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Abstract
Bragg-gratings have been frequently used to design compact and high extinction ratio (ER) on-chip polarizers. However, the strong reflection of the unwanted polarization may deteriorate the performance of the light source or cause unwanted interferences. In this paper, we propose a Bragg-grating-based all-silicon TM-pass polarizer with low reflection, low insertion loss (IL) and high ER. Unlike previously reported polarizers based on single mode waveguides, we construct the Bragg grating with a multimode waveguide, which not only acts as a Bragg reflector, but also a mode-order converter to convert the reflected TE light into higher order modes to be eventually filtered out by utilizing a tapered transition. On the other hand, the grating has little adverse influence on the TM input light since it works at sub-wavelength-guided wave propagation regime. Finally, the polarizer obtained has a length of 30µm, an ER of 51.83dB, an IL of 0.08dB, and an operating bandwidth of ∼61nm for ER > 30dB at the wavelength of 1.55µm. More importantly, the reflection of the unwanted polarization is suppressed to −12.6dB, which can be further lowered via additional design optimization. Our work points to a new direction for making better on-chip polarizers.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Silicon-on-insulator (SOI) platform has drawn a lot of attention worldwide for building high density photonic integrated circuits (PICs) as it enables sub-wavelength optical field confinement due to the high refractive index contrast between the waveguide cores and the surrounding media. In addition, SOI based photonic devices are completely compatible with the mature complementary metal-oxide-semiconductor (CMOS) process, which not only ensure the high yield and low cost of PICs but also make the optoelectronic hybrid integration easier [1]. However, a waveguide with high refractive index contrast usually presents very large birefringence between orthogonally polarized modes (TE and TM), which may deteriorate the performance of many practical devices that are highly sensitive to light polarization. Therefore, polarization management is one of the key requirements in SOI-based PICs, especially for a chip including polarization-dependent devices. A polarization diversity scheme [2] involving several polarization beam splitters [3–5] and rotators [6–8] is an effective solution to this issue, but it increases the complexity and footprint of the PICs. Alternatively, a polarizer to simply eliminate the unwanted polarization is a low-cost and efficient solution for an optical communication or sensor system [9,10] that does not involve polarization division multiplexing (PDM).
Over the years, various approaches have been proposed to realize on-chip silicon polarizers with reasonable performances. The most obvious approach is to design a waveguide structure with extremely strong polarization-dependent loss (PDL). Generally, two guiding principles can be used to enlarge the PDL of a rib silicon waveguide. One is to let the unwanted polarized light leak (or radiate) to the substrate or cladding, which can be easily realized with sharp bending [11] or shallow etching of the rib waveguide [12]. However, these polarizers generally have relatively large footprint since the PDL is limited in the all-silicon waveguide. The other is using additional lossy materials such as graphene [13,14], transparent conducting oxides [15,16] and metal [17] to form a waveguide with high absorption loss for the unwanted polarized light. However, not only such an approach will inevitably increase the insertion loss (IL) for the wanted polarized light, but also increase the complexity of the fabrication process. To reduce the IL, several polarizers based on the polarization-dependent asymmetric directional couplers (ADCs) have been proposed [18–21]. However, these polarizers usually have relatively narrow operating bandwidth and are sensitive to fabrication errors due to the critical phase matching condition.
Waveguide gratings are also promising candidates for building compact polarizers, because their optical properties can be engineered by simply changing the grating parameters [22]. Recent reported on-chip polarizers usually are based on two types of waveguide gratings, i.e., the sub-wavelength gratings (SWGs) and the Bragg gratings. In Refs [23,24], SWGs are utilized to realize TE-pass polarizer with ultra-broad operating bandwidth over 200nm. However, the IL of these polarizers is relatively large and the underlining principle may not work for a TM-pass polarizer on a standard SOI wafer, limited by the guided wave condition. Over the years, the Bragg gratings have been widely used for designing compact and high ER TE- and TM-pass polarizers [25–28]. Guan et al. proposed an all-silicon Bragg-grating-based TM-pass polarizer of 9μm in length with an ER of ∼27dB and IL of ∼0.5dB [25]. Bai et al. recently demonstrated a hybrid plasmonic Bragg-grating-based TE-pass polarizer of only 6 μm in length with a high ER of 33.7 dB and IL of ∼4.6 dB [28]. However, all these Bragg-grating-based polarizers suffer from a serious drawback that the undesired polarization is almost completely reflected to the input port, which may cause unwanted interferences and de-stabilize the light sources, particularly, in the cases where a light source is included on a chip. Therefore, it is of great importance to overcome this issue because on-chip isolators are unfavorably large and lossy presently [29,30]. Recently, a compact hybrid plasmonic TM-pass polarizer of 10μm in length with a reflection removal technique based on polarization-dependent mode conversion was proposed, which could effectively suppress the unwanted reflection to −14dB, seeming to be a promising solution [31]. Therefore, if polarization-dependent mode conversion can be simultaneously realized through a Bragg-grating-based reflector, practical polarizers may be obtained.
