Abstract

We derive a formula for the light field of a monochromatic plane wave that is truncated and reflected by a spherical mirror. Within the scalar field approximation, our formula is valid even for deep mirrors, where the aperture radius approaches the radius of curvature. We apply this result to micro-fabricated mirrors whose size scales are in the range of tens to hundreds of wavelengths, and show that sub-wavelength focusing (full-width at half-maximum intensity) can be achieved. This opens up the possibility of scalable arrays of tightly focused optical dipole traps without the need for high-performance optical systems.

©2008 Optical Society of America

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References

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  1. N. Schlosser, G. Reymond, I. Protsenko, and P. Grangier, “Sub-poissonian loading of single atoms in a microscopic dipole trap,” Nature (London) 411, 1024–1027 (2001).
    [Crossref]
  2. N. Schlosser, G. Reymond, and P. Grangier, “Collisional Blockade in Microscopic Optical Dipole Traps,” Phys. Rev. Lett. 89, 023005 (2002).
    [Crossref] [PubMed]
  3. Y. R. P. Sortais, H. Marion, C. Tuchendler, A. M. Lance, M. Lamare, P. Fournet, C. Armellin, R. Mercier, G. Messin, A. Browaeys, and P. Grangier, “Diffraction-limited optics for single-atom manipulation,” Phys. Rev. A 75, 013406 (2007).
    [Crossref]
  4. B. Darquié, M. P. A. Jones, J. Dingjan, J. Beugnon, S. Bergamini, Y. Sortais, G. Messin, A. Browaeys, and P. Grangier, “Controlled Single-Photon Emission from a Single Trapped Two-Level Atom,” Science 309, 454–456 (2005).
    [Crossref] [PubMed]
  5. M. Weber, J. Volz, K. Saucke, C. Kurtsiefer, and H. Weinfurter, “Analysis of a single-atom dipole trap,” Phys. Rev. A 73, 043406 (2006).
    [Crossref]
  6. M. K. Tey, Z. Chen, S. A. Aljunid, B. Chng, F. Huber, G. Maslennikov, and C. Kurtsiefer, “Strong interaction between light and a single trapped atom without a cavity,” arXiv:0802.3005v3 (2008), http://uk.arxiv.org/abs/0802.3005v3.
  7. B. Lounis and M. Orrit, “Single-photon sources,” Rep. Prog. Phys. 68, 1129–1179 (2005).
    [Crossref]
  8. E. Waks, C. Santori, and Y. Yamamoto, “Security aspects of quantum key distribution with sub-Poisson light,” Phys. Rev. A 66, 042315 (2002).
    [Crossref]
  9. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature (London) 409, 46–52 (2001).
    [Crossref]
  10. M. Oxborrow and A. G. Sinclair, “Single-photon sources,” Contemp. Phys. 46, 173–206 (2005).
    [Crossref]
  11. M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B 89, 489–482 (2007).
    [Crossref]
  12. S. J. van Enk and H. J. Kimble, “Strongly focused light beams interacting with single atoms in free space,” Phys. Rev. A 63, 023809 (2001).
  13. M. K. Tey, S. A. Aljunid, F. Huber, B. Chng, Z. Chen, G. Maslennikov, and C. Kurtsiefer, “Interfacing light and single atoms with a lens,” arXiv:0804.4861v2 (2008), http://uk.arxiv.org/abs/0804.4861v2.
  14. S. Eriksson, M. Trupke, H. F. Powell, D. Sahagun, C. D. J. Sinclair, E. A. Curtis, B. E. Sauer, E. A. Hinds, Z. Moktadir, C. O. Gollasch, and M. Kraft, “Integrated optical components on atom chips,” Eur. Phys. J. D 35, 135–139 (2005).
    [Crossref]
  15. M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, “Microfabricated high-finesse optical cavity with open access and small volume,” Appl. Phys. Lett. 87, 211106 (2005).
    [Crossref]
  16. For a recent review of atom chip experiments, see J. Fortágh and C. Zimmermann, “Magnetic microtraps for ultracold atoms,” Rev. Mod. Phys. 79, 235 (2007).
    [Crossref]
  17. M. Born and E. Wolf, Principles of Optics, 7th edition, (Cambridge University Press, Cambridge,1999).
  18. The software used in this work is available from http://ab-initio.mit.edu/wiki/index.php/Meep.
  19. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, (Artech, Norwood, MA,2000).
  20. A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. Burr, “Improving accuracy by subpixel smoothing in the finite-difference time domain,” Opt. Lett. 31, 2972–2974 (2006).
    [Crossref] [PubMed]
  21. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [Crossref]
  22. G. P. Agrawal and M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
    [Crossref]

2007 (3)

