Games and surfaces complete article on website Examples of collision excitation symmetries In the examples Consider the cross sections σ for different magnetic sublevels All m L are equivalentequal ID: 684175
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Slide1
Atomic Physics
Atoms with dipoles and other symmetriesSlide2
Games and surfaces
(complete article on website)Slide3
Examples of collision excitation symmetries
In the examples,
Consider the cross sections
σ
for different magnetic sublevels…
All m
L
are equivalent/equal
σ
(m
L
) =
σ
(-m
L
)
- EBIT
ditto, t=0
ditto
σ
(mL)
≠
σ
(-mL)
Note: (c), (d) & (e) may be time-resolved in the observation…Slide4
Removing cylindrical symmetrye.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
the angular momentum properties of these wavefunctions,
the links between such wavefunctions (e.g. their angular momentum properties) and the optical polarization properties of the light emitted.Slide5
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…
Photon emission and angular momentumSlide6
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 7
th
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
┴
*) = I
RH
- I
LHSlide7
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)Slide8
Analysis:
First term – independent of
β
;
second
term depends on
2
β
,
last
terms depend on
4
β
Take a Fourier transform of data (which is
parametrized
in
β) the phase plate rotation angleObserved intensity is:Slide9
Phase variation ofRetardation plate with wavelengthSlide10
Example: High Linear polarizationSlide11
Example: Low Linear polarizationSlide12
Example: Linear and Circular polarizationSlide13
The density matrix, and State Multipoles
The density matrix of the excited state can be expanded in terms of spherical harmonics/
multipole
moments ρ
k
q
.
[
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
ρ
0
0
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. Slide14
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…
There are 2 independent cross-sections e.g
σ
(m
L
=1) =
σ
(mL=-1) &
σ
(m
L
=0)
Looking perpendicularly to the beam z-axis, and measuring the light intensity with a polarizer in 2 directions, parallel and perpendicular to z gives:
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)(
e
λ
*
∙d
), with
e
λ
defining the state of polarization of the observed light and N
1
the population of mixed state
ρ
(t) so that
I(
e
λ
, P, t) = I
0
∑
(
e
λ
∙d
)(
e
λ
*
∙d
)
ρ
(t) (P=propagation vector)Slide15
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)Slide16Slide17
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.Slide18
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…
d
σ/dt = -Γσ
and thus
σ
(t) =
σ
(0) exp(-
Γ
t)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
= 90
0
Spherical symmetry
Only
σ00 nonzero-> M=C=S=0Cylindrical symmetryσ02 nonzero, -> M≠0 C=S=0Slide19
General form for photon emission
Just a note on nomenclature – the “Fano-Macek” “Orientation” and “Alignment” parameters O
1-
and A
0
, A
1+
and A
2+
are now the norm for describing anisotropic production and decay of atoms.
Using these parameters avoids most of the “Clebsch-Gordan” algebra.Slide20
Portable dynamic light scattering instrument and method for the measurement of blood platelet suspensionsElisabeth Maurer-Spurej et al 2006 Phys. Med. Biol.
51
3747-3758
MODELING OF LIGHT SCATTERING BY SINGLE RED BLOOD CELLS WITH THE FDTD METHODAlfons Hoekstra, et al Optics of Biological Particles 10.1007/978-1-4020-5502-7_7A 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
Some examples of the use of Stokes parameters in Light ScatteringSlide21
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.
Ultra-fast Holographic Stokesmeter for Polarization Imaging
in Real Time by . S. Shahriar et al
Hologram
Exit
Surface
Front
Surface
Mueller matrix representation of lightSlide22
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.Slide23
Optical observations
The Stokes parameters at (
θ
,
φ
) are:
Observing at
θ
=
φ
= 90
0
For an LS(J) coupled stateSlide24
Attempts to show that the maximum spin (S/I) is associated with the specularly reflected ions.Slide25Slide26
Quantum beat measurements of hyperfine structureSlide27
Fourier transform of the residuals of the decay curve – M/I quantum beats
(after fitting with smooth exponentials).Slide28
S/I quantum beats after surface scatteringSlide29
Results and Fourier transformSlide30
Example of pulsed laser excitation