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SECTION I

Slot waveguides can strongly confine light to the low-index sub-wavelength scale region [1], [2], which can enhance light-matter interactions. Due to the selectiveness of covered low index materials, slot waveguides are used in several applications, such as optical modulators [3], [4], optical nonlinear devices [5], [6], [7], biosensing circuits [8], [9], [10], and athermal waveguides [11], [12]. To efficiently integrate these devices, transmission-efficient bent slot waveguides and crossing slot waveguides need to be developed. Anderson *et al.* improved bending efficiency using asymmetric slot-based structures, but did not consider losses due to transitions from a straight waveguide to a bent slot waveguide [13]. Ma *et al.* presented right-angle bends based on a corner mirror and different resonant cavities, but their numerical results were limited to two-dimensional (2-D) analysis [14]. Su *et al.* proposed a design method for transmission-efficient 2-D crossing slot waveguides [15]. In [15], by filling up slots around the crossing, the transmission efficiency was improved compared with that of direct crossings. The numerical results of Su *et al.* were obtained by 2-D simulations based on the finite-difference time-domain (FDTD) method, which assumes infinite core height. In our previous work, we revealed that when considering the core height of the crossing slot waveguide in [15], a finite height caused a large coupling loss between an air slot and a Si-filled region due to the mode field mismatch [16].

In this paper, we present a transmission-efficient bent silicon slot waveguide and crossing silicon slot waveguide. To evaluate the transmission efficiency, we use the three-dimensional (3-D) vector finite-element method (VFEM) for the waveguide discontinuity problem [16]. First, we investigate the transmission characteristics of a bent silicon slot waveguide that has straight input and output waveguides. To construct the bent waveguide, an asymmetric geometry is employed [13]. We introduce a taper to efficiently connect the bent region to the straight waveguides. Numerical results show that the transmission efficiency is increased by optimizing parameters of the taper geometry. When the bending radius is 1 $\mu\hbox{m}$, the transmission efficiency is greatly improved from 33.8% to 84.2%. Next, we investigate the transmission characteristics of a 3-D crossing slot waveguide that has a vertical coupler. A comparison between the proposed structure and the conventional crossing structure [16] shows that the maximum transmission efficiency increases from 78.2% to 95.9%.

SECTION II

Fig. 1(a) and (b) shows schematics of a conventional silicon slot waveguide bend and a silicon slot waveguide bend that has an asymmetric slot, respectively. Typically, the slot region is filled with certain materials, for example, electro-optic material [3], nonlinear material [5], aqueous solution [8], and materials for athermal waveguides [11], [12]. In our work, the silicon slot waveguide is embedded with benzocyclobutene (BCB), which can be used for athermal on-chip interconnects [11]. We assume that the refractive indexes of Si, $\hbox{SiO}_{2}$, and BCB are 3.47, 1.45, and 1.54, respectively. In the structure, the length of the straight input/output waveguide is 2 $\mu\hbox{m}$. We define the slot width, the waveguide height, and the Si-width as $w_{s}$, $h$, and $w_{h}$, respectively. The bending radius is defined as $R$. In addition, we define the taper length and the offset distance as $t$ and $\Delta x$, respectively.