Y. R. P. Sortais, H. Marion, C. Tuchendler, A. M. Lance, M. Lamare, P. Fournet, C. Armellin, R. Mercier, G. Messin, A. Browaeys, and P. Grangier, “Diffraction-limited optics for single-atom manipulation,” Phys. Rev. A 75, 013406 (2007).
[Crossref]

M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B 89, 489–482 (2007).
[Crossref]

For a recent review of atom chip experiments, see J. Fortágh and C. Zimmermann, “Magnetic microtraps for ultracold atoms,” Rev. Mod. Phys. 79, 235 (2007).
[Crossref]

2006 (2)

2005 (5)

B. Lounis and M. Orrit, “Single-photon sources,” Rep. Prog. Phys. 68, 1129–1179 (2005).
[Crossref]

B. Darquié, M. P. A. Jones, J. Dingjan, J. Beugnon, S. Bergamini, Y. Sortais, G. Messin, A. Browaeys, and P. Grangier, “Controlled Single-Photon Emission from a Single Trapped Two-Level Atom,” Science 309, 454–456 (2005).
[Crossref] [PubMed]

M. Oxborrow and A. G. Sinclair, “Single-photon sources,” Contemp. Phys. 46, 173–206 (2005).
[Crossref]

S. Eriksson, M. Trupke, H. F. Powell, D. Sahagun, C. D. J. Sinclair, E. A. Curtis, B. E. Sauer, E. A. Hinds, Z. Moktadir, C. O. Gollasch, and M. Kraft, “Integrated optical components on atom chips,” Eur. Phys. J. D 35, 135–139 (2005).
[Crossref]

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, “Microfabricated high-finesse optical cavity with open access and small volume,” Appl. Phys. Lett. 87, 211106 (2005).
[Crossref]

2002 (2)

N. Schlosser, G. Reymond, and P. Grangier, “Collisional Blockade in Microscopic Optical Dipole Traps,” Phys. Rev. Lett. 89, 023005 (2002).
[Crossref] [PubMed]

E. Waks, C. Santori, and Y. Yamamoto, “Security aspects of quantum key distribution with sub-Poisson light,” Phys. Rev. A 66, 042315 (2002).
[Crossref]

2001 (3)

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature (London) 409, 46–52 (2001).
[Crossref]

N. Schlosser, G. Reymond, I. Protsenko, and P. Grangier, “Sub-poissonian loading of single atoms in a microscopic dipole trap,” Nature (London) 411, 1024–1027 (2001).
[Crossref]

S. J. van Enk and H. J. Kimble, “Strongly focused light beams interacting with single atoms in free space,” Phys. Rev. A 63, 023809 (2001).

1983 (1)

G. P. Agrawal and M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[Crossref]

1975 (1)

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

Agrawal, G. P.

G. P. Agrawal and M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[Crossref]

Aljunid, S. A.

M. K. Tey, Z. Chen, S. A. Aljunid, B. Chng, F. Huber, G. Maslennikov, and C. Kurtsiefer, “Strong interaction between light and a single trapped atom without a cavity,” arXiv:0802.3005v3 (2008), http://uk.arxiv.org/abs/0802.3005v3.

M. K. Tey, S. A. Aljunid, F. Huber, B. Chng, Z. Chen, G. Maslennikov, and C. Kurtsiefer, “Interfacing light and single atoms with a lens,” arXiv:0804.4861v2 (2008), http://uk.arxiv.org/abs/0804.4861v2.

Armellin, C.

Y. R. P. Sortais, H. Marion, C. Tuchendler, A. M. Lance, M. Lamare, P. Fournet, C. Armellin, R. Mercier, G. Messin, A. Browaeys, and P. Grangier, “Diffraction-limited optics for single-atom manipulation,” Phys. Rev. A 75, 013406 (2007).
[Crossref]

Bergamini, S.

B. Darquié, M. P. A. Jones, J. Dingjan, J. Beugnon, S. Bergamini, Y. Sortais, G. Messin, A. Browaeys, and P. Grangier, “Controlled Single-Photon Emission from a Single Trapped Two-Level Atom,” Science 309, 454–456 (2005).
[Crossref] [PubMed]

Bermel, P.

Beugnon, J.

B. Darquié, M. P. A. Jones, J. Dingjan, J. Beugnon, S. Bergamini, Y. Sortais, G. Messin, A. Browaeys, and P. Grangier, “Controlled Single-Photon Emission from a Single Trapped Two-Level Atom,” Science 309, 454–456 (2005).
[Crossref] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th edition, (Cambridge University Press, Cambridge,1999).

Browaeys, A.

