## Abstract

Waveguide superlattices with a subwavelength pitch and low crosstalk can significantly increase the waveguide integration density and are beneficial for many chip-scale applications. Bending of such high-density waveguide superlattices is necessary for flexible signal routing. However, tight bending tends to induce high crosstalk between guided modes, as witnessed in multimode waveguide bends. Here we explore the mechanisms of light guiding and coupling in a subwavelength-pitch waveguide superlattice bend and analyze how bending further modifies the already “renormalized” parameters of superlattice modes via various physical effects. Particularly, bending can skew the phase mismatch in a waveguide superlattice, sometimes producing a near phase-matching condition and causing salient crosstalk spikes among non-first-nearest neighbors. Interestingly, a waveguide superlattice with less pristine phase mismatch may be more robust against such skew of phase mismatch and can suppress crosstalk spikes by $\sim 10\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$. Bending with 5–15 μm radii and subwavelength pitches has been demonstrated with crosstalk lower than $-19.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$. The scaling of the footprint of waveguide superlattice bending is analyzed, and significant footprint reduction can be achieved for chip-scale applications. The rationale for footprint reduction of superlattice bending under crosstalk constraint differs markedly from that of a single waveguide bend.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Optical interconnects are invaluable in today’s high-performance computers and data centers, meeting bandwidth demands and enabling exponential data growth. Space-division multiplexing (SDM) in optical fibers is expected to be the next step in pushing the capacity limits of optical interconnects [1,2], and recent experiments have proven the feasibility of this technology in both fiber optics and photonic integrated circuits [3]. In the future, optical interconnects have been envisioned to be deployed on the chip scale [4–6]. On-chip SDM via many parallel waveguide channels may provide a useful solution to achieving the desired large bandwidths, but area usage may be a major concern for ordinary SDM approaches, as illustrated in the example of the contemplated 100-core future chip interconnected by tens of thousands of waveguides [7,8]. Shrinking the area was fundamentally limited by the waveguide density and inter-waveguide crosstalk. To overcome this limitation, waveguide superlattices (WGSLs) with a subwavelength pitch and low crosstalk can be introduced to significantly increase the waveguide density and reduce the area [9]. Densely packed waveguides have also been studied with atomic physics insight [10]. Nanophotonic cloaking structures enabled by inverse design [11] and periodic silicon nano-strip arrays between neighboring waveguides [12] reveal new possibilities for increasing the waveguide density and reducing area usage. Joint SDM and polarization-division multiplexing have also been achieved in densely packed waveguides [13]. Skin-depth engineering has been explored for dense waveguide integration [14].

Waveguide bends are essential in many applications to provide flexible routing of optical signals. Bending of high-density WGSLs may create excess crosstalk due to light leakage at bends. A trivial idea would be to temporarily increase the inter-waveguide pitch to 3–4 μm in the bending region to minimize excess crosstalk. However, with on-chip estate being increasingly precious, there should be every effort to decrease the footprint of each component in the optical interconnect system. Alternative approaches must be sought. It should be noted that crosstalk due to strong mode-mixing/coupling in tight bends of multimode waveguides has been a concern for multimode waveguide interconnects. Recently, reduced crosstalk through a multimode waveguide bend has been demonstrated for a 78.8 μm radius, by utilizing transformation optics-based design and advanced three-dimensional lithography [15]. In our SDM approach here, we delve into the original design principles of straight WGSLs [9] and examine how the curved structure alters these principles to find ways to achieve low crosstalk for a WGSL bend.

