# Broadband three-mode converter and multiplexer based on cascaded symmetric Y-junctions and subwavelength engineered MMI and phase shifters

David González-Andrade, Irene Olivares, Raquel Fernández de Cabo, Jaime

Vilas, Antonio Dias, Aitor V. Velasco

► To cite this version:

David González-Andrade, Irene Olivares, Raquel Fernández de Cabo, Jaime Vilas, Antonio Dias, et al.. Broadband three-mode converter and multiplexer based on cascaded symmetric Y-junctions and subwavelength engineered MMI and phase shifters. Optics and Laser Technology, 2023, 164, pp.109513. 10.1016/j.optlastec.2023.109513. hal-04610248

HAL Id: hal-04610248

https://hal.science/hal-04610248v1

Submitted on 9 Jul 2025

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# Broadband three-mode converter and multiplexer based on cascaded symmetric Y-junctions and subwavelength engineered MMI and phase shifters

David González-Andrade<sup>a,*</sup>, Irene Olivares<sup>b</sup>, Raquel Fernández de Cabo<sup>c</sup>, Jaime Vilas<sup>b</sup>, Antonio Dias<sup>b</sup>, Aitor V. Velasco<sup>c</sup>

## ARTICLE INFO

### Keywords:

Silicon photonics

Mode-division

multiplexing

Subwavelength

metamaterial

MMI coupler

Phase shifter

Y-junction

## ABSTRACT

Mode-division multiplexing has emerged as a promising route for increasing transmission capacity while maintaining the same level of on-chip integration. Despite the large number of on-chip mode converters and multiplexers reported for the silicon-on-insulator platform, scaling the number of multiplexed modes is still a critical challenge. In this paper, we present a novel three-mode architecture based on multimode interference couplers, passive phase shifters and cascaded symmetric Y-junctions. This architecture can readily operate up to the third-order mode by including a single switchable phase shifter. Moreover, we exploit subwavelength grating metamaterials to overcome bandwidth limitations of multimode interference couplers and phase shifters, resulting in a simulated bandwidth of 161 nm with insertion loss and crosstalk below 1.18 dB and -20 dB, respectively.

## 1. Introduction

The relentless growth of global Internet traffic has been driven in recent years by the emergence of data-hungry services and their mass adoption by an increasingly interconnected society [1-3]. Moreover, the cloud nature of many new applications such as machine learning or artificial intelligence require large data sets to be processed on internal servers or transferred between data centers. This resource-intensive paradigm for accessing, computing, and storing data has led to the creation of hyperscale data centers consisting of thousands of servers located in the same physical facility [4]. To cope with the resulting zetta scale of annual data flow, modern data centers have been relying on optical technologies for both long-haul and few-meter interconnects. Compared to their electronic counterparts, these optical technologies offer higher processing speeds, broader bandwidths and lower latency and energy consumption. Silicon photonics, leveraging the mature fabrication facilities of the microelectronics industry, plays a key role in the optical interconnect industry due to its capacity for high-yield and low-cost mass production of high-performance optoelectronic circuits [5,6].

However, the development of next-generation datacenters for Tbps communications and exascale computing systems is not feasible by scaling infrastructures alone and requires increasingly efficient optical interconnects for short-reach distances [7]. As single-mode transmission approaches its fundamental

limits, space-division multiplexing has emerged as a promising way to further improve the transmission capacity of optical interconnects through the use of multicore or multimode waveguides [8]. The latter, which is also called mode-division multiplexing (MDM), has attracted an increasing interest as it leverages the orthogonality of the eigenmodes supported by a single multimode waveguide, thus allowing to maintain the same level of on-chip integration [9,10]. That is, MDM enables encoding different data channels into specific spatial modes, increasing capacity proportionally to the number of modes used.

Numerous on-chip mode converters and multiplexers/demultiplexers (MCMD) have been proposed for the silicon-on-insulator (SOI) platform to date. Asymmetric Y-junctions are based on the principle of mode evolution in adiabatic structures, which results in broad operating bandwidths but also in long device lengths [11-13]. The minimum feature size of current lithography processes also has a significant impact in these devices, since the finite resolution at which the tip can be fabricated severely hampers their performance. Asymmetric directional couplers (ADCs) [14], relying on evanescent coupling between adjacent waveguides, are well suited for implementing high-channel count MDM systems, but they typically exhibit narrow bandwidths, and their performance is highly susceptible to fabrication errors. Shortcuts to adiabaticity have been successfully applied to reduce the device length and achieve broadband operation and

