Efficient Optical Coupling between Dielectric Strip Waveguides and a Plasmonic Trench Waveguide

Abstract

Buttcoupling is the most efficient way to excite surface plasmon polariton (SPP) waves at dielectric/metal interfaces in order to realize applications in broadband and ultra-compact integrated circuits (IOCs). We propose a reasonable waveguide structure to efficiently excite and collect the SPP waves supported in a plasmonic trench waveguide in the long-haul telecommunication wavelength range. Our simulation results show that the coupling efficiency between the dielectric strip waveguides and a plasmonic trench waveguide can be optimized, which is dominated by the zigzag propagation path length in the dielectric strip loaded on the metal substrate. It is noted that nearly a 100% coupling efficiency can be achieved when the distance between the excitation source and the plasmonic waveguide is about 6.76 μm.

1. Introduction

At dielectric/metal interfaces, the electromagnetic (EM) waves can be bounded and thereby form surface plasmon polariton (SPP) waves when the operating frequency is lower than the plasma frequency of the metal [1]. During the propagation of SPP waves, the electrons in the top region of the metal collectively interact with the EM waves, which forms two exponential decay waveforms. The light waves are localized near the dielectric/metal interfaces and thereby allow for a smaller bending radius in the waveguide structures [2]. Various sub-wavelength plasmonic-waveguide-based components have been designed and/or proposed in order to realize broadband and ultra-compact integrated optical circuits (IOCs), such as polarization beam splitters [3,4], super prisms [5,6], Mach–Zehnder interferometers [7,8], and band pass filters [9,10]. However, the intrinsic ohmic loss of metals hinders developments in plasmonic-waveguide-based OICs.

Previously, various metallic waveguide structures were proposed to increase the propagation distance while keeping sub-wavelength modal sizes, such as V-groove, wedge, and trench-based plasmonic waveguides [11,12,13,14]. When the lateral size of the plasmonic waveguides is smaller than 200 nm, the propagation length is about 100 μm in the long-haul telecommunication wavelength range. According to the wave-interference mechanisms, two or three plasmonic components can be integrated within a propagation length of 100 μm. Therefore, it is necessary to connect plasmonic components with low-loss dielectric waveguides [15] and to increase the optical signals with dielectric-waveguide-based amplifier devices [16]. The optical coupling efficiency between a dielectric waveguide and a plasmonic waveguide has been numerically investigated in order to minimize the insertion loss owing to the modal-field and modal-index mismatches [17,18]. However, these proposed efficient couplers are not easy to use in low-loss plasmonic waveguides, which limits the development of plasmonic-based OICs.

Among these low-loss sub-wavelength plasmonic waveguides, the plasmonic trench waveguides have a relatively long propagation length in the long-haul telecommunication wavelength range. Therefore, it is worthwhile investigating the coupling efficiency between the input (output) dielectric waveguide and the plasmonic trench waveguide. To consider the fabrication of the input and output couplers, the two dielectric strips are partially loaded on the metal substrate in order to excite and collect the SPP waves supported in the plasmonic trench waveguide via the buttcoupling method [19,20], as shown in Figure 1. The fabrication methods are illustrated in Figure S1, which contain lithography, dry etching, thermal evaporation, oxidation, and sputter processes. The electron-beam lithography method allows for the fabrication of nano-scaled structures in the lateral directions. The spatial resolution of e-beam lithography is about 2 nm [21]. The dry etching, thermal evaporation, oxidation, and sputter methods allow for the fabrication of nano-scaled structure in the vertical direction. The etching, deposition, and oxidation rates can be precisely controlled at about 0.1 nm/s, which results in accurate fabrications in terms of depth, thickness, and height [22,23,24,25]. In addition, the proposed fabrication methods have been used to fabricate photonic devices at sub-wavelength scales [26]. In other words, it is possible to realize plasmonic-based OICs when the proposed couplers are used to excite and collect the SPP waves. In this paper, the main finding of our work is that there is an optimal distance between the excitation source and the entrance of the Au plasmonic trench waveguide.

