## Surface plasmons amplifications in single Ag nanoring

Optics Express, Vol. 18, Issue 5, pp. 4006-4011 (2010)

http://dx.doi.org/10.1364/OE.18.004006

Acrobat PDF (196 KB)

### Abstract

The stimulated amplifications of surface plasmons (SPs) propagating along a single silver nanoring is theoretically investigated by considering the interactions between SPs and activated semiconductor quantum dots (SQDs). Threshold condition for the stimulated amplifications, the SP density as a function of propagation length and the maximum SP density are obtained. The SPs can be nonlinearly amplified when the pumping rate of SQDs is larger than the threshold, and the maximum value of SP density increases linearly with the pumping rate of SQDs.

© 2010 OSA

## 1. Introduction

1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature **424**(6950), 824–830 (2003). [CrossRef] [PubMed]

3. S. A. Maier, “Plasmonics: The Promise of Highly Integrated Optical Devices,” IEEE J. Sel. Top. Quantum Electron. **12**(6), 1671–1677 (2006). [CrossRef]

4. K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. **78**(9), 1667–1670 (1997). [CrossRef]

5. S. Lal, S. Link, and N. S. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics **1**(11), 641–648 (2007). [CrossRef]

6. W. Srituravanich, N. Fang, C. Sun, Q. Luo, and X. Zhang, “Plasmonic nanolithography,” Nano Lett. **4**(6), 1085–1088 (2004). [CrossRef]

7. I. I. Smolyaninov, J. Elliott, A. V. Zayats, and C. C. Davis, “Far-field optical microscopy with a nanometer-scale resolution based on the in-plane image magnification by surface plasmon polaritons,” Phys. Rev. Lett. **94**(5), 057401 (2005). [CrossRef] [PubMed]

8. Z. W. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science **315**(5819), 1686 (2007). [CrossRef] [PubMed]

9. 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]

10. N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing Spaser,” Nat. Photonics **2**(6), 351–354 (2008). [CrossRef]

22. A. Kumar, S. F. Yu, X. F. Li, and S. P. Lau, “Surface plasmonic lasing via the amplification of coupled surface plasmon waves inside dielectric-metal-dielectric waveguides,” Opt. Express **16**(20), 16113–16123 (2008). [CrossRef] [PubMed]

*et al*. [23

23. D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. **97**(5), 053002 (2006). [CrossRef] [PubMed]

24. D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Strong coupling of single emitters to surface plasmons,” Phys. Rev. B **76**(3), 035420 (2007). [CrossRef]

25. A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature **450**(7168), 402–406 (2007). [CrossRef] [PubMed]

26. H. M. Gong, L. Zhou, X. R. Su, S. Xiao, S. D. Liu, and Q. Q. Wang, “Illuminating Dark Plasmons of Silver Nanoantenna Rings to Enhance Exciton–Plasmon Interactions,” Adv. Funct. Mater. **19**(2), 298–303 (2009). [CrossRef]

27. S. D. Liu, Z. S. Zhang, and Q. Q. Wang, “High sensitivity and large field enhancement of symmetry broken Au nanorings: effect of multipolar plasmon resonance and propagation,” Opt. Express **17**(4), 2906–2917 (2009). [CrossRef] [PubMed]

28. T. H. Stievater, X. Li, D. G. Steel, D. Gammon, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “Rabi oscillations of excitons in single quantum dots,” Phys. Rev. Lett. **87**(13), 133603 (2001). [CrossRef] [PubMed]

31. X. Li, Y. Wu, D. Steel, D. Gammon, T. H. Stievater, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “An all-optical quantum gate in a semiconductor quantum dot,” Science **301**(5634), 809–811 (2003). [CrossRef] [PubMed]

## 2. Theoretical model and analysis

*ε*

_{1}.

*d*is the center-to-center distance between a SQD and the Ag nanowire, and all the SQDs have the same distance to the Ag nanowire. The SQDs spatial distribution is designed as the largest linear density surrounding the Ag nanoring for a given distance

*d*. Assuming the largest linear population densities of SQDs is

*η*, which depends on the structure parameters of the system with the relationship

*η*=

*R*

_{SQD}is the radius of the sphere SQD. Figure 1(c) shows the energy-level diagram of the two-level SQD coupling with SPs.

