Manipulating the effective index of the hybrid plasmonic

waveguide based on subwavelength grating

Rui Zhanga, Bowen Baia, Zhiping Zhou*a

a State Key Laboratory of Advanced Optical Communication Systems and Networks,

School of Electronics Engineering and Computer Science, Peking University, Beijing, China 100871

ABSTRACT

In this paper, we propose and numerically study a subwavelength grating based hybrid plasmonic waveguide. The

metal layer on top of the waveguide enables unique features compared with conventional silicon based waveguide. Since

the field distribution in this structure is different, traditional homogeneous medium approximation is not applicative.

Therefore, we develop a new effective index calculation method. This method is suitable for metal-existing waveguide as

well as structures with multiple medium. Effective index of this waveguide depends on grating period, duty ratio and width,

respectively. By modifying duty ratio and period of the waveguide, the relationship between effective index and waveguide

width can be concave function or convex function and the slope can be similar to TM mode of silicon based waveguide,

which opens up possibilities for SPPs based applications.

Keywords: Hybrid plasmonic waveguide, effective index manipulating, grating.

1. INTRODUCTION

Integrated photonic devices have been widely deployed for communication, computing, and clinical applications [1,2]

due to their large bandwidth, CMOS-compatibility, and enhanced sensitivity. Advances in Silicon-on-Insulator (SOI)

platform for integrated photonics paved the way for the rapid development of arrayed waveguide gratings [3] in the

emerging field of silicon photonics. Based on effective-medium theory, P. Cheben proposed a sub-wavelength Grating

(SWG) structure [4] with grating period smaller than the 1𝑠𝑡 Bragg period, given by:

∧𝐵𝑟𝑎𝑔𝑔= 𝜆/2𝑛𝑒𝑓𝑓

Where 𝜆 is the wavelength of light, 𝑛𝑒𝑓𝑓 is the effective index of the Bloch mode propagating through the structure.

At subwavelength scale (∧< 𝜆), composite medium can be approximated as a homogeneous medium. Pérez-Galacho et al.

presented a systematic method [5] for taper design to achieve adiabatic transitions between SWGs and conventional

waveguide.

Surface plasmon polaritions (SPPs) are waves propagating along a metal-dielectric interface with an exponentially

decaying field in both sides [6], have attracted attentions and research efforts in recent years for its potential for overcoming

diffraction limit in the subwavelength range as well as chip size reduction. A variety of SPP structures such as hybrid

plasmonic waveguides [7] and couplers [8] have been investigated numerically. However, the effective indices of TE mode

in either hybrid plasmonic waveguide or conventional silicon waveguide are very sensitive to waveguide width [9]. Our

calculation shows that the effective indices of hybrid plasmonic SWGs (HPSWG) are insensitive to width, which is of

great potential for solving this problem and achieving fabrication tolerate TE coupler.

In this paper, we introduced and numerically studied the proposed HPSWG waveguide. An effective index calculation

method which is suitable for metal-existing structures is developed. By modifying grating period (∧) and duty ratio(DR),

we are able to manipulate the slope of the relationship between effective index and waveguide width. The final coupling

result validates our calculation method.

2. STRUCTURE AND SIMULATION METHOD

In the first step, we employ three-dimensional Finite-Division-Time-Domain (FDTD) method to optimize the taper

on a SOI wafer with a Si top layer to obtain the maximum power transmission as shown in Fig. 1. The thickness of the Si

layer is defined as ℎ𝑆𝑖 and thin low-index layer is SiO2 with a thickness of ℎ𝑆𝑖𝑂2. The operation wavelength is 1.55 nm,

and corresponding refractive indices for SiO2 , and Si are 1.444, and 3.478. 𝐶𝑜𝑟𝑒𝐷𝐶1(𝑥) = 𝑤3(𝑥)/𝑤1 and

𝐶𝑜𝑟𝑒𝐷𝐶2(𝑥) = 𝑤2(𝑥)/𝑤1 are functions of the propagation direction. The duty ratio of taper is 𝐷𝑅1 = 𝑎1/𝑏1 and ∧1=

