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Abstract
We propose and demonstrate theoretically both broadband and ultra-compact waveguide mode-order converters and polarization rotators based on asymmetrical nanostructures fabricated in silicon nanowires. A TE0-like to TE1-like mode-order converter with the footprint 0.8 × 5.3μm2 is realized by two cascaded trenches fully etched inside silicon nanowires. Within the wavelength interval between 1420nm and 1620nm, the transmittance of the device is larger than 0.95. The incident TE0-like mode is almost completely converted into TE1-like mode with purity larger than 0.9. Moreover, a polarization rotator or TE0-like to TM0-like mode converter with the footprint 0.4 × 6.4μm2 is realized by two cascaded trenches partially etched on the edges of silicon nanowires. Within the wavelength interval between 1360nm and 1560nm, the transmittance of the device is larger than 0.9. The incident TE0-like mode is almost completely converted into TM0-like mode with purity larger than 0.9. Our designs may lead to a new gateway towards manipulating the spatial modes and polarizations of waveguides in small footprint.
© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The development of silicon photonics provides an attractive platform for high-density photonics integration and on-chip optical information processing. However, the larger refractive index contrast of silicon nanowires results in high mode and polarization dependences for silicon photonic devices. Efficient routers to eliminate such dependences are mode-diversity and polarization-diversity schemes [1–7], respectively. Based on these two schemes, mode-division multiplexing [3–6] and polarization-division multiplexing [7] introduce two novel channels for the on-chip optical communications and catch great attentions.
As key components for the realization of mode-diversity and polarization-diversity schemes, broadband and ultra-compact silicon-waveguide mode-order converters and polarization rotators (PR) are in great demand. The typical modes converters for TE0-like to TE1-like mode of silicon waveguides proved by experiments [8,9] and theoretical simulations [10, 11] were the topology optimized photonic-crystal waveguide [8], nanoscale dielectric metasurface etched into silicon waveguides [9], taper-structure waveguide [10] and computation-designed waveguide [11]. An obvious drawback of the reported modes converters [8–11] was the relatively narrow operation bandwidths, which were only several tens of nanometers. To design silicon-waveguide polarization rotators, both the horizontal and vertical symmetry of waveguide structures are required to be broken. Reported experiments have demonstrated that a 10-μm-long dual-trench waveguide [12], 23-μm-long double-stair waveguide [13], and 300-μm-long partial-rib waveguide [14] could work as silicon-waveguide polarization rotators with bandwidths (defined as measurable polarization extinction ratio (PER) larger than 10dB [13]) equal to 47nm, 80nm, and 200nm, respectively. In addition, silicon-based polarization splitter-rotators (PSR) [15] as a more powerful PR have been widely studied by theories [15–17] and experiments [18–20]. However, the footprints of the reported broadband PRs [12–20] were relatively large. Recently, telecom and broadband waveguide mode-order converters and PRs with low loss [21] were realized by integrating silicon metasurface structures into LiNbO3 and Si3N4 waveguides. But, the introducing of dielectric metasurfaces increases the complexity of their fabrications.
Here, focusing on the near-infrared wavelengths, we propose a novel waveguide mode-order converter and PR just by two cascaded trenches fully etched inside silicon nanowires and partially etched on the edges of silicon nanowires, respectively. The two trenches along the waveguide have mirror symmetry in the horizontal direction, which provides new degrees of freedom for efficiently manipulating the spatial modes and polarizations of waveguides in small footprint.
2. Mode-order converter
For TE0-like to TE1-like mode-order converter, we study a free-standing silicon nanowire (Fig. 1(a), inset) with thickness and width equal to 260nm and 800nm. Within the wavelength interval between 1200nm and 1700nm, the nanowire supports both TE0-like and TE1-like mode in Fig. 1(a). Due to the different mirror symmetry of the two modes, as shown in Figs. 1(b) and 1(c), the mode-order converter can adopt asymmetric structures only in the horizontal direction (X direction). A simple structure is a slot nanowire with an off-centered trench fully etched in the core, which divides it into two smaller nanowires with different widths. Figures 1(d)-1(f) show that the slot nanowire (Fig. 1(d), inset) supports two TE-like modes with their electric fields mainly localized at the right (Fig. 1(e), named by TER) and left (Fig. 1(f), named by TEL) smaller nanowire, respectively. The different propagation constants (Fig. 1(d)) and the spatial distributions (Figs. 1(e)-1(f)) of them in the slot nanowire make it possible to effectively convert TE0-like into TE1-like waveguide mode. From Fig. 1(d), it can be seen that, in a fairly large wavelength interval around 1550nm, the propagation constant difference between the TER and TEL mode is weakly dispersive, suggesting that we can realize a TE0-like to TE1-like mode-order converter with wide bandwidth.
