Rotational Doppler Effect: A Review

Generally, the linear motion between the source of a wave and an observer leads to a linear Doppler effect. It is associated with the linear momentum of the wave. For electromagnetic beams having a circular polarization or an azimuthal phase distribution, the rotation between the source and the observer results in a less well-known rotational Doppler effect. It is associated with the angular momentum of the wave. This is particularly the case for vortex beams. Here, the various physical insights that are given to explain the origin of the rotational Doppler effect is reviewed. The focus is on different cases where such an effect gives information on the rotational nature of the probed systems, and also on cases where the rotational Doppler effect is useless. Still debated issues and possible applications are then presented.     

The Doppler effect in NMR spectroscopy

An NMR sample may be subject to motions, such as those due to sample spinning or to liquid flow. Is the spectrum of such a sample affected by the Doppler effect? The question arises because, instrumental dimensions being much shorter than the wavelength, it is the near-field of the precessing magnetic moment which couples to the receiver coil, rather than the radiated far-field. We expand the near-field into plane propagating waves. For each such wave there is another one with the same amplitude, propagating in the opposite direction. The Doppler shifts are therefore equal and opposite. In the model case of a small fluid sample moving with constant velocity, this leads to a distribution of Doppler shifts which is symmetrical with respect to the unshifted frequency: there is no net spectral shift. We examine the possibility of observing the Doppler distribution in this case. We also consider the case of thermal motion of a gas. We draw attention to the resolved Doppler splitting of molecular rotational transitions in a supersonic burst as observed in a microwave resonator. We also mention briefly the Doppler effect in molecular beam spectroscopy.     

Novel properties of wave propagation in biaxially anisotropic left‐handed materials

Some physically interesting properties and effects (including the quantum effects) of wave propagation in biaxially anisotropic left‐handed materials are investigated in this paper: (i) we show that in the biaxially gyrotropic left‐handed material, the left‐right coupling of circularly polarized light arises due to the negative indices in permittivity and permeability tensors of gyrotropic media; (ii) it is well known that the geometric phases of photons inside a curved fiber in previous experiments often depend on the cone angles of solid angles subtended by a curve traced by the direction of wave vector of light, at the center of photon momentum space. Here, however, for the light propagating inside certain anisotropic left‐handed media we will present a different geometric phase that is independent of the cone angles; (iii) the extra phases of electromagnetic wave resulting from the instantaneous helicity inversion at the interfaces between left‐ and right‐handed (LRH) media is also studied in detail by using the Lewis‐Riesenfeld invariant theory. Some interesting applications (e.g., controllable position‐dependent frequency shift, detection of quantum‐vacuum geometric phases and helicity reversals at the LRH interfaces etc.) of above effects and phenomena in left‐handed media is briefly discussed.     

Spectroscopic observation of the rotational Doppler effect

We report on the first spectroscopic observation of the rotational Doppler shift associated with light beams carrying orbital angular momentum. The effect is evidenced as the broadening of a Hanle electromagnetically induced transparency coherence resonance on Rb vapor when the two incident Laguerre-Gaussian laser beams have opposite topological charges. The observations closely agree with theoretical predictions.     

The relativistic Doppler effect: when a zero‐frequency shift or a red shift exists for sources approaching the observer

It is shown without making use of Lorentz transformation that there exists a phenomenon of relativistic zero‐frequency shift in Doppler effect for a plane wave in free space, observed in two inertial frames of relative motion, and the zero shift takes place at a maximum aberration of light. When it is applied to analysis of a moving point light source, two unconventional physical implications result: (1) a light source, when it is approaching (moving closer to) the observer, may cause a red shift; (2) a zero‐frequency‐shift observation does not necessarily mean that the light source is not moving closer, and in contrast, the light source may be moving closer to the observer at a high speed. This fundamental result of special relativity may provide an alternative way to experimentally examine the principle of relativity, and might have a significant application in astrophysics.     

Rotational dynamics of molecules in crystals: Extended langevin-description and simulation

In this paper we study the rotational dynamics of molecules in a two-dimensional model crystal above the orientational ordering temperature. There the rotational motion is diffusive. Memory effects turn out to be very important, leading to a pronounced nonexponential decay of the angular momentum autocorrelation function. Unfortunately and the quantities measurable by neutron or light scattering seem to be rather insensitive to memory effects.     

