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The transmission characteristics of surface

plasmon polaritons in ring resonator

Tong-Biao Wang, Xie-Wen Wen, Cheng-Ping Yin, and He-Zhou Wang*

State Key Laboratory of Optoelectronic Materials and Technologies, Zhongshan (Sun Yat-Sen) University,

Guangzhou 510275, China

*stswhz@mail.sysu.edu.cn

Abstract: A two-dimensional nanoscale structure which consists of two

metal-insulator-metal (MIM) waveguides coupled to each other by a ring

resonator is designed. The transmission characteristics of surface plasmon

polaritons are studied in this structure. There are several types of modes in

the transmission spectrum. These modes exhibit red shift when the radius of

the ring increases. The transmission properties of such structure are

simulated by the Finite-Difference Time-Domain (FDTD) method, and the

eignwavelengths of the ring resonator are calculated theoretically. Results

obtained by the theory of the ring resonator are consistent with those from

the FDTD simulations.

©2009 Optical Society of America

OCIS codes: (240.6680) Surface plasmons; (140.4780) Optical resonators; (130.7408)

Wavelength filter

References and links

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operating at telecom wavelengths,” Appl. Phys. Lett. 85(24), 5833–5835 (2004).

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(C) 2009 OSA

Received 6 Oct 2009; revised 17 Nov 2009; accepted 7 Dec 2009; published 17 Dec 2009

21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24096

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1. Introduction

Surface plasmons are the electromagnetic surface waves that travel along the interface

between metal and dielectric with an exponentially decay to the both sides. Surface plasmons

polaritons (SPPs) have been considered as energy and information carriers in nanoscale optics

for their ability of overcoming diffraction limit of light in conventional optics. Because of

their potential applications to guide and manipulate light at deep subwavelength scales [1–3],

SPPs have received plenty of attention in recent years. Several metal-insulator-metal (MIM)

waveguides based on SPPs, for example, bends and splitters, Mach-Zehnder interferometers,

and couplers have been designed theoretically [4–6] and demonstrated experimentally [7,8].

Some photonic band gap structures have also been proposed, such as plasmon Bragg

reflectors [9], Bragg gratings with periodically varied width [10,11].

Plasmonic add/drop filters and tooth-shaped plasmonic waveguide filters have been

investigated recently [12,13]. The characteristics of these filters are that many wavelengths of

the light are allowed to pass through the structure while one or several wavelengths are

stopped, i.e. there are one or several dips in their transmission spectra. As a matter of fact,

another types of filters which only allow light at given wavelengths to transmit, while other

wavelengths are forbidden, are also important in nanoscale optics. Therefore, it is necessary

and meaningful to find out a SPPs filter with simple structure and whose transmission

spectrum can have one or several peaks.

In this paper, we propose a two-dimensional (2D) nanoscale structure which is composed

of two MIM waveguides and a ring resonator. We employ the 2D FDTD method to study the

transmission characteristics of SPPs, and find that several types of modes can appear in the

transmission spectrum when light passing through the structure. The wavelengths of the

modes can be easily modulated by changing the radius of the ring. The eigenwavelengths of

the ring have also been studied theoretically, which agrees with the FDTD simulations very

well.

2. Theory model

It is shown in Fig. 1(a) that the 2D nanoscale structure is studied in this paper. The structure is

considered invariant along y direction. The gray and white areas are silver and air,

respectively. Two waveguides are coupled by a ring resonator whose outer radius is ra, and

inner radius is ri. The radius of the ring is r, which is the average of the inner and the outer

radius, r = (ra + ri)/2, as depicted by the dashed circle in Fig. 1(a). d are the widths of the

waveguides and the ring. w are the coupling lengths between the waveguides and the ring. As

is well known, the dispersion equation of SPPs in the MIM structure can be written as [14]

