## Abstract

In this paper, a kind of surface plasmonic waveguide (SPW) with three circular air cores is presented. Based on the finite-difference frequency-domain (FDFD) method, dependence of the distribution of energy flux density, effective index, propagation length and mode area of the fundamental mode on the geometrical parameters and the working wavelengths is analyzed firstly. Then, comparison with the SPW which was proposed in our previous work has been carried out. Results show that this kind of three cores structure has better propagation properties than the double cores structure. To investigate the relative advantages of this kind of SPW over other previous reported SPWs, comparison with the SPW with a single wedge has been carried out. Results show that this kind of SPW has shorter propagation length and larger mode area. Finally, the possibility to overcome the large propagation loss by using a gain medium as core material is investigated. Since the propagation properties can be adjusted by the geometrical and electromagnetic parameters, this kind of surface plasmonic waveguide can be applied to the field of photonic components in the integrated optical circuits and sensors.

© 2009 Optical Society of America

## 1. Introduction

Recently, there has been an increase of interest in surface plasmon polaritons (SPPs) [1]. SPPs are electromagnetic wave that are bound to a metal-dielectric interface and are coupled to the oscillations of the free electrons in the metal [2]. Because SPPs have lateral dimensions on the order of subwavelength, it overcomes the diffraction limit that exists in conventional or photonic crystal waveguides, and fulfills the further miniaturization of photonic devices and high integration density of photonic chips. So during the past few years, SPPs-based waveguides have also been a subject of intensive research [3–6].

Up to now, several kinds of SPWs structures have been proposed, such as nanoparticle SPWs [7,8], film SPWs [9,10], rod SPWs [11,12], gap SPWs [13–15], slot SPWs [16,17], wedge SPWs [18–21], channel SPWs [22–25], heterostructured SPWs [26,27], and mixed SPWs [28].

In our previous work [21], a SPW with double elliptical air cores was proposed and analyzed. Results show that the energy flux density distributes mainly in the two wedged corners which are formed by two elliptical air cores, and the closer to the corners the stronger the longitudinal energy flux density. The effective index and propagation length of the fundamental mode can be adjusted by the centric distance of two ellipses as well as the size of the two semiaxis.

In this paper, an altered SPW with three circular air cores is proposed. The FDFD method is used to study the distribution of energy flux density, effective index, propagation length and mode area of the fundamental mode supported by this SPW structure. Then, comparison with the SPW in our previous work [21] and a SPW with a single wedge have been carried out.

Due to the limitation of the high absorption loss of the metal material, the propagation length of SPPs on SPWs is much shorter than that of the conventional dielectric optical waveguides. There is a tradeoff between modal confinement and propagation length in all SPWs without assisted gain [29]. This shortcoming can be overcome by using optical gain medium [30–35]. The possibility to overcome the large propagation loss by using the gain medium as core material is investigated finally.

## 2. Structure and simulation method

The cross section of our proposed surface plasmonic waveguide in this paper is shown in Fig. 1. It is composed of three circular air cores with the same radius *r* in the silver cladding, and 2*a* is the centric distance of the upper two air cores, 2*h* is the vertical distance from the centre of the nether circular air core to the horizontal line which is passed through the centres of the upper two air cores. Obviously, it can be separated into three cases of 2*a*>*r*, 2*a*=*r* and 2*a*<*r*, when 2*h* is a fixed value.

Noble metals are usually used to make SPWs. At optical frequencies, the dielectric constants of these metals are complex. In calculation, the dielectric constant of silver *ε _{clad}* is chosen as - 18.0550+0.4776j (

*λ*=632.8 nm), - 23.4046+0.3870j (

*λ*=705.0 nm), and - 31.0784+0.4118j (

*λ*=800.0 nm) [36] respectively. Here,

*λ*is the working wavelength in the vacuum.

In this paper, the 2D full-vectorial FDFD method [37–39] is used to study the propagationproperties of our proposed SPW structure. This is a simple and effective numerical simulation approach. Setting the geometrical parameters, electromagnetic parameters and working wavelength, an eigenvalue equation can be obtained. Solving the eigenvalue equation by Arnoldi arithmetic [40], which can deal with large matrix eigenvalue problem with complex coefficient matrix, propagation constant and the distribution of field of each mode at the working wavelength can be obtained. In our calculation, 601×601 Yee’s lattices are adopted to discretize the whole computational domain, and 20 layers of them are perfectly matched layer absorbing boundary layers (APML) that used to truncate the lattices. Spatial discretization distance is Δ*x*=Δ*y*=1.0 nm.

