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).

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.

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).

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.

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.

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.