Abstract

We give a theoretical treatment of single-atom detection in a compound optical microcavity. The cavity consists of a single-mode semiconductor waveguide with a gap to allow atoms to interact with the optical field in the cavity. Optical losses, both in the semiconductor and induced by the gap, are considered, and we give an estimate of the cavity finesse. We also compute the cooperativity parameter and show how it depends on the gap width and cavity length. Maximization of the cooperativity does not always correspond to maximization of the coupling.

© 2012 Optical Society of America

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  1. J. Fortagh, A. Grossmann, C. Zimmermann, and T. W. Hänsch, “Miniaturized wire trap for neutral atoms,” Phys. Rev. Lett. 81, 5310–5313 (1998).
    [CrossRef]
  2. J. Reichel, W. Hänsel, and T. W. Hänsch, “Atomic micromanipulation with magnetic surface traps,” Phys. Rev. Lett. 83, 3398–3401 (1999).
    [CrossRef]
  3. J. Denschlag, D. Cassettari, and J. Schmiedmayer, “Guiding neutral atoms with a wire,” Phys. Rev. Lett. 82, 2014–2017 (1999).
    [CrossRef]
  4. W. Hänsel, D. Hommelhoff, T. W. Hänsch, and J. Reichel, “Bose-Einstein condensation on a microelectronic chip,” Nature 413, 498–501 (2001).
    [CrossRef]
  5. H. Ott, J. Fortagh, G. Schlotterbeck, A. Grossmann, and C. Zimmermann, “Bose-Einstein condensation in a surface microtrap,” Phys. Rev. Lett. 87, 230401 (2001).
    [CrossRef]
  6. S. Schneider, A. Kasper, C. vom Hagen, M. Bartenstein, B. Engeser, T. Schumm, I. Bar-Joseph, R. Folman, L. Feenstra, and J. Schmiedmayer, “Bose-Einstein condensation in a simple microtrap,” Phys. Rev. A 67, 023612 (2003).
    [CrossRef]
  7. S. J. Enk, H. J. Kimble, and H. Mabuchi, “Quantum information processing in cavity-QED,” Quantum Inf. Process. 3, 75–90 (2004).
    [CrossRef]
  8. Y. J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, “Atom Michelson interferometer on a chip using a Bose-Einstein condensate,” Phys. Rev. Lett. 94, 090405 (2005).
    [CrossRef]
  9. T. Schumm, S. Hofferberth, L. M. Andersson, S. Wildermuth, S. Groth, I. Bar-Joseph, J. Schmiedmayer, and P. Krüger, “Matter-wave interferometry in a double well on an atom chip,” Nat. Phys. 1, 57–62 (2005).
    [CrossRef]
  10. P. Horak, B. G. Klappauf, A. Haase, R. Folman, J. Schmiedmayer, P. Domokos, and E. A. Hinds, “Possibility of single-atom detection on a chip,” Phys. Rev. A 67, 043806 (2003).
    [CrossRef]
  11. R. Poldy, B. C. Buchler, and J. D. Close, “Single-atom detection with optical cavities,” Phys. Rev. A 78, 013640(2008).
    [CrossRef]
  12. T. Steinmetz, Y. Colombe, D. Hunger, T. W. Hänsch, A. Balocchi, R. J. Warburton, and J. Reichel, “Stable fiber-based Fabry-Pérot cavity,” Appl. Phys. Lett. 89, 111110 (2006).
    [CrossRef]
  13. M. Trupke, J. Goldwin, B. Darquié, G. Dutier, S. Eriksson, J. Ashmore, and E. A. Hinds, “Atom detection and photon production in a scalable, open, optical microcavity,” Phys. Rev. Lett. 99, 063601 (2007).
    [CrossRef]
  14. Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose–Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007).
    [CrossRef]
  15. M. Kohnen, M. Succo, P. G. Petrov, R. A. Nyman, M. Trupke, and E. A. Hinds, “An array of integrated atom–photon junctions,” Nat. Photon. 5, 35–38 (2011).
    [CrossRef]
  16. H. J. Kimble, “Strong interactions of single atoms and photons in cavity QED,” Phys. Scr. 1998, 127–132 (1998).
    [CrossRef]
  17. S. Gleyzes, A. El Amili, R. A. Cornelussen, P. Lalanne, C. I. Westbrook, A. Aspect, J. Estève, G. Moreau, A. Martinez, X. Lafosse, L. Ferlazzo, J. C. Harmand, D. Mailly, and A. Ramdane, “Towards a monolithic optical cavity for atom detection and manipulation,” Eur. Phys. J. D 53, 107–111(2009).
    [CrossRef]
  18. J. P. Hugonin and P. Lalanne, “Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A 22, 1844–1849 (2005).
    [CrossRef]
  19. H. van de Stadt and J. M. Muller, “Multimirror Fabry–Perot interferometers,” J. Opt. Soc. A 2, 1363–1370 (1985).
    [CrossRef]

2011

M. Kohnen, M. Succo, P. G. Petrov, R. A. Nyman, M. Trupke, and E. A. Hinds, “An array of integrated atom–photon junctions,” Nat. Photon. 5, 35–38 (2011).
[CrossRef]

2009

S. Gleyzes, A. El Amili, R. A. Cornelussen, P. Lalanne, C. I. Westbrook, A. Aspect, J. Estève, G. Moreau, A. Martinez, X. Lafosse, L. Ferlazzo, J. C. Harmand, D. Mailly, and A. Ramdane, “Towards a monolithic optical cavity for atom detection and manipulation,” Eur. Phys. J. D 53, 107–111(2009).
[CrossRef]

2008

R. Poldy, B. C. Buchler, and J. D. Close, “Single-atom detection with optical cavities,” Phys. Rev. A 78, 013640(2008).
[CrossRef]

