Bound modes analysis of symmetric dielectric loaded surface plasmon-polariton waveguides

Bound modes analysis of symmetric dielectric loaded surface plasmon-polariton waveguides

Yun Binfeng, Hu Guohua, and Cui Yiping*

Advanced Photonics Center, Southeast University, Nanjing, China, 210096 *Corresponding author: [email protected]

Abstract: A symmetric dielectric loaded surface plasmon polariton

waveguide is proposed and numerically analyzed. The characteristics of the

symmetric and asymmetric bound modes, including the effective mode

indices, propagation lengths, mode sizes and mode shapes at telecom

wavelength 1.55 mµ are investigated in detail. The simulation results show

that the sub-wavelength confinement (about1.45 mµ ) and a long propagation

(about 820 mµ ) can be realized. Although the mode sizes are a bit larger

than that of the dielectric loaded surface plasmon polariton waveguide, an

order longer propagation length can be achieved. The proposed symmetric

dielectric loaded surface plasmon polariton waveguide provides a potential

for low loss and high density photonic integration.

©2009 Optical Society of America

OCIS codes: (130.2790) Guide waves; (240.6680) Surface plasmons; (250.5300) Photonic

integrated circuits

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#105767 - $15.00 USD Received 2 Jan 2009; revised 4 Feb 2009; accepted 4 Feb 2009; published 23 Feb 2009

(C) 2009 OSA 2 March 2009 / Vol. 17, No. 5 / OPTICS EXPRESS 3610

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1. Introduction

Surface plasmon polaritons (SPP) are electromagnetic waves coherently coupled to electron

oscillations and propagating at the interface between a dielectric and a conductor, evanescently

confined in the perpendicular direction[1]. Recently, surface plasmon polariton waveguides

have been received considerable attention for their ability to simultaneously guiding and

controlling light in optical waveguides. A variety of integrated optical devices based on surface

plasmon waveguides, such as optical attenuators[2,3], modulators[4], switches[5], Bragg

gratings[6], etc., have been demonstrated.

Since the SPP waveguides are the key elements to build these optical components, many

kinds of SPP waveguides geometries have been analyzed theoretically and experimentally. For

two dimensional (2D) planar SPP waveguides, the insulator-metal-insulator (IMI) or the

metal-insulator-metal (MIM) heterosturctures were investigated a long time ago. In symmetric

IMI heterostructures, when the metal thickness is thin enough, the SPP waves guided by the

two metal-dielectric interfaces can coupled efficiently and two kinds of SPP modes can arise:

the long-range SPP (LRSPP) mode and the short-range SPP (SRSPP) mode. Due to the

limitation of confinement only in one dimension for these 2D SPP waveguides, the three

dimensional (3D) SPP waveguides which can confine the SPP wave in two dimensions while

propagating are investigated greatly. The two typical SPP waveguide geometries are the metal

strip waveguides and the channel plasmon polariton (CPP) waveguides. Theoretical studies

show that the metal strip SPP waveguides can support LRSPP modes with two dimensional

confinement and a very low propagation loss[7-9]. Recently, a few low loss metal strip

waveguides have been realized and propagation loss of < 2dB/cm (propagation length of a few

centimeters) in telecom wavelength (1550nm) was achieved[10-13]. Although with the relative

low loss, both simulation and experimental results show that the mode sizes of the metal strip



SPP waveguides are on the order of a few micrometers which do not suit for high photonic

integration. This is because most optical fields are penetrated into the dielectric cladding and

substrate. The merits of the metal strip SPP waveguides compared to the conventional

dielectric waveguides are their capability of guiding and controlling light with the same metal

strip simultaneously. While the CPP waveguides are very promising for high density

integration because they can confine the light in sub-wavelength scale. Recently, various types

of CPP geometries have been investigated theoretically and experimentally, which including

the metal grooves[14-16], metal gap[18,19], and meal hole[20,21]. Generally, the losses of

these CPP waveguides are high because more optic fields reside in the metal. It is obviously

that the SPP confinement is achieved primarily by decreasing the SPP field into the dielectric,

thereby increasing the SPP power being absorbed by metal, so there is a trade-off between SPP

loss and SPP mode confinement. In order to optimize this trade-off, the dielectric-loaded

surface plasmon-polariton (DLSPP) waveguide with reduced mode size based on high index

contrast dielectric waveguide on metal film was demonstrated recently. Sub-wavelength

confinement (915nm) and moderate propagation length (a few tens of micrometers) at telecom

wavelength (1550nm) was achieved[22-25].