To validate our FDFD code, an optical fiber with a finite metallic core [41] is analyzed. The working wavelength is chosen as *λ*
_{0}=633*nm*. The relative dielectric constant of the metal and the cladding is taken as *ε _{core}*=19.00-0.53

*j*and

*ε*=4.00 respectively. In the calculation, the radius of the core is assumed from 0.05

_{clad}*λ*

_{0}to 0.25

*λ*

_{0}. Here we have taken the notation of the working wavelength in reference [41] as

*λ*

_{0}. In the later text, we will still use

*λ*as the working wavelength in the vacuum. For clarity, we show results in the figure form. As shown in Fig. 2, the analytic [41] and simulated propagation constants for the 0th-order TM mode exhibit good agreement. This fact shows that our FDFD code can be used to simulate plasmonic structures.

Yan and Qiu claimed that the corner angle and the corner tip sharpness would affect the modal properties of a Λ -wedge waveguide considerably [42]. Although the FDFD method use orthogonal mesh and corner tips are not rounded in this paper. We find that, in our calculation, the corner angles are not too small and the corner tips are not too sharp. So the situation is in the safe case.

## 3. Results and discussion

#### 3.1. Modal characteristics

Characteristics of the modes supported by the surface plasmonic waveguide with three circular air cores shown in Fig. 1 are investigated firstly. We find that there is a kind of mode which have longer propagation length among many modes supported by this kind of SPW, and call this mode the fundamental mode. The distribution of field *H _{x}, H_{y}*, and energy flux density

*S*in the cases of

_{z}*a*=60 nm,

*h*=85 nm, and

*r*=120 nm at

*λ*=705.0 nm is shown in Fig. 3. Here the energy flux density is defined as

*S*=

_{z}*R*(

_{e}*E*),

_{x}H_{y}-E_{y}H_{x}*E*and

_{x}*E*,

_{y}*H*and

_{x}*H*are components of electrical and magnetic field respectively.

_{y}It can be seen from Fig. 3(a) that the component of field *x H* is symmetric distribution with y axes, however, component of field *H _{y}* shown in Fig. 3(b) is antisymmetric distribution with y axes. From Fig. 3(c), we can find that the energy flux density mainly distributes near the left and right wedged corner which are formed by three circular air cores. Among the three cases of 2

*a*>

*r*, 2

*a*=

*r*and 2

*a*<

*r*, the two corner angles are smallest and the two corner tips are sharpest in the case of 2

*a*>

*r*, so the degree of localization of field is the largest.

#### 3.2. Effect of geometrical parameters of the waveguide

Since the effective index *Re*(*n _{eff}*), propagation length

*L*and mode area

_{prop}*A*are the three most important physical quantities that describe the propagation properties of surface plasmonic waveguides, then we investigate the dependence of

_{m}*Re*(

*n*),

_{eff}*L*and

_{prop}*A*of the fundamental mode shown in Fig. 3 on the geometrical parameters. Here

_{m}*Re*(

*n*) is defined as

_{eff}*Re*(

*β*)

*λ*/2

*π*,

*L*is defined as 1/

_{prop}*Im*(

*β*), and

*A*is defined as the area in which the energy flux density

_{m}*S*descend from 100% to 10% of its maximum value.

_{z}The relational graphs of *Re*(*n _{eff}*),

*L*and

_{prop}*A*varied with

_{m}*h*in the three cases of

*r*=2

*a*-5 nm, 2

*a*and 2

*a*+5 nm, and 2

*a*=100 nm, 120 nm, 140 nm at

*λ*=705.0 nm are shown in Fig. 4(a–c), Fig. 4(d–f) and Fig. 4(g–i) respectively. It can be seen from these figures that curves can be obviously separated into three groups according to different

*a*. In each group,

*Re*(

*n*) increases as

_{eff}*h*increases, however

*L*on the whole lessly varied with

_{prop}*h*, and

*A*increases as

_{m}*h*decreases. The position of curves is influenced by parameter

*r*. We can also find that curves corresponding to

*r*=2

*a*+5 nm are commonly on the upward side of each group of curves. The above phenomena can be explained by the different field distribution with different geometrical parameters. Because the field is mainly centralized near the left and right wedged corner which are formed by three circular air cores. Relative to the case of

*r*=2

*a*, when

*r*is small, the area of field distribution is small. Namely the degree of localization of field is large. In this instance, the interaction of field and silver is strong. The effective index increases. Then the propagation length and mode area decrease. However, when

*r*is large, the area of field distribution is large. Namely the degree of localization of field is small. In this instance, the interaction of field and silver becomes weak. The effective index decreases. Then the propagation length and mode area increase.