<footnote><sup>a</sup> Centre de Nanosciences et de Nanotechnologies, CNRS, Université Paris-Saclay, Palaiseau 91120, France</footnote>

<footnote><sup>b</sup> Alcyon Photonics S.L., Madrid 28004, Spain</footnote>

<footnote><sup>c</sup> Instituto de Óptica Daza de Valdés, Consejo Superior de Investigaciones Científicas (CSIC), Madrid 28006, Spain</footnote>

<footnote>* Corresponding author: david.gonzalez-andrade@c2n.upsaclay.fr</footnote>
relaxed fabrication tolerances [15,16]. Adiabatic tapers have also been employed in the coupling region of ADCs to improve the bandwidth and the resilience against fabrication deviations [17]. MCMDs building upon multimode interference (MMI) couplers and other auxiliary components such as phase shifters (PSs) and symmetric Y-junctions have been proposed as well [18,19], yielding low losses and low crosstalk over a relatively broad wavelength range (~100 nm).

The patterning of silicon at the subwavelength scale has proven to be a simple yet powerful tool to tailor the medium optical properties while inhibiting diffractive effects [20]. More specifically, subwavelength (SWG) metamaterials can behave as a homogeneous metamaterial that provides flexible dispersion and anisotropy engineering, non-feasible in conventional strip and rib waveguides. These properties have led to the realization of Si devices with unprecedented performance over the past 15 years [21-23]. In the MDM field, MCMDs based on subwavelength pixelated structures have demonstrated ultra-compact footprints [24]. SWGs have also been applied to ADCs and triple-waveguide couplers to improve fabrication tolerances and extend the operation bandwidth of conventional counterparts [25-27]. Furthermore, low losses and low crosstalk within ultra-broad bandwidths have also been reported using subwavelength engineered MMI couplers and PSs, and SWG-slot-assisted adiabatic couplers [28-32].

Despite the large number of available two-mode MCMDs, scaling the number of multiplexed modes beyond the fundamental and first-order modes is of great importance to multiply the capacity of next-generation datacom systems. Although it is fairly straightforward to extend operation to a larger number of modes in asymmetric Y-junctions and conventional

and tapered ADCs [33], three- and four-mode MCMD based on MMI couplers have only recently been reported [34-38]. However, the proposed architectures are still limited by narrow operating bandwidths and high crosstalk values.

In this work, we propose a novel MCMD architecture based on a 4x4 MMI, three phase shifters and four symmetric 1x2 Y-junctions arranged in a conventional cascaded configuration. The device operates as a three-mode MCDM with passive phase shifters but can readily convert up to the third-order mode by including a single switchable phase shifter. Moreover, we demonstrate loss and crosstalk reduction in a broad bandwidth by SWG-engineering of both the MMI coupler and phase shifters. Simulations show operation bandwidth of 161 nm with insertion loss and crosstalk below 1.18 dB and -20 dB, respectively.

## 2. Principle of operation and device design

To explain the operation principle and the device design, let us first focus on the nanophotonic structure shown in Fig. 1(a) consisting of a conventional 4x4 MMI, three phase shifters (PS1, PS2 and PS3) and four symmetric 1x2 Y-junctions (three identical Y1 and a different one Y2). SWG enhancement of the proposed architecture, shown in Fig. 1(b) and Fig. 1(d) will be discussed in epigraphs 4 and 5. An SOI platform with a thin Si wire surrounded by SiO₂ bottom layer and upper cladding are considered. A schematic view of the waveguide cross-section is shown in Fig. 1(c) for clarity.

In order to illustrate the operation of the MCMD, let us focus on the mode evolution and phase relations in each individual constituent of the MCMD. Here, we aim at mode conversion and multiplexing of the first four modes for transverse-electric (TE) polarization, that is, the fundamental mode (TE₀), the first-order

<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Three-dimensional schematic of the proposed three-mode converter and multiplexer/demultiplexer comprising a 4x4 MMI, three phase shifters and four symmetric Y-junctions implemented with (a) conventional homogeneous and (b) SWG metamaterial waveguides. (c) Cross-sectional view of the SOI strip waveguides with a SiO₂ cladding. (d) Top view of the SWG waveguides with their main geometrical parameters.</figcaption></figure>
mode (TE₁), the second-order mode (TE₂) and the third-order mode (TE₃).