2007

M. Trupke, J. Goldwin, B. Darquié, G. Dutier, S. Eriksson, J. Ashmore, and E. A. Hinds, “Atom detection and photon production in a scalable, open, optical microcavity,” Phys. Rev. Lett. 99, 063601 (2007).
[CrossRef]

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose–Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007).
[CrossRef]

2006

T. Steinmetz, Y. Colombe, D. Hunger, T. W. Hänsch, A. Balocchi, R. J. Warburton, and J. Reichel, “Stable fiber-based Fabry-Pérot cavity,” Appl. Phys. Lett. 89, 111110 (2006).
[CrossRef]

2005

Y. J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, “Atom Michelson interferometer on a chip using a Bose-Einstein condensate,” Phys. Rev. Lett. 94, 090405 (2005).
[CrossRef]

T. Schumm, S. Hofferberth, L. M. Andersson, S. Wildermuth, S. Groth, I. Bar-Joseph, J. Schmiedmayer, and P. Krüger, “Matter-wave interferometry in a double well on an atom chip,” Nat. Phys. 1, 57–62 (2005).
[CrossRef]

J. P. Hugonin and P. Lalanne, “Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A 22, 1844–1849 (2005).
[CrossRef]

2004

S. J. Enk, H. J. Kimble, and H. Mabuchi, “Quantum information processing in cavity-QED,” Quantum Inf. Process. 3, 75–90 (2004).
[CrossRef]

2003

P. Horak, B. G. Klappauf, A. Haase, R. Folman, J. Schmiedmayer, P. Domokos, and E. A. Hinds, “Possibility of single-atom detection on a chip,” Phys. Rev. A 67, 043806 (2003).
[CrossRef]

S. Schneider, A. Kasper, C. vom Hagen, M. Bartenstein, B. Engeser, T. Schumm, I. Bar-Joseph, R. Folman, L. Feenstra, and J. Schmiedmayer, “Bose-Einstein condensation in a simple microtrap,” Phys. Rev. A 67, 023612 (2003).
[CrossRef]

2001

W. Hänsel, D. Hommelhoff, T. W. Hänsch, and J. Reichel, “Bose-Einstein condensation on a microelectronic chip,” Nature 413, 498–501 (2001).
[CrossRef]

H. Ott, J. Fortagh, G. Schlotterbeck, A. Grossmann, and C. Zimmermann, “Bose-Einstein condensation in a surface microtrap,” Phys. Rev. Lett. 87, 230401 (2001).
[CrossRef]

1999

J. Reichel, W. Hänsel, and T. W. Hänsch, “Atomic micromanipulation with magnetic surface traps,” Phys. Rev. Lett. 83, 3398–3401 (1999).
[CrossRef]

J. Denschlag, D. Cassettari, and J. Schmiedmayer, “Guiding neutral atoms with a wire,” Phys. Rev. Lett. 82, 2014–2017 (1999).
[CrossRef]

1998

J. Fortagh, A. Grossmann, C. Zimmermann, and T. W. Hänsch, “Miniaturized wire trap for neutral atoms,” Phys. Rev. Lett. 81, 5310–5313 (1998).
[CrossRef]

H. J. Kimble, “Strong interactions of single atoms and photons in cavity QED,” Phys. Scr. 1998, 127–132 (1998).
[CrossRef]

1985

H. van de Stadt and J. M. Muller, “Multimirror Fabry–Perot interferometers,” J. Opt. Soc. A 2, 1363–1370 (1985).
[CrossRef]

Anderson, D. Z.

Y. J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, “Atom Michelson interferometer on a chip using a Bose-Einstein condensate,” Phys. Rev. Lett. 94, 090405 (2005).
[CrossRef]

Andersson, L. M.

T. Schumm, S. Hofferberth, L. M. Andersson, S. Wildermuth, S. Groth, I. Bar-Joseph, J. Schmiedmayer, and P. Krüger, “Matter-wave interferometry in a double well on an atom chip,” Nat. Phys. 1, 57–62 (2005).
[CrossRef]

Ashmore, J.

M. Trupke, J. Goldwin, B. Darquié, G. Dutier, S. Eriksson, J. Ashmore, and E. A. Hinds, “Atom detection and photon production in a scalable, open, optical microcavity,” Phys. Rev. Lett. 99, 063601 (2007).
[CrossRef]

Aspect, A.

S. Gleyzes, A. El Amili, R. A. Cornelussen, P. Lalanne, C. I. Westbrook, A. Aspect, J. Estève, G. Moreau, A. Martinez, X. Lafosse, L. Ferlazzo, J. C. Harmand, D. Mailly, and A. Ramdane, “Towards a monolithic optical cavity for atom detection and manipulation,” Eur. Phys. J. D 53, 107–111(2009).
[CrossRef]

Balocchi, A.

T. Steinmetz, Y. Colombe, D. Hunger, T. W. Hänsch, A. Balocchi, R. J. Warburton, and J. Reichel, “Stable fiber-based Fabry-Pérot cavity,” Appl. Phys. Lett. 89, 111110 (2006).
[CrossRef]

Bar-Joseph, I.

T. Schumm, S. Hofferberth, L. M. Andersson, S. Wildermuth, S. Groth, I. Bar-Joseph, J. Schmiedmayer, and P. Krüger, “Matter-wave interferometry in a double well on an atom chip,” Nat. Phys. 1, 57–62 (2005).
[CrossRef]

S. Schneider, A. Kasper, C. vom Hagen, M. Bartenstein, B. Engeser, T. Schumm, I. Bar-Joseph, R. Folman, L. Feenstra, and J. Schmiedmayer, “Bose-Einstein condensation in a simple microtrap,” Phys. Rev. A 67, 023612 (2003).
[CrossRef]

Bartenstein, M.

S. Schneider, A. Kasper, C. vom Hagen, M. Bartenstein, B. Engeser, T. Schumm, I. Bar-Joseph, R. Folman, L. Feenstra, and J. Schmiedmayer, “Bose-Einstein condensation in a simple microtrap,” Phys. Rev. A 67, 023612 (2003).
[CrossRef]

Bright, V. M.

