Surface transformation multi-physics for controlling electromagnetic and acoustic waves simultaneously

Fei Sun; Fei Sun; Yichao Liu; Sailing He; Sailing He

doi:10.1364/OE.379817

1. Introduction

Since the theory of transformation optics was proposed in 2006 [1,2], it has been successfully applied to manipulate various physical fields including electromagnetic fields [3–5], acoustic fields [6,7], thermal fields [8–11], water waves [12], and magnetostatics [13,14]. These coordinate-transformation-based methods are based on the formality-invariance of physical equations for various physical phenomena (e.g. fields, waves, fluxes) under coordinate transformations [15,16]. Meta-materials with artificial subwavelength units can provide unique properties that cannot be found in natural materials, which have been widely applied to realize novel devices designed by coordinate-transformation-based methods [17,18].

In recent years, “transformation multi-physics”, i.e. controlling more than one physical wave/field by a single device simultaneously, has become an important branch of transformation optics and meta-materials [19–28]. For example, bi-functional cloak for both DC electric field and thermal field has been theoretically designed and experimentally demonstrated by bi-functional meta-materials [19–24]. However, limited by the current bi-functional meta-materials, most studies on bi-functional devices with some coordinate-transformation-based methods are mainly focused on tailoring DC electric field and thermal field [19–28]. Although there are some studies on bi-functional meta-materials for other kinds of physical waves/fields [29–34], it still lacks some general theory and systematical studies on controlling electromagnetic waves and acoustic waves simultaneously by a single device. In this study, a general designing and realization method, “surface transformation multi-physics”, is proposed to tailor both electromagnetic waves and acoustic waves simultaneously in a graphical way with some naturally available materials. Surface transformation multi-physics can be derived from coordinate-transformation-based methods. In previous studies, we have introduced optical surface transformation and acoustic surface transformation for EM waves [35] and acoustic waves [36] separately. Optic-null medium and acoustic-null medium are essential materials to realize devices designed by optical surface transformation and acoustic surface transformation, respectively. Transmission metallic gratings used for extraordinary optical transmission (EOT) can be treated as a reduced null-medium for both EM waves and acoustic waves [37–44], which pave the way for the surface transformation multi-physics method.

2. Theoretical method

The transformed material parameters in transformation optics and transformation acoustics can be summarized by [1,6]:

(1)$$\left\{ \begin{array}{l} \varepsilon ' = \frac{{J\varepsilon {J^T}}}{{\det (J)}}\\ \mu ' = \frac{{J\mu {J^T}}}{{\det (J)}} \end{array} \right.,\ \left\{ \begin{array}{l} \rho {'^{ - 1}} = \frac{{J{\rho ^{ - 1}}{J^T}}}{{\det (J)}}\\ \kappa ' = \det (J)\kappa \end{array} \right..$$

Here ɛ, μ, ρ, к are permittivity, permeability, mass density and modulus of a medium in the reference space. Quantities with superscript prime are corresponding physical quantities in the real space. J=∂(x’, y’, z’)/∂(x, y, z) is Jacobian matrix whose determinant is det(J). Next, we will introduce the key element ‘null medium’ in our surface transformation multi-physics method by applying an extremely spatial compression transformation. As shown in Fig. 1, the yellow region between two surfaces of arbitrary shapes S₁ and S₂ in the real space is greatly compressed into an extremely thin strip of thickness Δ in the reference space. If Δ→0, the volume between two surfaces S₁ and S₂ in the real space is reduced into a surface in the reference space. That is the reason why the medium between two surfaces S₁ and S₂ in the real space is referred to as ‘null medium’ (its corresponding region in the reference space is a surface but not an area/volume [35,36]). Assuming the reference space is air, the required material parameters of the null medium can be calculated by dividing the yellow regions in the real space and in the reference space into many small trapezoid regions of height h_i by inserting infinite parallel lines (parallel to the x’ axis indicated by the dashed blue lines in Figs. 1(a) and 1(b)). The curved surfaces S₁ and S₂ are divided into many small elements ΔS_1i and ΔS_2i. When h_i is small enough, the elements on curved surfaces ΔS₁ⁱ and ΔS₂ⁱ can be treated as planes. We can use the following coordinate transformation to relate each trapezoid region in the real space (see Fig. 1(c)) with the corresponding thin trapezoid in the reference space (see Fig. 1(d)):

