# Doppler-Based Acoustic Gyrator
## 2 Acoustic Phase Shifter
We further remark that in the following, we neglect the Poiseuille flow originated from non-slippery walls, as our simulations show that its effect is indeed negligible. To measure the scattering parameters of the structure, we excite the waveguide from one end with a plane-wave sound field of unit amplitude and measure the phase of the signal receiving at the other end. Figure 2b shows the phase of the corresponding scattering parameters as a function of the fluid velocity. The results of this figure show a perfect agreement with those of Figure 2a, which were obtained by a simple argument based on Doppler shifts. To further assess the non-reciprocal behavior of the proposed device, we perform finite difference time domain (FDTD) simulation with a time step of \(10^{-5}\) s. The insets of Figure 3a represent the excitation (solid blue line) and the transmitted (dashed red line) signals when the waveguide is excited from the left side and when a stationary fluid motion is considered (\(\
u_{0}=0\)). As observed, the injected signal is received after a time delay of \(\Delta t=\frac{L}{c}=2.1\) ms. However, the received signal is still in phase with the excitation since the waveguide is multiple-wavelengths long. Now assume we impart a bias velocity of \(\
u_{0}=23\) m/s to the fluid within the waveguide, corresponding to an additional Doppler phase shift of 180 degrees. Figure 3b represents the time profiles of the excitation and transmitted fields. As expected, the transmitted signal is led by 180 degrees with respect to the excitation. The promising feature of the device, however, relies not on this phase shift, but on the non-reciprocal phase shift that it provides for the backward propagation. Hence, we repeat our analysis when the waveguide is excited from the other side. Figure 3c,d depict the temporal evolution of the corresponding excitation and transmitted signals in the cases of \(\
u_{0}=0\) and \(\
u_{0}=23\) m/s, respectively. We notice that, when \(\
u_{0}=0\), the transmitted field had exactly the same time profile as that of Figure 3a. It is because of the fact that, as previously explained, the structure is reciprocal for the stationary fluid velocity. When \(\
u_{0}=23\) m/s, however, the transmitted field is lagged (not led) by 180 degrees with respect to the excitation, clearly confirming the non-reciprocity of the structure. Such simple non-reciprocal phase shifters can be utilized as the building blocks for the synthesis of more complex non-reciprocal devices such as gyrators, isolators, and circulators, among which the realization of the gyrator is discussed in detail in the following section. Figure 2: Scattering parameters of the proposed non-reciprocal acoustic phase shifters. (**a**) Employing Doppler spectral analysis, the variations of \(\varphi_{1}\) and \(\varphi_{2}\) versus the bias velocity \(\
u_{0}\) are calculated. When the fluid inside the waveguide is stationary, the forward and backward waves are subject to the same amount of phase shift (\(\varphi_{1}=\varphi_{2}=0\)). However, they start acquiring different amounts of transmission phase shifts (\(\varphi_{1}\
eq\varphi_{2}\)) by increasing the bias velocity from zero. (**b**) Variation of the phase of the scattering parameters (S\({}_{12}\) and S\({}_{21}\)) of the structure versus the bias velocity, calculated via full-wave numerical simulations. Numerical simulations validate our Doppler spectral analysis.