Reprogrammable metasurfaces,which establish a fascinating bridge between physical and information domains,can dynamically control electromagnetic(EM)waves in real time and thus have attracted great attentions from res...Reprogrammable metasurfaces,which establish a fascinating bridge between physical and information domains,can dynamically control electromagnetic(EM)waves in real time and thus have attracted great attentions from researchers around the world.To control EM waves with an arbitrary polarization state,it is desirable that a complete set of basis states be controlled independently since incident EM waves with an arbitrary polarization state can be decomposed as a linear sum of these basis states.In this work,we present the concept of complete-basis-reprogrammable coding metasurface(CBR-CM)in reflective manners,which can achieve independently dynamic controls over the reflection phases while maintaining the same amplitude for left-handed circularly polarized(LCP)waves and right-handed circularly polarized(RCP)waves.Since LCP and RCP waves together constitute a complete basis set of planar EM waves,dynamicallycontrolled holograms can be generated under arbitrarily polarized wave incidence.The dynamically reconfigurable metaparticle is implemented to demonstrate the CBR-CM’s robust capability of controlling the longitudinal and transverse positions of holograms under LCP and RCP waves independently.It’s expected that the proposed CBR-CM opens up ways of realizing more sophisticated and advanced devices with multiple independent information channels,which may provide technical assistance for digital EM environment reproduction.展开更多
We propose a switchable vortex beam polarization state terahertz multi-layer metasurface,which consists of threelayer elliptical metal crosses,four-layer dielectrics,and two-layer hollow metal circles,which are altern...We propose a switchable vortex beam polarization state terahertz multi-layer metasurface,which consists of threelayer elliptical metal crosses,four-layer dielectrics,and two-layer hollow metal circles,which are alternately superimposed.Under the normal incidence of left-handed circularly polarized(LCP)wave and the right-handed circularly polarized(RCP)waves,the proposed structure realizes three independent control functions,i.e.,focused and vortex beam,vortex beam with different topological charges,and polarization states switching,and azimuth switching of two vortex beams with different polarization states.The results show that the proposed metasurface provides a new idea for investigating the multifunctional terahertz wave modulation devices.展开更多
We demonstrate theoretically that the epsilon-near-zero materials can be utilized to control effectively the polarization conversion of an electromagnetic wave through reflection. The significant feature differing fro...We demonstrate theoretically that the epsilon-near-zero materials can be utilized to control effectively the polarization conversion of an electromagnetic wave through reflection. The significant feature differing from all other means based on whatever natural materials or metamaterials is that for TM incident wave, the reflected phase is a constant, while for TE wave, the reflected phase is a linear function of the incident angle. The phase difference between them covers the range from -180°to 0°, and the polarization conversions from linear states to elliptical or circular states can be obtained by only adjusting the incident angle. Because no complex structures are employed, our proposal promises a simple approach for manipulating polarization conversion at both terahertz and optical frequencies.展开更多
The vectorial structure of an optical field with hybrid states of polarization （SOP） in the near-field is studied by using the angular spectrum method of an electromagnetic beam. Physical images of the longitudinal ...The vectorial structure of an optical field with hybrid states of polarization （SOP） in the near-field is studied by using the angular spectrum method of an electromagnetic beam. Physical images of the longitudinal components of evanescent waves are illustrated and compared with those of the transverse components from the vectorial structure. Our results indicate that the relative weight integrated over the transverse plane of the evanescent wave depends strongly on the number of the polarization topological charges. The shapes of the intensity profiles of the longitudinal components are different from those of the transverse components, and it can be manipulated by changing the initial SoP of the field cross-section. The longitudinal component of evanescent wave dominates the near-field region. In addition, it also leads to three-dimensional shape variations of the optical field and the optical spin angular momentum flux density distributions.展开更多
Using two typical types of polarization controller, this paper analyses theoretically and experimentally the fact that it is necessary to adjust at least three instead of two waveplates in order to transform any state...Using two typical types of polarization controller, this paper analyses theoretically and experimentally the fact that it is necessary to adjust at least three instead of two waveplates in order to transform any state of polarization to any other output covering the entire Poincar6 sphere. The experimental results are exactly in accordance with the theory discussed in this paper. It has corrected the conventional and inaccurate point of view that two waveplates of a polarization controller are adequate to complete the transformation of state of polarization.展开更多
