Reflection of Cylindrical Surface Waves - OSA Publishing

Reflection of Cylindrical Surface Waves Reuven Gordon Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, Canada V8W3P6 [email protected]

Abstract: The reflection of the radially polarized surface wave on a metal wire at an abrupt end is derived. This theory allows for straightforward calculation of the reflection coefficient, including the phase and the amplitude, which will prove useful to the many applications in nanoplasmonics and terahertz spectroscopy. The theory shows excellent quantitative agreement with past comprehensive numerical simulations for small wires and for predicting the minima in reflection for larger wires. Using this theory, the wavelength dependent reflection is calculated for gold rods of diameter 10 nm, 26 nm and 85 nm, from which the Fabry-Perot resonance wavelengths are found. The Fabry-Perot resonances show good agreement with experimentally measured surface plasmon resonances in nanorods. This demonstrates the predictive ability of the theory for applications involving widely-used nanorods, optical antennas and plasmonic resonators. © 2009 Optical Society of America OCIS codes: (230.7370) Waveguides; (240.6680) Surface plasmons; (240.6690) Surface waves; (280.1415) Biological sensing and sensors; (300.6495) Spectroscopy, terahertz; (310.6628) Subwavelength structures, nanostructures; (350.4238) Nanophotonics and photonic crystals

References and links 1. A. Sommerfeld, “Ueber die Fortpflanzung elektrodynamischer Wellen lngs eines Drahtes,” Annalen der Physik und Chemie 303, 233–290 (1899). URL http://dx.doi.org/10.1002/andp.18993030202. 2. G. Goubau, “Surface Waves and Their Application to Transmission Lines,” J. Appl. Phys. 21, 1119–1128 (1950). URL http://link.aip.org/link/?JAP/21/1119/1. 3. K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature 432, 376–379 (2004). URL http://dx.doi.org/0.1038/nature03040. 4. M. Wächter, M. Nagel, and H. Kurz, “Frequency-dependent characterization of THz Sommerfeld wave propagation on single-wires,” Opt. Express 13, 10815–10822 (2005). URL http://www.opticsexpress.org/ abstract.cfm?URI=oe-13-26-10815. 5. M. Walther, M. R. Freeman, and F. A. Hegmann, “Metal-wire terahertz time-domain spectroscopy,” Appl. Phys. Lett. 87, 261107 (2005). URL http://link.aip.org/link/?APL/87/261107/1. 6. Q. Cao and J. Jahns, “Azimuthally polarized surface plasmons as effective terahertz waveguides,” Opt. Express 13, 511–518 (2005). URL http://www.opticsexpress.org/abstract.cfm?URI= oe-13-2-511. 7. K. Wang and D. M. Mittleman, “Dispersion of Surface Plasmon Polaritons on Metal Wires in the Terahertz Frequency Range,” Phys. Rev. Lett. 96, 157401 (2006). URL http://link.aps.org/abstract/PRL/ v96/e157401. 8. J. A. Deibel, N. Berndsen, K. Wang, D. M. Mittleman, N. C. van der Valk, and P. C. M. Planken, “Frequency-dependent radiation patterns emitted by THz plasmons on finite length cylindrical metal wires,” Opt. Express 14, 8772–8778 (2006). URL http://www.opticsexpress.org/abstract.cfm?URI= oe-14-19-8772. 9. S. A. Maier, S. R. Andrews, L. Mart´ın-Moreno, and F. J. Garc´ıa-Vidal, “Terahertz Surface Plasmon-Polariton Propagation and Focusing on Periodically Corrugated Metal Wires,” Phys. Rev. Lett. 97, 176805 (2006). URL http://link.aps.org/abstract/PRL/v97/e176805.

