## Abstract

Surface plasmon polariton reflector (SPPR) based on metal-insulator-metal (MIM) Bragg grating waveguide is numerically studied. A quasi-chirped technique is applied to the engraved grooves in the surface of the MIM waveguide, and a new kind of broad-bandgap SPPR is achieved. Meanwhile, by optimizing the profile of gap width between the metal and dielectric, the spectral sidelobe of SPPR is effectively suppressed and thus the performance of the SPPR is further improved.

©2009 Optical Society of America

## 1. Introduction

In response to the needs of realization of highly integrated optical devices and circuits, surface plasmon polaritons (SPPs) are considered as one of the most promising ways for overcoming the diffraction limit and manipulating light at a subwavelength scale [1]. The SPPs are electromagnetic modes coupled to the oscillations of conduction electrons propagating along the interface between dielectric and metallic materials. The electromagnetic field associated with the SPPs is bounded along the interface and decreases exponentially in the direction perpendicular to the interface, which can thereby result in a subwavelength confinement of the electromagnetic mode [2]. One of the most attractive aspects of SPPs is the way they help us concentrate and channel light using subwavelength structures. Some photonic devices based on SPPs-assisted light control, such as lenses [3], modulators [4], sensors [5], filters [6, 7], waveguides [8–11], splitters [12], resonators [13, 14], interferometers [15,16], and cavities [17], have been theoretically proposed and experimentally demonstrated.

For SPPs-based photonic devices, the insulator-metal-insulator (IMI) waveguide [11, 15, 18–26] and the metal-insulator-metal (MIM) waveguide [27–30] are two types of fundamental and basic structures. The IMI-type waveguide has less loss and longer propagation length, but suffers from its poor ability of confining light into subwavelength scale, and is not suitable for the purpose of high integration. The MIM-type waveguide has been shown to be efficient for subwavelength manipulation of light with an acceptable propagation length [31], thus has recently attracted more and more attention. Wang *et al*. reported a planar heterowaveguide constructed by alternately stacking two kinds of MIM waveguides [32], which can improve the efficiency of energy transfer at the range of subwavelength because it provides spatial light confinement with lateral dimensions of less than the free-space wavelength. However, their MIM waveguides have big ohmic losses by introducing a lossy material such as aluminum. To deal with this problem, low-loss MIM SPP Bragg reflectors have been realized based on periodically changing the dielectric materials between the MIM waveguide [33, 34]. S-shaped Bragg cells to suppress sidelobe of the MIM SPP reflectors are also discussed [35]. Quite recently, Liu *et al*. has investigated a MIM SPP wide bandgap reflector [36], because photonic element with a wide bandgap is very important in applications [37, 38]. This reflector with relatively wide bandgap is realized by modulating the MIM waveguide width and inserting the dielectric materials. However, since the contrasts of the waveguide width and the dielectric constant in different layers cannot be sufficiently large in their design, the contrast of the effective refractive index is actually not large enough. Thereby, the bandgap of their reflector can not be as wide as expected.

It is well know that chirp technique can increase the photonic bandwidth in Bragg gratings [20, 39]. In this paper, by means of a quasi-chirped technique, the Bragg-grating-based MIM surface plasmon polariton reflector (SPPR) with a broad bandgap is successfully obtained. To ensure reflectors with nice filtering characters, we should guarantee them to have near-zero transmission in the band and small sidelobe out the band [40, 41]. Therefore, the MIM SPP waveguide to suppress spectral sidelobe is also investigated by the apodization technique. This paper is organized as follows. In Section 2, the model of the MIM SPP waveguide is presented and its optical properties are studied. Section 3 describes the quasi-chirped technique to realize broad bandgap MIM SPPR. In section 4, the SPP waveguides with apodized and micro-cavity structure are both demonstrated. Finally, conclusion and discussion are made in section 5.