Figure 1 Eigenmodes of silicon nanowire and slot nanowire. (a)–(c) are propagation constants β of TE0-like and TE1-like modes supported by the 800nm-width (X direction) and 260nm-thickness (Y direction) silicon nanowire ((a), inset) and their X-component electric-field distributions at the wavelength of 1550 nm. (d)-(f) are those of the two TE-like eigenmodes supported by the slot nanowire which is the same nanowire ((a), inset) with a 70nm-width and −30nm off-center (in the X direction) trench etched in the core ((d), inset). The eigen-modes are calculated by using COMSOL MULTIPHYSICS software, the refractive index of Silicon is 3.46.
To completely convert TE0-like into TE1-like mode at the half-beat length of the two modes [12], the electric-fields of the two eigen-modes supported by the slot nanowire should satisfy
where, c1, c2 and η are complex constants. Based on the symmetries of TE0-like and TE1-like modes under the horizontal mirror reflection σ(ex) which is defined aswe could deduce from Eq. (1) thatEquation (3) means that electric-fields andare mirror image of each other. However, in an asymmetric slot structure as shown in the inset of Fig. 1(d), the electric-field of the mode TER is not the mirror image of the mode TEL. So, a single slot structure cannot completely convert TE0-like into TE1-like mode even if its propagation loss was zero.To solve the issue, we propose a mode-order converter shown in Fig. 2(a), which cascades the slot nanowire (Fig. 1(d), inset) with its horizontal mirror-image structure which is the nanowire with a 70nm-width and 30nm-off-center trench etched in the core. The symmetry conditions are manifestly satisfied. Another obvious advantage of this structure is that the interference between the two asymmetric modes shown in Figs. 1(e)-1(f) and that between their mirror-image modes can be independently controlled by optimizing the lengths of the two slot nanowires, which guarantees a complete inter-conversion between TE0-like and TE1-like mode. Here, we firstly utilize the scattering-matrix method (SMM) [22] to obtain the lengths of the two slot nanowires. The scattering process in the mode-order converter (Fig. 2(a)) is schematically described by Fig. 2(b). Where, denote the transverse components of electric-magnetic fields [Ex, Ey, Hx, Hy] of the ± z directionally propagating TE0-like (TER) and TE1-like (TEL) modes in the nanowire (the slot nanowire). is that of the corresponding mode in the mirror-slot nanowire. For the incident TE0-like mode at the wavelength of 1550nm, Fig. 2(c) shows the SMM-calculated purity of TE1-like mode in the transmitted light through the mode-order converter with varying lengths of slot nanowire (h1) and mirror-slot nanowires (h2). The two lengths (h1, h2) equal to (1.3μm, 3.4μm) are chosen in our design.
Fig. 2 Schematic illustration and theoretical demonstration of the TE0-like to TE1-like mode-order converter. Along the 800nm-width and 260nm-thickness silicon nanowire, the proposed mode-order converter (a) consists of one linear inverse taper trench, one 70nm-width and 30nm-off-center trench (slot nanowire), one 70nm-width and 30nm-off-center trench (mirror-image slot nanowire), and one taper trench. The lengths of the fully etched trenches are 0.3μm,1.3μm,3.4μm and 0.3μm, respectively. (b)A schematic illustration of SMM. (c) Purity of TE1-like mode in the transmitted light through the mode-order converter with varying lengths of slot nanowire (h1) and mirror-slot nanowires (h2). The star marks the point of (h1 = 1.3μm, h2 = 3.4μm) (d) A 3D Lumerical FDTD simulation of the X-component electric-field distributions in the device when a TE0-like mode at the wavelength of 1550 nm incidents from the left side. (e) and (f) are total transmittance and purities of TE0- and TE1-like mode in the transmitted light for the incident TE0-like mode with varying wavelengths.