Our MCMD includes two types of symmetric multimode 1×2 Y-junctions: Y1, with a stem supporting up to two modes; and Y2, with a wider stem supporting up to four modes. In general, multimode symmetric 1×2 Y-junctions transform the two in-phase $m^{th}$-order modes in the arms into the $(2m)^{th}$-order mode in the stem when $m$ is even, and into the $(2m+1)^{th}$-order mode in the stem when $m$ is odd [39]. Likewise, two anti-phase $m^{th}$-order modes in the arms are transformed into the $(2m+1)^{th}$-order mode in the stem when $m$ is even, and into the $(2m)^{th}$-order mode in the stem when $m$ is odd. Figure 2(a) illustrates how this principle affects Y1 operation. Since only two modes are supported by the Y1 stem, a TE₀ (red) mode at the stem results in two in-phase TE₀ modes at the arms, whereas TE₁ (orange) mode at the stem results in two anti-phase TE₀ modes at the arms. Figure 2(b) shows the extension of this behavior to four mode operation in Y2. Operation for TE₀ (red) and TE₁ (orange) is the same as in Y1, whereas injection of TE₂ (green) and TE₃ (purple) modes through the stem waveguide generates two anti-phase TE₁ or two in-phase TE₁ modes at the arms, respectively. Therefore, by cascading Y1 and Y2, and judiciously tailoring the phase relations induced by the rest of the MCMD, mode conversion and multiplexing between up to four modes can be achieved. We will hence study the phase shift induced by the 4×4 MMI coupler, and subsequently design a phase shifter architecture that satisfies the phase distributions imposed by the cascaded Y-junctions.

Bachmann *et al.* already derived a set of equations to calculate the phase relations of N×N MMI couplers [40]. At this point, it is important to mention that the definition of the phase in this work is $\varphi = \beta x - \omega t$, where $\beta$ is the phase constant (also known as propagation constant), $x$ is the propagation direction

<figure><img src="image_3.png" /><figcaption>**Fig. 2.** Schematic and principle of operation of a multimode symmetric 1×2 Y-junction for (a) a two-mode stem, and (b) a four-mode stem.</figcaption></figure>

and the term $-\omega t$ corresponds to the temporal dependence. As in [40] the authors used the opposite phase convention, i.e., $= \omega t - \beta x$, equations can be rewritten as follows:

$$i + j \text{ even: } \varphi_{ij} = -\varphi_0 - \pi - \frac{\pi(j-i)(2N-j+i)}{4N}, \quad (1)$$

$$i + j \text{ odd: } \varphi_{ij} = -\varphi_0 - \frac{\pi(j+i-1)(2N-j-i+1)}{4N}, \quad (2)$$

where $\varphi_0$ is a constant phase, $i$ and $j$ are the indices of the N inputs and outputs, respectively. Using Eqs. (1) and (2), the phase relations of a 4×4 MMI coupler can be calculated as shown in Table 1. Please note the input and output numbering in Fig. 3.

<table><caption>**Table 1**

Calculated phase relations $\varphi_{ij}$ of a 4×4 MMI coupler.</caption><thead><tr><th></th><th>j</th><th>1</th><th>2</th><th>3</th><th>4</th></tr></thead><tbody><tr><th>i</th><td></td><td></td><td></td><td></td><td></td></tr><tr><th>1</th><td></td><td>-π</td><td>-3π/4</td><td>-7π/4</td><td>-π</td></tr><tr><th>2</th><td></td><td>-3π/4</td><td>-π</td><td>-π</td><td>-7π/4</td></tr><tr><th>3</th><td></td><td>π/4</td><td>-π</td><td>-π</td><td>-3π/4</td></tr><tr><th>4</th><td></td><td>-π</td><td>π/4</td><td>-3π/4</td><td>-π</td></tr></tbody></table>

We then calculate, for each input port, the resulting phase difference at the two upper output ports ($\Delta\varphi_{12}$) and the two lower ports ($\Delta\varphi_{34}$) as:

$$\Delta\varphi_{12} = \varphi_{i1} - \varphi_{i2}, \quad (3)$$

$$\Delta\varphi_{34} = \varphi_{i3} - \varphi_{i4}, \quad (4)$$

Calculated phase differences are shown in Table 2. Since phase evolution at both the MMIs and Y-splitters are fixed, we then need to design a combination of PSs (placement and phase shift values), that results in the required phase relations. As shown in Figure 1(a), we achieve this goal by including a first phase shifter (PS1) between inputs 3 and 2 of the MMI, with a phase shift of $\pi/2$; a second phase shifter (PS2) between outputs 2 and 1, with a phase shift of $-\pi/4$; and a third phase shifter (PS3) between outputs 3 and 4 with a phase shift of $3\pi/4$. An additional two-mode Y-junction (Y1) is included at MCMD port 2 to satisfy even-order modes phase conditions, as discussed hereunder.

