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In anisotropic media, the existence of leaky surface acoustic waves is a well-known phenomenon. Very recently, their analogs at the apex of an elastic silicon wedge have been found in experiments using laser-ultrasonics. In addition to a wedge-wave (WW) pulse with low speed, a pseudo-wedge wave (p-WW) pulse was found with a velocity higher than the velocity of shear bulk waves, propagating in the same direction. With a probe-beam-deflection technique, the propagation of the WW pulses was monitored on one of the faces of the wedge at variable distance from the apex. In this way, their depth structure and the leakage of the p-WW could be visualized directly. Calculations were carried out using a method based on a representation of the displacement field in Laguerre functions. This method has been validated by calculating the surface density of states in anisotropic media and comparing the results with those obtained from the surface Green's tensor. The approach has then been extended to the continuum of acoustic modes in infinite wedges with fixed wave-vector along the apex. These calculations confirmed the measured speeds of the WW and p-WW pulses.
Laser pulses focused near the tip of an elastic wedge generate acoustic waves guided at its apex. The shapes of the acoustic wedge wave pulses depend on the energy and the profile of the exciting laser pulse and on the anisotropy of the elastic medium the wedge is made of. Expressions for the acoustic pulse shapes have been derived in terms of the modal displacement fields of wedge waves for laser excitation in the thermo-elastic regime and for excitation via a pressure pulse exerted on the surface. The physical quantity considered is the local inclination of a surface of the wedge, which is measured optically by laser-probe-beam deflection. Experimental results on pulse shapes in the thermo-elastic regime are presented and confirmed by numerical calculations. They pertain to an isotropic sharp-angle wedge with two wedge-wave branches and to a non-reciprocity phenomenon at rectangular silicon edges.
The existence of acoustic waves with displacements localized at the tip of an isotropic elastic wedge was rigorously proven by Kamotskii, Zavorokhin and Nazarov. This proof, which is based on a variational approach, is extended to rectangular anisotropic wedges. For two high-symmetry configurations of rectangular edges in elastic media with tetragonal symmetry, a criterion is derived that allows identifying the boundary between the regions of existence for wedge modes of even and odd symmetry in regions of parameter space, where even- and odd-symmetry modes do not exist simultaneously. Furthermore, rectangular edges with non-equivalent surfaces are analyzed, and it is shown that at rectangular edges of cubic elastic media with one (110) surface and one (001) surface, a tip-localized guided wave always exists, apart from special cases that are characterized.
The laser ultrasound (LU) technique has been used to determine dispersion curves for surface acoustic waves (SAW) propagating in AlScN/Al2O3 systems. Polar and non-polar Al0.77Sc0.23N thin films were prepared by magnetron sputter epitaxy on Al2O3 substrates and coated with a metal layer. SAW dispersion curves have been measured for various propagation directions on the surface. This is easily achieved in LU measurements since no additional surface structures need to be fabricated, which would be required if elastic properties are determined with the help of SAW resonators. Variation of the propagation direction allows for efficient use of the system’s anisotropy when extracting information on elastic properties. This helps to overcome the complexity caused by a large number of elastic constants in the film material. An analysis of the sensitivity of the SAW phase velocities (with respect to the elastic moduli and their dependence on SAW propagation direction) reveals that the non-polar AlScN films are particularly well suited for the extraction of elastic film properties. Good agreement is found between experiment and theoretical predictions, validating LU as a non-destructive and fast technique for the determination of elastic constants of piezoelectric thin films.
Acoustic waves are investigated which are guided at the edge (apex line) of a wedge-shaped elastic body or at the edge of an elastic plate. The edges contain a periodic sequence of modifications, consisting either of indentations or inclusions with a different elastic material which gives rise to high acoustic mismatch. Dispersion relations are computed with the help of the finite element method. They exhibit zero-group velocity points on the dispersion branches of edge-localized acoustic modes. These special points also occur at Bloch-Floquet wavenumbers away from the Brillouin zone boundary. Deep indentations lead to flat branches corresponding to largely non-interacting, Einstein-oscillator like vibrations of the tongues between the grooves of the periodic structure. Due to the nonlinearity of the elastic media, quantified by their third-order elastic constants, an acoustic mode localized at a periodically modified edge generates a second harmonic which partly consists of surface and plate modes propagating into the elastic medium in the direction vertical to the edge. This acoustic radiation at the second-harmonic frequency is investigated for an elastic plate and a truncated sharp-angle wedge with periodic inclusions at their edges. Unlike nonlinear bulk wave generation by surface acoustic waves in an interdigital structure, surface and plate mode radiation by edge-localized modes can be visualized directly in laser-ultrasound experiments.
