## Complex modes and near-zero permittivity in 3D arrays of plasmonic nanoshells: loss compensation using gain [Invited] |

Optical Materials Express, Vol. 1, Issue 6, pp. 1077-1089 (2011)

http://dx.doi.org/10.1364/OME.1.001077

Acrobat PDF (1038 KB)

### Abstract

We report on the possibility of adopting active gain materials (specifically, made of fluorescent dyes) to mitigate the losses in a 3D periodic array of dielectric-core metallic-shell nanospheres. We find the modes with complex wavenumber in the structure, and describe the composite material in terms of homogenized effective permittivity, comparing results from modal analysis and Maxwell Garnett theory. We then design two metamaterials in which the epsilon-near-zero frequency region overlaps with the emission band of the adopted gain media, and we show that metamaterials with effective parameters with low losses are feasible, thanks to the gain materials. Even though fluorescent dyes embedded in the nanoshells’ dielectric cores are employed in this study, the formulation provided is general, and could account for the usage of other active materials, such as semiconductors and quantum dots.

© 2011 OSA

## 1. Introduction

1. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**(18), 3966–3969 (2000). [CrossRef] [PubMed]

2. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science **314**(5801), 977–980 (2006). [CrossRef] [PubMed]

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]

4. M. I. Stockman, “Spasers explained,” Nat. Photonics **2**(6), 327–329 (2008). [CrossRef]

5. M. C. Gather, K. Meerholz, N. Danz, and K. Leosson, “Net optical gain in a plasmonic waveguide embedded in a fluorescent polymer,” Nat. Photonics **4**(7), 457–461 (2010). [CrossRef]

6. I. De Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nat. Photonics **4**(6), 382–387 (2010). [CrossRef]

7. G. Strangi, A. De Luca, S. Ravaine, M. Ferrie, and R. Bartolino, “Gain induced optical transparency in metamaterials,” Appl. Phys. Lett. **98**(25), 251912 (2011). [CrossRef]

8. A. De Luca, M. P. Grzelczak, I. Pastoriza-Santos, L. M. Liz-Marzán, M. La Deda, M. Striccoli, and G. Strangi, “Dispersed and encapsulated gain medium in plasmonic nanoparticles: a multipronged approach to mitigate optical losses,” ACS Nano **5**(7), 5823–5829 (2011). [CrossRef] [PubMed]

17. A. Fang, Z. Huang, T. Koschny, and C. M. Soukoulis, “Overcoming the losses of a split ring resonator array with gain,” Opt. Express **19**(13), 12688–12699 (2011). [CrossRef] [PubMed]

18. Y. Sivan, S. Xiao, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Frequency-domain simulations of a negative-index material with embedded gain,” Opt. Express **17**(26), 24060–24074 (2009). [CrossRef] [PubMed]

19. A. D. Boardman, V. V. Grimalsky, Y. S. Kivshar, S. V. Koshevaya, M. Lapine, N. M. Litchinitser, V. N. Malnev, M. Noginov, Y. G. Rapoport, and V. M. Shalaev, “Active and tunable metamaterials,” Laser Photon. Rev. **5**(2), 287–307 (2011). [CrossRef]

## 2. Simulation model

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

### 2.1 Modal analysis for periodic arrays of plasmonic nanoshells

31. K. Tanabe, “Field enhancement around metal nanoparticles and nanoshells: a systematic investigation,” J. Phys. Chem. C **112**(40), 15721–15728 (2008). [CrossRef]

*a*,

*b*and

*c*are the periodicities along

*x*-,

*y*- and

*z*-direction, respectively [21,22,34

34. A. L. Fructos, S. Campione, F. Capolino, and F. Mesa, “Characterization of complex plasmonic modes in two-dimensional periodic arrays of metal nanospheres,” J. Opt. Soc. Am. B **28**(6), 1446–1458 (2011). [CrossRef]

