Page 1

Gap plasmon mode of eccentric coaxial metal

waveguide

Reuven Gordon,* Asif I. K. Choudhury and Tao Lu

Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, V8W 3P6

*rgordon@uvic.ca

Abstract: The gap plasmon mode of an eccentric coaxial waveguide is

analyzed by the effective index method. The results agree-well with fully-

vectorial numerical calculations. In the eccentric structure, there is extreme

subwavelength field localization around the narrowest gap due to the gap

plasmon. Furthermore, the effective index of the lowest-order waveguide

mode increases considerably, for example, to 3.7 in the structure considered

with a 2 nm minimum gap. The nanostructure waveguide geometry and

wavelength (4 µm) are comparable with recent experiments on coaxial

structures, except that that position of the center island is shifted for the

eccentric coaxial structure; therefore, the proposed structure is a good

candidate for future fabrication and experiments. In the visible regime, the

effective index increases to over 10 for the same structure. The influence of

symmetry-breaking in the eccentric coaxial structure is discussed as a way

to enhance the local field and improve optical coupling.

2009 Optical Society of America

OCIS codes: (240.6680) Surface plasmons; (230.7370) Waveguides

References and links

1.

2.

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

K. J. K. Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, "Strong influence of hole

shape on extraordinary transmission through periodic arrays of subwavelength holes," Phys. Rev. Lett. 92,

183901 (2004).

H. Cao and A. Nahata, "Influence of aperture shape on the transmission properties of a periodic array of

subwavelength apertures," Opt. Express 12, 3664-3672 (2004).

M. Sarrazin and J. P. Vigneron, "Polarization effects in metallic films perforated with a bidimensional array

of subwavelength rectangular holes," Opt. Commun. 240, 89-97 (2004).

K. L. van der Molen, F. B. Segerink, N. F. van Hulst, and L. Kuipers, "Influence of hole size on the

extraordinary transmission through subwavelength hole arrays," Appl. Phys. Lett. 85, 4316-4318 (2004).

K. L. van der Molen, K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers,

"Role of shape and localized resonances in extraordinary transmission through periodic arrays of

subwavelength holes: Experiment and theory," Phys. Rev. B 72, 045421 (2005).

R. Gordon and A. G. Brolo, "Increased cut-off wavelength for a subwavelength hole in a real metal," Opt.

Express 13, 1933-1938 (2005).

F. J. Garcia-Vidal, E. Moreno, J. A. Porto, and L. Martin-Moreno, "Transmission of light through a single

rectangular hole," Phys. Rev. Lett. 95, 103901 (2005).

F. J. Garcia-Vidal, L. Martin-Moreno, E. Moreno, L. K. S. Kumar, and R. Gordon, "Transmission of light

through a single rectangular hole in a real metal," Phys. Rev. B 74, 153411 (2006).

10. A. Mary, S. G. Rodrigo, L. Martin-Moreno, and F. J. Garcia-Vidal, "Theory of light transmission through

an array of rectangular holes," Phys. Rev. B 76, 195414 (2007).

11. W. Zhang, "Resonant terahertz transmission in plasmonic arrays of subwavelength holes," European

Physical Journal-Applied Physics 43, 1-18 (2008).

12. A. Roberts and R. C. McPhedran, "Bandpass grids with annular apertures," IEEE Trans. Antennas Propag.

36, 607-611 (1988).

13. F. I. Baida and D. Van Labeke, "Light transmission by subwavelength annular aperture arrays in metallic

films," Opt. Commun. 209, 17-22 (2002).

14. F. I. Baida and D. Van Labeke, "Three-dimensional structures for enhanced transmission through a metallic

film: Annular aperture arrays," Phys. Rev. B 67, 155314 (2003).

3.

4.

5.

6.

7.

8.

9.