Y. R. P. Sortais, H. Marion, C. Tuchendler, A. M. Lance, M. Lamare, P. Fournet, C. Armellin, R. Mercier, G. Messin, A. Browaeys, and P. Grangier, “Diffraction-limited optics for single-atom manipulation,” Phys. Rev. A 75, 013406 (2007).
[Crossref]

B. Darquié, M. P. A. Jones, J. Dingjan, J. Beugnon, S. Bergamini, Y. Sortais, G. Messin, A. Browaeys, and P. Grangier, “Controlled Single-Photon Emission from a Single Trapped Two-Level Atom,” Science 309, 454–456 (2005).
[Crossref] [PubMed]

Burr, G.

Chen, Z.

M. K. Tey, Z. Chen, S. A. Aljunid, B. Chng, F. Huber, G. Maslennikov, and C. Kurtsiefer, “Strong interaction between light and a single trapped atom without a cavity,” arXiv:0802.3005v3 (2008), http://uk.arxiv.org/abs/0802.3005v3.

M. K. Tey, S. A. Aljunid, F. Huber, B. Chng, Z. Chen, G. Maslennikov, and C. Kurtsiefer, “Interfacing light and single atoms with a lens,” arXiv:0804.4861v2 (2008), http://uk.arxiv.org/abs/0804.4861v2.

Chng, B.

M. K. Tey, S. A. Aljunid, F. Huber, B. Chng, Z. Chen, G. Maslennikov, and C. Kurtsiefer, “Interfacing light and single atoms with a lens,” arXiv:0804.4861v2 (2008), http://uk.arxiv.org/abs/0804.4861v2.

M. K. Tey, Z. Chen, S. A. Aljunid, B. Chng, F. Huber, G. Maslennikov, and C. Kurtsiefer, “Strong interaction between light and a single trapped atom without a cavity,” arXiv:0802.3005v3 (2008), http://uk.arxiv.org/abs/0802.3005v3.

Curtis, E. A.

S. Eriksson, M. Trupke, H. F. Powell, D. Sahagun, C. D. J. Sinclair, E. A. Curtis, B. E. Sauer, E. A. Hinds, Z. Moktadir, C. O. Gollasch, and M. Kraft, “Integrated optical components on atom chips,” Eur. Phys. J. D 35, 135–139 (2005).
[Crossref]

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, “Microfabricated high-finesse optical cavity with open access and small volume,” Appl. Phys. Lett. 87, 211106 (2005).
[Crossref]

Darquié, B.

B. Darquié, M. P. A. Jones, J. Dingjan, J. Beugnon, S. Bergamini, Y. Sortais, G. Messin, A. Browaeys, and P. Grangier, “Controlled Single-Photon Emission from a Single Trapped Two-Level Atom,” Science 309, 454–456 (2005).
[Crossref] [PubMed]

Dingjan, J.

B. Darquié, M. P. A. Jones, J. Dingjan, J. Beugnon, S. Bergamini, Y. Sortais, G. Messin, A. Browaeys, and P. Grangier, “Controlled Single-Photon Emission from a Single Trapped Two-Level Atom,” Science 309, 454–456 (2005).
[Crossref] [PubMed]

Eriksson, S.

S. Eriksson, M. Trupke, H. F. Powell, D. Sahagun, C. D. J. Sinclair, E. A. Curtis, B. E. Sauer, E. A. Hinds, Z. Moktadir, C. O. Gollasch, and M. Kraft, “Integrated optical components on atom chips,” Eur. Phys. J. D 35, 135–139 (2005).
[Crossref]

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, “Microfabricated high-finesse optical cavity with open access and small volume,” Appl. Phys. Lett. 87, 211106 (2005).
[Crossref]

Farjadpour, A.

Fortágh, J.

For a recent review of atom chip experiments, see J. Fortágh and C. Zimmermann, “Magnetic microtraps for ultracold atoms,” Rev. Mod. Phys. 79, 235 (2007).
[Crossref]

Fournet, P.

Y. R. P. Sortais, H. Marion, C. Tuchendler, A. M. Lance, M. Lamare, P. Fournet, C. Armellin, R. Mercier, G. Messin, A. Browaeys, and P. Grangier, “Diffraction-limited optics for single-atom manipulation,” Phys. Rev. A 75, 013406 (2007).
[Crossref]

Gollasch, C. O.

S. Eriksson, M. Trupke, H. F. Powell, D. Sahagun, C. D. J. Sinclair, E. A. Curtis, B. E. Sauer, E. A. Hinds, Z. Moktadir, C. O. Gollasch, and M. Kraft, “Integrated optical components on atom chips,” Eur. Phys. J. D 35, 135–139 (2005).
[Crossref]

Grangier, P.

