transmitted beam 1 fluctuates periodically as the wavelength increases. The period is half of the designed wavelength: 0.5λ₀. When the wavelength of the incident beam is smaller than the designed wavelength (λ < λ₀), the EAS cannot perform good splitting (the total transmittance will drop), as the metallic plates cannot be treated as the effective medium in this case.

In recent years, various structures with different functions for controlling electromagnetic/acoustic waves simultaneously have been proposed [22–25]. However, these studies have mainly focused on designing invisibility cloaks or stealth coats for electromagnetic/acoustic waves simultaneously, which cannot be extended to the design of an electromagnetic-acoustic splitter. For topology-optimized design methods, the advantage is that devices can work for both polarization states of electromagnetic waves and for acoustic waves [25]. However, if the topology optimization method is used for the electromagnetic-acoustic splitter, the main limitation is that the performance of the splitter may change significantly when the incident angle changes, i.e., the beam splitting ratio cannot be tuned using the incident angle.

In this work, a novel EAS based on copper plates has been proposed, which can simultaneously produce identical beam-splitting effects for TM-polarized electromagnetic and acoustic waves. In contrast to previous beam splitters, the proposed EAS can create two transmitted split beams (no reflected beam) instead of using reflection to separate the beams. In addition, the beam-splitting ratio of the EAS can be tuned by changing the incident angle of the input beam or the geometrical shear angle of the structure, which is a passive tunable splitter. The proposed EAS may have applications in double-wave interference, double-wave detection, and other fields that require the simultaneous splitting of electromagnetic waves and acoustic waves.