Abstract

We propose a lens-free nondestructive imaging method for cold atomic clouds using a Gaussian beam accompanied with phase shifting interferometry. This scheme requires no imaging lens. Hence, aberrations associated with it are completely eliminated and mechanical focusing can be avoided. Compared with the common single-beam nondestructive means, our proposed scheme lowers the energy per probe pulse delivered to the cold samples by almost three orders of magnitude, due to signal enhancement inherently provided in the two-beam configuration. Moreover, higher image resolution is attainable by magnifying the far-field interference distribution using a divergent Gaussian beam. We examine this novel lensless detection means for in situ imaging on typical cold atomic clouds under experimentally achievable conditions. Our simulations show the cloud position can be precisely determined, depending upon the cloud size and probe parameters, with an uncertainty from a few hundreds of micrometers to only a few micrometers, and the spatial resolution of the retrieved phase image can reach the diffraction limit.

© 2013 Optical Society of America

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References

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  1. M. R. Andrews, M.-O. Mewes, N. J. van Druten, D. S. Durfee, D. M. Kurn, W. Ketterle, “Direct, nondestructive observation of a Bose condensate,” Science 273, 84–87 (1996).
    [CrossRef]
  2. C. C. Bradley, C. A. Sackett, R. G. Hulet, “Bose–Einstein condensation of lithium: observation of limited condensate number,” Phys. Rev. Lett. 78, 985–989 (1997).
    [CrossRef]
  3. S. Kadlecek, J. Sebby, R. Newell, T. G. Walker, “Nondestructive spatial heterodyne imaging of cold atoms,” Opt. Lett. 26, 137–139 (2001).
    [CrossRef]
  4. T.-P. Ku, C.-Y. Huang, B.-W. Shiau, D.-J. Han, “Phase shifting interferometry of cold atoms,” Opt. Express 19, 3730–3741 (2011).
    [CrossRef]
  5. E. Cuche, P. Marquet, C. Depeursinge, “Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,” Appl. Opt. 38, 6994–7001 (1999).
    [CrossRef]
  6. C. G. J. Ollinger, “A waveguide-based lens-less x-ray microscope,” Ph.D. dissertation (University of Gottingen, Germany, 2006).
  7. L. D. Turner, “Holographic imaging of cold atoms,” Ph.D. dissertation (University of Melbourne, Australia, 2004).
  8. I. Yamaguchi, T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22, 1268–1270 (1997).
    [CrossRef]
  9. U. Schnars, W. Jueptner, Digital Holography (Springer-Verlag, 2005).
  10. L. D. Turner, K. F. E. M. Domen, R. E. Scholten, “Diffraction-contrast imaging of cold atoms,” Phys. Rev. A 72, 031403(R) (2005).
    [CrossRef]
  11. E. B. Champagne, N. G. Massey, “Resolution in holography,” Appl. Opt. 8, 1879–1885 (1969).
    [CrossRef]
  12. S. Lagomarsino, A. Cedola, P. Cloetens, S. Di Fonzo, W. Jark, G. Soullie, C. Riekel, “Phase contrast hard x-ray microscopy with submicron resolution,” Appl. Phys. Lett. 71, 2557–2560 (1997).
    [CrossRef]
  13. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, 1988).
  14. A plane wave is a special case of a Gaussian beam when the beam size is large.
  15. P. P. Banerjee, T.-C. Poo, eds. Principles of Applied Optics, (Irwin, 1991).
  16. C. Fuhse, C. Ollinger, T. Salditt, “Waveguide-based off-axis holography with hard x rays,” Phys. Rev. Lett. 97, 254801 (2006).
    [CrossRef]
  17. H.-S. Chen, “Lensless phase-shifting imaging of cold atoms,” Masters thesis (National Chung Cheng University, Taiwan, 2012).
  18. The DC phase π comes from the arctangent term in Eq. (5).
  19. F. L. Pedrotti, L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, 1993).
  20. S. F. Ray, “The geometry of image formation,” in The Manual of Photography, 8th ed. (Focal, 1988).
  21. K. Nelson, X. Li, D. Weiss, “Imaging single atoms in a three-dimensional array,” Nat. Phys. 3, 556–560 (2007).
    [CrossRef]
  22. Y.-J. Lin, I. Teper, C. Chin, V. Vuletic, “Impact of the Casimir–Polder potential and Johnson noise on Bose–Einstein condensate stability near surfaces,” Phys. Rev. Lett. 92, 050404 (2004).
    [CrossRef]
  23. P. Treutlein, P. Hommelhoff, T. Steinmetz, T. W. Hänsch, J. Reichel, “Coherence in microchip traps,” Phys. Rev. Lett. 92, 203005 (2004).
    [CrossRef]
  24. C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell, C. E. Wieman, “Production of two overlapping Bose–Einstein condensates by sympathetic cooling,” Phys. Rev. Lett. 78, 586–589 (1997).
    [CrossRef]
  25. A. S. Zibrov, M. D. Lukin, L. Hollberg, D. E. Nikonov, M. O. Scully, H. G. Robinson, V. L. Velichansky, “Experimental demonstration of enhanced index of refraction via quantum coherence in Rb,” Phys. Rev. Lett. 76, 3935–3938 (1996).
    [CrossRef]