Y. R. P. Sortais, H. Marion, C. Tuchendler, A. M. Lance, M. Lamare, P. Fournet, C. Armellin, R. Mercier, G. Messin, A. Browaeys, and P. Grangier, “Diffraction-limited optics for single-atom manipulation,” Phys. Rev. A 75, 013406 (2007).
[Crossref]

B. Darquié, M. P. A. Jones, J. Dingjan, J. Beugnon, S. Bergamini, Y. Sortais, G. Messin, A. Browaeys, and P. Grangier, “Controlled Single-Photon Emission from a Single Trapped Two-Level Atom,” Science 309, 454–456 (2005).
[Crossref] [PubMed]

N. Schlosser, G. Reymond, and P. Grangier, “Collisional Blockade in Microscopic Optical Dipole Traps,” Phys. Rev. Lett. 89, 023005 (2002).
[Crossref] [PubMed]

N. Schlosser, G. Reymond, I. Protsenko, and P. Grangier, “Sub-poissonian loading of single atoms in a microscopic dipole trap,” Nature (London) 411, 1024–1027 (2001).
[Crossref]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, (Artech, Norwood, MA,2000).

Hinds, E. A.

S. Eriksson, M. Trupke, H. F. Powell, D. Sahagun, C. D. J. Sinclair, E. A. Curtis, B. E. Sauer, E. A. Hinds, Z. Moktadir, C. O. Gollasch, and M. Kraft, “Integrated optical components on atom chips,” Eur. Phys. J. D 35, 135–139 (2005).
[Crossref]

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, “Microfabricated high-finesse optical cavity with open access and small volume,” Appl. Phys. Lett. 87, 211106 (2005).
[Crossref]

Huber, F.

M. K. Tey, S. A. Aljunid, F. Huber, B. Chng, Z. Chen, G. Maslennikov, and C. Kurtsiefer, “Interfacing light and single atoms with a lens,” arXiv:0804.4861v2 (2008), http://uk.arxiv.org/abs/0804.4861v2.

M. K. Tey, Z. Chen, S. A. Aljunid, B. Chng, F. Huber, G. Maslennikov, and C. Kurtsiefer, “Strong interaction between light and a single trapped atom without a cavity,” arXiv:0802.3005v3 (2008), http://uk.arxiv.org/abs/0802.3005v3.

Ibanescu, M.

Joannopoulos, J. D.

Johnson, S. G.

Jones, M. P. A.

B. Darquié, M. P. A. Jones, J. Dingjan, J. Beugnon, S. Bergamini, Y. Sortais, G. Messin, A. Browaeys, and P. Grangier, “Controlled Single-Photon Emission from a Single Trapped Two-Level Atom,” Science 309, 454–456 (2005).
[Crossref] [PubMed]

Kimble, H. J.

S. J. van Enk and H. J. Kimble, “Strongly focused light beams interacting with single atoms in free space,” Phys. Rev. A 63, 023809 (2001).

Knill, E.

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature (London) 409, 46–52 (2001).
[Crossref]

Konermann, H.

M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B 89, 489–482 (2007).
[Crossref]

Kraft, M.

S. Eriksson, M. Trupke, H. F. Powell, D. Sahagun, C. D. J. Sinclair, E. A. Curtis, B. E. Sauer, E. A. Hinds, Z. Moktadir, C. O. Gollasch, and M. Kraft, “Integrated optical components on atom chips,” Eur. Phys. J. D 35, 135–139 (2005).
[Crossref]

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, “Microfabricated high-finesse optical cavity with open access and small volume,” Appl. Phys. Lett. 87, 211106 (2005).
[Crossref]

Kukharenka, E.

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, “Microfabricated high-finesse optical cavity with open access and small volume,” Appl. Phys. Lett. 87, 211106 (2005).
[Crossref]

Kurtsiefer, C.

M. Weber, J. Volz, K. Saucke, C. Kurtsiefer, and H. Weinfurter, “Analysis of a single-atom dipole trap,” Phys. Rev. A 73, 043406 (2006).
[Crossref]

M. K. Tey, Z. Chen, S. A. Aljunid, B. Chng, F. Huber, G. Maslennikov, and C. Kurtsiefer, “Strong interaction between light and a single trapped atom without a cavity,” arXiv:0802.3005v3 (2008), http://uk.arxiv.org/abs/0802.3005v3.

M. K. Tey, S. A. Aljunid, F. Huber, B. Chng, Z. Chen, G. Maslennikov, and C. Kurtsiefer, “Interfacing light and single atoms with a lens,” arXiv:0804.4861v2 (2008), http://uk.arxiv.org/abs/0804.4861v2.

Laflamme, R.

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature (London) 409, 46–52 (2001).
[Crossref]

Lamare, M.

Y. R. P. Sortais, H. Marion, C. Tuchendler, A. M. Lance, M. Lamare, P. Fournet, C. Armellin, R. Mercier, G. Messin, A. Browaeys, and P. Grangier, “Diffraction-limited optics for single-atom manipulation,” Phys. Rev. A 75, 013406 (2007).
[Crossref]

Lance, A. M.

