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Transcript

Atomic Physics Atoms with dipoles and other symmetries Games and surfaces (complete article on website) Examples of collision excitation symmetries In the examples, Consider the cross sections σ for different magnetic sublevels… (a) All mL are equivalent/equal (b) σ(mL) = σ(-mL) - EBIT (c) ditto, t=0 (d) ditto (e) σ(mL) ≠ σ(-mL) Note: (c), (d) & (e) may be time-resolved in the observation… Removing cylindrical symmetry e.g. in a surface collision As we tilt the surface, we remove “cylindrical symmetry” in the excitation system? (reflection or transmission) (a) How does the loss of symmetry of the excitation process affect the symmetry of the wavefunction formed? (b) What happens in the decay processes for such wavefunctions? The general answer: Angular Momentum We need to understand (a) the angular momentum properties of these wavefunctions, (b) the links between such wavefunctions (e.g. their angular momentum properties) and the optical polarization properties of the light emitted. Photon emission and angular momentum Emission of a photon corresponds to a net change in angular momentum of one, its direction being determined by the polarization and direction of the emitted photon. The Stokes parameters define the polarization/angular momentum direction of the emitted photons… What are the Stokes parameters of a beam of light? Let’s go back to 1853…when Stokes determined that there are 7 types of light, and proposed how to measure the types… WHAT ARE THE SEVEN TYPES?? The types of emission depend on the angular momentum character of the photon (in optical cases, always a dipole) – no longer true!! – see higher multipole decays – EBIT, etc… Stokes (1853): There are 7 types of polarized light: Light can be linearly polarized or circularly polarized or elliptically polarized, with axes in any set of directions perpendicular to the observation direction. In addition each of these can have an unpolarized component – that makes 6 possibilities – the 7th is totally unpolarized light. How to distinguish the types (Stokes) – pass the light through first a 1/4 –wave plate and then through a linear polarizer – by rotating one and/or the other one can separate out all the components. There are actually only 4 independent parameters, e.g. the major and minor axes of the ellipse, the angle relative to a given axis, and the intensity of the unpolarized component. The modern definitions are called the Stokes parameters I, M, C, S which are : I = |E|||2 + | E┴ |2 = I(0) + I(90) M = |E|| |2 - | E┴|2 = I(0) – I(90) C = 2 Re (E||E┴*) = I(45) – I(135) S = 2 Im (E||E┴*) = IRH - ILH Practical method for measuring Stokes parameters ( a rotating phase plate, followed by a fixed linear polarizer) Fix the polarizer axis (α), rotate the retardation plate angle (β) which has a known retardation phase (δ) - measure with a standard source – rotate the waveplate in steps (digitally) through successive 2π sets of collection. (added together) Observed intensity is: Analysis: First term – independent of β; second term depends on 2β, last terms depend on 4β Take a Fourier transform of data (which is parametrized in β) the phase plate rotation angle Phase variation of Retardation plate with wavelength Example: High Linear polarization Example: Low Linear polarization Example: Linear and Circular polarization The density matrix, and State Multipoles The density matrix of the excited state can be expanded in terms of spherical harmonics/multipole moments ρkq. [Remember the expansion of the hyperfine interaction in state multipoles…k=0, 1, 2,… q = ±k, ±k-1, ±k-2,..0] For an isotropic state, (case a) only the zero order multipole moment ρ00 is non-zero. In the case of cylindrical symmetry, (cases b, c, d) one “alignment” parameter ρ20 can also be non-zero. In the case of reflection symmetry, without cylindrical symmetry, (case e) one independent first order (1st rank tensor) component can be non-zero – this is the “orientation” of the atomic state ρ10 - corresponding to <J> while two alignment parameters (2nd rank tensors) ρ20 , ρ21 , ρ22 can be non-zero and independent. These are combinations of <(J2 – 3Jz2)>, <JxJz> , etc. Photon emission from non-isotropic states 1. The Simplest Case • Observation of a 1P state decaying to a 1S state in a beam (cylindrical