Y. J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, “Atom Michelson interferometer on a chip using a Bose-Einstein condensate,” Phys. Rev. Lett. 94, 090405 (2005).
[CrossRef]

Buchler, B. C.

R. Poldy, B. C. Buchler, and J. D. Close, “Single-atom detection with optical cavities,” Phys. Rev. A 78, 013640(2008).
[CrossRef]

Cassettari, D.

J. Denschlag, D. Cassettari, and J. Schmiedmayer, “Guiding neutral atoms with a wire,” Phys. Rev. Lett. 82, 2014–2017 (1999).
[CrossRef]

Close, J. D.

R. Poldy, B. C. Buchler, and J. D. Close, “Single-atom detection with optical cavities,” Phys. Rev. A 78, 013640(2008).
[CrossRef]

Colombe, Y.

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose–Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007).
[CrossRef]

T. Steinmetz, Y. Colombe, D. Hunger, T. W. Hänsch, A. Balocchi, R. J. Warburton, and J. Reichel, “Stable fiber-based Fabry-Pérot cavity,” Appl. Phys. Lett. 89, 111110 (2006).
[CrossRef]

Cornell, E. A.

Y. J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, “Atom Michelson interferometer on a chip using a Bose-Einstein condensate,” Phys. Rev. Lett. 94, 090405 (2005).
[CrossRef]

Cornelussen, R. A.

S. Gleyzes, A. El Amili, R. A. Cornelussen, P. Lalanne, C. I. Westbrook, A. Aspect, J. Estève, G. Moreau, A. Martinez, X. Lafosse, L. Ferlazzo, J. C. Harmand, D. Mailly, and A. Ramdane, “Towards a monolithic optical cavity for atom detection and manipulation,” Eur. Phys. J. D 53, 107–111(2009).
[CrossRef]

Darquié, B.

M. Trupke, J. Goldwin, B. Darquié, G. Dutier, S. Eriksson, J. Ashmore, and E. A. Hinds, “Atom detection and photon production in a scalable, open, optical microcavity,” Phys. Rev. Lett. 99, 063601 (2007).
[CrossRef]

Denschlag, J.

J. Denschlag, D. Cassettari, and J. Schmiedmayer, “Guiding neutral atoms with a wire,” Phys. Rev. Lett. 82, 2014–2017 (1999).
[CrossRef]

Diot, Q.

Y. J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, “Atom Michelson interferometer on a chip using a Bose-Einstein condensate,” Phys. Rev. Lett. 94, 090405 (2005).
[CrossRef]

Domokos, P.

P. Horak, B. G. Klappauf, A. Haase, R. Folman, J. Schmiedmayer, P. Domokos, and E. A. Hinds, “Possibility of single-atom detection on a chip,” Phys. Rev. A 67, 043806 (2003).
[CrossRef]

Dubois, G.

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose–Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007).
[CrossRef]

Dutier, G.

M. Trupke, J. Goldwin, B. Darquié, G. Dutier, S. Eriksson, J. Ashmore, and E. A. Hinds, “Atom detection and photon production in a scalable, open, optical microcavity,” Phys. Rev. Lett. 99, 063601 (2007).
[CrossRef]

El Amili, A.

S. Gleyzes, A. El Amili, R. A. Cornelussen, P. Lalanne, C. I. Westbrook, A. Aspect, J. Estève, G. Moreau, A. Martinez, X. Lafosse, L. Ferlazzo, J. C. Harmand, D. Mailly, and A. Ramdane, “Towards a monolithic optical cavity for atom detection and manipulation,” Eur. Phys. J. D 53, 107–111(2009).
[CrossRef]

Engeser, B.

S. Schneider, A. Kasper, C. vom Hagen, M. Bartenstein, B. Engeser, T. Schumm, I. Bar-Joseph, R. Folman, L. Feenstra, and J. Schmiedmayer, “Bose-Einstein condensation in a simple microtrap,” Phys. Rev. A 67, 023612 (2003).
[CrossRef]

Enk, S. J.

S. J. Enk, H. J. Kimble, and H. Mabuchi, “Quantum information processing in cavity-QED,” Quantum Inf. Process. 3, 75–90 (2004).
[CrossRef]

Eriksson, S.

M. Trupke, J. Goldwin, B. Darquié, G. Dutier, S. Eriksson, J. Ashmore, and E. A. Hinds, “Atom detection and photon production in a scalable, open, optical microcavity,” Phys. Rev. Lett. 99, 063601 (2007).
[CrossRef]

Estève, J.

S. Gleyzes, A. El Amili, R. A. Cornelussen, P. Lalanne, C. I. Westbrook, A. Aspect, J. Estève, G. Moreau, A. Martinez, X. Lafosse, L. Ferlazzo, J. C. Harmand, D. Mailly, and A. Ramdane, “Towards a monolithic optical cavity for atom detection and manipulation,” Eur. Phys. J. D 53, 107–111(2009).
[CrossRef]

Feenstra, L.

S. Schneider, A. Kasper, C. vom Hagen, M. Bartenstein, B. Engeser, T. Schumm, I. Bar-Joseph, R. Folman, L. Feenstra, and J. Schmiedmayer, “Bose-Einstein condensation in a simple microtrap,” Phys. Rev. A 67, 023612 (2003).
[CrossRef]

Ferlazzo, L.

S. Gleyzes, A. El Amili, R. A. Cornelussen, P. Lalanne, C. I. Westbrook, A. Aspect, J. Estève, G. Moreau, A. Martinez, X. Lafosse, L. Ferlazzo, J. C. Harmand, D. Mailly, and A. Ramdane, “Towards a monolithic optical cavity for atom detection and manipulation,” Eur. Phys. J. D 53, 107–111(2009).
[CrossRef]

Folman, R.

