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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 22 — Oct. 25, 2010
  • pp: 23035–23040
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Low-loss plasmonic waveguide based on gain-assisted periodic metal nanosphere chains

Haixi Zhang and Ho-Pui Ho  »View Author Affiliations


Optics Express, Vol. 18, Issue 22, pp. 23035-23040 (2010)
http://dx.doi.org/10.1364/OE.18.023035


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Abstract

We propose a low-loss nanoscale waveguide based on gain-assisted plasmonic resonance metallic nanosphere chain. We demonstrate that by employing a gain material or even an appropriate dielectric for the host environment, waveguide loss can be reduced dramatically. A highly efficient pseudo-orthonormal basis expansion method for obtaining the complex dielectric spectra of the low-loss transmission has been developed. Eigenmode analysis revealed the physical origin of those low-loss waveguiding modes, which opens the possibility to achieve waveguiding other than using conventional dipolar resonances of individual particles. A scheme based on electron beam lithography and chemically synthesized nanoparticles has been proposed to fabricate the device.

© 2010 OSA

1. Introduction

The ability to guide optical fields through a prescribed route is critically important for realization of compact integrated optical devices and circuits. Strictly speaking, a nano-photonic circuit should have its transverse dimension in the range of sub-100 nanometers. To achieve such length scales, various strategies have been investigated such as array of quantum dots (QDs) [1

1. C. J. Wang, L. Huang, B. A. Parviz, and L. Y. Lin, “Subdiffraction photon guidance by quantum-dot cascades,” Nano Lett. 6(11), 2549–2553 (2006). [CrossRef] [PubMed]

], plasmonic fibers based on metallic nanowires [2

2. A. W. Sanders, D. A. Routenberg, B. J. Wiley, Y. Xia, E. R. Dufresne, and M. A. Reed, “Observation of plasmon propagation, redirection, and fan-out in silver nanowires,” Nano Lett. 6(8), 1822–1826 (2006). [CrossRef] [PubMed]

]. There also has been long-standing interest in employing metallic nanosphere (MN) chains to function as optical waveguides below the diffraction limit, in which wave propagation is via near-field coupling [3

3. 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]

6

6. S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, “Plasmonics-A route to nanoscale optical devices,” Adv. Mater. (Deerfield Beach Fla.) 13(19), 1501–1505 (2001). [CrossRef]

]. The MN chains waveguides should be superior to QD waveguides in terms of crosstalk because of better field confinement generated from MN’s whirlpool-like nanoscale optical vortices [7

7. M. Bashevoy, V. Fedotov, and N. Zheludev, “Optical whirlpool on an absorbing metallic nanoparticle,” Opt. Express 13(21), 8372–8379 (2005). [CrossRef] [PubMed]

]. Moreover, compared to the plasmonic fibers, it is more feasible from the nanofabrication viewpoint to realize energy splitter and perform assembly of optical circuit nanoelements where plasmonic and non-plasmonic nanospheres are utilized [8

8. N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett. 95(9), 095504 (2005). [CrossRef] [PubMed]

]. However, it is well accepted that radiation and absorption losses in MN waveguide can be serious [5

5. K. H. Fung and C. T. Chan, “Plasmonic modes in periodic metal nanoparticle chains: a direct dynamic eigenmode analysis,” Opt. Lett. 32(8), 973–975 (2007). [CrossRef] [PubMed]

].

2. Device design and theoretical discussion

We consider an infinite gain-assited Ag nanosphere chain waveguide. A schematic of the configuration and a proposed fabrication scheme are shown in Fig. 1
Fig. 1 Schematic of the Ag nanosphere waveguide. (a) to (d) depict a fabrication based on electron beam lithography. (e) is a 3-dimensional illustration of the device with geometrical parameters included.
.

