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Compact silicon three-mode multiplexer by refractive-index manipulation on a multi-mode interferometer

Zhen Wang, Chunhui Yao, Yong Zhang, and Yikai Su

Open Access Open Access

Abstract

Mode-division multiplexing can scale the capacity of optical communications and optical interconnects. We demonstrate an ultra-compact and fabrication-error tolerant silicon three-mode multiplexer by shallowly etching rectangular trenches on a multi-mode interferometer. Depending on the selected input port, the TE0 mode is converted to the eigenmodes of the bus waveguide. These modes are coupled to each other owing to the refractive-index perturbation induced by the shallow trenches and finally converted to a selected spatial mode at the output. A three-mode multiplexing device is experimentally demonstrated with a footprint of 2.00 × 17.05 µm2. The bandwidths for the three channels are >70 nm with crosstalk values below –10 dB.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The demand for communication capacity has grown exponentially, and various physical dimensions of an optical carrier can be employed in multiplexing technologies [1]. Mode-division multiplexing (MDM) scales the number of channels in fiber communications and has been widely exploited. Silicon mode multiplexers are essential components in on-chip MDM systems [2]. A series of on-chip mode multiplexers have been reported, mainly based on four approaches: 1) (cascaded) asymmetric directional couplers (DCs) [3–9]; 2) multi-mode interferometers (MMIs) with phase shifters [10–12]; 3) asymmetric Y branches [13,14]; and 4) 1 × N junctions based on inverse design [15,16].

Recently, refractive-index manipulation on a silicon waveguide has provided a new solution to achieve a compact device with a high fabrication error tolerance [17–21], which can be employed to realize mode multiplexers with high performances. In [20], we reported a two-mode multiplexer based on refractive-index manipulation, and the number of channels can be further scaled.

In this paper, we experimentally demonstrate a silicon three-mode multiplexer based on a shallowly-etched MMI. Compared to the one-step mode conversion in our previous work [20], here a two-step mode conversion process is proposed with a larger feature size of the device to relax the fabrication resolution requirement. Three inputs are firstly mapped to different intermediate states, and in the second step these intermediate states are converted to three eigenmodes of the bus waveguide. These eigenmodes are manipulated by carefully designing the refractive index distributions. Attributed to the large field overlapping in the shallow etched regions on the bus waveguide, the couplings between the TE modes on the silicon waveguide are much stronger than that between the evanescent fields in conventional DCs, which results in a compact footprint. The number of channels of the mode multiplexer can be further scaled using a multi-step mode conversion process.

2. Device structure and operation principle

The schematic configuration of the proposed three-mode multiplexer is shown in Fig. 1. A silicon MMI with three input ports is placed on a silicon-on-insulator (SOI) wafer. We design the input waveguides tapered from 0.4 µm to 0.6 µm to broaden the mode spots and reduce the coupling losses at the interface between the tapers and the multi-mode waveguide. The width of the multi-mode waveguide is Wbus = 2 µm to support four guided modes because at least N+1 = 4 guided modes are needed in an N = 3 ports MMI. Three rectangle trenches realize the refractive-index perturbations on guided modes. The depths of the shallowly etched regions are fixed to be 60 nm, which equals the depth of the TE grating coupler used for vertical coupling, thus saving one lithography process.

 figure: Fig. 1.

Fig. 1. 3D schematic configuration of the proposed three-mode multiplexer.

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The MMI based on the self-imaging effect consists of a multi-mode waveguide and several single-mode waveguides. As shown in Fig. 2(a), if an MMI is excited by an incident light, the field profile is decomposed into the eigenmodes of the multi-mode waveguide. This process can be expressed as

$${E_{(0 )}} = \alpha {E_{\textrm{in}}}, $$
where α represents the transmission matrix of the interface between the input waveguides and the MMI. There is no exchange of energy between the eigenmodes in the MMI,
$${E_{\textrm{out}}} = {I_4}\alpha {E_{\textrm{in}}} = {I_4}{E_{(0)}}. $$

 figure: Fig. 2.

Fig. 2. Illustration of the mode conversions in (a) the MMI and (b) the three-mode multiplexer. Insets show the processes of mode conversion for each input.