To investigate the influence of the parameter selection, we evaluated the effective refractive index $n_{eff}$, bending loss, and the confinement factor when changing slot width and waveguide height. Fig. 2(a) and (b) shows the effective refractive index and the bending loss of the silicon slot waveguide as a function of $w_{h}$, respectively. We can see that the effective refractive index increases as $w_{h}$ becomes larger. From Fig. 2(a) and (b), we can see that the bending loss is related to the effective refractive index. Specifically, it has been noted that a high effective refractive index leads to reduction of the bending loss. Fig. 2(c) shows the confinement factor in the cover medium (slot region) when changing $w_{h}$. The confinement factor $\Gamma$ is evaluated using the following equation [17]: TeX Source $$\Gamma = {\mathop{\int\!\!\!\int}\limits_{\rm slot}{\rm Re}({\mmb E} \times {\mmb H}^{\ast}) \cdot i_{z}dxdy \over \mathop{\int\!\!\! \int}\limits_{\rm total}{\rm Re}({\mmb E} \times {\mmb H}^{\ast}) \cdot i_{z}dxdy}\eqno{\hbox{(1)}}$$ where ${\mmb E}$ and ${\mmb H}$ are the electric field and the magnetic field, respectively, and ∗ denotes the complex conjugate. If we choose parameters that result in a high effective refractive index, we can obtain a low bending loss. However, the effect of the slot waveguide, namely, the contribution of the strong light confinement in the slot region, becomes weaker if the confinement factor is small. In other words, there exists a trade-off between the bending loss and the effect of the light confinement in the slot region. Here, we set the slot width of 0.07 $\mu\hbox{m}$ and the waveguide height of 0.3 $\mu\hbox{m}$, which have been used for the conventional slot waveguides [15], [16]. Then, considering the trade-off relation, we set $w_{h}$ to 0.2 $\mu\hbox{m}\ (n_{eff} = 1.939)$ for the bent silicon slot waveguide. Indeed, several studies set the effective refractive index of less than 2.0 [12], [18].

To investigate the influence of the coupling loss between a bent region and straight waveguides, we evaluated the transmission characteristics of the bent silicon slot waveguide having straight input/output ports [Fig. 1(a)] and a uniform 90-degree bend without straight input/output ports, as shown in Fig. 3(b). In the 3-D simulation, the normalized transmission power $P$ is obtained by the mode overlap integral: TeX Source $$P = \left\vert{\mathop{\int\!\!\!\int}\left({\mmb E}_{out} \times {\mmb H}_{0}^{\ast}\right) \cdot i_{z}dzdy \over \mathop{\int\!\!\!\int} \left({\mmb E}_{in} \times {\mmb H}_{in}^{\ast}\right) \cdot i_{x}dxdy}\right\vert^{2},\eqno{\hbox{(2)}}$$ where ${\mmb E}_{in}$ and ${\mmb H}_{in}$ are the electric field and the magnetic field at the input port, respectively. ${\mmb E}_{out}$ and ${\mmb H}_{0}$ are the propagated electric field and the intrinsic magnetic field at the output port, respectively. On the other hand, the transmission efficiency of the uniform 90-degree bend [Fig. 3(a)] is evaluated using the cylindrical coordinate based 2-D VFEM [19], from which the bending loss $[\hbox{dB}/\mu\hbox{m}]$ is obtained. The transmission efficiency $P_{uniform}$ of the uniform 90-degree bend is calculated using the following equation: TeX Source $$10\log P_{uniform} = \left(Bending\ loss [\hbox{dB}/\mu\hbox{m}]\right){\pi R \over 2}.\eqno{\hbox{(3)}}$$ In Fig. 3(b), the difference of transmission efficiency at the same bending radius corresponds to the coupling loss between the bent region and the straight input and output waveguides. When $R = 5\ \mu\hbox{m}$ and $R = 2\ \mu\hbox{m}$, the coupling losses are 0.11 dB and 0.69 dB, respectively.

We consider the silicon slot waveguide bend with an asymmetric slot [Fig. 1(b)]. We introduce a taper to connect the asymmetric slot bend to the straight input/output waveguides. At the junction points, the slot geometry is not changed, while the outer geometry near the cladding is transformed because the light intensity in the cladding is weaker than that in the slot. Fig. 4(a) shows the transmission efficiency with respect to $\Delta x$ when $t = 0.5\ \mu\hbox{m}$. Results of the normalized transmission power at $\Delta x = 0\ \mu\hbox{m}$ correspond to the case of the conventional silicon slot waveguide bend. We can see that the transmission efficiency at each bending radius is improved by optimizing $\Delta x$. Compared with the conventional silicon slot waveguide bend $(\Delta x = 0)$, the transmission efficiency increases from 88.0% to 94.9% when $R = 3\ \mu \hbox{m}$ and from 97.1% to 98.7% when $R = 5\ \mu\hbox{m}$. When $R = 1\ \mu\hbox{m}$, the transmission efficiency is drastically improved from 33.8% to 84.2%. We can see that the improvement of the transmission efficiency becomes larger as the bending radius decreases. The offset distance that results in the maximum transmission efficiency also becomes larger as the bending radius decreases. These comparisons are summarized in Table 1. To optimize the taper geometry, we evaluated the normalized transmission power as functions of the taper length $t$ and the offset distance $\Delta x$, as shown in Fig. 4(b)–(d). In the case of $R = 1\ \mu\hbox{m}$ and $R = 2\ \mu\hbox{m}$, the transmission is improved as the taper length becomes shorter. On the other hand, when $R = 3\ \mu\hbox{m}$, the transmission is slightly improved as the taper length becomes shorter. In other words, the influence of the taper length is low when the bending radius is large. Compared with the right-angle slot waveguide bend whose maximum transmission was 94.3% [14] (state-of-the-art), our bent structure can achieve a high transmission when $R\ >\ 3\ \mu\hbox{m}$.

