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.

©2013 Optical Society of America

1. Introduction

Recent developments of nanotechnologies incorporating plasmonic structures have led to the design of a new generation of components for optoelectronics operating in the deep subwavelength regime. Such devices include plasmonic waveguides, spectrometers and microscopes, in which local fields are enhanced by plasmonic resonance [13]. In 2003 Bergman and Stockman [4] proposed a quantum-plasmonic device, the spaser, that was demonstrated experimentally [5, 6]. Schematically, the spaser is an inversely populated two-level system (TLS), e.g. an atom, a molecule, or a quantum dot, placed near a plasmonic nanoparticle (NP) [4, 7]. Due to small length scales (d < 20 nm), energy transfer from the TLS to the NP is provided by near-fields [810]. Spasers have never been considered as efficient sources of light radiation but rather as systems that create high local intensity of the electric field and enhance nonlinear effects [11].

Boosting energy extraction from spasers is of special interest [12]. It is mainly concerned with excitation and support of eigenmodes of plasmonic transmission lines [2, 1321]. In particular, a chain of plasmonic NPs with dipole-dipole interaction supports travelling waves of NP dipole moments [22, 23]. The electromagnetic field of such eigenmodes is strongly localized on the chain. The synchronized modes with zero and near-zero wave numbers radiate leaky waves, which cause the far-field interaction of NPs. However, this interaction changes the dispersion curve and destroys the synchronization [17]. When the wavenumber decreases and tends to that of free space, the eigenfrequency tends to zero and does not reach the plasmonic resonance.

Chains of spasers also support travelling waves [24]. Due to the nonlinear nature of such systems, mode competition leads to survival of only one wave with fixed frequency and wavenumber. For weak TLS-NP interactions, this is a solution with a high wavenumber q (qω/c=k0). In this case, only near-field interactions need to be taken into account [17]. For high values of the TLS-NP coupling constant, the theory predicts in-phase oscillations accompanied by radiation of leaking waves. A consistent description of such synchronized states requires taking into account radiation and retardation effects which is done in this paper. We show that in a large 2D array, spasers can be mutually synchronized despite radiation and the far-field interaction. This synchronization arises due to the interaction of the spaser’s TLS with plasmonic particles of neighboring spasers. Synchronization results in superradiance. For arrays smaller than the free space wavelength, the interference of radiated fields is constructive in all directions, and the radiation intensity power depends on the number of spasers, N, as N2. For larger systems, the interference becomes destructive for almost all directions and the total radiation power is linear in N. Only in the direction perpendicular to the array plane the interference is still constructive. In this direction, the power of radiation per solid angle is proportional to N2.

2. Synchronization of the spaser array

As the 2D array of spasers, we consider inversely populated TLS’s positioned near nano-holes in a metallic film, as shown in Fig. 1. A nano-hole with a dipole mode plays role of the NP. The spacing between holes Δ is much smaller than the wavelength. We consider a square array with sides L so that the number of spasers in the array is N=(L/Δ)2. The TLS can be electrically pumped via either a p-n junction or a quantum well parallel to the film. However, discrete TLS’s are more convenient for modeling.

 figure: 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.

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The dynamics of a single spaser is described by the system of three equations for operators of the plasmon amplitude an, the TLS polarization σn and the population inversion Dn [25, 26]. To properly describe radiation of the array, we have to take into account retardation while considering the interaction between the spasers. A spaser located at the point n={nx,ny} “feels” the local field Ωnmam=Ω((nm)Δ)am of a spaser, located at the point m={mx,my}. This field includes both far and near-fields:

Ω(eR)=τR132(3(eex)21(k0R)3i3(eex)21(k0R)2(eex)21k0R)exp(ikR),
where k0=ω/c is the wave number of radiation in vacuum, ex is the unit vector parallel to the dipole moments, eR is the vector connecting the dipoles. The local field given by Eq. (1) is the x-projection of the electric field created by a unitary dipole ex pointed in x direction, expressed in units of the inverse radiation relaxation rate, τR1. It can be shown that at the dipole resonance frequency ωr the value of τR1 for a NP of any shape is related to polarizability α(ω) as τR1=2k03/3(α1/ω)|ω=ωr. The expression for the polarizability of a hole in metal films was obtained in [27].