*z*direction with wavenumber

### 2.2 Effective parameters

### 2.3 Modeling of the gain material

17. A. Fang, Z. Huang, T. Koschny, and C. M. Soukoulis, “Overcoming the losses of a split ring resonator array with gain,” Opt. Express **19**(13), 12688–12699 (2011). [CrossRef] [PubMed]

37. A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antenn. Propag. **46**(3), 334–340 (1998). [CrossRef]

38. D. Faubert, S. L. Chin, M. Cormier, and M. Boloten, “Numerical analysis of short laser pulse superposition in a fluorescent dye medium,” Can. J. Phys. **57**(2), 160–167 (1979). [CrossRef]

39. G. Grönninger and A. Penzkofer, “Determination of energy and duration of picosecond light pulses by bleaching of dyes,” Opt. Quantum Electron. **16**(3), 225–233 (1984). [CrossRef]

*i-*th state,

*i*to the lower state

*j*, and

37. A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antenn. Propag. **46**(3), 334–340 (1998). [CrossRef]

40. S.-H. Chang and A. Taflove, “Finite-difference time-domain model of lasing action in a four-level two-electron atomic system,” Opt. Express **12**(16), 3827–3833 (2004). [CrossRef] [PubMed]

41. A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B **79**(24), 241104 (2009). [CrossRef]

## 3. Results of epsilon near zero composite materials with mitigated loss

### 3.1 Case with silver shells

45. D. Magde, R. Wong, and P. G. Seybold, “Fluorescence quantum yields and their relation to lifetimes of rhodamine 6G and fluorescein in nine solvents: improved absolute standards for quantum yields,” Photochem. Photobiol. **75**(4), 327–334 (2002). [CrossRef] [PubMed]

45. D. Magde, R. Wong, and P. G. Seybold, “Fluorescence quantum yields and their relation to lifetimes of rhodamine 6G and fluorescein in nine solvents: improved absolute standards for quantum yields,” Photochem. Photobiol. **75**(4), 327–334 (2002). [CrossRef] [PubMed]

#### 3.1.1 Mode analysis and effective parameters computation

46. A. Alù, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express **14**(4), 1557–1567 (2006). [CrossRef] [PubMed]

47. I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B **62**(23), 15299–15302 (2000). [CrossRef]

*z*direction, for three cases: (i) accounting for metal losses, (ii) ideal lossless case (i.e.,

48. M. Meier and A. Wokaun, “Enhanced fields on large metal particles: dynamic depolarization,” Opt. Lett. **8**(11), 581–583 (1983). [CrossRef] [PubMed]

#### 3.1.2 Epsilon-near-zero region for silver shells

*f*= 526 THz, in the epsilon-near-zero frequency band, for the highest concentration considered.

### 3.2 Case with gold shells

49. P. Sperber, W. Spangler, B. Meier, and A. Penzkofer, “Experimental and theoretical investigation of tunable picosecond pulse generation in longitudinally pumped dye-laser generators and amplifiers,” Opt. Quantum Electron. **20**(5), 395–431 (1988). [CrossRef]

50. S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. **105**(12), 127401 (2010). [CrossRef] [PubMed]

#### 3.2.1 Mode analysis and effective parameters computation

*(*

51. N. K. Grady, N. J. Halas, and P. Nordlander, “Influence of dielectric function properties on the optical response of plasmon resonant metallic nanoparticles,” Chem. Phys. Lett. **399**(1-3), 167–171 (2004). [CrossRef]

53. S. Campione, S. Steshenko, and F. Capolino, “Description and characterization of the complex modes in a linear chain of gold metal nanospheres,” Proc. SPIE **7946**, 79461V (2011). [CrossRef]

*z*direction, are shown in Fig. 5 for three different cases: (a) accounting for metal losses, (b) ideal lossless case (i.e.,

#### 3.2.2 Epsilon-near-zero region for gold shells

*f*= 422 THz, in the epsilon-near-zero frequency band, for the highest concentration considered.