In this paper, we propose a low reflection all-silicon TM-pass polarizer based on Bragg-gratings. Different from those based on single mode waveguides [25–28] in the previously reports, we construct a Bragg grating with a multimode waveguide. By optimizing the grating parameters, the input fundamental TE mode (TE0) is strongly reflected and simultaneously converted into a higher order mode (TE2). To filter out undesired reflection from TE2, two identical tapered transitions are deployed between the input (output) waveguide and the multimode polarizer section. Owning to the strong birefringence of SOI waveguide, the gratings act as a uniform medium for TM input light since it works at sub-wavelength-guided wave propagation regime, and therefore allows the TM light to pass through the polarizer section with little loss. The device not only possesses the advantage of high ER of the Bragg-grating-based polarizers, but also shows very low IL since it is all-silicon. More importantly, it overcomes the strong back-reflection, the key drawback of the Bragg-grating-based polarizers. Moreover, fully etched gratings are used in the design, which can be fabricated simultaneously with the silicon waveguide by a single step etching. Numerical analysis shows that a polarizer of our design has a length of 30µm, an ER of 51.83dB and IL of 0.08dB at the wavelength of 1.55µm, with an operating bandwidth of ∼61nm for an ER > 30dB. Finally, the reflection from the undesired polarization is suppressed to −12.6dB.
2. Device structure and principle
Figures 1(a) and 1(b) show the three-dimensional (3D) schematic and the top view of the proposed TM-pass polarizer, respectively. The polarizer is composed of an input waveguide, two tapered transitions, a grating engineered waveguide section, and an output waveguide. The whole silicon part is symmetrical with respect to the central x axis and z axis. The input and output sections are two identical single mode waveguides with a width of W1. Two linear tapered transitions are incorporated with the input and output waveguides, where the silicon wire near the input (output) port has a core width tapered from W1 (W2) to W2 (W1) with a longitudinal length of L1. The central waveguide section with an engineered grating has a larger width (W2) relative to the input (output) waveguide so that higher-order modes are supported. Moreover, two rows of periodically distributed rectangle holes with a width of W4 are fully etched in this section, forming two lines of longitudinal waveguide gratings. The separation between the two periodic rows is W3, and the pitch width and duty ratio of these holes are Λ and a/Λ, respectively.
Fig. 1. The three-dimensional (3D) schematic (a) and the top view (b) of the proposed TM-pass polarizer.
The operating principle of the presented polarizer is also schematically illustrated in Fig. 1(a). When TM (TE) polarized light is injected into the single mode waveguide, it will first be converted into TM0 (TE0) mode of the multimode waveguide with the engineered grating by the tapered transition. Then, by properly selecting the grating parameters, TM0 mode will directly pass through the multimode section with very low loss, while TE0 mode will be reflected and converted into TE2 mode. Note that unlike the previously reported Bragg-grating-based polarizers [25–28] with almost all the light power of the undesired polarization directly reflected into the input port, here the TE2 mode is further transferred into radiated modes and gradually leaks into the cladding since it is not supported by the input single mode waveguide. Therefore, the present polarizer has an advantage of low reflection.
