Abstract

We show that due to near-field interaction of plasmonic particles via gain particles, a two-dimensional array of incoherently pumped spasers can be self-synchronized so that the dipole moments of all the plasmonic particles oscillate in phase and in parallel to the array plane. The synchronized state is established as a result of competition with the other possible modes having different wavenumbers and it is not destroyed by radiation of leaking waves, retardation effects, and small disorder. Such an array produces a narrow beam of coherent light due to continuous-wave superradiance. Thus, spasers, which mainly generate near-fields, become an efficient source of far-field radiation when the interaction between them is sufficiently strong.

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  1. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006), p. 558.
  2. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).
  3. V. M. Shalaev and S. Kawata, eds., Nanophotonics with Surface Plasmons, Advances in Nano-Optics and Nano-Photonics (Elsevier, 2007).
  4. D. J. Bergman and M. I. Stockman, “Surface Plasmon amplification by stimulated emission of radiation: Quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett.90(2), 027402 (2003).
    [CrossRef] [PubMed]
  5. M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature460(7259), 1110–1112 (2009).
    [CrossRef] [PubMed]
  6. Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic Nanolaser using epitaxially grown silver film,” Science337(6093), 450–453 (2012).
    [CrossRef] [PubMed]
  7. I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys. Rev. A71(6), 063812 (2005).
    [CrossRef]
  8. A. A. Kolokolov and G. V. Skrotskii, “Interference of reactive components of an electromagnetic field,” Sov. Phys. Usp.35(12), 1089–1093 (1992).
    [CrossRef]
  9. V. S. Zuev and G. Y. Zueva, “Very slow surface plasmons: Theory and practice (Review),” Opt. Spectrosc.107(4), 614–628 (2009).
    [CrossRef]
  10. A. P. Vinogradov and A. V. Dorofeenko, “Destruction of the image of the Pendry lens during detection,” Opt. Commun.256(4-6), 333–336 (2005).
    [CrossRef]
  11. M. I. Stockman, “Nanoplasmonics: past, present, and glimpse into future,” Opt. Express19(22), 22029–22106 (2011).
    [CrossRef] [PubMed]
  12. M. Premaratne and G. P. Agrawal, Light Propagation in Gain Medium (Cambridge University Press, 2011).
  13. L. C. Davis, “Electostatic edge modes of a dielectric wedge,” Phys. Rev. B14(12), 5523–5525 (1976).
    [CrossRef]
  14. A. Eguiluz and A. A. Maradudin, “Electrostatic edge modes along a parabolic wedge,” Phys. Rev. B14(12), 5526–5528 (1976).
    [CrossRef]
  15. J. B. Pendry, J. B. Holden, D. J. Robbins, and W. J. Stewart, “Low frequency plasmons in thin-wire structures,” J. Phys. Condens. Matter10(22), 4785–4809 (1998).
    [CrossRef]
  16. I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B66(3), 035403 (2002).
    [CrossRef]
  17. W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B70(12), 125429 (2004).
    [CrossRef]
  18. T. Okamoto, F. H’Dhili, and S. Kawata, “Towards plasmonic band gap laser,” Appl. Phys. Lett.85(18), 3968–3970 (2004).
    [CrossRef]
  19. A. Banerjee, R. Li, and H. Grebel, “Surface plasmon lasers with quantum dots as gain media,” Appl. Phys. Lett.95(25), 251106 (2009).
    [CrossRef]
  20. Z.-G. Dong, H. Liu, T. Li, Z.-H. Zhu, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Modeling the directed transmission and reflection enhancements of the lasing surface plasmon amplification by stimulated emission of radiation in active metamaterials,” Phys. Rev. B80(23), 235116 (2009).
    [CrossRef]
  21. D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics4(2), 83–91 (2010).
    [CrossRef]
  22. M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett.23(17), 1331–1333 (1998).
    [CrossRef] [PubMed]
  23. S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B67(20), 205402 (2003).
    [CrossRef]
  24. E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Stationary behavior of a chain of interacting spasers,” Phys. Rev. B85(16), 165419 (2012).
    [CrossRef]
  25. E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Forced synchronization of spaser by an external optical wave,” Opt. Express19(25), 24849–24857 (2011).
    [CrossRef] [PubMed]
  26. E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Dipole Response of Spaser on an External Optical Wave,” Opt. Lett.36(21), 4302–4304 (2011).
    [CrossRef] [PubMed]
  27. A. V. Klyuchnik, S. Y. Kurganov, and Y. E. Lozovik, “Plasmons at a hole in a screen,” Phys. Solid State45(9), 1793–1797 (2003).
    [CrossRef]
  28. A. N. Oraevsky, “Resonant properties of a system comprising a cavity mode and two-level atoms and frequency bistability,” Quantum Electron.29(11), 975–978 (1999).
    [CrossRef]
  29. R. H. Pantell and H. E. Puthoff, Fundamentals of Quantum Electronics (Wiley, 1969).
  30. M. Sargent and P. Meystre, Elements of Quantum Optics (Springer-Verlag Berlin Heidelberg, 2007), p. 508.
  31. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, 1997).
  32. Y. I. Khanin, Fundamentals of Laser Dynamics (Cambridge Int Science Publishing, 2006).
  33. R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev.93(1), 99–110 (1954).
    [CrossRef]
  34. M. Gross and S. Haroche, “Superradiance: An essay on the theory of collective spontaneous emission,” Phys. Rep.93(5), 301–396 (1982).
    [CrossRef]
  35. J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature484(7392), 78–81 (2012).
    [CrossRef] [PubMed]
  36. V. N. Pustovit and T. V. Shahbazyan, “Cooperative emission of light by an ensemble of dipoles near a metal nanoparticle: The plasmonic Dicke effect,” Phys. Rev. Lett.102(7), 077401 (2009).
    [CrossRef] [PubMed]
  37. C. A. Balanis, Antenna Theory - Analysis and Design, 3rd Ed. (Willey-Interscience, 2005).