S. Schneider, A. Kasper, C. vom Hagen, M. Bartenstein, B. Engeser, T. Schumm, I. Bar-Joseph, R. Folman, L. Feenstra, and J. Schmiedmayer, “Bose-Einstein condensation in a simple microtrap,” Phys. Rev. A 67, 023612 (2003).
[CrossRef]

P. Horak, B. G. Klappauf, A. Haase, R. Folman, J. Schmiedmayer, P. Domokos, and E. A. Hinds, “Possibility of single-atom detection on a chip,” Phys. Rev. A 67, 043806 (2003).
[CrossRef]

Fortagh, J.

H. Ott, J. Fortagh, G. Schlotterbeck, A. Grossmann, and C. Zimmermann, “Bose-Einstein condensation in a surface microtrap,” Phys. Rev. Lett. 87, 230401 (2001).
[CrossRef]

J. Fortagh, A. Grossmann, C. Zimmermann, and T. W. Hänsch, “Miniaturized wire trap for neutral atoms,” Phys. Rev. Lett. 81, 5310–5313 (1998).
[CrossRef]

Gleyzes, S.

S. Gleyzes, A. El Amili, R. A. Cornelussen, P. Lalanne, C. I. Westbrook, A. Aspect, J. Estève, G. Moreau, A. Martinez, X. Lafosse, L. Ferlazzo, J. C. Harmand, D. Mailly, and A. Ramdane, “Towards a monolithic optical cavity for atom detection and manipulation,” Eur. Phys. J. D 53, 107–111(2009).
[CrossRef]

Goldwin, J.

M. Trupke, J. Goldwin, B. Darquié, G. Dutier, S. Eriksson, J. Ashmore, and E. A. Hinds, “Atom detection and photon production in a scalable, open, optical microcavity,” Phys. Rev. Lett. 99, 063601 (2007).
[CrossRef]

Grossmann, A.

H. Ott, J. Fortagh, G. Schlotterbeck, A. Grossmann, and C. Zimmermann, “Bose-Einstein condensation in a surface microtrap,” Phys. Rev. Lett. 87, 230401 (2001).
[CrossRef]

J. Fortagh, A. Grossmann, C. Zimmermann, and T. W. Hänsch, “Miniaturized wire trap for neutral atoms,” Phys. Rev. Lett. 81, 5310–5313 (1998).
[CrossRef]

Groth, S.

T. Schumm, S. Hofferberth, L. M. Andersson, S. Wildermuth, S. Groth, I. Bar-Joseph, J. Schmiedmayer, and P. Krüger, “Matter-wave interferometry in a double well on an atom chip,” Nat. Phys. 1, 57–62 (2005).
[CrossRef]

Haase, A.

P. Horak, B. G. Klappauf, A. Haase, R. Folman, J. Schmiedmayer, P. Domokos, and E. A. Hinds, “Possibility of single-atom detection on a chip,” Phys. Rev. A 67, 043806 (2003).
[CrossRef]

Hänsch, T. W.

T. Steinmetz, Y. Colombe, D. Hunger, T. W. Hänsch, A. Balocchi, R. J. Warburton, and J. Reichel, “Stable fiber-based Fabry-Pérot cavity,” Appl. Phys. Lett. 89, 111110 (2006).
[CrossRef]

W. Hänsel, D. Hommelhoff, T. W. Hänsch, and J. Reichel, “Bose-Einstein condensation on a microelectronic chip,” Nature 413, 498–501 (2001).
[CrossRef]

J. Reichel, W. Hänsel, and T. W. Hänsch, “Atomic micromanipulation with magnetic surface traps,” Phys. Rev. Lett. 83, 3398–3401 (1999).
[CrossRef]

J. Fortagh, A. Grossmann, C. Zimmermann, and T. W. Hänsch, “Miniaturized wire trap for neutral atoms,” Phys. Rev. Lett. 81, 5310–5313 (1998).
[CrossRef]

Hänsel, W.

W. Hänsel, D. Hommelhoff, T. W. Hänsch, and J. Reichel, “Bose-Einstein condensation on a microelectronic chip,” Nature 413, 498–501 (2001).
[CrossRef]

J. Reichel, W. Hänsel, and T. W. Hänsch, “Atomic micromanipulation with magnetic surface traps,” Phys. Rev. Lett. 83, 3398–3401 (1999).
[CrossRef]

Harmand, J. C.

S. Gleyzes, A. El Amili, R. A. Cornelussen, P. Lalanne, C. I. Westbrook, A. Aspect, J. Estève, G. Moreau, A. Martinez, X. Lafosse, L. Ferlazzo, J. C. Harmand, D. Mailly, and A. Ramdane, “Towards a monolithic optical cavity for atom detection and manipulation,” Eur. Phys. J. D 53, 107–111(2009).
[CrossRef]

Hinds, E. A.

M. Kohnen, M. Succo, P. G. Petrov, R. A. Nyman, M. Trupke, and E. A. Hinds, “An array of integrated atom–photon junctions,” Nat. Photon. 5, 35–38 (2011).
[CrossRef]

M. Trupke, J. Goldwin, B. Darquié, G. Dutier, S. Eriksson, J. Ashmore, and E. A. Hinds, “Atom detection and photon production in a scalable, open, optical microcavity,” Phys. Rev. Lett. 99, 063601 (2007).
[CrossRef]

P. Horak, B. G. Klappauf, A. Haase, R. Folman, J. Schmiedmayer, P. Domokos, and E. A. Hinds, “Possibility of single-atom detection on a chip,” Phys. Rev. A 67, 043806 (2003).
[CrossRef]

Hofferberth, S.

T. Schumm, S. Hofferberth, L. M. Andersson, S. Wildermuth, S. Groth, I. Bar-Joseph, J. Schmiedmayer, and P. Krüger, “Matter-wave interferometry in a double well on an atom chip,” Nat. Phys. 1, 57–62 (2005).
[CrossRef]

Hommelhoff, D.