In order to realize such a polarizer, we take advantage of the high birefringence of the standard SOI waveguides and construct polarization-dependent gratings that can achieve the functions mentioned above in the multimode section. As is demonstrated in previous work [22], the periodic longitudinal gratings can work at three entirely different regimes, i.e., sub-wavelength-guided wave propagation, Bragg reflection and radiation, when specific conditions are satisfied. In our design, for TM polarized light, the central multimode waveguide is expected to support Bloch mode (i.e., working at sub-wavelength-guided wave propagation regime). In this regime, the gratings behave as homogenous media so that TM polarized light passes through with very low loss and reflection. To realize this, the following condition should be satisfied [25]
where λ0 is the central wavelength, n1 and n2 are the effective indices of TM polarization modes calculated in the etched and not-etched regions, respectively.On the other hand, for TE polarized light, the multimode waveguide section with the engineered grating works as a contra-mode converter which convert TE0 to higher order modes (TE2) and simultaneously reflect them backward. According to the coupled mode theory [32], efficient contra-directional coupling takes place at the wavelength that satisfies the following phase-match conditions
The proposed polarizer is designed on a standard SOI wafer with a 220-nm-thick top silicon layer covered by SiO2 cladding. The birefringence of a 220nm SOI waveguide is usually very large, which is beneficial for designing such a polarizer. The full-vectorial finite difference frequency-domain (FDFD) method [33] is utilized to study the modal characteristics and obtain the preliminary structural parameters. The 3D finite difference frequency-domain (FDTD) method [34] is employed to further optimize and evaluate the device. In the following simulation, if not specified, the refractive indices of SiO2 and Si at the wavelength of 1.55μm are taken as 1.445 and 3.455, respectively [35].
3. Design and optimization
In order to choose the suitable waveguide dimensions, modal characteristics of the waveguides are studied using the FDFD mode solver. Figure 2 shows the calculated effective indices at the wavelength of 1.55μm for a silicon wire waveguide as a function of waveguide width. In this simulation, the computational window is set to be 4μm×2.5μm (x×y) and the grid size in both x and y directions is selected to be 5nm, which are determined after convergence test. From Fig. 2, one sees that the number of modes the waveguide can support gradually increase as the waveguide width increases. For the presented polarizer, there are two kinds of waveguides with different widths (W1 and W2) to be determined. From the principle described in part 2, higher order TE modes (especially the TE2 mode) are required to be supported in the center waveguide with a width of W2 and cut-off in the input/output waveguide with a width of W1. To ensure single mode operation of the input/output waveguide, the waveguide width W1 should be narrower than TE1 cutoff point (∼550nm). On the other hand, when the waveguide width W2 is greater than ∼800nm, TE2 mode can be supported. Here, as an example, we simply select two proper points W1= 340nm and W2= 1000nm as shown by the pink dashed line in Fig. 2.
Fig. 2. The calculated effective indices at the wavelength of 1.55μm for a silicon wire waveguide as a function of waveguide width
Next, other two width parameters W3 and W4 are to be determined. Here, for the sake of simplicity, we choose these two width parameters intuitively from the mode characteristics, as explained below. Even though there are five modes are supported in the multimode section, only three modes (TE0, TE2 and TM0) are involved for the present device since the linear tapered transition just transforms the input light into fundamental modes (TE0 and TM0) and higher order mode conversion merely occurs between TE0 and TE2. Therefore, we show the field distributions of the dominant electric field component of these three modes for a multimode waveguide with a width of W2=1000nm determined above in Fig. 3. From the figure, one sees that the power of two fundamental modes (TE0 and TM0) mainly concentrates near the center of the waveguide and gradually decays toward the waveguide sides, while the power of TE2 mode have three peaks locating at the center and two sides of the waveguide. In addition, most power in TM0 spread into the top and bottom SiO2 layer near the Si core. Therefore, to make the TE2 mode strongly reflected while simultaneously reduce the influence on the TM0 mode, the gratings is preferred to be placed near the two side-peak of TE2 mode with a relatively narrow width (W4). It is noted that the gratings should not cover the zero points of TE2 as shown by enlarged field pattern in Fig 3. (c). Here, for a preliminary design, we set W3=540nm and W4=60nm, where the two side-peaks of the TE2 mode approximatively locate, and the influence of the width variations will be further analyzed in the next part.
Fig. 3. The field distributions of the dominant electric field component of these three modes for a multimode waveguide with a width of W2= 1000nm.
