Low-frequency noise mitigation and bandgap engineering using seismic metamaterials for terrestrial gravitational wave observatories

Abstract

Gravitational waves have become one of the most important observational methods for studying the Universe since their first detection. However, ground-based observatories have an inherent barrier to their detection frequency band due to the seismic and gravity gradient noises near the perturbation of the surroundings. A recent intriguing development of artificial structures for media called metamaterial is opening a new branch of wave mechanics and its application in various fields, in particular suggesting a novel way of mitigating noises by controlling the media structure for propagating waves. In this paper, we propose a novel framework for handling noises in ground-based gravitational wave detectors by using wave mechanics under metamaterial media. Specifically, we suggest an application of the bandgap engineering technique for mitigating the underground effects of acoustic noises resulting from seismic vibration in the KAGRA gravitational wave observatory.

1. Introduction

The Laser Interferometer Gravitational-wave Observatory (LIGO) was the first to detect gravitational waves (GWs) produced by merging binary black holes on September 14, 2015 [1]. This suggests that we now have a new tool for investigating the Universe by detecting gravitational changes in the history of observing the Universe, which previously depended on optical and electromagnetic (EM) approaches, culminating in the creation of a new science known as GW astronomy. As a result of this discovery, numerous international efforts to study GWs are now accelerating to create novel ideas for future GW telescopes, including ground-based GW observatories like the Cosmic Explorer [2] and Einstein Telescope [3] as well as space-based GW antennas like LISA [4].

While space-based GW antennas are made to detect signals such as merging supermassive black holes in the submillimeter Hz frequency band, ground-based GW telescopes are primarily made to detect GWs from binary collisions in the 30–2000 Hz frequency range. The difference between ground-based GW telescopes and space-based GW antennas is the presence of a seismic noise barrier brought about by the activity of the Earth. Seismic noise below 30 Hz resulting from the activities of the ground on Earth appears to be an inherent limitation that ground-based GW telescopes encounter.

Since terrestrial GW telescopes are installed on the ground, a variety of environmental conditions, including earthquakes, lightning, wind, and temperature, have an effect on their ability to be detected [5,6]. Furthermore, it is necessary to include noise effects from gravity gradients such as microseismic variations and acoustic disturbances caused by the Earth’s activity and the continuous change in air pressure, temperature, and other factors [7–10]. Therefore, overcoming the limitations of terrestrial GW telescopes needs new paradigms for understanding and reducing the surrounding noise factors affecting them. Installing the GW telescope underground to limit the transmission of surface vibrations, such as with the Einstein Telescope and KAGRA [11], is an effort to avoid some locations of the seismic noise barrier. Because there is significantly less seismic activity and no gravity perturbation resulting from convection flow by the atmosphere and ocean, another new concept against this limitation is to put a GW antenna on the Moon’s surface [12].

Apart from the aforementioned efforts, ground-based GW telescopes must decrease environmental noise like seismic vibrations to improve data quality and GW detection efficiency. Various noise sources impacting the GW telescope propagate as waves; e.g., a strong transient seismic wave near a GW observatory breaks the locked state of the laser interferometer, causing the observation mode to halt. In the network observation of GW, this interruption of the observation mode is a drawback for both the combined signal-to-noise ratio (SNR) and the sky localization efficiency of the GW event. From this point of view, it is of great importance to understand the various noise sources and mitigate them to improve the GW detection performance. In this study, we address the possibilities of minimizing this ambient noise by modulating wave propagation from the point of view of metamaterial (MM) science.