## 2. LIGHT GUIDING IN A WAVEGUIDE SUPERLATTICE BEND

In a WGSL, each supercell consists of a sub-array of waveguides of different propagation constants. For a straight WGSL, the principles given in Ref. [9] indicate that crosstalk can be reduced significantly by combinatorial control of inter-waveguide phase mismatch over inter-waveguide coupling strength across all the channels. To achieve low crosstalk, the elements of the effective coupling matrix shall satisfy

which is essentially a relation between the phase mismatch (left-hand side) and the coupling constants (right-hand side). Here the matrix is defined through $[K]={[B]}^{-1}[\mathrm{\Delta}A]+[\beta ]$, where $[\mathrm{\Delta}A]$ describes the change of the dielectric function due to the presence of surrounding waveguides, $[B]$ is a metric matrix, and $[\beta ]$ is a diagonal matrix containing the propagation constants ${\beta}_{n}$ of each waveguide mode. Note that this theory is not based on the approximation that the elements of $[\mathrm{\Delta}A]$ are small. Considering the coupling terms among all waveguide channels, we can expect to achieve low crosstalk if the ratio of the two sides of Eq. (1) is $>10$. For waveguide pitches not too small (e.g., $>0.7\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$), the diagonal term of $[K]$ is reasonably close to the propagation constant of the waveguide mode, and the off-diagonal term is reasonably close to the coupling constant of between two waveguides. In a dense WGSL, the elements of $[K]$ include contributions from overlap coupling integrals from many other modes. In some sense, Eq. (1) can be regarded as requiring a “renormalized” phase mismatch between any two waveguides (left-hand side) be substantially larger than “renormalized” coupling—a certain summation of collective coupling effects (right-hand side). For two waveguides, Eq. (1) is reduced to ${\beta}_{1}-{\beta}_{2}\gg 2\kappa $, consistent with directional coupler theory [16]. In Eq. (1), the appearance of phase mismatch (as ${K}_{mm}-{K}_{nn}$) is easy to grasp, and the “renormalization” of the coupling constant is mathematically more complicated and tends to be overlooked. However, it should be emphasized that the phase mismatch and coupling constant play equally important intertwined roles in determining the crosstalk.A WGSL bend can be locally approximated by small straight segments; therefore the essential ideas of Eq. (1) can be useful for studying light guiding and coupling in a WGSL bend and assessing its crosstalk. However, several modifications must be made to the “renormalized” propagation constant ${K}_{nn}$ and “renormalized” coupling constants ${K}_{mk}$. First, bending can change the effective index (${n}_{\mathrm{eff}}$) of a mode, thereby affecting ${K}_{nn}$. Our calculation indicates that for an isolated waveguide, the change of ${n}_{\mathrm{eff}}$ is small ($<0.002$) for typical waveguide widths and for $R>3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$. It turns out that the change of ${n}_{\mathrm{eff}}$ remains small ($<1\%$ for silicon waveguides with $R>3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$) for a waveguide embedded in a WGSL bend. Such change of ${n}_{\mathrm{eff}}$ (hence ${K}_{nn}$) is negligible compared to the ${n}_{\mathrm{eff}}$ difference between different waveguide modes in a supercell.

Second, light can leak out appreciably when the bending radius is small. This may cause additional coupling, thereby affecting ${K}_{mk}$. To understand this effect, we start by considering the fields in each homogeneous region (along one of the waveguide cores, or in one cladding region). In each region, the dielectric function is a constant; therefore one can readily show that the Helmholtz equation ${\nabla}^{2}{H}_{i}+\epsilon {k}_{0}^{2}{H}_{i}=0$ $(i=x,y,z)$ holds for the magnetic field in each homogeneous region. Here ${k}_{0}$ is the free-space wavevector. For convenience, we consider the ${H}_{z}$ component, where $z$ is normal in the plane of the waveguide array (similar results can be obtained for the in-plane $H$ components). The effective dielectric constant approach can be applied [17]. Let ${H}_{z}(\rho ,z)={\rho}^{-1/2}u(\rho )$; one can readily obtain

It turns out that a potentially significant effect of bending may stem from the varying angular rate of the change of phase at different radii. The angular dependence of a mode field for a waveguide bend is given by $\mathrm{exp}(i\beta R\varphi )$, where $\varphi $ is the angle of the cylindrical coordinates, $\beta $ is the propagation constant, and $R$ is the bending radius. The crosstalk between two waveguides in a bend depends on a certain overlap integral $I$ between the mode fields, which contains phase term $\mathrm{exp}(i{\beta}_{n}{R}_{n}\varphi )\mathrm{exp}(-i{\beta}_{m}{R}_{m}\varphi )$:

