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Abstract
We propose and experimentally demonstrate a polarization independent subwavelength grating (SWG) waveguide Bragg grating (WBG) by using an SWG waveguide with tilted segments. By optimizing the tilting angle and other geometry parameters, such as the width and the length of the loading segments used to create the BG, we can obtain a zero birefringence tilted SWG waveguide and consequently, a polarization independent SWG WBG. In our simulations, the optimal tilting angle is ∼ 58°, whereas the optimal angle obtained in fabrication is ∼ 46°. This deviation is mainly due to fabrication errors, e.g., on the sidewall angle of the silicon segments. For the optimal tilting angle of 46°, the characterized Bragg wavelengths of the TE and TM modes are both ∼ 1517 nm. We believe that the proposed device can have applications in optical communications and interconnections.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Waveguide Bragg gratings (WBGs) are essential building blocks for optical communications [1], photonic sensing [2], and microwave photonics [3]. The implementation of a WBG usually involves a periodic arrangement of lower and higher effective index sections. In the past, numerous WBG designs have been demonstrated with the silicon photonics (SiP) platform. For example, we can realize constructive Bragg reflection by perturbating the width of the solid waveguide core (i.e., sidewall corrugations) [4,5] or etching nanoholes in the waveguide [6]. By using a grating-assisted contra-directional coupler, wavelength division multiplexing (WDM) can be further implemented [7]. Although these designs allow us to readily adjust the Bragg wavelength and the bandwidth, the responses are still polarization-dependent, namely, the Bragg wavelengths or bandwidths for the fundamental transverse electric (TE) and transverse magnetic (TM) modes are inherently different. For polarization division multiplexing (PDM), a single polarization independent optical filter can have the benefit of reducing footprint for on-chip interconnections. On the other hand, for non PDM applications, a polarization independent optical filter will reduce impairments associated with polarization dependent loss.
To realize a polarization independent WBG, different approaches have been explored and demonstrated. These include using equal width and height for strip waveguides [8,9], optimizing the design of rib and ridge waveguides [10,11], or using other materials to reduce the polarization sensitivity, e.g., polymers [12,13]. The latter approach requires materials which may not be readily available in common silicon photonic (SiP) foundries. For strip waveguides, it is possible to obtain zero birefringence when the width and height of the waveguide are equal [10,11]. However, in common SiP foundries, the height of the waveguide is fixed, e.g., at 220 nm, and zero birefringence can only be achieved by reducing the waveguide width. Consequently, most of the energy of the TE mode will be distributed at the sidewall, and the propagation loss caused by the surface roughness can be significant, e.g., as high as ∼ 30 dB/cm for a width of 400 nm [14]. For ridge waveguides (e.g., [11]), the design involves 4 different effective indices or the use 4 different materials which, again, may not be readily available in common SiP foundries. Another approach involves cascading two WBGs whose responses are optimized separately for the TE and TM modes so that their reflection bands overlap [15]. Overall, it is still difficult to realize a polarization independent filter using a single WBG in SiP.
Subwavelength grating (SWG) waveguides, which have flexibility in tailoring the effective index and the potential for lower propagation loss, have been considered to develop passive SiP devices such as SWG assisted contra-directional couplers [16] or narrow band SWG WBGs [17]. Recently, tilted SWG waveguides have been investigated. By tilting the SWG segments, we can decrease the effective index of the TE mode while maintaining the effective index of the TM mode relatively constant [18]. Following this principle, it is possible to realize a zero birefringence SWG waveguide by optimizing the tilting angle, namely, the effective index of the TE mode can be made equal to that of the TM mode [19]. Recently, we have demonstrated polarization-dependent tuning of the Bragg wavelength using tilted SWG WBGs [20]. In particular, the blue shift in the Bragg wavelength of the TE mode can be up to ∼ 30 nm while that of the TM mode can be kept within 0.5 nm. Moreover, the tilted SWG WBG can be readily fabricated with a single etch in common SiP foundries. These characteristics of the tilted SWG waveguide provides an opportunity for us to realize a zero birefringence SWG WBG.
