Book•
Wave propagation and group velocity
01 Jan 1960-Vol. 8, pp 1-154
About: The article was published on 1960-01-01 and is currently open access. It has received 1106 citations till now. The article focuses on the topics: Wave propagation & Group velocity.
Citations
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Book•
01 Jan 1990
TL;DR: In this paper, the authors describe the derivation of conservation laws and apply them to linear systems, including the linear advection equation, the Euler equation, and the Riemann problem.
Abstract: I Mathematical Theory- 1 Introduction- 11 Conservation laws- 12 Applications- 13 Mathematical difficulties- 14 Numerical difficulties- 15 Some references- 2 The Derivation of Conservation Laws- 21 Integral and differential forms- 22 Scalar equations- 23 Diffusion- 3 Scalar Conservation Laws- 31 The linear advection equation- 311 Domain of dependence- 312 Nonsmooth data- 32 Burgers' equation- 33 Shock formation- 34 Weak solutions- 35 The Riemann Problem- 36 Shock speed- 37 Manipulating conservation laws- 38 Entropy conditions- 381 Entropy functions- 4 Some Scalar Examples- 41 Traffic flow- 411 Characteristics and "sound speed"- 42 Two phase flow- 5 Some Nonlinear Systems- 51 The Euler equations- 511 Ideal gas- 512 Entropy- 52 Isentropic flow- 53 Isothermal flow- 54 The shallow water equations- 6 Linear Hyperbolic Systems 58- 61 Characteristic variables- 62 Simple waves- 63 The wave equation- 64 Linearization of nonlinear systems- 641 Sound waves- 65 The Riemann Problem- 651 The phase plane- 7 Shocks and the Hugoniot Locus- 71 The Hugoniot locus- 72 Solution of the Riemann problem- 721 Riemann problems with no solution- 73 Genuine nonlinearity- 74 The Lax entropy condition- 75 Linear degeneracy- 76 The Riemann problem- 8 Rarefaction Waves and Integral Curves- 81 Integral curves- 82 Rarefaction waves- 83 General solution of the Riemann problem- 84 Shock collisions- 9 The Riemann problem for the Euler equations- 91 Contact discontinuities- 92 Solution to the Riemann problem- II Numerical Methods- 10 Numerical Methods for Linear Equations- 101 The global error and convergence- 102 Norms- 103 Local truncation error- 104 Stability- 105 The Lax Equivalence Theorem- 106 The CFL condition- 107 Upwind methods- 11 Computing Discontinuous Solutions- 111 Modified equations- 1111 First order methods and diffusion- 1112 Second order methods and dispersion- 112 Accuracy- 12 Conservative Methods for Nonlinear Problems- 121 Conservative methods- 122 Consistency- 123 Discrete conservation- 124 The Lax-Wendroff Theorem- 125 The entropy condition- 13 Godunov's Method- 131 The Courant-Isaacson-Rees method- 132 Godunov's method- 133 Linear systems- 134 The entropy condition- 135 Scalar conservation laws- 14 Approximate Riemann Solvers- 141 General theory- 1411 The entropy condition- 1412 Modified conservation laws- 142 Roe's approximate Riemann solver- 1421 The numerical flux function for Roe's solver- 1422 A sonic entropy fix- 1423 The scalar case- 1424 A Roe matrix for isothermal flow- 15 Nonlinear Stability- 151 Convergence notions- 152 Compactness- 153 Total variation stability- 154 Total variation diminishing methods- 155 Monotonicity preserving methods- 156 l1-contracting numerical methods- 157 Monotone methods- 16 High Resolution Methods- 161 Artificial Viscosity- 162 Flux-limiter methods- 1621 Linear systems- 163 Slope-limiter methods- 1631 Linear Systems- 1632 Nonlinear scalar equations- 1633 Nonlinear Systems- 17 Semi-discrete Methods- 171 Evolution equations for the cell averages- 172 Spatial accuracy- 173 Reconstruction by primitive functions- 174 ENO schemes- 18 Multidimensional Problems- 181 Semi-discrete methods- 182 Splitting methods- 183 TVD Methods- 184 Multidimensional approaches
3,827 citations
IBM1
TL;DR: An over 300-fold reduction of the group velocity on a silicon chip via an ultra-compact photonic integrated circuit using low-loss silicon photonic crystal waveguides that can support an optical mode with a submicrometre cross-section is experimentally demonstrated.