In the rotating wave approximation [2831], the system of interacting spasers is described by the system of equations

a˙n+τa1an=iΩRσniΩR1|mn|=1σm+imnΩnmam,
σ˙n+τσ1σn=iΩRanDn+iΩR1|mn|=1amDm,
D˙n+(DnD0)τD1=2iΩR(anσnσnan)+2iΩR1|mn|=1(amσnamσn),
where τσ and τD denote relaxation times of the polarization and the population inversion, respectively, D0 describes pumping of a TLS and corresponds to the population inversion in absence of NPs. In Eqs. (2)-(4), ΩR and ΩR1 (we consider ΩR=110τR1 and ΩR1=0.3ΩR) characterizes the interaction of a TLS with the adjacent and neighboring NPs, respectively. ΩR and ΩR1 are real-valued quantities which correspond to neglecting retardation and taking into account only near-fields in the TLS-NP interaction. The latter is justified by the small distance between the TLS and the NP. This interaction synchronizes the spaser array [24]. To proceed, we neglect quantum correlations and fluctuations and substitute operators by c-numbers. The decay rate of plasmonic modes is determined by the Joule loss in the NP and radiation, τa1=τJ1+τR1 (in our calculations τJ1=27τR1).

The local field given by Eq. (1) is characteristic for free space. The only effect of the metal film that we take into account by having an imaginary part in the wavevector, k=k+ik, is attenuation of the wave in space (for calculations we assume that k=k0 and k/k=0.2).

Figure 2 shows phase distributions for small and large arrays, in which TLS’s interact with NPs of neighboring spasers (ΩR10). These distributions are obtained by solving dynamic Eqs. (2)(4) numerically until the system reaches a stationary state. For both small and large arrays, the phase distributions are almost uniform. In the large system, a considerable deviation from the uniform distribution only occurs near boundaries of the array in the x-direction, which is parallel to the direction of oscillations of the dipole moments [Fig. 2(b)]. This is a boundary effect for which the length scale is of the order of the free space wavelength. In addition, the phase exhibits weak spatial oscillations along the direction perpendicular to the direction of dipoles. These oscillations increase in lossless systems. Both of these effects are due to a non-uniform change of the effective relaxation times of spasers and they do not lead to significant changes in the radiation pattern of large lossy spaser arrays.

 figure: 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.

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The effect of synchronization is a result of competition of collective modes in which the oscillations of spasers in the array are modulated by the wavenumber q={qx,qy}. The solution of the dynamic equations found by computer simulation shows that the mode with q = 0 corresponding to the synchronized state wins the competition with other modes.

The winning mode is determined by the minimum value of pumping threshold, Dth(q) [24, 32]. For the spaser array the threshold pumping may be written in the form Dth(q)=ΩR_eff2τa_eff1τσ1(1+δeff2τa_eff2), which is identical to that for a single spaser [25], with the only difference that ΩR, τa, and frequency detuning, δ, are replaced by the values modified by interactions of a NP with TLS’s and NPs of neighboring spasers:

ΩR_eff=ΩR+2ΩR1(cosqxΔ+cosqyΔ),τa_eff1=τJ1+Δ2Ω(eR)d2R,δeff3τR1(k0Δ)3(2cosqxΔcosqyΔ).
The integral in the second equation can be calculated in the case q=0. In addition, if k0, one obtains τa_eff1τJ1+3πτR1k02Δ2.

The value of the Dth(q) is determined by three factors, ΩR_eff2, τa_eff1, and 1+δeff2τa_eff2, given in Eqs. (5). The factor ΩR_eff2 has a minimum at q=0. Indeed, in the synchronized mode there is a constructive interference of the fields induced on a TLS by the neighboring NPs. Due to this interference, the corresponding coupling and the energy transfer are the most efficient. This mechanism of synchronization was considered in [24]. On the other hand, the second factor, τa_eff1, increases with vanishing k due to radiation losses, which causes increase of Dth. Due to this factor a desynchronized mode with minimal radiation should win in the mode competition. At the same time, increase of τa_eff1 makes the third factor, 1+δeff2τa_eff2, insignificant, because it becomes of the order of unity for any q for large arrays.

Thus, we have two competing mechanisms: (1) the synchronized mode extracts energy from TLS’s more efficiently than the modes with q0. This synchronizes the system and maximizes the radiation. (2) The synchronized mode has larger radiative losses and larger τa_eff1, which desynchronizes the system and minimizes radiation.

