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Abstract
It is shown that the thermo-optic (TO) coefficients of various waveguide modes of a sub-wavelength grating (SWG)-assisted strip waveguide is closely dependent on the various waveguide parameters with different dependencies, including the SWG width, strip waveguide width, duty cycle, and pitch. This offers what we believe to be new degrees of freedom in the design of TO coefficients for integrated-optic waveguides, opening the door to engineering the TO coefficients of individual spatial modes or polarization states using sub-wavelength structures. Such a capability is expected to offer new design possibilities for a variety of integrated photonic, thermo-optic devices. To demonstrate the application of the concept, a mode-insensitive switch on silicon-on-insulator using a TO coefficient-engineered SWG as a mode-independent, thermo-optic phase shifter is designed and experimentally demonstrated. The experimental results show that the switching powers of the TE0-TE2 modes are only ∼29 mW, and the maximum extinction ratios for the cross (bar) states are 38.2 dB (31 dB), 37.9 dB (37 dB), and 31.9 dB (20.5 dB) for the TE0-TE2 modes, respectively, at the wavelength of 1550 nm.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Tunable silicon photonic devices, such as optical switches [1–3], tunable filters [4,5], and variable optical attenuators [6], have been fundamental building blocks in reconfigurable photonic integrated circuits [7–9]. The key to the tuning is to control the effective index of the waveguide to alter the phase of the light, which can be realized by using e.g., thermo-optic (TO) effect [10–12], the electro-optic effect [13–15], and the acoustic-optic effect [16]. Among them, the TO effect has been a popular option for tunable silicon photonic devices owing to the large TO coefficient and high thermal conductivity of silicon, simple structures, and easy fabrication. The conventional silicon photonic thermo-optic phase shifters are based on strip or rib silicon-on-insulator (SOI) waveguides covered by metal micro-heaters. The TO coefficient of a specific waveguide mode can be defined as the change rate of the effective index with respect to local temperature, which will be denoted by $\mathrm {d}n_{\mathrm {eff}} /dT$.
So far, a large number of studies have focused on improving the heating efficiency and thereby decreasing the power consumption of tunable silicon photonic devices, which can be achieved by optimizing the waveguide structures [17,18] and heater structures [19,20], and using special heater materials [12,21]. On the other hand, however, an interesting and promising direction is to flexibly engineer $\mathrm {d}n_{\mathrm {eff}} /dT$ of the transmission modes of a waveguide, and even to modify $\mathrm {d}n_{\mathrm {eff}} /dT$ of individual waveguide modes or polarization states. This novel capability will provide new design possibilities for a variety of integrated photonic devices. For example, it can be immediately used to achieve mode- or polarization-independent thermo-optic phase shifters to realize mode- or polarization-insensitive, thermally-tunable devices for mode- or polarization-division multiplexing systems, such as optical switches, variable attenuators, etc. Other potential applications of flexible $\mathrm {d}n_{\mathrm {eff}} /dT$ engineering include designing athermal devices (e.g., by engineering $\mathrm {d}n_{\mathrm {eff}} /dT$ of different parts of an integrated-optics circuit to be the same for temperature effect cancellation [22]), optimizing the thermal sensitivity of specific waveguide modes or polarization states, and decreasing the power consumption of mode or polarization switches (by maximizing the difference in $\mathrm {d}n_{\mathrm {eff}} /dT$ between the switching modes or polarization states) [23].
