1. Introduction

Recently dielectric-loaded surface plasmon polariton waveguides (DLSPPW) constituted by dielectric ridges on top of metal film have attracted increased interests of researcher community in the field of nanophotonics and plasmonics [1–11]. DLSPPW show apparent advantages of relatively low bend and propagation loss, ease for large-scale fabrication with now existing lithography techniques and compatibility with different thermal-optical, electrical-optical and nonlinear Kerr materials for the development of active plasmonics components [3,6,7], and thus they are promising candidates to guide and manipulate surface plasmon polaritons (SPP), which are gradually recognized as a prospective optical information carrier in highly integrated photonic devices [8,12–14]. Several DLSPPW-based photonic fundamental components have been proposed and extensively studied, including waveguides and bends [1–8], Y-like beam splitters [3,8], directional couplers [3,4,8,11], Mach-Zehnder interferometers [6,8], ring resonators [9,10] and in-line Bragg gratings reflectors [8,10] for wavelength selection. However, multimode interference (MMI) devices, which are key components to offer flexible and reconfigurable functionality used for signal routing and coupling in integrated optical circuits and components [15,16], have not been explored so far.

In this paper, we numerically investigate the 1 × N MMI splitter based on DLSPPW. This study is organized as follows: In Sec. 2, the configuration and principle of MMI splitter used in this study are proposed. The functionalities of each part of the device as well as the MMI self-imaging theory are also briefly discussed in this section. In Sec. 3, the dependences of effective index, propagation length and transversal mode profiles of such SPP modes supported by DLSPPW on the geometrical parameters are discussed in detail, and the single mode condition is verified. In Sec. 4, the simulated near-field distributions and energy loss of the MMI splitter are given. Concluding remarks appear in Sec. 5.

2. Configuration and principle of MMI splitter based on DLSPPW

The schematic diagram of the proposed dielectric-loaded MMI structure is presented in Fig. 1 , where a polymethyl-methacrylate (PMMA) layer with predefined geometrical parameters can be fabricated on the top of a gold film using standard photolithography or electron-beam lithography technique. The incident wavelength is chosen at the telecommunication wavelength 1550 nm owing to the relatively low damping loss and long propagation length of SPP in this infrared spectral range. The structure is divided into three main regions and their functionalities are depicted in the following:

 

Fig. 1 Schematic diagram of the 1 × N MMI splitter based on dielectric-loaded surface plasmon polariton waveguides under investigation: t, PMMA layer thickness; W₁, PMMA width in the single-mode input and output waveguides; W and L, PMMA width and length in the multimode region; W_gap, distance between the adjacent output waveguides; W_taper and L_taper, hemline width and length of the coupling taper structure, respectively.

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Input region: SPP can be excited by a focused p-polarized Gaussian beam (wavelength 1550 nm) with normal incidence along the negative y direction on a subwavelength gold grating which is fabricated on the extended metal film. Practically, SPP can be excited by well-known Kretschmann method at the resonant angle, with a thin metal film (thickness ~50 nm) deposited on a glass substrate. Here we use a 200-nm thick gold film instead to reduce the numerical complexity by reducing the computer memory and simulation time, which would not significantly affect the physics involved in the MMI devices below. The generated SPP are then coupled into a single mode waveguide with taper structures. The coupling efficiency has a trade-off between adiabatic focusing and the absorption loss as the mode propagates along the taper [5,7]. Since the penetration depth of SPP in gold film at gold/air interface is only approximately 21 nm at 1550-nm incident wavelength, the gold film with thickness over 100 nm can be seen as infinite. In this case, the gold film thickness will not affect the coupling efficiency. However, for gold film thickness below 50 nm, it is noted that the propagation loss of SPP increases due to the leakage and low confinement of SPP from the PMMA ridge to the glass substrate, so the coupling efficiency will decrease accordingly. The thickness t and width W₁ of the input waveguide are both 600 nm to satisfy the single mode condition of DLSPPW which will be verified in Sec. 3.

MMI region: A wide PMMA waveguide is subsequently connected to the input single mode waveguide. Wider waveguide can sustain more high-order modes with different effective indices, and they interference with each other in this region which results in the formation of several images of the input single mode at certain distance away from the entrance of the multimode section. The imaging distance depends on the geometrical parameters (t and W) once the incident wavelength is fixed. It is well known from the MMI self-imaging theory [15] that the N-fold images can be obtained at distance$L_N=\genfrac{}{}{0.1ex}{}{p}{N}\left(\genfrac{}{}{0.1ex}{}{3L_{\pi }}{4}\right)$ for symmetrical structure, where $p\ge 0$ and $N\ge 1$ are integers having no common divisor, and $L_{\pi }=\lambda /[2(n_{eff0}-n_{eff1})]$ is the beat length of the two lowest-order modes, n_eff0 and n_eff1 are their effective indices, respectively.

