Doing Arithmetic With Nonlinear Acoustic Vortices

Phase singularities of wave-front-like screw dislocations or vortices possess a well-defined quantity that can only take integer value: the topological charge. In the nonlinear regime, it has been demonstrated that optical or acoustical vortices interact and the topological charge follows a conservation law. Here this facility is used in nonlinear parametric interaction of two vortices shifted in frequency to perform sums and subtractions of the topological charge. Thus, we experimentally demonstrate a new technique to perform wave computation in the group of integer $\mathbb{Z}$. When the two vortices have commensurable frequencies, different combinations give the same frequencies but different tolopological charges may occur. We show that an energy criterion can be used to predict the outcome. A corollary is that a modulation of amplitude of the vortices switches from one result to the other.

Figure 1

Experimental setup.

Figure 2

Amplitude of the pressure field for configuration 1 (states ($\mathbf{F}|f_1⟩=0.725\mathrm{MHz}$, $\mathbf{M}|f_1⟩=1$) and ($\mathbf{F}|f_2⟩=1.25\mathrm{MHz}$, $\mathbf{M}|f_2⟩=1$)). The left-hand figure is the instantaneous pressure and the right-hand figure is the rms pressure.

Figure 3

Measured phase patterns for the first eight linear combinations of two noncommensurable primary frequencies ($f_1/f_2=29/50$). The displayed states are (a) $\mathbf{F}|f_2-f_1⟩=0.525\mathrm{MHz}$, $\mathbf{M}|f_2-f_1⟩=0$, (b) $\mathbf{F}|f_1⟩=0.725\mathrm{MHz}$, $\mathbf{M}|f_1⟩=1$ (initial state 1), (c) $\mathbf{F}|f_2⟩=1.25\mathrm{MHz}$ $\mathbf{M}|f_2⟩=1$ (initial state 2), (d) $\mathbf{F}|2f_1⟩=1.45\mathrm{MHz}$, $\mathbf{M}|2f_1⟩=2$, (e) $\mathbf{F}|2f_2⟩=2.5\mathrm{MHz}$, $\mathbf{M}|2f_2⟩=2$, (f) $\mathbf{F}|f_1+f_2⟩=1.975\mathrm{MHz}$, $\mathbf{M}|f_1+f_2⟩=2$, (g) $\mathbf{F}|2f_1+f_2⟩=2.7\mathrm{MHz}$, $\mathbf{M}|2f_1+f_2⟩=3$, (h) $\mathbf{F}|f_1+2f_2⟩=3.225\mathrm{MHz}$, $\mathbf{M}|f_1+2f_2⟩=3$.

Figure 4

Measured phase patterns for the first eight linear combinations of two noncommensurable primary frequencies ($f_1/f_2=19/33$). The displayed states are (a) $\mathbf{F}|f_2-f_1⟩=0.7\mathrm{MHz}$, $\mathbf{M}|f_2-f_1⟩=-1$, (b) $\mathbf{F}|f_1⟩=0.95\mathrm{MHz}$, $\mathbf{M}|f_1⟩=2$ (initial state 1), (c) $\mathbf{F}|f_2⟩=1.65\mathrm{MHz}$ $\mathbf{M}|f_2⟩=1$ (initial state 2), (d) $\mathbf{F}|2f_1⟩=1.9\mathrm{MHz}$, $\mathbf{M}|2f_1⟩=4$, (e) $\mathbf{F}|2f_2⟩=3.3\mathrm{MHz}$, $\mathbf{M}|2f_2⟩=2$, (f) $\mathbf{F}|f_1+f_2⟩=2.6\mathrm{MHz}$, $\mathbf{M}|f_1+f_2⟩=3$, (g) $\mathbf{F}|2f_1+f_2⟩=3.55\mathrm{MHz}$, $\mathbf{M}|2f_1+f_2⟩=5$, (h) $\mathbf{F}|f_1+2f_2⟩=4.25\mathrm{MHz}$, $\mathbf{M}|f_1+2f_2⟩=4$.

Figure 5

Measured phase patterns for eight linear combinations of two commensurable primary frequencies ($f_1/f_2=2/3$). The displayed states are (a) $\mathbf{F}|f_2-f_1⟩=0.5\mathrm{MHz}$, $\mathbf{M}|f_2-f_1⟩=0$, (b) $\mathbf{F}|f_1⟩=1\mathrm{MHz}$, $\mathbf{M}|f_1⟩=1$ (initial state 1), (c) $\mathbf{F}|f_2⟩=1.5\mathrm{MHz}$, $\mathbf{M}|f_2⟩=1$ (initial state 2), (d) $\mathbf{F}|f_1+f_2⟩=2.5\mathrm{MHz}$, $\mathbf{M}|f_1+f_2⟩=2$, (e) $\mathbf{F}|2f_2⟩=3\mathrm{MHz}$, $\mathbf{M}|2f_2⟩=2$, (f) $\mathbf{F}|3f_1⟩=3\mathrm{MHz}$, $\mathbf{M}|3f_1⟩=3$, (g) $\mathbf{F}|3f_2⟩=4.5\mathrm{MHz}$, $\mathbf{M}|3f_2⟩=3$, (h) $\mathbf{F}|3f_1+f_2⟩=4.5\mathrm{MHz}$, $\mathbf{M}|3f_1+f_2⟩=4$.