Drexhage’s Experiment for Sound

Drexhage’s seminal observation that spontaneous emission rates of fluorophores vary with distance from a mirror uncovered the fundamental notion that a source’s environment determines radiative linewidths and shifts. Further, this observation established a powerful tool to determine fluorescence quantum yields. We present the direct analogue for sound. We demonstrate that a Chinese gong at a hard wall experiences radiative corrections to linewidth and line shift, and extract its intrinsic radiation efficiency. Beyond acoustics, our experiment opens new ideas to extend the Drexhage experiment to metamaterials, nanoantennas, and multipolar transitions.

Figure 1

(a) Image charge construction in optics for a vertical dipole above a mirror. (b) In acoustics, the mirror gong is not along, but opposite to, the source gong. Panel (c) shows the decay rate enhancement predicted by image theory for an acoustic dipole perpendicular to (red), and parallel to the interface (blue). The symbols and thin line show the electrodynamic case.

Figure 2

(a) Sketch of the experiment. A wooden ball launched on a rail generates a $\delta$ excitation at the gong center. The gong displacement is picked up by a small magnet glued to the back of the gong, and a pickup coil. Panel (b) [enlargement in (c)] shows a time-domain trace, showing a ring-down with many frequency components. Panel (d): the Fourier transform of the transient shows distinct resonances. The main resonances [enlargement shown for mode 1, 306 Hz in panel (e)] have a Lorentzian line shape. (f) Acoustic eigenmode profile for modes 1 and 2, the lowest order modes of zero angular quantum number. (g) Far-field radiation patterns for the gong in free space for modes 1 (blue circles, FEM result) and 2 (red squares, FEM result) indicate dipolelike emission (black curves indicating ${\cos }^2\theta$). In terms of integrated radiated flux for modes 1 and 2, respectively, over 99% and 95%, are in the dipole mode.

Figure 3

(a),(b) Fitted damping rate for mode 1 (306 Hz) and mode 2 (561 Hz) versus distance to the wall. Open orange points indicate individual measurement points, while solid blue ones show their averages, binned in 5 cm intervals. Overplotted is Eq. (2) with parameters ${\gamma }_{\infty }=0.255\mathrm{Hz}$ and $\eta =0.09$, respectively $\gamma =0.654\mathrm{Hz}$ and $\eta =0.20$. Panels (c),(d) show the line shift for each mode, where the theory contains no further adjustable parameters.

Figure 4

(a),(b) Linewidth and line shift normalized to the linewidth in free space of mode 1, assuming no viscoelastic damping. Dark circles (light diamonds) are for modes 1 and 2, respectively, from FEM. The solid line is the image-theory prediction. (c) Points show finite-element simulations of the linewidth for loss tangents $\tan \delta =0.1$, 0.2, 0.3, 0.5, and $1.0\times {10}^{-4}$. Lines show Eq. (2) with radiation efficiencies of $\eta =14.2$, 7.6, 5.2, 3.2, and 1.6%.