(2)$$x^{\prime} = \left\{ {\begin{array}{{c}} {{d_1}/({\Delta /2} )x,x^{\prime} \in [0,{d_1}]}\\ {\tan (\pi - {\alpha_1})(x - \frac{\Delta }{2})/\tan (\pi - {\theta_1}) + {d_1},x^{\prime} \in [{d_1},{d_1} + {h_i}/\tan (\pi - {\theta_1})]}\\ {{d_2}/({\Delta /2} )x,x^{\prime} \in [ - {d_2},0)}\\ {\tan {\alpha_2}(x + \frac{\Delta }{2})/\tan {\theta_2} - {d_2},x^{\prime} \in [ - {d_2} - {h_i}/\tan {\theta_2}, - {d_2}]}\\ {x,else} \end{array}} \right.;y^{\prime} = y;z^{\prime} = z.$$

Fig. 1. The coordinate transformation relation between the real space (a) and the reference space (b). The yellow region between surfaces S₁ and S₂ of arbitrarily shapes is compressed into a thin yellow slab of thickness Δ in the reference space. The yellow regions in the real space and in the reference space are divided into many small trapezoid regions of height h_i. Representative small trapezoid regions in the real space (c) and in the reference space (d).

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With the help of material parameter relations given by Eq. (1), we can calculate the required materials for coordinate transformation in Eq. (2):

(3)$$\frac{{\varepsilon '}}{{{\varepsilon _0}}} = \frac{{\mu '}}{{{\mu _0}}} = \frac{{{\rho _0}}}{{\rho '}} = \left\{ {\begin{array}{{c}} {diag(2{d_1}/\Delta ,\Delta /2{d_1},\Delta /2{d_1}),x' \in [0,{d_1}]}\\ {\begin{array}{{c}} diag(\tan (\pi - {\alpha _1})/\tan (\pi - {\theta _1}),\tan (\pi - {\theta _1})/\tan (\pi - {\alpha _1}),\tan (\pi - {\theta _1})/\tan (\pi - {\alpha _1})),\\ x' \in [{d_1},{d_1} + {h_i}/\tan (\pi - {\theta _1})]\end{array}}\\ \begin{array}{l} diag(2{d_2}/\Delta ,\Delta /2{d_2},\Delta /2{d_2}),x' \in [ - {d_2},0)\\ diag(\tan {\alpha _2}/\tan {\theta _2},\tan {\theta _2}/\tan {\alpha _2},\tan {\theta _2}/\tan {\alpha _2}),x' \in [ - {d_2} - {h_i}/\tan {\theta _2}, - {d_2}]\\ 1,esle \end{array} \end{array}} \right..$$

(4)$$ \kappa '/{\kappa _0} = \left\{ {\begin{array}{*{20}{c}} {2{d_1}/\Delta ,x' \in [0,{d_1}]}\\ {\tan (\pi - {\alpha _1})/\tan (\pi - {\theta _1}),x' \in [{d_1},{d_1} + {h_i}/\tan (\pi - {\theta _1})]}\\ \begin{array}{l} 2{d_2}/\Delta ,x' \in [ - {d_2},0)\\ \tan {\alpha _2}/\tan {\theta _2},x' \in [ - {d_2} - {h_i}/\tan {\theta _2}, - {d_2}]\\ 1,esle \end{array} \end{array}} \right..$$

θ₁, θ₂, d₁ and d₂ can take arbitrary values, which are related with trapezoid shapes formed by surfaces S₁ and S₂. If α₁→π/2, α₂→π/2 and Δ→0, the small trapezoid region in the reference space will be reduced to a small surface element. In this case, the medium in the trapezoid region of the real space is the null medium, which can be obtained by taking α₁→π/2, α₂→π/2 and Δ→0 in Eqs. (3) and (4):

(5)$$\frac{{\varepsilon '}}{{{\varepsilon _0}}} = \frac{{\mu '}}{{{\mu _0}}} = \frac{{{\rho _0}}}{{\rho '}}\mathop \to \limits^{\scriptstyle{\alpha _1} \to \pi /2\atop {\scriptstyle{\alpha _2} \to \pi /2\atop \scriptstyle{\Delta } \to 0}} \left\{ {\begin{array}{c} {diag(\infty ,0,0), - {d_1} \le x' \le {d_2}}\\ {1,else} \end{array}} \right..$$