Doppler effect in a rotating frame

Transformations of complex wave vectors decribing both plane and helical waves are examined on the basis of new complex relativistic transformations including orbital and rotational motions along with the usual. A consequence of these transformations is a generalized Doppler effect, which takes into account possible rotations of the light source. As a result of the motion and rotation of the light source, its emission spectrum will be shifted either to the red or to the blue, depending on the magnitudes of the translational and rotational velocities and the angle between their directions. An experiment is suggested to verify the effect of rotation of the light source on its emission spectrum.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 19–22, July, 1990.     

光子高阶轨道角动量制备、调控及传感应用研究进展

光子既是经典信息也是量子信息的理想载体. 单个光子不仅可以携带自旋角动量(与光波的圆偏振相关), 还可以携带轨道角动量(与光波的螺旋相位相关). 而轨道角动量的重要意义在于可利用单个光子的量子态构建一个高维的Hilbert空间, 从而实现高维量子信息的编码. 自Allen等于1992年确认光子轨道角动量的物理存在以来, 轨道角动量在经典光学和量子光学领域展现了诸多诱人的应用前景, 目前已成为国际光学领域的研究热点之一. 本综述将着重介绍高阶轨道角动量光束的制备与调控技术, 特别是高阶轨道角动量的量子纠缠态操控、旋转Doppler 效应测量及其在远程传感和精密测量技术中的应用.     

Energy and angular momentum transfers from an electromagnetic wave to a copper ring in the UHF band

Electromagnetic waves could carry orbital angular momentum. Such momentum can be transferred to macroscopic objects and can make them rotate under a constant torque. Based on experimental observations, we investigate the origin of orbital angular momentum and energy transfer. Due to angular momentum and energy conservation, we show that angular momentum transfer is due to the change in the sign of angular momentum upon reflection. This leads to a rotational Doppler shift of the electromagnetic wave frequency, ensuring energy conservation.     

Lorentz-Lorenz shift in gases

Effects of motion and collisions on the Lorentz-Lorenz shift in a gas of identical atoms are investigated. Within the binary collision regime and straight path approximation, no significant effect is observed on the shift of an isolated electric dipole transition between two nondegenerate levels. Most of the contribution to the shift arises from the collisions with large impact parameters and, hence, with large duration of collisions. This result seems to justify a generalized Lorentz-Lorenz model for gases, in which the collisional and the Doppler broadenings can also be taken into account.     

Poincaré-sphere equivalent for light beams containing orbital angular momentum

The polarization state of a light beam is related to its spin angular momentum and can be represented on the Poincaré sphere. We propose a sphere for light beams in analogous orbital angular momentum states. Using the Poincaré-sphere equivalent, we interpret the rotational frequency shift for light beams with orbital angular momentum [Phys. Rev. Lett. 80, 3217 (1998)] as a dynamically evolving geometric phase.     

The vacuum friction paradox and related puzzles

The frequency of light emitted by a moving source is shifted by a factor proportional to its velocity. We find that this Doppler shift requires the existence of a paradoxical effect: that a moving atom radiating in otherwise empty space feels a net or average force acing against its direction motion and proportional in magnitude to is speed. Yet there is no preferred rest frame, either in relativity or in Newtonian mechanics, so how can there be a vacuum friction force?     

Invariance of the Doppler bandwidth with flow displacement in the illuminating field.

It is known that if single frequency continuously transmitted ultrasound or electromagnetic energy is reflected from "straight line flow," defined here as one or more scatters moving with constant velocity along an infinite straight line, the Doppler effect will shift the echo spectrum center frequency from the transmitted value, and broaden its bandwidth. It is proved that if such straight line flow is shifted laterally or in range anywhere in the field, i.e., without change of orientation, its Doppler bandwidth remains unchanged. (The "Doppler bandwidth" is here defined as the frequency difference between the extrema of the echo power spectrum.) The theorem holds true even though the time domain echo changes dramatically with motion of the flow path, and is believed to be valid for electromagnetic as well as ultrasound waves. Its implications with respect to flow measurement, as well as preliminary experimental and computational confirmation, will be discussed.     