W. Hänsel, D. Hommelhoff, T. W. Hänsch, and J. Reichel, “Bose-Einstein condensation on a microelectronic chip,” Nature 413, 498–501 (2001).
[CrossRef]

Horak, P.

P. Horak, B. G. Klappauf, A. Haase, R. Folman, J. Schmiedmayer, P. Domokos, and E. A. Hinds, “Possibility of single-atom detection on a chip,” Phys. Rev. A 67, 043806 (2003).
[CrossRef]

Hugonin, J. P.

Hunger, D.

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose–Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007).
[CrossRef]

T. Steinmetz, Y. Colombe, D. Hunger, T. W. Hänsch, A. Balocchi, R. J. Warburton, and J. Reichel, “Stable fiber-based Fabry-Pérot cavity,” Appl. Phys. Lett. 89, 111110 (2006).
[CrossRef]

Kasper, A.

S. Schneider, A. Kasper, C. vom Hagen, M. Bartenstein, B. Engeser, T. Schumm, I. Bar-Joseph, R. Folman, L. Feenstra, and J. Schmiedmayer, “Bose-Einstein condensation in a simple microtrap,” Phys. Rev. A 67, 023612 (2003).
[CrossRef]

Kimble, H. J.

S. J. Enk, H. J. Kimble, and H. Mabuchi, “Quantum information processing in cavity-QED,” Quantum Inf. Process. 3, 75–90 (2004).
[CrossRef]

H. J. Kimble, “Strong interactions of single atoms and photons in cavity QED,” Phys. Scr. 1998, 127–132 (1998).
[CrossRef]

Kishimoto, T.

Y. J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, “Atom Michelson interferometer on a chip using a Bose-Einstein condensate,” Phys. Rev. Lett. 94, 090405 (2005).
[CrossRef]

Klappauf, B. G.

P. Horak, B. G. Klappauf, A. Haase, R. Folman, J. Schmiedmayer, P. Domokos, and E. A. Hinds, “Possibility of single-atom detection on a chip,” Phys. Rev. A 67, 043806 (2003).
[CrossRef]

Kohnen, M.

M. Kohnen, M. Succo, P. G. Petrov, R. A. Nyman, M. Trupke, and E. A. Hinds, “An array of integrated atom–photon junctions,” Nat. Photon. 5, 35–38 (2011).
[CrossRef]

Krüger, P.

T. Schumm, S. Hofferberth, L. M. Andersson, S. Wildermuth, S. Groth, I. Bar-Joseph, J. Schmiedmayer, and P. Krüger, “Matter-wave interferometry in a double well on an atom chip,” Nat. Phys. 1, 57–62 (2005).
[CrossRef]

Lafosse, X.

S. Gleyzes, A. El Amili, R. A. Cornelussen, P. Lalanne, C. I. Westbrook, A. Aspect, J. Estève, G. Moreau, A. Martinez, X. Lafosse, L. Ferlazzo, J. C. Harmand, D. Mailly, and A. Ramdane, “Towards a monolithic optical cavity for atom detection and manipulation,” Eur. Phys. J. D 53, 107–111(2009).
[CrossRef]

Lalanne, P.

S. Gleyzes, A. El Amili, R. A. Cornelussen, P. Lalanne, C. I. Westbrook, A. Aspect, J. Estève, G. Moreau, A. Martinez, X. Lafosse, L. Ferlazzo, J. C. Harmand, D. Mailly, and A. Ramdane, “Towards a monolithic optical cavity for atom detection and manipulation,” Eur. Phys. J. D 53, 107–111(2009).
[CrossRef]

J. P. Hugonin and P. Lalanne, “Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A 22, 1844–1849 (2005).
[CrossRef]

Linke, F.

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose–Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007).
[CrossRef]

Mabuchi, H.

S. J. Enk, H. J. Kimble, and H. Mabuchi, “Quantum information processing in cavity-QED,” Quantum Inf. Process. 3, 75–90 (2004).
[CrossRef]

Mailly, D.

S. Gleyzes, A. El Amili, R. A. Cornelussen, P. Lalanne, C. I. Westbrook, A. Aspect, J. Estève, G. Moreau, A. Martinez, X. Lafosse, L. Ferlazzo, J. C. Harmand, D. Mailly, and A. Ramdane, “Towards a monolithic optical cavity for atom detection and manipulation,” Eur. Phys. J. D 53, 107–111(2009).
[CrossRef]

Martinez, A.

S. Gleyzes, A. El Amili, R. A. Cornelussen, P. Lalanne, C. I. Westbrook, A. Aspect, J. Estève, G. Moreau, A. Martinez, X. Lafosse, L. Ferlazzo, J. C. Harmand, D. Mailly, and A. Ramdane, “Towards a monolithic optical cavity for atom detection and manipulation,” Eur. Phys. J. D 53, 107–111(2009).
[CrossRef]

Moreau, G.

S. Gleyzes, A. El Amili, R. A. Cornelussen, P. Lalanne, C. I. Westbrook, A. Aspect, J. Estève, G. Moreau, A. Martinez, X. Lafosse, L. Ferlazzo, J. C. Harmand, D. Mailly, and A. Ramdane, “Towards a monolithic optical cavity for atom detection and manipulation,” Eur. Phys. J. D 53, 107–111(2009).
[CrossRef]

Muller, J. M.

H. van de Stadt and J. M. Muller, “Multimirror Fabry–Perot interferometers,” J. Opt. Soc. A 2, 1363–1370 (1985).
[CrossRef]

Nyman, R. A.

M. Kohnen, M. Succo, P. G. Petrov, R. A. Nyman, M. Trupke, and E. A. Hinds, “An array of integrated atom–photon junctions,” Nat. Photon. 5, 35–38 (2011).
[CrossRef]

Ott, H.

