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even in the presence of absorption losses and in arbitrarily inhomogeneous media. The proof of this result is provided in volume II of Rayleigh's The Theory of Sound (1878), and it is typically discussed in most books on acoustics (e.g., Morse and Ingard, 1968; Pierce, 1981). Reciprocity is indeed an im- portant property that is widely exploited in measurement techniques, for instance, in scattering and radiation patterns measurements and transducer calibration.
Reciprocity is not a property specific to the field of acoustics. In electromagnetism, for instance, the equivalent statement is known as the Lorentz reciprocity theorem (Landau and Lifshitz, 1960). Indeed, a reciprocity theorem can be formu- lated for many physical systems supporting wave propaga- tion because it is simply related to time-reversal symmetry through the application of a fundamental result known as the Onsager-Casimir principle of microscopic reversibility (Casimir, 1945). This elementary physical result provides a clear picture into the conditions for which reciprocity holds.
Consider a linear time-invariant (LTI), possibly inhomo- geneous, medium in which time-harmonic perturbations are excited at frequency ω. Waves can be excited at source points and detected at receive points, which we call ports (for instance, an acoustic source at point A and a sensor at point B in Figure 1 constitute two ports). It is important to determine how the medium behaves under time rever- sal, i.e., whether it is invariant or not on reversing the time flow (changing the sign of the time parameter in all equa- tions describing the state of the medium). In a linear time- invariant medium, there can be only two different sources of broken time-reversal symmetry. The first one is the presence of absorption losses, represented by the material loss factor η, which macroscopically describes microscopic absorption processes that take energy out of wave propagation. At the macroscopic level, an absorptive medium is not invariant under a time-reversal operation because a time-reversed wave absorption process is equivalent to wave amplification. The second source of broken time-reversal symmetry, which appears at both the microscopic and macroscopic levels, is the dependence of the properties of the medium on a pa- rameter or set of parameters B, which is oddly symmetric on time reversal (i.e., its sign flips when a time-reversal op- eration is applied). A typical example of such a parameter is the static magnetic field bias. Because magnetic fields are the result of charges moving along circular paths, they flip their sign under time reversal as the circular motion of charges is reversed. In a general linear medium, both η and B may
be nonzero. In such a case, the Onsager-Casimir principle tells us that tAB(ω,η,B)=tBA(ω,η,-B) in general, where tAB is the complex field transmission coefficient between points A and B (Casimir, 1945). This relationship between the trans- mission from A to B and from B to A indicates that, in the presence of an odd bias, B ≠ 0, the reciprocity statement tAB(ω,η,B)=tBA(ω,η,B) does not hold unless B = 0. Reciproc- ity is therefore related to microscopic reversibility, defined as B = 0, and holds even if time-reversal symmetry is macro- scopically broken (η ≠ 0). This result shows how reciprocity, which is equivalent to microscopic reversibility, is intimately related to time-reversal symmetry.
From the above discussion, it is clear that it is possible to find situations in which reciprocity between two points in space may not hold. In a LTI medium, only the use of an external bias, i.e., B ≠ 0, odd with respect to time-reversal symmetry, allows one to break reciprocity. Magnetic fields can indeed break electromagnetic reciprocity in ferrites (Lax and But- ton, 1962) and acoustic reciprocity in magnetoelastic crys- tals (Kittel, 1958). Another possibility is to break some of the basic assumptions of the Onsager-Casimir principle, such as linearity or time invariance. Reciprocity is indeed broken in the presence of nonlinear processes (for instance, at large acoustic intensities) or time-dependent material properties (for instance, in a time-modulated medium or geometry).
Figure 1. Rayleigh reciprocity theorem states that the velocity poten- tial (φ) induced at point B by a time-harmonic source placed at point A (a) is the same in both magnitude and phase as the one induced at point A if the same source is placed at point B (b).
Nonreciprocal Electromagnetic Devices
Although it is known that reciprocity does not hold in every situation, it is not trivial to engineer compact and practi- cal devices that are capable of strongly breaking reciproc- ity, thereby isolating two different points, A and B, with tAB= 1 and tBA= 0. Until recently, highly nonreciprocal devices have existed for electromagnetic waves based on an external magnetic bias, B ≠ 0, but their acoustic counterparts have been absent. To better grasp the functionality, applicability,
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