In this paper, we design, simulate, and demonstrate experimentally a polarization independent SWG WBG in SiP by using a tilted SWG waveguide. To simulate the grating spectral responses, we combine the anisotropic model [18,19] with the transfer matrix method (TMM). By optimizing the tilting angle, the Bragg wavelengths of the TE and TM modes can be made to coincide. In our simulations, the optimal tilting angle is 58° and the spectral responses of the TE and TM modes are almost identical. However, the optimal angle of the fabricated devices is ∼ 46° and the extinction ratio of the TE mode is much smaller than that of the TM mode. Although there are some differences between the simulated (designed) and fabricated devices, our approach provides the possibility to realize a zero birefringence or polarization independent WBG filter.
2. Principle and simulation
The tilted SWG WBG can be implemented by interleaving the tilted SWG waveguide and the ‘loaded’ tilted SWG waveguide, as shown in Fig. 1. This periodic arrangement of loading segments alongside the tilted SWG waveguide can enable a periodic perturbation of the effective index (i.e., cladding modulation) leading to Bragg reflection [20]. To model the tilted SWG WBG, the effective indices of the tilted SWG waveguides (with or without loading segments) are calculated in advance; then, the grating responses can be simulated using the TMM. The design flow is summarized in Fig. 2.
Fig. 1. Schematic of (a) tilted SWG waveguide, (b) tilted SWG waveguide with loading segments, (c) tilted SWG WBG, and (d) corresponding cross sections at different positions highlighted by the vertical dashed lines in (c) of the tilted SWG WBG, where ${w^,}$ is the width of the silicon segment at this position.
2.1 Zero birefringence tilted SWG waveguide
The effective indices of the tilted SWG waveguides are simulated using the anisotropic model since it is faster than the 3-dimensional (3D) finite difference time domain (FDTD) algorithm [18]. The characteristic of the SWG structure is similar to that of a homogenous but anisotropic uniaxial medium, where tilting the SWG structure will have a greater impact on the optical energy distribution in the horizontal direction compared to the vertical direction; namely, the TE mode will be affected more by tilting the silicon segments [18]. In the anisotropic model, we assume the SWG structure is equivalent to the anisotropic material; then, this equivalent material can be created in Lumerical MODE and the effective index of the tilted SWG waveguide can be calculated using the eigenmode solver. Further details of the anisotropic model can be found in [18]. We target the Bragg wavelength for both TE and TM modes within the C band. From simulations, the effective index of the SWG waveguide at 1550 nm is ∼1.5. Then, from the Bragg condition ${\lambda _{BG}} = 2{\mathrm{\Lambda }_{BG}}{n_{BG}}$, we have ${\mathrm{\Lambda }_{BG}} = 512$ nm (since ${n_{BG}}$ ∼ 1.5) and correspondingly, ${\mathrm{\Lambda }_{SWG}} = 256\; $nm (since ${\mathrm{\Lambda }_{SWG}} = {\mathrm{\Lambda }_{SWG}}/2)$. Note that we use a duty cycle (DC) = 0.5; this SWG period and DC are also used in our previous work [20]. As illustrated in Fig. 2, we first need to optimize the tilting angle $\theta $ and width w of the SWG waveguide. After obtaining the optimal tilting angle and width, the length of the loading segments and the gap distance of the ‘loaded’ SWG waveguide need to be determined. The detailed notations of the parameters for the tilted SWG WBG are listed in Table 1. Note further that ${\mathrm{\Lambda }_{SWG\_t}} = {\; }{\mathrm{\Lambda }_{SWG}}\textrm{cos}(\theta )$.
Table 1. Tilted SWG WBG parameters.