Y. R. P. Sortais, H. Marion, C. Tuchendler, A. M. Lance, M. Lamare, P. Fournet, C. Armellin, R. Mercier, G. Messin, A. Browaeys, and P. Grangier, “Diffraction-limited optics for single-atom manipulation,” Phys. Rev. A 75, 013406 (2007).
[Crossref]

Lax, M.

G. P. Agrawal and M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[Crossref]

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

Leuchs, G.

M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B 89, 489–482 (2007).
[Crossref]

Lindlein, N.

M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B 89, 489–482 (2007).
[Crossref]

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

Lounis, B.

B. Lounis and M. Orrit, “Single-photon sources,” Rep. Prog. Phys. 68, 1129–1179 (2005).
[Crossref]

Maiwald, R.

M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B 89, 489–482 (2007).
[Crossref]

Marion, H.

Y. R. P. Sortais, H. Marion, C. Tuchendler, A. M. Lance, M. Lamare, P. Fournet, C. Armellin, R. Mercier, G. Messin, A. Browaeys, and P. Grangier, “Diffraction-limited optics for single-atom manipulation,” Phys. Rev. A 75, 013406 (2007).
[Crossref]

Maslennikov, G.

M. K. Tey, S. A. Aljunid, F. Huber, B. Chng, Z. Chen, G. Maslennikov, and C. Kurtsiefer, “Interfacing light and single atoms with a lens,” arXiv:0804.4861v2 (2008), http://uk.arxiv.org/abs/0804.4861v2.

M. K. Tey, Z. Chen, S. A. Aljunid, B. Chng, F. Huber, G. Maslennikov, and C. Kurtsiefer, “Strong interaction between light and a single trapped atom without a cavity,” arXiv:0802.3005v3 (2008), http://uk.arxiv.org/abs/0802.3005v3.

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

Mercier, R.

Y. R. P. Sortais, H. Marion, C. Tuchendler, A. M. Lance, M. Lamare, P. Fournet, C. Armellin, R. Mercier, G. Messin, A. Browaeys, and P. Grangier, “Diffraction-limited optics for single-atom manipulation,” Phys. Rev. A 75, 013406 (2007).
[Crossref]

Messin, G.

Y. R. P. Sortais, H. Marion, C. Tuchendler, A. M. Lance, M. Lamare, P. Fournet, C. Armellin, R. Mercier, G. Messin, A. Browaeys, and P. Grangier, “Diffraction-limited optics for single-atom manipulation,” Phys. Rev. A 75, 013406 (2007).
[Crossref]

B. Darquié, M. P. A. Jones, J. Dingjan, J. Beugnon, S. Bergamini, Y. Sortais, G. Messin, A. Browaeys, and P. Grangier, “Controlled Single-Photon Emission from a Single Trapped Two-Level Atom,” Science 309, 454–456 (2005).
[Crossref] [PubMed]

Milburn, G. J.

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature (London) 409, 46–52 (2001).
[Crossref]

Moktadir, Z.

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, “Microfabricated high-finesse optical cavity with open access and small volume,” Appl. Phys. Lett. 87, 211106 (2005).
[Crossref]

S. Eriksson, M. Trupke, H. F. Powell, D. Sahagun, C. D. J. Sinclair, E. A. Curtis, B. E. Sauer, E. A. Hinds, Z. Moktadir, C. O. Gollasch, and M. Kraft, “Integrated optical components on atom chips,” Eur. Phys. J. D 35, 135–139 (2005).
[Crossref]

Orrit, M.

B. Lounis and M. Orrit, “Single-photon sources,” Rep. Prog. Phys. 68, 1129–1179 (2005).
[Crossref]

Oxborrow, M.

M. Oxborrow and A. G. Sinclair, “Single-photon sources,” Contemp. Phys. 46, 173–206 (2005).
[Crossref]

Peschel, U.

M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B 89, 489–482 (2007).
[Crossref]

Powell, H. F.

S. Eriksson, M. Trupke, H. F. Powell, D. Sahagun, C. D. J. Sinclair, E. A. Curtis, B. E. Sauer, E. A. Hinds, Z. Moktadir, C. O. Gollasch, and M. Kraft, “Integrated optical components on atom chips,” Eur. Phys. J. D 35, 135–139 (2005).
[Crossref]

Protsenko, I.

N. Schlosser, G. Reymond, I. Protsenko, and P. Grangier, “Sub-poissonian loading of single atoms in a microscopic dipole trap,” Nature (London) 411, 1024–1027 (2001).
[Crossref]

Reymond, G.