2011 (1)

2007 (1)

K. Nelson, X. Li, D. Weiss, “Imaging single atoms in a three-dimensional array,” Nat. Phys. 3, 556–560 (2007).
[CrossRef]

2006 (1)

C. Fuhse, C. Ollinger, T. Salditt, “Waveguide-based off-axis holography with hard x rays,” Phys. Rev. Lett. 97, 254801 (2006).
[CrossRef]

2005 (1)

L. D. Turner, K. F. E. M. Domen, R. E. Scholten, “Diffraction-contrast imaging of cold atoms,” Phys. Rev. A 72, 031403(R) (2005).
[CrossRef]

2004 (2)

Y.-J. Lin, I. Teper, C. Chin, V. Vuletic, “Impact of the Casimir–Polder potential and Johnson noise on Bose–Einstein condensate stability near surfaces,” Phys. Rev. Lett. 92, 050404 (2004).
[CrossRef]

P. Treutlein, P. Hommelhoff, T. Steinmetz, T. W. Hänsch, J. Reichel, “Coherence in microchip traps,” Phys. Rev. Lett. 92, 203005 (2004).
[CrossRef]

2001 (1)

1999 (1)

1997 (4)

I. Yamaguchi, T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22, 1268–1270 (1997).
[CrossRef]

C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell, C. E. Wieman, “Production of two overlapping Bose–Einstein condensates by sympathetic cooling,” Phys. Rev. Lett. 78, 586–589 (1997).
[CrossRef]

S. Lagomarsino, A. Cedola, P. Cloetens, S. Di Fonzo, W. Jark, G. Soullie, C. Riekel, “Phase contrast hard x-ray microscopy with submicron resolution,” Appl. Phys. Lett. 71, 2557–2560 (1997).
[CrossRef]

C. C. Bradley, C. A. Sackett, R. G. Hulet, “Bose–Einstein condensation of lithium: observation of limited condensate number,” Phys. Rev. Lett. 78, 985–989 (1997).
[CrossRef]

1996 (2)

M. R. Andrews, M.-O. Mewes, N. J. van Druten, D. S. Durfee, D. M. Kurn, W. Ketterle, “Direct, nondestructive observation of a Bose condensate,” Science 273, 84–87 (1996).
[CrossRef]

A. S. Zibrov, M. D. Lukin, L. Hollberg, D. E. Nikonov, M. O. Scully, H. G. Robinson, V. L. Velichansky, “Experimental demonstration of enhanced index of refraction via quantum coherence in Rb,” Phys. Rev. Lett. 76, 3935–3938 (1996).
[CrossRef]

1969 (1)

Andrews, M. R.

M. R. Andrews, M.-O. Mewes, N. J. van Druten, D. S. Durfee, D. M. Kurn, W. Ketterle, “Direct, nondestructive observation of a Bose condensate,” Science 273, 84–87 (1996).
[CrossRef]

Bradley, C. C.

C. C. Bradley, C. A. Sackett, R. G. Hulet, “Bose–Einstein condensation of lithium: observation of limited condensate number,” Phys. Rev. Lett. 78, 985–989 (1997).
[CrossRef]

Burt, E. A.

C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell, C. E. Wieman, “Production of two overlapping Bose–Einstein condensates by sympathetic cooling,” Phys. Rev. Lett. 78, 586–589 (1997).
[CrossRef]

Cedola, A.