Abstract: It is known that light can be slowed down in dispersive materials near resonances. Dramatic reduction of the light group velocity-and even bringing light pulses to a complete halt-has been demonstrated recently in various atomic and solid state systems, where the material absorption is cancelled via quantum optical coherent effects. Exploitation of slow light phenomena has potential for applications ranging from all-optical storage to all-optical switching. Existing schemes, however, are restricted to the narrow frequency range of the material resonance, which limits the operation frequency, maximum data rate and storage capacity. Moreover, the implementation of external lasers, low pressures and/or low temperatures prevents miniaturization and hinders practical applications. Here we experimentally demonstrate an over 300-fold reduction of the group velocity on a silicon chip via an ultra-compact photonic integrated circuit using low-loss silicon photonic crystal waveguides that can support an optical mode with a submicrometre cross-section. In addition, we show fast (approximately 100 ns) and efficient (2 mW electric power) active control of the group velocity by localized heating of the photonic crystal waveguide with an integrated micro-heater.
1,307 citations
TL;DR: Gain-assisted linear anomalous dispersion is used to demonstrate superluminal light propagation in atomic caesium gas and is observed to be a direct consequence of classical interference between its different frequency components in an anomalously dispersion region.
Abstract: Einstein's theory of special relativity and the principle of causality imply that the speed of any moving object cannot exceed that of light in a vacuum (c) Nevertheless, there exist various proposals for observing faster-than-c propagation of light pulses, using anomalous dispersion near an absorption line, nonlinear and linear gain lines, or tunnelling barriers However, in all previous experimental demonstrations, the light pulses experienced either very large absorption or severe reshaping, resulting in controversies over the interpretation Here we use gain-assisted linear anomalous dispersion to demonstrate superluminal light propagation in atomic caesium gas The group velocity of a laser pulse in this region exceeds c and can even become negative, while the shape of the pulse is preserved We measure a group-velocity index of n(g) = -310(+/- 5); in practice, this means that a light pulse propagating through the atomic vapour cell appears at the exit side so much earlier than if it had propagated the same distance in a vacuum that the peak of the pulse appears to leave the cell before entering it The observed superluminal light pulse propagation is not at odds with causality, being a direct consequence of classical interference between its different frequency components in an anomalous dispersion region
1,211 citations
TL;DR: In this article, exact and approximate expressions for the bandwidth and Q of a general single-feed (one-port) lossy or lossless linear antenna tuned to resonance or antiresonance were derived.
Abstract: To address the need for fundamental universally valid definitions of exact bandwidth and quality factor (Q) of tuned antennas, as well as the need for efficient accurate approximate formulas for computing this bandwidth and Q, exact and approximate expressions are found for the bandwidth and Q of a general single-feed (one-port) lossy or lossless linear antenna tuned to resonance or antiresonance. The approximate expression derived for the exact bandwidth of a tuned antenna differs from previous approximate expressions in that it is inversely proportional to the magnitude |Z'/sub 0/(/spl omega//sub 0/)| of the frequency derivative of the input impedance and, for not too large a bandwidth, it is nearly equal to the exact bandwidth of the tuned antenna at every frequency /spl omega//sub 0/, that is, throughout antiresonant as well as resonant frequency bands. It is also shown that an appropriately defined exact Q of a tuned lossy or lossless antenna is approximately proportional to |Z'/sub 0/(/spl omega//sub 0/)| and thus this Q is approximately inversely proportional to the bandwidth (for not too large a bandwidth) of a simply tuned antenna at all frequencies. The exact Q of a tuned antenna is defined in terms of average internal energies that emerge naturally from Maxwell's equations applied to the tuned antenna. These internal energies, which are similar but not identical to previously defined quality-factor energies, and the associated Q are proven to increase without bound as the size of an antenna is decreased. Numerical solutions to thin straight-wire and wire-loop lossy and lossless antennas, as well as to a Yagi antenna and a straight-wire antenna embedded in a lossy dispersive dielectric, confirm the accuracy of the approximate expressions and the inverse relationship between the defined bandwidth and the defined Q over frequency ranges that cover several resonant and antiresonant frequency bands.
