Abstract

New counterpropagating geometries are presented for localizing ultracold atoms in the dark regions created by the interference of Laguerre–Gaussian laser beams. In particular dark helices, an “optical revolver,” axial lattices of rings, and axial lattices of ring lattices of rings are considered and a realistic scheme for achieving phase stability is explored. The dark nature of these traps will enable their use as versatile tools for low-decoherence atom interferometry with zero differential light shifts.

© 2012 Optical Society of America

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References

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  1. M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, Science 275, 637 (1997).
    [CrossRef]
  2. M. E. Zawadzki, P. F. Griffin, E. Riis, and A. S. Arnold, Phys. Rev. A 81, 043608 (2010).
    [CrossRef]
  3. K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Phys. Rev. Lett. 84, 806 (2000).
    [CrossRef]
  4. I. Bloch, Nat. Phys. 1, 23 (2005).
    [CrossRef]
  5. K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, Phys. Rev. A 63, 031602(R) (2001).
    [CrossRef]
  6. S. Franke-Arnold and A. S. Arnold, Am. Sci. 96, 226 (2008).
  7. R. Ozeri, L. Khaykovich, and N. Davidson, Phys. Rev. A 59, R1750 (1999).
    [CrossRef]
  8. J. Arlt and M. J. Padgett, Opt. Lett. 25, 191 (2000).
    [CrossRef]
  9. S. E. Olson, M. L. Terraciano, M. Bashkansky, and F. K. Fatemi, Phys. Rev. A 76, 061404(R) (2007).
    [CrossRef]
  10. S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Öhberg, and A. S. Arnold, Opt. Express 15, 8619 (2007).
    [CrossRef]
  11. X. He, P. Xu, J. Wang, and M. Zhan, Opt. Express 17, 21007 (2009).
  12. E. Courtade, O. Houde, J.-F. Clément, P. Verkerk, and D. Hennnequin, Phys. Rev. A 74, 031403 (2006).
    [CrossRef]
  13. L. Amico, A. Osterloh, and F. Cataliotti, Phys. Rev. Lett. 95, 063201 (2005).
    [CrossRef]
  14. M. Bhattacharya, Opt. Commun. 279, 219 (2007).
    [CrossRef]
  15. S. Palzer, C. Zipkes, C. Sias, and M. Köhl, Phys. Rev. Lett. 103, 150601 (2009), and references therein.
    [CrossRef]
  16. P. F. Griffin, E. Riis, and A. S. Arnold, Phys. Rev. A 77, 051402(R) (2008), and references therein.
    [CrossRef]
  17. A. Ramanathan, K. C. Wright, S. R. Muniz, M. Zelan, W. T. Hill, C. J. Lobb, K. Helmerson, W. D. Phillips, and G. K. Campbell, Phys. Rev. Lett. 106, 130401 (2011).
    [CrossRef]
  18. S. Moulder, S. Beattie, R. P. Smith, N. Tammuz, and Z. Hadzibabic, arXiv: 1112.0334.
  19. J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. Kuhr, Nature 467, 68 (2010).
    [CrossRef]

2011

A. Ramanathan, K. C. Wright, S. R. Muniz, M. Zelan, W. T. Hill, C. J. Lobb, K. Helmerson, W. D. Phillips, and G. K. Campbell, Phys. Rev. Lett. 106, 130401 (2011).
[CrossRef]

2010

J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. Kuhr, Nature 467, 68 (2010).
[CrossRef]

M. E. Zawadzki, P. F. Griffin, E. Riis, and A. S. Arnold, Phys. Rev. A 81, 043608 (2010).
[CrossRef]

2009

S. Palzer, C. Zipkes, C. Sias, and M. Köhl, Phys. Rev. Lett. 103, 150601 (2009), and references therein.
[CrossRef]

X. He, P. Xu, J. Wang, and M. Zhan, Opt. Express 17, 21007 (2009).

2008

P. F. Griffin, E. Riis, and A. S. Arnold, Phys. Rev. A 77, 051402(R) (2008), and references therein.
[CrossRef]

S. Franke-Arnold and A. S. Arnold, Am. Sci. 96, 226 (2008).

2007

M. Bhattacharya, Opt. Commun. 279, 219 (2007).
[CrossRef]

S. E. Olson, M. L. Terraciano, M. Bashkansky, and F. K. Fatemi, Phys. Rev. A 76, 061404(R) (2007).
[CrossRef]

S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Öhberg, and A. S. Arnold, Opt. Express 15, 8619 (2007).
[CrossRef]