*Γ*

_{rad}and

*Γ*

_{nonrad}are the rates of spontaneous emission into free space and the non-radiative emission into Ag nanoring, respectively.

*W*

_{01}is the pumping rate of the SQD. In the following part, we first discuss the stimulated interactions of a single SQD and SPs. Then we consider the interactions of many SQDs with SPs, and discuss the steady state solutions of this nano-system.

24. D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Strong coupling of single emitters to surface plasmons,” Phys. Rev. B **76**(3), 035420 (2007). [CrossRef]

*i*= 1, 2 represent the electric fields outside and inside the cylinder, respectively.

*k*

_{i⊥}and

*k*

_{//}, are transverse and longitudinal wave vectors,

*J*

_{0}and

*H*

_{0}are Bessel function and Hankel function of the first kind, respectively.

*a*is a constant.

32. P. W. Milonni, “Field quantization and radiative processes in dispersive dielectric media,” J. Mod. Opt. **42**(10), 1991–2004 (1995). [CrossRef]

*ω*

_{sp}is the frequency of SPs, which is approximately equal to the dipole transitions frequency

*ω*

_{10}of the SQD under the resonant excitation condition

*ω*

_{sp}≈

*ω*

_{10}≈

*ω*.

*g*is the coupling factor of the SQDs and SPs.

*g*=

*ω*

^{1/2}

**μ**

_{10}·

**Q**

_{1}(

**r**) / (2

*cћ*

^{1/2}) with

**μ**

_{10}being the transition matrix element of SQDs. The SQDs emission rate into SPs is obtained from the Fermi's golden rule,

*D*

_{SP}(

*ω*) is the SP density of states, and (

*n*+1) is the total number of SPs attributed to the stimulated and spontaneous emissions of SQDs. The value of

*Γ*

_{rad}and the non-radiative emission into Ag nanoring

*Γ*

_{nonrad}are ignored. In Ref. 24

24. D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Strong coupling of single emitters to surface plasmons,” Phys. Rev. B **76**(3), 035420 (2007). [CrossRef]

*P*= (

*n*+ 1)

*d*between the SQD and the Ag nanowire is ~60nm, which means

^{0}. When

*n*>>1,

*P*is very large and

*Γ*

_{rad}and

*Γ*

_{nonrad}can be ignored in the SP rate equations.

*N*

_{sp}(

*l*) is the number of SPs per unit length after the SPs propagate the distance

*l*along the ring. The total number of the SPs satisfies

*n*=

*N*

_{sp}(

*l*)

*L*', where

*L*' is the quantization length, which is assumed to be equal to the circumference

*L*of the ring.

*υ*

_{sp}is the group velocity of the SPs.

*ρ*

_{eff}=

*ρ*

_{1}-

*ρ*

_{0}is the population inversion with

*ρ*

_{1}being the linear population densities of SQDs in the excited states and

*ρ*

_{0}in the ground states. The linear population density of SQDs is

*η = ρ*

_{1}+

*ρ*

_{0}. In Eq. (2)

*ρ*

_{eff}can be obtained from Eq. (3),where

*N'*is a constant for a given

*W*

_{01}in our system with the dimension of linear population density. Defining the gain coefficient

*G'*=

*dN*

_{sp}(

*l*)/[

*N*(

_{sp}*l*)

*dl*] and substituting Eq. (4) into Eq. (2) yields the gain coefficient

*G'*is decreasing with the increasing of

*N*

_{sp}(

*l*). Under small-signal condition, namely

*N*

_{sp}(

*l*) <<

*N'*, the small-signal gain coefficient is written as

*α*= 2Im

*k*

_{//}, the net gain coefficient is given by

*G*

^{0}-

*α.*It should be noted that

*G*

^{0}should at least satisfy

*G*

^{0}>

*α*to realize the amplification of SPs, thus

*G*

^{0}=

*α*is interpreted as the threshold condition. The solution of

*W*

_{01}from the equation

*G*

^{0}=

*α*is the threshold value of pumping rate denoted by

*dN*

_{sp}(

*l*)/[

*N*

_{sp}(

*l*)