To better evaluate the difference ($\mathrm{\Delta }n$) of the effective indices of the TE and TM modes, we define $\mathrm{\Delta }n = {\bar{n}_{TE}} - {\bar{n}_{TM}}$, where ${\bar{n}_{TE}}$ and ${\bar{n}_{TM}}$ represent the averaged effective indices along the wavelength. Following the anisotropic model in [18], we can obtain the effective indices of the tilted SWG waveguide with different tilting angles and widths. We sweep w from 450 nm to 550 nm in steps of 5 nm and $\theta $ from 55° to 60° in steps of 2° (we first swept the angle in steps of 10° and found that the tilting angle should be around 55° to 60°; we then swept the angle in smaller steps within this range). The data is then further interpolated in MATLAB with a higher resolution. We use a wavelength range from 1500 nm to 1600 nm with 5,000 points in our simulations. The results are summarized in Fig. 3: (a-c) show three examples of how ${n_{TE}}$ and ${n_{TM}}$ varies over the wavelength range for fixed values of w and $\theta $. It is clear that ${n_{TE}}$ decreases when the tilting angle increases while ${n_{TM}}$ remains almost constant. When setting $w$ = 485 nm and $\theta $ = 58°, ${n_{TE}}$ and ${n_{TM}}$ almost overlap; hence, we can obtain a zero-birefringence tilted SWG waveguide. The contour plot of $\mathrm{\Delta }n$ as a function of tilting angle and waveguide width is shown in Fig. 3(d); we indicate one optimal set of values if tilting angle and waveguide width in this contour plot where $\mathrm{\Delta }n$ = 0 (the blue dot). According to the ‘0’ contour line in Fig. 3(d), we can realize zero birefringence by using a narrower width and smaller tilting angle. The contour plot of $\mathrm{\Delta }n$ obtained from the anisotropic model is very close to that obtained using 3D FDTD. The optimal width and tilting angle obtained from the 3D FDTD are $w$ = 504 nm and $\theta $ = 58°. However, to save computation time, we use the anisotropic model to further optimize the length of the loading segments and the gap distance.
Fig. 3. (a-c) Effective indices of tilted SWG waveguides with different widths and tilting angles. (d) The contour plot of $\mathrm{\Delta }n$ in terms of w and $\theta $.
With the optimal tilting angle and width, we can further optimize the length of the loading segments and the gap distance for the ‘loaded’ SWG waveguide. In particular, we sweep ${l_{SWG}}$ from 80 nm to 120 nm in steps of 10 nm and the gap distance g from 100 nm to 400 nm in steps of 5 nm. Again, the data is further interpolated in MATLAB with a higher resolution. Three different scenarios are compared in Fig. 4(a-c). It turns out that the effective indices remain almost the same when $g$ = 245 nm, regardless of the value of ${l_{SWG}}$. This is also shown in Fig. 4(d) where the contour plot of $\mathrm{\Delta }n$ indicates that the effective indices of the TE and TM modes are almost equivalent near $g = 245\; $nm even as ${l_{SWG}}$ is varied from 80 nm to 120 nm.
Fig. 4. (a-c) Effective indices of ‘loaded’ tilted SWG waveguides with different ${l_{SWG}}$ and g. (d) The contour plot of $\mathrm{\Delta }n$ in terms of ${l_{SWG}}$ and g.
2.2 Polarization independent SWG WBG
Based on the results shown in Figs. 3 and 4, we optimize the parameters of the polarization independent tilted SWG WBG as follows. We begin with an SWG period of ${\mathrm{\Lambda }_{SWG}}$ = 256 nm and a duty cycle DC = 0.5. We then choose the waveguide width to be $w$ = 485 nm with a tilting angle of $\theta $ = 58° and then vary ${l_{SWG}}$ and g. Figure 5(a) shows the simulated grating responses for ${l_{SWG}}$ = 100 nm, $g$ = 245 nm, and $N$ = 1,000 grating periods. Clearly, the transmission spectra of the TE and TM modes overlap. We then evaluate the difference ($\Delta \lambda = |{{\lambda_{TE}} - {\lambda_{TM}}} |$) between the Bragg wavelengths (${\lambda _{TE}}\; \textrm{and}\; {\lambda _{TM}}$) of the TE and TM modes for different gap distances and lengths of the loading segments. The contour plot for $\Delta \lambda $ is shown in Fig. 5(b). We can readily obtain a polarization independent tilted SWG WBG by setting the gap near 245 nm. Note that the maximum $\Delta \lambda $ is around 4 nm when $g = 100$ nm and ${l_{SWG}} = 120{\; }$nm.