(6)$$\frac{{\kappa '}}{{{\kappa _0}}}\mathop \to \limits^{\scriptstyle{\alpha _1} \to \pi /2\atop {\scriptstyle{\alpha _2} \to \pi /2\atop \scriptstyle{\Delta } \to 0}} \left\{ {\begin{array}{c} {\infty , - {d_1} \le x' \le {d_2}}\\ {1,else} \end{array}} \right..$$

When each small trapezoid region (see Fig. 1(c)) in the real space is filled with the null medium given by Eqs. (5) and (6), the corresponding region in the reference space is a small surface (α₁→π/2, α₂→π/2 and Δ→0 in Fig. 1(d)). This means that small plane elements ΔS₁ⁱ and ΔS₂ⁱ are equivalent surfaces for both electromagnetic wave and acoustic wave when they are linked by the multi-physics null medium given by Eqs. (5) and (6). We can summarize the null medium in the reference space given by Eqs. (5) and (6) as:

(7)$$\left\{ \begin{array}{l} \varepsilon^{\prime} = \mu^{\prime} = diag(\infty ,0,0)\\ \rho^{\prime} = diag(0,\infty ,\infty )\\ \kappa^{\prime} = \infty \end{array} \right., - {d_1} \le x^{\prime} \le {d_2}.$$

From Eq. (7), we can see the permittivity and permeability of the ideal null medium in the real space are extremely anisotropic tensors, whose values approach infinity along their main axis (x’ direction here) and are close to zero in other orthogonal directions. The mass density is also extremely anisotropic tensor, whose value approaches zero along its main axis and infinity in other orthogonal directions. The modulus of the ideal null medium is infinitely large. Once each plane element pair ΔS₁ⁱ and ΔS₂ⁱ in the small trapezoid region in the real space is linked by the null medium in Eq. (7), they are equivalent surfaces (they correspond to the same surface in the reference space: electromagnetic/acoustic field on one point of surface element ΔS₁ⁱ will be identically projected to its corresponding point on surface element ΔS₂ⁱ). As surfaces S₁ and S₂ in the real space can be treated as the superposition of infinite small planes ΔS₁ⁱ and ΔS₂ⁱ, surfaces S₁ and S₂ are also equivalent surfaces (they correspond to the same surface in the reference space). Since S₁ and S₂ are equivalent surfaces (any point on S₁ has one corresponding point on S₂), the electromagnetic/acoustic field on one point of surface S₁ can be identically projected onto its corresponding point on surface S₂ along the null medium’s main axis (the x’ direction here). Although in the above calculations we assumed the main axis of the null medium in Eq. (7) is along the x’ direction, the directional projecting feature of the multi-physics null medium also works when the null medium’s main axis is along other directions. Similar calculations can be made by setting a local Cartesian coordinate system with the x’ axis along the null medium’s main axis to obtain the same required material parameters. This corresponding relation between two equivalent surfaces can also be transmitted to more surfaces by introducing null media of various main axes to connect these surfaces.

In summary, the ideal null medium described in Eq. (7) performs like a perfect endoscope, which can identically project electromagnetic/acoustic field distribution from one surface it linked to another surface it linked along its main axis. From the perspective of transformation optics, all surfaces linked by the ideal null medium are equivalent surfaces, which correspond to the same surface in the reference space. In practice, the ideal null medium in Eq. (7) does not exist. However, the reduced multi-physics null medium (permittivity and permeability are very large along its main axis and very small in other orthogonal directions, mass density is very small along its main axis and very large in other orthogonal directions, and modulus is very large) can still keep a certain directional projecting property (e.g. a reduced endoscope) for both electromagnetic and acoustic waves. Before we show how to design novel multi-physics devices in a graphical way with the help of multi-physics null medium’s directional projection feature, we need to introduce how to realize the reduced multi-physics null medium by natural materials.