2012 (3)

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic Nanolaser using epitaxially grown silver film,” Science337(6093), 450–453 (2012).
[CrossRef] [PubMed]

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature484(7392), 78–81 (2012).
[CrossRef] [PubMed]

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Stationary behavior of a chain of interacting spasers,” Phys. Rev. B85(16), 165419 (2012).
[CrossRef]

2011 (3)

2010 (1)

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics4(2), 83–91 (2010).
[CrossRef]

2009 (5)

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature460(7259), 1110–1112 (2009).
[CrossRef] [PubMed]

V. S. Zuev and G. Y. Zueva, “Very slow surface plasmons: Theory and practice (Review),” Opt. Spectrosc.107(4), 614–628 (2009).
[CrossRef]

V. N. Pustovit and T. V. Shahbazyan, “Cooperative emission of light by an ensemble of dipoles near a metal nanoparticle: The plasmonic Dicke effect,” Phys. Rev. Lett.102(7), 077401 (2009).
[CrossRef] [PubMed]

A. Banerjee, R. Li, and H. Grebel, “Surface plasmon lasers with quantum dots as gain media,” Appl. Phys. Lett.95(25), 251106 (2009).
[CrossRef]

Z.-G. Dong, H. Liu, T. Li, Z.-H. Zhu, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Modeling the directed transmission and reflection enhancements of the lasing surface plasmon amplification by stimulated emission of radiation in active metamaterials,” Phys. Rev. B80(23), 235116 (2009).
[CrossRef]

2005 (2)

A. P. Vinogradov and A. V. Dorofeenko, “Destruction of the image of the Pendry lens during detection,” Opt. Commun.256(4-6), 333–336 (2005).
[CrossRef]

I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys. Rev. A71(6), 063812 (2005).
[CrossRef]

2004 (2)

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B70(12), 125429 (2004).
[CrossRef]

T. Okamoto, F. H’Dhili, and S. Kawata, “Towards plasmonic band gap laser,” Appl. Phys. Lett.85(18), 3968–3970 (2004).
[CrossRef]

2003 (3)

A. V. Klyuchnik, S. Y. Kurganov, and Y. E. Lozovik, “Plasmons at a hole in a screen,” Phys. Solid State45(9), 1793–1797 (2003).
[CrossRef]

D. J. Bergman and M. I. Stockman, “Surface Plasmon amplification by stimulated emission of radiation: Quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett.90(2), 027402 (2003).
[CrossRef] [PubMed]