Output region: If N output single-mode waveguides are placed at the N-fold imaging distance, the incident SPP mode is automatically divided into N parts. Here the taper structures are also used to reduce the scattering loss at the connection points and accordingly increase the collection efficiency of the divided modes. The distance between adjacent output waveguides W_gap is a key parameter which should be investigated carefully. If the distance is small enough, there is strong coupling between the adjacent waveguides. The energy will transfer form one waveguide to another, and vice versa. This phenomenon has been used to design wavelength-selective directional couplers of SPP [3,11]. Here we choose W_gap as 2.0 μm at which the tunneling and crosstalk between the adjacent output waveguides is extremely weak. The coupling length defined as the distance at which energy is transferred from one waveguide to another is approximately 300 μm, and is larger enough than the working distance of our proposed MMI splitter in the following.

3. Single mode condition and lateral multimode profiles

We first use a rigorous finite element method (FEM) to simulate the mode effective index, propagation length and electric field profiles in the cross section of the DLSPPW. The FEM technique is a full-vector implementation that can calculate both the propagating and leaky waveguide modes of an arbitrary structure on a non-uniform mesh [17]. The refractive indices of PMMA layer and gold are 1.5 and 0.55 + 11.5i, respectively [18]. The real part of effective index and propagation length of SPP modes propagating on a gold film covered by a PMMA ridge of different width W are calculated and plotted in Fig. 2 for a fixed ridge thickness t = 600 nm. It is seen that the real part of effective index augments and the DLSPPW support more modes as the width increases. The cut-off width of PMMA ridge of TM₀₁ and TM₀₂ modes are about 0.75 μm and 1.5 μm, respectively. Hence the width of the input and output waveguide W₁ is chosen as 0.6 μm throughout this paper to satisfy the single mode condition.

 

Fig. 2 The dependences of real part of effective index (a) and propagation length (b) of the dielectric-loaded surface plasmon polariton waveguides on the waveguide width. The thickness of PMMA layer is fixed at 600 nm and the incident wavelength is 1550 nm. The inset in the graph (a) shows the index difference of the two lowest-order TM₀₀ and TM₀₁ modes and (b) gives a sketch of the transversal geometry under investigation.

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Since the index difference between the TM₀₀ and TM₀₁ modes is a key parameter that determines the working distance of the MMI splitter mentioned above, we plot this index difference versus the width of the multimode waveguide in the inset of Fig. 2(a). It monotonously decreases from 0.302 to 0.0057 as the width increases from 1 μm to 10 μm. For W = 4 μm, n_eff0 = 1.43585 and n_eff1 = 1.40364, so the theoretical one-, two- and three-fold image distances are L₁ = 18.05 μm, L₂ = 9.03 μm, L₃ = 6.02 μm or 12.04 μm, respectively. For practical applications, we should also consider the propagation length of SPP modes which is determined by the imaginary part of the effective index in the multimode waveguide owing to the finite conductivity and intrinsic loss of metal (shown in Fig. 2(b)). When the width increases to a certain value (4 μm), the propagation length remains approximately constant (41 μm) for the first three modes. For very small ridge width, the TM₀₁ and TM₀₂ modes disappear at their corresponding cut-off width, while the effective index and propagation length of the fundamental mode TM₀₀ approaches that of the SPP at the infinite gold/air interface.

To clearly distinguish the modes with different orders, we investigate the lateral mode profiles of the ridge and calculate the normalized field distributions of magnitude of the major component of the electric field |E_y| for different ridge dimensions. Figure 3(a) shows the single mode case for W = 0.6 μm, and Figs. 3(b)-3(d) depict the two odd modes and one even mode for W = 4 μm, respectively. The white lines show the interfaces of PMMA ridge and gold film in Fig. 3. It is clearly shown that the field is strongly confined to the gold/PMMA interface and decays exponentially in the normal direction inside the ridge as we expected. The refractive index for SPP wave on the dielectric/metal interface is significantly higher than that on the outer air/metal interface, which makes SPP wave strongly guided in this type of DLSPPW similar to the optical mode confinement in conventional optical fiber and dielectric waveguide with high refractive index contrast. The field enhancement at the dielectric/metal interface can promote the nonlinear properties when the dielectrics are doped with active materials, and make DLSPPW-based active elements applicable [19].

 

Fig. 3 Normalized field distributions of the magnitude of the vertical component of the electric field |Ey|, calculated by FEM: (a) shows the fundamental TM₀₀ mode for W = 0.6 μm, (b) (c) (d) depict the two odd and one even modes TM₀₀, TM₀₁ and TM₀₂ for W = 4 μm, respectively. The white lines show the interfaces of PMMA ridge and gold film.