This paper reports on an experiment about a novel method of polarization stabilization. The polarization stabilizer proposed here has an additional function of polarization transformation from any state of polarizatio...This paper reports on an experiment about a novel method of polarization stabilization. The polarization stabilizer proposed here has an additional function of polarization transformation from any state of polarization into any others. The particle swarm optimization is introduced as a control algorithm in the process of either searching or endless tracking. The tracking speed of the stabilizer is obtained up to 12.6 krad/s by using hardware we have in the laboratory, which means that we can achieve a higher speed practical polarization stabilizer if we have faster hardware.展开更多
We extend the third perturbation theory to study the polarization control behavior of the intermediate state absorption in Nd^（3＋）ions. The results show that coherent interference can occur between the single-photo...We extend the third perturbation theory to study the polarization control behavior of the intermediate state absorption in Nd^（3＋）ions. The results show that coherent interference can occur between the single-photon and three-photon excitation pathways, and depends on the central frequency of the femtosecond laser field. Moreover,single-photon and three-photon absorptions have different polarization control efficiencies, and the relative weight of three-photon absorption in the whole excitation processes can increase with increasing the laser intensity.Therefore, the enhancement or suppression of the intermediate state absorption can be realized and manipulated by properly designing the intensity and central frequency of the polarization modulated femtosecond laser field.This research can not only enrich theoretical research methods for the up-conversion luminescence manipulation of rare-earth ions, but also can provide a clear physical picture for understanding and controlling multi-photon absorption in a multiple energy level system.展开更多
Electrons in graphene have fourfold spin and valley degeneracies owing to the unique bipartite honeycomb lattice and an extremely weak spin-orbit coupling,which can support a series of broken symmetry states.Atomic-sc...Electrons in graphene have fourfold spin and valley degeneracies owing to the unique bipartite honeycomb lattice and an extremely weak spin-orbit coupling,which can support a series of broken symmetry states.Atomic-scale defects in graphene are expected to lift these degenerate degrees of freedom at the nanoscale,and hence,lead to rich quantum states,highlighting promising directions for spintronics and valleytronics.In this article,we mainly review the recent scanning tunneling microscopy(STM)advances on the spin and/or valley polarized states induced by an individual atomicscale defect in graphene,including a single-carbon vacancy,a nitrogen-atom dopant,and a hydrogen-atom chemisorption.Lastly,we give a perspective in this field.展开更多
Polarization is a basic characteristic of electromagnetic waves that conveys much optical information owing to its many states.The polarization state is manipulated and controlled for optical information security,opti...Polarization is a basic characteristic of electromagnetic waves that conveys much optical information owing to its many states.The polarization state is manipulated and controlled for optical information security,optical encryption,and optical communication.Metasurface devices provide a new way to manipulate wave-fronts of light.A single ultrathin metasurface device can generate and modulate several differently polarized light fields,and thus carries optical information in several different channels.Terahertz(THz)waves have become widely used as carrier waves for wireless communication.Compact and functional metasurface devices are in high demand for THz elements and systems.This paper proposes a tri-layer metallic THz metasurface for multi-channel polarization generation and phase modulation with a high efficiency of approximately 80%.An azimuthally polarized THz vectorial beam generator is realized and characterized for use as a THz polarization analyzer.The incident polarization angle can be observed graphically with high accuracy.Moreover,a vectorial hologram with eight channels for different linear polarization states is demonstrated experimentally.The information in different holograms can be hidden by choosing the polarization channel for detection.This work contributes to achieving multi-functional metasurface in the THz band and can benefit THz communication and optical information security.展开更多