*linear*phase mismatch (unit: ${\mathrm{\mu m}}^{-1}$)

## 3. DESIGN AND SIMULATION

Based on the analysis in the previous section, we design and simulate waveguide superlattice bending with subwavelength pitches. The WGSL unit cell consists of five waveguides (supercell-5 or SC5) placed in an interlacing-recombination configuration [9], with corresponding widths of 450, 390, 330, 420, and 360 nm and a thickness of 260 nm. As shown in the prior study [9], for a well-designed WGSL that is sufficiently large (such as a SC5), appreciable intra-supercell crosstalk may sometimes occur between the second-nearest neighbors and possibly farther ones. However, the inter-supercell crosstalk, if any, generally occurs only between the nearest supercells. Hence the study of two supercells as illustrated in Fig. 3(a) is usually sufficient. We simulate the bending performance for two different inter-waveguide pitches of 0.78 and 1.0 μm forming a U-bend.

Typically, in a well-designed straight WGSL, high crosstalk occurs in the first-nearest neighbors, but at certain bending conditions when the radii cause a large skew of phase mismatch, high crosstalk could potentially occur in farther neighbors. For example, simulation results of a WGSL bend with an inter-waveguide pitch of 0.78 μm and minimum bending radius of 15 μm are shown in Fig. 3(b). We use three-dimensional finite difference time domain (3D-FDTD) methods to obtain the spectral transmission of the WGSL from 1500 to 1580 nm and determine the maximum possible relative crosstalk through the bend. The transmission loss of a WGSL bend is less than 0.1 dB for minimum bending radius $>5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$. One can readily see that the crosstalk between some second-nearest neighbors, such as CT(5,3), has comparable value to that of the first-nearest neighbors. Estimation based on Eq. (6) indicates that the “renormalized” phase mismatch ${({K}_{55}-{K}_{33})}_{\mathrm{bend}}$ is very close to zero. As such, the crosstalk CT(5,3) can become quite sensitive to small variation of the structures, which tends to occur during fabrication. Such uncertainty can cause the second-nearest neighbor crosstalk to escalate very high, potentially above that of the first-nearest neighbors in experiments.

Interestingly, this problem can be solved if the minimum difference of waveguide widths is reduced from $\mathrm{\Delta}w=30\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$ to 25 nm (new widths corresponding to 450, 400, 350, 425, and 375 nm). Although a small decrease of $\mathrm{\Delta}w$ will slightly reduce the phase mismatch, there are significant positive effects. First, it can help to reduce the inter-waveguide coupling constants. As we fix ${w}_{1}=450\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$ to stay within the single-mode regime, the widths of all other waveguides are increased (e.g., 330 nm increased to 350 nm) when $\mathrm{\Delta}w$ is reduced to 25 nm. For the narrow waveguides, this helps to better confine the mode and reduce the coupling strength with other waveguides. Second, the renormalized phase mismatch ${({K}_{55}-{K}_{33})}_{\mathrm{bend}}$ can now be shifted far away from zero. As shown in Fig. 3(d), the crosstalk CT(5,3) is reduced by about 10 dB, which provides enough margin for fabrication-induced structure variation. Note that our prior work indicated that small variation of the width ($<5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$) in a well-designed straight WGSL results in merely a slight increase of crosstalk (typically 1–2 dB) [9], which is far less than the benefit of the 10 dB reduction of second-nearest neighbor crosstalk in bending observed here.