When the wavelength is much larger than the periodicity of a metal plate array in Fig. 2(a) (i.e. λ₀>>d > a) and satisfies the Fabry-Pérot resonance condition (i.e. h =mλ₀, m = 1, 2, 3…), this structure can perform as perfect endoscope for both electromagnetic waves [37–42] and acoustic waves [43,44], which can be utilized to realize the 2D reduced null medium. For electromagnetic waves in the microwave frequencies, the metal can be modeled as perfect electric conductor (PEC), the effective electromagnetic parameters of the metal plate array in Fig. 2(a) can be expressed by [40,42]:

(8)$$\left\{ \begin{array}{l} {\varepsilon_y} = \frac{d}{a}{\varepsilon_h},{\varepsilon_x} = {\varepsilon_z} \to \infty \\ {\mu_y} = {\mu_h},{\mu_x} = {\mu_z} = \frac{a}{d}{\mu_h} \end{array} \right..$$

For TM polarization waves (magnetic field only contains z component), Eq. (8) reduced to:

(9)$${\varepsilon _y} = \frac{d}{a}{\varepsilon _h},{\mu _z} = \frac{a}{d}{\mu _h},{\varepsilon _x} \to \infty .$$

Fig. 2. The structures of metal plate array (a) and a metal plate with periodic sub-wavelength square holes (b) to effectively realize the 2D and 3D reduced null media, respectively. The metals are indicated by the blue color. d is the lattice constant. a is the height of the air gaps in (a) and side length of square air cylinders.

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Here ɛ_h and µ_h are permittivity and permeability of the host medium inside the metal plate array in Fig. 2(a). As shown in Eq. (9), we can achieve a reduced null medium of the main axis along the x direction by choosing appropriate geometrical parameters (a and d) and the host medium (ɛ_h and µ_h). The best choice is to use some near-zero refractive index materials as the host medium [45]. Some studies show the metal plate array in Fig. 2(a) can still perform as a reduced optic-null medium for electromagnetic wave even if the host medium is air (e.g. d = 2a, ɛ_y=2ɛ₀, µ_z=0.5μ₀, ɛ_x→∞) [38].

For acoustic waves, the effective modulus and mass density for the metal plate array in Fig. 2(a) can be calculated with the help of the effective medium theory [43]:

(10)$$\frac{1}{{{\rho _x}}} = \frac{{{f_h}}}{{{\rho _h}}} + \frac{{{f_m}}}{{{\rho _m}}},\textrm{ }{\rho _y} = {f_h}{\rho _h} + {f_m}{\rho _m},\textrm{ }\frac{1}{\kappa } = \frac{{{f_h}}}{{{\kappa _h}}} + \frac{{{f_m}}}{{{\kappa _m}}}.$$

The subscripts h and m means host medium and metal, respectively. f_h=a/d and f_m=(d-a)/d are the filling factors of the host medium and metal, respectively. We also choose air as the host medium (ρ_h=1.29 kg/m³, к_h=0.15MPa), brass as the metal (ρ_m=8500 kg/m³, к_m=104GPa), and geometrical parameter d = 2a (i.e., f_h=f_m=0.5). In this case, the effective modulus and mass density of the metal plate array in Fig. 2(a) can be calculated by Eq. (10): ρ_x=2.58kg/m³, ρ_y=4250.6kg/m³, к=0.3MPa, which can be treated as a reduced 2D null medium for an acoustic wave. There have also been some separate studies on the structure in Fig. 2(b) for electromagnetic waves [46] and acoustic waves [47]. Actually, the structure in Fig. 2(b) can perform as a perfect endoscope for both electromagnetic wave and acoustic wave simultaneously, and thus it is possible to realize 3D reduced multi-physics null medium. For convenience of simulations, we only consider 2D reduced multi-physics null medium in this study. As previously analyzed, the metal plate array by layered brass and air can perform as a 2D reduced multi-physics null medium for both electromagnetic waves and acoustic waves when both the effective medium approximation (λ₀>>d > a) and the Fabry-Pérot resonance condition (i.e. h =mλ₀, m = 1, 2, 3…) are satisfied. If the geometrical parameters of the brass plate array satisfy d = 2a, the effective parameters of the metal plate array by layered brass and air can be summarized by:

(11)$$ \left\{ \begin{array}{l} {\varepsilon _y} = 2{\varepsilon _0},{\mu _z} = 0.5{\mu _0},{\varepsilon _x} \to \infty \\ {\rho _x} = 2.58kg/{m^3},{\rho _y} = 4250.6kg/{m^3}, = 0.3MPa \end{array} \right..$$