2006

E. Courtade, O. Houde, J.-F. Clément, P. Verkerk, and D. Hennnequin, Phys. Rev. A 74, 031403 (2006).
[CrossRef]

2005

L. Amico, A. Osterloh, and F. Cataliotti, Phys. Rev. Lett. 95, 063201 (2005).
[CrossRef]

I. Bloch, Nat. Phys. 1, 23 (2005).
[CrossRef]

2001

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, Phys. Rev. A 63, 031602(R) (2001).
[CrossRef]

2000

K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Phys. Rev. Lett. 84, 806 (2000).
[CrossRef]

J. Arlt and M. J. Padgett, Opt. Lett. 25, 191 (2000).
[CrossRef]

1999

R. Ozeri, L. Khaykovich, and N. Davidson, Phys. Rev. A 59, R1750 (1999).
[CrossRef]

1997

M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, Science 275, 637 (1997).
[CrossRef]

Amico, L.

L. Amico, A. Osterloh, and F. Cataliotti, Phys. Rev. Lett. 95, 063201 (2005).
[CrossRef]

Andrews, M. R.

M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, Science 275, 637 (1997).
[CrossRef]

Arlt, J.

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, Phys. Rev. A 63, 031602(R) (2001).
[CrossRef]

Arnold, A. S.

M. E. Zawadzki, P. F. Griffin, E. Riis, and A. S. Arnold, Phys. Rev. A 81, 043608 (2010).
[CrossRef]

S. Franke-Arnold and A. S. Arnold, Am. Sci. 96, 226 (2008).

P. F. Griffin, E. Riis, and A. S. Arnold, Phys. Rev. A 77, 051402(R) (2008), and references therein.
[CrossRef]

S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Öhberg, and A. S. Arnold, Opt. Express 15, 8619 (2007).
[CrossRef]

Bashkansky, M.

S. E. Olson, M. L. Terraciano, M. Bashkansky, and F. K. Fatemi, Phys. Rev. A 76, 061404(R) (2007).
[CrossRef]

Beattie, S.

S. Moulder, S. Beattie, R. P. Smith, N. Tammuz, and Z. Hadzibabic, arXiv: 1112.0334.

Bhattacharya, M.

M. Bhattacharya, Opt. Commun. 279, 219 (2007).
[CrossRef]

Bloch, I.

J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. Kuhr, Nature 467, 68 (2010).
[CrossRef]

I. Bloch, Nat. Phys. 1, 23 (2005).
[CrossRef]

Bongs, K.

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, Phys. Rev. A 63, 031602(R) (2001).
[CrossRef]

Burger, S.

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, Phys. Rev. A 63, 031602(R) (2001).
[CrossRef]

Campbell, G. K.

A. Ramanathan, K. C. Wright, S. R. Muniz, M. Zelan, W. T. Hill, C. J. Lobb, K. Helmerson, W. D. Phillips, and G. K. Campbell, Phys. Rev. Lett. 106, 130401 (2011).
[CrossRef]

Cataliotti, F.

L. Amico, A. Osterloh, and F. Cataliotti, Phys. Rev. Lett. 95, 063201 (2005).
[CrossRef]

Cheneau, M.

J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. Kuhr, Nature 467, 68 (2010).
[CrossRef]

Chevy, F.

K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Phys. Rev. Lett. 84, 806 (2000).
[CrossRef]

Clément, J.-F.

E. Courtade, O. Houde, J.-F. Clément, P. Verkerk, and D. Hennnequin, Phys. Rev. A 74, 031403 (2006).
[CrossRef]

Courtade, E.

E. Courtade, O. Houde, J.-F. Clément, P. Verkerk, and D. Hennnequin, Phys. Rev. A 74, 031403 (2006).
[CrossRef]

Dalibard, J.

K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Phys. Rev. Lett. 84, 806 (2000).
[CrossRef]

Davidson, N.

R. Ozeri, L. Khaykovich, and N. Davidson, Phys. Rev. A 59, R1750 (1999).
[CrossRef]

Dettmer, S.

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, Phys. Rev. A 63, 031602(R) (2001).
[CrossRef]

Durfee, D. S.

M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, Science 275, 637 (1997).
[CrossRef]

Ellinas, D.

Endres, M.

J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. Kuhr, Nature 467, 68 (2010).
[CrossRef]

Ertmer, W.