*dl*] into the left-hand side of Eq. (5), and integrating from 0 to

*l*yields the variation of

*N*

_{sp}(

*l*) with propagating length:where

*G*decreases with the increasing of

*N*

_{sp}(

*l*), and when

*N*

_{sp}(

*l*) reaches to a certain value,

*G*turns to be almost zero, which means that the number of SPs becomes saturated and reaches the maximum. The maximum value of SP number per unit length denoted by

*G*

^{0}>>

*α*, Eq. (7) can be approximately written as

*η*(-

*W*

_{01})/(2

*αυ*

_{sp}), where it is assumed that the SP dispersion relation satisfies

*ω*=

*υ*

_{sp}

*k*

_{//}, and it shows clearly the relationships between

## 3. Numerical results

*D*

_{W}=100nm, the wavelength in vacuum

*λ*

_{0}=1000nm,

*d*=60nm,

*L*=5

*λ*

_{sp}= 2.99μm.

*ε*

_{1}=2,

*ε*

_{2}= −50+0.6

*i*and

*μ*

_{10}= 1.9×10

^{17}

*esu*[9

9. 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]

33. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**(12), 4370–4379 (1972). [CrossRef]

^{−1}. We take

*R*

_{SQD}=5nm for the SQD radius, and then we obtain

*η*= 4.8 × 10

^{3}μm

^{−1},

*ρ*

_{eff}

^{0}=1.4×10

^{1}μm

^{−1},

*N*

_{sp}(0) is assumed to be far smaller than

*N'*to ensure that the small signal approximation is satisfied at the starting point. Here we assume

*N*

_{sp}(0)/

*N'*= 0.01 at

*N*

_{sp}(0) = 0.033μm

^{−1}.

*N*

_{sp}(

*l*) within a very short propagating length

*l*. Figure 2 (a) shows that when the pumping rate is larger than the pumping rate threshold

_{sp}(

*l*) will increase exponentially with the propagating length, which means that the SPs amplification is realized. The larger the pumping rate, the faster the SPs number increases. For the pumping rate

*N*

_{sp}(

*l*) at first increases dramatically with the propagating length

*l*, which leads to the decrease of the gain coefficient

*G*, then

*N*

_{sp}(

*l*) turns to be saturated when the gain equals to the loss, as shown in Fig. 3(a) . Therefore, at first the SP field is first amplified as propagating along the nanoring, then reaches to its saturated value after circling for several rounds and at last the SP field in the nanoring is equivalent everywhere. As the pumping rate increases, the maximum value of SP number becomes larger. Figure 3(b) shows the maximum value of

*N*

_{sp}(

*l*) as a function of the pumping rate

*W*

_{01}. It shows that when

*W*

_{01}.

## 4. Conclusion

*W*

_{01}is larger than the pumping threshold

## Acknowledgments

## References and links

1. | W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature |

2. | E. Ozbay, “Plasmonics: Merging Photonics and Electronics at Nanoscale Dimensions,” Science |

3. | S. A. Maier, “Plasmonics: The Promise of Highly Integrated Optical Devices,” IEEE J. Sel. Top. Quantum Electron. |

4. | K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. |

5. | S. Lal, S. Link, and N. S. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics |

6. | W. Srituravanich, N. Fang, C. Sun, Q. Luo, and X. Zhang, “Plasmonic nanolithography,” Nano Lett. |

7. | I. I. Smolyaninov, J. Elliott, A. V. Zayats, and C. C. Davis, “Far-field optical microscopy with a nanometer-scale resolution based on the in-plane image magnification by surface plasmon polaritons,” Phys. Rev. Lett. |

8. | Z. W. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science |

9. | 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. |

10. | N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing Spaser,” Nat. Photonics |

11. | E. Plum, V. A. Fedotov, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Towards the lasing spaser: controlling metamaterial optical response with semiconductor quantum dots,” Opt. Express |

12. | 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 |

13. | M. A. Noginov, G. Zhu, M. Mayy, B. A. Ritzo, N. Noginova, and V. A. Podolskiy, “Stimulated emission of surface plasmon polaritons,” Phys. Rev. Lett. |

14. | M. A. Noginov, V. A. Podolskiy, G. Zhu, M. Mayy, M. Bahoura, J. A. Adegoke, B. A. Ritzo, and K. Reynolds, “Compensation of loss in propagating surface plasmon polariton by gain in adjacent dielectric medium,” Opt. Express |