Hall effect of light

We derive the semiclassical equation of motion for the wave packet of light taking into account the Berry curvature in momentum-space. This equation naturally describes the interplay between orbital and spin angular momenta, i.e., the conservation of the total angular momentum of light. This leads to the shift of wave-packet motion perpendicular to the gradient of the dielectric constant, i.e., the polarization-dependent Hall effect of light. An enhancement of this effect in photonic crystals is also proposed.     

Universality in heat conduction theory: weakly nonlocal thermodynamics

A linear irreversible thermodynamic framework of heat conduction in rigid conductors is introduced. The deviation from local equilibrium is characterized by a single internal variable and a current multiplier. A general constitutive evolution equation of the current density of the internal energy is derived by introducing a linear relationship between the thermodynamic forces and fluxes. The well‐known Fourier, Maxwell–Cattaneo–Vernotte, Guyer–Krumhansl, Jeffreys‐type, and Green–Naghdi‐type equations of heat conduction are obtained as special cases. The universal character of the approach is demonstrated by two examples. Solutions illustrating the properties of the equation with jump boundary conditions are given.     

Quantum carpets woven by cold atoms in a laser field

Propagation of wave packets of cold two-level atoms in a standing-wave laser field can be interpreted in the dressed-state basis as motion in two optical potentials. The three distinct regimes of the wavepacket motion are specified by the ratio of the squared atom–laser detuning to the normalized Doppler shift. We calculate the momentum and position probability densities, which form patterns with minima and maxima of probability both in the momentum and the position spaces known as quantum carpets. At small and large detunings, the atomic motion is substantially adiabatic, and the quantum carpets have a simple form. At intermediate detunings, the wave packet moves nonadiabatically, splitting at each node of the standing wave, which causes a proliferation or branching of atomic trajectories with a single atom. Nonadiabatic transitions produce beautiful quantum carpets with a rich structure.     

Attosecond light pulses for probing two-electron dynamics of helium in the time domain

Using attosecond light pulses to doubly ionize a two-electron wave packet of helium, we showed that the time-resolved correlated motion of the two electrons can be probed by measuring their six-dimensional momentum distributions. For simple wave packets, we showed that the measured momenta, when analyzed in appropriate coordinates, can reveal the stretching, the rotational, and the bending vibrational modes of their joint motion in momentum space, in spite of the Coulomb distortion in the final states.     

声子角动量与手性声子

在经典的物理学理论中,声子广泛地被认为是线极化的、不具有角动量的.最近的理论研究发现,在具有自旋声子相互作用的磁性体系(时间反演对称性破缺)中,声子可以携带非零的角动量,在零温时声子除了具有零点能以外还带有零点角动量;非零的声子角动量将会修正通过爱因斯坦-德哈斯效应测量的回磁比.在非磁性材料中,总的声子角动量为零,但是在空间反演对称性破缺的六角晶格体系中,其倒格子空间的高对称点上声子具有角动量,并具有确定的手性;三重旋转对称操作给予声子量子化的赝角动量,赝角动量的守恒将决定电子谷间散射的选择定则;此外还理论预测了谷声子霍尔效应.     

Multi-scale continuum mechanics: from global bifurcations to noise induced high-dimensional chaos

Many mechanical systems consist of continuum mechanical structures, having either linear or nonlinear elasticity or geometry, coupled to nonlinear oscillators. In this paper, we consider the class of linear continua coupled to mechanical pendula. In such mechanical systems, there often exist several natural time scales determined by the physics of the problem. Using a time scale splitting, we analyze a prototypical structural-mechanical system consisting of a planar nonlinear pendulum coupled to a flexible rod made of linear viscoelastic material. In this system both low-dimensional and high-dimensional chaos is observed. The low-dimensional chaos appears in the limit of small coupling between the continua and oscillator, where the natural frequency of the primary mode of the rod is much greater than the natural frequency of the pendulum. In this case, the motion resides on a slow manifold. As the coupling is increased, global motion moves off of the slow manifold and high-dimensional chaos is observed. We present a numerical bifurcation analysis of the resulting system illustrating the mechanism for the onset of high-dimensional chaos. Constrained invariant sets are computed to reveal a process from low-dimensional to high-dimensional transitions. Applications will be to both deterministic and stochastic bifurcations. Practical implications of the bifurcation from low-dimensional to high-dimensional chaos for detection of damage as well as global effects of noise will also be discussed.