Extension of coherent perfect absorption-lasing effect and super-collimation effect in an airborne phononic crystal with space-coiling structure

Abstract

Extending attractive phenomena in non-Hermitian systems is crucial for advancing wave manipulation properties. In this study, we extend the phenomena of coherent perfect absorption-lasing (CPAL) as well as super-collimation, which were generally achieved at specific angles and frequencies, to broad-angle and broadband, respectively. In an airborne two-dimensional phononic crystal, the combination of band folding and gain-loss modulation induces a parity−time phase transition, resulting in parity−time broken phase as well as a slab of exceptional points along one of the Brillouin zone boundaries. Based on the analysis of Hamiltonian, we design a Hilbert fractal space-coiling structure that minimizes the dispersion along this boundary. This approach significantly broadens the range of incident angles for CPAL and extends the frequency range for super-collimation. Our findings provide a design strategy for exploring wave manipulation phenomena in two-dimensional parameter spaces.

Graphical abstract

Keywords

acoustic metamaterials / coherent perfect absorption / phonoic crystal / parity−time

1 Introduction

Parity−time (PT) symmetric systems have garnered significant research interest in past decades [1−4]. Such kind of non-Hermitian system remains unchanged after the synergy of parity and time reversal [5−7], and enables the occurrence of real eigenvalues spectra [8−10]. There are many fascinating phenomena associated with PT-symmetric systems, such as unidirectional invisibility [11−13], whispering-gallery-mode resonating [14], and lasing [15, 16]. One of peculiar phenomena in a PT-symmetric system, coherent perfect absorption-lasing (CPAL) effect, draws extensive attention in electromagnetic and acoustic systems [17−22]. Although PT-symmetry has been extensively studied in one-dimensional and two-dimensional systems [17−22], CPAL is almost exclusively discussed in one-dimensional parameter space, e.g., for normal incident wave or specific working frequencies [1, 2]. It is partly because the CPAL is often related to the eigenvalues of scattering matrix associated with PT-symmetric systems, which are not straightforward to design directly through the geometry of artificial materials [23].

Acoustic metamaterial is a type of artificial material that has the capacity of flexibly control, direct, and manipulate phonons or acoustic waves [24, 25]. It offers distinctive functionalities, such as zero or negative refractive index [26−30], acoustic cloaking [31−33], acoustic subwavelength imaging [34−36], and acoustic self-bending effect [37]. Recently, space-coiling metamaterials, which enables broadband slowing-down of acoustic waves, have been theoretically and experimentally investigated for airborne acoustics [38−42]. The mechanism of the space-coiling metamaterial is to make the acoustic wave propagate within the structure, thereby increasing the distance that sound wave travels, reducing the effective phase velocity or increasing the effective refractive index, and thus enables the enhancement of emission at low frequencies by resonances [41]. Inspired by fractal geometry [43], metamaterials with self-similar fractal structures have been widely investigated in mechanics [44], electromagnetics [45], and acoustics [46], for their unique properties. In the past decade, a type of acoustic metamaterial with Hilbert fractal has yielded minimal dissipation of energy during transmission and elevated refractive index via their quasi-fractal geometries. The effective refractive index of such metamaterials can be enhanced by increasing the order of the Hilbert fractal, thereby enriches the design strategies of ultra-thin acoustic devices, such as lenses and absorbers [47, 48].

In this study, we propose a PT-symmetric two-dimensional (2D) airborne phononic crystal (PC) to extend the CPAL effect over a broad range of incident angles and a super-collimation effect over a broad frequency. Our approach creates doubly degeneracies along one of the Brillouin zone (BZ) boundaries by exploiting a band folding mechanism. By implementing periodic PT-symmetric gain-loss modulation, the PC experiences a thresholdless transition from the PT-symmetric phase to the PT-broken phase, deforming the double degeneracy into a slab of exceptional points (EPs) [49, 50]. Based on the analysis of Hamiltonian, we engineer the structure by using space-coiling structure with Hilbert fractal to minimize the dispersion of the EP slab as well as the band structure nearby, resulting in directional negligible dispersion. The directional dispersionless feature extends CPAL and super-collimation effect in 2D parameter space.