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, Phys. Rev. A 63, 031602(R) (2001).
[CrossRef]

Fatemi, F. K.

S. E. Olson, M. L. Terraciano, M. Bashkansky, and F. K. Fatemi, Phys. Rev. A 76, 061404(R) (2007).
[CrossRef]

Franke-Arnold, S.

Girkin, J. M.

Griffin, P. F.

M. E. Zawadzki, P. F. Griffin, E. Riis, and A. S. Arnold, Phys. Rev. A 81, 043608 (2010).
[CrossRef]

P. F. Griffin, E. Riis, and A. S. Arnold, Phys. Rev. A 77, 051402(R) (2008), and references therein.
[CrossRef]

Hadzibabic, Z.

S. Moulder, S. Beattie, R. P. Smith, N. Tammuz, and Z. Hadzibabic, arXiv: 1112.0334.

He, X.

Hellweg, D.

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, Phys. Rev. A 63, 031602(R) (2001).
[CrossRef]

Helmerson, K.

A. Ramanathan, K. C. Wright, S. R. Muniz, M. Zelan, W. T. Hill, C. J. Lobb, K. Helmerson, W. D. Phillips, and G. K. Campbell, Phys. Rev. Lett. 106, 130401 (2011).
[CrossRef]

Hennnequin, D.

E. Courtade, O. Houde, J.-F. Clément, P. Verkerk, and D. Hennnequin, Phys. Rev. A 74, 031403 (2006).
[CrossRef]

Hill, W. T.

A. Ramanathan, K. C. Wright, S. R. Muniz, M. Zelan, W. T. Hill, C. J. Lobb, K. Helmerson, W. D. Phillips, and G. K. Campbell, Phys. Rev. Lett. 106, 130401 (2011).
[CrossRef]

Houde, O.

E. Courtade, O. Houde, J.-F. Clément, P. Verkerk, and D. Hennnequin, Phys. Rev. A 74, 031403 (2006).
[CrossRef]

Ketterle, W.

M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, Science 275, 637 (1997).
[CrossRef]

Khaykovich, L.

R. Ozeri, L. Khaykovich, and N. Davidson, Phys. Rev. A 59, R1750 (1999).
[CrossRef]

Köhl, M.

S. Palzer, C. Zipkes, C. Sias, and M. Köhl, Phys. Rev. Lett. 103, 150601 (2009), and references therein.
[CrossRef]

Kuhr, S.

J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. Kuhr, Nature 467, 68 (2010).
[CrossRef]

Kurn, D. M.

M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, Science 275, 637 (1997).
[CrossRef]

Leach, J.

Lembessis, V. E.

Lobb, C. J.

A. Ramanathan, K. C. Wright, S. R. Muniz, M. Zelan, W. T. Hill, C. J. Lobb, K. Helmerson, W. D. Phillips, and G. K. Campbell, Phys. Rev. Lett. 106, 130401 (2011).
[CrossRef]

Madison, K. W.

K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Phys. Rev. Lett. 84, 806 (2000).
[CrossRef]

Miesner, H.-J.

M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, Science 275, 637 (1997).
[CrossRef]

Moulder, S.

S. Moulder, S. Beattie, R. P. Smith, N. Tammuz, and Z. Hadzibabic, arXiv: 1112.0334.

Muniz, S. R.

A. Ramanathan, K. C. Wright, S. R. Muniz, M. Zelan, W. T. Hill, C. J. Lobb, K. Helmerson, W. D. Phillips, and G. K. Campbell, Phys. Rev. Lett. 106, 130401 (2011).
[CrossRef]

Öhberg, P.

Olson, S. E.

S. E. Olson, M. L. Terraciano, M. Bashkansky, and F. K. Fatemi, Phys. Rev. A 76, 061404(R) (2007).
[CrossRef]

Osterloh, A.

L. Amico, A. Osterloh, and F. Cataliotti, Phys. Rev. Lett. 95, 063201 (2005).
[CrossRef]

Ozeri, R.

R. Ozeri, L. Khaykovich, and N. Davidson, Phys. Rev. A 59, R1750 (1999).
[CrossRef]

Padgett, M. J.

Palzer, S.

S. Palzer, C. Zipkes, C. Sias, and M. Köhl, Phys. Rev. Lett. 103, 150601 (2009), and references therein.
[CrossRef]

Phillips, W. D.