H. Ott, J. Fortagh, G. Schlotterbeck, A. Grossmann, and C. Zimmermann, “Bose-Einstein condensation in a surface microtrap,” Phys. Rev. Lett. 87, 230401 (2001).
[CrossRef]

Petrov, P. G.

M. Kohnen, M. Succo, P. G. Petrov, R. A. Nyman, M. Trupke, and E. A. Hinds, “An array of integrated atom–photon junctions,” Nat. Photon. 5, 35–38 (2011).
[CrossRef]

Poldy, R.

R. Poldy, B. C. Buchler, and J. D. Close, “Single-atom detection with optical cavities,” Phys. Rev. A 78, 013640(2008).
[CrossRef]

Prentiss, M.

Y. J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, “Atom Michelson interferometer on a chip using a Bose-Einstein condensate,” Phys. Rev. Lett. 94, 090405 (2005).
[CrossRef]

Ramdane, A.

S. Gleyzes, A. El Amili, R. A. Cornelussen, P. Lalanne, C. I. Westbrook, A. Aspect, J. Estève, G. Moreau, A. Martinez, X. Lafosse, L. Ferlazzo, J. C. Harmand, D. Mailly, and A. Ramdane, “Towards a monolithic optical cavity for atom detection and manipulation,” Eur. Phys. J. D 53, 107–111(2009).
[CrossRef]

Reichel, J.

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose–Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007).
[CrossRef]

T. Steinmetz, Y. Colombe, D. Hunger, T. W. Hänsch, A. Balocchi, R. J. Warburton, and J. Reichel, “Stable fiber-based Fabry-Pérot cavity,” Appl. Phys. Lett. 89, 111110 (2006).
[CrossRef]

W. Hänsel, D. Hommelhoff, T. W. Hänsch, and J. Reichel, “Bose-Einstein condensation on a microelectronic chip,” Nature 413, 498–501 (2001).
[CrossRef]

J. Reichel, W. Hänsel, and T. W. Hänsch, “Atomic micromanipulation with magnetic surface traps,” Phys. Rev. Lett. 83, 3398–3401 (1999).
[CrossRef]

Saravanan, R. A.

Y. J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, “Atom Michelson interferometer on a chip using a Bose-Einstein condensate,” Phys. Rev. Lett. 94, 090405 (2005).
[CrossRef]

Schlotterbeck, G.

H. Ott, J. Fortagh, G. Schlotterbeck, A. Grossmann, and C. Zimmermann, “Bose-Einstein condensation in a surface microtrap,” Phys. Rev. Lett. 87, 230401 (2001).
[CrossRef]

Schmiedmayer, J.

T. Schumm, S. Hofferberth, L. M. Andersson, S. Wildermuth, S. Groth, I. Bar-Joseph, J. Schmiedmayer, and P. Krüger, “Matter-wave interferometry in a double well on an atom chip,” Nat. Phys. 1, 57–62 (2005).
[CrossRef]

S. Schneider, A. Kasper, C. vom Hagen, M. Bartenstein, B. Engeser, T. Schumm, I. Bar-Joseph, R. Folman, L. Feenstra, and J. Schmiedmayer, “Bose-Einstein condensation in a simple microtrap,” Phys. Rev. A 67, 023612 (2003).
[CrossRef]

P. Horak, B. G. Klappauf, A. Haase, R. Folman, J. Schmiedmayer, P. Domokos, and E. A. Hinds, “Possibility of single-atom detection on a chip,” Phys. Rev. A 67, 043806 (2003).
[CrossRef]

J. Denschlag, D. Cassettari, and J. Schmiedmayer, “Guiding neutral atoms with a wire,” Phys. Rev. Lett. 82, 2014–2017 (1999).
[CrossRef]

Schneider, S.

S. Schneider, A. Kasper, C. vom Hagen, M. Bartenstein, B. Engeser, T. Schumm, I. Bar-Joseph, R. Folman, L. Feenstra, and J. Schmiedmayer, “Bose-Einstein condensation in a simple microtrap,” Phys. Rev. A 67, 023612 (2003).
[CrossRef]

Schumm, T.

T. Schumm, S. Hofferberth, L. M. Andersson, S. Wildermuth, S. Groth, I. Bar-Joseph, J. Schmiedmayer, and P. Krüger, “Matter-wave interferometry in a double well on an atom chip,” Nat. Phys. 1, 57–62 (2005).
[CrossRef]

S. Schneider, A. Kasper, C. vom Hagen, M. Bartenstein, B. Engeser, T. Schumm, I. Bar-Joseph, R. Folman, L. Feenstra, and J. Schmiedmayer, “Bose-Einstein condensation in a simple microtrap,” Phys. Rev. A 67, 023612 (2003).
[CrossRef]

Segal, S. R.

Y. J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, “Atom Michelson interferometer on a chip using a Bose-Einstein condensate,” Phys. Rev. Lett. 94, 090405 (2005).
[CrossRef]

Steinmetz, T.

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose–Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007).
[CrossRef]

T. Steinmetz, Y. Colombe, D. Hunger, T. W. Hänsch, A. Balocchi, R. J. Warburton, and J. Reichel, “Stable fiber-based Fabry-Pérot cavity,” Appl. Phys. Lett. 89, 111110 (2006).
[CrossRef]

Succo, M.

M. Kohnen, M. Succo, P. G. Petrov, R. A. Nyman, M. Trupke, and E. A. Hinds, “An array of integrated atom–photon junctions,” Nat. Photon. 5, 35–38 (2011).
[CrossRef]

Trupke, M.

M. Kohnen, M. Succo, P. G. Petrov, R. A. Nyman, M. Trupke, and E. A. Hinds, “An array of integrated atom–photon junctions,” Nat. Photon. 5, 35–38 (2011).
[CrossRef]

M. Trupke, J. Goldwin, B. Darquié, G. Dutier, S. Eriksson, J. Ashmore, and E. A. Hinds, “Atom detection and photon production in a scalable, open, optical microcavity,” Phys. Rev. Lett. 99, 063601 (2007).
[CrossRef]

van de Stadt, H.