## 4. Conclusion

## Acknowledgments

## References and links

1. | J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

2. | D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science |

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. |

4. | M. I. Stockman, “Spasers explained,” Nat. Photonics |

5. | M. C. Gather, K. Meerholz, N. Danz, and K. Leosson, “Net optical gain in a plasmonic waveguide embedded in a fluorescent polymer,” Nat. Photonics |

6. | I. De Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nat. Photonics |

7. | G. Strangi, A. De Luca, S. Ravaine, M. Ferrie, and R. Bartolino, “Gain induced optical transparency in metamaterials,” Appl. Phys. Lett. |

8. | A. De Luca, M. P. Grzelczak, I. Pastoriza-Santos, L. M. Liz-Marzán, M. La Deda, M. Striccoli, and G. Strangi, “Dispersed and encapsulated gain medium in plasmonic nanoparticles: a multipronged approach to mitigate optical losses,” ACS Nano |

9. | J. A. Gordon and R. W. Ziolkowski, “CNP optical metamaterials,” Opt. Express |

10. | N. M. Lawandy, “Localized surface plasmon singularities in amplifying media,” Appl. Phys. Lett. |

11. | S. Anantha Ramakrishna and J. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B |

12. | S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature |

13. | A. K. Sarychev and G. Tartakovsky, “Magnetic plasmonic metamaterials in actively pumped host medium and plasmonic nanolaser,” Phys. Rev. B |

14. | M. A. Noginov, G. Zhu, M. Bahoura, J. Adegoke, C. Small, B. A. Ritzo, V. P. Drachev, and V. M. Shalaev, “The effect of gain and absorption on surface plasmons in metal nanoparticles,” Appl. Phys. B |

15. | R. Bardhan, N. K. Grady, J. R. Cole, A. Joshi, and N. J. Halas, “Fluorescence enhancement by Au nanostructures: nanoshells and nanorods,” ACS Nano |

16. | J. Zhang, I. Gryczynski, Z. Gryczynski, and J. R. Lakowicz, “Dye-labeled silver nanoshell-bright particle,” J. Phys. Chem. B |

17. | A. Fang, Z. Huang, T. Koschny, and C. M. Soukoulis, “Overcoming the losses of a split ring resonator array with gain,” Opt. Express |

18. | Y. Sivan, S. Xiao, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Frequency-domain simulations of a negative-index material with embedded gain,” Opt. Express |

19. | A. D. Boardman, V. V. Grimalsky, Y. S. Kivshar, S. V. Koshevaya, M. Lapine, N. M. Litchinitser, V. N. Malnev, M. Noginov, Y. G. Rapoport, and V. M. Shalaev, “Active and tunable metamaterials,” Laser Photon. Rev. |

20. | C. F. Bohren and D. R. Huffman, |

21. | S. Steshenko and F. Capolino, “Single dipole approximation for modeling collections of nanoscatterers,” in |

22. | S. Campione and F. Capolino, “Linear and planar periodic arrays of metallic nanospheres: fabrication, optical properties and applications,” in |

23. | A. Sihvola, |

24. | A. Sihvola, “Mixing rules,” in |

25. | M. G. Silveirinha, A. Alu, B. Edwards, and N. Engheta, “Overview of theory and applications of epsilon-near-zero materials,” in |

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

27. | L. Landau and E. M. Lifschitz, |

28. | K. Ohta and H. Ishida, “Comparison among several numerical integration methods for Kramers-Kronig transformation,” Appl. Spectrosc. |

29. | K. E. Peiponen and E. M. Vartiainen, “Kramers-Kronig relations in optical data inversion,” Phys. Rev. B Condens. Matter |

30. | S. A. Ramakrishna and T. M. Grzegorczyk, |

31. | K. Tanabe, “Field enhancement around metal nanoparticles and nanoshells: a systematic investigation,” J. Phys. Chem. C |

32. | A. Vallecchi, M. Albani, and F. Capolino, “EM characterization of Raspberry-like nanocluster metamaterials,” in |