Nevertheless, engineering $\mathrm {d}n_{\mathrm {eff}} /dT$ of various modes for silicon photonic devices remains challenging, and limited work has yet been conducted on this subject. In principle, $\mathrm {d}n_{\mathrm {eff}} /dT$ of a waveguide mode is dependent on the waveguide parameters. For conventional strip or rib waveguide-based phase shifters, the designable structural parameters are mainly the waveguide thickness and width. The thickness can modify the difference in $\mathrm {d}n_{\mathrm {eff}} /dT$ between the TE and TM polarization states. When the waveguide cross section is square, $\mathrm {d}n_{\mathrm {eff}} /dT$ of the fundamental TE and TM polarization states will be close to each other, which can be used to realize a polarization-insensitive phase shifter [24,25]. However, the waveguide thickness generally cannot be customized and is usually determined by the foundry. Regarding the waveguide width, it has been found that, as the width becomes larger, $\mathrm {d}n_{\mathrm {eff}} /dT$ of the waveguide modes will be increased initially, then decreased, and finally remain unchanged at a value close to the material TO coefficient of silicon [26]. The width of the turning point is larger as the mode order is higher. This trend was exploited to design a mode-insensitive switch by simply using sufficiently wide strip waveguides to implement mode-insensitive thermo-optic phase shifters [27]. However, the minimum waveguide width required was as large as 4 um, which could reduce the heating efficiency, and also potentially increase the required radii of the bent waveguides and thus enlarge the size of the system. The change trend of $\mathrm {d}n_{\mathrm {eff}} /dT$ with waveguide width was also be utilized to achieve a temperature insensitive Mach–Zehnder interferometer, by choosing proper waveguide widths of the two different arms such that $\mathrm {d}n_{\mathrm {eff}} /dT$ of the arms are equal to each other for the thermal effect cancellation [22]. Nevertheless, only with this single design dimension of waveguide width for conventional silicon photonic phase shifters, it is impractical to engineer $\mathrm {d}n_{\mathrm {eff}} /dT$ of individual waveguide modes or polarizations states for further applications .
On the other hand, a sub-wavelength grating (SWG) is periodic waveguide structure with a pitch smaller than the light wavelength. The light propagating in such waveguides can be treated as it would be in a regular waveguide without gratings that had an equivalent effective index. Thanks to their high design flexibility, bringing the SWG into optical-waveguide technologies has provided new degrees of freedom to control the flow of light in integrated photonic devices. Such structures have been exploited to tailor the refractive index, dispersion properties, birefringence, and mode distributions of waveguides [28,29]. Also, a plethora of advanced integrated-optics devices with unprecedented performance have been demonstrated using subwavelength-engineered structures [30–38].
In this paper, the possibility of engineering the TO coefficients of various waveguide modes using sub-wavelength-engineered structures is suggested for silicon photonic devices. It is shown that $\mathrm {d}n_{\mathrm {eff}} /dT$ of different modes for SWG waveguides are closely related to the various waveguide parameters with different relationships. This indicates large degrees of freedom in the design of $\mathrm {d}n_{\mathrm {eff}} /dT$, which can potentially allow $\mathrm {d}n_{\mathrm {eff}} /dT$ of individual spatial modes or polarization states to be flexibly engineered. Such a capability will offer new insights into the design of a range of integrated photonic, thermo-optic devices, as discussed before. Then, by engineering $\mathrm {d}n_{\mathrm {eff}} /dT$ of multiple spatial modes of a SWG-assisted strip waveguide, a mode-insensitive silicon photonic switch for the TE0-TE2 modes is designed and experimentally demonstrated. The measured switching powers of the three modes are only $\sim$29 mw. The maximum extinction ratios (ERs) of the mode-insensitive switch for the cross (bar) states are 38.2 dB (31 dB), 37.9 dB (37 dB), and 31.9 dB (20.5 dB) for the TE0-TE2 modes, respectively, at the wavelength of 1550 nm.
2. Principle
2.1 Basic principle of $\mathrm {d}n_{\mathrm {eff}} /dT$ engineering using SWGs
The devices described in this paper are developed on 220-nm-high SOI waveguides with a silicon dioxide cladding layer deposited on the waveguides. A top view schematic of a SWG-assisted strip waveguide (SWG-SW) studied in this work is shown in Fig. 1(a), which consists of a narrower strip waveguide and wider symmetric sub-wavelength fins. The SWG-SW can be designed such that the operation band for the used waveguide modes are located in the sub-wavelength regime. Thus the light can be transmitted through the SWG-SW losslessly as Bloch modes. The critical parameters of a SWG-SW include the strip waveguide width ($W_{strip}$), SWG width ($W_{SWG}$), duty cycle ($\eta$), and pitch ($\Lambda$). Figure 1(b) plots the transmission spectra of the TE0-TE2 modes for a SWG-SW with $W_{SWG} = 1.2$ um, $W_{strip} = 0.4$ um, $\eta = 0.7$, and $\Lambda = 0.24$ um. It can be seen that within the operation band of 1500-1600 nm, the TE0-TE2 modes are in the sub-wavelength regime and can propagate through the waveguide losslessly. By combing a micro-heater, such as placing a metal heater on the top of the waveguide, the effective indices of the Bloch modes of the SWG-SW can be modulated by varying the local temperature, thereby realizing a thermo-optic phase-shifter.