#106425 - $15.00 USD

(C) 2009 OSA

Received 15 Jan 2009; revised 27 Feb 2009; accepted 17 Mar 2009; published 19 Mar 2009

30 March 2009 / Vol. 17, No. 7 / OPTICS EXPRESS 5311

Page 2

15. J. Salvi, M. Roussey, F. I. Baida, M. P. Bernal, A. Mussot, T. Sylvestre, H. Maillotte, D. Van Labeke, A.

Perentes, I. Utke, C. Sandu, P. Hoffmann, and B. Dwir, "Annular aperture arrays: study in the visible region

of the electromagnetic spectrum," Opt. Lett. 30, 1611-1613 (2005).

16. Y. Poujet, M. Roussey, J. Salvi, F. I. Baida, D. Van Labeke, A. Perentes, C. Santschi, and P. Hoffmann,

"Super-transmission of light through subwavelength annular aperture arrays in metallic films: Spectral

analysis and near-field optical images in the visible range," Photon. Nanostruct. Fundam. Appl. 4, 47-53

(2006).

17. F. I. Baida, A. Belkhir, D. Van Labeke, and O. Lamrous, "Subwavelength metallic coaxial waveguides in

the optical range: Role of the plasmonic modes," Phys. Rev. B 74, 205419 (2006).

18. W. J. Fan, S. Zhang, K. J. Malloy, and S. R. J. Brueck, "Enhanced mid-infrared transmission through

nanoscale metallic coaxial-aperture arrays," Opt. Express 13, 4406-4413 (2005).

19. W. J. Fan, S. Zhang, B. Minhas, K. J. Malloy, and S. R. J. Brueck, "Enhanced infrared transmission through

subwavelength coaxial metallic arrays," Phys. Rev. Lett. 94, 033902 (2005).

20. A. Moreau, G. Granet, F. I. Baida, and D. Van Labeke, "Light transmission by subwavelength square

coaxial aperture arrays in metallic films," Opt. Express 11, 1131-1136 (2003).

21. S. Wu, Q. J. Wang, X. G. Yin, J. Q. Li, D. Zhu, S. Q. Liu, and Y. Y. Zhu, "Enhanced optical transmission:

Role of the localized surface plasmon," Appl. Phys. Lett. 93, 3 (2008).

22. H. Y. Yee and N. F. Audeh, "Cutoff frequencies of eccentirc waveguides," IEEE Trans. Microwave Theory

Tech. IEEE Trans. Microwave Theory Techniq. 14, 487 (1966).

23. E. Abaka and W. Baier, "TE and TM modes in transmission lines with circular outer conductor and

eccentric circular innder conductor," Electron. Lett. 5, 251 (1969).

24. B. N. Das and S. B. Chakrabarty, "Evaluation of cut-off frequencies of higher order modes in eccentric

coaxial line," IEE Proc. Microwaves Ant. Propag. 142, 350-356 (1995).

25. B. N. Das and S. B. Chakrabarty, "Electromagnetic analysis of eccentric coaxial cylinders of finite length,"

J. Inst. Electron. Telecom. Engin. 42, 63-68 (1996).

26. S. C. Zhang, "Eigenfrequency shift of higher-order modes in a coaxial cavity with eccentric inner rod," Int.

J. Infrared Millim. Waves 22, 577-583 (2001).

27. R. Gordon, A. G. Brolo, A. McKinnon, A. Rajora, B. Leathem, and K. L. Kavanagh, "Strong polarization in

the optical transmission through elliptical nanohole arrays," Phys. Rev. Lett. 92, 037401 (2004).

28. J. Y. Chu, T. J. Wang, J. T. Yeh, M. W. Lin, Y. C. Chang, and J. K. Wang, "Near-field observation of

plasmon excitation and propagation on ordered elliptical hole arrays," Appl. Phys. A-Mat. Scie. Proc. 89,

387-390 (2007).

29. R. Gordon, M. Hughes, B. Leathem, K. L. Kavanagh, and A. G. Brolo, "Basis and lattice polarization

mechanisms for light transmission through nanohole arrays in a metal film," Nano Lett. 5, 1243-1246

(2005).