S. Lagomarsino, A. Cedola, P. Cloetens, S. Di Fonzo, W. Jark, G. Soullie, C. Riekel, “Phase contrast hard x-ray microscopy with submicron resolution,” Appl. Phys. Lett. 71, 2557–2560 (1997).
[CrossRef]

Champagne, E. B.

Chen, H.-S.

H.-S. Chen, “Lensless phase-shifting imaging of cold atoms,” Masters thesis (National Chung Cheng University, Taiwan, 2012).

Chin, C.

Y.-J. Lin, I. Teper, C. Chin, V. Vuletic, “Impact of the Casimir–Polder potential and Johnson noise on Bose–Einstein condensate stability near surfaces,” Phys. Rev. Lett. 92, 050404 (2004).
[CrossRef]

Cloetens, P.

S. Lagomarsino, A. Cedola, P. Cloetens, S. Di Fonzo, W. Jark, G. Soullie, C. Riekel, “Phase contrast hard x-ray microscopy with submicron resolution,” Appl. Phys. Lett. 71, 2557–2560 (1997).
[CrossRef]

Cornell, E. A.

C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell, C. E. Wieman, “Production of two overlapping Bose–Einstein condensates by sympathetic cooling,” Phys. Rev. Lett. 78, 586–589 (1997).
[CrossRef]

Cuche, E.

Depeursinge, C.

Di Fonzo, S.

S. Lagomarsino, A. Cedola, P. Cloetens, S. Di Fonzo, W. Jark, G. Soullie, C. Riekel, “Phase contrast hard x-ray microscopy with submicron resolution,” Appl. Phys. Lett. 71, 2557–2560 (1997).
[CrossRef]

Domen, K. F. E. M.

L. D. Turner, K. F. E. M. Domen, R. E. Scholten, “Diffraction-contrast imaging of cold atoms,” Phys. Rev. A 72, 031403(R) (2005).
[CrossRef]

Durfee, D. S.

M. R. Andrews, M.-O. Mewes, N. J. van Druten, D. S. Durfee, D. M. Kurn, W. Ketterle, “Direct, nondestructive observation of a Bose condensate,” Science 273, 84–87 (1996).
[CrossRef]

Fuhse, C.

C. Fuhse, C. Ollinger, T. Salditt, “Waveguide-based off-axis holography with hard x rays,” Phys. Rev. Lett. 97, 254801 (2006).
[CrossRef]

Ghrist, R. W.

C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell, C. E. Wieman, “Production of two overlapping Bose–Einstein condensates by sympathetic cooling,” Phys. Rev. Lett. 78, 586–589 (1997).
[CrossRef]

Han, D.-J.

Hänsch, T. W.

P. Treutlein, P. Hommelhoff, T. Steinmetz, T. W. Hänsch, J. Reichel, “Coherence in microchip traps,” Phys. Rev. Lett. 92, 203005 (2004).
[CrossRef]

Hollberg, L.

A. S. Zibrov, M. D. Lukin, L. Hollberg, D. E. Nikonov, M. O. Scully, H. G. Robinson, V. L. Velichansky, “Experimental demonstration of enhanced index of refraction via quantum coherence in Rb,” Phys. Rev. Lett. 76, 3935–3938 (1996).
[CrossRef]

Hommelhoff, P.

P. Treutlein, P. Hommelhoff, T. Steinmetz, T. W. Hänsch, J. Reichel, “Coherence in microchip traps,” Phys. Rev. Lett. 92, 203005 (2004).
[CrossRef]

Huang, C.-Y.

Hulet, R. G.

C. C. Bradley, C. A. Sackett, R. G. Hulet, “Bose–Einstein condensation of lithium: observation of limited condensate number,” Phys. Rev. Lett. 78, 985–989 (1997).
[CrossRef]

Jark, W.

S. Lagomarsino, A. Cedola, P. Cloetens, S. Di Fonzo, W. Jark, G. Soullie, C. Riekel, “Phase contrast hard x-ray microscopy with submicron resolution,” Appl. Phys. Lett. 71, 2557–2560 (1997).
[CrossRef]

Jueptner, W.

U. Schnars, W. Jueptner, Digital Holography (Springer-Verlag, 2005).

Kadlecek, S.

Ketterle, W.

M. R. Andrews, M.-O. Mewes, N. J. van Druten, D. S. Durfee, D. M. Kurn, W. Ketterle, “Direct, nondestructive observation of a Bose condensate,” Science 273, 84–87 (1996).
[CrossRef]

Ku, T.-P.