## 2. Fundamental properties of the MIM SPP waveguide

The MIM SPP waveguide consists of a dielectric core with a width w surrounded by two semi-infinite metallic claddings, as shown in Fig.1 (a). There are two possible plasmonic modes in the MIM waveguide, i.e., the anti-symmetric mode and the symmetric mode. The symmetric mode in MIM waveguide exhibits cut-off when the width decreases to a cut-off thickness which is about hundreds of nanometers. Since the gap width w in our structure is bellow the cut-off thickness, the anti-symmetric mode is considered here. When a light is injected to the MIM waveguide, it will be transformed into the coupled SPPs on the metal interface and propagate along the waveguide with a complex propagation constant *β _{spp}*. According to the Maxwell equations and boundary conditions, the dispersion relation of the SPPs can be obtained as:

where *k _{d}* and

*k*are transverse propagation constants of the dielectric and metal, respectively.

_{m}*k*=2π/λ is the propagation constant in vacuum,

_{0}*n*is the effective refractive index of SPP mode,

_{eff}*ε*and

_{d}*ε*are the dielectric constant of metal cladding and dielectric core, respectively. In the MIM waveguide, we assume that the metal to be silver and whose permittivity can be described by Drude formula [32]

_{m}where *w _{0}*,

*w*and

_{p}*γ*are the frequency of incident light, buck plasmon and electron collision, respectively. For sliver, the value of above parameters can be chosen as:

*w*=1.37×10

_{p}^{16}Hz, and

*γ*=0.15×10

^{13}Hz [42].

The light propagation along the MIM waveguide is modeled by the two-dimensional finite-difference-time-domain (FDTD) method [43]. In the MIM waveguide, the metal is dispersive and determined by the Drude formula, so electric filed *E* for FDTD update equation in the metal can be obtained as:

The magnetic filed *H* for FDTD update equation in the metal is the same as that in the dielectric and is determined by

In the FDTD simulation, the source is *S*=sin(*w _{0}t*), the gap width

*w*=20 nm, spatial and temporal cell size are set as Δ

*x*=Δ

*y*=5nm and Δ

*t*=Δ

*x*/(2

*c*), respectively, where

*c*is the velocity of light in vacuum, and

*t*is the time step. Meanwhile, the perfectly matched layer (PML) is used as absorbing boundary condition [43]. The calculated magnetic filed distribution |

*Hy*|

^{2}in the MIM structure is depicted in Fig. 1(b), as shown the light is highly localized along the two metal surfaces and propagating along the dielectric. In Fig. 1(c), the effective refractive index variations with gap width

*w*, wavelength

*λ*, and dielectric constant

*ε*are displayed. It can be obtained that the light propagation of the SPPs through the MIM waveguide shows little wavelength dispersion, and is mainly controlled by the gap width

_{d}*w*and the dielectric constant

*ε*. A smaller

_{d}*w*and a larger

*ε*tend to generate a larger effective refractive index. Therefore, by periodically changing the dielectric material and gap width, periodical effective refractive index can be formed. This personality provides us with an effective tool to design Bragg-grating-based MIM SPP devices in analogous to conventional Bragg grating to deal with classical optics.

_{d}## 3. Broad bandgap SPPR based on a quasi-chirped technique

In this section, a new structure to obtain broad bandgap MIM SPPR is proposed as shown in Fig. 2 (a), where grooves are periodically engraved on the two metal surfaces and different dielectrics are filled in the gap. In Fig. 2(b), the *n*-th period of the proposed structure containing a layer *A _{n}* and a layer

*B*is depicted. Here,

_{n}*LA*and

_{n}*LB*are the length of the layer

_{n}*A*and the layer

_{n}*B*,

_{n}*WA*and

_{n}*WB*are the gap width of the layer

_{n}*A*and the layer

_{n}*B*, respectively. In the Bragg gratings, the photonic bandwidth can be effectively increased by the chirp technique [20, 37]. Here, we applied the quasi-chirped technique to the MIM structure to enlarge the bandgap of the SPPR. In the proposed quasi-chirped MIM structure, there are