## 4. EXPERIMENTAL RESULTS

#### A. Crosstalk Analysis

WGSL bend structures have been fabricated on a silicon-on-insulator wafer and characterized by a waveguide coupling and measurement setup. The details of fabrication procedures and loss characterization can be found in Supplement 1. We illustrate experimental data statistically for each input–output channel combination of a fabricated WGSL bend with an inter-waveguide pitch of 0.78 μm and minimum bending radius of 15 μm in Figs. 4(a) and 4(b). At each wavelength, we show the averages and standard deviations (statistics over index $m$) for the direct transmission $T(m,m)$, the first-nearest neighbor crosstalk $T(m,m\pm 1)$, and the second-nearest neighbor crosstalk $T(m,m\pm 2)$. In addition, Figs. 4(a) and 4(b) show the worst crosstalk (thick black curve) for each particular wavelength $T{(m,n)}_{\text{worst}}$ and summarize the maximum overall relative crosstalk for key waveguide channels. The optimized design in Fig. 4(b) shows less loss, which is due to lower coupling loss to neighboring waveguides. Note that as the crosstalk is reduced, the coupling loss is also reduced. The relative crosstalk is determined by the difference of their transmissions to the direct transmission of the input channel, $\mathrm{CT}(m,n)=T(m,n)-T(m,m$), as shown in Figs. 4(c) and 4(d).

The effectiveness of changing $\mathrm{\Delta}w=30\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$ to 25 nm can be seen in Fig. 4(e). As expected, the crosstalk to some second-nearest neighbors such as CT(5,3) exhibits salient spikes for $\mathrm{\Delta}w=30\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$, and it decreases significantly as $\mathrm{\Delta}w$ changes to 25 nm, with a small increase in crosstalk to first-nearest neighbors. The crosstalk to third-nearest neighbors $T(m,m\pm 3)$ is very low and approaches the noise floor of our measurement setup, and the crosstalk of even farther neighbors is sheer noise. Some high-crosstalk second-nearest neighbors, especially CT(5,3), have their crosstalk suppressed by $\sim 10\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$, as predicted by theory and simulation. Overall, the modified WGSL design pulls down the maximum relative crosstalk from $-16.0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$ in the original design to $-19.9\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$ in the reduced $\mathrm{\Delta}w$ design, which is a significant improvement for many applications.

The maximum relative crosstalk for each WGSL bend configuration is summarized in Fig. 5(a) along with a scanning electron microscope (SEM) image of a fabricated structure. Evidently, the maximum relative crosstalk decreases with increasing bending radius. In general, a maximum relative crosstalk of $-20\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$ is sufficient for most applications. Based on this value, the minimum bending radius for a 0.78 μm pitch is around 15 μm. Many applications can have relaxed pitch requirements, e.g., allowing for a 1 μm pitch. Under such conditions, Fig. 5(a) shows that we can achieve a maximum relative crosstalk as low as $-26.4\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$ at a 10 μm minimum bending radius (more details in Fig. S3 of Supplement 1). Furthermore, for 1 μm pitch, we have demonstrated good performance for bending radii at 5 μm, where we achieved respective maximum relative crosstalk of $-19.6\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$ (more details in Fig. S4 of Supplement 1).

#### B. Chip Area Considerations and Discussions

For one single-mode waveguide bend, the bending radius is the only parameter. For a waveguide array, however, the bending footprint depends on both the pitch and the radius. Hence the rationale for footprint minimization for a waveguide superlattice bend can differ markedly from that of a single waveguide bend.

For straight waveguide arrays, the footprint scales with inter-waveguide pitch linearly. In bends, however, both the inter-waveguide pitch and the minimum bending radius contribute to the footprint with nonlinear scaling. The footprint of a bend can be estimated as

for any array bending at $\theta =(0,\pi ]$ radians with ($N+1$) waveguides of inter-waveguide pitch $a$ and minimum bending radius ${R}_{\mathrm{min}}$. Interestingly, pitch contributes both quadratic and linear terms to the footprint, while the minimum bending radius contributes a linear term. The footprint of an 11-channel array of various pitches and bending radii is depicted in Fig. 5(b). Due to the quadratic dependence on the pitch, in the regime of large pitches, primary space savings come from minimizing the pitch, while secondary space savings come from minimizing the bending radius. For example, suppose we take an 11-channel waveguide array with a minimum bending radius of 15 μm and compare the overall footprint reduction of a 0.78 μm pitch WGSL to a 3.0 μm pitch uniform waveguide array. The area reduction is more than six times as shown in the first row of Table 1. If the pitch is already fairly small, the area reduction due to minimizing the bending radius can also be important. For example, we consider a 1.0 μm pitch WGSL, which allows the minimum bending radius to reach $\sim 5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ and results in a smaller area than a 0.78 μm pitch WGSL with ${R}_{\mathrm{min}}=15\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$, as shown in Table 1.Based on these trends, it can be easily observed that a WGSL can provide significant savings of on-chip estate in bends. The several cases presented here illustrate a variety of options to be picked from based on system-level considerations in specific applications. Note that for a large array ($N>11$), the outer $N$-11 generally have much larger bending radii $R>{R}_{\mathrm{min}}+10a$, and hence much lower crosstalk than inner waveguides. Thus, adding more waveguides at the outer side of a superlattice bend can be readily done to scale up the array size.

