Observation of Temporal Solitons in Second-Harmonic Generation with Tilted Pulses
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By eliminating the group-velocity mismatch and enhancing the group-velocity dispersion via pulse-front tilting, we obtained the formation of temporal solitary waves in phase-mismatched second-harmonic generation. Experimental and numerical results match well in a wide range of intensities and $\ensuremath{\Delta}k$'s. In a 7 mm BBO crystal, 58 fs, 527 nm pulses are obtained starting from 200 fs pump, at the same wavelength.Keywords:
Group velocity
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Phase velocity
Relations between the phase velocity and group velocity of guided waves in homogeneous media and the restrictions under which they can be generalized to inhomogeneous media are discussed. Dirichlet, Neumann, and mixed boundary conditions are considered, including the case of frequency dependent media and boundary conditions.
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Dispersion relation
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The simplest dispersion laws for dissipative media, density of electromagnetic energy, phase and group velocities been considered. It has been shown that group velocity may exceed the velocity of light. Also it has been shown that for polar dielectrics with abnormal positive dispersion described by Deby's formula the velocity of energy coincides with phase velocity
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For the wave-particle duality of microscopic particles,the relationship of the velocity and the velocity of phase and the velocity of group for the microscopic particles is analyzed,it is pointed out that the velocity relating to the frequency and wavelength of two De Broglie expressions is not the velocity of group but the velocity of phase.On the contrary,the velocity in the defining expression of momentum is the real speed of particles,which doesn't equal to the velocity of phase but equals to the velocity of group.
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The phase velocity surface for waves propagating in a uniform cold plasma is sometimes misinterpreted as having the shape of a wave-front. A summary is presented of the correct interpretations of the phase velocity, ray velocity, and group velocity surfaces. A full set of computer generated plots of such surfaces are presented. These are intended as an aid to visualization of wave propagation in such a medium.
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Thermal velocity
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The motivation of this work is to study the group velocity of Lamb waves in anisotropic plates in order to understand the characteristics of the propagation of wave packets in arbitrary directions. The group velocity equations have been derived, taking into account the spatial dispersion (variation of the velocity with the direction of propagation) as well as the frequency dispersion. For bulk waves, it is already well known that anisotropy implies a difference between group velocity and phase velocity. It is shown here that this phenomenon occurs for Lamb waves as well, affecting both the velocity and the direction of propagation. An experimental demonstration is also given, using a plate composed of a unidirectional Glass Epoxy composite. The obtained experimental curves are in good agreement with the theoretical calculations.
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Lamb waves
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Relationship between group and phase velocities of electromagnetic waves in a drifting uniaxial plasma is discussed. When determining whether propagation between two points is possible, the allowed direction of the group velocity vector and not that of the phase velocity vector is important.
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The propagation characteristics of Lamb waves in a solid plate are typically represented by a set of dispersion curves, which describe the Lamb-wave phase velocity as a function of the product fd, where f is the acoustic frequency and d is the plate thickness. For certain modes, within a range of phase velocity and fd, it has been theoretically predicted that the associated group velocity could be negative, i.e., the energy transport is in the opposite direction to the phase velocity. In the present study, Lamb waves are generated via mode conversion from a water-borne sound beam incident onto a flat brass plate. Measurement of the phase and group velocities of the Lamb waves of the S1 mode is performed for the fd range of 2.0–2.3 MHz-mm. Comparison of the measured and computed values of phase and group velocities shows good agreement and clearly demonstrates that S1-mode Lamb waves have a negative group velocity for fd=2.08–2.24 MHz-mm.
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Experimentally measured Lamb wave group velocities in composite materials with anisotropic characteristics are not accorded with the theoretical group velocities as calculated with the Lamb wave dispersion equation. This discrepancy arises from the fact that the angle between the group velocity direction and the phase velocity direction in anisotropic materials exists. Wave propagation in a composite material with anisotropic characteristics should be considered with respect to magnitude in addition to direction. In this study, mode phase velocity dispersion corves are depicted with the variation of degree with respect to the fiber direction using a Lamb wave dispersion relation in the unidirectional, bidirectional, and quasi-isotropic composite plates. Slowness surface is sketched by the reciprocal value of the phase velocity curves. The magnitude and direction of the group velocity are calculated from the slowness surface. The theoretically determined group velocity, which is calculated from the slowness surface, Is compared with experimentally measured group velocities. The proposed method shows good agreements with theoretical and experimental results.
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Closed-form expressions are derived for phase velocity and group velocity in cylindrical and spherical sound waves. These are plotted and compared for orders 0, 1 and 2, but the expressions are general and may be applied to waves of any order. Dispersion characteristics of these waves are examined and discussed. The implications for thermodynamic applicability of the wave equation and for application of Huygens’ principle are discussed.
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