2 PT phase transition in an airborne 2D phononic crystal

We consider a 2D PC consisting of a hypothetical material and an air background. Fig.1(a) shows the unit cell, composed of two square rods with mass density {\rho }_r=1.21\mathrm{k}\mathrm{g}/{\mathrm{m}}^3 and side length a=61\mathrm{m}\mathrm{m}. The lattice constants are a_x=300\mathrm{m}\mathrm{m} along the x axis and a_y=150\mathrm{m}\mathrm{m} along the y axis. The centers of the rods are spaced by a_x/2. The mass density and sound speed of air are set as {\rho }_b=1.21\mathrm{k}\mathrm{g}/{\mathrm{m}}^3 and c=343\mathrm{m}/\mathrm{s}, respectively.

Doubly degeneracies along the BZ boundary are generated by a band-folding mechanism in the unit cell [51−53]. In addition, we consider PT-symmetric gain-loss modulation in the square rods, represented by the imaginary components of their sound velocities v_g=(140-\mathrm{i})\mathrm{m}/\mathrm{s} and v_l=\left(140+\mathrm{i}\right)\mathrm{m}/\mathrm{s}, respectively. We use COMSOL Multiphysics to calculate the band structure of the PC. In terms of dimensionless frequency 2\pi c/a_x, the real and imaginary parts of the band structure are shown in Fig.1(b) and (c). PT-symmetric non-Hermiticity deforms the original degeneracy into a slab of exceptional points (EPs) near the BZ boundary \mathrm{X}\mathrm{M}.

An attractive phenomenon in non-Hermitian systems, CPAL, is always discussed for incident waves at specific frequencies and incident angles and is related to the PT-broken phase [54]. This implies that the band structure should meet certain specific conditions to achieve CPAL in high-dimensional parameter space, i.e., broad band or broad angle. For instance, to ensure the effectiveness of CPAL for wide incident angles, a directional dispersionless PT-broken phase is expected. For this purpose, we need to investigate the Hamiltonian near the \mathrm{X}\mathrm{M} boundary of the BZ and engineer the square rods in the unit cell.

3 Minimize the dispersion of the EP slab along the boundary of BZ

For the lowest bands, the acoustic pressure field is mainly localized in the region of low sound speed. Thus, the PC can be modeled as a 2D non-Hermitian lattice (as illustrated in Appendix A). The square rods with loss and gain can be seemed as two sites A and B in one unit cell. The real-space Hamiltonian can be written as

(1)\begin{array}{rl}H=& \sum _{m,n}\left[\left(\omega -\mathrm{i}\beta \right)a_{m,n}^{†}{a}_{m,n}+\left(\omega +\mathrm{i}\beta \right)b_{m,n}^{†}{b}_{m,n}\right]\\ & +\sum _{m,n}\left[t_x{(a}_{m,n}^{†}{b}_{m,n}+a_{m+1,n}^{†}{b}_{m,n}\right)+h.c.]\\ & +\sum _{m,n}\left[t_y{(a}_{m,n+1}^{†}{a}_{m,n}+b_{m,n+1}^{†}{b}_{m,n}\right)+h.c.],\end{array}

where m and n denote the indices of unit cells in the x and y directions, \omega is the onsite energy associated with the frequency of eigenstates, t_x and t_y are the hopping strengths along x and y directions, a^{†}\left(a\right) and b^{†}\left(b\right) are the creation (annihilation) operators, and h.c. denotes the Hermitian conjugate, respectively. We apply the Fourier transform