From Eq. (11), the effective permittivity is infinitely large and very small along x and y directions, respectively. The permeability is also very small in z direction, which can be treated as a reduced null medium with the main axis along the x direction for TM polarized electromagnetic wave. The mass density is extremely large and very small in y and x direction, respectively. At the same time, its modulus is very large, which can be treated as a reduced null medium for acoustic waves. In summary, the metal plate array by layered brass and air of size d = 2a performs as a reduced 2D multi-physics null medium with the main axis along the orientation of metallic plates for both TM polarized electromagnetic waves and acoustic waves when the effective medium approximation and the Fabry-Pérot resonance condition are satisfied. It means that the metal plate array by layered brass and air can perform as a reduced multi-physics endoscope, which can one-to-one correspondingly project TM polarized electromagnetic fields and acoustics fields from one surface it linked to another surface it linked along the direction of the orientation of metallic plates. Next, we will show how to design some novel multi-physics devices based on the directional projecting feature of this reduced 2D multi-physics null medium.

3. Design examples and numerical simulations

The first example is the multi-physics beam shifter in Fig. 3(a), which can shift electromagnetic wave and acoustic wave simultaneously by a fixed pre-defined displacement. The input surface and output surface of the shifter are S₁ and S₂, respectively, which are connected by the brass plate array. The direction of the brass plane is the main axis of the reduced multi-physics null medium, which can project both electromagnetic field and acoustic field distribution from the input surface to the output surface. Figures 3(b)–3(e) show that the shifter can shift the position of the incident beam along y direction by a fixed pre-defined displacement h sinα without changing its wave-front. The multi-physical beam shifter can steer electromagnetic wave and acoustic wave simultaneously, which can work as a building block to achieve many other multi-physics illusions (e.g., imaging and scattering reduction). Multi-physics shifter can work not only for normal incidence but also oblique incidence (see Visualization 1 and Visualization 2).

Fig. 3. (a) Structure of the brass plate array used to achieve a multi-physics shifter for both electromagnetic wave and acoustic wave simultaneously (the thick black lines are brass and all other white regions are air). The length of each brass plate is h =mλ₀ (the Fabry-Pérot resonance condition) and m = 2 here. The thickness of brass plate and air gap are both λ₀/10, which means the filling factor of brass and air is 0.5. The angle between the brass plates and x axis is α=30 degree. The structure is designed at working wavelength λ₀=3cm: corresponding frequencies are 10 GHz for electromagnetic wave and 11.433kHz for acoustic wave, respectively. (b) and (c) are distributions of normalized magnetic field’s z component for the TM polarized waves when a Gaussian electromagnetic beam incidents onto the brass plate array by 0 degree and -30 degree, respectively (see Visualization 1). (d) and (e) are normalized acoustic pressure distributions when a Gaussian acoustic beam incidents onto the brass plate array by 0 degree and -30 degree, respectively (see Visualization 2). (e) and (f) show the reflectivity of the multi-physics shifter in (a) when the wavelength deviates from the designed value λ₀=3 cm for EM wave and acoustic wave, respectively (see Visualization 3 and Visualization 4).

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The reduced null medium by brass plate array may introduce some reflections when the Fabry-Pérot condition is not satisfied. The reflectivity of the multi-physics shifter changes as the wavelength deviates from the designed value λ₀=3cm in Figs. 3(f) and 3(g) for EM wave and acoustic wave, respectively (see Visualization 3 and Visualization 4). If the thickness of the device does not satisfy the Fabry-Pérot condition, the multi-physics shifter can still maintain its multi-physical shifting effect, while some unexpected reflections may appear.

Next, we can use the above designed multi-physics shifter as the building block to create a scattering reduction device that can simultaneously guide both the electromagnetic wave and acoustic wave smoothly around the strongly scattering object and restore the wave-front. We can use four shifters of Fig. 3(a) with the configuration shown in Fig. 4(a) to achieve a scattering reduction device. The detecting beam from the left is firstly split by the front two shifters, and then combined by the two rear shifters. In the central red region we put some strongly scattering object (e.g. PEC boundary for electromagnetic waves and hard wall boundary for acoustic waves). Figures 4(b)–4(g) shows the good scattering reduction performance of the device by the brass plate array for the electromagnetic wave and acoustic wave simultaneously. If we remove the scattering reduction device (i.e., only the central strongly scattering object is left in air), the scattering is obvious (see Fig. 5 for comparison). There were some separate studies on invisibility cloaking to achieve scattering reduction for electromagnetic waves by transformation optics [1–5,48–50] and acoustic waves by transformation acoustic [6,51–53]. However, previous studies can work only for electromagnetic waves or acoustic waves. Our scattering reduction device can effectively reduce the scattering cross section for both the electromagnetic wave and acoustic wave (not only for normal incidence but also oblique incidence; see Visualization 5 and Visualization 6), which may have some potential applications in stealth technology for both radar and sonar. The multi-physical scattering reduction device can still show good scattering reduction effect for both EM wave and acoustic wave when the working wavelength is scanned to some frequencies at which the Fabry-Pérot condition is not satisfied (see Visualization 7 and Visualization 8), compared with the cases when the scattering reduction device is removed (see Visualization 9 and Visualization 10).

