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Abstract
A compact and broadband silicon mode-order converter (MOC) scheme by employing reciprocal mode evolution between asymmetric input/output taper and bricked subwavelength gratings (BSWG) is proposed. In the proposed MOC, a quasi-TE0 mode is generated in the BSWG region, which can be regarded as an effective bridge between the two TE modes to be converted. Flexible mode conversion can be realized by only choosing appropriate structure parameters for specific mode transitions between input/output modes and the quasi-TE0 mode. By combing 3D finite difference time domain (FDTD) and particle swarm optimization (PSO) method, TE0-TE1 and TE0-TE2 MOCs are optimal designed, which can efficiently convert TE0 mode to TE1 and TE2 modes with lengths of 9.39 µm and 11.27 µm, respectively. Results show that the insertion losses of <1 dB and crosstalk of <-15 dB are achieved for both TE0-TE1 and TE0-TE2 MOCs, the corresponding working bandwidth are 128 nm (1511∼1639 nm) and 126 nm (1527∼1653 nm), respectively. Additionally, the MOCs can be fabricated with only single etch step with minimum feature size of 145 nm.
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1. Introduction
Photonic integrated circuit (PIC) is an enabling technology for high-density and cost-effective optical interconnect [1]. Facing with the increasing capacity requirements of on-chip optical interconnects, mode division multiplexing technology, which leverages the different spatial eigenmodes at the same wavelength as independent channels to transmit optical signals, has attracted tremendous interest [2]. Over past years, various mode-division multiplexing (MDM) devices have been demonstrated on the silicon-on-insulator (SOI) platform, such as mode (de) multiplexers [3,4], mode-order converters [5,6], multimode waveguide bends [7,8] and multimode waveguide crossings [9]. Among them, mode-order converters (MOCs) that can convert the fundamental mode to high-order modes are key components in MDM systems.
More recently, a variety of schemes for MOCs have been proposed, including asymmetric direction couplers (ADC) [10], Y-branch [11], Mach-Zehnder interferometer (MZI) [12], waveguide Bragg grating [13], refractive index perturbation structures [14,15], and subwavelength structure [16]. ADC can realize phase matching between the input fundamental mode and output high-order mode, but such device is limited by the relative narrow bandwidth and fabrication tolerances owing to the strict phase matching condition [10]. MZI is commonly used to realize the desired mode conversion by introducing an extra phase shift component between two arms of the MZI structure [12]. Waveguide Bragg grating, a normally employed component in the field of optical sensing system, introducing specified phase shift to achieve the efficient mode conversion [13]. However, all of them suffer from relatively large device lengths (>50 µm), which limits their high-density integration. In order to efficiently shrink the device size, polygonal etching slots and tilted slots are proposed to generate the required refractive index distributions with the conversion lengths of 24 µm [14] and 5.47 µm [15], respectively. In addition, periodic tilted subwavelength slots are introduced to realize the conversion between TE0 and TE1 modes with the device length of 5.75 µm [16]. However, two-step lithography and high-precision shallow etching processes are required. Besides, prevailing topology optimization [17], inverse design [18], and deep learning [19] methods can also be exploited to achieve the optimal refractive index distributions for mode conversion, while the formed irregular refractive index patterns are not easy for fabrication and the optimization processes are usually time-consuming.
Subwavelength gratings (SWGs), which can tailor the dispersion properties and refractive indices of photonic devices are widely used to achieve broadband operation and reduce device footprints [20–22]. For example, a high-order mode (de) multiplexer was proposed using SWG-based directional couplers, whose insertion loss lower than 5.2 dB in 1520 nm∼1570 nm and device footprint is less than 507×5.29 µm2 [23]. In Ref. [24], an ultra-broadband mode (de)multiplexer composed of three-waveguide coupler based on SWG has been demonstrated with working bandwidth exceeding 300 nm when the mode crosstalk <-15 dB, whose length is about 36.5 µm. In Ref. [25], Sun et al. have reported an ultra-broadband power coupling and mode-order conversion based on trapezoidal SWGs, whose conversion loss and crosstalk are below 1 dB and -12.5 dB, respectively. However, relative long length of 62.5 µm is required due to the adiabatic coupling between waveguides. Generally, it is imperative and challenging to realize mode conversion with a small footprint, low loss, low crosstalk, as well as broad working bandwidth. In addition, employing the full potential of SWGs is also challenging with current wafer-scale fabrication technologies. Recently, a bricked subwavelength grating (BSWG) which can synthesize a tailorable biaxial crystal with control over its anisotropy and dispersion has been proposed to alleviate the strict fabrication requirements of the conventional SWG structures [26].
