1. Introduction

High-index contrast dielectric waveguides have been in widespread use for many years as they offer a strong mode confinement, dense integration, low propagation loss (if smooth sidewalls are attainable), and the ease of fabrication through mature semiconductor technologies. However, further downscaling waveguiding structures for the realization of optical nanocircuitry is hampered by the natural diffraction limit and strong sub-wavelength field confinement thus becomes crucially important. Nanophotonics using surface plasmon polaritons (SPPs) that can be guided by metallic subwavelength structures beyond diffraction limit is now considered as a promising solution for the implementation of optical nanocircuitry [1, 2].

For either long-term developed dielectric- or emerging plasmonics-based photonic devices and circuits, the waveguide bend is one of the most important building blocks and has been attracted continued attention for the last 40 years [3–5]. Among various bend designs, air trenches etched into the substrate have been extensively investigated in high- and low-index-contrast material systems [6–13]. Corner mirrors formed by deeply-etched reflecting facets, on the other hand, are often found in rib/ridge waveguides [14–16]. Alternatively, low-Q resonant cavity [14, 17] and the use of a photonic crystal region as a 90° corner [18] have also been thoroughly studied. In contrast, plasmonics-based waveguide bends using a simple metal-dielectric-metal waveguide [19], metal micrograting as Bragg mirrors [20], V-groove at 30° in silver with a wedge pillar defect in the corner [21], dielectric-loaded SPP [22], and 3-D metallic slot waveguides [5] have also been reported in the literature.

Over the years, the research and developments in sharp waveguide bends have long been restricted exclusively to either dielectric- or plasmonic-based material systems. Because of their ultrasmall footprints, low-Q resonant cavity [17] and some corer mirror designs [14] may be most well-suited for high-density photonic integrations. However, they were investigated strictly for TE polarization and according to our studies [3-nm grid size and a slab mode excitation in two-dimensional (2-D) numerical simulations at λ₀ = 1550 nm], the respective TE and TM power transmissions (end-to-end) are < 77% and < 10% for the optimized structure given in [17]. As the high-index-contrast structures can be potentially low-loss, if properly fabricated, while plasmonic-based waveguides are inherently operational beyond the diffraction limit but are lossy in nature, it may be interesting and perhaps of real value to incorporate plasmonic-based structure into the conventional dielectric waveguides in implementing sharp bends for future photonic nanocircuitry.

In this paper, a new polarization-insensitive sharp waveguide bend in metal/multi-insulator configuration is analytically analyzed and numerically investigated through 2-D finite-difference-time-domain (FDTD) simulations. The introduction of the metal in Si-based sharp waveguide bends presents a radical departure from the conventional air trenches and resonant transmission effect reported in the literature. In addition, the insertion of low-index material (e.g. silica) in between the metal and high-index core sustains the low-loss propagation of transverse electric (TE) and transverse magnetic (TM) electromagnetic energy. The design, analysis, optimization, and physical insights behind them are elaborated in detail.

2. Structure Description

Figure 1 depicts the 2-D schematic of the proposed sharp waveguide bend in metal/multi-insulator configuration. It consists of a silica gap region sandwiched by a 100-nm-wide silver strip and a Si waveguide core region. The remaining areas are assumed to be air. The refractive indexes of silica, Si, and silver are 1.46, 3.5, and 0.1441 – j11.2141 at a free-space wavelength λ₀ = 1550 nm. The metal/multi-insulator configuration starts with (terminates at) the beginning (end) of the input (output) linearly-tapered section. The propagating (eigen)mode supported by the composite waveguide is asymmetrically excited by a Si waveguide of 450 nm in width followed by a linear input taper of length l_in. The Si core width along the linear sections $\overline{AB}$ and $\overline{CD}$ is 200 nm. This was chosen such that, in the absence of the silver region, the fundamental TM mode is below cutoff while its TE counterpart is above cutoff at λ₀ = 1550 nm.

 

Fig. 1 Schematic of the proposed sharp waveguide bend in metal/multi-insulator configuration, where δ_offset,in ≡ x₁ – x_in and δ_offset,out ≡ y₂ – y_out.

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The existence of plasmonic mode in 3-D metal/multi-insulator configuration with a thickness of 300 nm was first confirmed by 3-D Finite Element Method computations. Its excitation through the linear taper was further numerically demonstrated by 3-D FDTD method (FullWAVE^TM, RSOFT Design Group). Although being the most memory efficient numerical method, the 3-D FDTD simulation is still computational costly; thus the proposed sharp bend was investigated using 2-D FDTD with a minimum grid size of 3 nm in order for the convergence of numerical results.

