Reflection of Cylindrical Surface Waves

Reflection of Cylindrical Surface Waves

Reuven GordonDepartment of Electrical and Computer Engineering, University of Victoria, Victoria, BC,

Canada [email protected]

Abstract: The reflection of the radially polarized surface wave on a metalwire at an abrupt end is derived. This theory allows for straightforward cal-culation of the reflection coefficient, including the phase and the amplitude,which will prove useful to the many applications in nanoplasmonics andterahertz spectroscopy. The theory shows excellent quantitative agreementwith past comprehensive numerical simulations for small wires and forpredicting the minima in reflection for larger wires. Using this theory, thewavelength dependent reflection is calculated for gold rods of diameter10 nm, 26 nm and 85 nm, from which the Fabry-Perot resonance wave-lengths are found. The Fabry-Perot resonances show good agreement withexperimentally measured surface plasmon resonances in nanorods. Thisdemonstrates the predictive ability of the theory for applications involvingwidely-used nanorods, optical antennas and plasmonic resonators.

© 2009 Optical Society of America

OCIS codes: (230.7370) Waveguides; (240.6680) Surface plasmons; (240.6690) Surfacewaves; (280.1415) Biological sensing and sensors; (300.6495) Spectroscopy, terahertz;(310.6628) Subwavelength structures, nanostructures; (350.4238) Nanophotonics and photoniccrystals

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

(C) 2009 OSA 12 October 2009 / Vol. 17, No. 21 / OPTICS EXPRESS 18621

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1. Introduction

The Sommerfeld surface wave of a cylindrical metal wire, originally proposed in 1899 [1] andsubsequently studied for microwave transmission lines [2], has received growing research at-tention since 2004 for frequencies ranging from the THz to visible. In the THz regime, thissurface 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 addressedthis THz regime, considering, for example, waveguiding [6, 7], radiation patterns [8], and mod-ified geometries, such as corrugated wires [9]. In the visible-IR regime, the surface wavesreflected from the ends of a rod can produce compact resonantors [10], with applications foroptical antennas [11, 12, 13, 14, 15], nonlinear optics [11], imaging [16], sensors [17], Ramanspectroscopy [18, 19], optical manipulation [20], metamaterials [21], and near-field imagingand spectroscopy [22, 23]. These surface waves also have highly-efficient coupling to quan-tum emitters, which has been exploited for single plasmon sources [24] and detectors [25]. Ineach of these applications, the amplitude and phase of reflection of the surface wave at thewire-ends is critical to strength and wavelength of the resonances or coupling. It is well-knownthat wire resonators are shorter than expected from conventional antenna theory because ofthe phase of reflection at the end-faces [12, 13, 14, 15, 21, 22]. For quantum-emitters, thephase-constructive placement of the quantum emitter with respect to the end-face is criticalfor enhanced self-coupling emission and detection [24, 25]. In the past, the reflection has beencalculated a posteri from numerical calculations or experimental observations.

In this work, a theory for surface wave reflection for a cylindrical metal wire is derived. Thistheory provides a priori calculation of the complex reflection coefficient. The analytic resultsshow excellent agreement with comprehensive numerical simulations for small wires, and forminima in reflection in larger wires. The wavelength dispersion is studied for 10 nm, 26 nmand 85 nm wires in the visible-IR region. It is also shown that the theory allows for accuratelycalculating the resonances of experimentally measured gold nanorods. This work will providenew physical insight to the many aforementioned emerging applications of cylindrical surfacewaves 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 singlemode on one side of the boundary to the continuum of modes of free-space on the other side ofthe boundary. The approximation of this method is to neglect, to first order, the reflected waves



on the incident side of the boundary, which allows for a closed-form calculation of the reflec-tion and transmission coefficients. The approach has proven to be accurate for subwavelengthsystems, 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 negativez−axis and terminates at z = 0. The single lowest-order radially polarized mode of the wireis 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 magneticfield can be approximated by:

Eρ(ρ,φ ,z = 0−) � (1+ r)β

ωε0εr(ρ)F(ρ) (1)

andHφ (ρ,φ ,z = 0−) � (1− r)F(ρ) (2)

where r is the reflection coefficient, β is the propagation constant of the mode, ω is the angularfrequency of the electromagnetic wave, ε0 is the permittivity of free-space, εr(ρ < a) = εm therelative permittivity of the metal wire, εr(ρ > a) = εd the relative permittivity of the surround-ing dielectric medium, and