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B67(20), 205402 (2003).
[CrossRef]

2002 (1)

I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B66(3), 035403 (2002).
[CrossRef]

1999 (1)

A. N. Oraevsky, “Resonant properties of a system comprising a cavity mode and two-level atoms and frequency bistability,” Quantum Electron.29(11), 975–978 (1999).
[CrossRef]

1998 (2)

J. B. Pendry, J. B. Holden, D. J. Robbins, and W. J. Stewart, “Low frequency plasmons in thin-wire structures,” J. Phys. Condens. Matter10(22), 4785–4809 (1998).
[CrossRef]

M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett.23(17), 1331–1333 (1998).
[CrossRef] [PubMed]

1992 (1)

A. A. Kolokolov and G. V. Skrotskii, “Interference of reactive components of an electromagnetic field,” Sov. Phys. Usp.35(12), 1089–1093 (1992).
[CrossRef]

1982 (1)

M. Gross and S. Haroche, “Superradiance: An essay on the theory of collective spontaneous emission,” Phys. Rep.93(5), 301–396 (1982).
[CrossRef]

1976 (2)

L. C. Davis, “Electostatic edge modes of a dielectric wedge,” Phys. Rev. B14(12), 5523–5525 (1976).
[CrossRef]

A. Eguiluz and A. A. Maradudin, “Electrostatic edge modes along a parabolic wedge,” Phys. Rev. B14(12), 5526–5528 (1976).
[CrossRef]

1954 (1)

R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev.93(1), 99–110 (1954).
[CrossRef]

Andrianov, E. S.

Atwater, H. A.

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B67(20), 205402 (2003).
[CrossRef]

Aussenegg, F. R.

Bakker, R.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature460(7259), 1110–1112 (2009).
[CrossRef] [PubMed]

Banerjee, A.

A. Banerjee, R. Li, and H. Grebel, “Surface plasmon lasers with quantum dots as gain media,” Appl. Phys. Lett.95(25), 251106 (2009).
[CrossRef]

Belgrave, A. M.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature460(7259), 1110–1112 (2009).
[CrossRef] [PubMed]

Bergman, D. J.

D. J. Bergman and M. I. Stockman, “Surface Plasmon amplification by stimulated emission of radiation: Quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett.90(2), 027402 (2003).
[CrossRef] [PubMed]

Bohnet, J. G.

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature484(7392), 78–81 (2012).
[CrossRef] [PubMed]

Bozhevolnyi, S. I.

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics4(2), 83–91 (2010).
[CrossRef]

Cao, J.-X.

Z.-G. Dong, H. Liu, T. Li, Z.-H. Zhu, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Modeling the directed transmission and reflection enhancements of the lasing surface plasmon amplification by stimulated emission of radiation in active metamaterials,” Phys. Rev. B80(23), 235116 (2009).
[CrossRef]

Chang, W.-H.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic Nanolaser using epitaxially grown silver film,” Science337(6093), 450–453 (2012).
[CrossRef] [PubMed]

Chen, H.-Y.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic Nanolaser using epitaxially grown silver film,” Science337(6093), 450–453 (2012).
[CrossRef] [PubMed]

Chen, L.-J.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic Nanolaser using epitaxially grown silver film,” Science337(6093), 450–453 (2012).
[CrossRef] [PubMed]

Chen, Z.

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature484(7392), 78–81 (2012).
[CrossRef] [PubMed]

Dabidian, N.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic Nanolaser using epitaxially grown silver film,” Science337(6093), 450–453 (2012).
[CrossRef] [PubMed]

Davis, L. C.

L. C. Davis, “Electostatic edge modes of a dielectric wedge,” Phys. Rev. B14(12), 5523–5525 (1976).
[CrossRef]

Dicke, R. H.

R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev.93(1), 99–110 (1954).
[CrossRef]

Dong, Z.-G.

Z.-G. Dong, H. Liu, T. Li, Z.-H. Zhu, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Modeling the directed transmission and reflection enhancements of the lasing surface plasmon amplification by stimulated emission of radiation in active metamaterials,” Phys. Rev. B80(23), 235116 (2009).
[CrossRef]

Dorofeenko, A. V.