MM is defined as an artificial structure of media that causes unconventional properties of wave propagation. Historically, it was first introduced by Veselago [13] that EM fields can unconventionally propagate under a specific media condition, possessing a special property such as a negative refraction index, |$n=\sqrt{\mu \epsilon }$|⁠, by controlling the dielectric constant, ϵ, and the magnetic permeability, μ. Because we have a dispersion relation, |$k=\sqrt{\mu \epsilon }\omega$|⁠, it was then found that the wave propagates when μϵ > 0, whereas it decays exponentially when μϵ < 0. Here, we have four possible cases of wave propagation, double positive medium (DPS), ϵ negative medium (ENG), double negative medium (DNG), and μ negative medium (MNG), as below: i) DPS material (μ > 0 and ϵ > 0): forward wave propagation of most dielectric media; ii) ENG material (μ > 0 and ϵ < 0): evanescent waves for plasma media; iii) MNG material (μ < 0 and ϵ > 0): evanescent waves for gyrotropic magnetic media; and iv) DNG material (μ < 0 and ϵ < 0): backward wave propagation but no such media in nature. The classification of the medium can be illustrated graphically, as shown in Fig. 1. Many intensive studies have been conducted for the practical realization of DNG for EM waves using an artificial circuit structure [14–16], leading to the emerging development of metamaterial devices in nanophotonics. The ENG and MNG states, which produce an evanescent wave, may be used to reduce wave amplitudes in a variety of fields in addition to the realization of DNG. See Ref. [17] and references therein for the basic theory and its application to MM.

On the other hand, similar wave properties with EM MM were found to also be valid when controlling the effective density, ρ, and the effective compressibility, χ, for acoustic waves [18–20], and the mass density, ρ, and the bulk modulus, B, for seismic waves [21–24]. The MM properties were understood by investigating the structure of the bandgap in photonic crystals [25] and photon localization [26]. For acoustic and elastic waves, a similar bandgap structure was studied in phononic crystals [27,28]. The theoretical formulation of bandgap structure for EM, elastic, and acoustic waves has been extensively studied in Refs. [29–32]. Many practical applications of metamaterials can be considered in acoustics and seismological engineering, such as acoustic insulation [28], a breakwater device for ocean waves [33], and earthquake engineering [21,34–36].

From this point of view, we investigate the feasibility of applying MM to reduce the low-frequency noise resulting from seismic and/or acoustic waves that affect the ground-based observatory GW, in particular the Japanese GW detector called KAGRA. The KAGRA detector is an underground 3 km laser interferometer with a cryogenic test mass configuration. In Ref. [37], it was addressed that the underground effect by wind can be affected by the air pressure perturbation in the KAGRA gravitational wave detector tunnel in the low-frequency band. More specifically, a close association between wind speed during the day on the Ikeno mountain has been reported and a nonlinear correlation found between GW and microphone channels at the same time. A hypothetical explanation is transformation from seismic wave propagation to acoustic vibration in the tunnel. As addressed in Ref. [37], the noises from the sound pressure inside the tunnel in the directions of the y and x arms are quite different from each other due to the configuration of the KAGRA detector inside the mountain; the y arm is parallel to the slope of the valley, while the x arm is positioned from the slope to deep inside the mountain. The strong wind in the valley of the mountain strikes the slope on the y-arm side; then the seismic wave propagates from the slope to the tunnel of the KAGRA detector. Its vibration can transform into an acoustic wave inside the tunnel, which can produce acoustic noise. A simplified finite element analysis has been shown in Ref. [37], which presents a plausible explanation for the coincidence during the windy period. Since low-frequency acoustic noise may be thought of as transforming from seismic waves, it is anticipated that decreasing the seismic vibration will necessarily result in a reduction in acoustic noise. In this background, we want to mitigate the seismic propagation by the wind that transforms into acoustic vibration in a low-frequency band through metamaterial structures and bandgap engineering.

2. 2D metamaterial structure and seismic noise reduction

Let us start with a simple case of a 2D plane with periodic structure as shown in Fig. 2A. The specification of each circle is one meter in diameter squarely arranged in 11 × 11 square meters at intervals of one meter each. Each circle is assumed to be structural steel with typical material properties of ρ = 7.85 × 10³ [kg/m³], B = 2.0 × 10¹¹ [Pa], and the Poisson ratio ν = 0.30. The residual domain is assumed to be soil with the material properties of ρ = 1.35 × 10³ [kg/m³], B = 1.0 × 10⁶ [Pa], and ν = 0.45. A unit cell containing structural steel in Fig. 2A describes an irreducible Brillouin zone (IBZ) with a Bloch–Floquet boundary condition. The boundary constrains the displacement at boundaries of the periodic structure as |$u_{2} = e^{-i\vec{k}_\mathrm{BF}\cdot (\vec{x}_2-\vec{x}_1)} u_{1}$|⁠, where |$\vec{k}_\mathrm{BF}$| is the Bloch–Floquet wave vector. We impose the periodic boundary condition on the edge of the unit cell, then the eigenvalue problem on this IBZ cell,

completely determines the dispersion relation and the bandgap structure, where |$\mathfrak {m}$| is a Lamé constant. The resulting bandgap structure between 6.126 and 7.598 Hz is shown in Fig. 2B.