30. A. Degiron and T. W. Ebbesen, "The role of localized surface plasmon modes in the enhanced transmission

of periodic subwavelength apertures," J. Opt. A-Pure Appl. Opt. 7, S90-S96 (2005).

31. Y. M. Strelniker, "Theory of optical transmission through elliptical nanohole arrays," Phys. Rev. B 76,

085409 (2007).

32. J. B. Masson and G. Gallot, "Coupling between surface plasmons in subwavelength hole arrays," Phys.

Rev. B 73, 121401 (2006).

33. R. C. Compton, R. C. McPhedran, G. H. Derrick, and L. C. Botten, "Diffraction properties of a bandpass

grid," Infrared Phys. 23, 239-245 (1983).

34. R. M. Roth, N. C. Panoiu, M. M. Adams, J. I. Dadap, and R. M. Osgood, "Polarization-tunable plasmon-

enhanced extraordinary transmission through metallic films using asymmetric cruciform apertures," Opt.

Lett. 32, 3414-3416 (2007).

35. C. Y. Chen, M. W. Tsai, T. H. Chuang, Y. T. Chang, and S. C. Lee, "Extraordinary transmission through a

silver film perforated with cross shaped hole arrays in a square lattice," Appl. Phys. Lett. 91, 063108

(2007).

36. Y. H. Ye, Z. B. Wang, D. S. Yan, and J. Y. Zhang, "Role of shape in middle-infrared transmission

enhancement through periodically perforated metal films," Opt. Lett. 32, 3140-3142 (2007).

37. E. Jin and X. Xu, "Radiation transfer through nanoscale apertures," J. Quant. Spectrosc. Radiat. Transfer

93, 163-173 (2005).

38. J. W. Lee, M. A. Seo, D. J. Park, D. S. Kim, S. C. Jeoung, C. Lienau, Q. H. Park, and P. C. M. Planken,

"Shape resonance omni-directional terahertz filters with near-unity transmittance," Opt. Express 14, 1253-

1259 (2006).

39. M. Sun, R. J. Liu, Z. Y. Li, B. Y. Cheng, D. Z. Zhang, H. F. Yang, and A. Z. Jin, "Enhanced near-infrared

transmission through periodic H-shaped arrays," Phys. Lett. A 365, 510-513 (2007).

40. L. K. S. Kumar and R. Gordon, "Overlapping double-hole nanostructure in a metal film for localized field

enhancement," IEEE J. Sel. Top. Quantum Electron. 12, 1228-1232 (2006).

41. L. K. S. Kumar, A. Lesuffleur, M. C. Hughes, and R. Gordon, "Double nanohole apex-enhanced

transmission in metal films," Appl. Phys. B-Lasers Opt. 84, 25-28 (2006).

42. M. Airola, Y. Liu, and S. Blair, "Second-harmonic generation from an array of sub-wavelength metal

apertures," J. Opt. A-Pure Appl. Opt. 7, S118-S123 (2005).

#106425 - $15.00 USD

(C) 2009 OSA

Received 15 Jan 2009; revised 27 Feb 2009; accepted 17 Mar 2009; published 19 Mar 2009

30 March 2009 / Vol. 17, No. 7 / OPTICS EXPRESS 5312

Page 3

43. J. H. Kim and P. J. Moyer, "Transmission characteristics of metallic equilateral triangular nanohole arrays,"

Appl. Phys. Lett. 89, 121106 (2006).

44. T. Xu, X. Jiao, G. P. Zhang, and S. Blair, "Second-harmonic emission from sub-wavelength apertures:

Effects of aperture symmetry and lattice arrangement," Opt. Express 15, 13894-13906 (2007).

45. R. Gordon, L. K. S. Kumar, and A. G. Brolo, "Resonant light transmission through a nanohole in a metal

film," IEEE Trans. Nanotech. 5, 291-294 (2006).