2. Simulation and Methodology

The modal field distribution and transmittance of the waveguide structures were computed by using a commercial software (Rsoft FullWAVE) based on the finite-difference time-domain (FDTD) method. In the FDTD simulations, the size of the Yee cells was 20 nm × 20 nm × 20 nm, and the time step followed the Courant–Friedrichs–Levy condition in order to minimize the numerical dispersion. The size of the Yee cells was far smaller than the excitation wavelength in a free space. We used 1550 nm as the excitation wavelength, which is the central wavelength of the long-haul telecommunication wavelength range. At the computational boundaries, perfectly matched layers (10 layers) [27] were used to minimize the reflections from the outgoing EM waves in order to simulate an infinite space. In other words, the input and output dielectric strips extended infinitely in the –z and +z directions, respectively. A two-dimensional (2D) Gaussian excitation (orange plate) was placed in the input TiO₂ dielectric strip in order to match the modal profile of the rectangular dielectric waveguide, as shown in Figure 1c. The SPP waves in the plasmonic trench could be excited via the buttcoupling method [19,20] with the waveguide mode in the TiO₂ dielectric strip when the electric field was perpendicular to the inner walls of the Au trench. The input and output TiO₂ dielectric strips did not touch the plasmonic Au structure, with a gap distance of 50 nm. The small gap distance was required but was limited by the controllable linewidth in the electron beam lithography process. The TiO₂ dielectric strips were loaded on the Au substrate. W_d and H_d are the width and height of the TiO₂ strips, respectively. The refractive index of the dielectric strip was fixed at 2.8 in order to form a small modal size that could effectively excite the plasmonic waveguide at the sub-wavelength scale. The refractive index values of TiO₂ crystals ranged from 2.6 to 2.8 in the transparent wavelength range [28]. The refractive index and extinction coefficient values of Au were 0.524 and 10.742 at λ₀ = 1550 nm, respectively [29]. The electric field of the excitation was along the y-direction. W_m and H_m are the width and height of the plasmonic trench waveguide, respectively, which were both smaller than the excitation wavelength. To form a low-loss Au plasmonic trench waveguide, a small aspect ratio of the trench width and trench height was used [11]. W_m and H_m are 180 nm and 1000 nm, respectively. This predicted that the shorter trench height corresponded to the larger propagation loss owing to increased ohmic loss from the Au substrate. A time monitor (green plate) was placed in the output TiO₂ dielectric strip in order to compute the transmittance, which could be used to evaluate the coupling efficiency between the dielectric waveguide and plasmonic waveguide. In addition, the instantaneous field distribution along the propagation direction of the waveguides could be used to compute the modal index values by using the fast Fourier transform technique.

3. Results and Discussion

Figure 2a presentsthe cross-sectional electric field (E_y) distributions of the dielectric strip loaded on the Au substrate and the Au plasmonic trench waveguide at the wavelength of 1550 nm, respectively. The excitations are located at z = 0. In the dielectric waveguide, the TiO₂ strip is loaded on the Au substrate, which results in a zigzag propagation path under the 2D Gaussian excitation. The W_d and H_d of the TiO₂ strip are 300 nm and 1000 nm, respectively. The period of the zigzag path is about 5 μm. In other words, the optical waves propagate along the TiO₂ strip from the top (bottom) region to the bottom (top) region in the half period of a zigzag propagation. In the plasmonic trench waveguide, the electric field is mainly distributed in the top region of the trench, which forms a stable modal profile along the propagation direction, as indicated in the region of the dotted line rectangle of Figure 2a. Figure 2b plots the fast Fourier transform spectra of the E_y distributions in Figure 2a. The main peaks (λ_peak) are located at 1.058 × 10⁶/m and 0.794 × 10⁶/m, respectively, which can be used to compute the modal index values of the two waveguides by using the relation n = λ₀/λ_WG, where λ₀ is the wavelength in a free space, and λ_WG is the wavelength in the waveguide. The calculated modal index values of the TiO₂ strip waveguide loaded on the Au substrate and the Au plasmonic trench waveguide are 1.640 and 1.230, respectively. The modal index in the TiO₂ strip is higher than that in the Au plasmonic trench even though the W_d is larger than the W_m. The main reason is that the dielectric waveguide mode and SPP mode mainly propagate in the high-index medium (TiO₂) and low-index medium (air), respectively. It was predicted that the zigzag path in the TiO₂ strip significantly influences the coupling efficiency between the dielectric waveguide and the plasmonic trench waveguide.