#115495 - $15.00 USD Received 10 Aug 2009; revised 28 Sep 2009; accepted 29 Sep 2009; published 30 Sep 2009

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10. H. Ditlbacher, A. Hohenau, D. Wagner, U. Kreibig, M. Rogers, F. Hofer, F. R. Aussenegg, and J. R. Krenn, “Silver Nanowires as Surface Plasmon Resonators,” Phys. Rev. Lett. 95, 257 – 403 (2005). URL http:// link.aps.org/doi/10.1103/PhysRevLett.95.257403. 11. P. Muhlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant Optical Antennas,” Science 308, 1607–1609 (2005). http://www.sciencemag.org/cgi/reprint/308/5728/1607. pdf, URL http://www.sciencemag.org/cgi/content/abstract/308/5728/1607. 12. L. Novotny, “Effective Wavelength Scaling for Optical Antennas,” Phys. Rev. Lett. 98, 266802 (2007). URL http://link.aps.org/abstract/PRL/v98/e266802. 13. T. Søndergaard, J. Beermann, A. Boltasseva, and S. I. Bozhevolnyi, “Slow-plasmon resonant-nanostrip antennas: Analysis and demonstration,” Phys. Rev. B 77, 115420 (2008). URL http://link.aps.org/abstract/ PRB/v77/e115420. 14. E. S. Barnard, J. S. White, A. Chandran, and M. L. 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Zhao, “Aligned silver nanorod arrays produce high sensitivity surface-enhanced Raman spectroscopy substrates,” Appl. Phys. Lett. 87, 031908 (2005). URL http://link. aip.org/link/?APL/87/031908/1. 19. H. V. Chu, Y. Liu, Y. Huang, and Y. Zhao, “A high sensitive fiber SERS probe based on silver nanorod arrays,” Opt. Express 15, 12230–12239 (2007). URL http://www.opticsexpress.org/abstract. cfm?URI=oe-15-19-12230. 20. K. C. Toussaint, M. Liu, M. Pelton, J. Pesic, M. J. Guffey, P. Guyot-Sionnest, and N. F. Scherer, “Plasmon resonance-based optical trapping of single and multiple Au nanoparticles,” Opt. Express 15, 12017–12029 (2007). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-15-19-12017. 21. F. J. Rodr´ıguez-Fortuño, C. Garc´ıa-Meca, R. Ortuño, J. Mart´ı, and A. Mart´ınez, “Modeling high-order plasmon resonances of a U-shaped nanowire used to build a negative-index metamaterial,” Phys. Rev. B 79, 075103 (2009). URL http://link.aps.org/abstract/PRB/v79/e075103. 22. T. H. Taminiau, R. J. Moerland, F. B. Segerink, L. Kuipers, and N. F. van Hulst, “Lambda/4 Resonance of an Optical Monopole Antenna Probed by Single Molecule Fluorescence,” Nano Lett. 7, 28–33 (2006). URL http://dx.doi.org/10.1021/nl061726h. 23. X.-W. Chen, V. Sandoghdar, and M. Agio, “Highly efficient interfacing of guided plasmons and photons in nanowires,” Nano Lett. ASAP (2009). URL http://pubs.acs.org/doi/full/10.1021/ nl9019424. 24. 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, 402–406 (2007). URL http://dx.doi.org/10.1038/nature06230. 25. A. L. Falk, F. H. L. Koppens, C. L. Yu, K. Kang, N. de Leon Snapp, A. V. Akimov, M.-H. Jo, M. D. Lukin, and H. Park, “Near-field electrical detection of optical plasmons and single-plasmon sources,” Nat. Phys. 5, 475–479 (2009). URL http://dx.doi.org/10.1038/nphys1284. 26. R. Gordon, “Vectorial method for calculating the Fresnel reflection of surface plasmon polaritons,” Phys. Rev. B 74, 153417 (2006). URL http://link.aps.org/abstract/PRB/v74/e153417. 27. R. Gordon, “Light in a subwavelength slit in a metal: Propagation and reflection,” Phys. Rev. B 73, 153405 (2006). URL http://link.aps.org/abstract/PRB/v73/e153405. 28. R. Gordon, “Near-field interference in a subwavelength double slit in a perfect conductor,” J. Opt. A 8, L1–L3 (2006). URL http://stacks.iop.org/1464-4258/8/L1. 29. R. Gordon, “Angle-dependent optical transmission through a narrow slit in a thick metal film,” Phys. Rev. B 75, 193401 (2007). URL http://link.aps.org/abstract/PRB/v75/e193401. 30. P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B 6, 4370–4379 (1972). URL http://link.aps.org/doi/10.1103/PhysRevB.6.4370. 31. S. J. Al-Bader and H. A. Jamid, “Diffraction of surface plasmon modes on abruptly terminated metallic nanowires,” Phys. Rev. B 76, 235410 (2007). URL http://link.aps.org/abstract/PRB/v76/ e235410. 32. R. Pregla, Analysis of Electromagnetic Fields and Waves: The Method of Lines (John Wiley and Sons Ltd., West Sussex (England), 2008). 33. A.-S. A.-M. Al-Sherbini, “Thermal instability of gold nanorods in micellar solution of water/glycerol mixtures,”