𝑎1 + 𝑏1 is taper period. According to the simulation, when 𝐶𝑜𝑟𝑒𝐷𝐶1 and 𝐶𝑜𝑟𝑒𝐷𝐶2 vary linearly with x and slope of

the linear functions of 𝐶𝑜𝑟𝑒𝐷𝐶1 and 𝐶𝑜𝑟𝑒𝐷𝐶2 are kept as −0.1 and 0.08, the number of the taper on each side are

kept constant as 7 . We obtain an optimal power transmission of 0.97015 with optimized parameters ℎ𝑆𝑖𝑂2, ℎ𝑆𝑖, 𝑤1 , 𝐷𝑅1

, and ∧1 at 0.05μm , 0.25μm, 0.45μm, 1, and 0.29μm, respectively.

Figure 1. Schematic of taper optimization structure. (a) three-dimensional view (b) the in-plane views.

Then we add HPSWG waveguide in the central region of the two tapers as Fig. 2 depicts. The Si layer thickness and

SiO2 layer thickness of the HPSWG waveguide are same to the taper. The metal cap of the HPSWG waveguide is Ag with

a thickness of ℎ𝐴𝑔. ∧2= a2 + b2 is the grating period, 𝐷𝑅2 = 𝑎2/𝑏2 is duty ratio of the grating and W is the width of

the grating. For the simulation, light is input to the left side of the structure along the x-axis. The operation wavelength is

still 1.55 nm, and corresponding refractive index for Ag is 0.143246 + 11.3639i [10]. ℎ𝐴𝑔 is kept constant as 0.1μm. The

input polarization was transverse electric (TE) and the 3D model is simulated using finite element method(FEM). Building

upon the effective index calculation result, we design and simulate the coupling structure, the results are shown in the next

sections.

Figure 2. A general schematic of the proposed structure. (a) three-dimensional view of the structure. (b) the in-plane views of the

structure; (c) The cross-sectional view of the HPSWG waveguide parallel to the chip plane y-z

3. EFFECTIVE INDEX CALCULATION AND MODIFICATION

Due to the top metal layer, the field distribution of HPSWG waveguide has changed, so the homogeneous medium

approximation method cannot be employed to calculate the effective index. In this case, we propose a new solution to this

problem by taking field distribution into consideration. Moreover, the new method is a general alternative to calculate the

effective index of these metal-existing structures or other multiple medium waveguides. As Fig 3 shows, electric field is

distributed periodically along the HPSWG waveguide. Therefore, we can obtain the wavelength 𝜆1 in the HPSWG

waveguide from the electric field distribution period and thus calculate the effective index. The effective index of the

HPSWG waveguide is given by:

𝑛𝑒𝑓𝑓 = (𝜆1

𝜆2

) ∗ 𝑛𝑎𝑖𝑟

where 𝜆2 is wavelength of the input light, 𝑛𝑎𝑖𝑟 is the refractive index of air.

Figure 3. Electric field distribution of HPSWG waveguide, for 1550 nm wavelength.

The simulation result shows that the effective index can be manipulated by altering 𝐷𝑅2, ∧2 and W. Table 1 and Fig.

4 illustrates the calculation result of the relationship of HPSWG waveguide effective indices and grating width. The

effective index is reduced by narrowing the width, and as 𝐷𝑅2 decreases, the effective index increases. Here we define

the relationship between effective index and width as 𝑛𝑒𝑓𝑓(𝑊). With the changes of 𝐷𝑅2 and 𝛬2, the curve of 𝑛𝑒𝑓𝑓(𝑊)

can be concave or convex, which is different from conventional hybrid plasmonic waveguide. As shown in Fig 4(b), when

∧2 is 0.29μm and 𝐷𝑅2 is 7/3, the function is concave while it is convex when ∧2= 0.29μm, 𝐷𝑅2 = 1.5. By controlling

𝐷𝑅2, ∧2 and W, the concavity as well as convexity of the curve can be modified flexibly.