In order to increase the transmittance, two 300nm-long linear (inverse) tapers are added at both ends of the mode-order converter. The total length of the device is then 5.3μm (0.3μm + 1.3μm + 3.4μm + 0.3μm). For an incident TE0-like mode at the wavelength of 1550 nm, the X-component electric-field propagation along the device (Fig. 2(d)) shows that the transmitted light is almost TE1-like mode with purity about 0.97. Varying the wavelength of incident TE0-like mode, Fig. 2(e) shows that within the wavelength interval between 1420 and 1620 nm, the device is low-loss with transmittance larger than 0.95. Meanwhile, benefiting from the introducing of mirror-image slot nanowire, the purity of TE1-like mode in the transmitted light (Fig. 2(f)) within the same wavelength interval is larger than 0.9.
3. Polarization Rotator (PR)
For waveguide PR or TE0-like to TM0-like mode converter, we study a free-standing silicon nanowire (Fig. 3(a), inset) with a thickness of 260nm and width of 400nm. Within the wavelength interval between 1200nm and 1700nm, the nanowire supports TE0-like and TM0-like modes as shown in Fig. 3(a). The different mirror symmetries of the two modes, as shown in Figs. 3(b) and 3(c), require PR to be an asymmetric structure in both the horizontal (X) and vertical (Y) directions. A straightforward structure is an L-shaped nanowire (Fig. 3(d), inset) [14]. Figures 3(d)-3(f) show that the L-shaped nanowire supports two orthogonal hybrid modes, named by HM1 (Fig. 3(e)) and HM2 (Fig. 3(f)), with their optical axes nonparallel with the horizontal and vertical directions. As a result of their different propagation constants (Fig. 3(d)) and different optical-axe directions of the two modes, the L-shaped nanowire could rotate the polarization vector of the incident TE0-like and TM0-like modes. To achieve 90° polarization rotation at the half-beat length, electric-fields of the two eigen-modes supported by the L-shaped nanowire should satisfy
where, d1, d2 and γ are complex constants. Similar to the deviation of Eq. (3), we can also get the symmetry relationship between HM1 and HM2 modes:Fig. 3 Eigenmodes of rectangular and L-shaped silicon nanowire. (a)–(c) are propagation constant β of the TE0-like and TM0-like modes supported by a 400nm-width and 260nm-thickness silicon nanowire ((a), inset) and their transverse electric-field distribution at the wavelength of 1450 nm. (d)-(f) are those of the two hybrid modes supported by the L-shaped silicon nanowire ((d), inset) which is the same nanowire ((a), inset) with a 140nm-width and 200nm-depth trench etched on the positive X direction. The small arrows in (b), (c), (e), (f) denote the directions of the transverse electric-field vectors, and the different color hues denote the relative amplitudes of the electric fields.
However, limited by the thickness-width ratio of the nanowire, the electric-field of the mode HM1 is not the horizontal mirror image of the mode HM2. Thus, a single L-shaped nanowire also cannot completely convert TE0-like into TM0-like mode.
Similar as in the design of mode-order converters, we propose a PR as shown in Fig. 4 (a), which cascades the L-shaped silicon nanowire (Fig. 3(d), inset) with its horizontal (X-direction) mirror-image structure. Due to the large geometric differences between the cross sections of L-shaped and rectangle nanowires, a 1.5μm-long linear (inverse) taper and taper mode adaptation sections are added at both ends of the PR. One ridge nanowire rotated 5° with respect to Z direction is inserted between the L-shaped silicon nanowire and its mirror-image structure. We also use the scattering-matrix method to obtain the lengths of the two L-shaped nanowires. Figure 4(b) schematically describes the scattering process in the PR. Where, the scattering matrixes of the taper trench, 5° ridge nanowire and inverse taper trench are obtained by the Comsol simulation. For the incident TE0-like mode at the wavelength of 1450nm, Fig. 4(c) shows the SMM-calculated purity of TM0-like mode in the transmitted light through the PR with varying lengths of slot nanowire (h1) and mirror-slot nanowires (h2). Hence, the two lengths (h1, h2) equal to (0.8μm, 1.0μm) are chosen in our design. That is, the total length of the device is 6.4μm. The X- and Y- component electric-field propagation along the designed PR (Fig. 4(d)) show that the transmitted light is almost TM0-like mode with purity about 0.99. Varying the wavelength of incident TE0-like mode, Fig. 4(e) shows that within the wavelength interval between 1360 and 1560 nm, the device is low-loss with the transmittance larger than 0.9. Meanwhile, the purity of TM0-like mode in the transmitted light (Fig. 4(f)) within the same wavelength interval is larger than 0.9. Similar to the performance of the TE0-TE1 mode-order converter, the wide operation bandwidth of the TE0-TM0 converter also originates from the fact that the propagation constant difference between the two orthogonal hybrid modes in the converter is weakly dispersive in a fairly large wavelength interval (e.g. 1300nm-1600nm in this case as shown in Fig. 3(d)).
