$\mathcal{P}\mathcal{T}$-Symmetric Acoustics

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$P T$ -Symmetric Acoustics

Xuefeng Zhu, Hamidreza Ramezani, Chengzhi Shi, Jie Zhu, and Xiang Zhang

Phys. Rev. X 4, 031042 – Published 5 September 2014

Abstract

We introduce here the concept of acoustic parity-time ( $P T$ ) symmetry and demonstrate the extraordinary scattering characteristics of the acoustic $P T$ medium. On the basis of exact calculations, we show how an acoustic $P T$ -symmetric medium can become unidirectionally transparent at given frequencies. Combining such a $P T$ -symmetric medium with transformation acoustics, we design two-dimensional symmetric acoustic cloaks that are unidirectionally invisible in a prescribed direction. Our results open new possibilities for designing functional acoustic devices with directional responses.

Received 9 May 2014

DOI:https://doi.org/10.1103/PhysRevX.4.031042

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Authors & Affiliations

Xuefeng Zhu¹, Hamidreza Ramezani¹, Chengzhi Shi¹, Jie Zhu¹, and Xiang Zhang^1,2,*

¹NSF Nanoscale Science and Engineering Center, University of California, Berkeley, California 94720, USA
²Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

^*Corresponding author. xiang@berkeley.edu

Popular Summary

Controlling acoustic and phononic transport at will is a long-sought-after goal in modern applied physics. Advancements in metamaterials so far have resulted in intriguing achievements in acoustic and phononic transport manipulation, primarily based on modulation of the real parts of acoustic parameters. We examine the acoustic characteristics in the complex domain where systems exhibit judicious balanced gain and loss [i.e., parity-time ( $P T$ ) symmetric systems]. Such systems possess unidirectional zero reflection at specific frequencies and act as acoustic one-way cloaks that react differently depending on the direction of wave propagation.

We introduce the concept of $P T$ symmetry into acoustics by judicious designs of acoustic materials with complex parameters featuring carefully balanced loss and gain regions. Based on exact analytical expressions and the acoustic transfer matrix method, we show that our design displays unique scattering characteristics, including asymmetric reflection of mm-length waves. We are able to produce a unidirectional transparent medium. Since the acoustic wave equation is invariant under coordinate transformations, we combine transformation acoustics with $P T$ symmetric grating materials and demonstrate a $P T$ symmetric, two-dimensional cloak that protects its inner information from being detected only along one side but not another—a one-way cloak.

We expect that $P T$ acoustics will prompt new investigations for designing functional acoustic systems with nonreciprocal responses, enabling explorations of fundamental concepts related to $P T$ -symmetric systems using acoustic nonlinearities or time-varying parameters. Examining the complex part of acoustic parameters may reveal new functionalities of metamaterials largely immune to frequency shifts. These metamaterials may have direct applications related to directional noise cancellation and engineering buildings for optimal acoustics.

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Vol. 4, Iss. 3 — July - September 2014

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Acoustics

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Figure 1
Schematic of the acoustic $P T$ -symmetric medium. The scattering domain is composed of 11 sections: three gain regions (red blocks), three loss regions (green blocks), and five passive regions in between. The length of the active region is 0.148 m, while the length of each passive region is 0.015 m. The transmission and reflection of acoustic waves incident from the left (blue curve) and right (red curve) are shown. This $P T$ -symmetric medium is designed to be reflectionless for acoustic waves incident from the left. Insets: The normalized amplitude of the pressure field for the left (left inset) and right (right inset) incidences at the exceptional point.
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Figure 2
Phases and amplitude of left-reflected, right-reflected, and transmitted waves of the acoustic $P T$ -symmetric medium are shown in (a) and (b), respectively. The absolute value of the eigenvalues and the second component of the eigenvectors of the corresponding scattering matrix for this acoustic $P T$ -symmetric medium are shown in (c) and (d). Here, we have normalized the eigenvectors such that the first component is always unitary.
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Figure 3
The pressure-field maps in (a),(b) the virtual space and (c),(d) the physical space for a one-way invisible $P T$ -symmetric cloak. In (a) and (c), where the plane wave is incoming from the left, a strong backscattered reflection can be observed due to the satisfied phase matching, and therefore, the observers on the left can sense the existence of the cloak. However, in (b) and (d), where the plane wave is incident from the right, the waves propagate without any scattering due to phase mismatching, and thus, an arbitrary object is perfectly concealed in the white region. The wavelengths of incident waves are $λ = 0.0572 m$ .
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Figure 4
(a) Scattering cross section versus frequency, where the blue (upper) and red (lower) curves describe the scattering cross section of the $P T$ -symmetric cloak when the light is incoming from the left and right, respectively. (b) Scattering cross section versus the incident angle at the frequency 6 kHz, where $θ = 0$ corresponds to the parallel propagation along the $z$ direction from left to right.
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Figure 5
The pressure-field maps in (a),(b) the virtual space and (c),(d) the physical space for a one-way invisible $P T$ -symmetric acoustic ground cloak. In (a) and (c), where the plane wave is incoming from the left, a strong backreflection can be observed due to the satisfied phase matching, and therefore, the observers on the left can sense the existence of the ground cloak. However, in (b) and (d), where the plane wave is incident from the right, the waves propagate without any scattering due to phase mismatching, and thus, an arbitrary object can be concealed in the bump region. The wavelengths of incident waves are $λ = 0.8 m$ .
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Physical Review X

PT-Symmetric Acoustics

Xuefeng Zhu, Hamidreza Ramezani, Chengzhi Shi, Jie Zhu, and Xiang Zhang

Phys. Rev. X 4, 031042 – Published 5 September 2014

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$P T$ -Symmetric Acoustics