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Journal of Physics B Atomic Molecular and Optical Physics

Characterizing emitted electromagnetic radiation: the vector-spherical-harmonic/Bessel-function description

R. Grinter

Published Mar 17, 2014• 6 citations
Publisher

Abstract

General expressions for the dependence of the polarization of the radiation emitted in an atomic electronic transition upon the angle of emission are derived on the basis of a vector-spherical-harmonic/Bessel-function description of the photon. The concept of a photon of a specific multipolarity, i.e. quadrupole, octupole, etc, and identifiable as such, is found to be unviable. The angular distribution and dependence upon distance from the source of the transverse and parallel components of the electric and magnetic fields arising from E1, E2, M1 and M2 atomic electronic emissions are elucidated.

DOI
10.1088/0953-4075/47/7/075004

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The admixture effect of dipole and quadrupole type atomic transitions on the Stokes parameters characterizing the emitted radiation in the presence of an external magnetic field is studied. The transition from an upper level with angular momentum Ju = 3/2 to a lower level with angular momentum Jl = 1/2 is considered to take account of the admixture effect. The analytical expressions showing the emitted Stokes parameters for the case of upper and lower levels having same parities are derived. It is shown that the radiative transition probabilities of magnetic dipole (M1) and electric quadrupole transition (E2) are more intense than those of electric dipole transition (E1) and magnetic quadrupole transition (M2). The polarization components that are not usually seen in the Stokes I; Q and V profiles appear due to the admixture effect. The interference of the polarization components with opposite signs is responsible for the smaller strength of Stokes I; Q; V in admixture transition compared to those observed in pure transition. In pure dipole transition case, U Stokes profile is always zero for all emitted radiation directions. However, the Stokes U profile arises only due to the admixture effect.