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4. Near-field electric field distributions of MMI splitter

Based on the above discussions, we then perform three-dimensional finite-difference time-domain (3D-FDTD) simulations to calculate the near-field electric field distributions and compare the results with the theoretical self-imaging property in the proposed MMI structures. Perfect matched layer boundary conditions are used at the three boundaries. All the simulation results of electric field intensity distributions are taken from the xz plane located at 300 nm above the gold film. Figure 4(a) gives an example of multimode interference pattern for 4-μm wide PMMA ridge. In addition to the one-fold image at distance 18 μm, multiple fold images are found as well, for example, two-fold images at distance 9 μm and three-fold images at distances 6 μm and 12 μm, respectively. The imaging distances of 3D-FDTD simulation results agree well with the theoretical predictions in Sec. 2.

 

Fig. 4 Electric field intensity distributions in the xz plane located at 300 nm above the gold film for (a) 4-μm wide straight waveguide, (b) 4-μm wide two-fold splitter and (c) 6-μm wide three-fold splitter with taper structures, respectively. The results are calculated by 3D-FDTD. The outlines of the PMMA layers are shown as solid white lines.

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1 × 2 beam splitter can be directly obtained if we put two output waveguides at the two-fold imaging distance L₂ = 9 μm, and the result is shown in Fig. 4(b). The solid white lines show the positions of gold/PMMA interface. Two 2-μm long taper structures with half taper angle 28 degree are used to increase the coupling efficiency from the divided SPP modes to the output waveguides. The weak standing wave pattern in the MMI region is caused by the mode reflection at the taper structure. One may intuitively think that 1 × 3 beam splitter can be achieved by putting three output waveguides at L₃ = 6 μm, but in this case the gap between the adjacent output waveguides is only 1.3 μm and the crosstalk between them will be introduced and play a negative role. However, we can achieve this goal by using a wider waveguide. For 6-μm wide ridge as an example, the effective indices of the first two modes are n_eff0 = 1.44144 and n_eff1 = 1.42625, the corresponding three-fold image distance is around 13.5 μm (shown in Fig. 4(c)), and the result is also in good agreement with the theoretical prediction. The functional size is in the range of several micrometers at telecommunication wavelength 1550 nm because of the relatively large index difference, and is much shorter than that of the traditional silicon- and metallic stripe-based MMI splitter of which the typical value is on the level of hundreds of micrometers even to several millimeters [15,16].

The electric field intensities along dashed lines AB in Fig. 4(b) and CD in Fig. 4(c) at distance 2 μm away form the taper structure in the x direction are shown in Fig. 5 . The mode profiles at every output arms are almost the same because of the single mode condition of the output waveguides. It is clear that the input mode energy is divided into two and three approximately equal parts. The slight non-uniformity of the output energy from three output waveguides may be due to the shorter optical path (and thus less attenuation) of the central divided modes. The discrepancy is less than 5% and is acceptable for practical applications. Moreover, the mode overlap between the adjacent waveguides is very weak, so the divided modes can independently propagate in the output waveguides.

 

Fig. 5 Electric field intensity distributions (a) along the dashed white line AB and (b) along the dashed white line CD in Fig. 4(b) and Fig. 4(c), respectively. The mode overlap between the adjacent waveguides is very weak.

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The overall energy loss, including the scattering loss at the connection points and the intrinsic damping loss of SPP, should be considered for possible applications in telecommunication networks and integrated photonics. This can be evaluated by the transmission efficiency defined as the ratio of the core intensity integrals at the end of output arms to that at the beginning of the input arms. For 4-μm and 6-μm wide PMMA ridges, the 1 × 2 and 1 × 3 splitting lengths are 9 μm and 12 μm, and the transmission efficiency is about 65% and 58%, respectively. Both the splitting length and transmission efficiency of MMI splitter are comparable to the Y-like splitter in [3]. However, MMI splitter proposed in this paper shows apparent advantages in comparison with Y-like splitter. First, the designs of Y-like splitter require fabrication accuracy around 300 nm [8]. The scattering loss would increase as the fabrication imperfection increases especially for S-bends. The fabrication precision required for other proposed splitters based on coupled DLSPP straight waveguides is even as high as tens of nanometers [8]. In the contrary, the fabrication precision required for MMI splitter here is much more feasible. Second, if we want to divide the input mode into N parts larger than 2, we have to utilize cascaded Y-like splitters. This leads to the simultaneous increase of radiative bend loss and propagation damping loss. However, MMI splitter is capable of achieving 1 × N splitter for arbitrary integer N at the same time and has no bend loss.

5. Conclusions

In summary, we have numerically studied 1 × N MMI splitter on the basis of dielectric-loaded SPP waveguides. The dependences of complex effective index and lateral mode profiles on the geometrical parameters are analyzed in detail. The transmission efficiency of the MMI splitter is as much as 60%, with their functional size being in the range of just several micrometers. Such MMI splitter is helpful for high density photonic integration and circuits on chip, and shows apparent advantages of low fabrication precision requirement and arbitrary parts splitting capability of SPP modes in comparison with Y-like DLSPPW-based splitter.