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Stationary behavior of a chain of interacting spasers,” Phys. Rev. B85(16), 165419 (2012).
[CrossRef]

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Forced synchronization of spaser by an external optical wave,” Opt. Express19(25), 24849–24857 (2011).
[CrossRef] [PubMed]

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Dipole Response of Spaser on an External Optical Wave,” Opt. Lett.36(21), 4302–4304 (2011).
[CrossRef] [PubMed]

A. P. Vinogradov and A. V. Dorofeenko, “Destruction of the image of the Pendry lens during detection,” Opt. Commun.256(4-6), 333–336 (2005).
[CrossRef]

Eguiluz, A.

A. Eguiluz and A. A. Maradudin, “Electrostatic edge modes along a parabolic wedge,” Phys. Rev. B14(12), 5526–5528 (1976).
[CrossRef]

Ford, G. W.

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B70(12), 125429 (2004).
[CrossRef]

Gramotnev, D. K.

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics4(2), 83–91 (2010).
[CrossRef]

Grebel, H.

A. Banerjee, R. Li, and H. Grebel, “Surface plasmon lasers with quantum dots as gain media,” Appl. Phys. Lett.95(25), 251106 (2009).
[CrossRef]

Gross, M.

M. Gross and S. Haroche, “Superradiance: An essay on the theory of collective spontaneous emission,” Phys. Rep.93(5), 301–396 (1982).
[CrossRef]

Gwo, S.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic Nanolaser using epitaxially grown silver film,” Science337(6093), 450–453 (2012).
[CrossRef] [PubMed]

H’Dhili, F.

T. Okamoto, F. H’Dhili, and S. Kawata, “Towards plasmonic band gap laser,” Appl. Phys. Lett.85(18), 3968–3970 (2004).
[CrossRef]

Haroche, S.

M. Gross and S. Haroche, “Superradiance: An essay on the theory of collective spontaneous emission,” Phys. Rep.93(5), 301–396 (1982).
[CrossRef]

Herz, E.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature460(7259), 1110–1112 (2009).
[CrossRef] [PubMed]

Holden, J. B.

J. B. Pendry, J. B. Holden, D. J. Robbins, and W. J. Stewart, “Low frequency plasmons in thin-wire structures,” J. Phys. Condens. Matter10(22), 4785–4809 (1998).
[CrossRef]

Holland, M. J.

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature484(7392), 78–81 (2012).
[CrossRef] [PubMed]

Kawata, S.

T. Okamoto, F. H’Dhili, and S. Kawata, “Towards plasmonic band gap laser,” Appl. Phys. Lett.85(18), 3968–3970 (2004).
[CrossRef]

Kik, P. G.

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B67(20), 205402 (2003).
[CrossRef]

Kim, J.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic Nanolaser using epitaxially grown silver film,” Science337(6093), 450–453 (2012).
[CrossRef] [PubMed]

Klyuchnik, A. V.

A. V. Klyuchnik, S. Y. Kurganov, and Y. E. Lozovik, “Plasmons at a hole in a screen,” Phys. Solid State45(9), 1793–1797 (2003).
[CrossRef]

Kolokolov, A. A.

A. A. Kolokolov and G. V. Skrotskii, “Interference of reactive components of an electromagnetic field,” Sov. Phys. Usp.35(12), 1089–1093 (1992).
[CrossRef]

Krenn, J. R.

Kurganov, S. Y.

A. V. Klyuchnik, S. Y. Kurganov, and Y. E. Lozovik, “Plasmons at a hole in a screen,” Phys. Solid State45(9), 1793–1797 (2003).
[CrossRef]

Leitner, A.

Li, B.-H.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic Nanolaser using epitaxially grown silver film,” Science337(6093), 450–453 (2012).
[CrossRef] [PubMed]

Li, R.

A. Banerjee, R. Li, and H. Grebel, “Surface plasmon lasers with quantum dots as gain media,” Appl. Phys. Lett.95(25), 251106 (2009).
[CrossRef]

Li, T.

Z.-G. Dong, H. Liu, T. Li, Z.-H. Zhu, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Modeling the directed transmission and reflection enhancements of the lasing surface plasmon amplification by stimulated emission of radiation in active metamaterials,” Phys. Rev. B80(23), 235116 (2009).
[CrossRef]

Lisyansky, A. A.