For a grating based polarizer, the pitch width Λ is usually the key parameter that decides the performance of the device. According to Eq. (3), in order to realize the reverse coupling between the TE0 mode and the TE2 mode at the center wavelength of 1.55μm, Λ should be properly selected. Here, for simplicity, the duty ratio is preliminarily set to be a/Λ=1/2 and the forward (TE0) and backward (TE2) effective indices in Eq. (3) are estimated by $({n_1} + {n_2})/2$ for corresponding mode (TE0 or TE2). Figure 4 shows the Bragg period with respect to the center wavelength that satisfies the reverse coupling condition between different modes. As shown in the figure, there are three curves that correspond to TE0 to TE0, TE1 and TE2 conversion, respectively, and the Bragg period gradually increases as the central wavelength increases for all of the three cases. To realize TE0 to TE2 conversion at the wavelength of 1.55μm, the calculated period is Λ=361.8nm which will be used in the following analysis. Moreover, by substituting Λ=361.8nm in Eq. (1), the calculated value for the left-hand side of Eq. (1) is 0.684μm which is lower than ${{{\lambda _0}} / 2}$ (0.775μm) at ${\lambda _0}$= 1.55μm, suggesting that sub-wavelength-guided wave propagation can be satisfied for the TM0 mode.
Fig. 4. The Bragg period with respect to the center wavelength that satisfy the reverse coupling between different modes
To achieve efficient mode conversion between the single mode waveguide and the multimode polarizer section, the length of tapered transitions should also be optimized. Since the mode conversion efficiency directly influences the insertion loss of the TM mode and residual light power of TE mode detected at the output port, we show the normalized transmission and reflection as a function of the length of the tapered transitions at the wavelength of 1.55μm in Figs. 5(a) and 5(b), respectively. For this simulation, the number of periods of the gratings is set to be N =20 with the structure parameters determined above. It can be seen from Fig. 5(a) that as the taper length increases, the transmission of the TM mode first rapidly increases and then stabilizes around 0.98 when L1 is greater than 4μm, while that of TE mode shows little change in the range. The reflection fluctuates slightly as the taper length increases. To balance between the compactness and the performance of the device, the length of the tapered transition is chosen to be L1=5μm.
Fig. 5. The normalized transmission (a) and reflection (b) as a function of the length of the tapered transitions at the wavelength of 1.55μm.
Usually, the performance of a Bragg-grating-based polarizer can be improved by increasing the period number N via enhancing the reflection. Figure 6 shows the normalized transmission (in dB scale) for TE and TM polarized light with respect to the period number at the wavelength of 1.55μm. Here, the increment of the period number is set to be ΔN = 5 and the total length of the waveguide section with the grating is L2=N×Λ. From Fig. 6 one sees that as the period number increases, the transmission of the TE mode gradually decreases, while that of the TM mode remains at a very high level with no noticeable changes. Therefore, the performance of the device can be improved by increasing the number of periods of the gratings. It can be seen that the decrement of TE transmission becomes slower as the period number reaches N=55, corresponding to a polarizer section L2 ≈ 20μm. At this length, the TE transmission is lower than −50dB, which is an excellent result.
Fig. 6. The normalized transmission (in dB scale) for TE and TM polarized light with respect to the period number at the wavelength of 1.55μm.
4. Results and discussion
In the following analysis, the performance and fabrication tolerances of the key structural parameters of the entire polarizer is further investigated using the 3D FDTD method. The waveguide structure is meshed by non-uniform grids and the maximum grid sizes in x, y, and z direction are chosen as 10 nm, 5 nm, and 20 nm, respectively. The computational window is set to be 6μm×2.5 μm (in x-y plane, and z is decided by the device length), including a 0.5μm thickness of perfectly matched layer boundary at each edge. Considering that the radiation reflected from the SOI substrate is inevitable, it is important to evaluate the level of reflection due to the SOI substrate. For this purpose, we have performed simulations with and without considering the SOI substrate and found that the influence is negligible for the accuracy we considered from comparing the results. In the simulation for the case considering the substrate, a simulation window of 6μm×5μm is used to cover the substrate (2um below the device). For a polarizer, ER, IL, and reflection loss (RL) are three crucial parameters, which are defined as follows:
Figure 7 shows the calculated spectra of the ER, IL and RL with the structure parameters described in the caption. From Fig. 7, one sees that both the IL and RL for TM are very low in the whole wavelength range as expected, indicating that the TM polarized light goes through the device with little loss. In contrast, the ER is very high near the center wavelength (1.55μm) due to the strong Bragg reflection of the TE polarized light. Unlike the previously reported Bragg-grating-based polarizers in which all the optical power is reflected to the input waveguide, here the RL of TE polarized light is greatly reduced. From the results, the reflection for TE input light is ↑-12.6 dB at the center wavelength and lower than −8.1 dB in the whole wavelength range. The ER and IL at the center wavelength is calculated to be 51.83 dB and 0.08 dB, respectively. The operation bandwidth for ER>30 dB is ∼61nm (from 1.521μm to 1.582μm), with the IL lower than 0.13 dB in this range. Figures 8(a) and (b) show the field evolution of the main components along the propagation distance through the polarizer at the wavelength of 1.55 µm for the TE and TM input lights, respectively. From Fig. 8(a), one sees that the TE input light is reflected by the Bragg grating and attenuates rapidly as the propagation distance increases. Moreover, the reflected TE light is further transferred into radiated waves and gradually leaks into the cladding when passing through the tapered transition connected to the input waveguide. In contrast, from Fig. 8(b), the TM polarized light directly passes through entire device with little reflection observed.