F(ρ) =

{I1(pmρ)I1(pma) if ρ < a

K1(pdρ)K1(pda) if ρ > a

(3)

with In and Kn being the modified Bessel functions of first and second kind of order n,

pm,d =√

β 2 − k20εm,d and k0 = ω/c, where c is the speed of light in free-space. The prop-

agation constant is found by matching the boundary conditions at the edge of the cylinder,which requires solving the following equation for β :

K0(pda)I1(pma)K1(pda)I0(pma)

= −εd pm

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 angularfrequency in general). Dispersion in this equation has important effects on the resonances ofnanorods [12] and the propagation of THz waves [7]. Here, similarly important effects of thedispersion in reflection will be considered.

For z = 0+, the electric and magnetic fields are expanded in terms of the free-space modesthat are rotationally invariant to preserve the symmetry of the problem:

Eρ(ρ,φ ,z = 0+) =∫ ∞

0t(k)

√k2

0ε ′d − k2

ωε0ε ′dJ1(kr)dk (5)

andHφ (ρ,φ ,z = 0+) =

∫ ∞

0t(k)J1(kr)dk (6)

where J1(kr) is the first order Bessel function of the first kind and ε ′d is the relative permittiv-ity of the region z > 0, with a prime to emphasize that it may be different from the mediumsurrounding the wire.

To solve the reflection coefficient, the transverse electric and magnetic fields are matched atz = 0 and the orthogonality of the free-space and wire modes is used. Considering the electric



field first, Eq. (1) is equated to Eq. (5), and the orthogonality of the free-space is invoked bymultiplying both sides by J1(k′ρ)ρ and integrating over ρ from 0 to ∞. Using the orthogonalityproperty of Bessel functions of the first kind gives:

t(k) = (1+ r)kaβε ′d

ωε0

√k2

0ε ′d − k2[A1(k)+A2(k)] (7)

with

A1(k) =pmI2(pma)J1(ka)+ kI1(pma)J2(ka)

I1(pma)εm(k2 + p2m)

(8)

and

A2(k) =pdK2(pda)J1(ka)− kK1(pda)J2(ka)

K1(pda)εd(k2 + p2d)

. (9)

Next, the magnetic field is considered, Eq. (2) is equated to Eq. (6), and the orthogonality ofthe wire mode is used by multiplying both sides by F(ρ)ρ/εr(ρ) and integrating over ρ from0 to ∞. This gives the final result:

r =1−G1+G

(10)

where G is given by:

G =

∫ ∞0

2βε ′dk√k20ε ′d−k2

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

I1(pma)2−I0(pma)I2(pma)εmI1(pma)2 − K1(pda)2−K0(pda)K2(pda)

εdK1(pda)2

. (11)

3. Discussion and Calculations

Here, the reflection amplitude and phase-of-reflection calculated using the above theory arecompared with past comprehensive numerical solutions to Maxwell’s equations for the samegeometry. The wavelength dispersion of the reflection is calculated for abruptly terminated goldcylinders. Finally, the calculated Fabry-Perot resonances are compared with experimentallymeasured gold nanorod surface plasmon extinction resonances for nanorods prepared by twodifferent 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 fromEq. (10) above for a cylindrical wire in free-space at the free-space wavelength of 632.8 nmand assuming a relative permittivity of the wire of -16 (which is of larger magnitude thanreported 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 ascarried out using the method-of-lines approach [32]. The reflection calculations here show goodquantitative agreement with those past comprehensive numerical calculations for wire radii ofup to a few tens of nanometers.

Interestingly, there are minima in the reflection that result from oscillations in the Besselfunction in the radial direction. The diameters with minima in reflection also agree very wellwith that past work [31]. As expected, the quantitative agreement is not as good with that workwhen there is substantial reflection into other modes (which is not shown here, but can befound in Ref. [31]); however, the qualitative features are retained. This last point is expectedsince the analysis method here does not account for additional modes on the reflection side ofthe interface.

For completeness, Fig. 1(b) shows the phase-of-reflection for same conditions as Fig. 1(a).