On the other hand, we consider the deformation of a solid platform by an external force for a given material, which is described by the continuum equation of motion as

where P is the first Piola–Kirchhoff stress tensor and F is the deformation gradient. Here, the divergence can be calculated with respect to the coordinate system in the material frame and F_v is the volume force vector. Note that Eq. (2) can be equivalently expressed in terms of the Cauchy stress tensor, σ, with P = JσF^−T with respect to the spatial coordinate, where J is the Jacobian of F, |$J=\det {\bf F}$|⁠. The relevant Lamé constant in this configuration for material properties is given as λ = 1.15 × 10¹¹ [Pa] and |${\mathfrak {m}} = 7.69\times 10^{10}~{\rm [Pa]}$|⁠. Here, we impose a low-reflecting boundary condition [38] for letting waves pass out from the soil domain, which is defined as σ · n = −iωd_imu in the edges except for the incident perturbation edge in Figs. 3A, B, and C. Here, n is a unit normal vector, σ is a stress tensor, and d_im is the mechanical impedance with d_im = d_im(ρ, c_s, c_p), where c_s and c_p are the speeds of shear waves and pressure in the material, respectively.

The simulation of the finite element method¹ for the frequency domain of {3.0, 6.5, 9.0} Hz with an incident displacement |$u_{x}^{0}=0.3~[\rm m]$| shows that the incident wave from the displacement perturbation can be detoured only at 6.5 Hz of the bandgap frequency in the periodic structure, while other results demonstrate the opposite. To investigate the attenuation of the displacement amplitude, we set the cut line of the probe at (x, 0). Because the periodic structure is placed at x [m] = (−11, 11), we may draw the root mean square (RMS) of the total displacement, |$D_{\rm RMS}=\sqrt{\Delta x^2+\Delta y^2}$|⁠, along the cut line of the probe as shown in Fig. 3D. As a result, it is shown that the incident disturbance may therefore be successfully reduced in the metamaterial-like periodic structure region (Fig. 3B and the green curve in Fig. 3D).

3. Application to multiphysics noise control for the KAGRA GW detector

Next, we consider a 3D design for the KAGRA GW detector. For simplicity, we use a 20 m part of the entire 3 km arm length. A soil domain is established in which the cylinder beams were buried with the periodic structure of cylindrical beams near the underground tunnel of the KAGRA GW detector. Similar periodic designs of cylinder beams have been considered and investigated in Refs. [24,35]. The basic design and setup that we considered are illustrated in Fig. 4A. In the soil domain, a 4 m diameter underground tunnel with a concrete wall is considered, and there are periodic 0.3 m radius structural steel cylinders close to the tunnel.

The eigenvalue problem can be treated with a simple cell block shown in the LHS of Fig. 4A; then the bandgap structure appears between |$5.39\times 10^{-13}$| and |$7.93\ \rm Hz$|⁠, as shown in Fig. 4B. Note that we impose the periodic boundary condition on all vertical surfaces and the low-reflecting boundary condition on the top/bottom surface of the cylinder.

Since we want to investigate the acoustic effect of seismic wave propagation, we need to solve the multiphysics analysis of structural and acoustic simulations. The solid mechanics for the soil domain with structural steel beams and the concrete tunnel is governed by Eq. (2) with low-reflecting and free boundary conditions.

On the other hand, the wave equation for sound waves in a lossless medium (no thermal conduction and no viscosity), in principle, is described by the inhomogeneous Helmholtz equation:

where p is the total pressure, ρ is the total density, and c is the speed of sound. Note that s_d and s_m denote the sources of the dipole and monopole domains, respectively, which are set to zero in this study.