Fig. 7. The calculated spectra of the ER, IL and RL with the structure parameters of W1=340nm, W2=1000nm, W3=540nm, W4=60nm, L1=5um, L2=20um and Λ=361.8nm.
Fig. 8. The field evolution of the main components along the propagation distance through the polarizer at the wavelength of 1.55 µm for TE and TM input light, respectively.
To further evaluate the performance of the proposed polarizer, the fabrication tolerances of the key structural parameters are also analyzed. Usually, the performance of a grating based polarizer is more likely to be influenced by the pitch width and duty ratio errors of the waveguide gratings. Therefore, we show the ER, IL of the TM light and RL of the TE light as a function of the pitch width deviation Δp and the duty ratio deviation Δa in Figs. 9(a) and 9(b), respectively. In the simulation, the nominal duty ratio is set to be a/Λ=1/2 when the pitch width error is considered, while the optimum pitch width is Λ=361.8nm when the duty ratio deviation is calculated. From Fig. 9(a), one sees that both the ER and the RL are sensitive to the pitch width deviation, especially for the ER. To avoid rapid deterioration of the ER, the pitch width deviation is preferred to be within ±10 nm, which ensures an ER over 42.6 dB and RL below −10.2 dB. The corresponding IL for Δp in the range (-10, 10) nm is lower than 0.11 dB. From Fig. 9(b), it is observed that the duty ratio errors Δa exhibits little influence on the performance of the device for all three crucial parameters, implicating that the polarizer is fabrication-tolerant to this deviation. In addition, the IL and RL is lower than 0.10 dB and −12.3 dB, respectively, for Δa in the range of (−30, 30) nm.
Fig. 9. The ER, IL of TM and RL of TE light as the functions of the pitch width deviation Δp (a) and the duty ratio deviation Δa (b), respectively.
According to the modal analysis above, the separation W3 between the two rows of longitudinal gratings and the width of the gratings W4 may also influence the performance of the device. Figures 10(a) and 10(b) show the ER, the IL of the TM light and the RL of the TE light as the functions of variations for W3 and W4, respectively. Here, we defined the variations of W3 and W4 as Δw3 and Δw4. As shown in Fig. 10(a), the performance of the proposed device is insensitive to the grating separation variation Δw3. From the results, even with a fabrication error as large as +30nm, an excellent performance (ER∼ 49.8dB, IL∼0.09 dB and RL= −11.7 dB) can still be obtained. From Fig. 10(b), one sees that the grating width error Δw4 has some effects on the ER, however it has little influence on the IL and RL. The ER first rises then descends with slight fluctuations as Δw4 increases. In particular, the ER has a value higher than 45dB when Δw4 is in a range of (−15, 70) nm, corresponding to a width from 45nm to 130nm, indicating that the device is tolerant to the variation of W4. Considering that a narrow single mode waveguide usually exhibits higher loss due to the sidewall roughness, it is important to study the ER, the IL and the RL of the TE light as a function of the width of the single mode waveguide W1. As is shown in Fig. 10(c), the ER fluctuates up and down slightly around a nominal high level of ∼52dB, while the IL and RL slightly decrease with the increase of W1 as expected.
Fig. 10. The ER, the IL of TM light and the RL of TE light as the functions of Δw3 (a) and Δw4 (b) and W1 (c).