The evolution in momentum space of bound states in the continuum(BICs)and circularly polarized states(CPSs)—as far-field polarization singularities—can be observed by controlling the geometric parameters of photonic...The evolution in momentum space of bound states in the continuum(BICs)and circularly polarized states(CPSs)—as far-field polarization singularities—can be observed by controlling the geometric parameters of photonic crystals.This offers significant potential in optics and photonics.Here,we reveal that in complex lattices far-field polarization singularities can be flexibly manipulated while preserving structural symmetry.A change in topological charge for the at-ΓBIC can generate new BICs or CPSs.At an off-Γpoint,a BIC can spawn from the collision of two CPSs.As the thickness of the structure increases,this BIC will meet the at-ΓBIC.The merging of BICs can induce topological charge transition and yield a large wavevector space around theΓpoint with ultra-high quality(Q)factors.Our findings provide a novel degree of freedom for manipulating polarization singularities,which holds great promise in radiation modulation and singular optics.展开更多
Polarization, an intrinsic ingredient of photon, plays a critical role in its interaction with matter. A general polarization state can be an appropriate superposition of two basic polarization states, say, the vertic...Polarization, an intrinsic ingredient of photon, plays a critical role in its interaction with matter. A general polarization state can be an appropriate superposition of two basic polarization states, say, the vertical and horizontal linear polarized state. Here we study spontaneous emission of a V-type three-level atom（with two upper states close in energy level）strongly coupled with a single-mode damped optical cavity. By defining a general polarization state of atom as a specific superposition of the two upper quantum states, we can prepare atoms with linear polarization at arbitrary direction, left and right circular polarization, and left and right elliptical polarization, similar to photons. We find that the spontaneous emission of light from these ＂polarized＂ three-level atoms shows very different profiles of side and axis spectra. This means that the polarization state of three-level atoms can become an active ingredient to manipulate its interaction with light and control the quantum interference effect. Exploitation of the coherent superposition and interference of quantum states in＂polarized＂ atoms would allow one to deeply explore new frontiers of light–matter interaction.展开更多
The one-step quantum secure direct communication(QSDC)(Sci.Bull.67,367(2022))can effectively simplify QSDC’s operation and reduce message loss.For enhancing its security under practical experimental condition,we prop...The one-step quantum secure direct communication(QSDC)(Sci.Bull.67,367(2022))can effectively simplify QSDC’s operation and reduce message loss.For enhancing its security under practical experimental condition,we propose two measurement-device-independent(MDI)one-step QSDC protocols,which can resist all possible attacks from imperfect measurement devices.In both protocols,the communication parties prepare identical polarization-spatial-mode two-photon hyperentangled states and construct the hyperentanglement channel by hyperentanglement swapping.The first MDI one-step QSDC protocol adopts the nonlinear-optical complete hyperentanglement Bell state measurement(HBSM)to construct the hyperentanglement channel,while the second protocol adopts the linear-optical partial HBSM.Then,the parties encode the photons in the polarization degree of freedom and send them to the third party for the hyperentanglementassisted complete polarization Bell state measurement.Both protocols are unconditionally secure in theory.The simulation results show the MDI one-step QSDC protocol with complete HBSM attains the maximal communication distance of about354 km.Our MDI one-step QSDC protocols may have potential applications in the future quantum secure communication field.展开更多
The best optimal initial reactant state and collision energy for observing the stereodynamical vector properties of the title reaction in the ground electronic state X2A’ potential energy surface （PES）[Zanchet et a...The best optimal initial reactant state and collision energy for observing the stereodynamical vector properties of the title reaction in the ground electronic state X2A’ potential energy surface （PES）[Zanchet et al. 2006 J. Phys. Chem. A 110 12017] are theoretically predicted using the quasi-classical trajectory （QCT） method for the first time. The calculated results reveal that the smallest value of the rotational quantum number j, larger vibrational quantum number v, and the lower strength of collision energy should be selected for offering the most obvious picture about the stereodynamical vector properties. Polarization-dependent differential cross sections and the angular momentum alignment distribution, P（θr） and P（Φr） in the center-of-mass frame, are obtained to gain an insight into the alignment and orientation of the product molecules. The rotational angular momentum vector j’ of CO is aligned to be perpendicular to reagent relative velocity k. The product polarizations align along the y axis, pointing to the positive direction of the y axis. A new method is developed to investigate massive reactions with various initial states and to further study the vector properties of the fundamental reactions in detail.展开更多
基金supported by the National Natural Science Foundation of China (62101588)the National Key Research and Development Program of China (SQ2022YFB3806200)+1 种基金the Young Talent Fund of Association for Science and Technology in Shaanxi (20240129)the Postdoctoral Fellowship Program of CPSF (GZC20242285)