A. Ramanathan, K. C. Wright, S. R. Muniz, M. Zelan, W. T. Hill, C. J. Lobb, K. Helmerson, W. D. Phillips, and G. K. Campbell, Phys. Rev. Lett. 106, 130401 (2011).
[CrossRef]

Ramanathan, A.

A. Ramanathan, K. C. Wright, S. R. Muniz, M. Zelan, W. T. Hill, C. J. Lobb, K. Helmerson, W. D. Phillips, and G. K. Campbell, Phys. Rev. Lett. 106, 130401 (2011).
[CrossRef]

Riis, E.

M. E. Zawadzki, P. F. Griffin, E. Riis, and A. S. Arnold, Phys. Rev. A 81, 043608 (2010).
[CrossRef]

P. F. Griffin, E. Riis, and A. S. Arnold, Phys. Rev. A 77, 051402(R) (2008), and references therein.
[CrossRef]

Sengstock, K.

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, Phys. Rev. A 63, 031602(R) (2001).
[CrossRef]

Sherson, J. F.

J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. Kuhr, Nature 467, 68 (2010).
[CrossRef]

Sias, C.

S. Palzer, C. Zipkes, C. Sias, and M. Köhl, Phys. Rev. Lett. 103, 150601 (2009), and references therein.
[CrossRef]

Smith, R. P.

S. Moulder, S. Beattie, R. P. Smith, N. Tammuz, and Z. Hadzibabic, arXiv: 1112.0334.

Tammuz, N.

S. Moulder, S. Beattie, R. P. Smith, N. Tammuz, and Z. Hadzibabic, arXiv: 1112.0334.

Terraciano, M. L.

S. E. Olson, M. L. Terraciano, M. Bashkansky, and F. K. Fatemi, Phys. Rev. A 76, 061404(R) (2007).
[CrossRef]

Townsend, C. G.

M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, Science 275, 637 (1997).
[CrossRef]

Verkerk, P.

E. Courtade, O. Houde, J.-F. Clément, P. Verkerk, and D. Hennnequin, Phys. Rev. A 74, 031403 (2006).
[CrossRef]

Wang, J.

Weitenberg, C.

J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. Kuhr, Nature 467, 68 (2010).
[CrossRef]

Wohlleben, W.

K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Phys. Rev. Lett. 84, 806 (2000).
[CrossRef]

Wright, A. J.

Wright, K. C.

A. Ramanathan, K. C. Wright, S. R. Muniz, M. Zelan, W. T. Hill, C. J. Lobb, K. Helmerson, W. D. Phillips, and G. K. Campbell, Phys. Rev. Lett. 106, 130401 (2011).
[CrossRef]

Xu, P.

Zawadzki, M. E.

M. E. Zawadzki, P. F. Griffin, E. Riis, and A. S. Arnold, Phys. Rev. A 81, 043608 (2010).
[CrossRef]

Zelan, M.

A. Ramanathan, K. C. Wright, S. R. Muniz, M. Zelan, W. T. Hill, C. J. Lobb, K. Helmerson, W. D. Phillips, and G. K. Campbell, Phys. Rev. Lett. 106, 130401 (2011).
[CrossRef]

Zhan, M.

Zipkes, C.

S. Palzer, C. Zipkes, C. Sias, and M. Köhl, Phys. Rev. Lett. 103, 150601 (2009), and references therein.
[CrossRef]

Am. Sci.

S. Franke-Arnold and A. S. Arnold, Am. Sci. 96, 226 (2008).

Nat. Phys.

I. Bloch, Nat. Phys. 1, 23 (2005).
[CrossRef]

Nature

J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. Kuhr, Nature 467, 68 (2010).
[CrossRef]

Opt. Commun.

M. Bhattacharya, Opt. Commun. 279, 219 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

S. E. Olson, M. L. Terraciano, M. Bashkansky, and F. K. Fatemi, Phys. Rev. A 76, 061404(R) (2007).
[CrossRef]

R. Ozeri, L. Khaykovich, and N. Davidson, Phys. Rev. A 59, R1750 (1999).
[CrossRef]

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer, and K. Sengstock, Phys. Rev. A 63, 031602(R) (2001).
[CrossRef]

M. E. Zawadzki, P. F. Griffin, E. Riis, and A. S. Arnold, Phys. Rev. A 81, 043608 (2010).
[CrossRef]

E. Courtade, O. Houde, J.-F. Clément, P. Verkerk, and D. Hennnequin, Phys. Rev. A 74, 031403 (2006).
[CrossRef]

P. F. Griffin, E. Riis, and A. S. Arnold, Phys. Rev. A 77, 051402(R) (2008), and references therein.
[CrossRef]

Phys. Rev. Lett.