The concept of incorporating a metallic material in Si-based sharp waveguide bends originates from different modal characteristics associated with TE and TM modes at the metal-dielectric boundary. The idea is to have the plasmonic mode carry a portion of the TM modal energy that is initially confined in the Si core and to confine the TE mode by means of a metal/low-index-dielectric boundary acting as a reflection plane. By investigating the structure parameters analytically and numerically, including input taper length l_in, lengths of linear sections l₁ and l₂, bending length l_bend, bending width w_bend, output taper length l_out, and the lateral offsets δ_offset,in and δ_offset,out associated with the respective input and output tapers, we show that a polarization-insensitive subwavelength sharp waveguide bend could be implemented with the present metal/multi-insulator configuration.

3. Dispersion relations of TE and TM eigenmodes

The TE and TM dispersion relations from which the effective mode indexes of guided modes ( $N_{\text{eff}}=\sqrt{{\varepsilon }_{\text{eff}}}$) are obtained are the most important equations in describing the guidance characteristics associated with the proposed waveguide. The 2-D schematic (top view) of a general metal/multi-layer structure is depicted in Fig. 2. Each layer is characterized by its thickness h_i and relative permittivity ɛ_i, i = {m, 1, 2,··· ,n}, where the subscript m stands for the metal. For the case presented in Fig. 1, ɛ₁, ɛ₂, and ɛ₃ correspond to the relative permittivities of silica, Si, and air, respectively (ɛ₁ = ɛ_silica, ɛ₂ = ɛ_Si, ɛ₃ = 1.0). For simplicity, both metal (ɛ_m) and the bottommost dielectric (ɛ_n) layers are assumed to be semi-infinite in extent.

 

Fig. 2 Schematic of a general metal/multi-insulator waveguiding structure.

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The general dispersion relation can be obtained conveniently based upon the transmission-line network approach incorporated with the transverse resonance condition

The input impedance seen looking downward from the reference plane can be calculated successively, starting from the bottommost, semi-infinite region, with the following general equation

Substituting Eqs. (2)–(4) into Eq. (1) and rearranging terms yield the dispersion relations associated with a general 2-D metal/multi-insulator waveguide. For TE polarization,

Equations (9) and (10) can then be numerically solved for the complex effective refractive index N_eff using the complex Newton’s method [23]. The effective mode index will be used in section 4 in developing the design rules for the proposed sharp bend.

4. Design and optimization

A systematic approach, starting with the input linearly-tapered section, was taken in the design and optimization of the proposed sharp bend. In this approach, the entire structure was divided into several building blocks, each of which was successively optimized. The design, optimization, and their physical reasonings are described below.

4.1. Input linearly-tapered section

The input taper is described by the taper length l_in and the lateral offset δ_offset,in defined as δ_offset,in = x₁ – x_in, which is the x-position difference between the center line of the linear section and that of the input waveguide (Fig. 1). Figure 3 shows the isometric plots of the transmission (coefficient) as a function of l_in and δ_offset,in for both mode types. For the TE mode, the silver-silica boundary effectively resembles a reflection plane with a minimal field penetration. Accordingly, TE behaviors along the linear and linearly-tapered sections would be very similar to that along an asymmetric, single-layer slab waveguide. Hence, as is seen in Fig. 3(a), for all l_in values under consideration, their respective transmissions are only slightly skewed toward negative δ_offset,in and are nearly identical at δ_offset,in = −10 nm, which is close to the zero-offset point. The highest TE transmission falls within δ_offset,in = [10, 35]nm for l_in ≤ 900 nm.

 

Fig. 3 Transmission coefficient of (a) TE and (b) TM modes with varying input taper length l_in and input lateral offset δ_offset,in ≡ x₁ – x_in (Fig. 1). The highest transmissions for TE and TM modes are 88.24% and 96.36% at (l_in,δ_offset,in) = (700, 30) and (l_in,δ_offset,in) = (900, 5) nm, respectively.