Figure 4 shows the operation of the device working in multiplexer configuration, including the value of the phase relations at different locations for clarity. It should be noted that phase values have been calculated with

<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Schematic of a 4×4 MMI coupler, illustrating port and phase notations.</figcaption></figure>
<caption>**Table 2**</caption>

Calculated phase differences between MMI output pairs for each input.

<table><thead><tr><th><i>i</i></th><th>Δφ<sub>12</sub></th><th>Δφ<sub>34</sub></th></tr></thead><tbody><tr><td>1</td><td>−π/4</td><td>−3π/4</td></tr><tr><td>2</td><td>π/4</td><td>3π/4</td></tr><tr><td>3</td><td>5π/4</td><td>−π/4</td></tr><tr><td>4</td><td>−5π/4</td><td>π/4</td></tr></tbody></table>

respect to the mode phase at the input ports, which is considered to be zero for simplicity. When light is injected through MCMD port 1 [Fig. 4(a)], the combination of all the aforementioned phase relations results in all modes arriving in-phase at the arms of the cascaded Y-junctions. Thus, two in-phase TE₀ modes are coupled into Y1 stems, which subsequently generate the desired TE₀ mode at the multimode stem waveguide of Y2 (MCMD port 4).

When light is injected through MCMD port 2 [Fig. 4(b)], the combination of Y1 and PS1 results in simultaneous light coupling to MMI input ports 2 and 3, but with a π/2 phase difference. The phase relations at the output of the 4×4 MMI are calculated by multiplying the transfer matrix of the 4×4 MMI, according to Table 1, by the complex amplitudes at the input port. This in turn generates in-phase modes in the upper arms that are in anti-phase with the two in-phase modes in the lower arms at their arrival at the cascaded Y-junctions. This combination results in TE₁ generation at the MCMD output.

Finally, when light is injected through MCMD port 3 [Fig. 4(c)], that is, MMI input port 4, in-phase modes are generated in the middle arms, which are in anti-phase with the two in-phase modes generated in the top and bottom arms, before the cascaded Y-junctions. This

<figure><img src="image_5.png" /><figcaption>**Fig. 4.** Principle of operation of the proposed three-mode converter and multiplexer/demultiplexer for (a) TE₀, (b) TE₁, (c) TE₂ and (d) TE₃ mode multiplexing.</figcaption></figure>

results in anti-phase TE₁ modes at the output of Y1 stems, which subsequently generate the TE₂ mode at the MCDM output.

So far, we have only considered passive PSs, that is, PSs with a fixed phase shift. However, if the phase introduced by PS1 is 3π/2 instead of π/2, it is possible to generate the TE₃ mode at MCMD output [Fig. 4(d)]. For illustrative purposes, we represent this phase shift by switching the position of the tapers in PS1. This feature opens the possibility of extending MCMD operation to four modes using a single switchable PS.

### 3. Proof-of-concept results

To verify the principle of operation explained in the previous section, we firstly optimized each constituent (i.e., MMI, phase shifters and Y-junctions) for a design wavelength of 1550 nm. We chose a standard silicon thickness of H = 220 nm and an interconnection waveguide width of W₁ = 400 nm. Thus, symmetric Y-junctions are designed with stem widths of 2W₁ = 800 nm for Y1 and 4W₁ = 1600 nm for Y2. Geometrical parameters of the 4×4 MMI coupler, the phase shifters and the symmetric Y-junction are summarized in Table 3.

In order to evaluate the performance of each constituent element, the figures of merit for the MMI are the excess loss (EL), imbalance (IB) and phase error (PE):

<caption>**Table 3**</caption>

Geometrical parameters of the three-mode converter and multiplexer/demultiplexer with homogeneous waveguides.

