## Abstract

A simple and low-crosstalk 1 × 4 silicon mode (de)multiplexer based on multimode grating-assisted-couplers is proposed. Mode transitions can be flexibly controlled by designing the grating period at the phase-matching condition. Due to the contra-directional coupling, precise control of the coupling strength and the coupling length are not needed in the system. Calculation results show that the insertion loss and the 3 dB bandwidths of the device are 0.2 dB and 3.7 nm, 0.34 dB and 7.6 nm, and 0.21 dB and 11.8 nm for the channels which (de)multiplex to the 1st, 2nd, and 3rd modes of the bus waveguide, respectively.

©2013 Optical Society of America

## 1. Introduction

On-chip optical interconnect has been regarded as a way to satisfy the exponentially increasing demand of high bandwidth for future massively-parallel chip multiprocessors [1]. Several means have been demonstrated to increase the transmission capacity. A straightforward way is spatial-division multiplexing (SDM), in which independent signals are carried by an array of identical parallel optical waveguides. However, this approach would increase the footprint and complexity of the circuit layout, and thus its application in on-chip optical interconnect systems is limited. Wavelength-division multiplexing (WDM) is another approach to expand the transmission capacity. As multi-lasers with different wavelengths have to be embedded in a chip, this will increase cost as well as system complexity.

Compared with SDW and WDM, Mode-division multiplexing (MDM) is able to expand the capacity of a single photonics link without increasing too much footprint and complexity of the system. This technique uses a multimode optical waveguide as the bus waveguide, of which each eigen-mode is exploited as an independent channel to transmit data. The transmission capacity can be expanded further by combining MDM with WDM, SDM or polarization-division multiplexing (PDM). One of the key components in a MDM system is a mode-(de)multiplexer, which excite,multiplexes, and demultiplexes optical fields of different mode orders. Recently, several kinds of mode-(de)multiplexers have been proposed. The designs based on Y-splitter [2] and multimode interference [3] can realize (de)multiplex two modes of the same polarization. However, these devices require large footprints and complicated waveguide cross-sections to obtain only two channels. On the other hand, a design based on asymmetrical co-directional couplers [4] is proposed to achieve more than four channels, which requires a very precise control of the coupling length and the waveguide dimensions at different sections of the bus waveguide for a complete transfer of optical power between the two coupled waveguides.

Integrated silicon photonic devices based on grating-assisted contra-directional couplers (GACCs) have enabled many novel applications such as on-chip optical-pulse compression [5], wide-bandwidth wavelength (de)multiplexers [6], single-longitudinal-mode micro-ring resonators [7], FSR-free add-drop filters [8] and 4-port photonic resonators [9]. However, conventional GACCs consist of two asymmetry single-mode waveguides, which support only two fundamental compound modes and thus one mode channel. Therefore, GACCs is only regarded as a wavelength filter for wavelength-division multiplexing.

In this paper, we propose a mode-(de)multiplexer on silicon-on-insulator (SOI) platform by taking advantage of cascaded multimode grating-assisted contra-directional couplers (MGACCs), where the contra-directional coupling occurs between a single-mode and a multi-mode waveguides rather than two single-mode waveguides. Benefited from the MGACCs, the signal carried by the fundamental mode of the narrow access waveguide can be coupled to higher-order modes in the multimode bus waveguide. Therefore, GACCs’ application is extended from the wavelength domain for WDM to the mode domain for MDM. And more importantly, the contra-directional coupler possesses a property that the transfer of optical power between the two coupled waveguides is not periodically but monotonically depended on the coupling length. This indicates that a MGACC based mode-(de)multiplexer demand no rigorous control of the coupling strength and the coupling length. As long as the coupling region is sufficiently long, light at certain wavelength can be completely coupled from one waveguide to its neighbor. This point differs from the co-directional coupler based device [4]. Therefore, the scheme we propose here would be quite robust and tolerant to any design and fabrication errors. Our current design supports four channels. More channels can be easily achieved by choosing a wider multimode waveguide which supports more guided modes. We numerically study the light propagating in the MGACCs by using the three dimensional finite difference time domain (3-D FDTD) technique and the spectral responses of the MGACCs are also calculated.

## 2. Principle of mode-(de)multiplexers based on MGACCs

A compact MGACCs-based mode-(de)multiplexer is illustrated in Fig. 1. The device is composed of a single-mode waveguide, a wider multimode waveguide and a Bragg grating formed by a spatially periodic refractive-index perturbation either on or between the waveguides along the longitudinal direction.

