# Doppler-Based Acoustic Gyrator
## 1 Introduction  
Microwave phase shifters are two-port components that provide an arbitrary and variable transmission phase angle with low insertion loss [1; 2; 3]. Since their discovery in the 19 th century, such phase shifters have found important applications in devices such as phased array antennas and receivers [4; 5], beam forming and steering networks [6; 7], measurement and testing systems [8; 9], filters [10; 11], modulators [12; 13], frequency up-convertors [14], power flow controllers [15], interferometers [16], and mode lockers [17]. Various types of microwave phase shifters have been proposed and demonstrated over years, all of which can be fundamentally categorized into two groups: reciprocal and non-reciprocal phase shifters. Non-reciprocal phase shifters providing different phase shift for forward and backward waves, however, are of more importance especially when it comes to the realization of devices like gyrators [18], isolators [19], and circulators [20], being critical for radar systems and networks. In its most traditional form, a non-reciprocal microwave phase shifter is composed of a rectangular waveguide partially filled with a ferrite slab submitted to a magneto-static bias field [1] (see Figure 1a). When the bias field is zero, the ferrite is demagnetized and the structure provides a reciprocal phase shift of \(\varphi_{0}\) for instance, being proportional to the length of the waveguide, for both forward and backward propagation. However, when the bias field is increased from zero, the ferrite slab starts magnetizing, giving rise to distinct and continuously variable phase shifts of \(\varphi_{1}\) and \(\varphi_{2}\) for forward and backward waves, respectively. In contrast to electromagnetism, realizing a non-reciprocal phase shifter in acoustics is challenging as sound waves only weakly interact with magnetic fields [21; 22; 23; 24; 25; 26; 27; 28; 29]. Recently, proposals to break the conventional wisdom of sound reciprocity have been reported, of which most rely on non-linear mechanisms [30; 31; 32; 33]. Despite their applicability, these nonlinear structures accompany with unwantedfeatures such as high-power consumption and geometrical bulky configuration, reducing their practical worth. A recent breakthrough, however, has opened exciting venues to break the reciprocity of acoustic waves in fully linear manners. In [22], it was shown that applying a time-independent bias velocity to stationary fluids allows one to strongly break the time reversal symmetry and to induce acoustic non-reciprocity. This salient feature was then leveraged to demonstrate a subwavelength circulator in airborne acoustics, allowing audible sound to be coupled to a certain port, while isolating transmission in another. On the basis of this idea, here we first demonstrate a fully linear non-reciprocal acoustic phase shifter. This is achieved in a simple acoustic waveguide that is filled with air on which a constant bias velocity is applied. Through analytical analysis and numerical simulations, we show how the proposed structure provides different phase shifts for forward and backward transmission. We further show that the associated differential phase shift can be simply tuned by changing the bias velocity. By setting a fluid bias velocity that corresponds to a differential phase shift of 180 degrees, we then implement an acoustic gyrator, which is an important two port component serving as the building block for a large variety of non-reciprocal acoustic devices such as isolators and circulators. We believe  
Figure 1: Concept of non-reciprocal phase shifters. (**a**) Schematic of a typical non-reciprocal phase shifter in microwave engineering. The device is composed of a waveguide that is partially filled with a ferrite slab on which a constant magnetic bias field is applied. The amount of differential phase shift between the forward and backward waves can be controlled by changing the magnetic field. (**b**) Acoustic equivalent of the microwave phase shifter shown in panel a.