A. Ramanathan, K. C. Wright, S. R. Muniz, M. Zelan, W. T. Hill, C. J. Lobb, K. Helmerson, W. D. Phillips, and G. K. Campbell, Phys. Rev. Lett. 106, 130401 (2011).
[CrossRef]

S. Palzer, C. Zipkes, C. Sias, and M. Köhl, Phys. Rev. Lett. 103, 150601 (2009), and references therein.
[CrossRef]

L. Amico, A. Osterloh, and F. Cataliotti, Phys. Rev. Lett. 95, 063201 (2005).
[CrossRef]

K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, Phys. Rev. Lett. 84, 806 (2000).
[CrossRef]

Science

M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, Science 275, 637 (1997).
[CrossRef]

Other

S. Moulder, S. Beattie, R. P. Smith, N. Tammuz, and Z. Hadzibabic, arXiv: 1112.0334.

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Figures (6)

Fig. 1.
Fig. 1.

A dark helix lattice arising from a superposition of 100  mW of LG 5 and a counterpropagating 170 mW LG 15 beam. (a) Contains radial [left ( 40 μm ) 2 ] and axial [right ( 4 λ = 2.13 μm ) × 40 μm ] slices through the beam. Intensity and phase are shown in the top and bottom images, respectively. At the maximum intensity shown (white zones) the scattering rate and potential depth are R = 0.19 Hz and U = 43 μK , respectively. (b) Extends the radial 2D intensity plot into a 3D contour plot indicating the intensity minima (blue) at 10% and 25% of the maximum of the scale in the radial intensity slice. Note the axial extension is only one wavelength 532 nm, i.e., exaggerated by a factor of 75 .

Fig. 2.
Fig. 2.

An “optical revolver” generated by a superposition of 100 mW of LG 3 and 290 mW of LG 25 with counterpropagating 190 mW of LG 11 , all other dimensions as for Fig. 1. Here, R = 0.26 Hz , U = 60 μK . The inner helical dark core has a lead angle of 38 mrad and leads to harmonic trap frequencies 183 kHz and 6.9 kHz in the axial and radial directions, respectively.

Fig. 3.
Fig. 3.

An axial lattice of rings formed by a 100 mW LG 0 beam (with the typical 5 μm waist) counterpropagating with 1.0 W of LG 0 beam (with a waist of 30 μm). Here, R = 0.18 Hz and U = 42 μK . Each dark ring has trap frequencies 370 kHz and 4.9 kHz in the axial and radial directions, respectively. This geometry is useful for storing multiple persistent currents or creating annular Tonks–Girardeau gases.

Fig. 4.
Fig. 4.

An axial lattice of ring lattices formed by a 100 mW LG 7 beam and 100 mW LG 7 (with the typical 5 μm waist) counterpropagating with 2.6 W of LG 0 beam (with a waist of 30 μm). Here, R = 0.60 Hz and U = 140 μK . Each dark trap has an axial trap frequency of 550 kHz, and the trap is quartic rather than harmonic in the radial and azimuthal directions—useful for exploring 2D quantum gases.

Fig. 5.
Fig. 5.

An axial lattice of ring lattices of rings, all parameters as in Fig. 4, except the counterpropagating Gaussian beam has power 650 mW. Here, R = 0.15 Hz and U = 34 μK . Each dark ring has a spatially varying axial (radial) trap frequency of 260–290 kHz (4.6–15 kHz).

Fig. 6.
Fig. 6.

An incident (red, k ̲ i ) laser beam diffracts from a 2D phase grating (with a lens) into counterpropagating (green, k ̲ d ) and discarded reflected (blue, k ̲ r ) orders.

Equations (2)

Equations on this page are rendered with MathJax. Learn more.

L G = 2 1 + | | P r 2 | | π w 2 ( 1 + | | ) | | ! exp [ i s k z ( 1 + r 2 2 ( z 2 + z R 2 ) ) ] e r 2 / w 2 i θ i ω t i s ( | | + 1 ) arctan ( z / z R ) ,
( R , U ) ħ Γ β ( Δ Γ 2 , ( k B Δ Γ ) 1 ) / 8 ,

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