In this work, we propose a compact silicon-based MOC scheme by employing BSWG connected with asymmetric input/output waveguide tapers that can convert the fundamental TE mode to high order TE modes. In the proposed MOC, a quasi-TE0 mode is generated in the BSWG region, which can be regarded as an effective bridge between the two TE modes to be converted. In addition, benefiting from the lower dispersion of BSWG compared with traditional SWG, broad working bandwidth is achieved. As a proof of concept, two MOCs that can convert TE0 mode to TE1 (TE2) mode are optimal designed by using the particle swarm optimization (PSO) method. For the TE0-TE1 MOC, the device length is 9.39 µm, the working bandwidth is 128 nm (1511∼1639 nm) when the insertion loss is lower than 1 dB and the crosstalk is below -15 dB. For the TE0-TE2 MOC, the device length is 11.27 µm, whose working bandwidth is 126 nm (1527∼1653 nm) when the insertion loss is lower than 1 dB and the crosstalk is below -15 dB.
2. Structure and principle
Figure 1 depicts the three-dimensional (3D) and 2D schematic of the proposed MOC, as well as part enlarged view of the BSWG. Here, the device involves three parts, including the Taper1, Taper2 and the BSWG. The input waveguide Taper1 is tapered from w1 = 500 nm to ws = 145 nm in a length of Ltp1 and the output waveguide Taper2 is tapered from w4 to w3 in a length of Ltp2. The position of Taper2 is denoted by the length of wp as shown in Fig. 1(b). The BSWG is composed of alternating strips of the Si and SiO2 arrayed with a subwavelength period Λz = 290 nm along the propagation direction (z-axis), the Si strips are periodically partitioned along the x-axis with a period Λx = 290 nm, and the resulting Si blocks are then alternately shifted by a distance Δz in the z-direction. The duty cycles in the x and z directions are defined as fx = ax/Λx = 0.5 and fz = az/Λz=0.5, respectively, as it can produce the largest pixel dimensions (145×145 nm2). The silicon wire width a in the BSWG is equal to ax and az, represented by a = ax = az. The thickness of silicon is 220 nm and the refractive indices of Si and SiO2 at λ=1550 nm are taken as 3.476 and 1.444, respectively. The whole device is covered by SiO2 claddings.
Fig. 1. (a) 3D view of the proposed MOC; (b) 2D view of the proposed MOC and part enlarged view of the BSWG.
The proposed device operates in the deep subwavelength regime, that is to say Λ<<λ. According to the Rytov formulas, the equivalent refractive indices of the conventional SWG can be calculated by [27]:
where n|| and n⊥ represent polarization parallel/perpendicular to z-direction, respectively. Here, a full-vectorial mode solver based on finite-difference frequency-domain (FDFD) method is employed to investigate the characteristics of the waveguide taper embedded in the BSWG, in which the BSWG is treated as a homogenous anisotropic medium, thus the equivalent refractive index of the BSWG nBSWG can be represented by its diagonal refractive index tensor ($\overline {\Delta z}$) [26]:Figure 2 depicts the working principle of the proposed device. Figure 2(a) shows the top view of theTaper2 and BSWG region and Fig. 2(b) shows cross-section view of the Taper2 embedded in the BSWG, which can be regarded as a silicon waveguide inserted between two BSWG regions. Figure 2(c) and Fig. 2(d) show the effective indices and the corresponding field profiles (Ex) vary as the Taper2’s width wtp for the TE1-to-TE0 MOC and the TE2-to-TE0 MOC under the condition of $\overline {\Delta z}$= 0.16, respectively. As shown in Fig. 2(c) and (d), the modal field of TE1 (TE2) closed to the tapering boundary is gradually squeezed out and enhanced at the interface between the Taper2 and the BSWG when wtp decreases. Then the extruded modal field is separated from the Taper2 and guided as a quasi-TE0 mode in the BSWG region. Moreover, explanations of such a mode evolution process from a single-mode like component into higher-order modes can be found in Ref. [28]. To demonstrate the mode evolution process for TE1 (TE2) mode to quasi-TE0 mode, three-dimensional finite difference time domain (3D FDTD) simulations are performed and the results are shown in Fig. 2(e) and Fig. 2(f). It can be seen that the injected TE1 (TE2) mode is gradually transferred to the quasi-TE0 mode from the Taper2 into the BSWG region when the Taper2 is narrowed from 0.9 µm to 0.2 µm (1.4 µm to 0.2 µm), which is consistent with mode analysis shown in Fig. 2(c) and Fig. 2(d). And the transition lengths for TE1 (TE2) mode to quasi-TE0 mode are about 3.3 µm and 4.1 µm, respectively. Based on the same mode evolution processes, the TE1-to-TE0 and TE2-to-TE0 mode-order conversion can be obtained by introducing another Taper1 in the other side of the BSWG region to collect the quasi-TE0 mode in the BSWG and then convert it to the TE0 mode of the Taper1, as shown in Fig. 2(g) and Fig. 2(h).