If there is only the fundamental mode in the narrow access waveguide, and the bus multimode waveguide supports more than two eigen-modes, which is quite different from the conventional GACCs, then the confined wave propagating in the composite waveguide system without perturbation can be represented as a linear combination of all-eigen modes,

where*φ*(x,y),

_{m}*β*and

_{m}*A*represent the field distribution, propagation constant and the amplitude of the

_{m}*m*th eigen-mode, respectively. As the narrow access waveguide and the wide bus waveguides have different propagation constants, co-directional coupling between the two waveguides is greatly suppressed. However, the contra-directional coupling occurs provided that the phase matching is achieved by a spatially periodic refractive-index perturbation [10]. We consider contra-directional coupling between one forward-propagating mode with propagation constant

*β*, amplitude

_{i}*A*and one backward-propagating mode with propagation constant

_{i}*β*, amplitude

_{l}*A*. The coupled-mode equations are given by [10] where ${\Delta}_{il}={\beta}_{i}+{\beta}_{l}-2\pi /\Lambda $,

_{l}*κ*(

_{il}*κ*=

_{il}*κ**) is the coupling coefficient and can be written as

_{li}*u*(x,y) is the first-order Fourier-expansion coefficient of the dielectric perturbation [10].

It can be seen from Eqs. (2) and (3), that the fundamental mode of the narrow access waveguide could be coupled to any mode in the bus multimode waveguide when the grating period is designed to satisfy the phase-match condition, i.e., *β _{0}* +

*β*=

_{l}*2π/Λ*, where

*β*and

_{0}*β*are the propagation constants of the fundamental mode of the narrow access waveguide and the

_{l}*l*th higher-order mode of the bus multimode waveguide, respectively. When a series of MGACCs with different periods are cascaded, it is possible to multiplex/demultiplex independent optical signals into/from each different eigen-modes in the bus multimode waveguide.

## 3. Structure design and analysis

Figure 2 shows the schematic configuration of a 1 × 4 mode-(de)multiplexer, which includes a MMI-based 1 × 4 power splitter, an optical modulator array with 4 identical elements, two adiabatic tapers between the access waveguide and the bus waveguide, three pairs of MGACCs for mode (de)multiplexing. The fundamental modes of the four input access waveguides are transited to the 0th, 1st, 2nd and 3rd modes of the bus multimode waveguide by the multiplexers, and then are coupled back to the fundamental modes of the four output access waveguides by the demultiplexers. The corresponding optical paths from input to output access waveguides are referred as channels 0-3 throughout the paper as shown in Fig. 2.

As shown in Fig. 2, the transition from the fundamental mode of the access waveguide to the 0th mode of the bus waveguide is achieved by an adiabatic taper between the two waveguides [4], while the mode couplings between the fundamental mode of the access waveguide and different higher-order modes of the bus waveguide are implemented by corresponding MGACCs. We design the gratings whose periods satisfy the phase-match condition by

In the MGACCs, the thickness of all silicon strip waveguides is 220 nm. The width of the access single-mode waveguide is 450 nm, whilst the bus multimode waveguide has a width of 1700 nm to support five eigen modes (only the first four eigen modes are used as the mode channels are four). The waveguides are covered by SiO2. The gap between the two waveguides G is 300 nm. The finite-element method is used to calculate the compound TE-like modes of the MGACCs. It is worth to point out that the device is also applicable for TM-like modes. Therefore, the transmission capacity doubles by a combination with PDM [4].

The calculated effective indices of the contra-directional coupling modes are sketched in Fig. 3. The solid and dotted lines in Fig. 3 are the plots of the right-hand side and left-hand side of the Eqs. (5), as a function of the wavelength, respectively. Therefore, the cross points of the solid and dotted lines represent the phase-match condition. We choose the values of *Λ* for effective contra-directional coupling at the wavelength of 1550 nm. The calculation results show that the grating periods for coupling from the fundamental mode of the access waveguide to the 1st, 2nd and 3rd modes of the bus waveguide are *Λ _{1}* = 303 nm,

*Λ*= 314 nm and

_{2}*Λ*= 333 nm, respectively.

_{3}The relationships between the coupling coefficients and the rectangle corrugation width D are illustrated in Fig. 4(a), with the dielectric perturbation at the center of the two waveguides. The calculation results show that the coupling coefficients increase with the corrugation width, and the coupling between the fundamental mode of the access waveguide and the 3rd mode of the bus waveguide is the strongest, since the overlap of the 3rd mode profiles with the dielectric perturbation is larger than that of the 1st and 2nd modes.

The relationships between the coupling coefficients and the displacement of the rectangle corrugation from the center of the gap G are illustrated in Fig. 4(b). The inset is the cross-section of the proposed device. It can be seen that the coupling coefficients increase when the corrugations close to the bus waveguide. A high coupling coefficient helps to reduce the device length. Considering the linewidth limitation of the state-of-the-art fabrication technology, we place the corrugation with width of 200 nm next the bus waveguide, so the gap between the corrugation and the access waveguide is 100 nm. As the coupling coefficients for different mode transitions are different, the coupling lengths should be considered. Figure 5 shows the calculated power at the through port and drop port as a function of the coupling length L. As aforementioned, the extinction ratio increases monotonically with the coupling length. Any variation of the corrugation width or the gap between the two waveguides will certainly change the coupling coefficients, however, this has little effect on the mode transitions if the coupling length is long enough [11]. Therefore, the MGACCs have good fabrication tolerance. In our simulation, we choose the coupling length of the 1st, 2nd and 3rd modes of the bus waveguide to be 250 μm, 150 μm, and 90 μm, respectively, with which the extinction ratios are about 18 dB.