#### 3.3. Effect of working wavelengths

In order to find out the dependence of field distribution of the mode shown in Fig. 3 on the working wavelength, the distribution of field *H _{x}, H_{y}*, and energy flux density

*S*in the cases of

_{z}*r*=2

*a*=120 nm,

*h*=85 nm at

*λ*=632.8 nm and

*λ*=800.0 nm is calculated respectively. Here we extend the calculation to longer wavelength 800 nm where the vertical-cavity surface-emitting lasers (VCSELs) are available. Results shown that field distribution is similar to the Fig. 3. Relative to the case of

*λ*=705.0 nm, in the case of

*λ*=632.8 nm, the area of field distribution is small. And the field is mainly confined near the left and right wedged corner which are formed by three circular air cores. However, in the case of

*λ*=800.0 nm, the area of field distribution is large, and the field is less confined.

Figure 5(a–c), Fig. 5(d–f) and Fig. 5(g–i) also show the dependence of *Re*(*n _{eff}*),

*L*and

_{prop}*A*on

_{m}*h*when radius

*r*=2

*a*=100 nm, 120 nm, 140 nm and

*λ*=632.8 nm, 705.0 nm, 800.0 nm respectively. It can be seen that curves of

*Re*(

*n*),

_{eff}*L*and

_{prop}*A*could also be separated into three groups according to different

_{m}*λ*. And in each group,

*Re*(

*n*) increases along with the increase of

_{eff}*h*, however,

*L*has many trends that varied with

_{prop}*h*, and

*A*decreases as

_{m}*h*increases. Parameter

*r*has effect on the position of curves. It can be seen that curves according to

*r*=2

*a*=140 nm are always on the upward side of each group of curves. The above phenomena can be explained by the different field distribution at different wavelength. Because the field is mainly centralized near the left and right wedged corner which are formed by three circular air cores. Relative to the case of

*λ*=705.0 nm, when

*λ*is small, the area of field distribution is small. Namely the degree of localization of the field is large. In this instance, the interaction of the field and silver is strong. The effective index increases. Then the propagation length and mode area decrease. However, when

*λ*is large, the area of field distribution is large. Namely the degree of localization of field is small. In this instance, the interaction of the field and silver becomes weak. The effective index decreases. Then the propagation length and mode area increase.

#### 3.4. Comparation with the SPW with two air cores

It can be found that, under certain conditions, the three circular air cores can be turned into two circular air cores which is similar to a case of the SPW with double elliptical air cores in our previous work [21]. Here, Let’s take the three cores structure and the double cores structure for comparation.

When the centric distance of the upper two air cores 2a (shown in Fig. 1) decreases to zero, the three circular air cores are turned into two circular air cores. In this case, the cross section is similar to the double cores structure in the case of *a=b*. Here the parameter *r* of the three cores structure is equal to *a* and *b*, and *h* is equal to *c* of the double cores structure. Dependence of *Re*(*n _{eff}*),

*L*and

_{prop}*A*on

_{m}*h*when

*a*=0 nm, 60 nm,

*r*=120 nm, and

*λ*=632.8 nm, 705.0 nm, 800.0 nm is shown in Fig. 6 respectively. From Fig. 6, we can see that

*Re*(

*n*) (Fig. 6a and Fig. 6b) increases as h increases, however Lprop (Fig. 6c and Fig. 6d) increases as h increases when

_{eff}*λ*=800.0 nm, and in the other two cases the propagation length not obviously varied with

*h*. And

*A*(Fig. 6e and Fig. 6f) increases as

_{m}*h*decreases. Parameter

*λ*has effect on the position of the curves in Fig. 6. Results also show that SPW with three circular air cores (Fig. 6b, Fig. 6d and Fig. 6f) has longer propagation length, smaller effective index and mode area compared with the two circular air cores (Fig. 6a, Fig. 6c and Fig. 6e) with the same parameters. In other words, the proposed SPW in this paper has better propagation properties than that of our previous SPW.

#### 3.5. Comparation with the SPW with a single wedge

As shown in subsection 3.4, the three cores structure is an improvement over the double cores structure. To investigate the relative advantages of the double cores structure over other previous reported SPWs, we choose a SPW with a single wedge as shown in Fig. 7(a). It is composed of a single wedged region which is formed by two circular air cores with centric distance 2*c* on the silver slab. This structure not only can be seen as a representative SPW with a single wedge [18–20] but also be comparable to our double cores structure [21]. Here the parameter *r* and *c* is equal to *r* and *c* of the double cores structure [21]. The corresponding distribution of *H _{x}, H_{y}* and

*z*

*S*of the fundamental mode supported by this single wedge structure is shown in Fig. 7(b), Fig. 7(c) and Fig. 7(d). It can be seen that the energy flux density mainly distributes in the wedged corner. The component of field

*H*is symmetric distribution with y axes, however, component of field

_{x}*H*is antisymmetric distribution with y axes.