<table><thead><tr><th>Constituent</th><th>Parameter</th><th>Value</th></tr></thead><tbody><tr><td>Waveguides</td><td>Width</td><td>W<sub>I</sub> 400 nm</td></tr><tr><td rowspan="4">MMI</td><td>Separation</td><td>W<sub>S</sub> 500 nm</td></tr><tr><td>Access width</td><td>W<sub>A</sub> 1.3 µm</td></tr><tr><td>Taper length</td><td>L<sub>T</sub> 6 µm</td></tr><tr><td>MMI width</td><td>W<sub>MMI</sub> 7.2 µm</td></tr><tr><td rowspan="2">Y1</td><td>MMI length</td><td>L<sub>MMI</sub> 91 µm</td></tr><tr><td>Arm width</td><td>W<sub>I</sub> 400 nm</td></tr><tr><td rowspan="4">Y2</td><td>Arm length</td><td>L<sub>Y1</sub> 5 µm</td></tr><tr><td>Stem width</td><td>2W<sub>I</sub> 800 nm</td></tr><tr><td>Arm width</td><td>2W<sub>I</sub> 800 nm</td></tr><tr><td>Arm length</td><td>L<sub>Y2</sub> 20 µm</td></tr><tr><td rowspan="2">PS1</td><td>Arm width</td><td>4W<sub>I</sub> 1.6 µm</td></tr><tr><td>PS width</td><td>W<sub>PS1</sub> 600 nm</td></tr><tr><td rowspan="2">PS2</td><td>PS length</td><td>L<sub>PS1</sub> 2.41 µm</td></tr><tr><td>PS width</td><td>W<sub>PS2</sub> 600 nm</td></tr><tr><td rowspan="2">PS3</td><td>PS length</td><td>L<sub>PS2</sub> 8.38 µm</td></tr><tr><td>PS width</td><td>W<sub>PS3</sub> 600 nm</td></tr><tr><td>PS length</td><td>L<sub>PS3</sub> 3.61 µm</td></tr></tbody></table>

$$EL_i \text{ [dB]} = -10\log_{10} \left( \sum_j |S_{ji}|^2 \right), \quad (5)$$

$$IB_i^{jk} \text{ [dB]} = 10\log_{10} \left( \frac{|S_{ji}|^2}{|S_{ki}|^2} \right), \quad (6)$$

$$PE_i^{jk}[\circ] = [\angle(S_{ji}/S_{ki}) - \varphi_{ideal}] \cdot 180/\pi, \quad (7)$$
<figure><img src="image_6.png" /><figcaption>**Fig. 5.** Electric field amplitude $|E|$ in the XY plane at the middle of the silicon layer for (a) TE₀, (b) TE₁, (c) TE₂ and (d) TE₃ mode multiplexing. Simulated transmittance to output port 4 as a function of the wavelength when TE₀ mode is launched into (e) input port 1, (f) input port 2 with PS1 = π/2, (g) input port 3 and (h) input port 2 with PS1 = 3π/2. Vertical lines indicate the bandwidth where IL < 1 dB (55 nm) and XT < -20 dB (60 nm) are achieved for all modes</figcaption></figure>

where $S_{ji}$ and $S_{ki}$ are the scattering parameters for input $i$ and outputs $j$ and $k$, and $\phi_{ideal}$ is the ideal phase relation depending on selected input and output ports as shown in Table 1. The designed 4x4 MMI exhibits EL < 0.54 dB, IB < ±0.4 dB and PE < ±0.32° at the wavelength of $\lambda_0$ = 1550 nm. Regarding the spectral response, EL < 2.15 dB, IB < ±8.1 dB and PE < ±46.03° are attained in the entire simulated wavelength range (1.45 – 1.65 µm).

Designed phase shifters introduce small phase deviations of only 0.12° for PS1, 0.13° for PS2 and 0.16° for PS3, with respect to their target phase difference at 1550 nm. However, considering the

simulated bandwidth of 200 nm, phase errors increase up to 9.84° for PS1, 22.28° for PS2, and 12.12° for PS3.

Symmetric Y-junctions Y1 and Y2 were also designed showing negligible excess losses and power imbalance between output ports at the design wavelength. More specifically, Y1 losses are lower than 0.01 dB for both TE₀ and TE₁ mode operation in the 1.45 – 1.65 µm wavelength range. Conversely, calculated excess losses for Y2 are below 0.15 dB for TE₀, TE₁, TE₂ and TE₃ mode operation within the same bandwidth.

Once all elements were optimized, two-dimensional finite-difference time-domain (FDTD) simulations of the whole MCMD were performed by applying the effective index method to the original three-dimensional structure [see Fig. 1(a)]. The simulated field distribution of the three-mode MCMD is shown in Fig. 5(a)-(d), demonstrating the successful implementation of the phase relations described in section 2.