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Colloids and Surfaces A 246, 61–69 (2004). URL 10.1016/j.colsurfa.2004.06.038. 34. J. Pérez-Juste, L. Liz-Marzán, S. Carnie, D. Chan, and P. Mulvaney, “Electric-Field-Directed Growth of Gold Nanorods in Aqueous Surfactant Solutions,” Advanced Functional Materials 14, 571–579 (2004). URL http: //dx.doi.org/10.1002/adfm.200305068. 35. K.-S. Lee and M. A. El-Sayed, “Dependence of the Enhanced Optical Scattering Efficiency Relative to That of Absorption for Gold Metal Nanorods on Aspect Ratio, Size, End-Cap Shape, and Medium Refractive Index,” J. Phys. Chem. B 109, 20,331–20,338 (2005). URL http://dx.doi.org/10.1021/jp054385p. 36. S. W. Prescott and P. Mulvaney, “Gold nanorod extinction spectra,” J. Appl. Phys. 99, 123504 (2006). URL http://link.aip.org/link/?JAP/99/123504/1. 37. C. Ungureanu, R. G. Rayavarapu, S. Manohar, and T. G. van Leeuwen, “Discrete dipole approximation simulations of gold nanorod optical properties: Choice of input parameters and comparison with experiment,” J. Appl. Phys. 105, 102032 (2009). URL http://link.aip.org/link/?JAP/105/102032/1. 38. Q. Min and R. Gordon, “Squeezing light into subwavelength metallic tapers: single mode matching method,” J. Nanophotonics 3, 033505 (2009). URL http://link.aip.org/link/?JNP/3/033505/1. 39. E. Verhagen, M. Spasenović, A. Polman, and L. K. Kuipers, “Nanowire Plasmon Excitation by Adiabatic Mode Transformation,” Phys. Rev. Lett. 102, 203904 (2009). URL http://link.aps.org/abstract/PRL/ v102/e203904.

1.

Introduction

The Sommerfeld surface wave of a cylindrical metal wire, originally proposed in 1899 [1] and subsequently studied for microwave transmission lines [2], has received growing research attention since 2004 for frequencies ranging from the THz to visible. In the THz regime, this surface wave allows a single wire to be a simple low-loss low-dispersion waveguide [3, 4], which may be used as a near-field probe for THz spectroscopy [5]. Many works have addressed this THz regime, considering, for example, waveguiding [6, 7], radiation patterns [8], and modified geometries, such as corrugated wires [9]. In the visible-IR regime, the surface waves reflected from the ends of a rod can produce compact resonantors [10], with applications for optical antennas [11, 12, 13, 14, 15], nonlinear optics [11], imaging [16], sensors [17], Raman spectroscopy [18, 19], optical manipulation [20], metamaterials [21], and near-field imaging and spectroscopy [22, 23]. These surface waves also have highly-efficient coupling to quantum emitters, which has been exploited for single plasmon sources [24] and detectors [25]. In each of these applications, the amplitude and phase of reflection of the surface wave at the wire-ends is critical to strength and wavelength of the resonances or coupling. It is well-known that wire resonators are shorter than expected from conventional antenna theory because of the phase of reflection at the end-faces [12, 13, 14, 15, 21, 22]. For quantum-emitters, the phase-constructive placement of the quantum emitter with respect to the end-face is critical for enhanced self-coupling emission and detection [24, 25]. In the past, the reflection has been calculated a posteri from numerical calculations or experimental observations. In this work, a theory for surface wave reflection for a cylindrical metal wire is derived. This theory provides a priori calculation of the complex reflection coefficient. The analytic results show excellent agreement with comprehensive numerical simulations for small wires, and for minima in reflection in larger wires. The wavelength dispersion is studied for 10 nm, 26 nm and 85 nm wires in the visible-IR region. It is also shown that the theory allows for accurately calculating the resonances of experimentally measured gold nanorods. This work will provide new physical insight to the many aforementioned emerging applications of cylindrical surface waves and enable improved design of cylindrical waveguide structures. 2.