Note that sinusoidal anti-coupling in alternating bends has also been explored to reduce crosstalk [18]. However, in most applications, the waveguide paths comprise long straight waveguide arrays plus only a few bends. Approximating long straight waveguides with many alternating mini-bends brings up the concern of accumulated bending loss over many bends. Also note that multimode strip waveguide bends designed via particle swarm optimization [19] and through Bezier curving [20] have also been demonstrated most recently for waveguides containing three modes. It would be interesting to explore multimode waveguide bends containing a large of number of modes and further evaluate their potential. Note that a multimode waveguide with many modes will have a large width, and the total bending area will depend on both the radius and the width in a way similar to Eq. (7). To meet the demand of high bandwidth density with limited on-chip estate, wavelength-division multiplexing (WDM) is also a useful option. Our experience with compact silicon photonic devices for WDM [21–23] indicates that the temperature sensitivity, waveguide channel nonlinearity, and complexity of WDM approaches may limit the use of WDM on chip. The combination of WDM and SDM can provide a promising solution. With a wide bandwidth, the SDM approach based on waveguide arrays is fully compatible with the WDM approach.

## 5. CONCLUSION

In conclusion, we have explored the mechanisms of guiding light through a waveguide superlattice bend and demonstrated that WGSLs can be bent with low crosstalk. Our analysis classifies various effects and identifies the dominant effect in a curved WGSL segment that modifies the crosstalk characteristics from a straight segment. Interestingly, our analysis shows that the leakage-induced crosstalk is a higher-order effect. The varying angular rate of change of phase at different radii can skew the propagation constant (phase) to a crucial level and cause salient crosstalk spikes. In the half-wavelength pitch regime, we have shown that the previous WGSL design can reach a minimum bending radius ${R}_{\mathrm{min}}=15\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ with a maximum crosstalk of $-16.0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$, limited by a crosstalk spike between certain second-nearest neighbors due to bending-induced phase skew. Interestingly, reducing the width difference $\mathrm{\Delta}w$ in the WGSL can help to overcome the problem in some cases. By changing $\mathrm{\Delta}w$ from 30 nm to 25 nm, we can suppress the worst second-nearest neighbor crosstalk by $\sim 10\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$, and push down the maximum crosstalk to $-19.9\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$ for $a=0.78\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ and ${R}_{\mathrm{min}}=15\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$. For applications that allow for 1 μm waveguide pitches, we can substantially reduce the minimum bending radius to 5 μm with $-19.6\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$ crosstalk. For a large array of waveguides, the scaling of the bending footprint depends on both the pitch and the minimum radius, along with the crosstalk constraint. Hence, the footprint reduction rationale here will differ markedly from that of a single waveguide bend, for which the goal is simply to minimize the radius. Both cases of WGSL bends ($a=0.78\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$, ${R}_{\mathrm{min}}=15\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ and $a=1.0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$, ${R}_{\mathrm{min}}=5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$) have proven that they can achieve low crosstalk with significant reduction of occupied on-chip estate, thus potentially enabling high-density, flexible signal routing for a multitude of chip-scale applications.

## Funding

Defense Advanced Research Projects Agency (DARPA) (N66001-12-1-4246); U.S. DOE Office of Science Facility (DE-SC0012704).

See Supplement 1 for supporting content.

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