Laser phase effect on asymmetric harmonic distribution in H_2^+

The laser phase effect on the spatial distribution of the molecular high-order harmonic generation（MHHG） spectrum from H_2~＋ is theoretically investigated through solving the Non-Bohn-Oppenheimer（NBO） time-dependent Schrodinger equation（TDSE）.The results are shown as follows,（i） The generated harmonics from the two nuclei each present an asymmetric distribution.Particularly,when the laser phases are chosen from 0.0π to 0.6π and from 1.7π to 2.0π,the contribution from the negative-H plays a main role in harmonic generation.When the laser phases are chosen from 0.7πto 1.6k,the contribution from the positive-H to the harmonic generation is remarkably enhanced and becomes greater than that from the negative-H.The electron localization,the time-frequency analyses of the harmonic spectrum and the time-dependent wave function are shown to explain the asymmetric harmonic distribution in H_2~＋,which provides us with a method to control the electron motion in molecules,（ⅱ） As the pulse duration increases,the asymmetric distributions of the MHHG in two H nuclei decrease,（ⅲ） Isotope investigation shows that the asymmetric harmonic distribution can be reduced by introducing the heavy nucleus（i.e.,D_2~＋）.

ANALYTIC AND HARMONIC CONTINUATIONS AND DISTRIBUTIONS

The present paper gives several theorems concerning analytic and harmonic continuations described by distributional limits. For distributions on spheres, comparatively complete results are given. Generalizing the Direchlet and Neumann problem so that the boundary values are distributions, we obtain the results on existence and uniqueness. For one complex variable, a generalized Riemann's mapping theorem is obtained.

Spectral redshift of high-order harmonics by adding a weak pulse in the falling part of the trapezoidal laser pulse

We investigate the spectral redshift of high-order harmonics of the H_2~＋（D_2~＋） molecule by numerically solving the non-Born–Oppenheimer time-dependent Schr ¨odinger equation（TDSE）. The results show that the spectral redshift of highorder harmonics can be observed by adding a weak pulse in the falling part of the trapezoidal laser pulses. Comparing with the H_2~＋ molecule, the shift of high-order harmonic generation（HHG） spectrum for the D_2~＋ molecule is more obvious.We employ the spatial distribution in HHG and time-frequency analysis to illustrate the physical mechanism of the spectral redshift of high-order harmonics.

Symmetry detection of rotating patterns based on rotational Doppler effect of light

We propose a method for detecting the symmetry of rotating patterns based on the rotational Doppler effect(RDE)of light.The basic mechanisms of the RDE are introduced,and the spiral harmonic distribution of rotating patterns is analyzed.By irradiating the rotating pattern using a superimposed optical vortex and analyzing the amplitude of the RDE signal,the spiral harmonic distribution of the pattern can be measured,and then its symmetry can be detected.We demonstrate this method experimentally by using patterns with different symmetries and shapes.As the method does not need to receive the scattered light completely and accurately,it promises potential application in detecting symmetrical rotating objects at a long distance.