We estimate the propagation loss of our proposed bent structure. According to [20], the measured propagation loss of the silicon slot waveguide that has $w_{s} = 0.1\ \mu\hbox{m}$, $h = 0.22\ \mu\hbox{m}$, and $w_{h} = 0.27\ \mu\hbox{m}$ was 16.7 dB/cm. After conducting five cycles of the RCA clean, the propagation loss of the silicon slot waveguide was reduced from 16.7 dB/cm to 7.0 dB/cm for the quasi-TE mode [20]. The propagation loss for the mode is approximately given by the product of the confinement factor and the scattering attenuation coefficient in the sidewalls [20]. The confinement factor of the reported silicon slot waveguide is 0.4, as shown Fig. 2(c). Therefore, the propagation loss of our structure is estimated to be about 25% larger than the reported silicon slot waveguide because the confinement factor of our structure is 0.5.

SECTION III

A method for designing efficiently transmitting crossing silicon slot waveguides was proposed [15]. By filling up the slots around the cross, the transmission was improved compared with conventional direct crossings. However, their numerical results were obtained by 2-D simulations based on the FDTD method. Therefore, in our previous work, we investigated the transmission characteristics of a crossing silicon slot waveguide with finite core height and revealed that this crossing structure has a large coupling loss between the slot waveguide and the Si-filled segment [16].

To avoid the coupling loss due to the mode field mismatch, we propose in this article a 3-D crossing silicon slot waveguide with a vertical coupler, as shown in Fig. 5(a) and (b). The slot waveguide is embedded with BCB, and a Si wire waveguide, which is the vertical coupler, is bonded on the BCB surface. We assume that the refractive indexes of Si, $\hbox{SiO}_{2}$, and BCB are 3.47, 1.45, and 1.54, respectively. In the proposed structure, the slot width and height are $w_{s} = 0.07\ \mu\hbox{m}$ and $h_{s} = 0.3\ \mu\hbox{m}$, respectively [16]. To make the coupling length as short as possible, we set $w_{h}$ to 0.18 $\mu\hbox{m}$. The coupling length is determined by the overlap of the evanescent fields in two parallel waveguides. To obtain a large overlap of the evanescent fields, a structure that has a low effective refractive index should be selected. We define the width and height of the Si wire as $w$ and $h$, respectively. The distance between the Si wire waveguide and the slot waveguide is defined as $h_{g}$, and the length of the parallel region is $L$. Our proposed crossing silicon slot waveguide can be fabricated as follows. A Si under layer is first etched to form slot waveguides, and BCB can be spin-coated on the slot waveguides [11]. Next, a Si wire waveguide can be bonded on the BCB surface [21]. Another possible fabrication way is to deposit amorphous silicon. After embedding BCB, amorphous silicon can be deposited on the BCB surface [22]. And then, the amorphous silicon layer can be etched to form a Si wire waveguide. Fig. 6 shows the effective refractive index of the Si wire waveguide as a function of the width of the Si wire when $h = 0.2\ \mu\hbox{m}$. To achieve the phase matching between modes of the slot waveguide and the Si wire, we set the width of the Si wire to 0.36 $\mu\hbox{m}$ because the effective refractive index of the slot waveguide is 1.847 [Fig. 2(a)]. To determine a value of $L$, we conducted an eigen mode analysis for an $x - y$ plane in the coupling region [Fig. 5(b)]. Using the obtained effective refractive indexes of supermodes, we can calculate the coupling length $L_{c}$ as follows: TeX Source $$L_{c} = {\lambda \over 2(n_{eff, 0} - n_{eff, 1})}, \eqno{\hbox{(4)}}$$ where $n_{eff, 0}$ and $n_{eff, 1}$ are the effective refractive indexes of the fundamental supermode and the first order supermode, respectively, and $\lambda$ is the operating wavelength. From Eq. (4), $L_{c}$ becomes 6.25 $\mu \hbox{m}$ when $h_{g} = 0.35\ \mu\hbox{m}$.