33. | M. Abramowitz and I. A. Stegun, |

34. | A. L. Fructos, S. Campione, F. Capolino, and F. Mesa, “Characterization of complex plasmonic modes in two-dimensional periodic arrays of metal nanospheres,” J. Opt. Soc. Am. B |

35. | S. Campione, S. Steshenko, M. Albani, and F. Capolino, “Complex modes in three dimensional periodic arrays of plasmonic nanospheres,” Opt. Express (to be submitted). |

36. | A. Siegman, |

37. | A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antenn. Propag. |

38. | D. Faubert, S. L. Chin, M. Cormier, and M. Boloten, “Numerical analysis of short laser pulse superposition in a fluorescent dye medium,” Can. J. Phys. |

39. | G. Grönninger and A. Penzkofer, “Determination of energy and duration of picosecond light pulses by bleaching of dyes,” Opt. Quantum Electron. |

40. | S.-H. Chang and A. Taflove, “Finite-difference time-domain model of lasing action in a four-level two-electron atomic system,” Opt. Express |

41. | A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B |

42. | O. Svelto, |

43. | M. A. R. C. Alencar, G. S. Maciel, C. B. de Araújo, R. Bertholdo, Y. Messaddeq, and S. J. L. Ribeiro, “Laserlike emission from silica inverse opals infiltrated with Rhodamine 6G,” J. Non-Cryst. Solids |

44. | D. Magde, G. E. Rojas, and P. G. Seybold, “Solvent dependence of the fluorescence lifetimes of xanthene dyes,” Photochem. Photobiol. |

45. | D. Magde, R. Wong, and P. G. Seybold, “Fluorescence quantum yields and their relation to lifetimes of rhodamine 6G and fluorescein in nine solvents: improved absolute standards for quantum yields,” Photochem. Photobiol. |

46. | A. Alù, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express |

47. | I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B |

48. | M. Meier and A. Wokaun, “Enhanced fields on large metal particles: dynamic depolarization,” Opt. Lett. |

49. | P. Sperber, W. Spangler, B. Meier, and A. Penzkofer, “Experimental and theoretical investigation of tunable picosecond pulse generation in longitudinally pumped dye-laser generators and amplifiers,” Opt. Quantum Electron. |

50. | S. Wuestner, A. Pusch, K. L. Tsakmakidis, J. M. Hamm, and O. Hess, “Overcoming losses with gain in a negative refractive index metamaterial,” Phys. Rev. Lett. |

51. | N. K. Grady, N. J. Halas, and P. Nordlander, “Influence of dielectric function properties on the optical response of plasmon resonant metallic nanoparticles,” Chem. Phys. Lett. |

52. | S. Campione, A. Vallecchi, and F. Capolino, “Closed form formulas and tunability of resonances in pairs of gold-dielectric nanoshells,” Proc. SPIE |

53. | S. Campione, S. Steshenko, and F. Capolino, “Description and characterization of the complex modes in a linear chain of gold metal nanospheres,” Proc. SPIE |

**OCIS Codes**

(160.1245) Materials : Artificially engineered materials

(160.3918) Materials : Metamaterials

(250.5403) Optoelectronics : Plasmonics

**ToC Category:**

Metamaterials

**History**

Original Manuscript: July 20, 2011

Revised Manuscript: August 25, 2011

Manuscript Accepted: August 26, 2011

Published: September 6, 2011

**Virtual Issues**

Nanoplasmonics and Metamaterials (2011) *Optical Materials Express*

**Citation**

Salvatore Campione, Matteo Albani, and Filippo Capolino, "Complex modes and near-zero permittivity in 3D arrays of plasmonic nanoshells: loss compensation using gain [Invited]," Opt. Mater. Express **1**, 1077-1089 (2011)