0

1 exp(

−

+

),

)1 exp(

m

p

kd

kkd

ε

ε

=

(1)

where

[15]. k0 = 2π/λ is the wave number of light in the air, λ is the wavelength of incident light. ε0

and εm are the dielectric functions of air and silver, respectively. For more accurately match

the experimental optical constant of silver, εm can be characterized by the Lorentz-Drude

model [16]

2

gspp

2 1 2

0

)

0

(

kk

βε

=−

,

2

gspp

2 1 2

0

)(

m

pk

βε

=−

. βgspp is the propagation constant of gap SPPs

2

m

5

2

m

2

0

,

m

m

m

m

G

ω

−

i

εε

ωω

∞

=

Ω

+

=−

Γ

∑

(2)

where ε∞ is the relative permittivity in the infinity frequency. Gm is the oscillator strengths, Ωm

is the plasma frequency, ωm is the resonant frequency, Γm is the damping factor, and ω is the

angular frequency of incident light. All the parameters of the Lorentz-Drude model can be

found in Ref [16]. Effective refraction index (ERI) of the MIM structure is defined as neff =

β/k0, which can be calculated by Eq. (1). Figure 1(b) shows the real part of neff as a function of

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(C) 2009 OSA

Received 6 Oct 2009; revised 17 Nov 2009; accepted 7 Dec 2009; published 17 Dec 2009

21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24097

Page 3

d and λ. As seen from the figure, the neff decreases as the width d increases with the same

wavelength λ, while neff varies very slowly when the value of λ changes with d fixed.

Fig. 1. (a) The structure of the nanoscale ring resonator. The gray and white areas are silver and

air, respectively. (b) Dependence of Re(neff) of MIM structure on the wavelength of incident

light and width d.

3. Results and discussions

We use the 2D FDTD method with perfectly matched layer boundary conditions to simulate

the transmission characteristics of SPPs. The SPPs are assumed to be excited by a dipole point

source in the left side of waveguide I, and propagate along z direction. They will travel

clockwise and anticlockwise simultaneously in the ring, and transmit from the right side of

waveguide II. Two monitors are, respectively, put at the points of M1 and M2 to detect the

incident and transmission fields for getting the incident amplitude of A1 and the transmitted

amplitude A2. The transmittance is defined to be T = A2/A1. Only those wavelengths satisfy the

resonance condition can be transported efficiently, while others are stopped. In Fig. 2, we plot

the transmission spectrum and the field distributions of the propagation of SPPs in the

structure. The parameters of the structure are set to be d = 50 nm, w = 10 nm, and r = 170 nm

in calculation. One can see in Fig. 2(a), that there are three transmission peaks corresponding

to the wavelength λ = 1599.2, λ = 817.92, and λ = 579.84 nm, respectively. Figures 2(b), 2(c),

and 2(d) depict the contour profiles of fields |Hy|2 for different wavelengths. The field

distributions in Figs. 2(b) and 2(d) correspond to the Modes I and II in Fig. 2(a), respectively.

One can see that the SPPs can propagate through the ring, and transmit from the waveguide II

for these two cases. The standing waves form in the ring at resonance. There are one and two

modes in the rings of Fig. 2(b) and 2(d), respectively. It must be noted here that the

wavelengths of the transmission peaks do not satisfy the simple relation λ = Lneff/N, where N

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(C) 2009 OSA

Received 6 Oct 2009; revised 17 Nov 2009; accepted 7 Dec 2009; published 17 Dec 2009

21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24098

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is the mode number in the ring, L = 2πr is the perimeter of the ring. The resonance condition

will be discussed later. In Fig. 2(c), the wavelength of the SPPs is λ = 1200 nm. As can be

seen in Fig. 2(c), the field in the ring is very weak, and almost no Spps exist in the right-side

waveguide, the transmission of SPPs is forbidden in this case.

Fig. 2. (a) The transmission spectrum of the nanoscale ring resonator. The contour profiles of

field Hy of the nanoscale ring resonator at different wavelengths (b) λ = 1599.2 nm, (c) λ =

1200 nm and (d) λ = 817.92 nm.