文摘Reprogrammable metasurfaces,which establish a fascinating bridge between physical and information domains,can dynamically control electromagnetic(EM)waves in real time and thus have attracted great attentions from researchers around the world.To control EM waves with an arbitrary polarization state,it is desirable that a complete set of basis states be controlled independently since incident EM waves with an arbitrary polarization state can be decomposed as a linear sum of these basis states.In this work,we present the concept of complete-basis-reprogrammable coding metasurface(CBR-CM)in reflective manners,which can achieve independently dynamic controls over the reflection phases while maintaining the same amplitude for left-handed circularly polarized(LCP)waves and right-handed circularly polarized(RCP)waves.Since LCP and RCP waves together constitute a complete basis set of planar EM waves,dynamicallycontrolled holograms can be generated under arbitrarily polarized wave incidence.The dynamically reconfigurable metaparticle is implemented to demonstrate the CBR-CM’s robust capability of controlling the longitudinal and transverse positions of holograms under LCP and RCP waves independently.It’s expected that the proposed CBR-CM opens up ways of realizing more sophisticated and advanced devices with multiple independent information channels,which may provide technical assistance for digital EM environment reproduction.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.61871355,61831012,and 62271460)the Talent Project of Zhejiang Provincial Department of Science and Technology,China(Grant No.2018R52043)+1 种基金the Zhejiang Key Research and Development Project of China(Grant Nos.2021C03153 and 2022C03166)the Research Funds for the Provincial Universities of Zhejiang Province,China(Grant No.2020YW20)。
文摘We propose a switchable vortex beam polarization state terahertz multi-layer metasurface,which consists of threelayer elliptical metal crosses,four-layer dielectrics,and two-layer hollow metal circles,which are alternately superimposed.Under the normal incidence of left-handed circularly polarized(LCP)wave and the right-handed circularly polarized(RCP)waves,the proposed structure realizes three independent control functions,i.e.,focused and vortex beam,vortex beam with different topological charges,and polarization states switching,and azimuth switching of two vortex beams with different polarization states.The results show that the proposed metasurface provides a new idea for investigating the multifunctional terahertz wave modulation devices.
基金Supported by the National Natural Science Foundation of China under Grant Nos 51406126 and 11604229
文摘We demonstrate theoretically that the epsilon-near-zero materials can be utilized to control effectively the polarization conversion of an electromagnetic wave through reflection. The significant feature differing from all other means based on whatever natural materials or metamaterials is that for TM incident wave, the reflected phase is a constant, while for TE wave, the reflected phase is a linear function of the incident angle. The phase difference between them covers the range from -180°to 0°, and the polarization conversions from linear states to elliptical or circular states can be obtained by only adjusting the incident angle. Because no complex structures are employed, our proposal promises a simple approach for manipulating polarization conversion at both terahertz and optical frequencies.
基金supported by the Zhejiang Provincial Natural Science Foundation,China(Grant Nos.LZ17A040001 and LY16A040014)the National Natural Science Foundation of China(Grant Nos.11574271 and 11574272)+1 种基金the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministrythe Science Research Foundation of Zhejiang Sci-Tech University(ZSTU),China(Grant No.14062078-Y)
文摘The vectorial structure of an optical field with hybrid states of polarization （SOP） in the near-field is studied by using the angular spectrum method of an electromagnetic beam. Physical images of the longitudinal components of evanescent waves are illustrated and compared with those of the transverse components from the vectorial structure. Our results indicate that the relative weight integrated over the transverse plane of the evanescent wave depends strongly on the number of the polarization topological charges. The shapes of the intensity profiles of the longitudinal components are different from those of the transverse components, and it can be manipulated by changing the initial SoP of the field cross-section. The longitudinal component of evanescent wave dominates the near-field region. In addition, it also leads to three-dimensional shape variations of the optical field and the optical spin angular momentum flux density distributions.
基金supported by the National Natural Science Foundation of China (Grant No 60577046)the Corporative Building Project of Beijing Educational Committee of China (Grant No XK100130737) Shandong High Technology Project of China (Grant No 2006GG2201002)
文摘Using two typical types of polarization controller, this paper analyses theoretically and experimentally the fact that it is necessary to adjust at least three instead of two waveplates in order to transform any state of polarization to any other output covering the entire Poincar6 sphere. The experimental results are exactly in accordance with the theory discussed in this paper. It has corrected the conventional and inaccurate point of view that two waveplates of a polarization controller are adequate to complete the transformation of state of polarization.