Fig. 1. (a) Top view schematic of a SWG-assisted strip waveguide (SWG-SW); $W_{SWG}$: SWG width; $W_{strip}$: strip waveguide width; $\Lambda$: pitch; $\eta$: duty cycle. (b) Transmission spectra of a SWG-SW for the TE0-TE2 modes; the SWG-SW has $W_{SWG}=1.2$ um, $W_{strip}=0.4$ um, $\Lambda =0.24$ um, and $\eta =0.7$.
An important benefit of a SWG is its more adjustable parameters than conventional strip or rib waveguides, which provides significantly more degrees of freedom to engineer the properties of a waveguide mode, such as the effective index, group index, birefringence, and mode TO coefficient focused in this work. The principle behind controlling the mode TO coefficients, $\mathrm {d}n_{\mathrm {eff}} /dT$, here is that any changes in structural parameters of a SWG-SW will vary the Bloch mode profiles, which can in turn affect $\mathrm {d}n_{\mathrm {eff}} /dT$. To illustrate this, Figs. 2(a)–2(c) show Bloch mode profiles of SWG-SWs with different parameters for the first three TE modes (TE0-TE2), respectively, at the wavelength of 1550 nm. It can be seen that, for each mode, the Bloch mode profiles are significantly different for SWG-SWs with various parameters, suggesting the feasibility of engineering $\mathrm {d}n_{\mathrm {eff}} /dT$ by modifying the SWG-SW structure. The reason why various Bloch mode profiles can cause different $\mathrm {d}n_{\mathrm {eff}} /dT$ is explained below. Changes in $n_{\mathrm {eff}}$ of a waveguide mode with temperature can be considered to mainly come from two sources: 1) the inherent TO coefficient of the materials, and 2) the change of the mode profile against temperature. Various Bloch mode profiles generally mean different ratios of the mode energy confined in the silicon to that in the silicon-dioxide, which lead to different $\mathrm {d}n_{\mathrm {eff}} /dT$, considered that silicon and silicon-dioxide have very different material TO coefficients. Also, the change rate of mode profile with respect to temperature can also be different for various mode profiles, which can also contribute to the difference in $\mathrm {d}n_{\mathrm {eff}} /dT$ between various Bloch mode profiles.
Fig. 2. Bloch mode profiles of SWG-SWs with different parameters at the wavelength of 1550 nm for (a) TE0, (b) TE1, and (c) TE2 modes. The SWG-SWs in each row of the figure are the same. For the first row, $W_{strip}=0.4$ um, $W_{SWG}=1.2$ um; $\eta =0.7$ , and $\Lambda =0.24$ um; for the second row, $W_{strip}=1$ um, $W_{SWG}=1.2$ um; $\eta =0.7$ , and $\Lambda =0.24$ um; and for the third row, $W_{strip}=0.6$ um, $W_{SWG}=1.5$ um; $\eta =0.6$ , and $\Lambda =0.2$ um.
According to the discussion above, we can also conclude that $\mathrm {d}n_{\mathrm {eff}} /dT$ of a specific mode is dependent both on the mode confinement and the change rate of the mode profile against temperature. Thus, although Bloch modes are typically less confined within the waveguide than strip waveguide modes, their $\mathrm {d}n_{\mathrm {eff}} /dT$ are not necessarily smaller because they may have larger change rates of the mode profile with respect to temperature. This can also explain why the higher-order modes in some cases can have larger $\mathrm {d}n_{\mathrm {eff}} /dT$ than those of lower-order modes [as we shall see in Fig. 3(d)].
Fig. 3. $\mathrm {d}n_{\mathrm {eff}} /dT$ as a function of (a) duty cycle, (b) pitch, (c) SWG width, and (d) strip waveguide width of a SWG-SW for the TE0-TE2 modes; the waveguide parameters used in the figures are summarized in Table 1.