In this work, a symmetric dielectric-loaded surface-polariton (SDLSPP) waveguide is

presented. Two types of bound modes (the symmetric mode and the asymmetric mode)

supported by this structure are analyzed. The mode effective indices, propagation lengths,

mode sizes and mode shapes of the bound modes are obtained using a fininte-element method

(FEM). In the FEM method, the region of interest is subdivided into enough small triangle

segments, and the partial differential equations are solved for the propagation constant β and

magnetic field components. The boundary condition used at the edge of computational window

is that of a perfect electric conductor. And at the interfaces between the core, cladding and

substrate, the continuities of the tangential components of magnetic and electric fields and

normal components of the electric and magnetic flux densities are used [24]. In the following

section, the bound modes’ characteristics of SDLSPP waveguides are obtained first. Then some

comparisons with DLSPP waveguides are presented. Finally, a discussion on the advantages of

our suggested configuration will be summarized. The results show that the symmetric mode

(LRSPP mode) with both of sub-wavelength confinement and relative low loss (Propagation

length of a few hundred micrometers) at telecom wavelength (1550nm) can be realized.

2. Bound modes’ characteristics of SDLSPP waveguides

(a) (b)

Fig. 1. The cross section of SDLSPP waveguide (a) and the cross section of DLSPP waveguide (b)

A schematic of the proposed SDLSPP waveguide structures is presented in Fig. 1(a). Also the

DLSPP waveguide cross section is given in Fig. 1(b) for comparison. The refractive indices of

core, cladding, substrate, and metal are nc , ncl

, ns and nm , respectively. The metal film



thickness is d, the width of core is w and height of the core is h . Also the coordinate is given

where x, y and z are the transverse, lateral, and propagation direction respectively. Compare to

the DLSPP waveguide, the SDLSPP waveguide has two main different: one is that the SDLSPP

has two dielectric core with high refractive index allocated at upper and bottom sides of a thin

metal film symmetrically; The other is their dielectric refractive index distribution: the

refractive indices relation of n n nc s cl> = is fulfilled in the SDLSPP waveguide while the

relation of n n ns c cl> > in the DLSPP waveguide. The merits of SDLSPP structure is

obviously: both the LRSPP modes (symmetric modes) with low loss and the asymmetric modes

can be supported by the symmetric refractive index distributions in region 1, 2 and 3. Also by

using the effective index method (EIM)[17], the effective index in region 2 is larger than that of

region 1 and region 3, so the SPP field can also be confined in the transverse direction. While in

DLSPP, only the asymmetric mode with high loss (propagation length of a few tens of

micrometers) is supported due to the high refractive index difference between the cladding and

substrate [10, 13]. Besides, the LRSPP mode size can be drastically reduced in the lateral

direction by employing the core with high refractive index just like the conventional dielectric

waveguides. With these modifications, both the mode size and propagation length can be

optimized. In the simulation results presented hereafter, a excitation wavelength 1550nmλ = ,

refractive index of cladding, substrate 1.46n nscl= = , core 1.65nc = ,

gold 0.55 11.5n im = + [24], the width and height of dielectric core 600w nm= , 300h nm= ,

and metal film thickness of 20d nm= , are used when simulating the SDLSPP bound mode

characteristics unless noted other wise. These refractive indices can be realized by polymer

materials easily [26, 27].



(a) (b)

(c)

Fig. 2. The real parts of symmetric (LRSPP) and asymmetric mode effective indices with

various metal film thicknesses (a). The propagation length of symmetric (LRSPP) mode with

various metal thicknesses (b). The propagation length of asymmetric mode with various metal

thicknesses (c).

Because the metal film thickness has great effect on the LRSPP mode loss, the mode

effective indices and the propagation lengths with different metal film thickness d are analyzed.