A potential route to realize the proposed waveguide begins with spin-coating R6G-doped polymethylmethacrylate (PMMA) substrate on a silicon/silicon dioxide substrate, then by using electron-beam lithography (EBL) to pattern trenches on the surface of R6G-doped PMMA, one can control the width of the trench and make it can just contain one coated MN in transverse dimension. The coated MN refers to chemically synthesized MN with dye-doped silica layer. One candidate for the doped dye is a composite material containing R6G [11

11. L. Wang and W. Tan, “Multicolor FRET silica nanoparticles by single wavelength excitation,” Nano Lett. 6(1), 84–88 (2006). [CrossRef] [PubMed]

]. This kind of core-shell nanoparticle (NP) may be readily synthesized using current solution-based chemical techniques [11

11. L. Wang and W. Tan, “Multicolor FRET silica nanoparticles by single wavelength excitation,” Nano Lett. 6(1), 84–88 (2006). [CrossRef] [PubMed]

,12

12. T. Ming, L. Zhao, Z. Yang, H. Chen, L. Sun, J. Wang, and C. Yan, “Strong polarization dependence of plasmon-enhanced fluorescence on single gold nanorods,” Nano Lett. 9(11), 3896–3903 (2009). [CrossRef] [PubMed]

]. The selective binding to the bottom of the PMMA trenches can be achieved by DNA-mediated self-assembly and two layer molecular self-assembly procedure as stated in Ref. 1

1. C. J. Wang, L. Huang, B. A. Parviz, and L. Y. Lin, “Subdiffraction photon guidance by quantum-dot cascades,” Nano Lett. 6(11), 2549–2553 (2006). [CrossRef] [PubMed]

. The precision of spacing and size of the MNs are guaranteed by the EBL and the uniformly chemical synthesized NP. Finally, the linear MN chain is buried by the depositing R6G-doped PMMA using a sol-gel process. With the pump laser (e.g. with λ=532nm,tpulse=10nsfor R6G to achieve the gain of 420cm1 [10

10. 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]

]), the doped dye can induce a gain to the host environment.

To discuss the proposed waveguide, we begin with the coupled point-dipole approximation witha=25nm, d=75nm. In ideal modelling, we represent the complex permittivities of the MN and the host medium withεp=εp+εpiand εh=εh+εhi respectively. Note that compared to the practical parameters, due to the substrate effect, εhhere should be considered as the effective permittivity with its value bigger than but very close to that of the dye-doped silica, considering the near-field coupling which is the major mechanism of MN chain waveguide occurs mainly in the doped silica and PMMA. The substrate should not affect εh very much because the high gain region should be in the vicinity of MNs [13

13. V. M. Shalaev, and S. Kawata, Nanophotonics with Surface Plasmons (Elsevier Science Ltd. 2007), Chap. 5.

]. The induced moment on the nth dipole equals to the polarizability α(ω) times the sum of electric field generated by other point-like particles [3

3. 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]

], that is

pn=α(ω)mn((1-iω|nm|dc)3r^·pmr^pm|nm|3d3+ω2c2pmr^·pmr^|nm|d)exp(iω|nm|d/c)
(1)

Here m is an integer depicting the position of the mth particle along the chain, r^ is the position vector pointing from the dipole to the field point, c is the speed of light. The quasi-static dipole polarizability has the formα(ω)=a3[ε(ω)-1)]/[ε(ω)+2], whereε(ω)=εp/εh. For better accuracy, we use experimental data from Johnson and Christy for the dielectric response of the MN [14

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

]. Meanwhile, we employ the modified long wavelength approximation (MLWA) to account for dynamic polarizability in a more rigorous manner [15

15. H. Y. Chung, P. T. Leung, and D. P. Tsai, “Dynamic modifications of polarizability for large metallic spheroidal nanoshells,” J. Chem. Phys. 131(12), 124122 (2009). [CrossRef] [PubMed]

]. In the spirit of MLWA, 1/α(ω) is replaced by1/α(ω)2ω3i/3c3ω2/ac2. For a chain of N MNs, Eq. (1) becomes a set of N coupled equations with its matrix formMP=0, where p is an N-rowed column vector of the dipole moments. The matrix M is defined by its elements
Mn,mn=Aa3d3(1iω|nm|dcBω2|nm|2d2c2)exp(iω|nm|d/c)|nm|3,Mn,n=a3/α(ω),
where n=1,...,N, and for TM, A = 1, B = 1, for LM, A = −2, B = 0 [3