Figure 3 presents the distance-dependent insertion loss (IL) of the proposed dielectric–metallic–dielectric waveguide structure. The distance (D) is from the excitation plate to the entrance of the plasmonic trench waveguide, as shown in Figure 1c. The IL is computed by using the relation IL = 10log₁₀(T), where T is transmittance. When the D values are 1.32 μm and 6.76 μm, the IL (T) values are 0.06 dB (101.4%) and 0.12 dB (102.6%). The difference between the two D values is 5.46 μm, which is close to the period of the zigzag propagation path in the TiO₂ strip. When the D value is 4.59 μm, the IL (T) value is −1.63 dB (68.8%). The highest IL (lowest transmittance) is owing to the largest shift between the two modal profiles in the TiO₂ strip and in the Au trench. In other words, the modal field overlap between the modal fields in the TiO₂ strip and the Au plasmonic trench largely influences the coupling efficiency. It is noted that the IL values can be higher than 0 dB because the propagation loss of the Au plasmonic trench waveguide is lower than that of the TiO₂ strip waveguide loaded on the Au substrate. The attenuation coefficient of the two waveguides can be computed by using the relation I(z) = I₀exp(−αz), where I₀ is the intensity at z = 0, and α is the absorption coefficient. In the waveguides, I(z) is proportional to |E_x(z)|². According to the |E_x|² distributions in Figure S2, the calculated α values of the TiO₂ strip waveguide loaded on the Au substrate and the Au plasmonic trench waveguide are 10.35 × 10⁴/m and 1.22 × 10⁴/m, respectively.

It is noted that the real part and imaginary part of the propagation constant (k_R and k_I) can be computed by using the relations k_R = 2π/λ_peak and k_I = α/2. In addition, the k_R/k_I can be defined as the figure of merit (FOM) of the waveguide. The propagation characteristics of the two waveguides are listed in

Table 1. Propagation characteristics of the waveguides at λ₀ = 1550 nm.

Propagation Characteristics	TiO2 Strip on Au Substrate	Au Trench
kR(×106/m)	6.648	4.706
kI(×104/m)	5.175	0.610
kR/kI	128.46	771.48

. The high FOM shows that the Au plasmonic trench waveguides can stand as the building blocks in the near-infrared OICs when the width and height are 180 nm and 1000 nm, respectively. The FOM of the Au plasmonic trench waveguide is about 771, which is higher than that of other plasmonic waveguides [30,31,32].

Figure 4 presents the IL of the proposed dielectric–metallic–dielectric waveguide structure with different W_d or H_d values while keeping the D value at 6.76 μm. When the W_d value increases from 200 nm (300 nm) to 300 nm (400 nm), the IL value changes from −1.36 dB (0.12 dB) to 0.12 dB (−1.13 dB). In Figure 4a, the curve is slightly left–right asymmetrical, centered at the W_d of 300 nm. When the W_d values are 400 nm and 200 nm, the IL values are −1.13 dB and −1.36 dB, respectively, which is due to the larger effective coupling area of the plasmonic waveguide [17,20]. The effect of the larger effective coupling also results in the anomalous light transmissions. Refs. [33,34] When the H_d value increases from 800 nm (1000 nm) to 1000 nm (1200 nm), the IL value changes from −3.05 dB (0.12 dB) to 0.12 dB (−0.87 dB). In Figure 4b, the curve has a strong left–right asymmetry centered at the H_d of 1000 nm. When the H_d value is 800 nm, the modal fields in the TiO₂ strip and the Au trench have a large shift in the vertical direction, thereby resulting in the low coupling efficiency. When the H_d value is 1200 nm, the modal fields in the TiO₂ strip and the Au trench can be better matched in the vertical direction, thereby resulting in the relatively higher coupling efficiency. When the D value is fixed at 6.76 μm, the optimal W_d and H_d values are 300 nm and 1000 nm, respectively, which is mainly due to the modal-field match between the TiO₂ strip and Au trench in the vertical direction. Conceptually, the period of the zigzag propagation path in the TiO₂ strip waveguide loaded on the Au substrate is related to the W_d and H_d. In other words, the optimal distance for the lowest IL is related to the W_d and H_d. Therefore, it is possible to increase the coupling efficiency after optimizing the D, W_d, and H_d values simultaneously. It is noted that the D, W_d, and H_d values must be re-optimized when the W_m or H_m is changed.

On the other hand, the modal field in the TiO₂ strip can be up-shifted by using a thin cladding layer on the TiO₂ strip in order to increase the coupling efficiency. However, the use of a cladding layer will change the period of the zigzag propagation path in the TiO₂ strip waveguide loaded on the Au substrate, which also influences the optimal distance for the lowest IL. In other words, the modal-field match in the vertical direction can be obtained after optimizing the thickness (refractive index) of the cladding layer and the optimal distance simultaneously.

4. Conclusions

In summary, we have theoretically investigated the coupling efficiency between TiO₂ strips on an Au substrate and an Au plasmonic trench waveguide at the wavelength of 1550 nm. When the width (height) of the dielectric strip is 300 nm (1000 nm), the optimal distance between the excitation source and the entrance to the plasmonic waveguide is 6.76 μm, which effectively reduces the coupling loss, thereby forming a nearly 100% coupling efficiency. It is noted that the optimal distance is highly related to the period of the zigzag propagation path in the input TiO₂ strip.