15. | M. A. Noginov, G. Zhu, M. Bahoura, J. Adegoke, C. E. Small, B. A. Ritzo, V. P. Drachev, and V. M. Shalaev, “Enhancement of surface plasmons in an Ag aggregate by optical gain in a dielectric medium,” Opt. Lett. |

16. | J. Seidel, S. Grafström, and L. Eng, “Stimulated emission of surface plasmons at the interface between a silver film and an optically pumped dye solution,” Phys. Rev. Lett. |

17. | M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Observation of Stimulated Emission of Surface Plasmon Polaritons,” Nano Lett. |

18. | M. Ambati, D. A. Genov, R. F. Oulton, and X. Zhang, “Active Plasmonics: Surface Plasmon Interaction With Optical Emitters,” IEEE J. Sel. Top. Quantum Electron. |

19. | D. A. Genov, M. Ambati, and X. Zhang, “Surface Plasmon Amplification in Planar Metal Films,” IEEE J. Quantum Electron. |

20. | R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature |

21. | I. D. Leon and P. Berini, “Theory of surface plasmon-polariton amplification in planar structures incorporating dipolar gain media,” Phys. Rev. |

22. | A. Kumar, S. F. Yu, X. F. Li, and S. P. Lau, “Surface plasmonic lasing via the amplification of coupled surface plasmon waves inside dielectric-metal-dielectric waveguides,” Opt. Express |

23. | D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. |

24. | D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Strong coupling of single emitters to surface plasmons,” Phys. Rev. B |

25. | A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature |

26. | H. M. Gong, L. Zhou, X. R. Su, S. Xiao, S. D. Liu, and Q. Q. Wang, “Illuminating Dark Plasmons of Silver Nanoantenna Rings to Enhance Exciton–Plasmon Interactions,” Adv. Funct. Mater. |

27. | S. D. Liu, Z. S. Zhang, and Q. Q. Wang, “High sensitivity and large field enhancement of symmetry broken Au nanorings: effect of multipolar plasmon resonance and propagation,” Opt. Express |

28. | T. H. Stievater, X. Li, D. G. Steel, D. Gammon, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “Rabi oscillations of excitons in single quantum dots,” Phys. Rev. Lett. |

29. | A. Zrenner, E. Beham, S. Stufler, F. Findeis, M. Bichler, and G. Abstreiter, “Coherent properties of a two-level system based on a quantum-dot photodiode,” Nature |

30. | Q. Q. Wang, A. Muller, M. T. Cheng, H. J. Zhou, P. Bianucci, and C. K. Shih, “Coherent control of a V-type three-level system in a single quantum dot,” Phys. Rev. Lett. |

31. | X. Li, Y. Wu, D. Steel, D. Gammon, T. H. Stievater, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “An all-optical quantum gate in a semiconductor quantum dot,” Science |

32. | P. W. Milonni, “Field quantization and radiative processes in dispersive dielectric media,” J. Mod. Opt. |

33. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

**OCIS Codes**

(230.7370) Optical devices : Waveguides

(240.6680) Optics at surfaces : Surface plasmons

(230.4480) Optical devices : Optical amplifiers

(250.5590) Optoelectronics : Quantum-well, -wire and -dot devices

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: December 18, 2009

Revised Manuscript: January 21, 2010

Manuscript Accepted: January 26, 2010

Published: February 16, 2010

**Citation**

Zhong-Jian Yang, Nam-Chol Kim, Jian-Bo Li, Mu-Tian Cheng, Shao-Ding Liu, Zhong-Hua Hao, and Qu-Quan Wang, "Surface plasmons amplifications in single Ag nanoring," Opt. Express **18**, 4006-4011 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-5-4006