46. A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, "Optical transmission properties of a single

subwavelength aperture in a real metal," Opt. Commun. 239, 61-66 (2004).

47. C. Yeh and F. I. Shimabukuro, The Essence of Dielectric Waveguides (Springer New York, 2008).

48. P. B. Johnson and R. W. Christy, "Optical Constants of the Noble Metals," Phys. Rev. B 6, 4370 (1972).

49. W. Fan, S. Zhang, N. C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck,

"Second harmonic generation from a nanopatterned isotropic nonlinear material," Nano Lett. 6, 1027-1030

(2006).

50. A. Lesuffleur, L. K. S. Kumar, and R. Gordon, "Enhanced second harmonic generation from nanoscale

double-hole arrays in a gold film," Appl. Phys. Lett. 88, 261104 (2006).

51. K. Liu, L. Zhan, Z. Y. Fan, M. Y. Quan, S. Y. Luo, and Y. X. Xia, "Enhancement of second-harmonic

generation with phase-matching on periodic sub-wavelength structured metal film," Opt. Commun. 276, 8-

13 (2007).

52. A. G. Brolo, E. Arctander, R. Gordon, B. Leathem, and K. L. Kavanagh, "Nanohole-enhanced Raman

scattering," Nano Lett. 4, 2015-2018 (2004).

53. K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, "Ultrasensitive chemical analysis by Raman

spectroscopy," Chem. Rev. 99, 2957 (1999).

54. A. Lesuffleur, L. K. S. Kumar, A. G. Brolo, K. L. Kavanagh, and R. Gordon, "Apex-enhanced Raman

spectroscopy using double-hole arrays in a gold film," J. Phys. Chem. C 111, 2347-2350 (2007).

55. Q. Min, M. J. L. Santos, E. M. Girotto, A. G. Brolo, and R. Gordon, "Localized Raman enhancement from

a double-hole nanostructure in a metal film," J. Phys. Chem. C 112, 15098-15101 (2008).

56. T. H. Reilly, S. H. Chang, J. D. Corbman, G. C. Schatz, and K. L. Rowlen, "Quantitative evaluation of

plasmon enhanced Raman scattering from nanoaperture arrays," J. Phys. Chem. C 111, 1689-1694 (2007).

57. R. Quidant, and C. Girard, "Surface-plasmon-based optical manipulation," Laser Photon. Rev. 2, 47-57

(2008).

1. Introduction

Hole-shape influences the waveguide mode propagation inside a hole in a metal. The

influence of hole-shape on the propagation and transmission of light through a single hole and

hole-arrays in metals has been studied for many different shapes including cylindrical [1],

square and rectangular [1-11], cylindrical coaxial [12-19], rectangular coaxial [20], rectangle-

in-cylinder coaxial [21], eccentric-coaxial [22-26], elliptical [27-32], cruciform [33-36], C-,

H-, and E-shaped [37-39], double-hole (overlapping [40,41] and separated [29]), triangular

[42,43] and star-shaped [44] holes. These hole-shapes have been investigated throughout the

electromagnetic spectrum; perfect electric conductors (PECs) approximate-well the behavior

at long wavelengths, whereas the plasmonic response of the metal plays an important role at

shorter wavelengths.

Gap plasmons have a strong influence on the waveguide mode. For example, the lowest

order mode of a rectangular hole has a cut-off wavelength increases dramatically as the hole is

made smaller along its shortest axis [7,9,45], as has been observed experimentally [46]. This

effect is the result of the effective index of the gap plasmon increasing as the metal edges are

brought closer together [7]. The waveguide modes of a cylindrical coaxial structure are

similarly-dependent upon the gap plasmon [15-17]. The eccentric coaxial structure has a

varying gap-width with angle, so the properties of the gap plasmon are of particular interest.