Fig. 4. (a) The structure of the brass plate array for multi-physics scattering reduction. The thick black lines are brass plates of thickness λ₀/10 and length h= 2λ₀. (b)-(d) The normalized magnetic field’s z component for the TM polarized waves when a Gaussian electromagnetic beam incidents onto the brass plate array by 0, 20 and 40 degrees (see Visualization 5). (e)-(g) The normalized acoustic pressure when a Gaussian acoustic beam incidents onto the brass plate array by 0, 20 and 40 degrees (see Visualization 6). The boundaries of the central object of strong scattering are set as PEC and hard walls for electromagnetic wave and acoustic wave, respectively. All other parameters are the same as those in Fig. 3. Visualization 7 and Visualization 8 show Gaussian beam normally incidents onto the multi-physics scattering reduction device in Fig. 4(a) when the wavelength of incident Gaussian beam changes from λ₀/3 to 2λ₀ for electromagnetic case and acoustic case, respectively.

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Fig. 5. Scattering when a Gaussian beam incidents onto the strongly scattering object by 0, 20 and 40 degrees. (a)-(c) The corresponding normalized magnetic field’s z component when the brass plates are removed from Figs. 4(b)–4(d) (only the central object of strong scattering is left in air). (d)-(f) The corresponding normalized acoustic pressure when the brass plates are removed from Figs. 4(e)–4(g). Other parameters are the same as those in Fig. 3. Visualization 9 and Visualization 10 show Gaussian beam normally incidents onto the strongly scattering object in the center of Fig. 4(a) (scattering reduction device is removed) when the wavelength of incident Gaussian beam changes from λ₀/3 to 2λ₀ for electromagnetic case and acoustic case, respectively.

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Since the multi-physics null medium (e.g., brass plate array) can guide both electromagnetic wave and acoustic wave along its main axis, we can gradually change the main axis of the brass plate array to steer the propagation or radiation for both the electromagnetic wave and acoustic wave simultaneously. As shown in Fig. 6(a), two cascaded brass plate arrays indicated in green and black with the length L₁=λ₀ and L₂=2λ₀, respectively, can steer the incident electromagnetic wave and acoustic wave by a pre-defined angle. The inner brass plates (colored green) and outer brass plates (colored black) are rotated by 10 degrees and 20 degrees from the radical direction, respectively. If the acoustic beam and electromagnetic wave incident onto the beam steering device from the right, the output beams are steered by 20 degrees in Figs. 6(b) and 6(c). There were some separate studies on how to steer electromagnetic waves by transformation optics [54–56] and acoustic waves by transformation acoustic [57,58]. With the help of the present surface transformation multi-physics, we can easily design the orientation of the brass plate arrays to achieve a desired beam steering effect for both the electromagnetic wave and acoustic wave simultaneously.

Fig. 6. (a) Structure of the beam steering device. (b) and (c) are normalized amplitude of acoustic pressure and z component of the magnetic field in 2D numerical simulations when acoustic wave and TM polarized electromagnetic wave incident onto the beam steering device, respectively.

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Numerical simulations in Figs. 3–6 are made by commercial software COMSOL Multiphysics 5.0 based on the finite element method (wave optics package for electromagnetic wave and acoustic package for acoustic wave).

4. Discussion

The filling factor of brass and air for the brass plate array in Figs. 3 and 4 are f_h=f_m=0.5, whose effective material parameters are given in Eq. (11). Compared with the ideal null medium whose material parameters are given in Eq. (7), the brass plate array performs as a reduced null medium. In addition, the degrees of reduction are different for acoustic wave and EM wave, and this is the reason why the performance of the devices in Figs. 3 and 4 appears to be poorer for the case of acoustic waves as compared with the case of electromagnetic waves.