Derivation of Theory of Reflection at Wire-End

The method of calculation is based on matching the incident and reflected waves of a single mode on one side of the boundary to the continuum of modes of free-space on the other side of the boundary. The approximation of this method is to neglect, to first order, the reflected waves


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on the incident side of the boundary, which allows for a closed-form calculation of the reflection and transmission coefficients. The approach has proven to be accurate for subwavelength systems, where a single mode dominates the overall behavior [26, 27, 28, 29]. In cylindrical coordinates (ρ , φ , z), the cylindrical wire of radius a is centered on the negative z−axis and terminates at z = 0. The single lowest-order radially polarized mode of the wire is incident from z = 0− and it is assumed that the reflection is entirely into the same mode (traveling in the opposite direction), so that the radial electric field and the azimuthal magnetic field can be approximated by: Eρ (ρ , φ , z = 0− ) (1 + r) and

β F(ρ ) ωε0 εr (ρ )

Hφ (ρ , φ , z = 0− ) (1 − r)F(ρ )

(1)

(2)

where r is the reflection coefficient, β is the propagation constant of the mode, ω is the angular frequency of the electromagnetic wave, ε0 is the permittivity of free-space, εr (ρ < a) = εm the relative permittivity of the metal wire, εr (ρ > a) = εd the relative permittivity of the surrounding dielectric medium, and I (p ρ ) 1 m if ρ < a I1 (pm a) F(ρ ) = (3) K1 (pd ρ ) if ρ > a K (p a) 1

d

with In and Kn being the modified Bessel functions of first and second kind of order n, pm,d = β 2 − k02 εm,d and k0 = ω /c, where c is the speed of light in free-space. The propagation constant is found by matching the boundary conditions at the edge of the cylinder, which requires solving the following equation for β :

εd pm K0 (pd a)I1 (pm a) =− . K1 (pd a)I0 (pm a) εm pd

(4)

The propagation constant, β , is a function of the angular frequency, ω , the cylinder radius, a, and the relative permittivities of the metal and dielectric, εm,d (that also depend on the angular frequency in general). Dispersion in this equation has important effects on the resonances of nanorods [12] and the propagation of THz waves [7]. Here, similarly important effects of the dispersion in reflection will be considered. For z = 0+ , the electric and magnetic fields are expanded in terms of the free-space modes that are rotationally invariant to preserve the symmetry of the problem: ∞ k02 εd − k2 Eρ (ρ , φ , z = 0+ ) = t(k) J1 (kr) dk (5) ωε0 εd 0 and Hφ (ρ , φ , z = 0+ ) =

∞ 0

t(k)J1 (kr) dk

(6)

where J1 (kr) is the first order Bessel function of the first kind and εd is the relative permittivity of the region z > 0, with a prime to emphasize that it may be different from the medium surrounding the wire. To solve the reflection coefficient, the transverse electric and magnetic fields are matched at z = 0 and the orthogonality of the free-space and wire modes is used. Considering the electric #115495 - $15.00 USD Received 10 Aug 2009; revised 28 Sep 2009; accepted 29 Sep 2009; published 30 Sep 2009

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field first, Eq. (1) is equated to Eq. (5), and the orthogonality of the free-space is invoked by multiplying both sides by J1 (k ρ )ρ and integrating over ρ from 0 to ∞. Using the orthogonality property of Bessel functions of the first kind gives: t(k) = (1 + r) with

kaβ ε d [A1 (k) + A2 (k)] ωε0 k02 εd − k2

(7)

A1 (k) =

pm I2 (pm a)J1 (ka) + kI1 (pm a)J2 (ka) I1 (pm a)εm (k2 + p2m )

(8)

A2 (k) =

pd K2 (pd a)J1 (ka) − kK1 (pd a)J2 (ka) . K1 (pd a)εd (k2 + p2d )

(9)

and

Next, the magnetic field is considered, Eq. (2) is equated to Eq. (6), and the orthogonality of the wire mode is used by multiplying both sides by F(ρ )ρ /εr (ρ ) and integrating over ρ from 0 to ∞. This gives the final result: 1−G (10) r= 1+G where G is given by: ∞ 0

G= 3.