Fig. 5. (a) Spectral responses of the tilted SWG WBG with the parameters listed in Table 2. (b) Difference in the Bragg wavelengths of the TE and TM modes.
Table 2. Parameters of the polarization independent tilted SWG WBG
We fabricated a number of devices with varying values of $\theta $ and two values of $g.$ Characterization of the fabricated devices showed that the optimal tilting angle is closer to ∼ 46° and that the Bragg wavelength of the TE mode is shorter than that of the TM mode (${\lambda _{TE}} < {\lambda _{TM}}$) when $\theta $ = 58°. The specific parameters for the devices that we show are listed in Table 2.
Figure 6 presents the layout of the fabricated device, which consists of three vertical grating couplers (VGCs) for the input and output coupling [21] and two SWG tapers for mode conversion [22]. All devices were fabricated using electron beam lithography with a single etch at Applied Nanotech Inc. To measure the spectral responses of the TE and TM modes, each structure is fabricated twice and side-by-side on the same chip; one is designed with TE VGCs and the second with TM VGCs, see Fig. 6. Note that the designed height of the silicon layer is 220 nm and the top cladding material is silica.
3. Experiment and analysis
The measured normalized transmission spectra of the tilted SWG WBGs with gap distances of 245 nm and 235 nm are shown in Figs. 7(a) and 8(a), respectively. These normalized transmission spectra are obtained by subtracting the spectral responses of the VGCs (obtained through characterization of separate test structures on the same chip) from the measured data which corresponds to the total spectral response comprising the VGCs, Y branch, SWG tapers, and SWG WBG. While the VGCs, Y branches, and SWG tapers have the same design, their spectral responses may have some differences because of nonuniformities in fabrication and processing; these differences can cause some irregularities in the normalized transmission spectra. Nevertheless, we expect that the grating responses and in particular, peak wavelengths, will be accurate. We also extract the Bragg wavelengths from the measured data, shown in Figs. 7(b) and 8(b). In general, the responses and the Bragg wavelengths of the TE modes have obvious blue shifts by increasing the tilting angle, while those of the TM mode have a slight red shift; this is consistent with the simulation of the effective indices. For example, the effective index of the TM mode is increased slightly but that of the TE mode decreases significantly when increasing the tilting angle, see Fig. 3(a-c), indicating a small red shift for the response of the TM mode but a large blue shift for that of the TE mode. The optimal tilting angle for the fabricated device is ∼ 46°, where the responses of the TE and TM can overlap. As the tilting angle is detuned from the optimal value of 46°, the difference in Bragg wavelengths for the two polarizations $|{{\lambda_{BG - TE}} - {\lambda_{BG - TM}}} |$ increases. On the other hand, the extinction ratio of the TE mode is ∼ 5 dB while that of the TM mode is ∼ 20 dB. We attribute these mismatches between simulation and characterization to fabrication errors, e.g., the sidewall angle of the silicon segments. From our previous measurements [23], the propagation loss of the TE mode in an SWG WBG is ∼ 6 dB/cm. Because here our device is 0.512 mm (1,000 × 0.512 $\mu $m), the calculated insertion loss is expected to be only, ∼ 0.3 dB. Moreover, the coupling loss of different VGCs having the same design but at different locations on the chip can have a variation of around 1 or 2 dB due to the nonuniformity and errors in fabrication and processing so that the exact losses of the VGCs connected with the gratings may be lower/higher than the reference VGCs. As such, it is difficult to measure accurately the insertion loss for either polarization and correspondingly, the polarization dependent loss. In order to estimate the polarization dependent loss, we refer to the results in [24] which are for a longer SWG WBG (with 9,700 periods or 4.7 mm in length) with no tilt; they observed that the insertion loss of the TM mode was 2 dB greater than that of the TE mode (or ∼ 0.4 dB/mm). This gives an estimated polarization dependent loss of ∼ 0.2 dB for our grating.
Fig. 7. Experimental results of tilted SWG WBGs with a gap distance of 245 nm: (a) normalized transmission spectra and (b) extracted Bragg wavelengths for different tilting angles.