3. 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]

]. Given the first MN is subjected to a small excitation (eiωt) as waveguide feed-in, it refers to the case of forced perturbation problem MP=|1,0...0. If a proper gain host εhis employed, certain eignemode will be activated and emerge as guiding mode. However, searching the complex root (i.e. the complex dielectric spectra εh) for the mode-matching condition in the complex plane is a tedious task. Here we developed a straightforward method to obtain the complex dielectric spectra εhand reveal the physical nature. If we renderM=[a3/α(ω)]IG(ω), where G(ω)=[a3/α(ω)]IM, and I is the unit matrix, then G(ω), contains only geometrical information. The diagonal elements of M contains only material information. For ease of representation, an operator system will be utilized hereinafter. The perturbation problem becomes:

[a3/α(ω)]Pn=1nanbn|n=|1,0...0
(2)

Here we use eigenstates of G(ω):|1,|2...|n, to get the expansion of P, and anis an expansion coefficient, bn is the eigenvalue corresponding to nth eigenstate. Note that the completeness of those eigenvectors, forming a complete set, is different from that of conventional Hermitian operators in quantum mechanics. Here G(ω) is a complex symmetric operator, whose eigenvectors form a complete, pseudo-orthonormal basis [16

16. V. A. Markel, “Antisymmetrical optical states,” J. Opt. Soc. Am. B 12(10), 1783 (1995). [CrossRef]

]. We use bra vector n¯|, where the bar denotes complex conjugation of all entries, to pre-multiply Eq. (2). In this work, this operation means a pseudo-inner product defined as the simple Euclidean type inner product [16

16. V. A. Markel, “Antisymmetrical optical states,” J. Opt. Soc. Am. B 12(10), 1783 (1995). [CrossRef]

], then we obtain an=n¯|1,0...0/[a3/α(ω)bn], hence

P=n=1n{n¯|1,0...0/[a3/α(ω)bn]}|n
(3)

The physical meaning of Eq. (3) is quite clear. For a certain excitation external field (eiωt), if proper ratio ε(ω)=εp/εh is fulfilled, the singularity in the denominator of an results in an infinite internal field and scattering field, followed by the emergence of a zero-loss (or ultra low-loss) guiding mode. The mode number equals to the MN number as stated in Ref. 3

3. 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]

. Hence we obtain the complex dielectric spectra rapidly with the mode-matching condition satisfied. The complex dielectric spectra for Ag NP waveguide (N = 50) at some wavelengths are shown in Fig. 2
Fig. 2 Complex dielectric spectra of Ag NP waveguide (N = 50) for mode-matching condition with excitation field of different wavelengths (a): 354nm; (b): 442nm; (c): 633nm; (d): 830nm. The solid dots and the hollow dots represent results for LM and TM respectively.
. It is not a coincidence that the imaginary parts in Fig. 2 always have negative values. To realize low-loss energy guiding, the MN should be accompanied by a host with negative εh representing optical gain. In practical experiments based on the aforesaid scheme, the gain can be contributed from doped R6G. Those modes as revealed in Fig. 2(b) with values of εh from 2.25 to 3.75 are within the range of effective permittivity offered by the dielectric environment.

It is generally believed that the optimized transmission wavelength is very close to the plasmonic resonance wavelength of single MN [3

3. 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]

,4

4. 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]

,6

6. S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, “Plasmonics-A route to nanoscale optical devices,” Adv. Mater. (Deerfield Beach Fla.) 13(19), 1501–1505 (2001). [CrossRef]

] (~375nm for Ag NP). From Fig. 2(a), interestingly, we found that for Ag NP waveguide (N = 50) in vacuum as what has been discussed in Ref. 3

3. 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]

, optimized TM transmission will never occur at transmission wavelength ~375nm, because εh is always less than 1.0. Although a guiding mode with complex permittivity εh=0.9892-0.1368i (approaches 1.0) does exist for LM, due to the considerable imaginary part, transmission performance is not very good (the decay will be addressed in later sessions).