Table 1. Curve fitting result of 𝑛𝑒𝑓𝑓(𝑊)

∧2 𝐷𝑅2 𝑛𝑒𝑓𝑓(𝑊)

0.28μm 1 0.0000004614𝑊2 + 0.0002849𝑊

0.28μm 1.5 −0.000001947𝑊2 + 0.003899𝑊

0.28μm 7/3 0.0000000512𝑊2 − 0.0001𝑊 + 0.0711

0.29μm 1 −0.00000004458𝑊2 + 0.0009912𝑊

0.29μm 1.5 −0.000002233𝑊2 + 0.004251𝑊

0.30μm 1 −0.00000004671𝑊2 + 0.00009126𝑊

− 0.05809

0.30μm 1.5 0.00000006014𝑊2 − 0.0001186𝑊 + 0.07875

0.30μm 7/3 −0.000001451𝑊2 + 0.003366𝑊

0.31μm 1 −0.000002271𝑊2 + 0.003908𝑊

0.31μm 1.5 0.00000003876𝑊2 − 0.00007497𝑊+0.04941

0.31μm 7/3 −0.0000000743𝑊2 + 0.001636𝑊

Figure 4. effective indices (𝑛𝑒𝑓𝑓) of HPSWG waveguide versus width (W) with different grating duty ratio (𝐷𝑅2) and grating

period (∧2). (a) 𝑛𝑒𝑓𝑓 versus W when ∧2= 0.28μm and DR2 = 1, 1.5,7/3. (b) 𝑛𝑒𝑓𝑓 versus W when ∧2= 0.29μm and 𝐷𝑅2 = 1,

1.5,7/3. (c) 𝑛𝑒𝑓𝑓 versus W when ∧2= 0.30μm and 𝐷𝑅2 = 1, 1.5,7/3. (d) 𝑛𝑒𝑓𝑓 versus W when ∧2= 0.31μm and 𝐷𝑅2 = 1,

1.5,7/3.

Moreover, as Fig.5 shows, when ∧2= 0.29μm and 𝐷𝑅2 = 1.5 , the slope of the T mode’s 𝑛𝑒𝑓𝑓(𝑊) curve in

proposed structure is similar to that of TM mode in silicon based waveguide, which shows great potential for fabricating

high-tolerant T coupler.

Figure 5. effective indices (𝑛𝑒𝑓𝑓) of HPSWG waveguide and silicon waveguide versus width (W)

4. COUPLING WITH TRADITIONAL SLICON WAVEGUIDE

When ∧2= 0.29μm, 𝐷𝑅2 = 1.5 and W = 700nm, the effective index of HPSWG waveguide matches that of a

silicon based waveguide with the width of 365nm. The thin low-index layer of silicon based waveguide is SiO2 with a

thickness of 0.05μm and the Si layer is 0.25μm thick. The match of effective index yields strong coupling between

HPSWG waveguide and silicon based waveguide according to the phase-matching theory [11].

Fig. 6 presents the coupling structure and the power distribution. Light is the input from port 1 and propagates to the

coupling region. The gap between the waveguides is kept as 150nm and the length of coupling region (L) is swept to

maximize the fraction of exchanged power between two waveguides. The calculation result shows that when 𝐿 = 4.06μm,

the transmission in port 2 is 0.867 while transmission of port 3 is 0.0517, which verifies our effective index calculation.

However, since the grating number can only be integer, 𝐿 is not changed continuously under a fixed grating period. Better

coupling result might be obtained by sweeping the grating period and grating number simultaneously.

Figure 6. Power distribution of our proposed coupling structure. Light is input from port 1 and the output power is monitored in port

2 and port 3.

5. CONCLUSION

We propose a new effective index calculation method to investigate and analyze the properties of HPSWG waveguide

in this paper. We can generally employ the new method to a variety of multiple medium structure as well as metal-existing

waveguides which cannot be applied to traditional calculation method. The effective index increases with the increasing

of grating width. The concavity and convexity of the function 𝑛𝑒𝑓𝑓(𝑊) can be manipulated by altering the duty ratio and

period. When the duty ratio and period are kept constant to a certain value, the slope of the curve between effective index

and width is similar to TM mode’s slope in silicon waveguide, which is of great potential for fabrication tolerant T mode

coupler application. We also calculate and simulate the coupling transmission between HPSWG waveguide and the result

verifies our calculation result. These unique features may provide more possibilities for SPP based coupler applications.

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