In this work, we study the gap-plasmon of an eccentric coaxial structure. The eccentric

coaxial structure has been studied before for PECs using conformal mapping [22-26]. In those

works, the cut-off wavelength was shown to be offset-dependent. Here, we focus on a

different effect: the influence of the gap plasmon. The effective index method is used, which

is adapted from the study of rectangular holes [7] to the cylindrical geometry. The structure

analyzed is comparable to recent structures produced by interference lithography [18,19],

which has the great advantage of allowing for mass production. The analytical results of the

effective index model agree-well with comprehensive finite-difference mode-solver (FDMS)

numerical calculations for the geometry considered for a wavelength of 4 µm. In the visible –

#106425 - $15.00 USD

(C) 2009 OSA

Received 15 Jan 2009; revised 27 Feb 2009; accepted 17 Mar 2009; published 19 Mar 2009

30 March 2009 / Vol. 17, No. 7 / OPTICS EXPRESS 5313

Page 4

near-IR region of the optical spectrum, the FDMS gives spurious results for smaller gaps;

however, the simple effective index theory agrees quantitatively with a fully-vectorial finite-

element method (FEM) in that regime.

2. Effective Index Model for Eccentric Coaxial Gap Plasmon

Past works on eccentric coaxial structures used conformal mapping to study the geometric

influence on mode propagation for PECs. Here, the geometric influence is accounted for by

the effective index variation from the gap plasmon, which is a good approximation for

plasmonic materials when the effective index is determined predominantly by the gap

variation. This approximation neglects conformal mapping effects that become important if

the tangent angles to the inner and outer cylinders vary significantly – the approximation is

suitable for relatively large inner island sizes or small offsets.

The effective index method is an approximation that assumes that separation of variables

is allowed. We consider first consider the radial dependence of the field to determine an

effective index as a function of angle – each angle has a different gap, which contributes to

the change in the effective index. These effective index calculations assume that there is no

angular dependence, so that the equations of a concentric structure can be used. The next step

is to determine the angular dependence by using the effective index as a function of angle to

replace the radial dependence. Therefore, the overall propagation constant of the mode can be

determined from the one dimensional angular calculation.

Figure 1 illustrates the effective index method as applied to the eccentric coaxial

structure, where the radial and angular co-ordinates are approximated to be separable. Figure

1(a) shows the structure under consideration – the metal cylinder island with radius a is offset

from the center by d and the outer cylinder has radius . b Figure 1(b) shows the two extreme

example angles where the rotationally symmetric structure is analyzed as if the island cylinder

was coaxial and had the same radial extent as the eccentric cylinder at that angle. Figure 1(c)

shows the equivalent structure given the effective index calculated in Fig.1(b), where the

metal is replaced with a perfect electric conductor (PEC) and the air region is replaced with a

dielectric. Figure 1(d) shows the angular dependence calculated with the same PEC-dielectric

structure; however, the dielectric is modified as a function of angle, using the values from the

part (c).

The Helmholtz equation for the axial electric field,

component is present in the gap-plasmon and is continuous at the boundaries:

∂

+

∂

∂

r

rr

r

θ

where ),( θε

r

z

is the relative permittivity of dielectric material in the annular region, ω is

the angular frequency and c is the speed of light in vacuum.

For the concentric cylindrical coaxial structure (i.e., no angular variation), the effective

index of the mode is found by equating the tangential electric and magnetic fields at the

boundary of island (radius a) and outer circle (radius b) to gives the solution [47]:

, 0

=−CDAB

where

)(

20

apI

ε

)(θ

z

E

, is used since this field

, 0

11

2

2

2

2

2

2

22

2

=+

∂

∂

+

∂

∂

+

∂

zz

zzzz

E

cz

EE

E

E

ε

ω

)(

)(

)(

1121

2112

10

apIp

apIp

apI

A

ε

ε

−=

)(

)(

)(

)(

30

20

3123

2112

bpK

bpK

b

ε

pKp

bpKp

B

−=ε

)(

)(

)(

)(

1121

2

a

112

10

2

a

0

pIp

apKp

pI

apK

C

ε

−=

(1)

(2)

(3)

(4)

(5)