Liu, H.

Z.-G. Dong, H. Liu, T. Li, Z.-H. Zhu, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Modeling the directed transmission and reflection enhancements of the lasing surface plasmon amplification by stimulated emission of radiation in active metamaterials,” Phys. Rev. B80(23), 235116 (2009).
[CrossRef]

Lozovik, Y. E.

A. V. Klyuchnik, S. Y. Kurganov, and Y. E. Lozovik, “Plasmons at a hole in a screen,” Phys. Solid State45(9), 1793–1797 (2003).
[CrossRef]

Lu, M.-Y.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic Nanolaser using epitaxially grown silver film,” Science337(6093), 450–453 (2012).
[CrossRef] [PubMed]

Lu, Y.-J.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic Nanolaser using epitaxially grown silver film,” Science337(6093), 450–453 (2012).
[CrossRef] [PubMed]

Maier, S. A.

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B67(20), 205402 (2003).
[CrossRef]

Maradudin, A. A.

I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B66(3), 035403 (2002).
[CrossRef]

A. Eguiluz and A. A. Maradudin, “Electrostatic edge modes along a parabolic wedge,” Phys. Rev. B14(12), 5526–5528 (1976).
[CrossRef]

Meiser, D.

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature484(7392), 78–81 (2012).
[CrossRef] [PubMed]

Narimanov, E. E.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature460(7259), 1110–1112 (2009).
[CrossRef] [PubMed]

Noginov, M. A.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature460(7259), 1110–1112 (2009).
[CrossRef] [PubMed]

Novikov, I. V.

I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B66(3), 035403 (2002).
[CrossRef]

O’Reilly, E. P.

I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys. Rev. A71(6), 063812 (2005).
[CrossRef]

Okamoto, T.

T. Okamoto, F. H’Dhili, and S. Kawata, “Towards plasmonic band gap laser,” Appl. Phys. Lett.85(18), 3968–3970 (2004).
[CrossRef]

Oraevsky, A. N.

A. N. Oraevsky, “Resonant properties of a system comprising a cavity mode and two-level atoms and frequency bistability,” Quantum Electron.29(11), 975–978 (1999).
[CrossRef]

Pendry, J. B.

J. B. Pendry, J. B. Holden, D. J. Robbins, and W. J. Stewart, “Low frequency plasmons in thin-wire structures,” J. Phys. Condens. Matter10(22), 4785–4809 (1998).
[CrossRef]

Protsenko, I. E.

I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys. Rev. A71(6), 063812 (2005).
[CrossRef]

Pukhov, A. A.

Pustovit, V. N.

V. N. Pustovit and T. V. Shahbazyan, “Cooperative emission of light by an ensemble of dipoles near a metal nanoparticle: The plasmonic Dicke effect,” Phys. Rev. Lett.102(7), 077401 (2009).
[CrossRef] [PubMed]

Qiu, X.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic Nanolaser using epitaxially grown silver film,” Science337(6093), 450–453 (2012).
[CrossRef] [PubMed]

Quinten, M.

Robbins, D. J.

J. B. Pendry, J. B. Holden, D. J. Robbins, and W. J. Stewart, “Low frequency plasmons in thin-wire structures,” J. Phys. Condens. Matter10(22), 4785–4809 (1998).
[CrossRef]

Samoilov, V. N.

I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys. Rev. A71(6), 063812 (2005).
[CrossRef]

Sanders, C. E.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic Nanolaser using epitaxially grown silver film,” Science337(6093), 450–453 (2012).
[CrossRef] [PubMed]

Shahbazyan, T. V.

V. N. Pustovit and T. V. Shahbazyan, “Cooperative emission of light by an ensemble of dipoles near a metal nanoparticle: The plasmonic Dicke effect,” Phys. Rev. Lett.102(7), 077401 (2009).
[CrossRef] [PubMed]

Shalaev, V. M.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature460(7259), 1110–1112 (2009).
[CrossRef] [PubMed]

Shih, C.-K.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic Nanolaser using epitaxially grown silver film,” Science337(6093), 450–453 (2012).
[CrossRef] [PubMed]

Shvets, G.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic Nanolaser using epitaxially grown silver film,” Science337(6093), 450–453 (2012).
[CrossRef] [PubMed]

Skrotskii, G. V.

