## Abstract

A vertical multimode waveguide enabling micro-bends is proposed for mode-multiplexed optical interconnect links. The multimode waveguide is designed to be singlemode in the lateral direction and support higher-order modes in the vertical direction. The characteristic analysis for an SOI (silicon-on-insulator)-based vertical multimode waveguide with a ~0.3μm × ~1.5μm cross section is given as an example. The theoretical pure bending loss is negligible for all the lowest eight modes when the bending radius is even less than 5μm. When light goes through the structure consisting of a straight section connected with a bent section, it is found that some inter-mode crosstalk is caused by the significant mode hybridization happening in the sharply bent multimode waveguide. For the designed SOI-based vertical multimode waveguide, the inter-mode crosstalk is lower than −20dB even when the bending radius is chosen as small as *R* = 10μm, which is one order smaller than that for the traditional lateral multimode waveguide (whose minimal bending radius is about 130μm). The inter-mode crosstalk can be even reduced to −30dB when choosing *R* = 30μm. Such a multimode optical waveguide microbend with low inter-mode crosstalk is promising for realizing compact mode-multiplexing links.

© 2014 Optical Society of America

## 1. Introduction

In order to satisfy the increasing demand for the optical communication capacity, in recent years the spatial-division-multiplexing (SDM) technique [1] has been re-activated by introducing multi-core fibers [2,3] as well as few-mode fibers [4–6] for the long-distance optical interconnects. The SDM channels carrying different data share the same wavelength and thus only one laser diode (LD) (i.e., a single fixed wavelength) is needed, which helps to reduce the system cost greatly. The SDM technology is also becoming more and more attractive for optical interconnects in data centers where it is convenient to upgrade an existing network or install a new network in comparison with the long-distance optical networks [7]. Particularly, for photonic networks-on-chip, optical signals propagate along planar optical waveguides and one can control the mode propagation / conversion conveniently by utilizing some specific on-chip waveguide structures. Consequently it becomes promising to develop the on-chip SDM technology. In comparison to the multi-core SDM, the multimode SDM [7,8] provides a way with more concise and compact photonic integrated circuits (PICs) because only one multimode bus waveguide is included and the guided-modes carrying different data are overlapped spatially in the multimode bus waveguide. In recent years, the key component like mode (de)multiplexer has been realized with various structures [9], including multimode-interferometer-based structures [10], adiabatic Y-branches [11–13], and adiabatic directional couplers [14–19]. For these multimode devices demonstrated previously, the bus waveguide is designed to be wide enough to support multiple modes in the lateral direction.

In order to achieve a flexible layout design, a bent multimode bus waveguide for a mode-multiplexed optical interconnect link is indispensable to change the propagation direction of light. A small bending radius is usually desired to achieve a compact photonic system/network-on-chip. As it well known, a singlemode SOI strip nanowire enables an extremely sharp bending (~several microns) with a low loss (including the pure bending loss and the transition loss) due to the ultra-high index contrast [20]. However, there is not much work on the mode propagation in bent multimode waveguides yet. As shown in Fig. 2 in our previous paper [21], the modal fields of a wide multimode SOI strip waveguide are squeezed to the outside sidewall (like a whispery-gallery mode) and becomes very asymmetric when the bending radius becomes small. As a consequence, when light propagates along a multimode straight waveguide connected to a multimode bent waveguide with a bending radius *R*, inter-mode coupling occurs, which introduces some transition loss and inter-mode crosstalk if *R* is small [21]. In [22] a smart design for a wide multimode waveguide bend with low inter-mode coupling is given with the assistance of transformation optics theory and low inter-mode coupling is achieved with a ~78μm bending radius. Alternatively, another possible simple method is using a bent section whose curvature varies from zero to a certain value gradually so that the mode of straight waveguide can be converted to the desired mode in the bent waveguide adiabatically, which has been used for singlemode SOI strip nanowires [23] and also been suggested for multimode waveguides recently [21].

In this paper, we propose to a vertical multimode waveguide enabling microbends for a mode-multiplexed optical interconnect link. The proposed waveguide is narrow and tall so that it is singlemode in the lateral direction while supports higher-order modes in the vertical direction. As an example, a systematic analysis on the mode properties for an SOI-based vertical multimode waveguide is given and the mechanism for generating the inter-mode crosstalk is studied. With the designed SOI-based vertical multimode waveguide, the minimal bending radius is as small as *R* = 10μm for guaranteeing the inter-mode crosstalk to be lower than −20dB.