_{n}*k*segments. Each segment consists of periodically stacking the layer

*A*and the layer

_{n}*B*with constant length

_{n}*LA*and

_{n}*LB*, i.e., each segment is uniform. But in different segment, the length

_{n}*LA*and

_{n}*LB*are different and determined by

_{n}*Segment* 1, *LA _{n}*=

*λ*

_{B}_{1}/(4×

*n*

_{Aeff}_{1}),

*LB*=

_{n}*λ*

_{B}_{1}/(4×

*n*

_{Beff}_{1}),

*n*=1,2…,

*n*

_{1},

*Segment* 2, *LA _{n}*=

*λ*

_{B}_{2}/(4×

*n*

_{Aeff}_{2}),

*LB*=

_{n}*λ*

_{B}_{2}/(4×

*n*

_{Beff}_{2}),

*n*=1,2…,

*n*

_{2},

……

*Segment*
*k*, *LA _{n}*=

*λ*/(4×

_{Bk}*n*),

_{Aeffk}*LB*=

_{n}*λ*/(4×

_{Bk}*n*),

_{Beffk}*n*=1,2…,

*n*.

_{k}Here, *λ _{Bk}* is the desired central wavelength of the

*k*-th bandgap.

*n*and

_{Aeffk}*n*are the effective refractive index in the dielectric

_{Beffk}*A*and

*B*at the wavelength

*λ*

_{B}_{k}, and are obtained by Eqs. (1)–(3) and demonstrated in Fig. 2(c). The total numbers of the periods along the MIM structure are

*N*=

*n*+

_{1}*n*…+

_{2}*n*.

_{k}Before analyzing the quasi-chirped MIM SPP structure, uniform MIM SPPR (we label it as uniform-design-1) is presented. It contains one segment (S2) with 40 uniform periods. The used parameters are shown in Table 1. The wavelength *λ _{B}*

_{2}is 1550 nm, the effective refractive index

*n*and

_{Aeff}*n*are 1.8263 and 1.1715, thus the length

_{Beff}*LA*and

_{n}*LB*(

_{s}*n*=1,2…40) is 212 nm and 330 nm, respectively. The transmission spectrum of the uniform-design-1 is shown in Fig. 3, with a bandgap centering at

*λ*

_{B}_{2}(1550 nm) and ranging from about 1360 nm to 1790 nm. In above analysis, the transmission spectrum is achieved by the transfer matrix method [44] which is effective to study the spectral properties of the Bragg-grating-based MIM SPP waveguide [32, 33]. Uniform-design-1 has uniform periods and is the same as the structure proposed in [36]. In order to show the feasibility of the proposed quasi-chirped technique to widen the bandgap, two quasi-chirped MIM SPPRs are designed. The first quasi-chirped SPPR (quasi-chirped-design-2) contains two segments. The first segment includes 20 periods,

*λ*

_{B}_{1}is 1200 nm, so the length

*LA*and

_{n}*LB*in each period are 163 nm and 255 nm, respectively. The second segment also includes 20 periods, but

_{n}*λ*2 is 1550 nm, so the

_{B}*LA*and

_{n}*LB*are 212 nm and 330 nm, respectively. The transmission spectrum of this reflector is shown in Fig. (3), as shown that the bandgap is effectively widened from about 1050 nm to 1790 nm comparing with the situation of uniform-design-1. The quasi-chirped MIM SPPR with three segments (quasi-chirped-design-3) is also designed. The three segments are all containing 15 periods,

_{n}*LA*and

_{n}*LB*for each segment are 163 nm and 255 nm, 212 nm and 330 nm, 260 nm 405 nm, respectively. The transmission spectrum is also plotted in Fig. (3), from which it can be obtained that the reflector’s bandgap is further enlarged.

_{n}From the Fig. 3 and the Table 1, it can be clearly noted that the quasi-chirped technique is effective to achieve MIM SPPR with a broad bandgap. The reason for the quasi-chirped technique to widen the bandgap can be understood intuitively by the band structure [45]. In the uniform-design-1, the MIM SPPR contains one segment, and its band structure is shown with red solid line in Fig. 4, where the Bloch wave number versus angular frequency is exhibited. In the quasi-chirped-design-2, the MIM SPPR contains two segments (S1 and S2), whose corresponding bands are centering at *λ _{B}*

_{1}(1200 nm) and

*λ*(1550 nm), respectively. When the two segments are combing together in quasi-chirped-design-2, their bandgap will be incorporated as depicted in Fig. 4, therefore, the bandgap of quasi-chirped-design-2 is wider than that of uniform-design-1. Bearing this in mind, it is not strange that quasi-chirped-design-3 has a broader bandgap.