Fig. 4 Schematic illustration and theoretical demonstration of the TE0-like to TM0-like mode converter (PR). Along the 400nm-width and 260nm-thickness silicon nanowire, the PR (a) consists of one taper trench, a 140nm-width trench etched on the positive X direction (L-shaped nanowire), a 5° ridge nanowire, a 140nm-width inverse trench etched on the negative X direction (mirror-image L-shaped nanowire), and one taper trench. Their lengths are 1.5μm, 0.8μm, 1.6μm, 1.0μm and 1.5μm, respectively. All the trenches and the ridge nanowire are partially etched with depths equal to 200nm. (b) A schematic illustration of SMM. (c) Purity of TM0 like mode in the transmitted light through the mode converter with varying lengths of L-shaped nanowire (h1) and mirror- L-shaped nanowires (h2). The star marks the point of (h1 = 0.8μm, h2 = 1.0μm) (d) A 3D Lumerical FDTD simulation of the X-and Y-component electric-field distributions in the device when a TE0-like mode at the wavelength of 1450 nm incident from the left side. (e) and (f) are total transmittance and purities of TE0- and TM0-like mode in the transmitted light from the PR for the incident TE0-like mode with the varying wavelengths.
It is worth noticing that though the proposed mode-order converter in Fig. 2 and PR in Fig. 4 are based on free-standing silicon nanowires, our designs can be directly used as high efficient mode converters with small footprints on silicon-on-insulator (SOI) wafer with the 260 nm-thickness top-silicon layer. Because the refractive index of the buried oxide layer is relatively low, the TE-like modes of free-standing slot nanowires are similar as those of slot waveguides on SOI. Thus, for TE0-like to TE1-like mode-order converters on SOI in Fig. 5(a), we choose the geometric structure and parameters of top-silicon layer to be same as those of the mode-order converter in Fig. 2. Figures 5(b) and 5(c) show that within the wavelength interval between 1410nm and 1600nm, both the transmittance and the purity of TE1-like mode in the transmitted light are larger than 0.9. For the PR on SOI in Fig. 5(d), we also choose the geometric structure and parameters of top-silicon layer to be same as those of PR in Fig. 4 except the lengths (h1, h2) of two L-shaped waveguides, because the propagation constants of L-shaped waveguide on SOI are different with those of free-standing L-shaped nanowire. With the lengths (h1, h2) equal to (0.4μm, 3.4μm), Figs. 5(e) and 5(f) show that within the wavelength interval between 1300nm and 1520 nm, both the transmittance and the purity of TM0-like mode in the transmitted light are larger than 0.9. Therefore, the performances of the mode-order converter and PR on SOI are similar as those on free-standing silicon.
Fig. 5 Schematic illustration and theoretical demonstration of the TE0-like to TE1-like mode-order converter (a) and PR (d) on SOI. For the incident TE0-like mode with varying wavelengths, (b) and (c) are total transmittance and purities of TE0- and TE1-like mode in the transmitted light from the mode-order converter, (e) and (f) are total transmittance and purities of TE0- and TM0-like mode in the transmitted light from the PR.
4. Conclusion
In conclusion, we have proved theoretically that low-loss and broadband mode-order converters and polarization rotators could be compactly realized by fully and partially etching trenchers in the nanowires, respectively. The key idea for achieving complete mode and polarization conversions in our designs is to cascade the asymmetrical structures (slot or L-shaped nanowires in the work) with their mirror-image structures along the nanowires. Meanwhile, the relatively weakly dispersive and large differences between propagation constants of the designed asymmetrical structures guarantee the proposed mode-order converters and polarization rotators with wide operation bandwidths and small footprints [23], which present obvious advantages for fabrication and on-chip integration.
Funding
National Natural Science Foundation of China (NSFC) (11374367, 61405056).
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