Experimental validation of the theoretical prediction for the optical $S$ matrix

Scattering of waves is present in many areas of physics. Within all these areas, in a great number of systems, the scattering can be separated in an averaged response that crosses rapidly the scattering region and a fluctuating delayed response. This fact is the basis of the optical model; the averaged response, represented by the optical matrix $\langle S\rangle$, is combined with the fluctuating part that can be taken as a random matrix. Although the optical model was developed more than 60 years ago, a theoretical prediction for the optical matrix was obtained only very recently. The validity of such a prediction is experimentally demonstrated here. This is done studying the scattering of torsional waves in a quasi-1D elastic system in which a locally periodic system is built; the full distribution of the scattering matrix is then calculated completely free of parameters. In contradistinction to all previous works, in microwaves and in elasticity, in which the value of $\langle S\rangle$ is obtained from the experiment, here the theoretical prediction is used to compare with the experiment. Numerical simulations show that the theoretical value is still valid when strong disorder is present. Several applications of the theoretical expression for the optical matrix in other areas of physics are proposed. Possible extensions of this work are also discussed.

Figure 1

(a) Aluminum beam ($\sqrt{G/\rho }=3104.7\mathrm{m}/\mathrm{s}$) that consists of a region with a locally periodic structure, composed of $N$ scatterers, and a semi-infinite uniform region of squared cross-section. The unit cell (inset) of length $D$ consists of bodies 1 and 2 of lengths $D-d$ and $d$, with polar moments of inertia $I_1$ and $I_2$, respectively. We use $w_1=h_1=w_2=D=2.54$ cm, $h_2=h=0.9525$ cm and $d=1.5875$ cm. (b) Setup: a beam consists of a structured zone, a wedge zone with an absorber, and a free region where the exciter and detector are located. (Inset) The coils of the EMAT exciter are connected in series and their polarity is such that the EMAT excites only torsional waves. The EMAT detector is composed of a magnet and a small coil.

Figure 2

(a) Phase of the scattering matrix, Eq. (5), as a function of the frequency, for $N=100$, is shown. The red line corresponds to the phase of the optical matrix $\langle S\rangle$ given in Eq. (6). (b) Zoom of panel (a) for low frequencies. (c) Histogram of the resonance highlighted in blue in (b). The (red) curve is Poisson's kernel (1) with the optical matrix $\langle S\rangle$ taken from Eq. (6).

Figure 3

Amplitude (a) and phase (b) of the $S$ matrix as a function of the frequency. Shifted and normalized $S$ matrix in the Argand plane (c), in a part of the allowed band. The histogram of the highlighted resonance (blue) at $f=2570$ Hz and its comparison to Poisson's kernel, Eq. (1) with $\langle S\rangle$ given by Eq. (6), is shown in (d). The dimensions of the unit cell are the same as in Fig. 1 but $h$ replaced by an effective value, $0.746h$, due to the punching of body 2 [28].

Figure 4

(a) Measured amplitude (in arbitrary units) as a function of the frequency. The analyzed resonances (b), (c), (d), and (e), and their respective histograms (f), (g), (h), and (i), are taken in the intervals (3244.0, 3399.0) Hz, (4859.8, 5019.8) Hz, (5792.0, 5998.0) Hz, and (9560.0, 9760.0) Hz, respectively. The continuous line in the lower panels is Poisson's kernel, Eq. (1), with $\langle S\rangle$ given by Eq. (6).

Figure 5

Dynamics of the map given in Eq. (A1): (a) $f=15000.0$, (b) $20272.824461676$, (c) 25180.0, and (d) $39735.6491631850$ Hz. The initial conditions are ${\theta }_0=2$ (green) and ${\theta }_0=5$ (purple).

Figure 6

(a) Bifurcation diagram of the map given in Eq. (A1). The red and green lines correspond to $w_{+}$ and ${\theta }_{+}$. (b) Lyapunov exponent $\mathrm{\Lambda }$ calculated numerically; a very low convergence is seen at $f=0$.

Figure 7

(Left) The last 15 iterations, of 1000, of Eq. (5), are plotted as a function of the frequency with an initial condition ${\theta }_0=\pi$ for different degrees of disorder: (a) 0%, (b) 10%, (c) 25%, (d) 50%, (e) 75%, and (f) 90%. The red line corresponds to the phase of $\langle S\rangle$ in Eq. (6); it is valid for all panels at the left because only indistinguishable changes appear. The respective histograms of the phase are given in the right panels. The continuous curve is Poisson's kernel, Eq. (1), with $\langle S\rangle$ given by Eq. (6) averaged over the disorder realizations.