H. van de Stadt and J. M. Muller, “Multimirror Fabry–Perot interferometers,” J. Opt. Soc. A 2, 1363–1370 (1985).
[CrossRef]

vom Hagen, C.

S. Schneider, A. Kasper, C. vom Hagen, M. Bartenstein, B. Engeser, T. Schumm, I. Bar-Joseph, R. Folman, L. Feenstra, and J. Schmiedmayer, “Bose-Einstein condensation in a simple microtrap,” Phys. Rev. A 67, 023612 (2003).
[CrossRef]

Wang, Y. J.

Y. J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, “Atom Michelson interferometer on a chip using a Bose-Einstein condensate,” Phys. Rev. Lett. 94, 090405 (2005).
[CrossRef]

Warburton, R. J.

T. Steinmetz, Y. Colombe, D. Hunger, T. W. Hänsch, A. Balocchi, R. J. Warburton, and J. Reichel, “Stable fiber-based Fabry-Pérot cavity,” Appl. Phys. Lett. 89, 111110 (2006).
[CrossRef]

Westbrook, C. I.

S. Gleyzes, A. El Amili, R. A. Cornelussen, P. Lalanne, C. I. Westbrook, A. Aspect, J. Estève, G. Moreau, A. Martinez, X. Lafosse, L. Ferlazzo, J. C. Harmand, D. Mailly, and A. Ramdane, “Towards a monolithic optical cavity for atom detection and manipulation,” Eur. Phys. J. D 53, 107–111(2009).
[CrossRef]

Wildermuth, S.

T. Schumm, S. Hofferberth, L. M. Andersson, S. Wildermuth, S. Groth, I. Bar-Joseph, J. Schmiedmayer, and P. Krüger, “Matter-wave interferometry in a double well on an atom chip,” Nat. Phys. 1, 57–62 (2005).
[CrossRef]

Wu, S.

Y. J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, “Atom Michelson interferometer on a chip using a Bose-Einstein condensate,” Phys. Rev. Lett. 94, 090405 (2005).
[CrossRef]

Zimmermann, C.

H. Ott, J. Fortagh, G. Schlotterbeck, A. Grossmann, and C. Zimmermann, “Bose-Einstein condensation in a surface microtrap,” Phys. Rev. Lett. 87, 230401 (2001).
[CrossRef]

J. Fortagh, A. Grossmann, C. Zimmermann, and T. W. Hänsch, “Miniaturized wire trap for neutral atoms,” Phys. Rev. Lett. 81, 5310–5313 (1998).
[CrossRef]

Appl. Phys. Lett.

T. Steinmetz, Y. Colombe, D. Hunger, T. W. Hänsch, A. Balocchi, R. J. Warburton, and J. Reichel, “Stable fiber-based Fabry-Pérot cavity,” Appl. Phys. Lett. 89, 111110 (2006).
[CrossRef]

Eur. Phys. J. D

S. Gleyzes, A. El Amili, R. A. Cornelussen, P. Lalanne, C. I. Westbrook, A. Aspect, J. Estève, G. Moreau, A. Martinez, X. Lafosse, L. Ferlazzo, J. C. Harmand, D. Mailly, and A. Ramdane, “Towards a monolithic optical cavity for atom detection and manipulation,” Eur. Phys. J. D 53, 107–111(2009).
[CrossRef]

J. Opt. Soc. A

H. van de Stadt and J. M. Muller, “Multimirror Fabry–Perot interferometers,” J. Opt. Soc. A 2, 1363–1370 (1985).
[CrossRef]

J. Opt. Soc. Am. A

Nat. Photon.

M. Kohnen, M. Succo, P. G. Petrov, R. A. Nyman, M. Trupke, and E. A. Hinds, “An array of integrated atom–photon junctions,” Nat. Photon. 5, 35–38 (2011).
[CrossRef]

Nat. Phys.

T. Schumm, S. Hofferberth, L. M. Andersson, S. Wildermuth, S. Groth, I. Bar-Joseph, J. Schmiedmayer, and P. Krüger, “Matter-wave interferometry in a double well on an atom chip,” Nat. Phys. 1, 57–62 (2005).
[CrossRef]

Nature

W. Hänsel, D. Hommelhoff, T. W. Hänsch, and J. Reichel, “Bose-Einstein condensation on a microelectronic chip,” Nature 413, 498–501 (2001).
[CrossRef]

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose–Einstein condensates in an optical cavity on a chip,” Nature 450, 272–276 (2007).
[CrossRef]

Phys. Rev. A

P. Horak, B. G. Klappauf, A. Haase, R. Folman, J. Schmiedmayer, P. Domokos, and E. A. Hinds, “Possibility of single-atom detection on a chip,” Phys. Rev. A 67, 043806 (2003).
[CrossRef]

R. Poldy, B. C. Buchler, and J. D. Close, “Single-atom detection with optical cavities,” Phys. Rev. A 78, 013640(2008).
[CrossRef]

S. Schneider, A. Kasper, C. vom Hagen, M. Bartenstein, B. Engeser, T. Schumm, I. Bar-Joseph, R. Folman, L. Feenstra, and J. Schmiedmayer, “Bose-Einstein condensation in a simple microtrap,” Phys. Rev. A 67, 023612 (2003).
[CrossRef]

Phys. Rev. Lett.