The results of the symmetric mode (LRSPP) and asymmetric mode supported by the SDLSPP

waveguide and the planar IMI waveguide are show in Fig. (2). The structure of planar IMI

waveguide is also shown in the inset in Fig. 2(b). And the refractive indices of insulator of

IMI(A) and IMI(B) structure are 1.46 and 1.65 respectively, which correspond to the cl

n and

cn value of SDLSPP waveguide. In Fig. 2(a), it is clear that the real parts of mode effective

indices ( Re( )eff

n ) of symmetric and asymmetric of SDLSPP waveguide hold the same trends

as that of IMI structures: The real parts of effective refractive indices of all symmetric modes

increase with increasing metal thickness while asymmetric modes have the contrary trend. Also

it is very reasonable that the Re( )eff

n of SDLSPP waveguide is between that of IMI(A) and

IMI(B) structure because the three regions of SDLSPP can be regarded to IMI(A) and IMI(B)



structures respectively. The propagation length is related to the image part of mode effective

index ( Im( )neff

) according to 4 Im

Lpropneff

λπ

=⋅

[24]. By using this relation, in Fig. 2(b)

and Fig. 2(c), it is obvious that the propagation lengths of symmetric and asymmetric modes of

SDLSPP waveguide also share the same trends as IMI structures, where the propagation

lengths of symmetric modes are much longer than that of asymmetric modes. Besides, it is very

reasonable that the propagation length of SDLSPP waveguide is shorter than those of both

IMI(A) and IMI(B) structures due to more SPP wave are resided in metal film which bring the

ohmic loss. This is because the SDLSPP waveguide adds the confinement of the SPP wave in

transverse (x) direction and also enhance the confinement in lateral (y) direction due to the high

refractive index core, which the IMI structures do not have. Although the symmetric mode

propagation length of SDLSPP waveguide is shorter than that of IMI structures, it is still much

larger (a few hundreds micrometers) than that of DLSPP waveguides (a few tens micrometers)

[24] when the metal film is thin enough.

(a) (b)

(c) (d)

Fig. 3. The mode size of symmetric mode of SDLSPP waveguide VS the core width (a); The

mode size of asymmetric mode of SDLSPP waveguide VS the core width (b); The propagation

length of symmetric mode and asymmetric mode of SDLSPP waveguide VS the core width (c);

Three mode shapes in both x and y directions under core with w=200, 500 and 800nm (d).



From Fig. 2(b), it is obvious that the symmetric mode propagation length of SDLSPP

waveguide increases rapidly when the metal film thickness is lower than 40nm. In addition to

achieve mode confinement, the mode size and corresponding propagation length of symmetric

mode and asymmetric mode under various core widths w is investaged. The results are show

in Fig. (3) and the mode sizes (including the mode height and mode width) are obtained by

measuring the height and width at 1e of the normalized maximum main electric field | |

yE .

In Fig. 3(a) and Fig. 3(b), the mode height, mode width of symmetric mode and asymmetric

mode can be decreased by increasing the core width. Also the mode sizes of these modes

approach the minimum values when the core width is large enough. Even enlarge the core

width, the mode sizes of these bound modes can be increased [24]. And from Fig. 3(c), the

propagation length of symmetric mode decreases with increasing core width, because more

SPP wave can be confined in the core region where metal loss is enhanced. Besides, the

propagation length of symmetric mode is much larger than that of asymmetric mode. Since

compare to the symmetric mode, the asymmetric SPP mode penetrates much deeper to the

metal film, where greate omhic loss can be arisen. Also the smaller mode size relative to the

symmetric mode in Fig. 3(b) proves this. Figure 3(d) shows a few mode shapes of symmetric

modes with various core width: 200,500,800w nm= . It is clear the mode shapes is very Gauss

like except when the core width is small enough. According to the above results, a trade off

between the mode size and propagation length with the core width exists, which is the essential

restrict of surface plasmon[1]. So the core width of 600w nm= is chosen in the following

optimization.