## 2. Structure and analysis

Figure 1(a) shows the proposed vertical multimode waveguide, whose core height is much larger than the core width so that vertical higher-order modes are supported. Figure 1(b) shows the calculated effective indices of all the guided-modes in the proposed vertical multimode waveguide as the core height hco varies. For this calculation, a commercial full-vectorial finite-difference method (FV-FDM) mode-solver from PhotonDesign was used. Here the core width is chosen as *w*_{co} = 0.3μm and the operation wavelength is λ = 1550 nm. The corresponding refractive indices for all the involved materials are *n*_{SiO2} = 1.445 and *n*_{Si} = 3.455. From Fig. 1(b), it can be seen that the first higher-order modes for both TE and TM polarizations appear around *w*_{co} = 0.44μm and more guided-modes are supported when choosing larger core height, as expected. The core height *h*_{co} should be large enough to support sufficient guided-modes for the mode channel number as required. For example, when eight mode-channels (i.e., *N*_{ch} = 8) are desired, the core height *h*_{co} should be larger than 1.0μm. And the core height should also be large enough to make the highest-order mode be supported well (not close to the cut-off boundary) so that it is possible to achieve a multimode bus waveguide enabling a low-loss sharp bend. On the other hand, however, the fabrication becomes not easy for an optical waveguide when it is with a very high aspect ratio (*h*_{co}/*w*_{co}). Therefore, it is desired to reduce the core height *h*_{co} while the guided-modes supported are sufficient. As an example, we choose *h*_{co} = 1.5μm in our design to have a reasonable aspect ratio (*h*_{co}/*w*_{co} = 5) and sufficient guided-modes (*N*>8). The calculated mode profiles for the lowest 8 guided-modes are presented in Fig. 1(c). It can be seen that there are four modes for TE as well as TM polarizations so that there are eight mode channels available in this multimode bus waveguide.

In order to have a multimode bus waveguide enabling an ultra-sharp bend, low pure bending losses are required for all the eigenmodes used for the mode channels (including the fundamental mode as well as the higher-order modes). Therefore, one should choose an ultra-high index-contrast (e.g., Si/SiO_{2}) and sufficiently large bending radius for the multimode bus waveguide. The mode analysis for bent SOI vertical multimode waveguides is given below. Figure 2(a) show the calculated normalized effective indices of the eigen-modes supported by the bent multimode waveguide with *w*_{co} = 0.3μm and *h*_{co} = 1.5μm as the bending radius *R* decreases. The *i*-th normalized effective index shown here is defined as *n*_{eff_}* _{i}*(

*R*)−

*n*

_{eff0_}

*, where*

_{i}*n*

_{eff_}

*(*

_{i}*R*) is the effective index of the

*i*-th eigenmode in the bending waveguide,

*n*

_{eff0_}

*is the effective index of the*

_{i}*i*-th eigenmode in the straight waveguide. From Fig. 2(a), it can be seen that the effective indices of most eigenmodes in the bending waveguide increases slightly as the bending radius

*R*decreases, which is because the peak of the modal field moves to the right side (away from the bend center) slightly. This is similar to those results for a singlemode bent waveguide reported previously [24,25]. However, for the 7-th mode (

*i*= 7, TE

_{02}), the result is completely different and it can be seen that the effective index

*n*

_{eff_7}(

*R*) decreases slightly as the bending radius

*R*decreases, which has not be reported before. In order to explain this “abnormal” result, we have a deep look at the field profiles for these modes. Our full-vectorial mode-solving calculation shows that the field profiles of the major components for all these modes have a slight peak shift to the right side (away from the bend center), which happens as usual. We note that the mode hybridization might happen notably when an optical waveguide is bent sharply and thus the minor component for the modal field could play an important role.

In order to evaluate the mode hybridization quantitatively, we calculate the fraction *f*_{TE} of the Poynting vector with horizontal electric field, which is defined as

*f*

_{TE}= 100% (for TE polarization) or 0 (for TM polarization). When there is mode hybridization, the fraction

*f*

_{TE}is between 0 and 100%. For the present case, our calculation shows that the mode hybridization for the TE

_{02}mode (

*i*= 7) discussed here is quite notable. For example, one has

*f*

_{TE}= 98%, 96%, 93%, and 89% for

*R*= ∞, 10, 5, and 3μm, respectively. It indicates that the major component

*E*(

_{x}*x*,

*y*) for the TE

_{02}mode (

*i*= 7) becomes less dominant while the minor component

*E*(

_{y}*x*,

*y*) becomes more dominant as the bending radius decreases. Therefore, we focus on the dependence of the field profile

*E*(

_{y}*x*,

*y*) for the minor component on the bending radius. Figures 2(b)-2(e) show the minor component