H. Ott, J. Fortagh, G. Schlotterbeck, A. Grossmann, and C. Zimmermann, “Bose-Einstein condensation in a surface microtrap,” Phys. Rev. Lett. 87, 230401 (2001).
[CrossRef]

J. Fortagh, A. Grossmann, C. Zimmermann, and T. W. Hänsch, “Miniaturized wire trap for neutral atoms,” Phys. Rev. Lett. 81, 5310–5313 (1998).
[CrossRef]

J. Reichel, W. Hänsel, and T. W. Hänsch, “Atomic micromanipulation with magnetic surface traps,” Phys. Rev. Lett. 83, 3398–3401 (1999).
[CrossRef]

J. Denschlag, D. Cassettari, and J. Schmiedmayer, “Guiding neutral atoms with a wire,” Phys. Rev. Lett. 82, 2014–2017 (1999).
[CrossRef]

Y. J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan, S. R. Segal, and S. Wu, “Atom Michelson interferometer on a chip using a Bose-Einstein condensate,” Phys. Rev. Lett. 94, 090405 (2005).
[CrossRef]

M. Trupke, J. Goldwin, B. Darquié, G. Dutier, S. Eriksson, J. Ashmore, and E. A. Hinds, “Atom detection and photon production in a scalable, open, optical microcavity,” Phys. Rev. Lett. 99, 063601 (2007).
[CrossRef]

Phys. Scr.

H. J. Kimble, “Strong interactions of single atoms and photons in cavity QED,” Phys. Scr. 1998, 127–132 (1998).
[CrossRef]

Quantum Inf. Process.

S. J. Enk, H. J. Kimble, and H. Mabuchi, “Quantum information processing in cavity-QED,” Quantum Inf. Process. 3, 75–90 (2004).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

(a) Transverse geometry of the waveguide. The waveguide is a rectangular ridge of AlGaAs, 4 μm wide. The false color image shows the result of a numerical calculation of the amplitude of the fundamental mode of the waveguide operating at λ=780nm. The mode area is calculated numerically, and we find A=9.9μm2. (b) Top view of the gapped waveguide. Atoms are to be guided into this gap to interact with the intracavity field. The gap constitutes a low-finesse cavity within a larger waveguide cavity, which is closed by reflecting surfaces at both ends of the waveguide.

Fig. 2.
Fig. 2.

(a) Amplitude of the transmission and reflection of the gap as a function of the gap width. (b) Phase of the transmission and reflection coefficients (the propagating term k0d has been subtracted). The solid lines correspond to the analytical model (r˜g, t˜g), and the square and circles correspond to the result of the full numerical simulation (r˜g,sim and t˜g,sim respectively; see text). (c) Losses induced by the gap. The solid line corresponds to 1|t˜g|2|r˜g|2, the dotted line with squares corresponds to 1|t˜g,sim|2|r˜g,sim|2. The losses are maximal for a gap width equal to an integer number of half-wavelengths.

Fig. 3.
Fig. 3.

(a) Cavity finesse as a function of the gap width d and waveguide length L3. (b) Finesse as a function of L3 for d=1.95μm (solid line), and d2.15μm (dotted line). (c) Finesse as a function of d for nL3/λ=1000.5 (solid line), and nL3/λ=1000.25 (dotted line).

Fig. 4.
Fig. 4.

Contour plot of the atom–field coupling g˜ as function d and L3. Insets, electric field profile inside the coupled cavity for special cases of d and L3 with a normalization |E1|+|E1|=1. (a) Electric field profile for d=1.95μm and L3 equal to an integer number of half wavelengths. (b) Electric field profile for d=1.95μm and L3=λ/4n (modλ/2n). (c) Electric field profile for d=1.75μm and L3 equal to an integer number of half wavelengths. (d) Electric field profile for d=2.05μm and L3=0.23λ (modλ/2n) (value that optimizes C in the lossless case).

Fig. 5.
Fig. 5.

Cooperativity of a Rb atom in the coupled cavity system as a function of d and L3, including propagation losses in the waveguides. The attenuation coefficient corresponds to that measured in [17]. The coupling is maximized for d=1.95μm and nL3=1000.5λ.

Fig. 6.
Fig. 6.

Cooperativity of a Rb atom in the coupled cavity system as a function of d and L3. The waveguides are assumed to be lossless.

Equations (21)

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

Q(z)=f*(x,y,0)f(x,y,z)dxdy|f(x,y,0)|2dxdy|f(x,y,z)|2dxdy.
f(0,0,z)=|f(0,0,z)|expi(kz+φ0(z)),
Ep+(x,y,z)=Ep+f(x,y,2pd+z),
Ep(x,y,z)=Epf(x,y,(2p+2)dz).
Et=tpQp+tr2pEi.
Er=rEi+tpQptr2p+1Ei,
E3=t˜g(d)E1+r˜g(d)E3,
E1=t˜g(d)E3+r˜g(d)E1,
E3=r0exp(2inkL3)E3,
E1=r˜eff(d,k,L3)E1,
Fd=πreff1reff1n(L1+L3)(nL1+ϕk).
Fd=πreff1reff1n(L1+L3)(nL1+nL3ϕ(nkδL3)),
Eg+(z)=[E1peventrp|f0(pd+z)|exp(ikpd+iφ0(pd+z))+E3poddtrp|f0(pd+z)|exp(ikpd+iφ0(pd+z))]exp(ikz)=E˜g+(z)exp(ikz),
Eg(z)=[E1poddtrp|f0(pd+dz)|exp(ikpd+iφ0(pd+dz))+E3peventrp|f0(pd+dz)|exp(ikpd+iφ0(pd+dz))]exp(ikz)=E˜g(z)exp(ikz),
|Eg(z*)|=|E˜g+(z*)|+|E˜g(z*)||E˜g+(d2)|+|E˜g(d2)|.
hν4n2ϵ0A(L1|E1|2+L3|E3|2),
E=hν(|Eg+|+|Eg|)4n2ϵ0A(L1|E1|2+L3|E3|2).
g0=3πcγk2n2AL,
g˜(d,L1,L3)=12|Eg+|+|Eg||E1|2+|E3|2.
κ=πc2n(L1+L3)F,
C=2σabsFπnAg˜2(d,L3),

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