Fig. 2. Working principle for the proposed device (a) Top view and geometrical parameters of the Taper2 and BSWG region; (b) Cross-section view of the Taper2 embedded in the BSWG; Effective indices and corresponding field profiles (Ex) vary as the Taper2’s width wtp for (c) TE1-to-TE0 MOC and (d) TE2-to-TE0 MOC; Simulated major component (Ex) of TE mode field evolutions when input (e) TE1 mode and (f) TE2 mode; Mode field evolutions of (g) TE1-to-TE0 and (h) TE2-to-TE0 mode-order conversions.
Here, to illustrate the advantages of BSWG comparing with traditional SWG, we take the TE0-TE1 MOC as an example. The optimized width of BSWG w2= 2.1 µm. Figure 3 shows the mode effective indices of the TE0 Floquet-Bloch modes propagating along the z-axis when $\overline {\Delta z} = 0$ and $\overline {\Delta z} \ne 0$, calculated by 3D FDTD method for photonic structures [29]. It should be noted that TE0 Floquet-Bloch mode here is different from the quasi-TE0 mode in Fig. 2. TE0 Floquet-Bloch mode is the eigenmode in the BSWG while the quasi-TE0 mode is a TE0-like mode, which is similar with the TE0 Floquet-Bloch mode generated in the mode evolution process. The optimized shifting value $\overline {\Delta z}$= 0.16 for the TE0-TE1 MOC is obtained by using the particle swarm optimization method and the details are discussed in the following part. As shown in Fig. 3, it is obviously that the mode effective index and the dispersion of the TE0 Floquet-Bloch mode are reduced when $\overline {\Delta z} \ne 0$ is introduced. Compared with conventional SWG (minimum feature size of ∼100 nm), the BSWG also has a uniform grid and pixel dimensions as large as 145×145 nm2 in the near-infrared band. Generally, the introduction of the BSWG decreases the wavelength dependence of the mode effective index, which is benefit to broadband working and can alleviate the fabrication difficulty.
Fig. 3. Calculated mode effective indices of the TE0 mode as $\overline {{\Delta }z}$ varies for the TE0-TE1 MOC.
3. Results and discussion
In order to optimize the proposed MOC with BSWG based on mode evolution and conversion, the structure parameters (w2, w3, w4, wp, Ltp1, Ltp2, Lg1, $\overline {{\Delta }z}$) shown in Fig. 1 has been optimized at the wavelength of 1.55 µm by combing 3D FDTD and PSO methods. The figure of merit (FOM) is defined as:
where i = 1 or 2, PTEi represents the optical power of TE mode obtained at the output waveguide.To simply illustrate the optimization process, we take TE0-TE1 MOC as an example. A TE0-TE1 MOC with a specific combination of parameters is represented as a “particle”. The particle’s position and velocity can be updated by utilizing the equations given below [30]:
Table 1. Optimized design parameters for the proposed MOCs
The electric field distributions of TE0-TE1 and the TE0-TE2 MOCs are simulated and shown in Fig. 6. It can be seen from Fig. 6(a) that the incident TE0 mode can be converted to the TE1 mode with a conversion length of 9.39 µm. To verify the scalability of our proposed structure, a TE0-TE2 MOC is also demonstrated as shown in Fig. 6(b), where the launched TE0 mode can be converted to TE2 mode with a conversion length of 11.27 µm.
