Imaging acoustic waves in microscopic wedges

Using an ultrafast optical technique we image near-gigahertz flexural acoustic waves propagating in a thin gold wedge deposited on an ultrathin silicon nitride slab. Temporal Fourier transforms of the imaged data in two dimensions allow individual frequencies to be accessed. The wave fronts associated with antisymmetric Lamb waves bend towards the thin end of the wedge. This behaviour is mimicked with an analytical model based on sections of a linear wedge. We also conduct numerical simulations which show good agreement with the experimental results.


Introduction
Acoustic waves of various kinds in wedges (Lagasse et al 1973, De Billy and Hladky-Hennion 1999, Shuvalov and Krylov 2000, Yang and Tsen 2006, Moss et al 1973, Krylov 1998, Lomonosov et al 2012 have spurred a great deal of research. Such research can also be applied to, for example, ultrasonic transducers (Bertoni andTamir 1975, Manneberg et al 2008), vibration damping (Bayod 2011, Bowyer et al 2012, nondestructive testing of blades (Joosse et al 2002, Liu and, moisture detection (Tung and Yang 2010), black-hole analogies (Krylov andTilman 2004, Krylov andWinward 2007), and even aquatic propulsion (Krylov and Pritchard 2007). Various kinds of acoustic modes are known to exist in wedges. Acoustic waves may, for example, be guided along the thin edge of a wedge. For perfect wedges with sharp edges-there being no characteristic length scale in this case-such waves are non-dispersive (Moss et al 1973, Jia andDe Billy 1992). In contrast, waves launched from inside the wedge towards the thin edge are highly dispersive and show a steadily decreasing sound velocity as they travel (Krylov and Tilman 2004). Plane waves launched from inside the wedge parallel to the edge of the wedge are therefore expected to follow curved ray paths (Shuvalov and Krylov 2000). Although laser ultrasonics had been used to generate and detect the waves in wedges for years (Hess et al 2014, Jia and De Billy 1992, Jia et al 2011, Edwards et al 2012, to our knowledge there has been no attempt to image them at neargigahertz frequencies or at micron lateral resolution.
In this paper we use an ultrafast optical technique to excite and to image near-gigahertz flexural acoustic waves in gold wedges deposited on thin silicon nitride slabs. The optical pulses are focused to a line source set perpendicular to the edge of the wedge. The wave propagation characteristics are imaged in the time domain by using scanned optical probing. By using temporal Fourier transforms, we extract constant-frequency images, also viewable as animations, and observe wave-front bending. This behaviour is explained with the help of a semi-analytical model based on the theory of flexural waves in a double-film layer.

Sample and method
A microscopic gold wedge was prepared by radio-frequency (rf) sputtering, making use of a programmable translation stage inside the vacuum chamber with a 0.5 μm positioning accuracy. The schematic of the sample structure is shown in figure 1(a). During the rf sputtering at a constant deposition rate of 0.4 nm s −1 , a thin flat mask plate attached to the translation stage is moved just above the substrate surface in the −x direction (see figure 1). The substrate is a commercial amorphous-SiN slab with a thickness of 100 nm. The spatial thickness distribution of the gold wedge was mapped by a confocal laser-scanning microscope, and the profile is shown in figure 1(b). The wedge thickness changes along the x-axis from 0 to 6 μm over a μ ∼150 m distance. Local thickness variations along the y-direction are within 10 nm over the imaged region.
The acoustic wave field is imaged by the use of an optical pump-and-probe technique combined with a common-path interferometer (Sugawara et al 2002, Tachizaki et al 2006. The pump (frequency-doubled, wavelength 415 nm, incident pulse energy 0.18 nJ) and probe (wavelength 830 nm, incident pulse energy 0.06 nJ) pulses originate from a mode-locked Ti: Sapphire laser with an 80.4 MHz repetition rate and a 0.2 ps pulse duration. The pump pulses, frequency-doubled by a BBO (β-BaB 2 O 4 ) crystal, modulated at 1 MHz for lock-in detection, reshaped by a cylindrical lens and a ×50 microscope objective lens, produce a quasi-line source with ∼60 μm length and ∼2 μm width (full width at half maximum) on the sample surface. The line source is perpendicular to the edge of the wedge, and the line centre is set at a distance of 45 μm from the edge (figure 1(a)). Broadband plane Lamb waves including both quasisymmetric and quasi-antisymmetric modes with near-GHz frequencies are thermoelastically generated. (Because of the presence of the thin silicon nitride slab, perfectly symmetric or antisymmetric modes do not exist.) Out-of-plane motion (∼10 pm in amplitude) is detected with ∼0.5 pm resolution by two probe pulses at an interval of 300 ps, using MHz lock-in detection at the frequency of the pump-beam chopping. The 2 μm diameter probe spot is scanned over a 120 μm × 160 μm region including the edge of the wedge, as shown in figure 1(a), and the optical phase difference Δϕ between the probe pulses is imaged at a fixed pump-probe delay time τ. Here Δϕ ∝ du dt z , the time derivative of the out-of-plane motion. By incrementing τ, animations of 24 frames are obtained within the 12.44 ns laser repetition period at 0.52 ns intervals.
To model and simulate the acoustic wave propagation in the sample, we fitted the measured thickness distribution by two quadratic curves: setting the x axis parallel to the wedge slope and with x = 0 set at the wedge edge, the thickness is fitted as an ellipse for the region < < x 20 120 μm, and as a parabola = z cx 2 for the region < < x 0 20 μm (figure 1(b)). Here = x 170 0 μm, a = 155 μm, b = 6.70 μm, and c = 0.0042 μm −1 . Further details of the simulation are given in the appendix.