The first mechanism is dominant for our, quite realistic, parameters. Hence, the q=0 mode provides minimum for Dth(q) and wins the competition. This makes the spaser array self-synchronize.

For a smaller value of the interaction constant ΩR1, the synchronization state is destroyed. The surviving mode becomes the one with q0. In this case, the second mechanism determines the winning mode as the one with lower radiation losses [13].

The key role of the interaction of a TLS with neighboring NPs in the effect of synchronization is confirmed by our computer simulation of Eqs. (2)(4). If ΩR1=0, almost ideally synchronized state turns into a state with the phase distribution that corresponds to absence of synchronization shown in Fig. 3.

 figure: Fig. 3

Fig. 3 Phase distribution of plasmon oscillations in the 100 × 100 spaser array with ΩR1=0.

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The considered mechanism of synchronization is principally different from the one studied by Dicke [33]. Indeed, in our case, the synchronization occurs due to the near-field dipole-dipole interaction of the spasers, which destroys Dicke’s synchronization.

3. Superradiance of the spaser array

The ability of the spaser array to self-synchronize solves the problem of the radiation extraction from spasers. In a synchronized array, interaction given by Eq. (1) forces spasers to emit radiation. This phenomenon is more evident for a system of a small size, k0R1, for which Eq. (1) becomes

Ω(eR)τR1(323(eex)21(k0R)3+i)
with ImΩ(eR)τR1. When all the dipoles oscillate with the same phase and amplitude, the last term in Eq. (2) can be split into two parts:
imnΩnmamianmnReΩnman(N1)τR1
The second term at the right hand side of Eq. (7) leads to an increase in the relaxation rate of the n-th plasmon by the factor of (N1)τR1. As a result, the effective relaxation rate of a plasmon becomes τJ1+NτR1. Equation (2) can now be rewritten as
a˙n+(τJ1+NτR1)an=iΩRσniΩR1|mn|=1σm+iRemnΩnmam
Thus, all plasmonic NPs, in the area, which size is much smaller than the wavelength, contribute equally to the radiation rate giving an effective radiation rate of τR_eff1=NτR1. The total intensity of radiation of N NPs is then proportional to N2, which is characteristic for superradiance [3436]. Let us note that to obtain this dependence we use the full dipole field, Eq. (1), including the retardation effects. Hence, all terms in Eq. (1) are required to correctly account for radiative damping in the dipole (spaser) array.

The radiation power of the array can be evaluated by using the energy balance equation, which follows from Eq. (2) for the stationary regime:

n[ΩRIm(anσn)+ΩR1|mn|=1Im(anσm)]=τJ1n|an|2+n,mIm(Ωnm)Re(anam).
According to Eq. (6), ImΩ(eR)τR1 for R0. Therefore, the term with n=m in the sum n,m, can be evaluated with the account of ImΩnm=τR1. The left-hand side of Eq. (9) is proportional to the power supplied into the system by TLS’s. The first term in the right hand side corresponds to Joule losses. It can be shown that the second term, up to a factor independent of the dipole amplitudes, equals the radiation power I,
I=mω2e2n,mIm(Ωnm)Re(anam).
This term normalized by the radiation power of a single spaser is shown in Fig. 4.

 figure: 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.

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As long as the array size is smaller than the half-wavelength (N < 100 for our parameters), all spasers are synchronized and oscillate in phase [Fig. 2(a)]. In this case, as one can see in Fig. 4, intensity growth with the array size is characteristic for superradiance, I/NN. Also, for a small array, radiation from the array nearly coincides with the radiation of a system of synchronized classical dipoles with uniform amplitudes over the array equal to the average amplitude of a spaser an (see inset in Fig. 4). The radiated intensity drops as the array size increases. The reason for this is a decrease of an effective system size due to boundary effects [see Fig. 2(b)]. An increase of the array size to Lλ leads to synchronization of spasers for the most of the array, as shown in Fig. 2(b). Now, radiation per spaser of the array tends to catch up with the radiation of the ideally synchronized system (N > 5000 in Fig. 4). Due to destructive interference of radiation emitted by different parts of a large array, radiations of both spaser and ideally synchronized dipole arrays saturate, so that total radiation becomes proportional to the array size, IN.