#106425 - $15.00 USD

(C) 2009 OSA

Received 15 Jan 2009; revised 27 Feb 2009; accepted 17 Mar 2009; published 19 Mar 2009

30 March 2009 / Vol. 17, No. 7 / OPTICS EXPRESS 5314

Page 5

)(

)(

)(

)(

30

20

3123

2112

bpK

bpI

bpKp

bpIp

D

−−=

ε

ε

and

11,,,

KIKI

oo

are modified Bessel functions of order zero and one,

1

)(

εε

=

′

< ar

z

,

2

),(

εε

=<

′

>

brar

z

,

3

)(

εε

=>br

z

, and

()

mm

n

c

p

εθ

(

ω

−=

)

2

eff

2

2

2

with . 3 , 2 , 1

=

m

Eqs. (2-6) are solved for varying inner-diameter, a′ , of concentric coaxial cylindrical

structures to determine the angle-dependent effective index,

effective index found in each case is used to solve the angular dependence of electric field in

the eccentric structure, and the propagation constant along the axial direction, β , of the

)(

effθ

n

. The angle-dependent

mode using:

, 0)(

1

2

eff

2

2

2

2

2

2

=+−

∂

∂

zz

z

En

c

E

E

r

θ

ω

β

θ

where we set

noted that

of θ and not the same as

b

is the effective index in the “effective index method”. It is only a function

r =

(

to solve the field at the rotationally invariant outer radius. It should be

)

effθ

n

z ε , which is a function of both r and θ . The parameters

contained within the calculation for (

n

While Eq. 7 is valid for concentric structures only, it is used here to approximate the

behavior of the eccentric structure within the approximation of the effective index method. In

Cartesian co-ordinates, the effective index method takes a similar approximation by replacing

the real problem of non-matching boundaries by one with a uniform effective index. In

cylindrical co-ordinates, we use a similar approach. For the radial electric field component,

this can be pictured as a thin slice surrounded by a PEC boundary (as shown in Fig. 1).

Approximate approaches to solve for the propagation constant may be used, such as the

Wentzel–Kramers–Brillouin method; however, due to the simplicity of this one-dimensional

differential equation, we solve it numerically using discretization in 1 degree angular steps

and the “eig” function in Matlab to find largest eigenvalue to give β and the corresponding

eigenvector for )(θ

z

E

of the lowest order mode, where the radial dependence considered

previously is assumed constant in the effective index method. The lowest order mode is the

one that has phase change in the radial electric field in the gap region.

bad

,,are

)

effθ

, so they do not appear explicitly in Eq. 7.

(6)

(7)

#106425 - $15.00 USD

(C) 2009 OSA

Received 15 Jan 2009; revised 27 Feb 2009; accepted 17 Mar 2009; published 19 Mar 2009

30 March 2009 / Vol. 17, No. 7 / OPTICS EXPRESS 5315

Page 6

Fig. 1. (a) Schematic of eccentric cylindrical coaxial waveguide in gold with air gap.

(b) Equivalent structures to calculate radial contribution to effective index assuming at each

angle that the structure is rotationally symmetric. (c) Effective index of the rotationally

symmetric structure is equivalent to a dielectric inside a coaxial perfect electric conductor

(PEC). (d) Angular dependence, using effective index values calculated from the radial

dependence at each angle.

3. Effective Index Increase and Field Localization for Eccentric Coaxial Structure

3.1 Infrared example comparable to recent experiments and FDMS calculations

We consider a coaxial structure with inner island radius of 224 nm, and outer radius of

286 nm, in gold for the free-space wavelength of 4 µm. These values were chosen to be

similar with experiments on structures created by interference lithography [18]. At this

wavelength the relative permittivity of gold is -600 + 130i. Fig. 2 shows the comparison

between effective index (using the common definition

of the eccentric coaxial structure as calculated by the effective index method, as outlined in

the previous section, and by a commercially available FDMS. Good agreement is seen

between the two methods. The commercial FDMS incorporates the loss of the material from

an imaginary part of the relative permittivity (not shown). In principle, this can be

incorporated in the effective index method; however, since the loss is small for this example,