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### References

- W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef] [PubMed]
- E. Ozbay, “Plasmonics: Merging Photonics and Electronics at Nanoscale Dimensions,” Science 311(5758), 189–193 (2006). [CrossRef] [PubMed]
- S. A. Maier, “Plasmonics: The Promise of Highly Integrated Optical Devices,” IEEE J. Sel. Top. Quantum Electron. 12(6), 1671–1677 (2006). [CrossRef]
- K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. Dasari, and M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78(9), 1667–1670 (1997). [CrossRef]
- S. Lal, S. Link, and N. S. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics 1(11), 641–648 (2007). [CrossRef]
- W. Srituravanich, N. Fang, C. Sun, Q. Luo, and X. Zhang, “Plasmonic nanolithography,” Nano Lett. 4(6), 1085–1088 (2004). [CrossRef]
- I. I. Smolyaninov, J. Elliott, A. V. Zayats, and C. C. Davis, “Far-field optical microscopy with a nanometer-scale resolution based on the in-plane image magnification by surface plasmon polaritons,” Phys. Rev. Lett. 94(5), 057401 (2005). [CrossRef] [PubMed]
- Z. W. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007). [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]
- N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing Spaser,” Nat. Photonics 2(6), 351–354 (2008). [CrossRef]
- E. Plum, V. A. Fedotov, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Towards the lasing spaser: controlling metamaterial optical response with semiconductor quantum dots,” Opt. Express 17(10), 8548–8551 (2009). [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]
- M. A. Noginov, G. Zhu, M. Mayy, B. A. Ritzo, N. Noginova, and V. A. Podolskiy, “Stimulated emission of surface plasmon polaritons,” Phys. Rev. Lett. 101(22), 226806 (2008). [CrossRef] [PubMed]
- M. A. Noginov, V. A. Podolskiy, G. Zhu, M. Mayy, M. Bahoura, J. A. Adegoke, B. A. Ritzo, and K. Reynolds, “Compensation of loss in propagating surface plasmon polariton by gain in adjacent dielectric medium,” Opt. Express 16(2), 1385–1392 (2008). [CrossRef] [PubMed]
- M. A. Noginov, G. Zhu, M. Bahoura, J. Adegoke, C. E. Small, B. A. Ritzo, V. P. Drachev, and V. M. Shalaev, “Enhancement of surface plasmons in an Ag aggregate by optical gain in a dielectric medium,” Opt. Lett. 31(20), 3022–3024 (2006). [CrossRef] [PubMed]
- J. Seidel, S. Grafström, and L. Eng, “Stimulated emission of surface plasmons at the interface between a silver film and an optically pumped dye solution,” Phys. Rev. Lett. 94(17), 177401 (2005). [CrossRef] [PubMed]
- M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Observation of Stimulated Emission of Surface Plasmon Polaritons,” Nano Lett. 8(11), 3998–4001 (2008). [CrossRef] [PubMed]
- M. Ambati, D. A. Genov, R. F. Oulton, and X. Zhang, “Active Plasmonics: Surface Plasmon Interaction With Optical Emitters,” IEEE J. Sel. Top. Quantum Electron. 14(6), 1395–1403 (2008). [CrossRef]
- D. A. Genov, M. Ambati, and X. Zhang, “Surface Plasmon Amplification in Planar Metal Films,” IEEE J. Quantum Electron. 43(11), 1104–1108 (2007). [CrossRef]
- R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009). [CrossRef] [PubMed]
- I. D. Leon and P. Berini, “Theory of surface plasmon-polariton amplification in planar structures incorporating dipolar gain media,” Phys. Rev. 78, 161401 (2008). [CrossRef]
- A. Kumar, S. F. Yu, X. F. Li, and S. P. Lau, “Surface plasmonic lasing via the amplification of coupled surface plasmon waves inside dielectric-metal-dielectric waveguides,” Opt. Express 16(20), 16113–16123 (2008). [CrossRef] [PubMed]
- D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. 97(5), 053002 (2006). [CrossRef] [PubMed]
- D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Strong coupling of single emitters to surface plasmons,” Phys. Rev. B 76(3), 035420 (2007). [CrossRef]
- A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature 450(7168), 402–406 (2007). [CrossRef] [PubMed]
- H. M. Gong, L. Zhou, X. R. Su, S. Xiao, S. D. Liu, and Q. Q. Wang, “Illuminating Dark Plasmons of Silver Nanoantenna Rings to Enhance Exciton–Plasmon Interactions,” Adv. Funct. Mater. 19(2), 298–303 (2009). [CrossRef]
- S. D. Liu, Z. S. Zhang, and Q. Q. Wang, “High sensitivity and large field enhancement of symmetry broken Au nanorings: effect of multipolar plasmon resonance and propagation,” Opt. Express 17(4), 2906–2917 (2009). [CrossRef] [PubMed]
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