N. Schlosser, G. Reymond, and P. Grangier, “Collisional Blockade in Microscopic Optical Dipole Traps,” Phys. Rev. Lett. 89, 023005 (2002).
[Crossref] [PubMed]

N. Schlosser, G. Reymond, I. Protsenko, and P. Grangier, “Sub-poissonian loading of single atoms in a microscopic dipole trap,” Nature (London) 411, 1024–1027 (2001).
[Crossref]

Rodriguez, A.

Roundy, D.

Sahagun, D.

S. Eriksson, M. Trupke, H. F. Powell, D. Sahagun, C. D. J. Sinclair, E. A. Curtis, B. E. Sauer, E. A. Hinds, Z. Moktadir, C. O. Gollasch, and M. Kraft, “Integrated optical components on atom chips,” Eur. Phys. J. D 35, 135–139 (2005).
[Crossref]

Santori, C.

E. Waks, C. Santori, and Y. Yamamoto, “Security aspects of quantum key distribution with sub-Poisson light,” Phys. Rev. A 66, 042315 (2002).
[Crossref]

Saucke, K.

M. Weber, J. Volz, K. Saucke, C. Kurtsiefer, and H. Weinfurter, “Analysis of a single-atom dipole trap,” Phys. Rev. A 73, 043406 (2006).
[Crossref]

Sauer, B. E.

S. Eriksson, M. Trupke, H. F. Powell, D. Sahagun, C. D. J. Sinclair, E. A. Curtis, B. E. Sauer, E. A. Hinds, Z. Moktadir, C. O. Gollasch, and M. Kraft, “Integrated optical components on atom chips,” Eur. Phys. J. D 35, 135–139 (2005).
[Crossref]

Schlosser, N.

N. Schlosser, G. Reymond, and P. Grangier, “Collisional Blockade in Microscopic Optical Dipole Traps,” Phys. Rev. Lett. 89, 023005 (2002).
[Crossref] [PubMed]

N. Schlosser, G. Reymond, I. Protsenko, and P. Grangier, “Sub-poissonian loading of single atoms in a microscopic dipole trap,” Nature (London) 411, 1024–1027 (2001).
[Crossref]

Sinclair, A. G.

M. Oxborrow and A. G. Sinclair, “Single-photon sources,” Contemp. Phys. 46, 173–206 (2005).
[Crossref]

Sinclair, C. D. J.

S. Eriksson, M. Trupke, H. F. Powell, D. Sahagun, C. D. J. Sinclair, E. A. Curtis, B. E. Sauer, E. A. Hinds, Z. Moktadir, C. O. Gollasch, and M. Kraft, “Integrated optical components on atom chips,” Eur. Phys. J. D 35, 135–139 (2005).
[Crossref]

Sondermann, M.

M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B 89, 489–482 (2007).
[Crossref]

Sortais, Y.

B. Darquié, M. P. A. Jones, J. Dingjan, J. Beugnon, S. Bergamini, Y. Sortais, G. Messin, A. Browaeys, and P. Grangier, “Controlled Single-Photon Emission from a Single Trapped Two-Level Atom,” Science 309, 454–456 (2005).
[Crossref] [PubMed]

Sortais, Y. R. P.

Y. R. P. Sortais, H. Marion, C. Tuchendler, A. M. Lance, M. Lamare, P. Fournet, C. Armellin, R. Mercier, G. Messin, A. Browaeys, and P. Grangier, “Diffraction-limited optics for single-atom manipulation,” Phys. Rev. A 75, 013406 (2007).
[Crossref]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, (Artech, Norwood, MA,2000).

Tey, M. K.

M. K. Tey, Z. Chen, S. A. Aljunid, B. Chng, F. Huber, G. Maslennikov, and C. Kurtsiefer, “Strong interaction between light and a single trapped atom without a cavity,” arXiv:0802.3005v3 (2008), http://uk.arxiv.org/abs/0802.3005v3.

M. K. Tey, S. A. Aljunid, F. Huber, B. Chng, Z. Chen, G. Maslennikov, and C. Kurtsiefer, “Interfacing light and single atoms with a lens,” arXiv:0804.4861v2 (2008), http://uk.arxiv.org/abs/0804.4861v2.

Trupke, M.

S. Eriksson, M. Trupke, H. F. Powell, D. Sahagun, C. D. J. Sinclair, E. A. Curtis, B. E. Sauer, E. A. Hinds, Z. Moktadir, C. O. Gollasch, and M. Kraft, “Integrated optical components on atom chips,” Eur. Phys. J. D 35, 135–139 (2005).
[Crossref]

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, “Microfabricated high-finesse optical cavity with open access and small volume,” Appl. Phys. Lett. 87, 211106 (2005).
[Crossref]

Tuchendler, C.

