Propagation and Imaging of Mechanical Waves in a Highly Stressed Single-Mode Acoustic Waveguide

E. Romero, R. Kalra, N.P. Mauranyapin, C.G. Baker, C. Meng, and W.P. Bowen

Phys. Rev. Applied 11, 064035 – Published 14 June 2019

Abstract

We demonstrate a single-mode acoustic waveguide that enables robust propagation of mechanical waves. The waveguide is a highly stressed silicon-nitride membrane that supports the propagation of out-of-plane modes. In direct analogy to rectangular microwave waveguides, there exists a band of frequencies over which only the fundamental mode is allowed to propagate, while multiple modes are supported at higher frequencies. We directly image the mode profiles using optical heterodyne vibration measurement, showing good agreement with theory. In the single-mode frequency band, we show low-loss propagation (approximately 1 $dB / cm$ ) for an approximately 5-MHz mechanical wave. This design is well suited for acoustic circuits interconnecting elements such as nonlinear resonators or optomechanical devices for signal processing, sensing, or quantum technologies.

Received 12 February 2019
Revised 10 May 2019

DOI:https://doi.org/10.1103/PhysRevApplied.11.064035

Physics Subject Headings (PhySH)

Acoustic phonons Acoustic wave phenomena Mechanical & acoustical properties Mechanical properties of membranes Mesoscopics Phonons

Waveguides

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

E. Romero, R. Kalra, N.P. Mauranyapin^*, C.G. Baker, C. Meng, and W.P. Bowen

ARC Centre for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, Brisbane, Queensland 4072, Australia

^*n.mauranyapin@uq.edu.au

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Vol. 11, Iss. 6 — June 2019

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Images

Figure 1
(a) Illustration of the acoustic waveguide with width $L_{x}$ and length $L_{y}$ . The four boundaries of the rectangular waveguide are fixed by the substrate. The waveguide is actuated electrostatically through a capacitor formed by an on-chip fabricated electrode on top of the membrane (shown in gold) and a probe electrode (in silver). The oscillations represent mechanical waves propagating along $y$ with wave vector ${\vec{k}}_{y}$ . These waves are detected using a lensed optical fiber through the phase modulation of the laser light reflected back into the fiber. (b) Image of a chip with 14 waveguides in parallel, four of which are intact. (c) False color cross-sectional scanning electron micrograph of a cut of the released ${Si}_{3} N_{4}$ membrane (blue) held by the $Si$ substrate (gray). (d) Calculated dispersion relation for the waveguide, showing the first four mode branches. Shaded in blue and gray are the frequency bands corresponding to the single-mode region and the band below cutoff, respectively. (e) Examples of two-dimensional amplitude mode profiles for a waveguide of finite length $|u (x, y)|$ calculated using Eq. (2). These resonance frequencies are indicated by dashed lines in (d) and correspond to the experimental results shown in Fig. 3.
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Figure 2
(a) Fabrication process for the acoustic waveguides described in Sec. 3a. The process requires silicon ( $Si$ ), silicon nitride ( ${Si}_{3} N_{4}$ ), aluminium ( $Al$ ), photoresist (PR) and gold ( $Au$ ). The perspective of the scheme represents the lateral cut along $x$ of a membrane waveguide with width $L_{x}$ . (b) Schematic of the optical heterodyne detection setup. The optical setup is composed by an acousto-optical modulator (AOM), polarization controllers (PC), and beam splitters (BS), used to measure the mechanical vibrations of the waveguide. The diffracted local oscillator (orange) and nondiffracted probe beams (red) are combined in a final beam splitter for balanced photodetection. The actuation voltage $V (Ω)$ is applied to the waveguide in the vacuum chamber via a feedthrough.
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Figure 3
(i)–(iv) Experimentally measured transverse-mode profiles of the acoustic waveguide when actuated at (i) $Ω / 2 π = 12.3 MHz$ , (ii) $Ω / 2 π = 8.1 MHz$ , (iii) $Ω / 2 π = 5.1 MHz$ , and (iv) $Ω / 2 π = 2.3 MHz$ . The photocurrent at frequency $ω_{AOM} + Ω$ is plotted as a function of the fiber’s lateral position $x$ while it is scanned across the waveguide of width $L_{x} \approx 76 μ m$ . The fiber is positioned approximately 3 mm from the actuation electrodes along the $y$ axis, which corresponds to the direction of propagation. The red curves are the theoretical fit for the transverse-mode profile $|u (x, y)|$ calculated with Eq. (2), which are displayed in Figs. 1 and 1. The color of each plot corresponds to the color of the dashed lines in Fig. 1, indicating the (iv) zero-, (iii) one-, (ii) two-, and (i) three-mode frequency bands.
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Figure 4
(a) Consecutive measurements of the transverse-mode profile of the waveguide actuated at a fixed frequency $Ω / 2 π = 10.12 MHz$ and position along the $y$ axis. The three mode profiles, (i) blue, (ii) red, and (iii) purple are measured in that chronological order. These measurements demonstrate that the mode shape changes over time. The solid lines are the theoretical fit to the transverse-mode profile $|u (x, y)|$ . (b) Theoretical lateral and transverse-mode profile $|u (x, y)|$ as a function of the relative phase between the $n = 1$ and $n = 2$ modes for a given position along $y$ . The relative amplitudes are $u_{2} / u_{1} = 3.3$ . The theoretical fit for each measurement in (a) correspond to a phase-difference value displayed here. The phase difference for each measurement is (i) $θ_{2} - θ_{1} = 4.27 rad$ , (ii) $θ_{2} - θ_{1} = 4.02 rad$ , and (iii) $θ_{2} - θ_{1} = 2.76 rad$ .
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Figure 5
(a) Network analysis of the mechanical waveguides. The mechanical signal (blue) decreases to the laser noise level (orange) in the forbidden band (shaded gray background), which corresponds to the frequency band below the cutoff frequency $Ω < Ω_{c, 1}$ . The single-mode frequency band is represented as the blue shaded region $Ω_{c, 1} < Ω < Ω_{c, 2}$ . (b) Frequency response of the waveguide highlighting a resonance at $Ω_{m} / 2 π = 4.754 MHz$ (orange dots). The peak is fit to a Lorentzian (in blue with the shading denoting uncertainty) from which we extract a quality factor $Q = 2100 \pm 200$ . (c) Response of the waveguide to a 5-ms pulse at 4.754 MHz (blue dots). The normalized photocurrent is plotted in decibels from which the ring-up and ring-down times are extracted using curve fitting. The resulting fitting curves are shown in red and green for the ring up and ring down, yielding quality factors of $Q = 2000_{- 500}^{+ 1000}$ and $1950 \pm 50$ , respectively.
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