Based on the principle of $\mathrm {d}n_{\mathrm {eff}} /dT$ engineering using SWGs described above, it can be expected that the concept can also be applied on other SWG structures to further increase the degrees of freedom in the design, such as multibox SWGs, multiple strip waveguide-assisted SWGs, etc [28].
It should be noted that in [39] SWG structures were also utilized to control $\mathrm {d}n_{\mathrm {eff}} /dT$ of the silicon waveguide, by which athermal operations of the waveguides were realized. However, in that work, the $\mathrm {d}n_{\mathrm {eff}} /dT$ engineering was realized by filling the periodical waveguide gaps of the silicon SWG with other materials (which was polymer in that case). This is fundamentally different from the current work where $\mathrm {d}n_{\mathrm {eff}} /dT$ is engineered by directly designing the waveguide parameters of the SWG to manipulate the Bloch mode profiles. The advantages of our work, compared with the previous one, are the simpler fabrication and significantly more degrees of freedom in the $\mathrm {d}n_{\mathrm {eff}} /dT$ engineering.
2.2 Investigation of $\mathrm {d}n_{\mathrm {eff}} /dT$ as a function of SWG-SW parameters
In this section, we investigate the relationship between $\mathrm {d}n_{\mathrm {eff}} /dT$ and the critical parameters of a SWG-SW for the TE0-TE2 modes. The investigation is first performed using the phase analysis method, and some of the results are then verified by employing the photonic band-structure approach. All of the theoretical calculations in this section are conducted by using the 3D-FDTD method.
The phase variation of an optical signal after propagating through a waveguide is dependent on the effective index of the mode. Thus, $\mathrm {d}n_{\mathrm {eff}} /dT$ of a SWG-SW can be calculated from the phase difference between the output optical signals from the SWG-SW under different temperatures. According to this principle, we build a transmission system for each characterized SWG-SW, and then calculate the phase of the output light at various temperatures. The system consists of the SWG-SW and two tapered SWG-SWs connected to the input and output ends of the SWG-SW. The tapered SWG-SWs are used to achieve adiabatic mode conversions between the strip waveguide mode and the Bloch mode. The lengths of the SWG-SW and tapered SWG-SW are both 9.6 um. We use the mode source at the input to provide the light source of the system, and calculate the phases of the output lights at two various simulation temperatures of 300 K and 370 K. The material TO coefficients of the silicon and silicon-dioxide in the SWG-SW region are set to $\mathrm 1.84\times {10}^{-4} K^{-1}$ and $\mathrm 1.84\times {10}^{-5} K^{-1}$, respectively. The materials within the two tapered SWG-SW regions are set to insensitive to temperature, to avoid unintended thermal-induced phase variations caused by these parts. The effective indices of the modes here are assumed to be changed linearly with temperature for the temperature range of 300-370 K. In this way, $\mathrm {d}n_{\mathrm {eff}} /dT$ can be calculated by the following equation:
Using the approach described above, we can study the dependence of $\mathrm {d}n_{\mathrm {eff}} /dT$ of different modes on the structural parameters of a SWG-SW, by analyzing the phases of the output signals of the corresponding transmission systems. Figures 3(a)–3(d) plot the calculated relationships between $\mathrm {d}n_{\mathrm {eff}} /dT$ and the various SWG-SW parameters of $\eta$, $\Lambda$, $W_{SWG}$, and $W_{strip}$, respectively, for the TE0-TE2 modes. The used parameters of the SWG-SWs for obtaining Figs. 3(a)–3(d) are summarized in Table 1. The results are obtained at the wavelength of 1550 nm. It can be seen that $\mathrm {d}n_{\mathrm {eff}} /dT$ of the TE0-TE2 modes are closely dependent on all SWG-SW parameters. Specifically, $\mathrm {d}n_{\mathrm {eff}} /dT$ for the TE0-TE2 modes shows an overall increase trend with each SWG-SW parameter. It can also be observed that the different modes exhibit various rates of change with respect to waveguide parameters. This can potentially enable us to engineer $\mathrm {d}n_{\mathrm {eff}} /dT$ of individual modes of a SWG-SW, which is unattainable for conventional strip waveguides due to the highly limited design space.