*E*(

_{y}*x*,

*y*) for the TE

_{02}mode (

*i*= 7) when

*R*= ∞, 10μm, 5μm, and 3μm, respectively. From these figures, it can be seen that the field profile

*E*(

_{y}*x*,

*y*) is symmetrical and has two equal peaks in the lateral direction when

*R*= ∞. As the bending radius decreases, both peaks move to the right side (away from the bend center) very slightly. More importantly, for the case with a smaller bending radius, the field profile becomes more asymmetrical and the peak at the left side (with smaller

*R*) becomes much more dominant than the peak at the right side (with larger

*R*). Such asymmetry is equivalent to a peak shift toward to the bend center (at the left side) for the minor component. As a consequence, this will partially compensates the peak shift to the right side for the major component. When the mode hybridization is strong and the asymmetry of the field profile for the minor component is also significant, the effective index for the mode will decrease with decreased bending radius. This is the case for the TE

_{02}mode, as shown in Fig. 2(a). For the other modes, one has larger effective indices when the bending radius is smaller because of the weak mode-hybridization or the small symmetry of the minor component.

Figure 3 shows the calculated pure bending losses *L* of the eigen-modes in the bent multimode waveguide with *w*_{co} = 0.3μm and *h*_{co} = 1.5μm as the bending radius *R* decreases. The pure bending loss *L* is calculated with the following formula *L* = 10log_{10}[exp(2*n*_{eff_im}*k*_{0}*l*_{0})] (dB/cm), where *l*_{0} = 1cm, *n*_{eff_im} is the imaginary part of the effective index *n*_{eff} of the eigen-modes of the bending waveguide, *k*_{0} is the wavenumber in vacuum. From this figure, it can be seen that the bending losses for all the eight eigen-modes are very small even when the bending radius is reduced to 5μm, which is attributed to the ultra-high index contrast (similar to singlemode SOI nanowires [26]). This provides the opportunity to have a multimode bus waveguide enabling an ultra-sharp bends.

When light goes along a structure consisting of a straight section connected with a bent section, the mode mismatch between the straight section and the bent section will introduce not only some mode-mismatching loss but also some inter-mode crosstalk, which definitely should be low enough regarding the requirement for the data transmission through the mode-multiplexed optical interconnects. One can estimate the mode-mismatching loss and the inter-mode crosstalk theoretically by calculating the mode excitation ratios *η _{ij}* with the overlapping integral method as follows [27]:

**E**

_{S}

*(*

_{i}*x*,

*y*) is the normalized electrical field for the

*i*-th eigen-mode of the straight section,

**H**

_{B}

*(*

_{j}*x*,

*y*) is the normalized magnetic field for the

*j*-th mode of the bent section. The mode-mismatching loss

*L*and the inter-mode crosstalk CT

_{i}*are then given by*

_{ij}*L*= 10log

_{i}_{10}|η

*|*

_{ii}^{2}, and CT

*= 10log*

_{ij}_{10}|η

*|*

_{ij}^{2}, respectively.

Figures 4(a)-4(h) respectively show the calculated mode excitation ratios *η _{ij}* for the lowest eight guided-modes in the bending section when the

*i*-th guided-mode of the straight waveguide is launched (

*i*= 1, 2, …, 8). In these figures, the right vertical axis is for the mode excitation ratios

*η*(which is the mode coupling ratio from the

_{ii}*i*-th eigen-mode of the straight section to the

*i*-th mode of the bent section), while the left vertical axis is for the mode excitation ratios

*η*(

_{ij}*j*≠

*i*) (which is the mode coupling ratio from the

*i*-th eigen-mode of the straight section to the

*j*-th mode of the bent section). From these figures, it can be seen that the mode excitation ratio

*η*decreases and the mode excitation ratios

_{ii}*η*(

_{ij}*j*≠

*i*) increase as the bending radius

*R*decreases. Even when the bending radius is as small as

*R*= 5μm, the mode excitation ratio

*η*(

_{ii}*i*= 1, …, 8) is still larger than 0.96 and correspondingly the mode-mismatching loss

*L*is very low (<0.2dB). Therefore, we focus on the analysis for the mode excitation ratios

_{i}*η*(

_{ij}*j*≠

*i*) corresponding to the inter-mode crosstalk. From Figs. 4(a)-4(h), it can be seen that the mode excitation ratios

*η*(

_{ij}*j*≠

*i*) for the cases of

*i*= 1, 2, and 4 are very small (<0.002) even when the bending radius is reduced to

*R*= 3μm. In contrast, for the cases of

*i*= 3, 5, 6, 7, and 8, the mode excitation ratios

*η*(

_{ij}*j*≠

*i*) are notable for a sharp bend (e.g.,

*R*<10μm). For example, both the mode excitation ratios

*η*

_{67}and

*η*

_{76}are about 0.01 when

*R*= 10μm, as shown in Figs. 4(f)-4(g). We note that the 6-th mode is the TM

_{03}mode while the 7-th mode is the TE

_{02}mode, which indicates that the notable inter-mode crosstalk happens between two modes with orthogonal polarizations.