Y. R. P. Sortais, H. Marion, C. Tuchendler, A. M. Lance, M. Lamare, P. Fournet, C. Armellin, R. Mercier, G. Messin, A. Browaeys, and P. Grangier, “Diffraction-limited optics for single-atom manipulation,” Phys. Rev. A 75, 013406 (2007).
[Crossref]

van Enk, S. J.

S. J. van Enk and H. J. Kimble, “Strongly focused light beams interacting with single atoms in free space,” Phys. Rev. A 63, 023809 (2001).

Volz, J.

M. Weber, J. Volz, K. Saucke, C. Kurtsiefer, and H. Weinfurter, “Analysis of a single-atom dipole trap,” Phys. Rev. A 73, 043406 (2006).
[Crossref]

Waks, E.

E. Waks, C. Santori, and Y. Yamamoto, “Security aspects of quantum key distribution with sub-Poisson light,” Phys. Rev. A 66, 042315 (2002).
[Crossref]

Weber, M.

M. Weber, J. Volz, K. Saucke, C. Kurtsiefer, and H. Weinfurter, “Analysis of a single-atom dipole trap,” Phys. Rev. A 73, 043406 (2006).
[Crossref]

Weinfurter, H.

M. Weber, J. Volz, K. Saucke, C. Kurtsiefer, and H. Weinfurter, “Analysis of a single-atom dipole trap,” Phys. Rev. A 73, 043406 (2006).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th edition, (Cambridge University Press, Cambridge,1999).

Yamamoto, Y.

E. Waks, C. Santori, and Y. Yamamoto, “Security aspects of quantum key distribution with sub-Poisson light,” Phys. Rev. A 66, 042315 (2002).
[Crossref]

Zimmermann, C.

For a recent review of atom chip experiments, see J. Fortágh and C. Zimmermann, “Magnetic microtraps for ultracold atoms,” Rev. Mod. Phys. 79, 235 (2007).
[Crossref]

Appl. Phys. B (1)

M. Sondermann, R. Maiwald, H. Konermann, N. Lindlein, U. Peschel, and G. Leuchs, “Design of a mode converter for efficient light-atom coupling in free space,” Appl. Phys. B 89, 489–482 (2007).
[Crossref]

Appl. Phys. Lett. (1)

M. Trupke, E. A. Hinds, S. Eriksson, E. A. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, “Microfabricated high-finesse optical cavity with open access and small volume,” Appl. Phys. Lett. 87, 211106 (2005).
[Crossref]

Contemp. Phys. (1)

M. Oxborrow and A. G. Sinclair, “Single-photon sources,” Contemp. Phys. 46, 173–206 (2005).
[Crossref]

Eur. Phys. J. D (1)

S. Eriksson, M. Trupke, H. F. Powell, D. Sahagun, C. D. J. Sinclair, E. A. Curtis, B. E. Sauer, E. A. Hinds, Z. Moktadir, C. O. Gollasch, and M. Kraft, “Integrated optical components on atom chips,” Eur. Phys. J. D 35, 135–139 (2005).
[Crossref]

Nature (London) (2)

E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature (London) 409, 46–52 (2001).
[Crossref]

N. Schlosser, G. Reymond, I. Protsenko, and P. Grangier, “Sub-poissonian loading of single atoms in a microscopic dipole trap,” Nature (London) 411, 1024–1027 (2001).
[Crossref]

Opt. Lett. (1)

Phys. Rev. A (5)

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

G. P. Agrawal and M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[Crossref]

Y. R. P. Sortais, H. Marion, C. Tuchendler, A. M. Lance, M. Lamare, P. Fournet, C. Armellin, R. Mercier, G. Messin, A. Browaeys, and P. Grangier, “Diffraction-limited optics for single-atom manipulation,” Phys. Rev. A 75, 013406 (2007).
[Crossref]

M. Weber, J. Volz, K. Saucke, C. Kurtsiefer, and H. Weinfurter, “Analysis of a single-atom dipole trap,” Phys. Rev. A 73, 043406 (2006).
[Crossref]

E. Waks, C. Santori, and Y. Yamamoto, “Security aspects of quantum key distribution with sub-Poisson light,” Phys. Rev. A 66, 042315 (2002).
[Crossref]

Phys. Rev. Lett. (1)

N. Schlosser, G. Reymond, and P. Grangier, “Collisional Blockade in Microscopic Optical Dipole Traps,” Phys. Rev. Lett. 89, 023005 (2002).
[Crossref] [PubMed]

Rep. Prog. Phys. (1)

B. Lounis and M. Orrit, “Single-photon sources,” Rep. Prog. Phys. 68, 1129–1179 (2005).
[Crossref]

Rev. Mod. Phys. (1)

For a recent review of atom chip experiments, see J. Fortágh and C. Zimmermann, “Magnetic microtraps for ultracold atoms,” Rev. Mod. Phys. 79, 235 (2007).
[Crossref]