http://www.opticsinfobase.org/ome/abstract.cfm?URI=ome-1-6-1077

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### References

- J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85(18), 3966–3969 (2000). [CrossRef] [PubMed]
- D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science314(5801), 977–980 (2006). [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]
- M. I. Stockman, “Spasers explained,” Nat. Photonics2(6), 327–329 (2008). [CrossRef]
- M. C. Gather, K. Meerholz, N. Danz, and K. Leosson, “Net optical gain in a plasmonic waveguide embedded in a fluorescent polymer,” Nat. Photonics4(7), 457–461 (2010). [CrossRef]
- I. De Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nat. Photonics4(6), 382–387 (2010). [CrossRef]
- G. Strangi, A. De Luca, S. Ravaine, M. Ferrie, and R. Bartolino, “Gain induced optical transparency in metamaterials,” Appl. Phys. Lett.98(25), 251912 (2011). [CrossRef]
- A. De Luca, M. P. Grzelczak, I. Pastoriza-Santos, L. M. Liz-Marzán, M. La Deda, M. Striccoli, and G. Strangi, “Dispersed and encapsulated gain medium in plasmonic nanoparticles: a multipronged approach to mitigate optical losses,” ACS Nano5(7), 5823–5829 (2011). [CrossRef] [PubMed]
- J. A. Gordon and R. W. Ziolkowski, “CNP optical metamaterials,” Opt. Express16(9), 6692–6716 (2008). [CrossRef] [PubMed]
- N. M. Lawandy, “Localized surface plasmon singularities in amplifying media,” Appl. Phys. Lett.85(21), 5040–5042 (2004). [CrossRef]
- S. Anantha Ramakrishna and J. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B67(20), 201101 (2003). [CrossRef]
- S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature466(7307), 735–738 (2010). [CrossRef] [PubMed]
- A. K. Sarychev and G. Tartakovsky, “Magnetic plasmonic metamaterials in actively pumped host medium and plasmonic nanolaser,” Phys. Rev. B75(8), 085436 (2007). [CrossRef]
- M. A. Noginov, G. Zhu, M. Bahoura, J. Adegoke, C. Small, B. A. Ritzo, V. P. Drachev, and V. M. Shalaev, “The effect of gain and absorption on surface plasmons in metal nanoparticles,” Appl. Phys. B86(3), 455–460 (2007). [CrossRef]
- R. Bardhan, N. K. Grady, J. R. Cole, A. Joshi, and N. J. Halas, “Fluorescence enhancement by Au nanostructures: nanoshells and nanorods,” ACS Nano3(3), 744–752 (2009). [CrossRef] [PubMed]
- J. Zhang, I. Gryczynski, Z. Gryczynski, and J. R. Lakowicz, “Dye-labeled silver nanoshell-bright particle,” J. Phys. Chem. B110(18), 8986–8991 (2006). [CrossRef] [PubMed]
- A. Fang, Z. Huang, T. Koschny, and C. M. Soukoulis, “Overcoming the losses of a split ring resonator array with gain,” Opt. Express19(13), 12688–12699 (2011). [CrossRef] [PubMed]
- Y. Sivan, S. Xiao, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Frequency-domain simulations of a negative-index material with embedded gain,” Opt. Express17(26), 24060–24074 (2009). [CrossRef] [PubMed]
- A. D. Boardman, V. V. Grimalsky, Y. S. Kivshar, S. V. Koshevaya, M. Lapine, N. M. Litchinitser, V. N. Malnev, M. Noginov, Y. G. Rapoport, and V. M. Shalaev, “Active and tunable metamaterials,” Laser Photon. Rev.5(2), 287–307 (2011). [CrossRef]
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
- S. Steshenko and F. Capolino, “Single dipole approximation for modeling collections of nanoscatterers,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), p. 8.1.
- S. Campione and F. Capolino, “Linear and planar periodic arrays of metallic nanospheres: fabrication, optical properties and applications,” in Selected Topics in Metamaterials and Photonic Crystals, A. Andreone, A. Cusano, A. Cutolo, and V. Galdi, eds. (World Scientific, 2011), pp. 141–194.
- A. Sihvola, Electromagnetic Mixing Formulas and Applications (IEEE Publishing, 1999).