In order to understand why this happens, we show the modal field profiles including the major component and the minor components (*E _{x}*(

*x*,

*y*),

*E*(

_{y}*x*,

*y*)) for the 6-th mode (the TM

_{03}mode) and 7-th mode (the TE

_{02}mode) in the straight waveguide (

*R*= ∞) as well as the bent waveguide (

*R*= 10μm) in Figs. 5(a)-5(d), respectively. From these figures, it can be seen that the major component (i.e.,

*E*(

_{x}*x*,

*y*) for TE polarization, or

*E*(

_{y}*x*,

*y*) for TM polarization) does not change almost when the waveguide is bended. In contrast, the minor component (

*E*(

_{y}*x*,

*y*) for TE polarization, or

*E*(

_{x}*x*,

*y*) for TM polarization) does change greatly when the bending radius is reduced to e.g. 10μm due to the mode hybridization cause by the bending. For example, the minor component

*E*(

_{x}*x*,

*y*) for the TM

_{03}mode (

*i*= 6) in the bent waveguide with

*R*= 10μm (see Fig. 5(c)) becomes to have three peaks in the square core region, which is slightly similar to the major component (

*E*(

_{x}*x*,

*y*) for the TE

_{02}mode (

*i*= 7) in the straight waveguide (see Fig. 5(b)). In this case, some coupling between them (i.e.,

*η*

_{76}) happens when the TE

_{02}mode (

*i*= 7) in the straight waveguide is launched. Similarly, when the TM

_{03}mode (

*i*= 6) in the straight waveguide is launched, some coupling to the TE

_{02}mode (

*i*= 7) in the bent waveguide (i.e.,

*η*

_{67}) happens because the minor component

*E*(

_{y}*x*,

*y*) of the TE

_{02}mode (

*i*= 7) in the bent waveguide distorts as shown in Fig. 5(d).

Regarding this kind of polarization coupling resulting from the mode hybridization and the field distortion in the bent waveguide, the bending radius should be large enough to minimize the inter-mode crosstalk sufficiently. For the present multimode waveguide, the bending radius can be as small as *R* = 10μm to make the inter-mode crosstalk lower than −20dB, i.e., *η _{ij}* (

*j*≠

*i*)<0.01. In contrast, the minimal bending radius for the conventional lateral multimode bus waveguide has to be as large as 130μm [21]. When a lower inter-mode crosstalk is required, e.g., −30dB (i.e.,

*η*(

_{ij}*j*≠

*i*)<0.001), the bending radius should be

*~*30μm, which is still one order smaller than that for the conventional lateral multimode bus waveguide (whose minimal bending radius is about 400μm [21]). Therefore, the present vertical multimode bus waveguide is promising to realize ultra-compact mode-multiplexed photonic integrated circuits.

The dependence of the mode excitation ratios *η _{ij}* on the core width

*w*

_{co}is also analyzed and shown in Figs. 6(a)-6(h). In this example, the core with varies from 0.25μm to 0.4μm while the core height is

*h*

_{co}= 1.5μm and the bending radius of the bent section connecting to the straight waveguide is fixed as

*R*= 10μm. From these figures, it can be seen that the mode excitation ratios

*η*

_{1}

*,*

_{j}*η*

_{2}

*, and*

_{j}*η*

_{3}

*are insensitive to the core width and one has very low ratios*

_{j}*η*

_{1}

*,*

_{j}*η*

_{2}

*, and*

_{j}*η*

_{3}

*(*

_{j}*j*≠

*i*) as well as very high

*η*

_{11},

*η*

_{22}, and

*η*

_{33}. In contrast, a very large undesired ratio

*η*(

_{ij}*j*≠

*i*) is observed for the cases of

*i*= 4, 5, 6, 7, and 8 when choosing the core width around

*w*

_{co}= 0.385μm, 0.385μm, 0.322μm, 0.322μm, and 0.275μm, respectively, as shown in Figs. 6(d)-6(h). For example, for the case of