Kurn, D. M.

M. R. Andrews, M.-O. Mewes, N. J. van Druten, D. S. Durfee, D. M. Kurn, W. Ketterle, “Direct, nondestructive observation of a Bose condensate,” Science 273, 84–87 (1996).
[CrossRef]

Lagomarsino, S.

S. Lagomarsino, A. Cedola, P. Cloetens, S. Di Fonzo, W. Jark, G. Soullie, C. Riekel, “Phase contrast hard x-ray microscopy with submicron resolution,” Appl. Phys. Lett. 71, 2557–2560 (1997).
[CrossRef]

Li, X.

K. Nelson, X. Li, D. Weiss, “Imaging single atoms in a three-dimensional array,” Nat. Phys. 3, 556–560 (2007).
[CrossRef]

Lin, Y.-J.

Y.-J. Lin, I. Teper, C. Chin, V. Vuletic, “Impact of the Casimir–Polder potential and Johnson noise on Bose–Einstein condensate stability near surfaces,” Phys. Rev. Lett. 92, 050404 (2004).
[CrossRef]

Lukin, M. D.

A. S. Zibrov, M. D. Lukin, L. Hollberg, D. E. Nikonov, M. O. Scully, H. G. Robinson, V. L. Velichansky, “Experimental demonstration of enhanced index of refraction via quantum coherence in Rb,” Phys. Rev. Lett. 76, 3935–3938 (1996).
[CrossRef]

Marquet, P.

Massey, N. G.

Mewes, M.-O.

M. R. Andrews, M.-O. Mewes, N. J. van Druten, D. S. Durfee, D. M. Kurn, W. Ketterle, “Direct, nondestructive observation of a Bose condensate,” Science 273, 84–87 (1996).
[CrossRef]

Myatt, C. J.

C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell, C. E. Wieman, “Production of two overlapping Bose–Einstein condensates by sympathetic cooling,” Phys. Rev. Lett. 78, 586–589 (1997).
[CrossRef]

Nelson, K.

K. Nelson, X. Li, D. Weiss, “Imaging single atoms in a three-dimensional array,” Nat. Phys. 3, 556–560 (2007).
[CrossRef]

Newell, R.

Nikonov, D. E.

A. S. Zibrov, M. D. Lukin, L. Hollberg, D. E. Nikonov, M. O. Scully, H. G. Robinson, V. L. Velichansky, “Experimental demonstration of enhanced index of refraction via quantum coherence in Rb,” Phys. Rev. Lett. 76, 3935–3938 (1996).
[CrossRef]

Ollinger, C.

C. Fuhse, C. Ollinger, T. Salditt, “Waveguide-based off-axis holography with hard x rays,” Phys. Rev. Lett. 97, 254801 (2006).
[CrossRef]

Ollinger, C. G. J.

C. G. J. Ollinger, “A waveguide-based lens-less x-ray microscope,” Ph.D. dissertation (University of Gottingen, Germany, 2006).

Pedrotti, F. L.

F. L. Pedrotti, L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, 1993).

Pedrotti, L. S.

F. L. Pedrotti, L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, 1993).

Ray, S. F.

S. F. Ray, “The geometry of image formation,” in The Manual of Photography, 8th ed. (Focal, 1988).

Reichel, J.

P. Treutlein, P. Hommelhoff, T. Steinmetz, T. W. Hänsch, J. Reichel, “Coherence in microchip traps,” Phys. Rev. Lett. 92, 203005 (2004).
[CrossRef]

Riekel, C.

S. Lagomarsino, A. Cedola, P. Cloetens, S. Di Fonzo, W. Jark, G. Soullie, C. Riekel, “Phase contrast hard x-ray microscopy with submicron resolution,” Appl. Phys. Lett. 71, 2557–2560 (1997).
[CrossRef]

Robinson, H. G.

A. S. Zibrov, M. D. Lukin, L. Hollberg, D. E. Nikonov, M. O. Scully, H. G. Robinson, V. L. Velichansky, “Experimental demonstration of enhanced index of refraction via quantum coherence in Rb,” Phys. Rev. Lett. 76, 3935–3938 (1996).
[CrossRef]

Sackett, C. A.

C. C. Bradley, C. A. Sackett, R. G. Hulet, “Bose–Einstein condensation of lithium: observation of limited condensate number,” Phys. Rev. Lett. 78, 985–989 (1997).
[CrossRef]

Salditt, T.