Клиновые акустические волны в твёрдом те-ле — это третий фундаментальный тип волн, после объёмных и поверхност-ных волн, импульсы которых распространяются без изменений своих форм (дисперсия отсутствует). Систему упругого клина можно получить из систе-мы упругого полупространства, “разрезав” его вдоль некоторой плоскости, а систему упругого полупространства можно получить из распределённой в пространстве упругой среды тем же методом, поэтому связи между поверх-ностными и объёмными волнами должны во многом повторяться при рас-смотрении клиновых и поверхностных волн. Например, существование быст-рых псевдоповерхностных волн в системе упругого полупространства, излу-чающих энергию при распространении в объёмные волны, имеет свой аналог и для системы упругого клина: совсем недавно были открыты псевдоклино-вые волны, излучающие как объёмные, так и поверхностные волны по мере своего распространения. С другой стороны, в этой же последовательности объёмных, поверхностных и клиновых волн должны выделяться и отличи-тельные особенности. Если поверхностные волны отличаются от объёмных волн тем, что они локализованы на двухмерной поверхности (объёмные вол-ны являются нелокализованными), то клиновые волны локализованы вдоль одномерной поверхности (линии) — кромки клина. Клиновые волны — это волноводные акустические волны, которые распространяются без дифракци-онных потерь, а также они не обладают дисперсией, поскольку в системе бесконечного упругого клина нет ни одного параметра размерности длины.
В заключении приведены основные результаты работы, которые со-стоят в следующем:
1. С помощью метода функций Лагерра была построена функция динами-ческого отклика на импульсный линейный источник (функция Грина) для задачи Лэмба в полупространстве, а также были изучены вопросы о сходимости и устойчивости данного построения. Было показано, что в предельном случае построенная функция динамического отклика совпа-дает с классической функцией Грина для этой задачи.
2. На основе результатов предыдущего пункта была построена функция Грина для упругого клина (и функция плотности состояния на кром-ке, совпадающая с диагональными компонентами функции Грина), с по-мощью которой удалось идентифицировать импульсы псевдоклиновых волн на экспериментальных кривых.
3. Для определённых клиновых конфигураций в анизотропных упругих средах (тетрагональных кристаллах) удалось получить критерий суще-ствования клиновых волн на основе характеристик поверхностных волн, распространяющихся на гранях исследуемых конфигураций, а также в некоторых случаях удалось классифицировать клиновые волны по типу симметрии.
4. Была разработана теория, описывающая формы импульсов клиновых волн при различных режимах генерации: абляционном и термоупругом.
5. Для клиновых волн была представлена нелинейная теория второго по-рядка. Были проведены численные расчёты функции ядра эволюцион-ного уравнения клиновых волн для кремниевых клиньев с одной гранью, совпадающей с поверхностью (111) (поверхность скола), и с произволь-ной ориентацией второй грани.
6. Были описаны фундаментальные отличия нелинейных линовых волн от нелинейных объёмных и поверхностных волн, а также было проведено численное моделирование эволюции импульса клиновых волн, которое показало соответствие теории эксперименту.
7. Получены решения солитонного типа для клиновых волн. Рассмотрены взаимодействия солитонов и свойства солитонного распада.
A laser-operated, angle-tunable transducer was employed to excite selectively elastic waves guided along the apex of a solid wedge. The propagation of wedge waves at anisotropic monocrystalline silicon edges with different symmetry properties was studied by optical detection. The reduced symmetry in crystals, as compared to isotropic media, causes a number of new features, such as the existence of supersonic leaky wedge waves, tilted spatial pulse profiles, and other peculiarities of their localization. Experimental and theoretical results are presented for three different types of symmetry configurations: the wedge symmetric about its midplane, the wedge symmetric about the plane normal to its apex line, and the wedge symmetric about one of its faces. The experiments include accurate measurements of the phase velocity and the wave field distribution, providing information on localization and coupling of wedge waves with other waves. Theoretically, the wedge waves were treated by the Laguerre function method, extended to modes that are not localized at the tip of the wedge. This approach allowed an accurate description of the observed localized and leaky wedge waves in anisotropic wedges.
Nonlinear acoustic waves are considered that have displacements localized at the tip of an elastic wedge. The evolution equation governing their propagation is discussed and compared with its analogues pertaining to nonlinear acoustic surface and bulk waves. Solitary wave solutions of the evolution equation have been determined numerically for the cases of two rectangular edges which may be viewed as generated by splitting a half-space, consisting of crystalline silicon, into two quarter-spaces. For these two geometries, the kernel in the nonlinear terms of the evolution equation has been calculated from the second-order and third-order elastic constants of silicon, and weak dispersion due to tip truncation has been considered. Solitary pulse shapes have been computed and collisions of solitary pulses have been simulated for various relative speeds of the two collision partners. Collision scenarios for the two wedge geometries were found to differ considerably. Special attention is paid to the peculiar interaction of two initially identical solitary pulses.
Surface acoustic waves are propagated toward the edge of an anisotropic elastic medium (a silicon crystal), which supports leaky waves with a high degree of localization at the tip of the edge. At an angle of incidence corresponding to phase matching with this leaky wedge wave, a sharp peak in the reflection coefficient of the surface wave was found. This anomalous reflection is associated with efficient excitation of the leaky wedge wave. In laser ultrasound experiments, surface acoustic wave pulses were excited and their reflection from the edge of the sample and their partial conversion into leaky wedge wave pulses was observed by optical probe-beam deflection. The reflection scenario and the pulse shapes of the surface and wedge-localized guided waves, including the evolution of the acoustic pulse traveling along the edge, have been confirmed in detail by numerical simulations.