*i*= 4 (or 5), when

*w*

_{co}= 0.385μm, the mode excitation ratio

*η*

_{45}(or

*η*

_{54}) is as large as ~0.442 meanwhile the mode excitation ratio

*η*

_{44}(or

*η*

_{55}) is ~0.556. For the case of

*i*= 8, there is mode coupling from the 8-th eigen-mode of the straight section to the 9-th mode of the bent section when the core width

*w*

_{co}is around 0.2755μm. When

*w*

_{co}= 0.2755μm, the mode excitation ratio

*η*

_{89}is as large as ~0.504 meanwhile the mode excitation ratio

*η*

_{88}is ~0.493. This means that the inter-mode crosstalk as well as the excess loss is seriously high. This is again due to the mode hybridization happening in the region around some special core width.

The mode hybridization can be seen clearly from Figs. 7(a)-7(b), which show the modal field profiles including the major component and the minor component (*E _{x}*,

*E*) for the TM

_{y}_{02}mode (

*i*= 4) and the TE

_{01}mode (

*i*= 5) of the bent section with

*R*= 10μm when

*w*

_{co}= 0.385μm, respectively. It can be seen that these two modes are hybridized notably and their fractions

*f*

_{TE}are 56% and 45%, respectively. Therefore, both of these two modes are excited when the TM

_{02}mode or the TE

_{01}mode in the straight waveguide is launched, which introduces significant inter-mode crosstalk. As a comparison, we also show the modal field profiles (

*E*,

_{x}*E*) for the TM

_{y}_{02}mode (

*i*= 4) and the TE

_{01}mode (

*i*= 5) of the bent section with

*R*= 10μm when

*w*

_{co}= 0.3μm (see Fig. 7(c) and Fig. 7(d)). It can be seen that the mode hybridization is very weak because the minor component is much smaller than the major component. Furthermore, the mode profile in the bent section is very similar to that for the straight section (with the same core dimension). Therefore, the mode excitation ratio

*η*

_{33}is 100% almost while

*η*

_{3}

*(*

_{j}*j*≠3) is very small, as shown in Figs. 6(d)-6(e).

From Figs. 6(a)-6(h), it can be seen that one should choose the core width appropriately to avoid any significant mode hybridization. For the present case (i.e., *h*_{co} = 1.5μm), the core width can be chosen to be around 0.35μm, 0.3μm, and 0.25μm so that low inter-mode crosstalk is achieved even when the bending radius is as small as 10μm.

## 4. Conclusion

In summary, we have proposed a vertical multimode waveguide for a mode-multiplexed optical interconnect link. The proposed waveguide is narrow and tall so that it is quasi-singlemode in the lateral direction while higher-order modes are supported in the vertical direction. The characteristic analysis for the eigenmodes in an SOI-based vertical multimode waveguide has been given in details as an example. For an SOI-based vertical multimode waveguide with a 0.3μm × 1.5μm cross section, the bending radius can be even less than 5μm while the theoretical pure bending loss is negligible for all the lowest eight modes. For the proposed vertical multimode waveguide, it has shown that there is some significant mode hybridization in the multimode waveguide when it is bending sharply and the enhanced minor component of the modal field in the bent multimode waveguide introduces some inter-mode crosstalk between some mode channels when light goes through the structure consisting of a straight section connected with a bent section. This is different from the case of the traditional lateral multimode waveguide bends (for which the inter-mode crosstalk is mainly from the significant distortion of the major component of the modal field). Fortunately, for the present 0.3μm × 1.5μm SOI-based vertical multimode waveguide, the inter-mode crosstalk can be also very low. For example, the bending radius can be as small as *R* = 10μm to guarantee the inter-mode crosstalk to be lower than −20dB, which is one order smaller than that for the traditional lateral multimode waveguide (whose minimal bending radius is about 130μm). The inter-mode crosstalk can be even as low as −30dB when the bending radius is chosen as 30μm. This is promising for realizing compact mode-multiplexing links in the future. It is also possible to reduce the bending radius further by designing the width and the height of the core (e.g., *w*_{co} = 250nm and *h*_{co} = 1.5μm). However, the higher aspect ratio makes the fabrication more difficult. In order to realize mode (de)multiplexing for the proposed vertical multimode waveguide, a potential design is using an asymmetrical directional coupler consisting of two optical waveguides with different core heights, or a vertical directional coupler similar to those developed for three-dimensional optical integration operating with the fundamental mode.

## Acknowledgments

This project was partially supported by the Nature Science Foundation of China (No. 11374263, 61422510), the Doctoral Fund of Ministry of Education of China (No. 20120101110094).

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