Fig. 6. Electric-field evolution with the simulated Ex distribution for the proposed (a) TE0-TE1 MOC and (b) TE0-TE2 MOC.
Figure 7 (a) and (b) show the transmission spectra of the proposed TE0-TE1 and TE0-TE2 MOCs, in which all the supported eigenmodes are listed. For the TE0-TE1 MOC, the TE0-TE1 mode conversion loss is lower than 1 dB and the crosstalk is below -15 dB in the wavelength range of 1511 nm∼1639 nm, including S, C, L and U band. For the TE0-TE2 MOC, the TE0-TE2 mode conversion loss is lower than 1 dB and the crosstalk is below -15 dB in the wavelength range of 1527 nm∼1653 nm, including S, C, L and U band. Concerning about the transmission responses in Fig. 7, the mode conversion loss drops and the crosstalk increases quickly at the short wavelength margin near 1500 nm. These are caused by Bragg reflection since the wavelength is getting close to the Bragg wavelength.
In order to investigate the fabrication tolerances of the proposed MOCs, we have simulated the transmission responses of the proposed device with the deviation of Taper1 tip width Δws, Taper2 tip width Δw4, silicon wire width Δa and shifting δz in the BSWG at λ=1.55 µm as shown in Fig. 8. Specially, the variation of shifting Δz is denoted by δz. For the TE0-TE1 MOC, the insertion losses are < 1.5 dB and the crosstalk are <-10 dB within the range of -50 nm <Δws < 50 nm, -100 nm<Δw4< 100 nm and -14 nm <Δa < 10 nm, - 13 nm <δz < 11 nm. For the TE0-TE2 MOC, in the range of -50 nm <Δws < 50 nm, -100 nm<Δw4< 100 nm and -8 nm <Δa < 10 nm, - 14 nm<δz< 9 nm the insertion losses are < 2 dB and the crosstalk are <-9 dB. The proposed device allows a relatively large fabrication tolerance and shows good potential for the large-scale fabrication.
Fig. 8. The transmission spectra by changing the variations of width (Δws, Δw4, Δa, δz) for the proposed (a-d) TE0-TE1 MOC and (e-h) TE0-TE2 MOC.
Table 2 compares our proposed structures with several reported MOCs. It shows that our work has greatly broadened the working bandwidth by introducing of the BSWG with low dispersion. Compared with Ref. [28], our device has larger minimum feature size of 145 nm, smaller lowest crosstalk <-20 dB at 1.55 µm as well as broader bandwidth >125 nm, as shown in Fig. 7. Meanwhile, our scheme can flexibly achieve fundamental mode to high-order modes with compact device size, low insertion loss, low crosstalk, as well as broad working bandwidth.
Table 2. Comparison of various MOCs.a
To further verify the scalability of our proposed scheme, we also demonstrate a TE1–TE2 MOC. The conversion between the TE1 and TE2 modes has been simulated using the optimized structure parameters listed in Table 3. The electric field distribution is shown in Fig. 9. It can be seen that the incident TE1 mode can be converted to the TE2 mode with a conversion length of 11.87 µm.
Table 3. Optimized design parameters for the proposed TE1-TE2 MOC
4. Conclusion
In conclusion, based on the mode evolution and coupling between the BSWG and the input/output asymmetric waveguide taper, a compact and broadband mode-order converter based on the BSWG is proposed. Two silicon-based MOCs employing the BSWG are demonstrated, which can convert the TE0 mode to the TE1 mode and TE2 mode, respectively. The conversion lengths are 9.39 µm and 11.27 µm for the TE0-TE1 and TE0-TE2 MOCs, respectively. The broad design bandwidth of 128 nm (1511∼1639 nm) and 126 nm (1527∼1653 nm) are achieved with the conversion loss of <1 dB and the crosstalk value of <-15 dB for the TE0-TE1 and TE0-TE2 MOCs, with pixel size dimensions as large as 145×145 nm2. Furthermore, our proposed scheme can be scaled to realize high-order waveguide mode conversions. We believe that the proposed mode order conversion scheme has potential applications in high-dense on-chip MDM systems and optical signal processing system.
Funding
National Key Research and Development Program of China (2018YFB2201800); National Natural Science Foundation of China (62105061, 62171118).
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.
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