- A. Sihvola, “Mixing rules,” in Theory and Phenomena of Metamaterials, F. Capolino, ed. (CRC Press, 2009), p. 9.1.
- M. G. Silveirinha, A. Alu, B. Edwards, and N. Engheta, “Overview of theory and applications of epsilon-near-zero materials,” in URSI General Assembly (Chicago, IL, 2008).
- P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972). [CrossRef]
- L. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media (Pergamon Press, 1984), Chap. IX.
- K. Ohta and H. Ishida, “Comparison among several numerical integration methods for Kramers-Kronig transformation,” Appl. Spectrosc.42(6), 952–957 (1988). [CrossRef]
- K. E. Peiponen and E. M. Vartiainen, “Kramers-Kronig relations in optical data inversion,” Phys. Rev. B Condens. Matter44(15), 8301–8303 (1991). [CrossRef] [PubMed]
- S. A. Ramakrishna and T. M. Grzegorczyk, Physics and Applications of Negative Refractive Index Materials (CRC Press and SPIE Press, 2009).
- K. Tanabe, “Field enhancement around metal nanoparticles and nanoshells: a systematic investigation,” J. Phys. Chem. C112(40), 15721–15728 (2008). [CrossRef]
- A. Vallecchi, M. Albani, and F. Capolino, “EM characterization of Raspberry-like nanocluster metamaterials,” in Antennas and Propagation Society International Symposium (Toronto, Canada, 2010).
- M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover Publications, 1965).
- A. L. Fructos, S. Campione, F. Capolino, and F. Mesa, “Characterization of complex plasmonic modes in two-dimensional periodic arrays of metal nanospheres,” J. Opt. Soc. Am. B28(6), 1446–1458 (2011). [CrossRef]
- S. Campione, S. Steshenko, M. Albani, and F. Capolino, “Complex modes in three dimensional periodic arrays of plasmonic nanospheres,” Opt. Express (to be submitted).
- A. Siegman, Lasers (University Science Books, 1986).
- A. S. Nagra and R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antenn. Propag.46(3), 334–340 (1998). [CrossRef]
- D. Faubert, S. L. Chin, M. Cormier, and M. Boloten, “Numerical analysis of short laser pulse superposition in a fluorescent dye medium,” Can. J. Phys.57(2), 160–167 (1979). [CrossRef]
- G. Grönninger and A. Penzkofer, “Determination of energy and duration of picosecond light pulses by bleaching of dyes,” Opt. Quantum Electron.16(3), 225–233 (1984). [CrossRef]
- S.-H. Chang and A. Taflove, “Finite-difference time-domain model of lasing action in a four-level two-electron atomic system,” Opt. Express12(16), 3827–3833 (2004). [CrossRef] [PubMed]
- A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B79(24), 241104 (2009). [CrossRef]
- O. Svelto, Principles of Lasers (Kluwer Academic Plenum Publishers, 1998).
- M. A. R. C. Alencar, G. S. Maciel, C. B. de Araújo, R. Bertholdo, Y. Messaddeq, and S. J. L. Ribeiro, “Laserlike emission from silica inverse opals infiltrated with Rhodamine 6G,” J. Non-Cryst. Solids351(21-23), 1846–1849 (2005). [CrossRef]
- D. Magde, G. E. Rojas, and P. G. Seybold, “Solvent dependence of the fluorescence lifetimes of xanthene dyes,” Photochem. Photobiol.70(5), 737–744 (1999). [CrossRef]
- D. Magde, R. Wong, and P. G. Seybold, “Fluorescence quantum yields and their relation to lifetimes of rhodamine 6G and fluorescein in nine solvents: improved absolute standards for quantum yields,” Photochem. Photobiol.75(4), 327–334 (2002). [CrossRef] [PubMed]
- A. Alù, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express14(4), 1557–1567 (2006). [CrossRef] [PubMed]
- I. El-Kady, M. M. Sigalas, R. Biswas, K. M. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B62(23), 15299–15302 (2000). [CrossRef]
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