C. Fuhse, C. Ollinger, T. Salditt, “Waveguide-based off-axis holography with hard x rays,” Phys. Rev. Lett. 97, 254801 (2006).
[CrossRef]

Schnars, U.

U. Schnars, W. Jueptner, Digital Holography (Springer-Verlag, 2005).

Scholten, R. E.

L. D. Turner, K. F. E. M. Domen, R. E. Scholten, “Diffraction-contrast imaging of cold atoms,” Phys. Rev. A 72, 031403(R) (2005).
[CrossRef]

Scully, M. O.

A. S. Zibrov, M. D. Lukin, L. Hollberg, D. E. Nikonov, M. O. Scully, H. G. Robinson, V. L. Velichansky, “Experimental demonstration of enhanced index of refraction via quantum coherence in Rb,” Phys. Rev. Lett. 76, 3935–3938 (1996).
[CrossRef]

Sebby, J.

Shiau, B.-W.

Soullie, G.

S. Lagomarsino, A. Cedola, P. Cloetens, S. Di Fonzo, W. Jark, G. Soullie, C. Riekel, “Phase contrast hard x-ray microscopy with submicron resolution,” Appl. Phys. Lett. 71, 2557–2560 (1997).
[CrossRef]

Steinmetz, T.

P. Treutlein, P. Hommelhoff, T. Steinmetz, T. W. Hänsch, J. Reichel, “Coherence in microchip traps,” Phys. Rev. Lett. 92, 203005 (2004).
[CrossRef]

Teper, I.

Y.-J. Lin, I. Teper, C. Chin, V. Vuletic, “Impact of the Casimir–Polder potential and Johnson noise on Bose–Einstein condensate stability near surfaces,” Phys. Rev. Lett. 92, 050404 (2004).
[CrossRef]

Treutlein, P.

P. Treutlein, P. Hommelhoff, T. Steinmetz, T. W. Hänsch, J. Reichel, “Coherence in microchip traps,” Phys. Rev. Lett. 92, 203005 (2004).
[CrossRef]

Turner, L. D.

L. D. Turner, K. F. E. M. Domen, R. E. Scholten, “Diffraction-contrast imaging of cold atoms,” Phys. Rev. A 72, 031403(R) (2005).
[CrossRef]

L. D. Turner, “Holographic imaging of cold atoms,” Ph.D. dissertation (University of Melbourne, Australia, 2004).

van Druten, N. J.

M. R. Andrews, M.-O. Mewes, N. J. van Druten, D. S. Durfee, D. M. Kurn, W. Ketterle, “Direct, nondestructive observation of a Bose condensate,” Science 273, 84–87 (1996).
[CrossRef]

Velichansky, V. L.

A. S. Zibrov, M. D. Lukin, L. Hollberg, D. E. Nikonov, M. O. Scully, H. G. Robinson, V. L. Velichansky, “Experimental demonstration of enhanced index of refraction via quantum coherence in Rb,” Phys. Rev. Lett. 76, 3935–3938 (1996).
[CrossRef]

Vuletic, V.

Y.-J. Lin, I. Teper, C. Chin, V. Vuletic, “Impact of the Casimir–Polder potential and Johnson noise on Bose–Einstein condensate stability near surfaces,” Phys. Rev. Lett. 92, 050404 (2004).
[CrossRef]

Walker, T. G.

Weiss, D.

K. Nelson, X. Li, D. Weiss, “Imaging single atoms in a three-dimensional array,” Nat. Phys. 3, 556–560 (2007).
[CrossRef]

Wieman, C. E.

C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell, C. E. Wieman, “Production of two overlapping Bose–Einstein condensates by sympathetic cooling,” Phys. Rev. Lett. 78, 586–589 (1997).
[CrossRef]

Yamaguchi, I.

Yariv, A.

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, 1988).

Zhang, T.

Zibrov, A. S.