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In the case of the three-mode multiplexer, we apply the shallow etchings on the MMI so that the original eigenmodes can couple to each other. The output of the three-mode multiplexer, Eout, can be written as

$${E_{\textrm{out}}} = {P_3}{E_{\textrm{in}}} = {P_3}{\alpha ^{ - 1}}\alpha {E_{\textrm{in}}} = {P_3}{\alpha ^{ - 1}}{E_{(0)}}, $$
where P3 is a three-order permutation matrix, P3α−1 is the transmission matrix of the three-mode multiplexer. In our previous work, a two-mode multiplexer was realized using one-step mode conversion. To verify the scalability of the proposed scheme, initially we applied the same one-step conversion method to the three-mode multiplexer. However, the optimized structure parameters showed that there was a slot with its width narrow than 100 nm. Due to lithography resolution, the depth of this slot would be significantly shallower than other trenches due to the slow etching rate. To avoid this practical issue, here we introduce a two-step mode conversion process with a larger feature size of the device to relax the fabrication resolution requirement. Firstly, we decompose the matrix, P3α−1, into a product of two new matrices, whose corresponding structures can be fabricated in practice. Thus, Eq. (3) can be rewritten as:
$$\begin{array}{c} {E_{out}} = {P_3}{\alpha ^{ - 1}}{E_{(0 )}}\\ = \gamma \beta {E_{(0 )}}\\ = \gamma {E_{(1 )}} \end{array}, $$
where E(1) is the intermediate state, β and γ are the transmission matrices of the first and second step, respectively, as shown in Fig. 2(b). The input state to the second stage is the output state of a previous stage. Through the introduction of two-step mode conversion, the dimension of the problem space is increased, and more matrix pairs that meet the requirements of Eq. (4) can be found. Among them, there exist solutions that can relax the fabrication requirements.

The first transmission matrix, α, is obtained by the eigenmode expansion method as shown in Table 1. For the device shown in Fig. 2(b), α represents the transmission coefficient between the input field and each eigenmode of the MMI. Note that the sums of the normalized power for different input ports are less than 1 because of the coupling losses.

Tables Icon

Table 1. The Complex Amplitudes and Powers of Different Modes at the Joint Between the Input Waveguides and the MMI

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To expound the mode conversion (β and γ) in the first and second steps, the coupled mode theory (CMT) is utilized to analyze the mode conversion along the device. The complex amplitudes of all guided modes along the propagation direction in the multi-mode waveguide can be calculated by a set of differential equations [18,19]:

$$- \frac{{\partial {A_p}}}{{\partial z}} = \sum\limits_{p = 0,p \ne q}^3 {j{\kappa _{pq}}{A_q}{e^{j({\beta _p} - {\beta _q})z}}} $$
where Ap is the complex amplitude of the TEp mode, βp is the propagation constant of the TEp mode (p = 0, 1, 2, or 3). The mode coupling coefficient κpq (k = 0, 1, 2, or 3) between the TEp and TEq mode, is defined as:
$${\kappa _{pq}}(z) = \frac{{\omega {\varepsilon _0}\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {{E}_p^\ast{\cdot} \Delta \varepsilon \cdot {{E}_q}dxdy} } }}{{\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {{{\mathbf u}_z} \cdot ({\mathbf{E}_p^\ast{\times} {{H}_q} + {{E}_p} \times {H}_q^\ast } )dxdy} } }}. $$
Here, Δɛ denotes the refractive-index perturbation.

Since Eq. (5) has no analytical solution, the refractive index perturbation Δɛ is optimized by particle swarm optimization (PSO) method in MATLAB to achieve the maximum output powers of TE0/TE1/TE2 modes [21–23]. We use rectangular shallow etchings on an MMI to form the two-step refractive-index perturbations and realize the multi-mode multiplexing. For simplicity, the etching patterns of the two steps are reversed, because their coupling coefficients κpq are of opposite signs due to the reversals of the signs of Δɛ. The eigenmode fields of the MMI, Ep, are calculated by Mode solutions (Lumerical). Since the depth of the shallow etching is fixed to 60 nm (160 nm < y < 220 nm), the width of the etching area in the first and the second steps can be determined by x1 and x2, respectively, and the corresponding lengths can be determined by z1 and z2, respectively. In the shallow etched area of step 1, Δɛ(x, y) = nSi2 − nAir2 (x1 < x < x2), while in the shallow etched area of step 2, Δɛ(x, y) = − (nSi2 − nAir2) (−1 µm < x < x1 and x2 < x <1 µm). In the unetched areas, Δɛ (x, y) = 0. Therefore, according to Eq. (5), the transmission matrices (β and γ) of the two steps can be calculated under the four parameters (x1, x2, z1, and z2). The figure-of-merit (FOM) is used to characterize the performance of the device:

$$FOM = \frac{1}{3}\sum\limits_{i = 1}^3 {{T_{i,i}}} ,$$
where T = abs(γβ)abs(γβ)T. Because of the conservation of energy, each column vector of abs(γβ) is close to the unit vector. If abs(γβ) is a transpose matrix, T is the unit matrix. In the optimization, 100 particles are used to search the parameters that make T close to the unit matrix. The problem space of parameters is −1 µm < x1, x2 <1 µm, 0 µm < z1, z2 < 20 µm. As shown in Fig. 3, after 20 iterations, the FOM converges to 0.96.

 figure: Fig. 3.

Fig. 3. The FOM vs iteration in the PSO.