Table 1. Waveguide parameters of the SWG-SWs in Fig. 3
For the verification purpose, we use the photonic band-structure method to validate the relationship between $W_{strip}$ and $\mathrm {d}n_{\mathrm {eff}} /dT$ in Fig. 3(d) obtained by the phase analysis method. The calculations are based on simulating one unit cell of the SWG-SW using Bloch boundary conditions through the 3D-FDTD method [Fig. 4(a)]. In this way, the frequency band functions, $f(k)$, which is the frequency as a function of Bloch wave vector, for the interested modes can be obtained. The effective index at a frequency of $f$ can then be calculated via
where $k$ is the wave vector at the frequency of $f$, and $c$ is the speed of light in vacuum. By calculating the band structure of the SWG-SW at various temperatures, $n_{\mathrm {eff}}$ for different temperatures can be obtained, which can then be used to calculate $\mathrm {d}n_{\mathrm {eff}} /dT$.Fig. 4. (a) Schematic illustration of the photonic band structure method. (b) TE band structure diagrams of the SWG-SW shown in (a). (c) Fitted frequency band curves of the three modes under different temperatures of 300 K and 370 K (solid and dashed lines, respectively). (d) Comparison of $\mathrm {d}n_{\mathrm {eff}} /dT$ with respect to $W_{strip}$ obtained by the phase analysis method and the photonic band structure approach.
Figure 4(b) shows the TE band structure diagram of a SWG-SW with $\Lambda =0.24$ um, $\eta =0.7$, $W_{SWG}=1.2$ um, and $W_{strip}=0.8$ um. The frequency bands of the TE0-TE2 modes can be seen in the band structure diagram. The band curves of the modes are then obtained by fitting the intensity peaks of the band structure diagram using 3-order polynomials, and the effective indices then can be more accurately extracted from the fitted data. Figure 4(c) shows the fitted frequency band curves of the three modes under different temperatures of 300 K and 370 K (solid and dashed lines, respectively). A small difference can be observed between the band curves at the various temperatures for each mode, confirming that the effective index variations due to the temperature change for a SWG-SW can be obtained from the band structure diagrams.
The relationship between $W_{strip}$ and $\mathrm {d}n_{\mathrm {eff}} /dT$ for the TE0 and TE1 modes calculated by using the photonic band structure method is shown in Fig. 4(d), where that obtained from the phase analysis method [the data in Fig. 3(d)] is also included for comparison. $W_{strip}$ is swept from 0.4 um to 0.7 um in steps of 33.3 nm in Fig. 4(d). The general trends of the data obtained by the two methods show good agreement with each other. The small differences between them are attributed to the limited accuracy of the photonic band structure method, which could be mainly due to the uncertainty resulting from the fitting of the band curves.
3. Design of mode-insensitive switch
To demonstrate the application of the TO coefficient engineering capability using SWG structures, we implement a mode-insensitive switch for the TE0-TE2 modes using a $\mathrm {d}n_{\mathrm {eff}}/dT$ engineered SWG-SW as a mode-independent, thermo-optic phase shifter. The current switch is designed on the SOI platform and thus can be compatible with the mature CMOS fabrication process. Nevertheless, the device can be potentially developed in other materials with higher material TO coefficients (such as polymers [40]) to achieve a smaller power consumption. The key to achieving this switch is to design the SWG-SW such that $\mathrm {d}n_{\mathrm {eff}}/dT$ of the three modes are equal to each other. Due to the different rates of change of $\mathrm {d}n_{\mathrm {eff}} /dT$ of the three modes with respect to the SWG-SW parameters, there will exist a set of waveguide parameters which can result in the target mode-insensitive thermo-optic properties. Note that as the relationships between $\mathrm {d}n_{\mathrm {eff}} /dT$ of the TE0-TE2 modes and the various parameters of the SWG-SW are complex and nonlinear, it is difficult to determine the parameters of the required SWG-SW directly. Thus, here we perform a number of parameter sweeps of a SWG-SW and track $\mathrm {d}n_{\mathrm {eff}} /dT$ of the TE0-TE2 modes using the phase analysis method described above to search the parameters of the target phase shifter. In future work, the design of SWG structures with specific thermo-optic properties may be accomplished more efficiently and effectively by using the artificial neural network [41,42].
