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 – Mode usually found in lasers ● Laguerre-Gaussian Beam – 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 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

Simultaneous Generation and characterization of LG beams

Experimental Realization of quadrupole transitions Schmiegelow, “Excitation of an Atomic Transition with a Vortex Laser Beam”

● State is easily probed – 3 2 D State is metastable ● Transition wavelengths accessible Calcium Quadrupole Transition

● 3 2 D state depopulated – 854nm transition ● 3 2 D state is metastable – Population vs. 729 nm LG pulse length gives Rabi frequency State Preparation

Population Detection ● 4 2 S 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