Science (1)

B. Darquié, M. P. A. Jones, J. Dingjan, J. Beugnon, S. Bergamini, Y. Sortais, G. Messin, A. Browaeys, and P. Grangier, “Controlled Single-Photon Emission from a Single Trapped Two-Level Atom,” Science 309, 454–456 (2005).
[Crossref] [PubMed]

Other (6)

M. K. Tey, Z. Chen, S. A. Aljunid, B. Chng, F. Huber, G. Maslennikov, and C. Kurtsiefer, “Strong interaction between light and a single trapped atom without a cavity,” arXiv:0802.3005v3 (2008), http://uk.arxiv.org/abs/0802.3005v3.

M. Born and E. Wolf, Principles of Optics, 7th edition, (Cambridge University Press, Cambridge,1999).

The software used in this work is available from http://ab-initio.mit.edu/wiki/index.php/Meep.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, (Artech, Norwood, MA,2000).

S. J. van Enk and H. J. Kimble, “Strongly focused light beams interacting with single atoms in free space,” Phys. Rev. A 63, 023809 (2001).

M. K. Tey, S. A. Aljunid, F. Huber, B. Chng, Z. Chen, G. Maslennikov, and C. Kurtsiefer, “Interfacing light and single atoms with a lens,” arXiv:0804.4861v2 (2008), http://uk.arxiv.org/abs/0804.4861v2.

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Figures (5)

Fig. 1.
Fig. 1. (Color online) Light incident along the path r′ is reflected at a point on the spherical mirror surface, then propagates along r to the point of observation (cartesian coordinates (ξRRR)).
Fig. 2.
Fig. 2. (Color online) Intensity distribution on axis with R=100λ for values of mirror aperture ρ=0.2 (bottom), 0.4 (middle), 0.6 (top). These are offset vertically for clarity. Labels give the peak intensity for unit incident intensity. (a) Prediction of Eq. (7). Curves are re-scaled to be equal in height. (b) Result of full numerical integration of Maxwell’s equations. Each curve has the same scale as the corresponding curve in (a). Dashed lines mark the geometrical focus at ζ=1/2.
Fig. 3.
Fig. 3. (Color online) Simulated intensity distributions of light outside concave spherical mirrors with radius of curvature R=100λ, etched in a plane substrate. The field of view is 10λ (vertical) ×40λ (horizontal), centered on z=R/2, with the mirror to the right. The calculations are done by numerical integration of Maxwell’s equations. Upper image: ρ=0.1. Lower image: ρ=0.4.
Fig. 4.
Fig. 4. (Color online) Normalized spot size r 1/2/λ as a function of aperture ρ with R=100λ. Solid line: radius given by Eq. (7) evaluated at the peak of the axial intensity distribution. Dashed line: radius given by Eq. (1) for an ideal optic with focal length R/2. Dots: full numerical integration of Maxwell’s equations.
Fig. 5.
Fig. 5. (Color online) Normalized spot volume, as defined in the text, for varying aperture sizes and R=100λ. Dots: numerical integration of Maxwell’s equations. Line: results from Eq. (7).

Equations (12)

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

r 1 2 diff λ = 1.62 R 2 2 π a = 0.13 ρ , ( ρ 1 2 )
ψ ( 0 ) ( x O ) = 1 4 π [ e ikr r n G ( r ) G ( r ) n e ikr r ] d A ,
n e ikr r ik cos ( n , r ) e ikr r ,
ψ ( 0 ) = ikψ 4 π e ik ( r + R cos θ ) r [ cos ( n, r ) cos ( n , r ) ] d A .
r cos ( n , r ) = R ( 1 ξ cos ϕ sin θ ζ cos θ ) ,
r R 1 + ζ 2 2 ζ cos θ ξ sin θ cos ϕ 1 + ζ 2 2 ζ cos θ
δ 0 + ξ δ 1 cos ϕ .
ψ ( 0 ) = k ψ 4 π i e i κ ( μ + δ 0 + ξ δ 1 cos ϕ ) R δ 0 ( μ + 1 ζ μ δ 0 ) d A
= κ ψ 2 i μ 0 1 e i κ ( μ + δ 0 ) δ 0 ( μ + 1 ζ μ δ 0 ) J 0 ( κ ξ δ 0 1 μ 2 ) d μ ,
ψ ( 0 ) ψ ( ρ q 2 ) 2 J 1 ( q ξ ) q ξ , q = κ ρ 1 ζ .
r 1 2 = 0.26 ( 1 ζ ) λ ρ ( when ρ 1 ) .
ψ z ( 1 ) ( ξ , ζ ) = i κ ξ [ ψ ( 0 ) ( ξ , ζ ) ] .

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