geometry along a z-axis). The final state is an s-state which by definition is isotropic, so that all the angular information is carried by the emitted photon… (a) There are 2 independent cross-sections e.g σ(mL=1) = σ(mL=-1) & σ(mL=0) (b) Looking perpendicularly to the beam z-axis, and measuring the light intensity with a polarizer in 2 directions, parallel and perpendicular to z gives: 2. The same transition with excitation of the k=1 and k=2 multipoles – e.g. in the “tilted target geometry”: We need to write both the excited state and the photon in multipole form: The light intensty is I10(t) = A10 N1(t) Where A10 for an electric dipole transition is proportional to (eλ∙d)(eλ*∙d), with eλ defining the state of polarization of the observed light and N1 the population of mixed state ρ(t) so that I(eλ, P, t) = I0 ∑ (eλ∙d)(eλ*∙d) ρ(t) (P=propagation vector) Excuse me – I have changed ρ to σ for the next few slides (too lazy to retype all the messy multipole tensors!) – see ref: H.G. Berry, Rep. Prog. Phys. 40, 155 (1977) Example 1 The 2 geometries, observing in the “z”-direction The Stokes parameter data -> Note that the grazing incidence data link up well with the tilted foil data, justifying the conclusion that the excited electron is picked up as the atom/ion leaves the surface. General form for photon emission For a single state (no sum of mixed states), in a field-free region, we have a simple exponential decay of all components of the density matrix… dσ/dt = -Γσ and thus σ(t) = σ(0) exp(-Γt) The Stokes parameters of light emission at any angle (θ,φ) are thus also unchanging in time, and can be derived from the above matrix elements… For the case of an initial state of angular momentum F to a state of angular momentum G (thus, this could be a single hyperfine transition), we get Notes: φ = 900 Spherical symmetry Only σ00 nonzero -> M=C=S=0 Cylindrical symmetry σ02 nonzero, -> M≠0 C=S=0 General form for photon emission Just a note on nomenclature – the “Fano-Macek” “Orientation” and “Alignment” parameters O1- and A0, A1+ and A2+ are now the norm for describing anisotropic production and decay of atoms. Using these parameters avoids most of the “Clebsch-Gordan” algebra. Some examples of the use of Stokes parameters in Light Scattering Portable dynamic light scattering instrument and method for the measurement of blood platelet suspensions Elisabeth Maurer-Spurej et al 2006 Phys. Med. Biol. 51 3747-3758 MODELING OF LIGHT SCATTERING BY SINGLE RED BLOOD CELLS WITH THE FDTD METHOD Alfons Hoekstra, et al Optics of Biological Particles 10.1007/978-1-4020-5502-7_7 A more general, long article: Particle Sizing by Static Laser Light Scattering Paul A. Webb, Micromeritics Instrument Corp. January 2000 Ultra-fast Holographic Stokesmeter for Polarization Imaging in Real Time by . S. Shahriar et al Ultra-fast Holographic Stokesmeter for Polarization Imaging in Real Time by . S. Shahriar et al We propose an ultra-fast holographic Stokesmeter using a volume holographic substrate with two sets of two orthogonal gratings to identify all four Stokes parameters of the input beam. We derive the Mueller matrix of the proposed architecture and determine the constraints necessary for reconstructing the complete Stokes vector. The speed of this device is determined primarily by the channel spectral bandwidth (typically 100 GHz), corresponding to a few psec. This device may be very useful in high-speed polarization imaging. Beamsplitter Holographic substrate SH SFS Ii Qi Si U i Vi Incident Image SES Hologram Front Surface Exit It Ii Qt Qi S t Mt U t U i Vt Vi Surface Mt = MES . MH . MFS Mueller matrix representation of light Quarter-wave plate Detectors Surface scattering “playing pool with atoms…” Example of argon ions (with E of a few MeV) hitting a surface. Note how most of them are specularly reflected at the most grazing angles. Optical observations The Stokes parameters at (θ,φ) are: For an LS(J) coupled state Observing at θ = φ = 900 Attempts to show that the maximum spin (S/I) is associated with the specularly reflected ions. Quantum beat measurements of hyperfine structure Fourier transform of the residuals of the decay curve – M/I quantum beats (after fitting with smooth exponentials). S/I quantum beats after surface scattering Results and Fourier transform Example of pulsed laser excitation