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Atomic Physics
Atoms with dipoles and other symmetries
Games and surfaces (complete article on website)
Examples of collision excitation
In the examples,
Consider the cross sections
σ for different magnetic
(a) All mL are
(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
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…
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
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:
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
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
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.
Holographic substrate
 Ii 
 
Si   
U i 
 
Vi 
Incident Image
 It 
 Ii 
 
 
S t     Mt  
U t 
U i 
 
 
Vt 
Vi 
Mt = MES . MH . MFS
Mueller matrix
representation of light
Quarter-wave plate
Surface scattering
“playing pool with
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
Results and Fourier transform
Example of pulsed laser