The important quantity for applications is the intensity per solid angle radiated in the direction normal to the array plane. This is the quantity measured by a detector of a small size. We calculate this intensity considering the spaser array as an antenna. The distribution of power radiated by the antenna into a solid angle dΩ in the direction of a unit vector e, IΩ(e), is referred to as the radiation pattern and can be determined by the Fourier transform of the field distribution in the antenna opening [37]. In our case, this is the dipole moment distribution in the array. Thus, the radiation pattern is equal to

IΩ(e)=IΩ(0)(e)|nanexp(ik0ern)|2,
where an is the stationary solution of Eqs. (2)(4), rn is the position vector of the n-th dipole in the array and IΩ(0)(e)=(8π)1ck04|e×ex|2 is the radiation pattern for a unitary dipole. The integration of Eq. (11) over directions of e gives Eq. (10). For the normal direction, IΩ(0)(e) is just the sum IΩ=IΩ(0)(ez)|nan|2, which gives IΩ/N~N for a synchronized array of any size.

Interestingly, the linear increase of IΩ/N is determined by two effects. For small arrays, L<λ, this happens due to the growth of the integral intensity, I/N, resulting from superradiance. When L>λ, I/N saturates and the growth of IΩ is caused by the narrowing of the radiation pattern. The latter is due to the growth of the aperture of the radiative system.

The considered mechanism of superradiance differs from ones that we are aware of because of its continuous-wave character. Dicke’s superradiance of atoms [33] is a spontaneous radiation of synchronized molecules. It is of pulse nature because after photon radiation, all the atoms return to the ground state. Further pumping brings atoms back to the excited state but the next spontaneous radiation is uncorrelated in phase. The continuous-wave regime for Dicke’s superradiance can be achieved only if pumping is coherent [35]. In a spaser, surface plasmons arise due to multi-quantum excitations. Therefore, after radiating a single photon, the spaser oscillations continue without phase change. Due to stimulated emission, the surface plasmon wasted for photon radiation is regenerated with the same phase. This phenomenon is closer to radiation of an array of semiconductor lasers. However, it differs from the latter because synchronization of lasers is performed by far-fields and requires additional optical equipment (lens, mirrors, etc.). In the array of spasers, the synchronization is performed by near-fields automatically. This kind of synchronization turns out to be stable; it is not destroyed by interference in the far-field and by effects of retardation. As a consequence, the synchronization of 2D array of spasers is not constrained by the array size and superradiance occurs even for large apertures.

The considered effects are rather robust. The presence of disorder affects both superradiance and synchronization of the spaser arrays. A small disorder only slightly suppress superradiance leading to an incoherent radiation component, which is proportional to the square of the spaser density fluctuation, Δn2. This creates effective loss in the coherent component. Our computer simulations show that the synchronization is stable against disorder. Because variation of the spaser positions shift the NP-QD interaction frequencies, the presence of aperiodicity is to some extent equivalent to variations of ΩR1 in a periodic array. Modeling of the array with randomly variable ΩR1 shows that the effects do not disappear up to value of mean square deviation of ΩR1 up to 30%.

4. Conclusions

To summarize, we propose a highly directional continuous-wave coherent light source. It is based on a 2D array of incoherently pumped spasers, which are shown to mutually synchronize each other. This leads to superradiance, which can increase the radiation intensity by two orders of magnitude compared to a single spaser. We hope that the intrinsic property of spasers to self-synchronize will find the applications in optical communications. In this field the use of nanolasers, because of their fast response, may be a good alternative to the conventional laser arrays.

Acknowledgments

The work is partly supported by the RFBR grants 12-02-01093-а, 13-02-00407_а, 13-08-01062_а, and by the Dynasty Foundation.

References and links

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,” Nature 460(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,” Science 337(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. A 71(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. Express 19(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. B 14(12), 5523–5525 (1976). [CrossRef]  

14. A. Eguiluz and A. A. Maradudin, “Electrostatic edge modes along a parabolic wedge,” Phys. Rev. B 14(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. Matter 10(22), 4785–4809 (1998). [CrossRef]  

16. I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B 66(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. B 70(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. B 80(23), 235116 (2009). [CrossRef]  

21. D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(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. B 67(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. B 85(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. Express 19(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 State 45(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,” Nature 484(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).