•Jan 1, 2019•0 citations
General expressions for the dependence of the polarization of the radiation emitted in an atomic electronic transition upon the angle of emission are derived on the basis of a vector-spherical-harmonic/Bessel-function description of the photon. The concept of a photon of a specific multipolarity, i.e. quadrupole, octupole, etc, and identifiable as such, is found to be unviable. The angular distribution and dependence upon distance from the source of the transverse and parallel components of the electric and magnetic fields arising from E1, E2, M1 and M2 atomic electronic emissions are elucidated.
A density-matrix approach has been developed for the angular distribution and polarization of radiative emission during single-photon atomic transitions for a general set of steady-state excitation processes in an arbitrary arrangement of static (or quasistatic) electric and magnetic fields. Particular attention has been directed at spectroscopic observations in the intense fields of the high-power ion diodes on the Particle Beam Fusion Accelerator II (PBFA II) and SABRE devices at Sandia National Laboratories and at magnetic-field measurements in tokamak plasmas. The field-dependent atomic eigenstates are represented as expansions in a complete basis set of field-free bound and continuum eigenstates. Particular emphasis has been given to directed-electron collisional excitations, which may be produced by an anisotropic incident-electron velocity distribution. We have allowed for the possibility of the coherent excitation of the nearly degenerate field-dependent atomic substates, which can give rise to a complex spectral pattern of overlapping Stark-Zeeman components. Coherent excitations may be produced by a beam of electrons that are spin-polarized at an angle with respect to the propagation direction or by nonparallel electric and magnetic fields. Our main result is a general expression for the matrix elements of the photon-polarization density operator representing the total intensity, angular distribution, and polarization of the atomic radiative emission. For the observation of radiative emission in the direction of the magnetic field, the detection of linearly polarized emission, in addition to the usual circularly polarized radiation, can reveal the presence of a perpendicular electric field or a coherent excitation mechanism.
The unitary transformation from linearly to elliptically polarized states of photons is determined in a scheme of the second quantization (\textsection{}1). It is given by ${\mathbf{e}}_{l}={\mathbf{e}}_{1}cos\ensuremath{\phi}+i{\mathbf{e}}_{2}sin\ensuremath{\phi}$, ${\mathbf{e}}_{r}=i{\mathbf{e}}_{1}sin\ensuremath{\phi}+{\mathbf{e}}_{2}cos\ensuremath{\phi}$ where e1{\mathbf{e}}_{1}e1​ and e2{\mathbf{e}}_{2}e2​ are orthogonal unit vectors corresponding to linearly polarized states of photons and el{\mathbf{e}}_{l}el​ and er{\mathbf{e}}_{r}er​ are those corresponding to left and right elliptically polarized states of photons. A general formula for a probability of an emission of light from an atom is deduced on the basis of the Dirac electron theory so as to include up to both a magnetic dipole and an electric quadrupole term (\textsection{}2). Its general form is improved as compared with the hitherto obtained one. The general theory is applied to electric dipole and quadrupole and magnetic dipole spectral lines (\textsection{}3 and \textsection{}4). Polarization states of the Zeeman components corresponding to all combinations of magnetic and orbital magnetic quantum numbers of an atom are completely determined for an arbitrary given direction of an observation on the basis of quantum mechanics.
An expression is derived for the intensity of electrons inelastically scattered from an atomic state prepared by optical pumping. The expression is a sum of state multipoles, the coefficient of each multipole being a spherical harmonic in the polar coordinates of the light direction for circularly polarized light, or the direction of the electric vector for linearly polarized light times a weight function depending on the population of the initial state. These weight functions are given in closed form for both linearly and circularly polarized light.
A simple way to calculate Rabi frequencies is outlined for interactions of atomic or nuclear multipole moments with laser fields that focuses on their relative geometry. The resulting expression takes the form of a dot product between the laser polarization and a vector spherical harmonic, thereby naturally connecting to the multipole's far-field spontaneous-emission pattern and providing a way to visualize the interaction. Since the vector spherical harmonics are not yet a standard tool in quantum science, their relevant properties are reviewed. This approach is illustrated in the calculation of a variety of beam effects, yielding both perturbative corrections and some nontrivial cases with non-vanishing coupling.
Summary form only given. Although it is often assumed that all light-matter interactions at optical frequencies are mediated by electric dipole transitions, strong optical-frequency magnetic dipoles do exist. In fact, we see magnetic dipole emission every day from the many lanthanide ions (such as erbium, europium, and terbium) that help to illuminate everything from fluorescent lighting to telecom fiber amplifiers. Higher-order processes such as magnetic dipole and electric quadrupole transitions also play an important part in the light emission from transition metal ions and semiconductor quantum dots. Nevertheless, most applications have overlooked the device implications of these electric-dipole-forbidden transitions throughout the visible and near-infrared regime, and their contributions to many important emitters have not been fully characterized.In this presentation, we will experimentally characterize the "forbidden" transitions in a range of solid-state emitters and investigate their applications and implications for nanophotonics. We will examine the electric dipole approximation commonly used to describe light-matter interactions and discuss naturally occurring systems that exhibit higher-order magnetic dipole and electric quadrupole emission. We will illustrate how these nanoscale quantum transitions can provide both a new way to probe magnetic light-matter interactions and a new degree of design freedom for active electronic and photonic devices. Specifically, we will demonstrate how the different symmetries of multipolar transitions can be exploited to identify, quantify, and control light emission, even at sub-lifetime scales. Despite similar radiation patterns, magnetic and electric dipole emitters have different symmetries with respect to polarization and phase. Thus, in an inhomogeneous environment, we can tailor interference effects and the local density of optical states to selectively enhance either electric or magnetic dipole emission [1,2]. To examine the scope of such higher-order transitions, we will present quantum mechanical calculations that identify all the magnetic dipole and electric quadrupole emission lines in the trivalent lanthanide series in the visible to near-infrared spectrum [3]. Then, we will present an energyand momentum-resolved spectroscopy technique to directly quantify the electric and magnetic dipole contributions from any mixed transition. [4] Using energy-momentum spectroscopy, we will experimentally examine the higher order transitions in lanthanide ions [4,5], transition metal ions [6], and epitaxial quantum dots [7]. If time permits, we will then show how the symmetry differences between magnetic and electric dipoles can be used to address specific electronic states [6] and to dynamically tune emission spectra at sub-lifetime-scales [8].
The admixture effect of dipole and quadrupole type atomic transitions on the Stokes parameters characterizing the emitted radiation in the presence of an external magnetic field is studied. The transition from an upper level with angular momentum Ju = 3/2 to a lower level with angular momentum Jl = 1/2 is considered to take account of the admixture effect. The analytical expressions showing the emitted Stokes parameters for the case of upper and lower levels having same parities are derived. It is shown that the radiative transition probabilities of magnetic dipole (M1) and electric quadrupole transition (E2) are more intense than those of electric dipole transition (E1) and magnetic quadrupole transition (M2). The polarization components that are not usually seen in the Stokes I; Q and V profiles appear due to the admixture effect. The interference of the polarization components with opposite signs is responsible for the smaller strength of Stokes I; Q; V in admixture transition compared to those observed in pure transition. In pure dipole transition case, U Stokes profile is always zero for all emitted radiation directions. However, the Stokes U profile arises only due to the admixture effect.