基金Project supported by the National High Technology Research and Development Program of China (Grant No. 2009AA01Z224)Huawei Technology Project,China (Grant No. YBON2008014)
文摘This paper reports on an experiment about a novel method of polarization stabilization. The polarization stabilizer proposed here has an additional function of polarization transformation from any state of polarization into any others. The particle swarm optimization is introduced as a control algorithm in the process of either searching or endless tracking. The tracking speed of the stabilizer is obtained up to 12.6 krad/s by using hardware we have in the laboratory, which means that we can achieve a higher speed practical polarization stabilizer if we have faster hardware.
基金Supported by the National Natural Science Foundation of China under Grant Nos 51132004,11474096,11604199,U1704145 and 11747101the Science and Technology Commission of Shanghai Municipality under Grant No 14JC1401500+1 种基金the Henan Provincial Natural Science Foundation of China under Grant No 182102210117the Higher Education Key Program of He’nan Province of China under Grant Nos 17A140025 and 16A140030
文摘We extend the third perturbation theory to study the polarization control behavior of the intermediate state absorption in Nd^（3＋）ions. The results show that coherent interference can occur between the single-photon and three-photon excitation pathways, and depends on the central frequency of the femtosecond laser field. Moreover,single-photon and three-photon absorptions have different polarization control efficiencies, and the relative weight of three-photon absorption in the whole excitation processes can increase with increasing the laser intensity.Therefore, the enhancement or suppression of the intermediate state absorption can be realized and manipulated by properly designing the intensity and central frequency of the polarization modulated femtosecond laser field.This research can not only enrich theoretical research methods for the up-conversion luminescence manipulation of rare-earth ions, but also can provide a clear physical picture for understanding and controlling multi-photon absorption in a multiple energy level system.
基金financial supported by the National Natural Science Foundation of China(Grant Nos.92163206 and 61725107)the National Key Research and Development Program of China(Grant No.2020YFA0308800)+1 种基金Beijing Natural Science Foundation(Grant No.Z190006)China Postdoctoral Science Foundation(Grant No.2021M700407)。
文摘Electrons in graphene have fourfold spin and valley degeneracies owing to the unique bipartite honeycomb lattice and an extremely weak spin-orbit coupling,which can support a series of broken symmetry states.Atomic-scale defects in graphene are expected to lift these degenerate degrees of freedom at the nanoscale,and hence,lead to rich quantum states,highlighting promising directions for spintronics and valleytronics.In this article,we mainly review the recent scanning tunneling microscopy(STM)advances on the spin and/or valley polarized states induced by an individual atomicscale defect in graphene,including a single-carbon vacancy,a nitrogen-atom dopant,and a hydrogen-atom chemisorption.Lastly,we give a perspective in this field.
基金supported by the National Natural Science Foundation of China(Grant Nos.11874132,1174243,11774246,12074094 and 121774271)the National Key R&D Program of China(Grant No.2019YFC1711905)+2 种基金the Beijing Talents Project(Grant No.2018A19)the Sino-German Mobility Program of the Sino-German Center for Science Funding(Grant No.M-0225)the Capacity Building for Science&Technology Innovation-Fundamental Scientific Research Funds(Grant No.00820531120017)。
文摘Polarization is a basic characteristic of electromagnetic waves that conveys much optical information owing to its many states.The polarization state is manipulated and controlled for optical information security,optical encryption,and optical communication.Metasurface devices provide a new way to manipulate wave-fronts of light.A single ultrathin metasurface device can generate and modulate several differently polarized light fields,and thus carries optical information in several different channels.Terahertz(THz)waves have become widely used as carrier waves for wireless communication.Compact and functional metasurface devices are in high demand for THz elements and systems.This paper proposes a tri-layer metallic THz metasurface for multi-channel polarization generation and phase modulation with a high efficiency of approximately 80%.An azimuthally polarized THz vectorial beam generator is realized and characterized for use as a THz polarization analyzer.The incident polarization angle can be observed graphically with high accuracy.Moreover,a vectorial hologram with eight channels for different linear polarization states is demonstrated experimentally.The information in different holograms can be hidden by choosing the polarization channel for detection.This work contributes to achieving multi-functional metasurface in the THz band and can benefit THz communication and optical information security.