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A positive δ_offset,in is required for the TM mode to compensate for a shorter l_in in order to maintain > 90% TM transmission at the end of input taper. However, its functionality becomes minimum for l_in ≥ 800 nm for both mode types. Further investigations show that, for l_in > 700 nm, the TM power channeled in the silica gap varies merely within 2.79% over the span of δ_offset,in (not shown in the figure). This indicates that the phase matching condition is fulfilled and is hardly altered with varying l_in if l_in is sufficiently large. The parameters l_in and δ_in are thus chosen to be 700 nm and 45 nm, respectively, for maintaining transmissions of 88.18% and 96.09% for the respective TE and TM modes at a position 300 nm away from the taper end. It should be emphasized that, physically, δ_offset,in is intended to adjust the equi-phase front associated with the y-directed complex Poynting vector so that it is nearly in parallel to that along the liner metal/multi-insulator waveguide section for minimizing the mode mismatch. This can be also perceived in Fig. 4.

 

Fig. 4 Phase plots of the y-directed complex Poynting vector along the input taper followed by the linear section of length l₁ for (a) TE and (b) TM excitations.

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4.2. Linear metal/multi-insulator section

The functionality of linear metal/multi-insulator sections (l₁ and l₂) before and after the bending section is to ensure the equi-phase front associated with the y-directed complex Poynting vector P_y is phase-matched (“well-aligned”) across each boundary and is (nearly) perpendicular to the y axis, as shown in Fig. 4. The phase mismatch across each interface along the taper section is clearly seen for the TM mode owing to the phase mismatch associated with the x-directed electric field. On the contrary, the TE phase front is always continuous across the boundaries and is perpendicular to the y axis right after the input taper end. Compared to the TE case, the TM mode requires approximately another 225 nm for its phase front to become well-aligned. Thus both l₁ and l₂ were set to 364 nm, which is approximately one half the TM guided wavelength.

4.3. Waveguide bending section

With a fixed silica gap width, the geometry of the bending section is completely described by its length l_bend and width w_bend and may be fully characterized by the bending efficiency η defined as

The dependency of the bending efficiency on w_bend and l_bend is illustrated in Fig. 5. Simulations were conducted with varying bending length at a 10-nm step for differing bending widths. Curves shown in Fig. 5 were obtained through curve fitting of the discrete data. To provide an insight into this figure, it may be necessary to express the physical dimensions in terms of the guided wavelength. Using the dispersion relations, Eqs. (9) and (10), the effective refractive indexes (N_eff) of the fundamental mode for w_bend ranging from 290 nm to 350 nm and their corresponding guided wavelengths at λ₀ = 1550 nm were obtained (Table 1). Notice that with the increase in w_bend, the imaginary part of the effective index (Im [N_eff]) decreases and the real part (Re [N_eff]) increases as more electromagnetic energy is confined within the Si core region. These results are shown to be significant in understanding the transmission behaviors discussed below.

 

Fig. 5 Bending efficiency as a function of the bending length for differing bending widths from 290 nm to 350 nm for (a) TE and (b) TM modes. The E_z field component in the case of (w_bend, l_bend) = (350, 250) nm is shown as an inset in (a).

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Table 1. Effective refractive index N_eff and the corresponding guided wavelength λ_g at the operating wavelength of 1550 nm for differing bending widths w_bend. w_bend and λ_g are in units of nm

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For the TE mode, transmission valleys exist for w_bend ≥ 310nm owing to the resonance-like field behavior. The z-component of the electric field E_z is shown as an inset in Fig. 5(a) for (w_bend, l_bend) = (350,250) nm as a representative case. It exhibits a standing-wave-like pattern between the beginning of the input waveguide and the end of the bending section. However, as the forward- and backward-propagating fields do not form a perfect standing wave, the field amplitude is nonzero at those node-like positions, yielding a nonzero transmission through the bend. On the other hand, the bending efficiency is larger than 90% for l_bend ≥ 0.85λ_g,TE, if w_bend ≥ 310 nm, which is one-fifth of the operating wavelength.

For the TM mode, an allowable range of l_bend between which the bending efficiency is above 95% is observed. For a fixed l_bend value, the bending efficiency is progressively increased as the bending width increases. This may be physically interpreted from the effective length standpoint where, with a larger w_bend value, a longer effective bending length in terms of the guided wavelength (l_bend/λ_g) is seen by the guided mode. In addition, the highest bending efficiency associated with each case in Fig. 5(b) occurs when l_bend ≈ 0.85λ_g,TM. However, the bending efficiency stays above 90% for l_bend > 0.40λ_g,TM if w_bend is larger than 0.2λ₀. It is apparent that, when w_bend ≥ 330 nm, the bending efficiency is larger than unity for certain ranges of l_bend. Further investigations into the time-average Poynting vectors along the Si-air boundaries reveal that this is determined by the combined net power flow across the boundaries $\overline{AB}$, $\overline{BC}$, and $\overline{CD}$ [24]. In general, with the increase in w_bend, a net power density is coupled across the joint boundary $\overline{ABCD}$ into the Si core. The arguments also applies to those TE cases having larger-than-unity bending efficiencies shown in Fig. 5(a).