A. S. Zibrov, M. D. Lukin, L. Hollberg, D. E. Nikonov, M. O. Scully, H. G. Robinson, V. L. Velichansky, “Experimental demonstration of enhanced index of refraction via quantum coherence in Rb,” Phys. Rev. Lett. 76, 3935–3938 (1996).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

S. Lagomarsino, A. Cedola, P. Cloetens, S. Di Fonzo, W. Jark, G. Soullie, C. Riekel, “Phase contrast hard x-ray microscopy with submicron resolution,” Appl. Phys. Lett. 71, 2557–2560 (1997).
[CrossRef]

Nat. Phys. (1)

K. Nelson, X. Li, D. Weiss, “Imaging single atoms in a three-dimensional array,” Nat. Phys. 3, 556–560 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (1)

L. D. Turner, K. F. E. M. Domen, R. E. Scholten, “Diffraction-contrast imaging of cold atoms,” Phys. Rev. A 72, 031403(R) (2005).
[CrossRef]

Phys. Rev. Lett. (6)

Y.-J. Lin, I. Teper, C. Chin, V. Vuletic, “Impact of the Casimir–Polder potential and Johnson noise on Bose–Einstein condensate stability near surfaces,” Phys. Rev. Lett. 92, 050404 (2004).
[CrossRef]

P. Treutlein, P. Hommelhoff, T. Steinmetz, T. W. Hänsch, J. Reichel, “Coherence in microchip traps,” Phys. Rev. Lett. 92, 203005 (2004).
[CrossRef]

C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell, C. E. Wieman, “Production of two overlapping Bose–Einstein condensates by sympathetic cooling,” Phys. Rev. Lett. 78, 586–589 (1997).
[CrossRef]

A. S. Zibrov, M. D. Lukin, L. Hollberg, D. E. Nikonov, M. O. Scully, H. G. Robinson, V. L. Velichansky, “Experimental demonstration of enhanced index of refraction via quantum coherence in Rb,” Phys. Rev. Lett. 76, 3935–3938 (1996).
[CrossRef]

C. Fuhse, C. Ollinger, T. Salditt, “Waveguide-based off-axis holography with hard x rays,” Phys. Rev. Lett. 97, 254801 (2006).
[CrossRef]

C. C. Bradley, C. A. Sackett, R. G. Hulet, “Bose–Einstein condensation of lithium: observation of limited condensate number,” Phys. Rev. Lett. 78, 985–989 (1997).
[CrossRef]

Science (1)

M. R. Andrews, M.-O. Mewes, N. J. van Druten, D. S. Durfee, D. M. Kurn, W. Ketterle, “Direct, nondestructive observation of a Bose condensate,” Science 273, 84–87 (1996).
[CrossRef]

Other (10)

C. G. J. Ollinger, “A waveguide-based lens-less x-ray microscope,” Ph.D. dissertation (University of Gottingen, Germany, 2006).

L. D. Turner, “Holographic imaging of cold atoms,” Ph.D. dissertation (University of Melbourne, Australia, 2004).

U. Schnars, W. Jueptner, Digital Holography (Springer-Verlag, 2005).

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, 1988).

A plane wave is a special case of a Gaussian beam when the beam size is large.

P. P. Banerjee, T.-C. Poo, eds. Principles of Applied Optics, (Irwin, 1991).

H.-S. Chen, “Lensless phase-shifting imaging of cold atoms,” Masters thesis (National Chung Cheng University, Taiwan, 2012).

The DC phase π comes from the arctangent term in Eq. (5).

F. L. Pedrotti, L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, 1993).

S. F. Ray, “The geometry of image formation,” in The Manual of Photography, 8th ed. (Focal, 1988).

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Figures (6)

Fig. 1.
Fig. 1.

Schematic of lensless far-field phase shifting interferometry on cold atomic cloud. The Gaussian beam waist and atomic cloud locate at positions z = 0 and z = z a , respectively. The CCD camera is set up at position z = z a + d , a distance d to the right of the cloud. A collimated phase-tuning beam is injected into the interferometer, with only a few millimeters separation from the probe beam, and does not affect the atoms in the cloud. The interference signal is sensed by a photodiode and further processed to give real-time feedback to the PZT for relative phase stabilization and control of the interferometer.

Fig. 2.
Fig. 2.

Complex field reconstruction at the CCD plane for a Rb 87 cloud of peak density 10 12 atoms / cm 3 and N = 8 × 10 6 . (a) Four simulated interference images at the CCD plane, shown in grayscale, for Φ equal to 0, π / 2 , π , and 3 π / 2 , respectively. (b) The retrieved 3D false-color phase distribution ϕ p H using the interferograms in (a). (c) The 3D intensity distribution of the probe beam, shown in false-color, at the CCD plane when atoms are loaded.