(a)

0 100 200 300 400 500 600 70040

60

80

100

120

140

radius (nm)

refle

ctio

n ph

ase

(deg

rees

)

(b)

ρ

z φ

2a

0 100 200 300 400 500 600 700radius (nm)

0

0.2

0.4

0.6

0.8

1

refle

ctio

n am

plitu

de

Fig. 1. (a) Reflection amplitude, R = |r|2, for the surface wave reflecting at a terminatedwire (εm = −16) in free-space for free-space wavelength of 632.8 nm, as calculated fromEq. (10). Dashed line extracted from Fig. 4 of Ref [31], where a comprehensive numericalsimulation 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 (re-fractive index of 1.33) and the relative permittivity values for gold are taken from experimentalmeasurements [30].

The reflection amplitude increases as the wire diameter is made smaller or as the wavelengthapproaches the plasma wavelength (i.e., is shortened). The reflection phase also increases asthe wavelength is shortened, but decreases as the wire is made narrower. The phase has animportant influence on the Fabry-Perot resonances of nanorods, as will be described in the next



section.

(a)

(b)

500 1000 1500 2000

0.6

0.7

0.8

0.9

1

wavelength (nm)

refle

ctio

n am

plitu

de

500 1000 1500 2000

0.8

1

1.2

1.4

1.6

1.8

2

wavelength (nm)

refle

ctio

n ph

ase

(rad

ians

)

Fig. 2. Reflection (a) amplitude and (b) phase of abruptly terminated gold nanowires inwater 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 thesurface wave along the rod and reflection at the ends. If a resonance is required at a particularwavelength, 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 nmgold nanorods of various lengths for resonance orders m = 1,2,3,4. Experimentally measured



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

(a)

(b)

0 5 10 15600

700

800

900

1000

1100

1200

1300

1400

1500

aspect ratio

reso

nanc

e w

avel

engt

h (n

m)

0 50 100 150600

700

800

900

1000

1100

1200

length (nm)

reso

nanc

e w

avel

engt

h (n

m)

Fig. 3. Comparison between calculated Fabry-Perot resonances and experimentally meas-ured plasmon resonances. (a) 85 nm diameter nanorods of varying length. Solid lines showresonances calculated using Eq. (12) for m = 1 (blue), 2 (red), 3 (green) and 4 (black). Datapoints 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 experimen-tally 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 and26 nm with experimentally measured extinction resonance and geometry values reported inRefs. [33] and [34] (see the supporting material of the latter work). The agreement in the slopeand the relative change when going from 10 nm to 26 nm is good; however, there seems to be asystematic offset between the theory and experiment. This is attributed to the rounded ends of



the nanorods, since these were fabricated by the electrochemical method. Rounded ends pro-duce shorter-wavelength resonances than abruptly terminated cylinders, with differences up to100 nm reported from calculations [35, 36, 37]. Inaccuracies in the relative permittivity valuesused may also contribute to part of the disagreement, since this has a greater influence for thin-ner rods. Using relative permittivity values from another commonly cited reference gives upto 30 nm shorter wavelength resonance values in comprehensive calculations for rod diametersless 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 wire-end was derived. This theory allows for rapid calculation of the reflection coefficient, givingboth the phase and the amplitude of reflection. This will prove useful to the many applicationsemploying nanorods and nanowires as basic plasmonic elements, including enhanced quantumemitters, nonlinear optics, spectroscopy, and plasmonic integrated circuits. It is also possible toapply the theory to the treatment of THz Sommerfeld waves, with applications in THz waveg-uiding and spectroscopy.

It is expected that this method can be combined with recently demonstrated single modematching method to calculate the reflection of a taper with an abrupt end [38]. Along the samelines, this work may find use in terminated adiabatic tapers, which have recently been demon-strated [39]. Furthermore, extensions of the theory to investigate the influence of losses, whichwere 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]), butthey were not considered here due to the ease in calculating the integral accurately using readilyavailable mathematical software.

Acknowledgements

RG thanks Drs. Mario Agio and Tim Taminiau for interesting conversations regardingnanorods. This work is supported by funding from the Natural Sciences and Engineering Re-search Council (NSERC) Canada: Discovery Grant and Strategic Network for BioplasmonicSystems (Biopsys).



Reflection of Cylindrical Surface Waves

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