Analysis of the eigenmodes appears in Fig. 3
Fig. 3 Analysis of eigenmodes. The excitation wavelength is 442nm. (a) (b) Normalized dipole moment and phase distribution of LM1, LM2 and TM1. (c) Normalized P2of LM3 and comparison with its ideal guiding case. (d) Phase distributions of LM3.
. We selected four typical modes as illustration. Here the mode profiles in terms of dipole moments in Fig. 3(a) are normalized to keep the sum of the real-time electromagnetic energy stored inside those dispersive spheres (i.e. the MN chain) constant [17

17. R. Ruppin, “Electromagnetic energy in dispersive spheres,” J. Opt. Soc. Am. A 15(2), 524 (1998). [CrossRef]

]. Different mode profiles induced by different oscillation phase distributions of MNs are revealed in Fig. 3(b) and 3(d). To the entire MN chain, some of the eigenmodes originate from in-phase oscillation of the MNs, such as LM1; some are from dipole oscillation, such as TM1 (LM1: εh=1.90030.5583i, TM1: εh=3.35070.4384i). The most common origin is multipole oscillation as demonstrated by LM2 and LM3 (LM2: εh=1.91090.5481i, LM3 εh=2.95900.0935i). We should point out that besides fundamental excitations that lead to propagation at the dipolar resonances of the individual particles, higher order multipole fields in the interparticle interactions will play more crucial role in MN chain based waveguides. A merit of MN chain waveguide is T or L splitting, in which two waveguides will be jointed perpendicularly for integrated nanooptical circuit [6

6. S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, “Plasmonics-A route to nanoscale optical devices,” Adv. Mater. (Deerfield Beach Fla.) 13(19), 1501–1505 (2001). [CrossRef]

]. The most favorable host environment for low-loss T or L splitting can also be derived using the aforementioned physical model. The ideal guiding modes, i.e. zero insertion loss, will exist with the appropriate choice of parameters. Since the main mechanism of MN waveguide is near-field coupling, one can consider that T or L splitting as two jointed waveguides. As a reasonable approximation, it can be expected that T or L splitting can be achieved by selecting a host medium that can provide good transmission performance for both TM and LM. This can be satisfied with the value ofεh located in the crossover regions of solid dots and hollow dots in Fig. 2. Meanwhile, the best splitting node can be estimated from the mode profile analysis. To the best of our knowledge, this is the first time that this issue has been discussed from this angle and we will continue to explore this point in the future.

To evaluate the decay of the transmission by plotting out the final distribution of P2, we substituted the calculated εhinto the matrixP=M1|1,0...0in order to find the solutions. For the “butterfly” mode (i.e. LM3), we demonstrated that when the mode-matching condition is fulfilled exactly (εh=2.95900.0935i), the loss approaches zero as the comparison shown in Fig. 3(c). Note that for the P2 distribution, the normalization means the first dipole moment should be normalized to 1. Figure 4(a)
Fig. 4 (a) Normalized P2with different ε at excitation wavelength 442nm for the “butterfly” mode. (b) Normalized P2for different dielectric host at different wavelengths. The dielectric constant for TM and LM at 633nm are 5.6143 and 7.3651; and at 830nm are 15.9860 and 15.5893 respectively. At 354nm, the medium is vacuum for both TM and LM.
shows the emergence of the “butterfly” mode when εh varies from 0 to −0.0785 with fixed valueεh=2.9590. It reveals that in practice some deviation from the fully satisfied mode-matching condition is tolerable for achieving low-loss transmission. The gain coefficient (2π/λ0)εh/(εh)12 for the “butterfly” mode is ~103cm1, which is within the limits of semiconductor polymers, laser dyes (highly concentrated, or absorbed onto MN) [13

13. V. M. Shalaev, and S. Kawata, Nanophotonics with Surface Plasmons (Elsevier Science Ltd. 2007), Chap. 5.

]. Actually, for many other modes, the required gain coefficients are all within the limits too, especially when the excitation wavelength has a red shift. In the absence of a gain medium, a material with only optimized εh can also lead to significantly reduced transmission loss as is shown in Fig. 4(b). The transmission decay of LM and TM at excitation wavelengths 633nm and 830 nm are plotted. As a comparative study, the results in Ref. 3 are also reproduced and shown for LM and TM in plasmonic resonance wavelength of single MN (~375nm for Ag).

