1. Prism Pairs for Elliptical Beams
Graham Edge

2. Why use elliptic beams
We sympathetically cool 40K with 87Rb, and so we want the two species to be in good thermal contact in the optical trap
Need tight confinement in the z-direction to counteract the different gravitational potential
Want to have weak confinement in the x-y plane so that the density is kept low, reducing 3-body losses

3. What we want:
Keep the ~70um waist that we currently have for the vertical cross section of the beam
Expand the beam size in the horizontal direction
Use the 2.7x prisms found in the lab

4. What we want:
We want the beam to be focused at the atoms, but now we don’t have control over the position of the focus for both small and large diameters of the beam
Force the small diameter to focus at the atoms
Measure how far the other profile of the beam is from its focus...
The farther from the focus the beam is, the less deep the resulting trap, and there will also be some gradient in the trapping force

5. Currently for both Dipole 1/2:
200mm Lens
2:1 Telescope
~70um Waist
Need to replace this telescope since it makes the beam too small, which we compensate by purposefully diverging the beam, which is not the best setup

6. Dipole 1
This is the beam that crosses the plug

7. Dipole 1 - Option 1: Vertical 200mm Lens 3:2 Telescope ~107um Waist
Pickoff
Telescope front and back lenses
Focusing lens
Lattice Chamber Window
Atoms
Lattice Chamber Window

8. Dipole 1 - Option 1: Horizontal 200mm Lens 3:2 Telescope ~290um Beam
0.86 Rayleigh lengths from focus
Prism pair
Focusing Lens

9. Dipole 1 - Option 2:
Vertical
200mm Lens
3:2 Telescope
~55um Waist

10. Dipole 1 - Option 2: Horizontal 200mm Lens 3:2 Telescope ~320um Beam
0.67 Rayleigh lengths from focus

11. Dipole 1 - Option 2:
size = 320um, z/zR = 0.67, wo = 265um
Seems like we might make the horizontal better by moving the prisms away...

12. Dipole 1 - Option 2:
size = 269um, z/zR = 2.8, wo = 91um
But the divergence of the small beam makes z/zR even worse

13. Option 1 had a collimated beam incident at the lens, and the wide part of the beam focused after the atoms
Option 2 had a diverging beam incident at the lens, and the wide part of the beam focused before the atoms
If we adjusted the divergence of the beam with a telescope, perhaps we could place the focus closer to the atoms -> Option 3

14. Dipole 1 - Option 3: Vertical 200mm Lens
3:2 Telescope (beam diverging)
~76um Waist

15. Dipole 1 - Option 3: Horizontal 200mm Lens
3:2 Telescope (beam diverging)
~400um Beam
0.16 Rayleigh lengths from focus

16. Use the prisms in the other direction?
We have been setting the beam parameters and lens position to get the vertical profile we want, then adding prisms to shrink the horizontal profile (before the lens)
Is it easier to set the horizontal beam size, and use the prisms to enlarge the vertical profile?
prisms
lens
prisms
lens

17. Dipole 1 - Option 4: Horizontal 200mm Lens 2:1 Telescope ~189um Beam
0.27 Rayleigh lengths from focus
1000um
186um
444um
449um

18. Dipole 1 - Option 4: Vertical 200mm Lens 2:1 Telescope ~70um Waist
2cm from Telescope to Prisms
972um
987um
1000um
70um
444um
720um
5.9cm
5.2cm

19. Dipole 1 - Option 4:
This is much better than the other way of using the prisms. The beam should definitely be expanded with the prisms!
Because the beam is not so divergent after the prisms (due to short zR), the prisms can be placed further away from the lenses, which will be beneficial
The focus properties could probably be made better if I tried moving the telescope as well, this was not one of the parameters that I varied so far

20. Dipole 1 - Option 4:
Because the angled faces of the prisms now face in the vertical plane, there will be scattered beams with ~10% beam power leaving the table
These need to be blocked very well!

21. Now try moving the telescope
With the mirrors where they are, the telescope can’t go more than about 20cm from the pickoff (currently 15cm)
Prisms need to be very close to the telescope to not interfere with the mirrors

22. Dipole 1 – Allow Telescope to Move:
Horizontal
200mm Lens
2:1 Telescope
~163um Beam
0.13 Rayleigh lengths from focus
1020um
465um
163um
462um

23. Dipole 1 – Allow Telescope to Move:
Vertical
200mm Lens
2:1 Telescope
~60um Waist
2cm from Telescope to prisms
1124um
1020um
60um
465um
1120um
820um
5.45cm
20cm
5.75cm

24. Dipole 2
I should have done this one first, since this is the more likely beam to be made elliptical
The path length from telescope to lens is longer here, so need to check how the focus is affected
Can always compensate for the longer path length by moving the telescope closer to the lens

25. Dipole 2 - Without Moving the Telescope :
Horizontal
200mm Lens
2:1 Telescope
~140um Beam
0.62 Rayleigh lengths from focus
TOO FAR
12cm

26. Dipole 2 - Without Moving the Telescope:
TOO SMALL
Vertical
200mm Lens
2:1 Telescope
~39um Waist

27. We will certainly need to move the telescope closer to the lens

28. Dipole 2 – After Moving Telescope:
Horizontal
200mm Lens
2:1 Telescope
~160um Beam
0.12 Rayleigh lengths from focus
1310um
608um
475um
160 um
40cm
5.54cm
5.7cm

29. Dipole 2 – After Moving Telescope:
Vertical
200mm Lens
2:1 Telescope
~58um Waist
Set to 30cm
1150um
1310um
58um
427um
608um
40cm
5.54cm
5.7cm
820um

30. Hard to place the prisms 30cm from the focusing lens, so redo calculations for the prisms 45cm away from this lens

31. Dipole 2 – After Moving Telescope:
Horizontal
200mm Lens
2:1 Telescope
~158um Beam
0.16 Rayleigh lengths from focus
1310um
608um
475um
158 um
40cm
5.54cm
5.4cm

32. Dipole 2 – After Moving Telescope:
Vertical
200mm Lens
2:1 Telescope
~56um Waist
Set to 45cm
1175um
1310um
56um
608um
450um
40cm
5.56cm
5.4cm
855um

33. Conclusions
By moving the telescopes and keeping them as 2:1 magnification, we can achieve:
Dipole 1:
60um Waist
163um, 0.13 Rayleigh lengths from focus
Dipole 2:
56um Waist
158um, 0.16 Rayleigh lengths from focus