A. A. Kolokolov and G. V. Skrotskii, “Interference of reactive components of an electromagnetic field,” Sov. Phys. Usp.35(12), 1089–1093 (1992).
[CrossRef]

Stewart, W. J.

J. B. Pendry, J. B. Holden, D. J. Robbins, and W. J. Stewart, “Low frequency plasmons in thin-wire structures,” J. Phys. Condens. Matter10(22), 4785–4809 (1998).
[CrossRef]

Stockman, M. I.

M. I. Stockman, “Nanoplasmonics: past, present, and glimpse into future,” Opt. Express19(22), 22029–22106 (2011).
[CrossRef] [PubMed]

D. J. Bergman and M. I. Stockman, “Surface Plasmon amplification by stimulated emission of radiation: Quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett.90(2), 027402 (2003).
[CrossRef] [PubMed]

Stout, S.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature460(7259), 1110–1112 (2009).
[CrossRef] [PubMed]

Suteewong, T.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature460(7259), 1110–1112 (2009).
[CrossRef] [PubMed]

Thompson, J. K.

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature484(7392), 78–81 (2012).
[CrossRef] [PubMed]

Uskov, A. V.

I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys. Rev. A71(6), 063812 (2005).
[CrossRef]

Vinogradov, A. P.

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Stationary behavior of a chain of interacting spasers,” Phys. Rev. B85(16), 165419 (2012).
[CrossRef]

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Forced synchronization of spaser by an external optical wave,” Opt. Express19(25), 24849–24857 (2011).
[CrossRef] [PubMed]

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Dipole Response of Spaser on an External Optical Wave,” Opt. Lett.36(21), 4302–4304 (2011).
[CrossRef] [PubMed]

A. P. Vinogradov and A. V. Dorofeenko, “Destruction of the image of the Pendry lens during detection,” Opt. Commun.256(4-6), 333–336 (2005).
[CrossRef]

Wang, C.-Y.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic Nanolaser using epitaxially grown silver film,” Science337(6093), 450–453 (2012).
[CrossRef] [PubMed]

Wang, S.-M.

Z.-G. Dong, H. Liu, T. Li, Z.-H. Zhu, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Modeling the directed transmission and reflection enhancements of the lasing surface plasmon amplification by stimulated emission of radiation in active metamaterials,” Phys. Rev. B80(23), 235116 (2009).
[CrossRef]

Weber, W. H.

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B70(12), 125429 (2004).
[CrossRef]

Weiner, J. M.

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature484(7392), 78–81 (2012).
[CrossRef] [PubMed]

Wiesner, U.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature460(7259), 1110–1112 (2009).
[CrossRef] [PubMed]

Wu, C.

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic Nanolaser using epitaxially grown silver film,” Science337(6093), 450–453 (2012).
[CrossRef] [PubMed]

Zaimidoroga, O. A.

I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys. Rev. A71(6), 063812 (2005).
[CrossRef]

Zhang, X.

Z.-G. Dong, H. Liu, T. Li, Z.-H. Zhu, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Modeling the directed transmission and reflection enhancements of the lasing surface plasmon amplification by stimulated emission of radiation in active metamaterials,” Phys. Rev. B80(23), 235116 (2009).
[CrossRef]

Zhu, G.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature460(7259), 1110–1112 (2009).
[CrossRef] [PubMed]

Zhu, S.-N.

Z.-G. Dong, H. Liu, T. Li, Z.-H. Zhu, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Modeling the directed transmission and reflection enhancements of the lasing surface plasmon amplification by stimulated emission of radiation in active metamaterials,” Phys. Rev. B80(23), 235116 (2009).
[CrossRef]

Zhu, Z.-H.

Z.-G. Dong, H. Liu, T. Li, Z.-H. Zhu, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Modeling the directed transmission and reflection enhancements of the lasing surface plasmon amplification by stimulated emission of radiation in active metamaterials,” Phys. Rev. B80(23), 235116 (2009).
[CrossRef]

Zuev, V. S.