Some ripples can be observed for TE₀ and TE₂ mode multiplexing in the stem waveguide of Y-junction Y2 [see Figs. 5(a) and 5(c)], which we attribute to a higher crosstalk between both modes compared to TE₁ and TE₃ mode multiplexing.

The transmittance as a function of the wavelength was computed for the complete MCMD [see Fig. 5(e)-(h)]. At the central wavelength of $\lambda_0$ = 1550 nm, insertion losses are lower than 0.53 dB, 0.79 dB and 0.59 dB for the generation of TE₀, TE₁ and TE₂ modes in the stem waveguide, respectively. Our device also exhibits a low crosstalk at the same wavelength with values below -21.61 dB for TE₀, -28.94 dB for TE₁ and -21.11 dB for TE₂.

By tuning the value of PS1 to 3π/2, TE₃ mode (instead of TE₁ mode) can be generated. In this case, insertion losses are below 0.75 dB, and the crosstalk is better than -28.79 dB, both at 1550 nm. These results corroborate the higher crosstalk for TE₂ mode operation, which leads to a slight ripple in the field distribution at port 4.

Regarding performance across the spectrum, insertion losses lower than 1 dB are attained for a 55 nm bandwidth (1542 – 1597 nm), whereas the crosstalk is below -20 dB for a 60 nm bandwidth (1537 – 1597 nm) as shown with vertical lines in Fig. 5. These results prove the correct operation of the proposed architecture, but the overall bandwidth is significantly limited by the narrow spectral response of both the MMI and PSs.

## 4. SWG performance enhancement
<figure><img src="image_7.png" /><figcaption>**Fig. 6.** Simulated performance of the 4x4 SWG MMI including (a) excess loss, (b) imbalance and (c) phase error between output ports. (d) Phase error of each SWG PSs as a function of the wavelength.</figcaption></figure>

To overcome these bandwidth limitations, we propose the MCMD with SWG metamaterials shown in Fig. 1(b). SWG waveguides are periodic dispositions of core and cladding materials that act as a homogeneous anisotropic metamaterial with an equivalent refractive index $\mathbf{n}_{\text{eq}} = \text{diag}[n_{yy}, n_{yy}, n_{xx}]$ that can be tailored by judiciously designing the period and duty cycle of the SWG structure [20,22]. Since the beat length of SWG MMIs is given by $L_{\pi}(\lambda) \approx (4W_e n_{xx}^2)/(3\lambda n_{yy})$ [41] and the phase shift of SWG PS is given by $\Delta\Phi(\lambda) \approx [(\pi\lambda n_{yy}L_{PS})(W_{e,D}^{-2} - W_{e,R}^{-2})]/(4n_{xx}^2)$ [43], where $W_e$ is the effective width of the corresponding SWG waveguide, the term $n_{xx}^2/n_{yy}$ can be engineered through

SWG design to be proportional to $\lambda$, hence greatly extending the operational bandwidth of both components. The design of each of the constituents of the SWG MCMD was performed by individual three-dimensional FDTD simulations. The three symmetric Y-junction labeled Y1 maintain the same geometrical dimensions as those used for the conventional multiplexer for the arm and stem widths (see Table 3), but arm length was shortened to $L_{Y1} = 2$ µm. Y-junction Y2 was slightly redesigned to reduce the crosstalk between TE₀ and TE₂ modes by increasing the length of the arms to $L_{Y2} = 40$ µm.