In the results from Ref. 3

3. 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]

in Fig. 4(b), the initial decay is rapid and non-exponential. This is mainly due to large deviation from the mode-matching condition as revealed in Fig. 2. In the terminal section of TM (354nm), energy transmission is mainly due to superposition of the far field. Even the decay is slow, the value is very small due to absence of sufficient near-field coupling. However, the transmission performance is much better with optimized dielectric host, such as for the case of LM (at 830nm,εh=15.5893) in Fig. 4(b), with a global exponential fitP2=P0exp(x/τ), where the 1/e damping length has a value of approximately 479nm. Note that the damping length here accounts for the global fitting instead of for arbitrarily chosen terminal section in Ref. 3

3. 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]

. The proposed optimization can be evaluated at the output terminal of the waveguide. Compared to the results from Ref. 3

3. 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]

, at least 3~4 orders higher energy output can be achieved at the 50th MN.

3. Conclusion

In conclusion, we have studied the gain-assisted MN chain waveguide and proposed a fabrication scheme. We demonstrate that the transmission loss can be dramatically reduced, and even completed removed in an ideal case upon choosing an appropriate gain medium. The mode-matching condition in terms of complex dielectric spectra has been identified through the pseudo-orthonormal basis expansion method. We also point out that besides dipolar resonances of individual MNs, majority of the energy guiding in the MN waveguide occurs in multipole collective oscillation mode. The proposed low-loss MN waveguide is also capable of performing efficient power splitting.

Acknowledgments

The authors would like to acknowledge the financial support from a Research Grants Council (RGC) Central Allocation Grant (HKUST 2/07C), which provides a research studentship (H.X. Zhang), and a General Research Grant under project number 412208.

References and links

1.

C. J. Wang, L. Huang, B. A. Parviz, and L. Y. Lin, “Subdiffraction photon guidance by quantum-dot cascades,” Nano Lett. 6(11), 2549–2553 (2006). [CrossRef] [PubMed]

2.

A. W. Sanders, D. A. Routenberg, B. J. Wiley, Y. Xia, E. R. Dufresne, and M. A. Reed, “Observation of plasmon propagation, redirection, and fan-out in silver nanowires,” Nano Lett. 6(8), 1822–1826 (2006). [CrossRef] [PubMed]

3.

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]

4.

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]

5.

K. H. Fung and C. T. Chan, “Plasmonic modes in periodic metal nanoparticle chains: a direct dynamic eigenmode analysis,” Opt. Lett. 32(8), 973–975 (2007). [CrossRef] [PubMed]

6.

S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, “Plasmonics-A route to nanoscale optical devices,” Adv. Mater. (Deerfield Beach Fla.) 13(19), 1501–1505 (2001). [CrossRef]

7.

M. Bashevoy, V. Fedotov, and N. Zheludev, “Optical whirlpool on an absorbing metallic nanoparticle,” Opt. Express 13(21), 8372–8379 (2005). [CrossRef] [PubMed]

8.

N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett. 95(9), 095504 (2005). [CrossRef] [PubMed]

9.

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]

10.

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]

11.

L. Wang and W. Tan, “Multicolor FRET silica nanoparticles by single wavelength excitation,” Nano Lett. 6(1), 84–88 (2006). [CrossRef] [PubMed]

12.

T. Ming, L. Zhao, Z. Yang, H. Chen, L. Sun, J. Wang, and C. Yan, “Strong polarization dependence of plasmon-enhanced fluorescence on single gold nanorods,” Nano Lett. 9(11), 3896–3903 (2009). [CrossRef] [PubMed]

13.