V. S. Zuev and G. Y. Zueva, “Very slow surface plasmons: Theory and practice (Review),” Opt. Spectrosc.107(4), 614–628 (2009).
[CrossRef]

Zueva, G. Y.

V. S. Zuev and G. Y. Zueva, “Very slow surface plasmons: Theory and practice (Review),” Opt. Spectrosc.107(4), 614–628 (2009).
[CrossRef]

Appl. Phys. Lett. (2)

T. Okamoto, F. H’Dhili, and S. Kawata, “Towards plasmonic band gap laser,” Appl. Phys. Lett.85(18), 3968–3970 (2004).
[CrossRef]

A. Banerjee, R. Li, and H. Grebel, “Surface plasmon lasers with quantum dots as gain media,” Appl. Phys. Lett.95(25), 251106 (2009).
[CrossRef]

J. Phys. Condens. Matter (1)

J. B. Pendry, J. B. Holden, D. J. Robbins, and W. J. Stewart, “Low frequency plasmons in thin-wire structures,” J. Phys. Condens. Matter10(22), 4785–4809 (1998).
[CrossRef]

Nat. Photonics (1)

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics4(2), 83–91 (2010).
[CrossRef]

Nature (2)

J. G. Bohnet, Z. Chen, J. M. Weiner, D. Meiser, M. J. Holland, and J. K. Thompson, “A steady-state superradiant laser with less than one intracavity photon,” Nature484(7392), 78–81 (2012).
[CrossRef] [PubMed]

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature460(7259), 1110–1112 (2009).
[CrossRef] [PubMed]

Opt. Commun. (1)

A. P. Vinogradov and A. V. Dorofeenko, “Destruction of the image of the Pendry lens during detection,” Opt. Commun.256(4-6), 333–336 (2005).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Opt. Spectrosc. (1)

V. S. Zuev and G. Y. Zueva, “Very slow surface plasmons: Theory and practice (Review),” Opt. Spectrosc.107(4), 614–628 (2009).
[CrossRef]

Phys. Rep. (1)

M. Gross and S. Haroche, “Superradiance: An essay on the theory of collective spontaneous emission,” Phys. Rep.93(5), 301–396 (1982).
[CrossRef]

Phys. Rev. (1)

R. H. Dicke, “Coherence in spontaneous radiation processes,” Phys. Rev.93(1), 99–110 (1954).
[CrossRef]

Phys. Rev. A (1)

I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys. Rev. A71(6), 063812 (2005).
[CrossRef]

Phys. Rev. B (7)

L. C. Davis, “Electostatic edge modes of a dielectric wedge,” Phys. Rev. B14(12), 5523–5525 (1976).
[CrossRef]

A. Eguiluz and A. A. Maradudin, “Electrostatic edge modes along a parabolic wedge,” Phys. Rev. B14(12), 5526–5528 (1976).
[CrossRef]

I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B66(3), 035403 (2002).
[CrossRef]

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B70(12), 125429 (2004).
[CrossRef]

Z.-G. Dong, H. Liu, T. Li, Z.-H. Zhu, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Modeling the directed transmission and reflection enhancements of the lasing surface plasmon amplification by stimulated emission of radiation in active metamaterials,” Phys. Rev. B80(23), 235116 (2009).
[CrossRef]

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B67(20), 205402 (2003).
[CrossRef]

E. S. Andrianov, A. A. Pukhov, A. V. Dorofeenko, A. P. Vinogradov, and A. A. Lisyansky, “Stationary behavior of a chain of interacting spasers,” Phys. Rev. B85(16), 165419 (2012).
[CrossRef]

Phys. Rev. Lett. (2)

V. N. Pustovit and T. V. Shahbazyan, “Cooperative emission of light by an ensemble of dipoles near a metal nanoparticle: The plasmonic Dicke effect,” Phys. Rev. Lett.102(7), 077401 (2009).
[CrossRef] [PubMed]

D. J. Bergman and M. I. Stockman, “Surface Plasmon amplification by stimulated emission of radiation: Quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett.90(2), 027402 (2003).
[CrossRef] [PubMed]

Phys. Solid State (1)

A. V. Klyuchnik, S. Y. Kurganov, and Y. E. Lozovik, “Plasmons at a hole in a screen,” Phys. Solid State45(9), 1793–1797 (2003).
[CrossRef]

