Transcript Optical Vortices and Electric Quadrupole transitions

Optical Vortices and Electric
Quadrupole transitions
James Bounds
Organization
●
Dipole and Quadrupole transitions
●
Special Laser beam types
●
Using special laser beams to induce Quadrupole
transitions
●
Experimental Realization
●
Possible extensions
E-M Fields
Gauge Freedom
Perturbing Term in Length Gauge
Classical correspondance of perturbing
term
Time-Dependent Perturbation Theory
Problem is reduced to finding expansion
coefficients
Separation of emission and absorption
terms
Dipole Matrix Element
Probability of being in state b
Relation to Einstein Coefficients
Classical Dipole Radiation
Higher Order terms
By including higher order terms, the field gradients become more important.
Selection Rules
Quadrupole selection Rules
●
Selection rules are then for hydrogen like systems:
Fundamental Laser Modes
●
Hermite-Gaussian Beam
–
●
Laguerre-Gaussian Beam
–
●
Mode usually found in lasers
Contains a sharp amount of orbital angular momentum
Bessel Beam
–
Diffraction Free
–
Not possible with finite aperture
Huygen's principle
Approximated Fresnel Integral
Helmholtz Equation
Relation to Schrodinger equation
Substitution into Helmholtz equation
Hermite-Gaussian Modes
●
Solution of the paraxial wave equation in cartesian
coordinates
Hermite-Gaussian Modes
Laguerre-Gaussian Modes
●
Arise when there is cylindrical symmetry
–
●
Usually not favored due to astigmatism
Carry sharply defined amount of orbital angular
momentum (OAM)
Leguerre-Gaussian Modes
Ince-Gaussian Modes
●
Solution of paraxial wave equation in elliptic
coordinates
–
Provides smooth connection between HG and LG beams
–
OAM not as sharply defined
The non-zero Leguerre-Gaussian modes
form optical vortices
Orbital Angular momentum
●
●
The Laguerre-Gaussian Beams are special in that
they carry a very sharp amount of orbital angular
momentum
The Poyting vector reperesents a helical spiral
Orbital Angular Momentum (OAM)
●
Property of individual photons and not just beam
●
Can be coupled to external systems
–
Rotation of Ion crystals
–
Forbidden transitions
–
Communication Systems
●
OAM Multiplexing
Generation of LG beams
●
Computer Generated Hologram
–
Diffracts plane wavefronts into helical wavefronts
–
Spatial light modulator
–
Laser etched gratings
Holographic Plates
Construction of Laser etched gratings
Phase-Amplitude modulation from
phase only grating
Difficulty for pulsed operation
●
●
Pulsed operation not
favored for
holographic plates
–
Angular chirp
–
Pulse front tilt
2f-2f setup
Spatial Light modulator
●
●
LCD Crystals respond to computer generated image
Can be used to not only generate, but characterize
LG beams
–
Work done by
–
James Strohaber
–
Holographic Knife
–
edge technique
Holographic Knife Edge
●
Similar to a mechanical knife edge technique, we
can use the SLM to diffract part of the beam away
Knife Edge for LG Beams
Knife Edge for LG Beams
Simultaneous Generation and
characterization of LG beams
Experimental Realization of quadrupole
transitions
Schmiegelow, “Excitation of an Atomic Transition with a Vortex Laser Beam”
Calcium Quadrupole Transition
●
State is easily
probed
–
●
32D State is
metastable
Transition
wavelengths
accessible
State Preparation
●
32D state
depopulated
–
●
854nm
transition
32D state is
metastable
–
Population vs.
729 nm LG
pulse length
gives Rabi
frequency
Population Detection
●
42S poulation
determined by
866nm
fluorescence
Zeeman Splitting of Ca+ Quadrupole
Transition at 729nm
Quadrupole Transition
Relative strengths of transitions
Large Gradient and zero field = electric
quadrupole transition
Conclusions
●
We have demonstrated the origin of the quadrupole
transition
–
●
●
Selection rules
Investigated fundamental beam modes
–
Generation and characterization
–
Orbital Angular momentum
Experimental realization of coupling of OAM to
atoms