_{y}Dependence of *Re*(*n _{eff}*),

*L*and

_{prop}*A*on

_{m}*c*when

*r*=100 nm and

*c*=85 nm at

*λ*=705.0 nm is shown in Fig. 8 respectively. From Fig. 8, we can see that

*Re*(

*n*) increases as

_{eff}*c*increases, however

*L*and

_{prop}*A*decrease as

_{m}*c*increase. Results also show that the single wedge SPW has longer propagation length and smaller mode area, but the double cores structure has shorter propagation length and larger mode area. This phenomenon can also be explained by the interactional intensity between the field and metal. For the single wedge SPW, the field is confined only on one single wedge. The interaction of the field and siver is small. So the propagation length is large, but the mode area is small. For the double cores structure, the field is coupled and confined on two wedges. The interaction of the field and siver is large. So the propagation length is short, but the mode area is large. This fact implies that the double cores structure as well as the three cores structure can be used as a key component of a sensor although it is difficult to fabricate than the single wedge SPW.

#### 3.6. Effect of Gain in the Core Dielectric

To investigate the effect of gain in the core dielectric, we fill the whole air cores of the structure shown in Fig. 1(b) with a kind of available gain medium [34,35,43], and the dielectric constants of the SPW are shown in Table 1. Here, the corresponding gain coefficirnt *γ*=298*cm*
^{-1} (εcore=11.56-0.025j) and 596*cm*
^{-1}(*ε _{core}*=11.56-0.050j) [34]. As an example, in the case of

*λ*=1550 nm,

*a*=60 nm,

*r*=120 nm, dependence of

*Re*(

*n*),

_{eff}*L*and

_{prop}*A*on

_{m}*h*and the gain of the core dielectric are shown in Fig. 9. The corresponding distribution of the field

*z S*are shown in Fig. 10.

As shown in Fig. 9(a), Fig. 9(b), and Fig. 9(c), the propagation length can be extended obviously with the help of the gain medium. It demonstrated that the presence of the gain medium result in an increase of the propagation length *L _{prop}*. However, the effective index is almost kept invariably with different

*ε*, here

_{core}*A*not obviously varied with

_{m}*ε*.

_{core}Since the core material has been changed from air to a gain medium with higher dielectric constant, confinement is expected to increase. As shown in Fig. 10, when the core is filled with gain medium (Fig. 10(a), Fig. 10(b) and Fig. 10(c)), the field *S _{z}* is more concentrated on the surface of the metal than that with air cores (Fig. 10(d)).

The presence of the gain medium to compensate for the absorption loss in propagating [30–32] SPPs, although very small, it has been a possible solution, and at the same time localized [33] the surface plasmons. The influence of the gain medium on SPPs propagation has received some attention previously. There are much work to do in later time.

Methods of how to fabricate this kind of SPW and how to couple SPPs efficiently from an external source to this kind of SPW have been suggested in our previous work [21].

## 4. Conclusions

In this paper, we have designed a kind of surface plasmonic waveguide with three circular air cores. The structure of the waveguide can be separated into three cases of 2*a*>*r*, 2*a*=*r* and 2*a*<*r*. Numerical calculation show that:

(1) The energy flux density distributes mainly in the left and right wedged corner which are formed by three circular air cores, and the closer to the corners the stronger modal energy flux density.

(2) At the certain working wavelength, in the case of 2*a*<*r*, the degree of localization of field is the smallest, and the interaction of field and silver becomes small. The effective index decreases. Then the propagation length and mode area become large.

(3) With the certain geometric parameters, in the case of *λ*=800.0 nm, the area of field distribution is large. Namely, the degree of localization of field is small, and the interaction of field and silver becomes small. The effective index decreases. Then the propagation length and mode area become large.

(4) Comparison with the double air cores structure, this kind of SPW has better propagation properties. Comparison with the SPW with a single wedge, it has shorter propagation length and larger mode area.

(5) The propagation length can be extended obviously with the help of the gain dielectric medium.

Since the effective index, propagation length and the mode area can be adjusted by the geometrical and electromagnetic parameters, this kind of surface plasmonic waveguide can be applied to the field of photonic components in the integrated optical circuits and sensors.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 60878008, 60771052) and the Natural Science Foundation of Shanxi Province (Grant No. 2008012002-1, 2006011029).

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