831 citations
TL;DR: This work investigated the propagation of femtosecond laser pulses through a metamaterial that has a negative index of refraction for wavelengths around 1.5 micrometers and directly inferred the phase time delay from the interference fringes of a Michelson interferometer.
Abstract: We investigated the propagation of femtosecond laser pulses through a metamaterial that has a negative index of refraction for wavelengths around 1.5 micrometers. From the interference fringes of a Michelson interferometer with and without the sample, we directly inferred the phase time delay. From the pulse-envelope shift, we determined the group time delay. In a spectral region, phase and group velocity are negative simultaneously. This means that both the carrier wave and the pulse envelope peak of the output pulse appear at the rear side of the sample before their input pulse counterparts have entered the front side of the sample.
758 citations
References
More filters
Book•
01 Jan 1990
TL;DR: In this paper, the authors describe the derivation of conservation laws and apply them to linear systems, including the linear advection equation, the Euler equation, and the Riemann problem.
Abstract: I Mathematical Theory- 1 Introduction- 11 Conservation laws- 12 Applications- 13 Mathematical difficulties- 14 Numerical difficulties- 15 Some references- 2 The Derivation of Conservation Laws- 21 Integral and differential forms- 22 Scalar equations- 23 Diffusion- 3 Scalar Conservation Laws- 31 The linear advection equation- 311 Domain of dependence- 312 Nonsmooth data- 32 Burgers' equation- 33 Shock formation- 34 Weak solutions- 35 The Riemann Problem- 36 Shock speed- 37 Manipulating conservation laws- 38 Entropy conditions- 381 Entropy functions- 4 Some Scalar Examples- 41 Traffic flow- 411 Characteristics and "sound speed"- 42 Two phase flow- 5 Some Nonlinear Systems- 51 The Euler equations- 511 Ideal gas- 512 Entropy- 52 Isentropic flow- 53 Isothermal flow- 54 The shallow water equations- 6 Linear Hyperbolic Systems 58- 61 Characteristic variables- 62 Simple waves- 63 The wave equation- 64 Linearization of nonlinear systems- 641 Sound waves- 65 The Riemann Problem- 651 The phase plane- 7 Shocks and the Hugoniot Locus- 71 The Hugoniot locus- 72 Solution of the Riemann problem- 721 Riemann problems with no solution- 73 Genuine nonlinearity- 74 The Lax entropy condition- 75 Linear degeneracy- 76 The Riemann problem- 8 Rarefaction Waves and Integral Curves- 81 Integral curves- 82 Rarefaction waves- 83 General solution of the Riemann problem- 84 Shock collisions- 9 The Riemann problem for the Euler equations- 91 Contact discontinuities- 92 Solution to the Riemann problem- II Numerical Methods- 10 Numerical Methods for Linear Equations- 101 The global error and convergence- 102 Norms- 103 Local truncation error- 104 Stability- 105 The Lax Equivalence Theorem- 106 The CFL condition- 107 Upwind methods- 11 Computing Discontinuous Solutions- 111 Modified equations- 1111 First order methods and diffusion- 1112 Second order methods and dispersion- 112 Accuracy- 12 Conservative Methods for Nonlinear Problems- 121 Conservative methods- 122 Consistency- 123 Discrete conservation- 124 The Lax-Wendroff Theorem- 125 The entropy condition- 13 Godunov's Method- 131 The Courant-Isaacson-Rees method- 132 Godunov's method- 133 Linear systems- 134 The entropy condition- 135 Scalar conservation laws- 14 Approximate Riemann Solvers- 141 General theory- 1411 The entropy condition- 1412 Modified conservation laws- 142 Roe's approximate Riemann solver- 1421 The numerical flux function for Roe's solver- 1422 A sonic entropy fix- 1423 The scalar case- 1424 A Roe matrix for isothermal flow- 15 Nonlinear Stability- 151 Convergence notions- 152 Compactness- 153 Total variation stability- 154 Total variation diminishing methods- 155 Monotonicity preserving methods- 156 l1-contracting numerical methods- 157 Monotone methods- 16 High Resolution Methods- 161 Artificial Viscosity- 162 Flux-limiter methods- 1621 Linear systems- 163 Slope-limiter methods- 1631 Linear Systems- 1632 Nonlinear scalar equations- 1633 Nonlinear Systems- 17 Semi-discrete Methods- 171 Evolution equations for the cell averages- 172 Spatial accuracy- 173 Reconstruction by primitive functions- 174 ENO schemes- 18 Multidimensional Problems- 181 Semi-discrete methods- 182 Splitting methods- 183 TVD Methods- 184 Multidimensional approaches
3,827 citations
IBM1
TL;DR: An over 300-fold reduction of the group velocity on a silicon chip via an ultra-compact photonic integrated circuit using low-loss silicon photonic crystal waveguides that can support an optical mode with a submicrometre cross-section is experimentally demonstrated.