Although we select TE0 to TE2 conversion as an example in above analysis, the principle can be extended to arbitrary mode order conversion. Here, without loss of generality, we construct polarizers for all the three cases (TE0 to TE0, TE1 and TE2) to make a simple comparison. For the sake of simplicity, we do not change the structure parameters except for the duty cycle which determines the mode conversion condition. It should be noted that TE1 mode is antisymmetric, which cannot be converted from the TE0 mode of a symmetrical structure. To realize TE0 to TE1 conversion, we moved one of the longitudinal grating by half of a duty cycle in the propagation direction. Figure 11 shows the ER and the IL of the TM light, and the RL of the TE light at the center wavelength for TE0 to TE0, TE1 and TE2 conversions, respectively. From Fig. 11, one sees that the key features show some improvement with the increase of the mode conversion order. Even though the ER of the former two cases can be improved through further optimizing, the variation of the IL and RL may provide a useful guide for designing a better polarizer. To further reduce the RL, properly utilizing higher order mode conversion may prove effective, although a well-designed tapered transition may also work.
Fig. 11. The ER, the IL of TM light and the RL of TE light for TE0 to TE0, TE1 and TE2 conversion, respectively. The structure parameters are W1=340nm, W2=1000nm, W3=540nm, W4=60nm, L1=5um, L2=20um.
The proposed polarizer can be fabricated with standard silicon processes [36,37] on a commonly used 220-nm SOI wafer. An electron beam lithography (EBL) process can be used for the photoresist-patterning. In addition, the silicon waveguide and gratings can be fully etched using a reactive ion etching (RIE) or an inductively coupled plasma (ICP) etching process. After washing away the residues resist, a silica layer is deposited over the sample using plasma-enhanced chemical vapor deposition (PECVD). From above analysis, the pitch width takes the main influence on the performance of device and the fabrication tolerance is about 20nm (from −10 to 10 nm) to ensure a relatively high ER over 42.6dB. Considering that the linewidth uniformity of the state-of-art CMOS technologies is usually at several nanometers [37], the proposed polarizer can be fabricated with high performance. Moreover, only single step etching process is needed since the gratings are fully etched.
Table 1 summarizes the performances of some recently reported TM-pass polarizers and our work. It can be noticed that our polarizer shows better performances in terms of ER and IL than those of other polarizers in the table. The polarizer proposed in [14] has the highest bandwidth, but its footprint a bit too large. Even though the footprints of the polarizers in [16,21,27,31] are more compact, their fabrication processes are more complicated. Comparing with the all silicon polarizer of [25], our device has the advantage of low reflection. Moreover, if a lower ER is sufficient, the footprint and IL of our device can be further reduced. Therefore, the principles proposed in this work point to a new direction for making better on-chip polarizers.
Table 1. Comparison of various recently reported TM-pass polarizers
5. Conclusion
In summary, we have proposed a novel all-silicon Bragg-grating based TM-pass polarizer with low reflection, low IL and high ER. Different from the previously reported grating polarizers based on the single mode waveguides, we construct the Bragg gratings with a multimode waveguide such that the gratings act both as a Bragg reflector and a mode-order (TE0 to TE2) converter for the TE input light. Consequently, the strong TE light reflection of the Bragg-grating-based polarizers, a key drawback of the device, is suppressed since TE2 is not supported in the input/output single mode waveguide. In contrast, the gratings act as a uniform medium for the TM light to pass through with little loss. The polarizer maintains the high ER and compact size expected of Bragg-grating-based polarizers, as well as the low loss of all-silicon waveguides. Numerical analysis shows that such a polarizer of 30µm in length can achieve an ER of 51.83dB and IL of 0.08dB at the wavelength of 1.55µm, with an operation bandwidth of ∼61nm for ER > 30dB. Moreover, the reflection from the undesired polarization is suppressed to −12.6dB. Further optimization is expected to reduce the return loss even further. Comparing with the recently reported polarizers, the present device shows excellent performances in terms of IL and ER with a comparable device footprint. Therefore, the proposed polarizer may have the potential for building various compact and high performance system-on-chip devices for optical communication and sensing systems.
Funding
National Natural Science Foundation of China (12004092, 61975049); Natural Science Foundation of Hebei Province (F2019201019); Science and Technology Project of Hebei Education Department (QN2020259); Key R & D project of Hebei Province (20542201D); Advanced Talents Program of Hebei University (521000981006, 521000981203).
Acknowledgement
We thank Prof. Ting Feng of the Photonics Information Innovation Center of Hebei University for the helpful discussions.
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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