Not only the geometric parameters can alter the mode size, but also the refractive index

difference between the core and cladding can affect the SPP mode size and propagation length

greatly just like the dielectric waveguide. By altering the cladding and substrate refractive

index n ncl s= , the mode sizes and propagation lengths are analyzed with the core

index 1.65nc = . As shown in Fig. 4(a), the mode heights, widths of symmetric and asymmetric

modes are both decreased with decreasing the cladding refractive index. This is very physical

reasonable because the mode confinement capability of the core region is enhanced by

increasing the refractive index difference between the core and cladding. And because this high

confinement, more SPP field are attenuated by the metal film, which cause the propagation

length decreases rapidly as in Fig. 4(c). Also it is very nice that the mode height and width of

symmetric mode can be almost equal when choosing the cladding and substrate refractive index

1.4n nc s= = , which can be physical realized by polymers. With these parameters, the

symmetric mode size can be very close in the lateral and transverse direction, which induces a

Gauss-like symmetric mode shape as show in Fig. 4(d). The mode height, width in Fig. 4(a) is

1452 nm, 1454nm respectively and the corresponding propagation length in Fig. 4(c) is about

850 mµ . The sub-wavelength confinement in two dimensions and a relative long propagation



length are achieved simultaneously. Besides, it is very clear that the propagation length of

asymmetric mode (a few tens micrometers) is much shorter than that of symmetric mode.

(a) (b)

(c) (d)

Fig. 4. The mode size of symmetric mode with various cladding index (a); The mode size of

asymmetric mode with various cladding index (b); The propagation length according to different

cladding index (c); The normalized Ey field distribution (mode shape) of symmetric mode

with 1.4n nscl= = (d).

3. Comparison with DLSPP waveguides

As shown in Fig. (1), the DLSPP waveguide is actually buildup by the asymmetric IMI

structures, while the SDLSPP waveguide is composed of several symmetric IMI structures.

Due to the large refractive index differences between the core and substrate, only the

asymmetric mode is supported for the DLSPP waveguide. On the contrary, the SDLSPP

waveguide can support both symmetric mode and asymmetric mode. The longer propagation

length can be achieved by the symmetric mode because comparing to the asymmetric mode, the

SPP wave inside the metal film is greatly reduced. In order to compare the mode shapes of

DLSPP and SDLSPP waveguide, a same core size of ( )600 600nm nm× is chosen for

simulation. The results in Figs. 5(a) and Fig. 5(b) shown the mode size of DLSPP waveguide is

smaller than that of SDLSPP waveguide especially in the lateral direction, which is because the

more SPP field can be resided in the metal film according to the asymmetric mode

characteristics. Also due to the asymmetry of the core region with respect to the metal film, the



mode shape is highly asymmetric. So the another merits of SDLSPP waveguide is that the

Gauss-like symmetric mode shape can be realized while the DLSPP waveguide can not

achieve. Comparing to the DLSPP waveguide, the mode size of SDLSPP waveguide is larger

than that of DLSPP waveguide. Also the bend loss of SDLSPP waveguide will be a bit larger

than that of the DLSPP waveguide. But still the sub-wavelength and relative long propagation

length (a few hundreds micrometers) can be achieved simultaneously by the SDLSPP

waveguide.

(a) (b)

Fig. 5. The Ey field distribution (mode shape) of DLSPP waveguide with metal thickness of

100nm, 1.46n nscl= = , 1.65nc = , 600w h nm= = (a); The normalized symmetric

mode Ey field distribution (mode shape) of SDLSPP waveguide with metal thickness of

20nm, 1.46n nscl= = and 1.65nc = , 600 , 300w nm h nm= = (b)

4. Conclusion

In this paper, the detail characteristics of bound modes supported by the symmetric dielectric

loaded surface plasmon polariton waveguide are analyzed. The simulation results show that the

characters of bound modes are just like those of IMI structures. And by the symmetric

geometric and refractive index distribution, the low loss symmetric mode (LRSPP) is supported

by the SDLSPP waveguide. By introducing the symmetric high refractive index dielectric

cores, the LRSPP can be confined in the lateral and transverse directions very well, also a much

longer propagation length than that of DLSPP waveguide can be achieved because of the

LRSPP mode characteristics. Based on these results, the sub-wavelength confinement

(about 1.45 mµ ) and long propagation length ( 820 mµ ) are realized simultaneously by

optimization. The proposed SDLSPP waveguide can be used to made passive optical

components for high density photonic integration.



Bound modes analysis of symmetric dielectric loaded surface plasmon-polariton waveguides

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