Fig. 3.
Fig. 3.

(a) 3D false-color phase image at the cloud plane reconstructed from Figs. 2(b) and 2(c). (b) One-dimensional x cut of the reconstructed phase distribution in (a), showing no discernible pedestal.

Fig. 4.
Fig. 4.

Retrieved 3D false-color phase images for clouds of peak density (a)  2 × 10 11 atoms / cm 3 and (b)  1 × 10 14 atoms / cm 3 . The peak phase is 29° in (a) and 27.1° in (b).

Fig. 5.
Fig. 5.

Two out-of-focus phase images retrieved using the same cloud parameters in Fig. 3. (a) 3D false-color phase image retrieved at a position of 7 σ z in front of the focal plane. (b) The one-dimensional cut through the center of the phase image in (a) along the x direction, showing a central peak on top of a pedestal. (c) 3D false-color phase image retrieved at a position of 7 σ z behind the focal plane. (d) The one-dimensional x cut through the center of the phase image in (c), showing a central peak on top of a inverted pedestal.

Fig. 6.
Fig. 6.

Magnitude of peak pedestal and DOF versus out-of-focus distance s . The red solid squares and blue dash line are the peak pedestal magnitudes, calculated by numerical simulation and a modified point source model, respectively. The green solid circles and black solid line are the DOF, obtained from numerical simulation and a modified point source model, respectively.

Equations (13)

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U 0 ( x , y , z a ) = A 0 · exp { i [ ϕ ( x , y ) + ϕ G ( x , y , z a ) ] } · δ ( x , y , z a ) ,
U ( x , y , z a + d ) = e i k d i λ d · U 0 ( x , y , z a ) · exp [ i k ( x x ) 2 + ( y y ) 2 2 d ] · d x d y ,
U ( x , y , z a + d ) = e i k d i λ d · A 0 exp [ i ϕ ( x , y ) + i ϕ G ( x , y , z a ) + i k ( x x ) 2 + ( y y ) 2 2 d ] · d x d y = A · exp { i [ ϕ P ( x , y , z a + d ) + ϕ G ( x , y , z a + d ) ] ,
U ( x , y , z a ) = e i k d i λ d · U ( x , y , z a + d ) · exp [ i k ( x x ) 2 + ( y y ) 2 2 d ] · d x d y .
ϕ ( x , y , z a ) = tan 1 { Im [ U ( x , y , z a ) ] Re [ U ( x , y , z a ) ] } ϕ G ( x , y , z a ) ,
U ( x , y , z a + d ) = e i k d i λ d · A 0 · exp [ i ( ϕ ( x 0 , y 0 ) + ϕ G ( x 0 , y 0 , z a ) + i k ( x x 0 ) 2 + ( y y 0 ) 2 2 d ] .
U 1 ( X , Y , Z ) = A 0 · exp [ i ( ϕ ( x 0 , y 0 ) + ϕ G ( x 0 , y 0 , z a ) ] · e i k d i λ d · e i k | Z ( z a + d ) | i λ | Z ( z a + d ) | · exp [ i k ( x x 0 ) 2 + ( y y 0 ) 2 2 d + i k ( X x ) 2 + ( Y y ) 2 2 [ Z ( z a + d ) ] ] d x d y .
U 1 ( x 0 , y 0 , z a ) = A 0 · exp ( i ϕ t ) · ( 1 λ 2 d 2 ) · α ,
U 1 ( x 0 , y 0 , z a + s ) A 0 · exp ( i ϕ t ) · ( 1 λ 2 d 2 ) · exp ( i π ) · exp ( i k x 2 + y 2 2 d · s d ) d x d y ,
θ e ( s ) = π 2 + θ , if s > 0 , = π 2 + θ , if s < 0 ,
DOF = 2 [ ( 2 σ z ) 2 + s min 2 ] 1 / 2 ,
n ( r⃗ ) = 1 + ρ ( r⃗ ) σ 0 λ 4 π i 2 Δ 1 + 4 Δ 2 ,
ϕ p H ( x , y ) = tan 1 { [ I π ( x , y ) I 0 ( x , y ) ] sin ( π ζ / 2 ) [ I π / 2 ( x , y ) I 3 π / 2 ( x , y ) ] [ I π ( x , y ) I 0 ( x , y ) ] cos ( π ζ / 2 ) } π ζ 2 ,

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