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 figure: Fig. 4.

Fig. 4. The amplitudes of the four lowest order modes (TE0, TE1, TE2, and TE3) along the propagation for (a) Input #1, (b) Input #2 and (c) Input #3, respectively.

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The optimized parameters are x1 = 435 nm, x2 = 767 nm, z1 = 8.403 µm and z2 = 17.056 µm, which means W31 = 435 nm, W32 = 332 nm, L31 = 8.403 µm and L31 = 8.653 µm for #1→TE1, #2→TE0, and #3→TE2, respectively. Consequently, the amplitudes of different modes along the device can be calculated according to Eq. (5). As shown in Figs. 4(a)–4(c), the mode purities of the TE1, TE0, or TE2 reach the maximum values at z = 18.05 µm, respectively, while the other modes are effectively suppressed.

Figure 5(a) gives the top view of the three-mode multiplexer with rectangle trenches. Applying this structure in the finite-difference time-domain method (FDTD solutions, Lumerical), the simulated Ey profiles of the device for different inputs are obtained and shown in Figs. 5(b)–5(d). The light of TE0 mode is injected from either Input #1, #2, or #3 and can be decomposed to be a set of quasi-TE modes in the multi-mode waveguide [22]. These excited modes couple to each other along the propagation direction owing to the refractive index perturbation induced by the partially etched structures. Depending on the input port, the light that consists of four quasi-TE modes eventually evolves to the TE1, TE0, or TE2 mode, respectively. Hence, three input TE0 signals can be converted to TE1/TE0/TE2 modes and multiplexed at the output.

 figure: Fig. 5.

Fig. 5. (a) Top view of the proposed three-mode multiplexer. Simulated Ey distribution of the three-mode multiplexer for (a) Input #1, (b) Input #2, and (c) Input #3 port at the wavelength of 1550 nm.

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3. Device fabrication and experimental results

A commercial SOI wafer with a 220-nm-thick silicon layer on top of a 3-µm-thick silica layer was used to fabricate the proposed three-mode multiplexer. E-beam lithography (Vistec EBPG 5200+) and inductively coupled plasma dry etching were used to define the three-mode multiplexer structures and grating couplers on the SOI wafer. We designed a three-mode multiplexing system with two multiplexers cascaded back to back to test the performance of the proposed multiplexer. Magnified optical micrograph and scanning electron microscope (SEM) images of the fabricated devices are provided in Fig. 6. Note that we extended the shallowly etched regions to cover both edges of the bus waveguide to compensate for the overlay misalignment. A tunable continuous-wave laser (Keysight 81960A) and an optical power meter (Keysight N7744A) were used to characterize the proposed mode multiplexer. The grating couplers were used to couple the light in and out of the device in the experiment, and the period and duty cycle of the grating are 630 nm and 50%, respectively. The coupling loss was 6.1 dB/port at 1550 nm.

 figure: Fig. 6.

Fig. 6. (a) Optical and (b) SEM micrograph of the fabricated three-mode multiplexer.

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Figures 7(a)–7(f) present the simulated and measured transmission responses (T) of three output ports when the light is injected into input port #1/#2/#3, respectively. As the simulation results shown in Figs. 7(a), 7(c), and 7(e), at the wavelength of 1550 nm, the insertion losses are 1.1 dB, 0.8 dB, and 0.9dB, and the crosstalk values (the max powers of the unwanted modes) are −19.1 dB, −13.9 dB, and −14.8 dB for #1→TE1, #2→TE0, and #3→TE2, respectively. Across the wavelength range of 90 nm (1510 nm∼1600 nm) in the simulation, the crosstalk values are less than −10 dB. In Figs. 7(b), 7(d), and 7(f), the experiments show that at the wavelength of 1550 nm, the insertion losses are 2.4 dB, 1.8 dB, and 2.3 dB, and the crosstalk values are −15.6 dB, −15.7 dB, and −14.2 dB for #1→TE1, #2→TE0, and #3→TE2, respectively. In a wavelength range of 70 nm (1510 nm∼1580 nm), the crosstalk values are less than −10 dB. All the transmission spectra are normalized to the response of a pair of grating couplers with a single-mode waveguide on the same wafer. The relatively large ILs and crosstalk values in the experiment can be attributed to the fabrication errors. Also, the ILs of all three channels in the experiment are 1∼2 dB larger than the simulated data, possibly due to the lower IL of the TE gratings used in the normalization process.

 figure: Fig. 7.

Fig. 7. (a), (c), and (e) Simulated and (b), (d), and (f) measured mode-conversion transmissions of the proposed device for Input #1, Input #2, and Input #3, respectively.