Fig. 8. Experimental results of tilted SWG WBGs with a gap distance of 235 nm: (a) normalized transmission spectra and (b) extracted Bragg wavelengths for different tilting angles.
To verify our hypothesis, we adjust the cross-section of the SWG waveguide and the loading segments in terms of the sidewall angle. Figure 9(a) compares the estimated cross-section with the ideal design. As a proof of concept, we apply this estimated cross section into the design of the polarization independent tilted SWG WBG. Note that we use the aforementioned parameters but with the change to the sidewall angle. For the sake of the accuracy, we use 3D FDTD to simulate the band structure and extract the effective index; then, the transmission spectra of the tilted SWG WBG can be calculated through the TMM. More details of the simulation are discussed in Ref. [16]. The energy distributions of the TE and TM modes for the ‘loaded’ tilted SWG waveguide with a sidewall angle of 83° are shown in Fig. 9(b), where the energy profile of the TE mode is much more confined than that of the TM mode. These plots show that the effective index of the TM mode is more sensitive to the loading segments as it expands and interacts further with them. Consequently, the effective index difference of the TE mode will be smaller than that of the TM mode, and the grating response of the TE mode is weaker and its extinction ratio will be smaller. According to the results from the TMM, the sidewall angle has an impact on the responses of the tilted SWG WBG. For example, by decreasing the sidewall angle, ${\lambda _{TE}}$ moves toward ${\lambda _{TM}}$ and the extinction ratios of both modes decrease, while the extinction ratio of the TE mode is smaller than that of the TM mode, which is in agreement with our measurements, see in Figs. 9(c) and (d). These simulations show that the sidewall angles can have a significant impact on the characterized grating responses, and it is possible that the optimal tilting angle is shifted to ∼ 46° when considering the sidewall angle. Moreover, there are other factors that can impact the response of the grating, such as the gap distances, the width, and the thickness of the silicon layer. It should be noted that the Bragg wavelength may have a significant shift ∼ 30 nm due to the etching process and adjustment of the sidewall angles [20,24]. To realize identical responses for the two modes, we need to optimize further other parameters, e.g., moving the loading segments horizontally to vary the extinction ratio [25]. On the other hand, we should consider the sidewall angle in the optimization process. Besides adjusting the design approach, we can also optimize the sidewall angle in the fabrication by controlling the feature dose and background dose [26].
Fig. 9. (a) Cross section of the SWG waveguide; red: ideal design, blue: estimated cross section in the fabrication; (b) the simulated energy distribution of the TE and TM modes, tilting angle: 46°, sidewall angle: 83°, the black box represents the silicon segments. (c) and (d), calculated transmission spectra of the tilted SWG WBGs with the tilting angle at 46° but different sidewall angles.
4. Conclusion
In this paper, we design, simulate, and characterize a polarization independent SWG WBG by using the tilting SWG waveguide. By tilting the SWG segments, we can shift the response of the TE mode towards shorter wavelengths while the response of the TM mode has a slight red shift. At a certain tilting angle, the responses of the TE and TM modes will overlap; consequently, a polarization independent SWG WBG can be achieved. In our simulation, the optimal tilting angle is ∼ 58°, where the responses of the TE and TM modes are almost identical. However, due to fabrication errors on the width, the thickness, and the sidewall angle, the optimal angle for the fabricated devices is ∼ 46°, where the Bragg wavelengths of both polarizations are almost same (∼ 1517 nm) but a large difference (∼ 15 dB) between the extinction ratios of both modes is observed. Thus, we need to take the fabrication errors into consideration in the design and optimization process. Although there exist some deviations between the simulations and fabrications, our proposal provides a novel approach to overcome the polarization dependent response in the conventional WBG design and can enable the polarization independent grating response. We believe that the tilted SWG WBG can be further applied in optical communications, especially in PDM.
Funding
Natural Sciences and Engineering Research Council of Canada.
Acknowledgments
We thank Applied Nanotools Inc. for their support in the fabrication.
Disclosures
The authors declare no conflict 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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