(2)\{\begin{array}{c}{a}_{m,n}={\displaystyle \frac{1}{\sqrt{N_x{N}_y}}}{{\displaystyle \sum }}_{\mathit{k}}{\mathrm{e}}^{\mathrm{i}(m{k}_x{a}_x+n{k}_y{a}_y)}{a}_{\mathit{k}}\\ b_{m,n}={\displaystyle \frac{1}{\sqrt{N_x{N}_y}}}{{\displaystyle \sum }}_{\mathit{k}}{\mathrm{e}}^{\mathrm{i}(m{k}_x{a}_x+n{k}_y{a}_y)}{b}_{\mathit{k}}\end{array},

where N_x and N_y are the total number of sites, k_x and k_y are the x and y components of wave vector \mathit{k}. Then we substitute Eq. (2) into the real-space Hamiltonian. In the basis {\psi }_{\mathit{k}}={(a_{\mathit{k}},b_{\mathit{k}})}^{\mathrm{T}}, the Hamiltonian in the momentum space can be expressed as

(3)H=\left(\begin{array}{cc}\omega +2t_y\cos \left(k_y{a}_y\right)-\mathrm{i}\beta & t_x(1+{\mathrm{e}}^{-\mathrm{i}{k}_x{a}_x})\\ t_x(1+{\mathrm{e}}^{\mathrm{i}{k}_x{a}_x})& \omega +2t_y\cos \left(k_y{a}_y\right)+\mathrm{i}\beta \end{array}\right).

The Hamiltonian describes a non-Hermitian phononic crystal with PT-symmetric gain and loss in the A and B sites, and anisotropic nearest-neighbor hopping characterized by t_x and t_y along x and y directions. The non-Hermitian term \pm \mathrm{i}\beta enables a PT phase transition, while the anisotropic hopping leads to the directional-dependent dispersion as well as the phenomena discussed in this work. The eigenvalues of the Hamiltonian can thus be calculated as

(4)\{\begin{array}{c}{\omega }_1=\omega +2t_y\cos \left(k_y{a}_y\right)+\sqrt{{2t}_x^2+{2t}_x^2\cos \left(k_x{a}_x\right)-{\beta }^2}\\ {\omega }_2=\omega +2t_y\cos \left(k_y{a}_y\right)-\sqrt{{2t}_x^2+{2t}_x^2\cos \left(k_x{a}_x\right)-{\beta }^2}\end{array}.

At the BZ boundary k_x=\pi /a_x, the eigenvalues are

(5)\{\begin{array}{c}{\omega }_1=\omega +2t_y\cos \left(k_y{a}_y\right)-\mathrm{i}\beta \\ {\omega }_2=\omega +2t_y\cos \left(k_y{a}_y\right)+\mathrm{i}\beta \end{array},

indicating that t_y must be significantly reduced in order to minimize the dispersion of the EP slab along k_y. Intuitively, a non-zero t_x is required to maintain non-zero dispersion along k_x when t_y approaches zero, which means that the hypothetical material in Fig.1 needs to be replaced by an anisotropic one.

Space-coiling structures are effective in reducing the phase velocity by increasing the propagation distance within a compact unit cell. However, many space-coiling structures simultaneously reduce the effective phase velocity in both x and y directions [38, 39], thereby configuring both t_x and t_y simultaneously. To achieve a significant suppression of t_y while maintaining a finite t_x, we consider a space-coiling structure with Hilbert fractal to enhance the anisotropy. A Hilbert curve is a continuous fractal curve constructed via a recursive iteration [55]. The order of the Hilbert curve refers to the level of iteration, determines how many times a square is divided. At the order n, the square is divided into 4^n smaller squares (a 2^n\times 2^n grid), and the Hilbert curve connects their centers. Fig.2 shows space-coiling structures with first-, second-, and third-order Hilbert fractal curves. In Fig.2(a), we divide a sound-hard square (colored blue) with side length a into 2^1\times 2^1 sections, and connect the center of the sections clockwisely to increase a first order fractal. An air channel with width b is opened, with both the inlet and outlet aligned horizontally, resulting in anisotropy in the PC as well as in the corresponding Hamiltonian. When such kind of structures are arranged periodically in plane, the orientation of channel determines that they can couple with their neighbors in the x direction, but cannot couple with their neighbors in the y direction. Fig.2(b) and (c) exhibit the second-order channel and the third-order channel, respectively, which can be built by dividing the squares continually into 2^2\times 2^2 and 2^3\times 2^3 sections [55].

