A representation theory of finite electromagnetic beams in free space is formulated by factorizing the field vector of the plane-wave component into a $3 times 2$ mapping matrix and a 2-component Jones-like vector. The mapping matrix has one degree o
f freedom that can be described by the azimuthal angle of a fixed unit vector with respect to the wave vector. This degree of freedom allows us to find out such a beam solution in which every plane-wave component is specified by the same fixed unit vector $mathbf{I}$ and has the same normalized Jones-like vector. The angle $theta_I$ between the fixed unit vector and the propagation axis acts as a parameter that describes the vectorial property of the beam. The impact of $theta_I$ is investigated on a beam of angular-spectrum field scalar that is independent of the azimuthal angle. The field vector in position space is calculated in the first-order approximation under the paraxial condition. A transverse effect is found that a beam of elliptically-polarized angular spectrum is displaced from the center in the direction that is perpendicular to the plane formed by the fixed unit vector and the propagation axis. The expression of the transverse displacement is obtained. Its paraxial approximation is also given.
We continue our study of entanglement entropy in the holographic superconducting phase transitions. In this paper we consider the holographic p-wave superconductor/insulator model, where as the back reaction increases, the transition is changed from
second order to first order. We find that unlike the s-wave case, there is no additional first order transition in the superconducting phase. We calculate the entanglement entropy for two strip geometries. One is parallel to the super current, and the other is orthogonal to the super current. In both cases, we find that the entanglement entropy monotonically increases with respect to the chemical potential.
We prove that for the two-dimensional steady complete compressible Euler system, with given uniform upcoming supersonic flows, the following three fundamental flow patterns (special solutions) in gas dynamics involving transonic shocks are all unique
in the class of piecewise $C^1$ smooth functions, under appropriate conditions on the downstream subsonic flows: $(rmnum{1})$ the normal transonic shocks in a straight duct with finite or infinite length, after fixing a point the shock-front passing through; $(rmnum{2})$ the oblique transonic shocks attached to an infinite wedge; $(rmnum{3})$ a flat Mach configuration containing one supersonic shock, two transonic shocks, and a contact discontinuity, after fixing the point the four discontinuities intersect. These special solutions are constructed traditionally under the assumption that they are piecewise constant, and they have played important roles in the studies of mathematical gas dynamics. Our results show that the assumption of piecewise constant can be replaced by some more weaker assumptions on the downstream subsonic flows, which are sufficient to uniquely determine these special solutions. Mathematically, these are uniqueness results on solutions of free boundary problems of a quasi-linear system of elliptic-hyperbolic composite-mixed type in bounded or unbounded planar domains, without any assumptions on smallness. The proof relies on an elliptic system of pressure $p$ and the tangent of the flow angle $w=v/u$ obtained by decomposition of the Euler system in Lagrangian coordinates, and a newly developed method for the $L^{infty}$ estimate that is independent of the free boundaries, by combining the maximum principles of elliptic equations, and careful analysis of shock polar applied on the (maybe curved) shock-fronts.
We present a method to distinguish the high harmonics generated in individual half-cycle of the driving laser pulse by mixing a weak ultraviolet pulse, enabling one to observe the cutoff of each half-cycle harmonic. We show that the detail informatio
n of the driving laser pulse, including the laser intensity, pulse duration and carrier-envelope phase, can be {it in situ} retrieved from the harmonic spectrogram. In addition, our results show that this method also distinguishes the half-cycle high harmonics for a pulse longer than 10 fs, suggesting a potential to extend the CEP measurement to the multi-cycle regime.
We study the non-thermal emission from old shell-type supernova remnants (SNRs) on the frame of a time-dependent model. In this model, the time-dependent non-thermal spectra of both primary electrons and protons as well as secondary electron/positron
($e^{pm}$) pairs can be calculated numerically by taking into account the evolution of the secondary $e^{pm}$ pairs produced from proton-proton (p-p) interactions due to the accelerated protons collide with the ambient matter in an SNR. The multi-wavelength photon spectrum for a given SNR can be produced through leptonic processes such as electron/positron synchrotron radiation, bremsstrahlung and inverse Compton scattering as well as hadronic interaction. Our results indicate that the non-thermal emission of the secondary $e^{pm}$ pairs is becoming more and more prominent when the SNR ages in the radiative phase because the source of the primary electrons has been cut off and the electron synchrotron energy loss is significant for a radiative SNR, whereas the secondary $e^{pm}$ pairs can be produced continuously for a long time in the phase due to the large energy loss time for the p-p interaction. We apply the model to two old SNRs, G8.7$-$0.1 and G23.3$-$0.3, and the predicted results can explain the observed multi-wavelength photon spectra for the two sources.
A unified description of the free-space cylindrical vector beams is presented, which is an integral transformation solution to the vector Helmholtz equation and the transversality condition. The amplitude 2-form of the angular spectrum involved in th
is solution can be arbitrarily chosen. When one of the two elements is zero, we arrive at either transverse-electric or transverse-magnetic beam mode. In the paraxial condition, this solution not only includes the known $J_1$ Bessel-Gaussian vector beam and the axisymmetric Laguerre-Gaussian vector beam that were obtained by solving the paraxial wave equations, but also predicts two new kinds of vector beam, called the modified-Bessel-Gaussian vector beam.