A procedure similar to those already reported in [41,42] was followed for the optimization of the 4x4 SWG MMI. We restrict the value of the duty cycle (DC = $a/\Lambda$) to 0.5 in order to maximize the minimum feature size for a given period ($\Lambda$) [see Fig. 1(d)]. We explored then different periods and found that $\Lambda = 222$ nm significantly flattens the beat length across the spectrum. Compared to the conventional MMI section design, the width $W_{SMMI}$ is increased by 0.8 µm but the length $L_{SMMI}$ is reduced by more than half to ~41.3 µm. To increase the quality of the interferometric patterns formed in the MMI, the access width is $W_B = 1.7$ µm and the separation is reduced to $W_R = 0.3$ µm. The transition between the interconnection waveguides ($W_I = 400$ nm) and the access to the MMI section ($W_B$) is performed by means of adiabatic SWG tapers with a length $L_{ST} = 13.32$ µm. The performance of the 4x4 SWG MMI is shown in Fig. 6(a)-(c). Owing to the symmetry of the structure, only the results obtained when injecting light into input ports 1 and 2 are depicted. It is observed that the device exhibits EL < 0.77 dB, IB < ±1 dB and PE < ±8.02° within a broad bandwidth of 200 nm (1.45 – 1.65 µm).
To drastically extend the operating bandwidth of the
nanophotonic phase shifters, we build upon the strategy
we recently reported in [43] to develop SWG phase
shifters SPS1, SPS2 and SPS3. Notwithstanding, here
we employ four parallel SWG waveguides of two
different widths to implement SPS2 and SPS3. That is,
each PS has three identical reference SWG waveguides
with width $W_D$, and one dissimilar SWG waveguide
with width $W_R$. Both the reference and dissimilar
waveguides have a length of $L_{SPS}$. Note that for SPS1
this configuration is not necessary as only two MMI
inputs are illuminated for $TE_1$ and $TE_3$ mode
generation. Analogous to the 4x4 SWG MMI, a flat
phase shift can be achieved by judicious selecting the
SWG period and duty cycle. A duty cycle of 0.5 was
fixed to maximize MFS, while a period of 200 nm
resulted in minimum phase shift deviation. In order to
induce $\pi/4$, $\pi/2$, and $3\pi/4$ phase shifts, we selected
respectively $W_{D2} = 1.8$ µm, $W_{R2} = 1.6$ µm, $L_{SPS2} =
6.2$ µm and $L_{ST2} = 3.0$ µm for SPS2; $W_{D1} = 1.8$ µm,
$W_{R1} = 1.6$ µm, $L_{SPS1} = 16.8$ µm and $L_{ST1} = 3.0$ µm
for SPS1; and $W_{D3} = 1.8$ µm, $W_{R3} = 1.6$ µm, $L_{SPS3} =
28.2$ µm and $L_{ST3} = 3.0$ µm for SPS3. The simulated
phase shifts are shown in Fig. 6(d). Negligible
deviations can be appreciated with phase shift errors as
small as 2.29° for SPS1, and 1.15° for SPS2 and SPS3
within the entire 1.45 – 1.65 µm wavelength range.

5. SWG results

The simulation of the entire MCMD is quite resource-
intensive and time-consuming due to the device
footprint and the need for a fine mesh to simulate SWG-
based devices (e.g., minimum mesh of 10 nm in x
direction, 20 nm in y direction, and 22 nm in z
direction). Thus, instead of performing the full device
simulation, we leverage the S-parameter matrices
calculated during the design process and concatenate all
of them using a circuit simulator to obtain the S-
parameter matrix and hence the spectral response of the
complete device. The circuit simulator enables
bidirectional signals to be accurately simulated,
including coupling of modes in the single elements.
Figure 7 shows the overall transmittance of the SWG
MCMD. Insertion losses (ILs) are lower than 0.37 dB,
0.47 dB and 0.37 dB for TE₀, TE₁ and TE₂
multiplexing, respectively, at the central wavelength of
λ₀ = 1550 nm. Moreover, low crosstalk (XT) is
achieved at the same wavelength with values below -
21.54 dB for TE₀, -32.89 dB for TE₁ and -21.24 dB for
TE₂ multiplexing.

When SPS1 takes the value of 3π/4, insertion
losses for TE₃ multiplexing reach a low value of 0.47
dB at 1550 nm, while crosstalk values are lower than -
39.48 dB for the same wavelength.

<figure><img src="image_8.png" /><figcaption>**Fig. 7.** Simulated transmittance as a function of the wavelength of the MCMD with SWG metamaterials when TE₀ mode is launched into (a) input port 1, (b) input port 2 with SPS1 = π/2, (c) input port 3 and (d) input port 2 with SPS1 = 3π/2. Vertical lines indicate the bandwidth where IL < 1 dB (183 nm) and XT < -20 dB (161 nm) are achieved for all modes simultaneously.</figcaption></figure>
<table><caption>**Table 4**
Performance comparison of state-of-the-art three- and four-MCMD based on MMI couplers.</caption><thead><tr><th>Ref.</th><th>IL [dB]</th><th>XT [dB]</th><th>BW [nm]</th><th>Length [µm]</th><th>No. of modes</th></tr></thead><tbody><tr><td>[34]</td><td>&lt;0.71</td><td>&lt;-18</td><td>100</td><td>400</td><td>3</td></tr><tr><td>[35]</td><td>&lt;0.9</td><td>&lt;-17</td><td>40</td><td>400</td><td>3</td></tr><tr><td>[36]</td><td>&lt;1.1</td><td>&lt;-10</td><td>90</td><td>17.05</td><td>3</td></tr><tr><td>[37]</td><td>&lt;1.3</td><td>&lt;-15</td><td>40</td><td>120</td><td>3</td></tr><tr><td>[38]</td><td>&lt;3.3</td><td>&lt;-14</td><td>70</td><td>173</td><td>4</td></tr><tr><td>[38]</td><td>&lt;2.25</td><td>&lt;-15</td><td>140</td><td>123</td><td>4</td></tr><tr><td>This work</td><td>&lt;1.18</td><td>&lt;-20</td><td>161</td><td>181.14</td><td>3</td></tr></tbody></table>