_{B}_{2}Notes: λ* _{Bk}* is central wavelength of the k-th segment,

*n*and

_{Aeffk}*n*are the effective refractive index in the dielectric A and B at the wavelength λ

_{Beffk}_{Bk},

*LA*and

_{n}*LB*are the length of layer

_{n}*A*and layer

_{n}*B*, respectively.

_{n}*n*is the numbers of periods in the

_{k}*k*-th segment, and

*N*is the total numbers of the periods in the MIM SPPR.

*S*1,

*S*2, and

*S*3 represent segment 1, segment 2, and segment 3, respectively.

## 4. Apodized MIM SPPR with small sidelobe

In this section, the apdization technique to suppress the sidelobe of the SPPR is investigated. The apodization profile is applied to the effective refractive index having a form of:

where *n* is the integer from -(*N*-1)/2 to (*N*-1)/2, *N* is the numbers of period, *n*
_{0}=*n _{A}*

_{0}+

*n*

_{B}_{0},

*f*(

*n*) is the apodization function with a Gaussian profile,

*n*

_{A}_{0}and

*n*

_{B}_{0}are the effective refractive index of SPP mode in the dielectric

*A*and

*B*at wavelength

*λ*, respectively. In our simulation,

_{B}*N*=21,

*λ*=1550 nm,

_{B}*n*=1.46 and

_{A}*n*=1.0, thus

_{B}*n*

_{A}_{0}=1.8263,

*n*

_{B}_{0}=1.1715,

*LA*=λ

_{n}*B*/(4

*n*

_{A}_{0})=212 nm, and

*LB*=λ

_{n}*B*/(4

*n*

_{B}_{0})=330 nm. The apodized effective refractive-index

*n*and

_{Aeff}*n*are shown in Fig. 5(a), from which the associated MIM SPP widths

_{Bff}*WA*and

_{n}*WB*are optimized and shown in Fig. 5(b). The transmission of the apodized SPPR is plotted in Fig. 5(c), as shown that that sidelobe of apodized reflector is effectively suppressed in contrast to the non-apodized reflector. In addition, the sidelobe can be further suppressed by decreasing

_{n}*F*in Eq. (10), whereas at the cost of slightly narrowing the bandgap.

In [33, 36], photonic nanocavities in uniform MIM structure are studied. Here, nanocavity in apodizd MIM structure is demonstrated. In the apodized MIM SPP waveguide, by introducing a defect at the center, i.e., by repeating the *B*11 layer in the 11-th period, the SPP nanocavity is formed. The transmission spectrum is depicted in the Fig. 6. It displays a sharp peak inside the bandgap and suggests the existence of the SPP defect mode, which can find important applications such as ultra-narrow threshold lasers and light-emitting diodes [46].

## 5. Discussion and conclusion

In conclusion, utilizing the FDTD method and the dispersion relation, we have numerically investigated Bragg-grating-based MIM SPP waveguide, and find that the propagation of the SPPs through the MIM waveguide is highly confined in the metal surfaces and is controlled by the gap width and dielectric. By introducing a finite array of periodic grooves on the two metal surfaces and applying the quasi-chirped technique to them, a broad bandgap SPPR is obtained successfully. The underlying physical mechanism for the quasi-chirped technique to enlarge photonic bandgap is explained by the transfer matrix method and the theory of band structure. The MIM SPP waveguide to suppress the spectral sidelobe of SPPR is also presented by optimally varying the gap width. At last, we demonstrate that by introducing a defect at the centre of the apodized MIM SPP waveguide, a SPP nanocavity can be formed. These designed reflectors have good filtering characteristics such as broadband, near-zero transmission in the band and small sidelobe out the band, and are expected to have important potential applications in the highly integrated SPP-based photonic networks.

## Acknowledgments

This work was supported by the “Hundreds of Talents Programs” of the Chinese Academy of Sciences and by the National Natural Science Foundation of China under Grant 10874239 and 10604066.

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