Abstract: It is known that light can be slowed down in dispersive materials near resonances. Dramatic reduction of the light group velocity-and even bringing light pulses to a complete halt-has been demonstrated recently in various atomic and solid state systems, where the material absorption is cancelled via quantum optical coherent effects. Exploitation of slow light phenomena has potential for applications ranging from all-optical storage to all-optical switching. Existing schemes, however, are restricted to the narrow frequency range of the material resonance, which limits the operation frequency, maximum data rate and storage capacity. Moreover, the implementation of external lasers, low pressures and/or low temperatures prevents miniaturization and hinders practical applications. Here we experimentally demonstrate an over 300-fold reduction of the group velocity on a silicon chip via an ultra-compact photonic integrated circuit using low-loss silicon photonic crystal waveguides that can support an optical mode with a submicrometre cross-section. In addition, we show fast (approximately 100 ns) and efficient (2 mW electric power) active control of the group velocity by localized heating of the photonic crystal waveguide with an integrated micro-heater.
1,307 citations
TL;DR: Gain-assisted linear anomalous dispersion is used to demonstrate superluminal light propagation in atomic caesium gas and is observed to be a direct consequence of classical interference between its different frequency components in an anomalously dispersion region.
Abstract: Einstein's theory of special relativity and the principle of causality imply that the speed of any moving object cannot exceed that of light in a vacuum (c) Nevertheless, there exist various proposals for observing faster-than-c propagation of light pulses, using anomalous dispersion near an absorption line, nonlinear and linear gain lines, or tunnelling barriers However, in all previous experimental demonstrations, the light pulses experienced either very large absorption or severe reshaping, resulting in controversies over the interpretation Here we use gain-assisted linear anomalous dispersion to demonstrate superluminal light propagation in atomic caesium gas The group velocity of a laser pulse in this region exceeds c and can even become negative, while the shape of the pulse is preserved We measure a group-velocity index of n(g) = -310(+/- 5); in practice, this means that a light pulse propagating through the atomic vapour cell appears at the exit side so much earlier than if it had propagated the same distance in a vacuum that the peak of the pulse appears to leave the cell before entering it The observed superluminal light pulse propagation is not at odds with causality, being a direct consequence of classical interference between its different frequency components in an anomalous dispersion region
1,211 citations
TL;DR: This work investigated the propagation of femtosecond laser pulses through a metamaterial that has a negative index of refraction for wavelengths around 1.5 micrometers and directly inferred the phase time delay from the interference fringes of a Michelson interferometer.
Abstract: We investigated the propagation of femtosecond laser pulses through a metamaterial that has a negative index of refraction for wavelengths around 1.5 micrometers. From the interference fringes of a Michelson interferometer with and without the sample, we directly inferred the phase time delay. From the pulse-envelope shift, we determined the group time delay. In a spectral region, phase and group velocity are negative simultaneously. This means that both the carrier wave and the pulse envelope peak of the output pulse appear at the rear side of the sample before their input pulse counterparts have entered the front side of the sample.
758 citations