ωβc

– not )(

effθ

n

as defined above)

#106425 - $15.00 USD

(C) 2009 OSA

Received 15 Jan 2009; revised 27 Feb 2009; accepted 17 Mar 2009; published 19 Mar 2009

30 March 2009 / Vol. 17, No. 7 / OPTICS EXPRESS 5316

Page 7

and it complicates the analysis by adding complex roots, the imaginary part of the relative

permittivity is ignored.

Fig. 2. Comparison of lowest order mode effective index (

index method (line) and calculated by a comprehensive vectorial FDMS (crosses). The

structure chosen is gold, with an air gap, an outer cylinder radius of 286 nm, and an inner

island radius of 224 nm. The inner island is offset to produce different narrowest gap values.

ωβc

) calculated by the effective

Figure 3 shows the square of the electric field amplitude of the lowest order mode

calculated by the effective index method. As described above,

from the eigenvector of Eq. (7).For a 60 nm shift in the center island, the field is strongly

localized to a FWHM of 56° in a 2 nm gap. As shown in Fig. 4, a similar localization is seen

for the comprehensive FDMS calculations (~60°), which also contain a radial dependence.

This example shows that the effective index method provides a reasonable approximation

for the field localization and effective index increase of the lowest order waveguide mode in

an eccentric-coaxial structure of a plasmonic metal. In the following section, we will discuss

the specific benefits of the eccentric coaxial structure.

z

E in Fig. 3 is determined

0

10

2030405060

1

2

3

4

5

6

7

narrowest gap (nm)

effective index (βc/ω)

#106425 - $15.00 USD

(C) 2009 OSA

Received 15 Jan 2009; revised 27 Feb 2009; accepted 17 Mar 2009; published 19 Mar 2009

30 March 2009 / Vol. 17, No. 7 / OPTICS EXPRESS 5317

Page 8

Fig. 3. Amplitude squared of electric field of lowest order mode calculated using calculated by

the effective index method for eccentric coaxial structure, described in Fig. 2, with offset of

0 nm, 45 nm, and 60 nm (black, red, blue). The 60 nm offset has a 2 nm narrowest gap, which

leads to strong field localization.

Fig. 4. Axial component of the electric field intensity for the same structures as in Fig. 3, with

offsets of d= 0 nm, 45 nm, and 60 nm (left to right). Normalized color scale: red – 1, blue – 0.

3.2 Extension to visible – near-IR region and comparison with FEM calculations

The values presented in Sec. 3.1 were chosen to be comparable to recent experiments. We

also investigated the extension of these results to the visible and near-IR regions of the optical

spectrum and for another metal (silver). We use the same structure, but vary the wavelength

and relative permittivity values appropriately [48]. In this region, there is significant

penetration of the electric field into the metal, which is not well-captured by the FDMS for d

> 40 nm; even for grid sizes of 0.2 nm, beyond which the calculation did not converge. (We

attribute the inaccuracy of the FDMS method to the poor representation of the curved

geometry using a Cartesian grid and linear interpolation).

We found better agreement for the effective index model calculations with a

commercially available FEM solver. As shown in Fig. 5, we compare with the effective index

method model for d = 55 nm and 60 nm, with good agreement between effective index

method and the FEM solver. That figure also shows the calculated FDMS values. It is

noteworthy that the FDMS calculated index actually reduces when the gap is made narrower

-180-90

090 180

0

5

10

15

θ (degrees)

|Ez|2

#106425 - $15.00 USD

(C) 2009 OSA

Received 15 Jan 2009; revised 27 Feb 2009; accepted 17 Mar 2009; published 19 Mar 2009

30 March 2009 / Vol. 17, No. 7 / OPTICS EXPRESS 5318

Page 9

from 7 nm to 2 nm, which contrary to the usual behavior of gap modes and is believed to be a

spurious result.