Figure 5 shows $\mathrm {d}n_{\mathrm {eff}} /dT$ versus $W_{strip}$ for a SWG-SW with $\eta =0.75$, $\Lambda =0.26$ um and $W_{SWG}=1.5$ um, respectively. It can be seen that $\mathrm {d}n_{\mathrm {eff}} /dT$ of the three modes are close to each other when $W_{strip}$ is near 0.69 um. The difference in $\mathrm {d}n_{\mathrm {eff}} /dT$ between the modes is calculated to be as small as $<\pm 0.5{\% }$ at this point. Thus, the SWG-SW with such parameters can be used as a mode-insensitive thermo-optic phase shifter. Compared with the mode-independent phase shifter recently reported in [27], the difference in $\mathrm {d}n_{\mathrm {eff}} /dT$ between the modes here is even smaller ($< \pm 0.5 {\% }$ versus $< \pm 1 {\% }$). Furthermore, the required waveguide width of 1.5 um in the present switch is also significantly smaller than 4 um needed by the previous work. Such narrower waveguides can 1) increase the heating efficiency and 2) decrease the required radii of the bent waveguide and thus potentially reduce the size of the system.
Fig. 5. $\mathrm {d}n_{\mathrm {eff}} /dT$ as a function of $W_{strip}$ for a SWG-SW with duty cycle, pitch, and $W_{SWG}$ of 0.75, 0.26 um, and 1.5 um, respectively; the results are calculated by using the phase analysis approach based on the 3D-FDTD method.
It can be seen in Fig. 5 that $\mathrm {d}n_{\mathrm {eff}} /dT$ of the TE1 and TE2 modes show increase trends as $W_{strip}$ is larger. This can be explained by the fact that these two modes become more confined in the strip waveguide as $W_{strip}$ is increased, which also means larger portions of the mode energy located in Si. This can generally result in higher $\mathrm {d}n_{\mathrm {eff}} /dT$, due to the larger material thermo-optic coefficient of Si than that of SiO2. It can also be noticed that the increase of $\mathrm {d}n_{\mathrm {eff}} /dT$ of the TE1 mode with increasing $W_{strip}$ is less significant than that of TE2 mode, and $\mathrm {d}n_{\mathrm {eff}} /dT$ of the TE0 mode remains relatively stable with respect to $W_{strip}$. This could be because the TE0 and TE1 modes in the characterized range of $W_{strip}$ (0.6 um-1.5 um) are already highly confined in the strip waveguide. Therefore, the variation of $W_{strip}$ would have a smaller impact on the mode energy distributions and thus on $\mathrm {d}n_{\mathrm {eff}} /dT$ of these two modes.
Now we describe the design of a multimode switch based on SWG-SWs. When this switch uses the mode-independent thermo-optic phase shifter designed above, it can offer a mode-insensitive switching behavior. A schematic illustration of the multimode switch is shown in Fig. 6, which is based on a balanced Mach–Zehnder interferometer. It consists of two multimode interference (MMI) couplers serving as mode-insensitive 3-dB power splitters, and two identical waveguide arms. Each arm contains a SWG-SW, and two tapered SWG-SWs placed at both ends of the SWG-SW, which are used for adiabatic mode conversions between the strip waveguide mode and the Bloch modes. Multimode S-bends, with horizontal and vertical lengths of 50 um and 20 um, respectively, are used to connect the MMI couplers to the tapered SWG-SWs. The waveguide widths of the S-bends are 1.45 um. A metal micro-heater is placed above one of the SWG-SW to control the local temperature and thus produce a phase difference between the two arms required for the light switch. When the mode-independent thermo-optic phase shifter based on the SWG-SW is used in such a switch, any local temperature change induced by the micro-heater will result in the same phase shifts for the TE0-TE2 modes, thereby realizing a mode-insensitive switching operation.