2β εd k k02 εd −k2

[A1 (k) + A2 (k)]2 dk

I1 (pm a)2 −I0 (pm a)I2 (pm a) εm I1 (pm a)2

0 (pd a)K2 (pd a) − K1 (pd a)ε −K K (p a)2 2

d 1

.

(11)

d

Discussion and Calculations

Here, the reflection amplitude and phase-of-reflection calculated using the above theory are compared with past comprehensive numerical solutions to Maxwell’s equations for the same geometry. The wavelength dispersion of the reflection is calculated for abruptly terminated gold cylinders. Finally, the calculated Fabry-Perot resonances are compared with experimentally measured gold nanorod surface plasmon extinction resonances for nanorods prepared by two different methods. 3.1.

Comparison with Comprehensive Numerical Simulations of Maxwell’s Equations

Figure 1(a) shows with a solid blue line the reflection amplitude R = |r|2 as calculated from Eq. (10) above for a cylindrical wire in free-space at the free-space wavelength of 632.8 nm and assuming a relative permittivity of the wire of -16 (which is of larger magnitude than reported elsewhere [30]). The dashed line, extracted for comparison from Fig. 4 of Ref. [31], shows comprehensive numerical solutions to Maxwell’s equations for the same geometry as carried out using the method-of-lines approach [32]. The reflection calculations here show good quantitative agreement with those past comprehensive numerical calculations for wire radii of up to a few tens of nanometers. Interestingly, there are minima in the reflection that result from oscillations in the Bessel function in the radial direction. The diameters with minima in reflection also agree very well with that past work [31]. As expected, the quantitative agreement is not as good with that work when there is substantial reflection into other modes (which is not shown here, but can be found in Ref. [31]); however, the qualitative features are retained. This last point is expected since the analysis method here does not account for additional modes on the reflection side of the interface. For completeness, Fig. 1(b) shows the phase-of-reflection for same conditions as Fig. 1(a). #115495 - $15.00 USD Received 10 Aug 2009; revised 28 Sep 2009; accepted 29 Sep 2009; published 30 Sep 2009

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1 (a) reflection amplitude

0.8 ρ

0.6

φ z

2a

0.4 0.2 0

0

100

200

300 400 radius (nm)

500

600

700

200

300 400 radius (nm)

500

600

700

140

reflection phase (degrees)

(b) 120 100 80 60 40 0

100

Fig. 1. (a) Reflection amplitude, R = |r|2 , for the surface wave reflecting at a terminated wire (εm = −16) in free-space for free-space wavelength of 632.8 nm, as calculated from Eq. (10). Dashed line extracted from Fig. 4 of Ref [31], where a comprehensive numerical simulation of Maxwell’s equations was used for similar parameters (except εm = −16 + 0.53i). (b) Reflection phase for the same conditions as (a).

3.2.

Reflection for Various Wire Diameters in the Visible to Near-IR

Figure 2 shows the reflection for abruptly terminated gold wires of diameter 10 nm, 26 nm, 85 nm for a range of wavelengths in the visible to near-IR region. The wires are in water (refractive index of 1.33) and the relative permittivity values for gold are taken from experimental measurements [30]. The reflection amplitude increases as the wire diameter is made smaller or as the wavelength approaches the plasma wavelength (i.e., is shortened). The reflection phase also increases as the wavelength is shortened, but decreases as the wire is made narrower. The phase has an important influence on the Fabry-Perot resonances of nanorods, as will be described in the next #115495 - $15.00 USD Received 10 Aug 2009; revised 28 Sep 2009; accepted 29 Sep 2009; published 30 Sep 2009

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section.

reflection amplitude

1

(a)

0.9 0.8 0.7 0.6

500

1000 1500 wavelength (nm)

2000

1000 1500 wavelength (nm)

2000

2 (b) reflection phase (radians)

1.8 1.6 1.4 1.2 1 0.8 500

Fig. 2. Reflection (a) amplitude and (b) phase of abruptly terminated gold nanowires in water for diameters of 10 nm (blue), 26 nm (red) and 85 nm (black).

3.3.