4.4. Output linearly-tapered section

The output linearly-tapered section is equally important as the input one whenever the plasmonic-to-photonic mode conversion is necessary in the photonic nanocircuitry. It is therefore described and optimized in a similar manner as is the input taper. The output lateral offset defined as δ_offset,out ≡ y₂ – y_out is, like the input lateral offset, meant to reshape the equi-phase front associated with the x-directed complex Poynting vector that would normally require a longer taper length l_out to “stabilize”. This is necessary since for both TE and TM modes the metal/multi-insulator structure is asymmetric with respect to the direction of propagation as opposed to the symmetric output Si waveguide. Thus the output taper serves as a transition for minimizing mode mismatch due to structure variations. The optimum values for l_out and δ_offset,in were found to be 660 nm and −85 nm, respectively.

5. Results and discussions

Figure 6 shows the time-average power flow for the optimized structure for TE and TM polarization. The normalized end-to-end transmission to that of a straight, 450-nm-wide Si waveguide of identical length is 98.07% for TE and 94.87% for TM, corresponding to the respective insertion losses of < 0.085 dB and < 0.229 dB introduced by the entire metal/multi-insulator structure. Also, it is worth mentioning that the insertion loss associated with the optimized bend design without the silver region is found to be > 0.65 dB for TE and > 1.61 dB for TM. This indicates that the metal/multi-insulator configuration can indeed significantly improve the transmission efficiency, in particular for the TM mode.

 

Fig. 6 Time average power flow of the optimized sharp waveguide bend for (a) TE and (b) TM modes with structure parameters w_gap = 80, l_in = 700, δ_offset,in = 45, l₁ = l₂ = 364, w_bend = 330, l_bend = 450, l_out = 660, and δ_offset,out = −85. All figures are in units of nm.

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The general design rules for the waveguide bend, in particular for the following five parameters {w_gap, l₁, w_bend, l_bend, l₂} were derived from the 1550-nm case and are given in Table 2. They were subsequently tested on the bend design at λ₀ = 1310 nm and produced immediate result of > 90% end-to-end transmission for both mode types without further optimization. Note that w_bend expressed in terms of the free-space wavelength uniquely determines l_bend through the calculation of TE effective mode index.

Table 2. General design rules for the proposed waveguide bend

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As is shown in Fig. 6(b), time-average power varies within the Si core and silica gap in the direction of propagation, suggesting power interchange exists between these two regions. The investigation of power interchange (or photonic-plasmonic mode conversion in the case presented) is therefore necessary in gaining physical insight into the waveguiding structures of this kind. However, before further investigations could be made on the sharp waveguide bend, it is helpful and perhaps equally important to first understand the mode conversions associated with its straight counterpart.

Figure 7 depicts the power interchange in a straight metal/multi-insulator waveguide with the total length and input/output tapers identical to those in the sharp bend. The lower case characters in the figures correspond to the structure discontinuities in the inset in Fig. 7(a). The computational domain in this study was set to 5.156 × 3.635 μm² and only the power along the direction of propagation in each region was considered. The propagation distance was measured along the Si-silica interface for the power in the Si core (P_Si) and along the silica-Ag interface for that in the silica gap (P_silica). For TM polarization [Fig. 7(b)], more than 56% of the input power is coupled to the silica gap at the input taper end (point b). No further significant photonic-plasmonic mode conversions are found to occur beyond point b. This indicates the propagating field solution of the eigenmode (which may also be treated as “supermode”) is nearly independent of the propagation coordinate y along the uniform, straight, “coupled” waveguide system. In this case, more than 54.74% of the input power is confined in the silica gap region while at least 36.61% is in the Si core. The combined power in silica and Si regions remains above 93% across the entire linear section and increases to > 96.0% at the end of the output taper (point d). For TE polarization [Fig. 7(a)], the amount of power penetrating into the silica region increases with decreasing input taper width and remains nearly unchanged before the start of the output taper (c – c′ cut) at which the combined modal and radiation power in the air region (P_air) is more than 12.30%. The increase in P_Si along the output taper results largely from the coupling of modal power within the 1/e penetration depth (P_1/e) and also from the radiation power (approximately 2% of the normalized power) in the air region. This is indicated by the monotonic decrease in P_air and P_1/e over the span of the output taper.