We choose the third-order fractal space-coiling structure to replace the square rods in Fig.1 and configure the lattice constants of the PC, and then minimize the dispersion of the EP slab along k_y. The new unit cell after engineering, with a_x=150\mathrm{m}\mathrm{m}, a_y=250\mathrm{m}\mathrm{m}, and a=61\mathrm{m}\mathrm{m}, is in the inset of Fig.3(a). Fig.3(a) and (b) display the real and imaginary components of the band structures along \mathrm{\Gamma }\mathrm{X} and the boundaries of the first BZ, respectively. Both of them exhibits nearly zero dispersion along the BZ boundary k_x=\pm \pi /a_x as predicted. Meanwhile, Fig.3(c) and (d) show the real and imaginary components of the dispersion surfaces near the BZ boundary k_x=\pi /a_x. Two straight trajectories of EP exist at k_x=0.965(\pi /a_x) and k_x=1.035(\pi /a_x), outline the PT-broken phase at and PT-exact phase at \left|k_x-\pi /a_x\right|>0.035(\pi /a_x).

As mentioned in previous works [1, 9, 54], the length L of a finite-sized system plays an important role in the emergence of CPAL effect. Since our Hamiltonian in the momentum space and the band structure in Fig.3 are obtained from periodic systems, they are not directly predicting the CPAL effect in a PC composed of finite layer of unit cells. The analysis of Hamiltonian is a qualitative design strategy to achieve dispersionless PT-broken phase, which is a necessary condition for achieving broad-angle CPAL in finite-sized systems. Moreover, the design strategy in this work is different from band engineering method based on the analysis of eigenfields [56]. It is because a general approach in the design of photonic crystal or water-based PCs is to use materials with lower wave speed (compared to that in the background medium) to construct scatterers. But for airborne phononic systems, the choice of materials with a wave speed smaller than air is extremely limited. Therefore, we consider sound-hard materials to construct the space-coiling structure, engineering the hopping terms in the Hamiltonian to flatten the dispersion. This approach is more efficient than band engineering and optimization algorithm with traditional models.

4 Extension of CPAL effect and super-collimation effect in a 2D parameter space

In a PT-symmetric system, CPAL effect can occur in the PT-broken phase as a result of complex interference. In previous literatures [20, 51, 57, 58], CPAL always appear at a specific incident angle and working frequency. In our PC, the directional dispersionless PT-broken phase implies that the CPAL effect can be extrapolated to a broad spectrum of incident angles.

We examine a PC slab consisting of 12 layers of unit cells in the horizontal direction, with periodicity in the vertical direction. The response of the PC to planar incident waves can be obtained by COMSOL Multiphysics in terms of both frequency and incident angle. The CPAL effect corresponds to two special cases of the eigenvalues of the following scattering matrix:

(6)S=\left(\begin{array}{cc}{r}_l& t\\ t& r_g\end{array}\right),

where t, r_l and r_g represent the transmission coefficient, the reflection coefficients when plane wave incident from loss side and gain side, respectively [23]. Fig.4(a) is the smaller eigenvalue (\left|{\lambda }_1\right|) of the scattering matrix, while Fig.4(b) is the larger one (\left|{\lambda }_2\right|). In a system undergoing a PT phase transition, the PT-exact phase displays a uni-modular behavior (\left|{\lambda }_1\right|=\left|{\lambda }_2\right|=1), while the PT-broken phase exhibits an inverse behavior (\left|{\lambda }_1\right|=1/\left|{\lambda }_2\right|) [59]. Due to the dispersionless nature in the band structure, there is a line of zeros in Fig.4(a) and a line of poles in Fig.4(b) for wide incident angles around the frequency of PT-broken phase \omega =0.179(2\pi c/a_x), implying an extension of CPA effect as well as lasing effect from specific incident angle to broad-angle. At \omega =0.179(2\pi c/a_x), Fig.4(c)−(j) show the response of the PC when two counter-propagating planar waves, with incident angles 0°, 30°, 45° and 60°, are incident onto opposite sides of the PC. For each incident angle, there exists CPA (Lasing) effect when there is a phase difference of 0.5\pi \left(1.5\pi \right) between coherent incident waves.