Quantum Electron. (1)

A. N. Oraevsky, “Resonant properties of a system comprising a cavity mode and two-level atoms and frequency bistability,” Quantum Electron.29(11), 975–978 (1999).
[CrossRef]

Science (1)

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic Nanolaser using epitaxially grown silver film,” Science337(6093), 450–453 (2012).
[CrossRef] [PubMed]

Sov. Phys. Usp. (1)

A. A. Kolokolov and G. V. Skrotskii, “Interference of reactive components of an electromagnetic field,” Sov. Phys. Usp.35(12), 1089–1093 (1992).
[CrossRef]

Other (9)

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, 2006), p. 558.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

V. M. Shalaev and S. Kawata, eds., Nanophotonics with Surface Plasmons, Advances in Nano-Optics and Nano-Photonics (Elsevier, 2007).

M. Premaratne and G. P. Agrawal, Light Propagation in Gain Medium (Cambridge University Press, 2011).

R. H. Pantell and H. E. Puthoff, Fundamentals of Quantum Electronics (Wiley, 1969).

M. Sargent and P. Meystre, Elements of Quantum Optics (Springer-Verlag Berlin Heidelberg, 2007), p. 508.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, 1997).

Y. I. Khanin, Fundamentals of Laser Dynamics (Cambridge Int Science Publishing, 2006).

C. A. Balanis, Antenna Theory - Analysis and Design, 3rd Ed. (Willey-Interscience, 2005).

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Figures (4)

Fig. 1
Fig. 1

Phase distribution of the plasmon oscillations in spaser arrays of (a) 5 × 5 and (b) 100 × 100 spasers. In all calculations, we use Δ = λ /20, where λ is the free space wavelength.

Fig. 2
Fig. 2

Phase distribution of the plasmon oscillations in spaser arrays of (a) 5 × 5 and (b) 100 × 100 spasers. In all calculations, we use Δ = λ /20, where λ is the free space wavelength.

Fig. 3
Fig. 3

Phase distribution of plasmon oscillations in the 100 × 100 spaser array with Ω R1 =0 .

Fig. 4
Fig. 4

The integral intensity of radiation per spaser for the array with calculated amplitude distribution (solid line) and the array of ideally synchronized dipoles (dashed line) as a function of the array size. The dependence for a small number of spasers is magnified in the inset.

Equations (11)

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

Ω( eR )= τ R 1 3 2 ( 3 (e e x ) 2 1 ( k 0 R ) 3 i 3 (e e x ) 2 1 ( k 0 R ) 2 (e e x ) 2 1 k 0 R )exp(ikR),
a ˙ n + τ a 1 a n =i Ω R σ n i Ω R1 | mn |=1 σ m +i mn Ω nm a m ,
σ ˙ n + τ σ 1 σ n =i Ω R a n D n +i Ω R1 | mn |=1 a m D m ,
D ˙ n +( D n D 0 ) τ D 1 =2i Ω R ( a n σ n σ n a n )+2i Ω R1 | mn |=1 ( a m σ n a m σ n ) ,
Ω R_eff = Ω R +2 Ω R1 ( cos q x Δ+cos q y Δ ), τ a_eff 1 = τ J 1 + Δ 2 Ω( eR ) d 2 R , δ eff 3 τ R 1 ( k 0 Δ ) 3 ( 2cos q x Δcos q y Δ ).
Ω( eR ) τ R 1 ( 3 2 3 (e e x ) 2 1 ( k 0 R ) 3 +i )
i mn Ω nm a m i a n mn Re Ω nm a n ( N1 ) τ R 1
a ˙ n +( τ J 1 +N τ R 1 ) a n =i Ω R σ n i Ω R1 | mn |=1 σ m +iRe mn Ω nm a m
n [ Ω R Im( a n σ n )+ Ω R1 | mn |=1 Im( a n σ m ) ] = τ J 1 n | a n | 2 + n,m Im( Ω nm )Re( a n a m ) .
I= m ω 2 e 2 n,m Im( Ω nm )Re( a n a m ) .
I Ω ( e )= I Ω ( 0 ) ( e ) | n a n exp( i k 0 e r n ) | 2 ,

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