Next, we would like to study the influence of the radius of the ring on the wavelengths of

the transmission peaks. Figure 3 shows the transmission spectra of the SPPs for different

radius of the ring. The transmission spectra plotted in black, blue, red, and green lines

correspond to the rings with radii 170 nm, 180 nm, 190 nm and 200 nm, respectively. As can

be seen from Fig. 3, the transmission peaks exhibit red shift as the radius increase. Since the

loss of the metal is inevitable, the transmittance cannot reach 1.0. The loss will be increase

when the propagation length increases.

In Fig. 4, we plot three different transmission modes as a function of radius of the ring.

The horizontal coordinate is the radii of the rings, and the vertical coordinate is the

corresponding wavelengths of the transmission peaks. The radius is set to vary from 100 nm

to 280 nm and the interval is 10 nm in the FDTD simulations. The transmission modes for

coupling length w = 10 nm and w = 20 nm are depicted in blue and red scattering lines,

respectively. We can see that the transmission modes exhibit blue shift as the coupling length

increase. Three modes are separated from each other, and intervals between them are large

enough. These results will provide the theoretical basis for designing band-pass filters at the

given wavelength. We will theoretically study the relationship between the wavelengths of the

transmission peaks and the radii of the rings in the following.

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Received 6 Oct 2009; revised 17 Nov 2009; accepted 7 Dec 2009; published 17 Dec 2009

21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24099

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Fig. 3. Transmission spectra of the structure for different radius. The black, blue, red, and green

lines correspond to the radii of 170 nm, 180 nm, 190nm, and 200 nm, respectively.

Now, we discuss the resonance condition according to the theory of ring resonator. The

resonating wavelength of a ring resonator can be obtained theoretically by the following

equation [17]:

()()

0,

()()

n

′

′

an

′

′

a

nini

J krN kr

J krN kr

−=

(3)

where k = ω(ε0εrµ0)½. µ0 is the permeability in the air.

effective relative permittivity. Jn is a Bessel function of the first kind and order n, and Nn is a

Bessel function of second kind and order n.

functions to the argument (kr). Equation (3) is a transcendent equation which can be

numerically solved. In principle, the ERI is wavelength dependent. However, we can see that,

from Fig. 1(b), the wavelength has little influence on ERI when the width of the ring is fixed.

Therefore, we can treat ERI as a constant in calculation for convenience. The real part of ERI

is 1.4325 for a thickness of the waveguide of d = 50 nm in the calculation, which can be

obtained from Fig. 1(b). The imaginary part can be neglected because it is small and only

affects the propagation length of SPPs. The calculation results are shown in Fig. 4 with black

solid lines. Modes I, II, and III correspond to the one, two, and three orders of Bessel

functions. One can see that there is a little difference between the theory result and the FDTD

ones. In the calculation, we assume that the effective relative permittivity is the same in the

whole ring, but this assumption is not exact. Since the coupling length between the ring and

the waveguides are finite, the effective relative permittivity in the coupling areas is different

from that in other areas of the ring. Only when the coupling length is infinite, the effective

relative permittivity will be the same with that of the standard MIM model. Consequently, we

can see that the FDTD results for w = 20 nm are closer to the theory results than those for w =

10 nm. When the radius of the ring gets larger, the effects of the coupling areas on the

effective relative permittivity is smaller or even can be neglected. In addition, the curvature of

the ring also have an effect on the effective permittivity, which means that for larger radius

(with smaller curvature), every part of arc in the ring is closer to the straight MIM structure.

Therefore, the theory results for larger radius are closer to FDTD results than that for smaller

radius, which can be seen in Fig. 4. From the analysis above, we can see that the theory results

match the FDTD results quite well.

2

eff

0

r

n

εµ

=

is the frequency-dependent

n J′ and

n

N′ are derivatives of the Bessel

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Received 6 Oct 2009; revised 17 Nov 2009; accepted 7 Dec 2009; published 17 Dec 2009

21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24100