 

Fig. 7 Power interchange of a straight silver-silica-Si waveguide for (a) TE and (b) TM modes with input/output tapers identical to those used in the optimized waveguide bend.

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The coupling of TE radiation modes into the output taper may seem, at the first glance, to disagree with the conventional description of the radiation modes in that their ray paths would transmit through at least one core-cladding interface of an asymmetric waveguide with the propagation angles smaller than the critical angle for that interface [25]. However, when both critical angles for the two core-cladding interfaces are exceeded, the ray associated with the radiation mode can be totally internally reflected at both interfaces and the guidance condition is satisfied. In the present case, since the structure is asymmetric and the radiation modes are incident from the lower-index region, it can be shown that if the total internal reflection (TIR) condition is satisfied at the silica-Si interface, then the TIR condition at the Si-air interface will also be satisfied. To have the TIR take place at the silica-Si interface, the following condition must be satisfied for the angle ϕ between the normals to the two sides of the output taper

In the presence of the bending section, a forced change in the power transfer between P_Si and P_silica is found to occur at each waveguide discontinuity for TE and TM modes (Fig. 8). The propagation distance is defined in the same manner as that in Fig. 7. In the TE case, the P_Si curve remains almost the same as that in the straight case before point F [shown in the inset of Fig. 8(a)] and gradually increases until 70 nm after point G. Further, the 9.34% increase in P_Si from B–F to B′–G cuts and the corresponding 3.14% decrease in P_silica from F – F′ to G–G′ cuts indicate the coupling of “air” modes back to the Si core. Between B′ – G and C′–H cuts, the bending section is transversely uniform, consequently the variation in P_Si is negligibly small (1.87%). However, the largest difference in P_silica from G′ to H′ in Fig. 8(a) is 3.56%. Further investigations show that the increase in P_silica is mainly supplied by the radiation from the air region as the TIR at Si-silica interface is not attainable. Similar arguments are also applicable to sections beyond C′–H and H–H′ cuts as the structure is symmetric from A–E′ to D–J′ cuts with respect to the middle of the bending section.

 

Fig. 8 Power interchange of the optimized sharp waveguide bend for (a) TE and (b) TM modes.

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For the TM mode, mode conversions in the form of the power transfer between P_Si and P_silica are clearly seen in Fig. 8(b). Before the A–E′ cut, both P_Si and P_silica curves are nearly identical to those in the straight case. As the Si core width increases from B–F to B′–G cuts, the power inside the core also increases while that in the silica gap decreases. Compared to that along the linear taper, a relatively weaker power transfer is observed over the span of the bending section. This is due to a larger phase mismatch as a direct consequence of a larger core width. Likewise, the decrease in P_Si and the increase in P_silica from C′–H to C–I cuts stem from the decreasing Si core width. Hence the power interchange between P_Si and P_silica merely reflects the inevitable energy redistribution in order to fulfill the guidance condition for the TM mode in each section. As a result, the structure-enabled successive mode conversions in reality increase the modal power within the Si core and have the plasmonic mode carry up to 42.6% of the input power along the bending section.

6. Summary

A new polarization-insensitive sharp waveguide bend in asymmetric, 2-D metal/multi-insulator configuration has been detailed. The transmission through the bending section is assisted to a large extent by the plasmonic mode for the TM-polarized field and by the metal/low-index-dielectric boundary that resembles a reflection plane for the TE. These could only be achieved by introducing a low-index dielectric between the metal and high-index core regions. The dispersion relations governing the guided TE and TM modes in a general metal/multi-insulator waveguide have been developed and subsequently used in seeking the design rules for the waveguide bend in terms of the free-space/TE/TM guided wavelengths. The insertion loss of the proposed sharp bend is shown to be < 0.085 dB for TE and < 0.229 dB for TM. Power interchange study reveals that the combined TE modal and radiation power in the air region can couple back to the conventional Si waveguide through the output taper section, leading to a high end-to-end TE transmission. Photonic-plasmonic mode conversions are shown to occur for the TM mode at each structure discontinuity along the bend, giving rise to an increase in the photonic mode confinement while having the plasmonic mode carry the rest of the input power that would otherwise give up in the absence of the metal.