Semi-analytical method for acoustic ray trajectories
The theoretical formula for the velocity of a flexural wave, i.e., one that corresponds to the lowfrequency approximation for the lowest-order quasi-antisymmetric Lamb mode in a bi-layered plate, is (Shuvalov et al 2005)  Here the notation σ i is introduced for the so-called flexural where E i denotes the Young modulus and ν i the Poisson ratio. It is instructive to rewrite equation (1) in the classical form (Graff 1975, Biryukov et al 1995 where, however, in contrast with the case of a single-layer plate, the effective thickness of the composite bi-layered plate d is introduced: In our experiments the effective thickness of the gold layer and, consequently, the velocity of the flexural wave, depend on the distance from the edge of the gold layer, i.e., To apply the theory of waves in linear wedges to our case, we approximate our wedge profile by a series of linear segments. At a position x inside a short segment between x N and , the effective thickness d e can be written as For our sample, flexural waves at frequencies ∼100 MHz can be accurately treated in the framework of geometrical acoustics, i.e., ray acoustics. Geometrical acoustics, or the Wentzel, Kramers, Brillouin (WKB) approximation, is theoretically valid when the quantity  k k | | | |, a measure of variation of the wave vector on spatial scales corresponding to the inverse wave vector, is much smaller than the wave vector k | | itself, i.e., when ≪  k k | | | | 1 2 (Brekhovskikh andGodin 1990, Kravtsov andOrlov 1990). We have verified that this condition holds in our sample not only at 80.4 MHz but also at other detectable frequencies. However, the condition ⩽ kd 1, which ensures that the low-frequency approximation equation (1) for the lowest-order quasi-antisymmetric Lamb mode is valid, is not satisfied when the frequency is large compared with 80.4 MHz. The trajectories of the acoustic rays on the x y ( , ) plane can be obtained from the relations x y y x 2 2 2 where the flexural wave velocity depends only on one of the following two coordinates: the wave number of the flexural wave x y 2 2 , which are coordinate-dependent. In contrast, k y is a constant depending on the starting point of the ray x y ( , ) 0 0 . In our experimental geometry, where the laser radiation is focused to a line on the x-axis (y = 0), the acoustic waves are emitted in the ±y directions (except near the ends of the line source): The integration of equation (4) provides an analytical expression for the ray trajectories corresponding to starting points x y ( , ) 0 0 in the segment between x N and + x N 1 : N N on the segment boundary, the ray trajectory for the next segment is given by By repeating this calculation (of equation 6) in each segment with a small enough Δx, semianalytical acoustic ray trajectories can be obtained. These analytical predictions are compared with our experimental results in the next section. quasi-antisymmetric Lamb-wave modes, and the strongest wave mode is the zeroth-order quasiantisymmetric mode, i.e., the flexural wave, which is identified by its approximate propagation velocity of ∼1200 m s −1 for our frequency range, as derived by following ripples in the time domain. As expected, the wave fronts bend toward the edge of the wedge. Other ripples propagate with components in the +x-direction with parabolic-like wave fronts. Since these ripples travel much faster (∼2300 m s −1 ) than the dominant ripples, we identify them as zerothorder quasi-symmetric Lamb waves, because their velocities agree with that of the S0 guided mode with the velocity

