1. Optical Vortices and Electric Quadrupole transitions James Bounds

2. Organization ● Dipole and Quadrupole transitions ● Special Laser beam types ● Using special laser beams to induce Quadrupole transitions ● Experimental Realization ● Possible extensions

3. E-M Fields

4. Gauge Freedom

5. Perturbing Term in Length Gauge

6. Classical correspondance of perturbing term

7. Time-Dependent Perturbation Theory

8. Problem is reduced to finding expansion coefficients

9. Separation of emission and absorption terms

10. Dipole Matrix Element

11. Probability of being in state b

12. Relation to Einstein Coefficients

13. Classical Dipole Radiation

14. Higher Order terms By including higher order terms, the field gradients become more important.

15. Selection Rules

16. Quadrupole selection Rules ● Selection rules are then for hydrogen like systems:

17. 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

18. Huygen's principle

19. Approximated Fresnel Integral

20. Helmholtz Equation

21. Relation to Schrodinger equation

22. Substitution into Helmholtz equation

24. Hermite-Gaussian Modes ● Solution of the paraxial wave equation in cartesian coordinates

25. Hermite-Gaussian Modes

26. Laguerre-Gaussian Modes ● Arise when there is cylindrical symmetry – Usually not favored due to astigmatism ● Carry sharply defined amount of orbital angular momentum (OAM)

27. Leguerre-Gaussian Modes

28. Ince-Gaussian Modes ● Solution of paraxial wave equation in elliptic coordinates – Provides smooth connection between HG and LG beams – OAM not as sharply defined

29. The non-zero Leguerre-Gaussian modes form optical vortices

30. 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

31. 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

32. Generation of LG beams ● Computer Generated Hologram – Diffracts plane wavefronts into helical wavefronts – Spatial light modulator – Laser etched gratings

33. Holographic Plates

34. Construction of Laser etched gratings

35. Phase-Amplitude modulation from phase only grating

36. Difficulty for pulsed operation ● Pulsed operation not favored for holographic plates – Angular chirp – Pulse front tilt ● 2f-2f setup

37. 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

38. Holographic Knife Edge ● Similar to a mechanical knife edge technique, we can use the SLM to diffract part of the beam away

39. Knife Edge for LG Beams

41. Simultaneous Generation and characterization of LG beams

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

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

44. ● 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

45. Population Detection ● 4 2 S poulation determined by 866nm fluorescence

46. Zeeman Splitting of Ca+ Quadrupole Transition at 729nm Quadrupole Transition

47. Relative strengths of transitions

48. Large Gradient and zero field = electric quadrupole transition

49. 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