This design also shows an excellent performance over a broad bandwidth (BW) of 200 nm with insertion loss lower than 1.18 dB and crosstalk below -16.53 dB. Insertion losses decrease to 1 dB when the bandwidth is restricted to 183 nm (1450 – 1633 nm), whereas a crosstalk below -20 dB is achieved over a 161 nm bandwidth (1489 – 1650 nm). For the sake of comparison, Table 4 summarizes the performance of other three- and four-mode MCMD that are based on MMI couplers and have been reported in the state of the art. To the best of our knowledge, it is the first time such low losses and crosstalk are achieved in an outstanding 161 nm wavelength range.

## 6. Conclusions

In this work, we have proposed a novel architecture to scale the number of multiplexed modes of mode converters and multiplexers based on MMI couplers. Unlike other reported architectures that use unconventional 1x4 Y-junctions or 1x3 Ψ-junctions, here we employ symmetric 1x2 Y-junctions arranged in a conventional cascaded configuration. The design methodology was proposed on the basis of a two-dimensional model with conventional homogenous components (i.e., without patterning the silicon waveguide). The conventional mode converter and multiplexer features sub-decibel insertion loss and crosstalk better than -20 dB in the 1542 – 1597 nm wavelength range. Once the principle of operation was verified, we redesigned and optimized the mode converter and multiplexer by incorporating subwavelength grating metamaterials to leverage the additional degrees of freedom they introduced into the design. A broad design bandwidth of 161 nm for insertion losses below 1.18 dB and crosstalk lower than -20 dB was confirmed by 3D FDTD simulations, comparing very favorably to state-of-the-art three- and four-mode converters and multiplexers. The effect of temperature variations of each device was also studied for a temperature range of ΔT = 20 K centered at 300 K, showing negligible performance degradation. Thermoptic coefficients of $dn_{si}/dT = 1.9 \times 10^{-4} K^{-1}$ and $dn_{si0_2}/dT = 9.3 \times 10^{-6} K^{-1}$ were considered for the silicon core and silica cladding respectively [44]. The crosstalk between TE₀ and TE₁ modes could be further

reduced by including optimized Y-junction geometries that mitigate the effect of the non-perfect tip at the junction [45-47]. Scaling the number of modes that can be converted and multiplexed using the proposed architecture will require adding a larger number (N) of cascaded Y-junction stages, which ultimately impose the number of inputs and outputs of the MMI (2^N), as well as the largest order mode that can be multiplexed (2^N-1). However, the synergy between the phase relations of the MMI and the cascaded Y-junctions must be specifically studied for each device to determine the number, position and induced phase shift of the phase shifters. We believe that our design strategy will open promising prospects for the development of high-performance mode converters and multiplexer based on MMI couplers with a high channel count.

## Credit authorship contribution statement

**David González-Andrade:** Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Writing – original draft, Visualization. **Irene Olivares:** Methodology, Software, Validation, Formal analysis, Data curation, Writing – review & editing. **Raquel Fernández de Cabo:** Software, Validation, Data curation, Writing – review & editing. **Jaime Vilas:** Writing – review & editing. **Antonio Dias:** Resources, Writing – review & editing, Project administration, Funding acquisition. **Aitor V. Velasco:** Resources, Writing – review & editing, Supervision, Project administration, Funding acquisition.

## Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

This work has been funded in part by the Spanish Ministry of Science and Innovation (MICINN) under grants RTI2018-097957-B-C33, PID2020-115353RA-100; the Spanish State Research Agency (MCIN/AEI/10.13039/501100011033); the Community of Madrid – FEDER funds (S2018/NMT-4326); the European Union – NextGenerationEU through the Recovery, Transformation and Resilience Plan (DIN2020-011488, PTQ2021-011974); and the European Union's Horizon Europe research and innovation program under the Marie Sklodowska-Curie grant agreement N° 101062518.

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