First, we set $L$ to 6.25 $\mu\hbox{m}$ and $h_{g}$ to 0.35 $\mu\hbox{m}$. Fig. 7(a) shows the wavelength dependence of the normalized transmission and reflection powers when light is inputted from port A. We find that the transmission efficiency is over 79.5% in the whole C-band, whereas the normalized reflection is less than 0.5% in the whole C-band. Typically, the crosstalk in a 3-D crossing waveguide is smaller than the back reflection [23]. Our structure can also achieve this condition. The maximum transmission efficiency increases from 78.2% for the conventional crossing structure [16] to 95.9% for our proposed structure. Fig. 7(b) shows the wavelength dependence of the normalized transmission and reflection powers when light is inputted from port B. We find that the transmission efficiency is over 95.3% in the whole C-band. However, the transmission efficiency when light is inputted from port B is somewhat low, even though light is not coupled into the Si wire waveguide. To understand this result, we evaluated the transmission efficiency with respect to $h_{g}$. Fig. 8(a) shows the normalized transmission power as a function of $h_{g}$ when light is inputted from port B. From this figure, we can see that the transmission efficiency exceeds 99% if $h_{g}$ is over 0.5 $\mu\hbox{m}$. A possible explanation is that the scattering loss occurs due to a collision with the Si wire waveguide when $h_{g}$ is small. Fig. 8(b) shows the wavelength dependence of the normalized transmission power with respect to $h_{g}$ when light is inputted from port A. Here, the values of $L$, which is given by Eq. (4), are summarized in Table 2. When choosing a large $h_{g}$, the transmission bandwidth tends to become narrower. On the other hand, the transmission bandwidth tends to become wider when choosing a small $h_{g}$, while the maximal transmission decreases. We can conclude that there exists a trade-off between the transmission bandwidth and the highest transmission power when light is inputted from port A. Recently, a crossing slot waveguide using the slot-to-strip mode converter and the strip-multimode-waveguide crossing has been proposed [24] (state-of-the-art). The reported structure has a large bandwidth and a high transmission of 98%, but results were obtained by the 2-D simulations. In the 3-D structure of the multimode-interference-based crossing waveguide, transmission was 95.3% at the wavelength of 1.55 $\mu\hbox{m}$ [25]. Therefore, although the bandwidth is reduced, our crossing structure achieves a high transmission compared with the state-of-the-art [24].

SECTION IV

We presented a transmission-efficient bent silicon slot waveguide with asymmetric geometry at the bending slot region and investigated the transmission efficiency using the 3-D VFEM. We introduced a taper to efficiently connect the asymmetric bent region to straight waveguides. Numerical results showed that the transmission efficiency is increased by optimizing parameters of the taper geometry. We found that the improvement of the transmission efficiency becomes larger as the bending radius becomes smaller. We also proposed a 3-D crossing silicon slot waveguide that has a vertical coupler and investigated its transmission efficiency using the 3-D VFEM. Numerical results showed that our crossing structure can achieve a high transmission efficiency compared with conventional structures. The presented transmission-efficient bent and crossing silicon slot waveguides will be able to promote the integration of slot waveguide-based circuits.

Corresponding author: Y. Ishizaka (e-mail: ishizaka@icp.ist.hokudai.ac.jp).

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