V. M. Shalaev, and S. Kawata, Nanophotonics with Surface Plasmons (Elsevier Science Ltd. 2007), Chap. 5.

14.

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

15.

H. Y. Chung, P. T. Leung, and D. P. Tsai, “Dynamic modifications of polarizability for large metallic spheroidal nanoshells,” J. Chem. Phys. 131(12), 124122 (2009). [CrossRef] [PubMed]

16.

V. A. Markel, “Antisymmetrical optical states,” J. Opt. Soc. Am. B 12(10), 1783 (1995). [CrossRef]

17.

R. Ruppin, “Electromagnetic energy in dispersive spheres,” J. Opt. Soc. Am. A 15(2), 524 (1998). [CrossRef]

OCIS Codes
(130.3120) Integrated optics : Integrated optics devices
(230.7370) Optical devices : Waveguides
(160.3918) Materials : Metamaterials
(250.5403) Optoelectronics : Plasmonics

ToC Category:
Integrated Optics

History
Original Manuscript: July 13, 2010
Revised Manuscript: October 6, 2010
Manuscript Accepted: October 8, 2010
Published: October 18, 2010

Citation
Haixi Zhang and Ho-Pui Ho, "Low-loss plasmonic waveguide based on 
gain-assisted periodic metal nanosphere chains," Opt. Express 18, 23035-23040 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-22-23035


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References

  1. C. J. Wang, L. Huang, B. A. Parviz, and L. Y. Lin, “Subdiffraction photon guidance by quantum-dot cascades,” Nano Lett. 6(11), 2549–2553 (2006). [CrossRef] [PubMed]
  2. A. W. Sanders, D. A. Routenberg, B. J. Wiley, Y. Xia, E. R. Dufresne, and M. A. Reed, “Observation of plasmon propagation, redirection, and fan-out in silver nanowires,” Nano Lett. 6(8), 1822–1826 (2006). [CrossRef] [PubMed]
  3. 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]
  4. 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]
  5. K. H. Fung and C. T. Chan, “Plasmonic modes in periodic metal nanoparticle chains: a direct dynamic eigenmode analysis,” Opt. Lett. 32(8), 973–975 (2007). [CrossRef] [PubMed]
  6. S. A. Maier, M. L. Brongersma, P. G. Kik, S. Meltzer, A. A. G. Requicha, and H. A. Atwater, “Plasmonics-A route to nanoscale optical devices,” Adv. Mater. (Deerfield Beach Fla.) 13(19), 1501–1505 (2001). [CrossRef]
  7. M. Bashevoy, V. Fedotov, and N. Zheludev, “Optical whirlpool on an absorbing metallic nanoparticle,” Opt. Express 13(21), 8372–8379 (2005). [CrossRef] [PubMed]
  8. N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett. 95(9), 095504 (2005). [CrossRef] [PubMed]
  9. 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]
  10. 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]
  11. L. Wang and W. Tan, “Multicolor FRET silica nanoparticles by single wavelength excitation,” Nano Lett. 6(1), 84–88 (2006). [CrossRef] [PubMed]
  12. T. Ming, L. Zhao, Z. Yang, H. Chen, L. Sun, J. Wang, and C. Yan, “Strong polarization dependence of plasmon-enhanced fluorescence on single gold nanorods,” Nano Lett. 9(11), 3896–3903 (2009). [CrossRef] [PubMed]
  13. V. M. Shalaev, and S. Kawata, Nanophotonics with Surface Plasmons (Elsevier Science Ltd. 2007), Chap. 5.
  14. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]
  15. H. Y. Chung, P. T. Leung, and D. P. Tsai, “Dynamic modifications of polarizability for large metallic spheroidal nanoshells,” J. Chem. Phys. 131(12), 124122 (2009). [CrossRef] [PubMed]
  16. V. A. Markel, “Antisymmetrical optical states,” J. Opt. Soc. Am. B 12(10), 1783 (1995). [CrossRef]
  17. R. Ruppin, “Electromagnetic energy in dispersive spheres,” J. Opt. Soc. Am. A 15(2), 524 (1998). [CrossRef]

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