There are also some other ways to design multi-physical devices [31,32]. A homogenization method [32] can be used to achieve a carpet cloak [31] and full space cloak [32] working for microwave, acoustic wave and water wave if the aluminum cloak can be designed for one kind of these waves, as aluminum cloak provides the same mathematical model (Helmholtz equation with Neumann boundary conditions) for transverse magnetic EM wave (perfect electric conductors in microwave frequencies), acoustic wave (rigid inclusions in air) and water wave (no flow conditions). However, we still need to design first a cloak or devices of other functions before we extend these devices to multi-physical fields by the homogenization method. The homogenization method provides a way to extend the designed device for one physical field to multi-physical fields, but not a general design method as the transformation optics and surface transformation multi-physics proposed in this study. Surface transformation multi-physics can be utilized directly to design multi-physical devices of various functions in a surface-mapping way. Another important difference between the surface transformation multi-physics and the homogenization method is that all multi-physical devices of various functions designed by our method only require one same material (i.e., null medium), which is homogenous (no gradient control is required). The same structure of metal plate array given in Fig. 2 can be utilized to realize all devices designed by our surface transformation multi-physics no matter how functions, geometrical shapes and sizes of these devices would change. However, devices designed by the homogenization method require different aluminum units when the function or geometrical shapes change. For example, full space cloak [32] and carpet cloak [31] are both designed by the homogenization method, but using different aluminum structures (perforated rotating aluminum disc with gradient small air tubes and aluminum meta-surfaces).

5. Summary

The designing process in surface transformation multi-physics is very simple. The first step is to choose the proper geometrical shapes of the input/output surfaces. The second step is to fill the brass plate array of the proper orientations that can link the input and output surfaces. The period of the brass plate array should be much smaller than the working wavelength and the length of each brass plate satisfies the Fabry-Pérot condition. Compared with coordinate-transformation-based methods, our surface transformation multi-physics method has many advantages, which can be summarized as: 1) the designing process is very simple. We only need to design the geometrical shapes of the input/output surfaces and the orientations of the brass plate array between them. There are no complex mathematical calculations during the whole designing process. 2) All devices of various functions designed by our method only require one natural material, e.g. the brass plate array, to realize, while conventional coordinate-transformation-based methods requires different materials for various devices. Note that some devices designed by coordinate-transformation-based methods can be implemented with the same materials by applying effective medium theories [48]. 3) All devices designed by our method can always work for both electromagnetic waves and acoustic waves simultaneously. The method (surface transformation multi-physics) proposed in this study is derived from the transformation optics, which is a more general method. Only a subset of devices of novel functions designed by the transformation optics can be achieved by the surface transformation method.

Exploring and designing functional devices of multi-physics is an interesting topic for transformation optics and artificial metamaterials. Controlling electromagnetic waves and acoustic waves simultaneously will promote the development of multi-physics technology. The present surface transformation multi-physics provides an effective, simple and realizable way to design novel multi-physics devices. Multi-physics devices that can control electromagnetic wave and acoustic wave simultaneously will have important applications in various fields (e.g., composite stealth technology and multi-physics biomedical imaging).

Funding

National Natural Science Foundation of China (11604292, 11621101, 11674239, 60990322, 61905208, 61971300, 91233208, 91833303); Scientific and Technological Innovation Programs (STIP) of Higher Education Institutions in Shanxi (2019L0146, 2019L0159); he National Key Research and Development Program of China (2017YFA0205700); the Program of Zhejiang Leading Team of Science and Technology Innovation.

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Surface transformation multi-physics for controlling electromagnetic and acoustic waves simultaneously

Abstract

1. Introduction

2. Theoretical method

3. Design examples and numerical simulations

4. Discussion

5. Summary

Funding

References

Supplementary Material (10)

Cited By

Figures (6)

Equations (11)

Optics Express

Name	Description
Visualization 1	Visualization 1
Visualization 2	Visualization 2
Visualization 3	Visualization 3
Visualization 4	Visualization 4
Visualization 5	Visualization 5
Visualization 6	Visualization 6
Visualization 7	Visualization 7
Visualization 8	Visualization 8
Visualization 9	Visualization 9
Visualization 10	Visualization 10