It should be emphasized that the directional flat bands in our PC are not directly induced by the PT-symmetric gain-loss modulation, but are generated by the unit cell design based on the analysis of Hamiltonian. The configuration of Hilbert fractal leads to directional dispersionless bands span across the PT-exact and PT-broken phases. In the PT-broken phase, the directional dispersionless bands are the necessary condition for broad-angle CPAL. In the PT-exact phase, the directional dispersionless bands are naturally related to other wave manipulation properties.

In previous literatures, isotropic flat bands are often associated with slow-wave effects or high-Q cavities due to their high density of states as well as low group velocity [60−65]. Some flat bands cover the whole BZ, while others appear in a restricted region in momentum space [64−67]. The directional flat bands in our work do not result in such a high density of states, but exhibits strong anisotropy: dispersive along k_x and dispersionless along k_y. Thus, the direction of group velocity is limited.

In Fig.5(a) and (b), we present the equal-frequency contours for the first and second bands, respectively. The dashed box in Fig.5(a) as well as the entire Fig.5(b) represent the frequency range of directional flat band, where the direction of the group velocity is marked by black arrows. The directional flat band along k_yover the entire BZ prevents acoustic wave from propagating along y direction, indicating a broadband super-collimation effect. To illustrate it clearly, we put a point source at the center of the PC. At \omega =0.175(2\pi c/a_x), \omega =0.18(2\pi c/a_x), and \omega =0.185(2\pi c/a_x), simulation results of acoustic pressure field distributions are plotted in Fig.6(a)−(c), respectively. In each case, wave propagates only in the horizontal direction because of the direction of group velocity. Fig.6(d)−(f) are the simulation results when the point source in Fig.6(a)−(c) is substituted with a normally incident Gaussian beam. The beam preserves its width, showing minimal divergence as it propagates through the PC, which is another evidence of the super-collimation phenomenon. The above results demonstrate that the acoustic wave propagates in our PC with strong directionality along the horizontal axis and minimal divergence.

Here are some discussions about experimental feasibility. While introducing losses in acoustic systems is relatively straightforward, realizing gain for acoustic waves remains highly constrained. Several studies have demonstrated gain in acoustic systems through active electric amplification in enclosed waveguides and cavities [68, 69]. However, our design of broad-angle CPAL is not based on a closed system. As a result, the broad-angle CPAL discussed here is not easy to be realized in experiment at this stage.

5 Conclusion

Based on the structural design and analysis of the effective Hamiltonian, we predict the extension of two typical phenomena: the CPAL effect associated with PT-symmetric systems, and super-collimation associated with flat bands, to broad-angle and broadband regimes, respectively. In an airborne PC, a transition from the PT-exact phase to the PT-broken phase is facilitated by the synergy of the band-folding mechanism and the PT-symmetric gain-loss modulation, resulting in a slab of EPs. We use a space-coiling structure with third-order Hilbert fractal in the construction of PC to minimize the dispersion along k_y near the BZ boundary. Interesting phenomena arise when the dispersionless nature is extended to both the PT-symmetric and the PT-broken phases. At the frequency of PT-broken phase, the CPAL effect exists independent of the incident angles. Furthermore, the PC supports super-collimation effect over a broad frequency range. Our work provides a design strategy to extend interesting phenomena in non-Hermitian systems to 2D parameter space.

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