When the MGACCs are used as mode multiplexers, light is launched from different access waveguides. A 3-D FDTD technique [12] based simulation tool is used to calculate the device behavior. In the 3-D FDTD simulations of Figs. 6(a)-6(d), the coupling length is set to be 28 μm, which is much shorter than the desired length for a complete power exchange, as larger coupling length will dramatically increase the computation time and memory. Figure 6(a) shows that power cannot transfer from the access waveguide to the bus waveguide without the corrugations. Figures 6(b)-6(d) show that the fundamental mode of the access waveguide can be coupled to the 1st, 2nd, 3rd modes of the bus waveguide with designed grating periods. Although the power transfer is incomplete due to the insufficient coupling length in our simulation, the operation principle of the device is proved. The insets in Figs. 6(b)-6(d) are the electric field distributions of the high-order modes in the bus waveguide.

The spectral responses for contra-directional power transfer from the access waveguides to the bus multimode waveguide are calculated by using RSoft Design Group’s GratingMOD software [13]. The results are plotted in Figs. 7(a)-7(c). Figures 7(a)-7(c) show the through and the drop ports spectra of the 1st, 2^{st}, and 3^{st} mode multiplexers, whose grating periods are 303 nm, 314 nm, and 333 nm, respectively. At the center wavelength of 1550 nm, the fundamental mode of the access waveguide is coupled to the targeted high-order mode of the bus waveguide by the corresponding MGACC. As the wavelength deviates from 1550 nm, the fundamental mode of the access waveguide then will be coupled to the other four modes of the bus waveguide provided that the corresponding phase matching conditions are satisfied. On the other hand, the periodic perturbations will also cause intra-waveguide backward reflection in the access waveguides at the Bragg condition, i.e., *λ = 2Λn _{0}*. Therefore, there should be six notches in the through-port spectra throughout the whole spectrum range for every mode multiplexer. We do not draw all the notches in the spectra, as the scale of the wavelengths is 150 nm in our calculation. To avoid the influence of the adjacent modes coupling, the difference between the propagation constants of the two interacting modes should be large enough to keep the two notches separated. The 3 dB bandwidths of the drop-port spectra in Figs. 7(a)-7(c) at the center wavelength of 1550 nm, are 4 nm, 7.6 and 12.8 nm, respectively. The multiplexer to high order mode offers a wider optical bandwidth and larger extinction ratio than that to low order mode. This is due to the fact that the coupling strength between the bus and the access waveguides increases with the mode order of the bus waveguide as shown in Figs. 4(a) and 4(b).

At the output terminal, different optical higher order modes in the bus waveguide are coupled to different access waveguides by mode demultiplexers. The parameters of the mode demultiplexers are the same as those of the mode multiplexers. Figure 8 shows the responses of the demultiplexer of the channel 2, which is supposed to demultiplex the 2nd mode at 1550 nm. Different peaks correspond to different modes in the bus waveguide. The spacing between the adjacent coupling center wavelengths is larger than 40nm. The cross talk is 22.6 dB, which could be improved by modulating corrugation width [14]. For the demultiplexers of the channel 1 and 3 whose responses are not presented here, the corresponding cross talks are 26.8 dB and 30.3 dB, respectively.

The final output responses of the channels 1-3 are shown in Fig. 9. Each channel includes two MGACCs and the bus waveguide. We can see that the insertion loss and the 3 dB bandwidths are 0.2 dB and 3.7 nm, 0.34 dB and 7.6 nm, 0.21 dB and 11.8 nm for the channels 1, 2, and 3, respectively.

## 4. Conclusion

We have proposed a silicon 1 × 4 mode-(de)multiplexer based on MGACCs to expand the capacity of a single-wavelength-carrier light by 4 times for future photonic network-on-chip. The device has a simple structure, low crosstalk and good fabrication tolerance. We can flexibly control mode transitions by designing the grating period at the phase-matching condition and more channels can be easily implemented by choosing a wider bus waveguide and cascading more MGACCs. Calculation results show that the cross talks of the demultiplexers of the channels 1, 2 and 3, are 22.6 dB, 26.8 dB and 30.3 dB, respectively. The insertion loss and the 3 dB bandwidths are 0.2 dB and 3.7 nm, 0.34 dB and 7.6 nm, 0.21 dB and 11.8 nm for the channels 1, 2, and 3, respectively.

## Acknowledgment

This work is supported the Natural Science Foundation of China (No. 61177055), the Nature Basic Research Program of China (No.2013CB632105), and the Fujian Province Education Department Foundation of China (JK2013053).

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