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The fabrication error tolerances of the device are also numerically investigated at 1550 nm by varying the width and length of the shallow trenches. As shown in Figs. 8(a)–8(c), the crosstalk value remains < −10 dB with a W32 variation range of 90 nm (−50 nm ∼ +40 nm). Figures 8(d)–8(f) and 8(g)–8(i) indicate that the variations (−50 nm ∼ +50 nm) of L31 and L32 can negligibly degenerate the performance of the device. The simulated results demonstrate that the proposed multiplexers based on a shallowly-etched MMI have high fabrication tolerances to the structural deformation.

 figure: Fig. 8.

Fig. 8. The fabrication tolerances of (a)-(c) ΔW32, (d)-(f) ΔL31and (g)-(i) ΔL32 of the shallow etchings for different input ports.

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Table 2 compares our proposed device with several silicon mode (de-)MUXs. It can be seen that the demonstrated device significantly reduces the device size while maintaining similar bandwidth and crosstalk performance relative to the ADC-based mode multiplexers. Compared to the mode multiplexer based on the inverse design, the proposed designing method consumes less computation time. The simulation time would not significantly increase when the mode multiplexer towards higher-order mode scale.

Tables Icon

Table 2. The Performance of Various Silicon Mode (de-)MUXs

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4. Summary

In summary, we experimentally demonstrated a compact silicon three-mode multiplexer by using a shallowly-etched MMI structure. Attributed to the refractive index perturbation induced by asymmetric partial etching on a multi-mode waveguide, the initially orthogonal guided modes can be coupled to each other during the propagation. All the excited modes in the multi-mode waveguide are converted to a high purity mode depending on the selected port, thus three-mode multiplexing can be achieved. The footprint of the proposed three-mode multiplexer is 2.00 × 17.05 µm2. The crosstalk value is less than −10 dB in the wavelength range of 1510 nm∼1580 nm. Furthermore, the fabrication-error tolerances of the three-mode multiplexer are numerically analyzed, and the simulation results show that the proposed device features high tolerance to overlapping misalignment within 90 nm. This device can find applications in integrated on-chip mode and polarization division multiplexing communication and optical signal processing systems.

Funding

National Natural Science Foundation of China (61835008, 61860206001, 61975115).

Acknowledgment

We thank the Center for Advanced Electronic Materials and Devices (AEMD) of Shanghai Jiao Tong University (SJTU) for the support of device fabrication.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (8)
Fig. 1.
Fig. 1. 3D schematic configuration of the proposed three-mode multiplexer.

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Fig. 2.
Fig. 2. Illustration of the mode conversions in (a) the MMI and (b) the three-mode multiplexer. Insets show the processes of mode conversion for each input.

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Fig. 3.
Fig. 3. The FOM vs iteration in the PSO.

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Fig. 4.
Fig. 4. The amplitudes of the four lowest order modes (TE0, TE1, TE2, and TE3) along the propagation for (a) Input #1, (b) Input #2 and (c) Input #3, respectively.

View in Article | Download Full Size | PDF

Fig. 5.
Fig. 5. (a) Top view of the proposed three-mode multiplexer. Simulated Ey distribution of the three-mode multiplexer for (a) Input #1, (b) Input #2, and (c) Input #3 port at the wavelength of 1550 nm.

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Fig. 6.
Fig. 6. (a) Optical and (b) SEM micrograph of the fabricated three-mode multiplexer.

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Fig. 7.
Fig. 7. (a), (c), and (e) Simulated and (b), (d), and (f) measured mode-conversion transmissions of the proposed device for Input #1, Input #2, and Input #3, respectively.

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Fig. 8.
Fig. 8. The fabrication tolerances of (a)-(c) ΔW32, (d)-(f) ΔL31and (g)-(i) ΔL32 of the shallow etchings for different input ports.

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Tables (2)

Tables Icon

Table 1. The Complex Amplitudes and Powers of Different Modes at the Joint Between the Input Waveguides and the MMI

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Tables Icon

Table 2. The Performance of Various Silicon Mode (de-)MUXs

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Equations (7)

Equations on this page are rendered with MathJax. Learn more.

E ( 0 ) = α E in ,
E out = I 4 α E in = I 4 E ( 0 ) .
E out = P 3 E in = P 3 α − 1 α E in = P 3 α − 1 E ( 0 ) ,
E o u t = P 3 α − 1 E ( 0 ) = γ β E ( 0 ) = γ E ( 1 ) ,
− ∂ A p ∂ z = ∑ p = 0 , p ≠ q 3 j κ p q A q e j ( β p − β q ) z
κ p q ( z ) = ω ε 0 ∫ − ∞ ∞ ∫ − ∞ ∞ E p ∗ ⋅ Δ ε ⋅ E q d x d y ∫ − ∞ ∞ ∫ − ∞ ∞ u z ⋅ ( E p ∗ × H q + E p × H q ∗ ) d x d y .
F O M = 1 3 ∑ i = 1 3 T i , i ,
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