References

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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,” Nature 460(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,” Science 337(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. A 71(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. Express 19(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. B 14(12), 5523–5525 (1976).
    [Crossref]
  14. A. Eguiluz and A. A. Maradudin, “Electrostatic edge modes along a parabolic wedge,” Phys. Rev. B 14(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. Matter 10(22), 4785–4809 (1998).
    [Crossref]
  16. I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B 66(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. B 70(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. B 80(23), 235116 (2009).
    [Crossref]
  21. D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(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. B 67(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. B 85(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. Express 19(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 State 45(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,” Nature 484(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,” Science 337(6093), 450–453 (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. B 85(16), 165419 (2012).
[Crossref]

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,” Nature 484(7392), 78–81 (2012).
[Crossref] [PubMed]

2011 (3)

2010 (1)

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010).
[Crossref]

2009 (5)

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. B 80(23), 235116 (2009).
[Crossref]

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

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,” Nature 460(7259), 1110–1112 (2009).
[Crossref] [PubMed]

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]

2005 (2)

I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys. Rev. A 71(6), 063812 (2005).
[Crossref]

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]

2004 (2)

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70(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)

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. B 67(20), 205402 (2003).
[Crossref]

A. V. Klyuchnik, S. Y. Kurganov, and Y. E. Lozovik, “Plasmons at a hole in a screen,” Phys. Solid State 45(9), 1793–1797 (2003).
[Crossref]

2002 (1)

I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B 66(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. Matter 10(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. B 14(12), 5523–5525 (1976).
[Crossref]

A. Eguiluz and A. A. Maradudin, “Electrostatic edge modes along a parabolic wedge,” Phys. Rev. B 14(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. B 67(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,” Nature 460(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,” Nature 460(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,” Nature 484(7392), 78–81 (2012).
[Crossref] [PubMed]

Bozhevolnyi, S. I.

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(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. B 80(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,” Science 337(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,” Science 337(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,” Science 337(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,” Nature 484(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,” Science 337(6093), 450–453 (2012).
[Crossref] [PubMed]

Davis, L. C.

L. C. Davis, “Electostatic edge modes of a dielectric wedge,” Phys. Rev. B 14(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. B 80(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. B 85(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. Express 19(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. B 14(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. B 70(12), 125429 (2004).
[Crossref]

Gramotnev, D. K.

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(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,” Science 337(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,” Nature 460(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. Matter 10(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,” Nature 484(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. B 67(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,” Science 337(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 State 45(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 State 45(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,” Science 337(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. B 80(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. B 80(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 State 45(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,” Science 337(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,” Science 337(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. B 67(20), 205402 (2003).
[Crossref]

Maradudin, A. A.

I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B 66(3), 035403 (2002).
[Crossref]

A. Eguiluz and A. A. Maradudin, “Electrostatic edge modes along a parabolic wedge,” Phys. Rev. B 14(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,” Nature 484(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,” Nature 460(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,” Nature 460(7259), 1110–1112 (2009).
[Crossref] [PubMed]

Novikov, I. V.

I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B 66(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. A 71(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. Matter 10(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. A 71(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,” Science 337(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. Matter 10(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. A 71(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,” Science 337(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,” Nature 460(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,” Science 337(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,” Science 337(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. Matter 10(22), 4785–4809 (1998).
[Crossref]

Stockman, M. I.

M. I. Stockman, “Nanoplasmonics: past, present, and glimpse into future,” Opt. Express 19(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,” Nature 460(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,” Nature 460(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,” Nature 484(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. A 71(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. B 85(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. Express 19(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,” Science 337(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. B 80(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. B 70(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,” Nature 484(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,” Nature 460(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,” Science 337(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. A 71(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. B 80(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,” Nature 460(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. B 80(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. B 80(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. Matter 10(22), 4785–4809 (1998).
[Crossref]

Nat. Photonics (1)

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(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,” Nature 484(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,” Nature 460(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. A 71(6), 063812 (2005).
[Crossref]

Phys. Rev. B (7)

L. C. Davis, “Electostatic edge modes of a dielectric wedge,” Phys. Rev. B 14(12), 5523–5525 (1976).
[Crossref]

A. Eguiluz and A. A. Maradudin, “Electrostatic edge modes along a parabolic wedge,” Phys. Rev. B 14(12), 5526–5528 (1976).
[Crossref]

I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B 66(3), 035403 (2002).
[Crossref]

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70(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. B 80(23), 235116 (2009).
[Crossref]

S. A. Maier, P. G. Kik, and H. A. Atwater, “Optical pulse propagation in metal nanoparticle chain waveguides,” Phys. Rev. B 67(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. B 85(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 State 45(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,” Science 337(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)

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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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