Indeed, the vibration in the seismic domains of the soil, steel beam, and concrete tunnel with a metamaterial structure can affect the acoustic domain inside the tunnel. In the acoustic–structure boundary in the concrete tunnel, we have the conditions

where n is a unit normal vector to the surface and F_A is the force projected on the area A. Then the Helmholtz equation for sound waves combined with a solid mechanical equation in this underground periodic structure can be solved in the frequency domain by a finite element approach with the aforementioned boundary conditions. As a result, the computation with an incident displacement |$u_{z}^{0}=1~[\rm m]$| has been done in the frequency values of {0.1, 1.0, 3.0, 5.5, 10.0} Hz and two simulation results of total sound pressure level inside the tunnel for 3 and 10 Hz are shown in Figs. 4C and D. The total level of sound pressure within the tunnel for 3 Hz within the bandgap frequency is a negative value, while the total level of sound pressure for 10 Hz outside the bandgap frequency range is quite loud inside the tunnel.

For a probe cut line at (15, y, 5) as shown in Fig. 4A, we draw the variation of the sound pressure level (top two figures of Fig. 5) and the total acoustic pressure (RMS) (bottom two figures of Fig. 5) through the tunnel inside. Here, the labels A and B in Fig. 5 represent the cases with a metamaterial-like beam and without the beam, respectively. Comparing Figs. 5A and B, it is found that the metamaterial-like structure can mitigate the sound pressure level and the total acoustic pressure. Furthermore, the mitigation appears to be effective even for the case of 10 Hz beyond the bandgap frequency range (the magenta curves in Fig. 5).

4. Discussion

In this study, we investigated the possibility of applying periodic metamaterial-like structures to the underground GW detector by virtue of mitigating a low-frequency noise below 10 Hz. The suggested problem in this study was exhibited in Sect. III.C of Ref. [37]: the nonlinear correlation between the wind speed meter at the entrance to KAGRA and the microphone channel in the physical environment monitor (PEM) system in the KAGRA detector during the daytime (Fig. 7 in Ref. [37]). A possible assumption to explain this correlation is as follows: i) high wind between mountains during the daytime strikes the slope of the mountain where KAGRA is located, ii) the seismic vibration caused by the wind propagates to the underground site, iii) the seismic vibration transforms to an acoustic pressure perturbation at the concrete boundary of KAGRA’s tunnel, and iv) the increased acoustic pressure perturbation produces the sound pressure noise inside the tunnel. A comparison between the x and y arms of the KAGRA detector for this assumption is shown in Fig. 8 of Ref. [37]. Since the y arm of the KAGRA detector is parallel to the slope of the mountain whereas the x arm of the detector stretches to a deeper site from the slope, the presumed seismic vibration will be attenuated in the x-arm direction. Therefore, the acoustic noise converted from the seismic vibration by the winds becomes more severe in the y-arm tunnel, which may affect the GW detection bands through the frequency up-conversion effect, which needs to be mitigated. Therefore, we investigated a possible way of using the metamaterial-like approach to mitigate seismic vibrations from the slope by inserting structured steel beams near the KAGRA tunnel.

As a result, we have shown that the bandgap structure of structural steel beams can reduce the appropriate sound pressure level compared to the case without the metamaterial-like structure. Moreover, it has been found that the mitigation effect appears even for the case beyond the bandgap frequency. Because the modeling and structure in this study are pretty simplified, further investigation with realistic and practical models and parameters is required. Because the terrestrial GW observatory is very sensitive to constant and transient seismic vibrations like an earthquake, this kind of low-frequency noise mitigation can be considered at the initial stage of construction. In addition, this approach is also worth considering as a possible alternate way to cancel Newtonian noises below 10 Hz (see Ref. [39] and references therein). Therefore, a relevant investigation into the mitigation of Newtonian noises and the improvement of noise sensitivity at the low-frequency band may be required in future work. Furthermore, this study may potentially be applied to overcome the low-frequency noise barrier in ground-based laser interferometer GW telescopes as well as new conceptual ground-based GW detectors with decihertz frequency bands like SOGRO [40,41].

Acknowledgement

The author thanks Piljong Jung, Edwin J. Son, Young-Min Kim, Chan Park, Muhan Choi, Dongwoo Lee, Hyung Won Lee, and Dong-Uk Hwang for helpful discussions and comments. This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2020R1I1A2054376) and the National Institute for Mathematical Sciences (NIMS).

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References