Results
(1 ) l 1 1 1 2 of a free gold plate. The gold film area and the SiN area have opposite image contrast. We presume this difference to be caused by multiple optical reflections inside the transparent SiN slab. Figure 3 shows acoustic images at different times obtained from the numerical simulations. These images also represent out-of-plane (z-direction) surface velocities to match the experimental results. (Animations of the simulated acoustic images are given in the supplementary data, available at stacks.iop.org/njp/16/103029/mmedia.) In contrast to the experimental images which are generated by a pump-pulse train, the simulated images exhibit ripples generated by a single pump pulse. The amplitude of the simulated ripples in the SiN area is bigger than in the gold wedge area because of the smaller thickness and the mass of the former. In contrast, the amplitude of the ripples in the SiN area in the experimental images are smaller than in the gold wedge area, because the probe-beam reflectance for the SiN is much lower than that for gold. (Our data collection is weighted by the probe-beam power.) To better understand the behaviour of the different acoustic modes, we calculated temporal Fourier transforms of the set of 24 experimental images (some of which are shown in figure 2) that cover one period of the laser pulses (12.44 ns). We also carried out the same analysis for the 286 simulation images (some of which are shown in figure 3) that cover a time corresponding to  figure 1(a). The image colour represents the z-directed surface velocity. eight periods of the laser pulses (99.1 ns) in our experiment. Figure 4 shows the thus-obtained constant-frequency acoustic images for both the experiment and the simulation. (Animations of the constant-frequency acoustic images at 80.4 MHz,160.8 MHz,241.2 MHz,321.6 MHz,402.0 MHz,and 482.4 MHz from both the experiment and the simulation-shown side by side -are given in the supplementary data, available at stacks.iop.org/njp/16/103029/mmedia.) Low-frequency images at 80.4 MHz and 160.8 MHz show strong energy-propagating directions in the ±y directions, but for higher frequencies these change to the to upper-and lower-right diagonal directions. The semi-analytical acoustic ray trajectories, calculated according to equations (5) and (6) with the segment width Δ = x 0.5 μm, are plotted on the 80.4 MHz images. These ray trajectories are found to be perpendicular to all wave fronts in both the experimental and simulated images, thus demonstrating the applicability of the ray theory for this frequency.

Conclusion
We have imaged propagating near-gigahertz flexural waves in the time-domain in microscopic gold wedges deposited on thin silicon nitride slabs by an ultrafast optical technique. The optical pump pulses are focused to a line source set perpendicular to the edge of the wedge. We observe wave-front bending, especially at low frequencies. This behaviour is explained with the help of an analytical model based on the theory of flexural wave propagation in a bi-layer film and on geometrical acoustics, and is reproduced with numerical simulations. This work should stimulate further studies on the propagation of waves in wedges. Other opto-acoustic experimental methods appropriate for probing or imaging unsupported thin films, such as Brillouin scattering (Grimsditch et al 1987, Cuffe et al 2012 or Impulsive Stimulated Thermal Scattering Nelson 1994, Rogers et al 2000), would be interesting to apply to flexural waves in wedges.  (Anderson 1965). To decrease computing time and memory size, the calculated region 120 μm × 80 μm is applied by incorporation of symmetric boundary conditions that cut through the centre of the line source. The calculation region is continued in the region <

Appendix. Simulation method
x 0 for a further 15 μm of SiN slab without gold film to reduce the effect of acoustic reflections from boundaries. On the outer boundaries absorbing boundary conditions are applied. The thermoelastic laser excitation is represented by a simplified model that makes use of a dipole-type horizontal force (y-directed force) with a spatial distribution given by − y e y d 2 2 applied over the line source region with length 60 μm, where d = 1 μm. The temporal variation of the excitation is chosen as a Gaussian function with a 1 ns e 1 full-width-at-half-maximum width. A sufficiently long calculation time (∼100 ns, which is 8 times longer than the laser repetition period) is chosen for the simulation. The results of the simulation are compared with our experimental results in the main text.