Comparison with Experimentally Measured Nanorod Resonances

Fabry-Perot resonances of a nanorod may be calculated by considering the propagation of the surface wave along the rod and reflection at the ends. If a resonance is required at a particular wavelength, the corresponding rod-length to produce such a resonance is given by: l=

mπ − φ β

(12)

where m is the whole-number resonance order, φ is the phase of reflection calculated from Eq. (10), and β is the propagation constant calculated from Eq. (4). Figure 3(a) shows with lines the Fabry-Perot resonances calculated using Eq. (12) for 85 nm gold nanorods of various lengths for resonance orders m = 1, 2, 3, 4. Experimentally measured #115495 - $15.00 USD Received 10 Aug 2009; revised 28 Sep 2009; accepted 29 Sep 2009; published 30 Sep 2009

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surface plasmon extinction maximum wavelengths for 85 nm wide gold rods formed by electrochemical deposition in anodic alumina templates are shown with markers (extracted from Ref. [17]). Good agreement is seen between the experimental extinction maxima and the calculated Fabry-Perot resonance wavelengths, over a wide range of wavelengths and for different resonance orders. 1500

(a)

resonance wavelength (nm)

1400 1300 1200 1100 1000 900 800 700 600 0

5

10

15

100

150

aspect ratio

resonance wavelength (nm)

1200

(b)

1100 1000 900 800 700 600 0

50 length (nm)

Fig. 3. Comparison between calculated Fabry-Perot resonances and experimentally measured plasmon resonances. (a) 85 nm diameter nanorods of varying length. Solid lines show resonances calculated using Eq. (12) for m = 1 (blue), 2 (red), 3 (green) and 4 (black). Data points are experimentally measured values attributed to those resonances [17]. (b) 10 nm (blue) and 26 nm (red) diameter nanorods of various lengths. Data points are experimentally measured values for diameters of 10 nm (blue) [33] and 26 nm (red) [34].

Figure 3(b) compares the calculated Fabry-Perot resonances for nanorods of 10 nm and 26 nm with experimentally measured extinction resonance and geometry values reported in Refs. [33] and [34] (see the supporting material of the latter work). The agreement in the slope and the relative change when going from 10 nm to 26 nm is good; however, there seems to be a systematic offset between the theory and experiment. This is attributed to the rounded ends of #115495 - $15.00 USD Received 10 Aug 2009; revised 28 Sep 2009; accepted 29 Sep 2009; published 30 Sep 2009

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the nanorods, since these were fabricated by the electrochemical method. Rounded ends produce shorter-wavelength resonances than abruptly terminated cylinders, with differences up to 100 nm reported from calculations [35, 36, 37]. Inaccuracies in the relative permittivity values used may also contribute to part of the disagreement, since this has a greater influence for thinner rods. Using relative permittivity values from another commonly cited reference gives up to 30 nm shorter wavelength resonance values in comprehensive calculations for rod diameters less than 20 nm [37]. 4.

Conclusions and Outlook

In summary, the theory of reflection for a surface wave on a cylindrical metal wire at the wireend was derived. This theory allows for rapid calculation of the reflection coefficient, giving both the phase and the amplitude of reflection. This will prove useful to the many applications employing nanorods and nanowires as basic plasmonic elements, including enhanced quantum emitters, nonlinear optics, spectroscopy, and plasmonic integrated circuits. It is also possible to apply the theory to the treatment of THz Sommerfeld waves, with applications in THz waveguiding and spectroscopy. It is expected that this method can be combined with recently demonstrated single mode matching method to calculate the reflection of a taper with an abrupt end [38]. Along the same lines, this work may find use in terminated adiabatic tapers, which have recently been demonstrated [39]. Furthermore, extensions of the theory to investigate the influence of losses, which were ignored in this work, may be possible. Finally, approximations to the integral in Eq. (11) may prove useful for more detailed analysis of scaling laws (for example, see Ref. [12]), but they were not considered here due to the ease in calculating the integral accurately using readily available mathematical software. Acknowledgements RG thanks Drs. Mario Agio and Tim Taminiau for interesting conversations regarding nanorods. This work is supported by funding from the Natural Sciences and Engineering Research Council (NSERC) Canada: Discovery Grant and Strategic Network for Bioplasmonic Systems (Biopsys).


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