632.8 850 1550

2

4

6

8

10

12

wavelength (nm)

effective index (βc/ω)

EIM 55 nm

EIM 60 nm

FEM 55 nm

FEM 60 nm

FDMS 55 nm

FDMS 60 nm

(a) Au

632.88501550

2

4

6

8

wavelength (nm)

effective index (βc/ω)

EIM 55 nm

EIM 60 nm

FEM 55 nm

FEM 60 nm

FDMS 55 nm

FDMS 60 nm

(b) Ag

Fig. 5. Effective index calculations for (a) gold and (b) silver in the visible – near-IR region.

EIM: effective index method; FEM: finite element method; FDMS: finite difference mode

solver.

4. Discussion

While FDMS and FEM methods can be used to study the eccentric structure, they are entirely

numerical and general. As a result, they do not contain any special insight into the physics of

strong field localization. Using the effective index method, it is clear that the localization is

physically the result of the gap mode having an increase in the local index. Furthermore, since

the behavior of the gap mode can be well-approximated by a simple parametric expression,

#106425 - $15.00 USD

(C) 2009 OSA

Received 15 Jan 2009; revised 27 Feb 2009; accepted 17 Mar 2009; published 19 Mar 2009

30 March 2009 / Vol. 17, No. 7 / OPTICS EXPRESS 5319

Page 10

the effective index method has the possibility of providing fully-analytical information about

the behavior of the eccentric structure (and other cylindrical geometry structures). This will

allow for rapid design and optimization of such structures. For example, all of the effective

index method calculations in this manuscript can be completed in less than a minute using

Matlab, whereas the FDMS and FEM methods take several minutes for each case.

As could be seen from the last section, the eccentric coaxial structure allows for strong

field localization in the region of the narrowest gap. This provides even greater field

localization than the concentric coaxial waveguide, which has already attracted great interest

from the plasmonics community.

An additional benefit from the eccentric coaxial structure comes from symmetry-

breaking, which introduces linear polarization to the lowest order mode. The lowest order

mode of the corresponding concentric structure is radially polarized. As a result, linearly

polarized light can be used to excite the lowest order mode of the eccentric structure, but not

the concentric structure. This is important to practical application, when it comes to actually

exciting these modes in real structures.

Past works have used focused-ion beam milling (e.g. Ref. 21) and interference

lithography (e.g., Ref. 18) to produce coaxial structures. Those approaches can be readily

extended to the eccentric structure described here; however, to obtain narrow gaps of only a

few nanometers, the interference lithography approach seems to be especially interesting. In

particular, it is possible to use adapt that method by using angled-evaporation (in step 6a of

Fig. 1, Ref. 18) to produce the central island with an offset, and at least in principle, the center

island can be made arbitrarily close to the outer cylinder by this method.

5. Conclusion

We have demonstrated that the gap plasmon mode of an eccentric coaxial structure allows for

an increased effective index, strong field localization and linear polarized field to benefit

coupling to linear polarized light. Each of these properties becomes more prominent as the

narrowest gap is reduced by shifting the center island, to make the structure more eccentric. It

is expected that proposed structures may be fabricated by focused-ion beam milling or

interference lithography methods. Several potential applications arise from the strong field

localization of this structure including: nonlinear optics [42,44,49-51], surface-enhanced

Raman scattering [52-56], and optical trapping [57].

This paper also showed an example of using the effective index method for a cylindrical

geometry to rapidly produce results that are in good agreement with more-cumbersome

comprehensive numerical calculations. This method may also be applied to other concentric

structures.

Acknowledgment

The authors acknowledge financial support from the NSERC (Canada) Discovery Grant.

#106425 - $15.00 USD

(C) 2009 OSA

Received 15 Jan 2009; revised 27 Feb 2009; accepted 17 Mar 2009; published 19 Mar 2009

30 March 2009 / Vol. 17, No. 7 / OPTICS EXPRESS 5320