# Doppler-Based Acoustic Gyrator
## 3 Acoustic Gyrator  
This section describes how an acoustic gyrator can be implemented making use of the non-reciprocal phase shifter that was demonstrated in the previous section. A gyrator is a fully passive, lossless two port device. Over the past decades, gyrators have been utilized not only for the network realization of two port components, but also as the building blocks of microwave filters, isolators, impedance converters, and equalizers [1]. The scattering matrix of an ideal gyrator is of the form  
\[\mathrm{S}=\left( \begin{array}{cc}0&1\\ -1&0\end{array} \right) \tag{2}\]  
which means that a gyrator transmits forward waves without any phase shift, whereas it shifts the phase of backward waves by 180 degrees. Suppose now that the velocity of the fluid filling the waveguide of Figure 1a is chosen such that it corresponds to a phase shift of 90 degrees for forward,  
Figure 3: Demonstration of the proposed non-reciprocal acoustic phase shifter. **(a)** An excitation pulse (the solid blue line) is considered to be injected to the left side of the waveguide of Figure 1b. The signal that is received at the other side (the red dashed line) is in phase with the excitation when \(v_{0}=0\). **(b)** Same as panel a, except that a bias velocity of \(v_{0}=23\) m/s, which corresponds to the phase shift of \(\varphi_{1}=\pi\), is imparted on the fluid. The transmitted signal is now led by 180 degrees with respect to the excitation. **(c,d)** The same process has been followed when the waveguide is excited from the right side. In the case of the stationary fluid (panel c), the time evolution of the transmitted signal is exactly identical to that of panel a, stemming from the reciprocity of the structure. When assuming \(v_{0}=23\) m/s, however, the received signal is lagged (not led) by 180 degrees with respect to the input field, manifesting the non-reciprocity of the structure in phase. but that of \(-90\) degrees for backward propagation (Figure 4a). The scattering matrix of the resulting phase shifter then reduces to  
\[\mathrm{S}=\mathrm{e}^{\mathrm{i}\frac{\pi}{2}}\left( \begin{array}{cc}0&1\\ -1&0\end{array} \right) \tag{3}\]  
which is quite similar to the scattering matrix of an ideal gyrator, apart from a phase factor of \(\mathrm{e}^{\mathrm{i}\frac{\pi}{2}}\). This constant phase factor, however, should not cause any concern as it can always be compensated by cascading the phase shifter with a regular reciprocal system that shifts the phase of the travelling waves by \(-90\) degrees for both propagation directions. Hence, our non-reciprocal phase shifter indeed works as an acoustic gyrator for this choice of bias velocity. To prove this, we again perform time domain simulations, assuming a bias velocity motion of \(v_{0}=12\) m/s. Figure 4b depicts the transmitted signals corresponding to the forward (the solid red line) and backward (the solid blue line) propagation. Notice that the excitation signal is also marked with the dashed black line in the figure. It is obvious that there exists a differential transmission phase shift of 180 degrees between the forward and backward waves. It is further observed that the transmitted signal is led (lagged) by 90 degrees with respect to the excitation for forward (backward) propagation. These objations are in accordance with the scattering matrix of Equation (3). The pressure field distributions corresponding to the forward and backward transmissions are also depicted in Figure 4c,d, respectively. The obtained results demonstrate how the proposed phase shifter realizes a gyrator for this choice of bias velocity. Figure 4: Demonstration of an acoustic gyrator via the proposed non-reciprocal acoustic phase shifter. (**a**) Bias velocity motion is set such that it corresponds to a transmission phase shift of \(\varphi_{1}=\pi/2\) for forward, and that of \(\varphi_{2}=-\pi/2\) for backward direction. (**b**) Time profile of the transmitted signals associated with forward (solid red line) and backward (solid blue line) waves.