基金Project supported by the National Natural Science Foundation of China(Grant No.12374205)。
文摘The evolution in momentum space of bound states in the continuum(BICs)and circularly polarized states(CPSs)—as far-field polarization singularities—can be observed by controlling the geometric parameters of photonic crystals.This offers significant potential in optics and photonics.Here,we reveal that in complex lattices far-field polarization singularities can be flexibly manipulated while preserving structural symmetry.A change in topological charge for the at-ΓBIC can generate new BICs or CPSs.At an off-Γpoint,a BIC can spawn from the collision of two CPSs.As the thickness of the structure increases,this BIC will meet the at-ΓBIC.The merging of BICs can induce topological charge transition and yield a large wavevector space around theΓpoint with ultra-high quality(Q)factors.Our findings provide a novel degree of freedom for manipulating polarization singularities,which holds great promise in radiation modulation and singular optics.
基金supported by the National Basic Research Foundation of China(Grant No.2011CB922002)
文摘Polarization, an intrinsic ingredient of photon, plays a critical role in its interaction with matter. A general polarization state can be an appropriate superposition of two basic polarization states, say, the vertical and horizontal linear polarized state. Here we study spontaneous emission of a V-type three-level atom（with two upper states close in energy level）strongly coupled with a single-mode damped optical cavity. By defining a general polarization state of atom as a specific superposition of the two upper quantum states, we can prepare atoms with linear polarization at arbitrary direction, left and right circular polarization, and left and right elliptical polarization, similar to photons. We find that the spontaneous emission of light from these ＂polarized＂ three-level atoms shows very different profiles of side and axis spectra. This means that the polarization state of three-level atoms can become an active ingredient to manipulate its interaction with light and control the quantum interference effect. Exploitation of the coherent superposition and interference of quantum states in＂polarized＂ atoms would allow one to deeply explore new frontiers of light–matter interaction.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11974189 and 12175106)the Natural Science Foundation of the Jiangsu Higher Education Institutions of China(Grant No.20KJB140001)the Postgraduate Research&Practice Innovation Program of Jiangsu Province,China(Grand No.KYCX22-0963)
文摘The one-step quantum secure direct communication(QSDC)(Sci.Bull.67,367(2022))can effectively simplify QSDC’s operation and reduce message loss.For enhancing its security under practical experimental condition,we propose two measurement-device-independent(MDI)one-step QSDC protocols,which can resist all possible attacks from imperfect measurement devices.In both protocols,the communication parties prepare identical polarization-spatial-mode two-photon hyperentangled states and construct the hyperentanglement channel by hyperentanglement swapping.The first MDI one-step QSDC protocol adopts the nonlinear-optical complete hyperentanglement Bell state measurement(HBSM)to construct the hyperentanglement channel,while the second protocol adopts the linear-optical partial HBSM.Then,the parties encode the photons in the polarization degree of freedom and send them to the third party for the hyperentanglementassisted complete polarization Bell state measurement.Both protocols are unconditionally secure in theory.The simulation results show the MDI one-step QSDC protocol with complete HBSM attains the maximal communication distance of about354 km.Our MDI one-step QSDC protocols may have potential applications in the future quantum secure communication field.
基金Project supported by the Program of Shenyang Key Laboratory of Optoelectronic Materials and Technology,China(Grant No.F12-254-1-00)the National Natural Science Foundation of China(Grant No.11274149)the Natural Science Foundation of Liaoning Province,China(Grant No.20111035)
文摘The best optimal initial reactant state and collision energy for observing the stereodynamical vector properties of the title reaction in the ground electronic state X2A’ potential energy surface （PES）[Zanchet et al. 2006 J. Phys. Chem. A 110 12017] are theoretically predicted using the quasi-classical trajectory （QCT） method for the first time. The calculated results reveal that the smallest value of the rotational quantum number j, larger vibrational quantum number v, and the lower strength of collision energy should be selected for offering the most obvious picture about the stereodynamical vector properties. Polarization-dependent differential cross sections and the angular momentum alignment distribution, P（θr） and P（Φr） in the center-of-mass frame, are obtained to gain an insight into the alignment and orientation of the product molecules. The rotational angular momentum vector j’ of CO is aligned to be perpendicular to reagent relative velocity k. The product polarizations align along the y axis, pointing to the positive direction of the y axis. A new method is developed to investigate massive reactions with various initial states and to further study the vector properties of the fundamental reactions in detail.
