The law of reciprocity in acoustics and elastodynamics codifies a relation of symmetry between action and reaction in fluids and solids. In its simplest form, it states that the frequency-response functions between any two material points remain the same after swapping source and receiver, regardless of the presence of inhomogeneities and losses. As such, reciprocity has enabled numerous applications that make use of acoustic and elastic wave propagation. A recent change in paradigm has prompted us to see reciprocity under a new light: as an obstruction to the realization of wave-bearing media in which the source and receiver are not interchangeable. Such materials may enable the creation of devices such as acoustic one-way mirrors, isolators and topological insulators. Here, we review how reciprocity breaks down in materials with momentum bias, structured space-dependent and time-dependent constitutive properties, and constitutive nonlinearity, and report on recent advances in the modelling and fabrication of these materials, as well as on experiments demonstrating nonreciprocal acoustic and elastic wave propagation therein. The success of these efforts holds promise to enable robust, unidirectional acoustic and elastic wave-steering capabilities that exceed what is currently possible in conventional materials, metamaterials or phononic crystals. Nonreciprocal acoustic and elastic wave propagation may enable the creation of devices such as acoustic one-way mirrors, isolators and topological insulators. This Review presents advances in the creation of materials that break reciprocity and realize robust, unidirectional acoustic and elastic wave steering.

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Nonreciprocity in acoustic and elastic materials

Investigating the interaction of electron beams with materials and light has been a field of research since more than a century. The field was advanced theoretically by the raise of quantum mechanics and technically by the introduction of electron microscopes and accelerators. It is possible nowadays to uncover a multitude of information from electron-induced excitations in matter by means of advanced techniques like holography, tomography, and most recently photon-induced near-field electron microscopy. The question is whether the interaction can be controlled in an even more efficient way in order to unravel important questions like modal decomposition of the electron-induced polarization, by performing experiments with better spatial, temporal, and energy resolutions. This review discusses recent advances in controlling the electron and light interactions at the nanoscale. Theoretical and numerical aspects of the interaction of electrons with nanostructures and metamaterials will be discussed, with the aim to understand mechanisms of radiation in interaction of electrons with even more sophisticated structures.

https://iopscience.iop.org/article/10.1088/2040-8986/aa8041/pdf

Interaction of electron beams with optical nanostructures and metamaterials: from coherent photon sources towards shaping the wave function

Interaction of electron beams with optical nanostructures and metamaterials: From coherent photon sources towards shaping the wave function

We present an exact mathematical framework for electromagnetic wave propagation in periodically time-modulated media, in which the permittivity is homogenous and modulated in a step-varying fashion. By using Hill’s equation theory, we show that this problem has analytical solutions. We connect the dispersion relation, which exhibits $k-$ gaps, with the Hill stability analysis, providing an alternative mathematical description for wave propagation in temporal crystals. Our analysis, which is exact and transposable to other kinds of waves or modulation schemes, provides general useful physical and mathematical insights, complementing the use of numerical techniques such as finite differences in the time domain, or harmonic balance schemes, with a more transparent and practical design tool. The present analytical transient mathematical analysis, in contrast with the existing frequency-domain numerical approaches, can exhibit the parametric properties of electromagnetic waves inside a time periodic medium. For this reason, it can be a useful tool for the design of active microwave and optical devices, which employ time periodic wave medium modulation to filter or parametrically amplify wave signals.

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Electromagnetic Waves in a Time Periodic Medium With Step-Varying Refractive Index

We explore the propagation and transformation of electromagnetic waves through spatially homogeneous yet smoothly time-dependent media within the framework of classical electrodynamics. By modelling the smooth transition, occurring during a finite period τ, as a phenomenologically realistic and sigmoidal change of the dielectric permittivity, an analytically exact solution to Maxwell׳s equations is derived for the electric displacement in terms of hypergeometric functions. Using this solution, we show the possibility of amplification and attenuation of waves and associate this with the decrease and increase of the time-dependent permittivity. We demonstrate, moreover, that such an energy exchange between waves and non-stationary media leads to the transformation (or conversion) of frequencies. Our results may pave the way towards controllable light–matter interaction in time-varying structures.

Electromagnetic wave propagation in spatially homogeneous yet smoothly time-varying dielectric media

Propagation of sound waves through a spatially homogeneous but smoothly time-dependent medium

We report on the possibility of diffracting electrons from light waves traveling inside a dielectric medium. We show that, in the frame of reference which moves with the group velocity of light, the traveling wave acts as a stationary diffraction grating from which electrons can diffract, similar to the conventional Kapitza–Dirac effect. To characterize the Kapitza–Dirac effect with traveling light waves, we make use of the Hamiltonian Analogy between electron optics and quantum mechanics and apply the Helmholtz–Kirchhoff theory of diffraction.

https://iopscience.iop.org/article/10.1088/1367-2630/17/8/082002/pdf

Kapitza-Dirac effect with traveling waves

The possibility of high-energy neutron diffraction in a crystal is shown by applying the solution of time-dependent Schrodinger equation for a neutron in the field of a standing laser wave. The scattering picture is examined within the framework of non-stationary S-matrix theory, where the neutron–laser field interaction is considered exactly and the neutron–crystal interaction is considered as a perturbation described by Fermi pseudopotential (Farri representation). The neutron–crystal interaction is elastic, and the neutron–laser field interaction has both inelastic and elastic behaviors which results in the observation of an analogous to the Kapitza–Dirac effect for neutrons. The neutron scattering probability is calculated and the analysis of the results are adduced. Both inelastic and elastic diffraction conditions are obtained and the formation of a “sublattice” is illustrated in the process of neutron–photon–phonon elastic interaction.

On the neutron diffraction in a crystal in the field of a standing laser wave

The diffraction of neutrons is considered in crystals under the influence of a standing sound wave. The scattering probability is calculated for the elastic neutron–crystal interaction, whereas the neutron-standing sound wave interaction can be either elastic and inelastic. The possibility of short-wave (high-energy) neutrons diffraction is illustrated. It is shown that the Debye–Waller factor can be changed and tuned. The analysis of conservation laws are adduced both for thermal and short-wave neutrons. The formation of a “sublattice” is shown in the process of neutrons elastic diffraction with respect to standing sound wave. The analogous to the Kapitza–Dirac effect is considered for neutrons. The problem is solved within the frame of non-stationary S-matrix theory, where the neutron–phonon interaction is described by the Fermi pseudopotential, which is considered as a perturbation.

K.K. Grigoryan

Papers

Electromagnetic wave propagation in spatially homogeneous yet smoothly time-varying dielectric media

Propagation of sound waves through a spatially homogeneous but smoothly time-dependent medium

Kapitza-Dirac effect with traveling waves

On the neutron diffraction in a crystal in the field of a standing laser wave

On the neutron diffraction in crystals in the field of a standing sound wave