The mode-insensitive 3-dB power splitter based on the MMI coupler used here is similar to that designed in [27]. The basic concept of the device is to design the MMI coupler such that the MMI two-fold images formed by general multimode interference happen at the output for all of the TE0-TE2 modes. To ensure good image qualities of the TE0-TE2 mode, the width of the multimode waveguide is selected to be 10 um. The beating length of the MMI coupler can be first estimated through [43]:
where $n_{\text {eff}}$ is the effective index, $W_m$ is the width of the multimode waveguide, and $\lambda _0$ is the desired wavelength. Then, the multimode waveguide length is further optimized using the eigenmode expansion (EME) method in Ansys Lumerical software, and the final length is 362 um. The widths of the ports of the MMI coupler are 3.8 um, and the center-to-center distance between the ports and the multimode waveguide is 2.95 um. Figures 7(a)–7(c) show the transmission spectra of the designed MMI coupler when inputting the TE0-TE2 modes respectively, calculated by using the 3D-FDTD method. Due to the imperfect imaging of the TE2 modes, the mode crosstalk for the input mode of TE2 is higher than those of the other cases. Nevertheless, the crosstalk is below −15 dB over most of the characterized wavelength range (1500-1600 nm) for all modes.Fig. 7. Simulated transmission spectra of the MMI coupler-based mode-insensitive 3-dB power splitter for the input modes of (a) TE0, (b) TE1, and (c) TE2.
4. Experimental results
The fabrication of the SOI-based devices in this work was conducted in Applied Nanotools, Inc. The silicon device layer was patterned using a 100 keV electron-beam lithography (EBL) followed by an inductively coupled plasma-induced reactive ion etching process. The Ti/W thin film as metal heater and aluminum thin film for metal routing were deposited using electron beam evaporation. A thin (300 nm) SiO2 passivation layer was deposited by chemical vapor deposition to protect the metal layers. A scanning electron microscope (SEM) of a fabricated SWG-SW is shown in Fig. 8(a). Vertical grating couplers from the foundry Process Design Kit (PDK), spaced on 127 $\mathrm{\mu}$m centers, were used to couple light into and out of the chip from a 8-degree polished single-mode optical fiber array with a 127-$\mathrm{\mu}$m fiber-to-fiber pitch.
Fig. 8. (a) Scanning electron microscope (SEM) of a fabricated SWG-SW (b) Optical microscope image of a multimode switch
To verify the capability of engineering the mode TO coefficients using SWGs, a number of multimode switches using SWG-SWs as thermo-optic phase shifters were fabricated and experimentally characterized. The switches are the same with each other except that the incorporated SWG-SWs have different parameters. In this way, $\mathrm {d}n_{\mathrm {eff}} /dT$ of the varying SWG-SWs can be characterized by measuring the switching powers of the switches. An optical microscope image of a fabricated switch is shown in Fig. 8(b). The multimode switch is composed of the balanced Mach–Zehnder interferometer structure shown in Fig. 6, and TE0-TE2 mode multiplexer and demultiplexer placed at the input and output of the interferometer, respectively, which allows the different modes to be characterized individually. The mode multiplexer and demultiplexer are based on asymmetric directional couplers (ADCs) consisting of two counter-tapered waveguides, which have broad operation bandwidths and large fabrication tolerance [44]. The SWG-SW in one of the arms is covered by the metal heater and thus acts as the thermo-optic phase shifter. The lengths of the SWG-SWs are 207 um, and the length and width of the metal heater are 157 um and 4.5 um, respectively. The 25 um long sections of the SWG-SW on both sides are not covered by the heater. This provides separations between the heaters and the tapered SWG-SWs to minimize unintended thermal effects on the tapered SWG-SWs, which could induce unexpected phase variations and thus affect the characterization results. The distances between the two arms of the switches are also set to 40 um to reduce thermal crosstalk between them. Calibration grating coupler pairs connected by uniform strip waveguides were fabricated and tested, and the results were used to calibrate the insertion losses of the grating couplers from the measured spectra of the switches.
A series of multimode switches using the SWG-SWs deigned in Fig. 5 were fabricated and tested. For the SWG-SWs used in these switches, $W_{strip}$ is varied from 0.62 um to 1.2 um, while $W_{SWG}$, $\Lambda$, and $\eta$ are fixed at 1.5 um, 0.26 um and 0.75, respectively. The measured electrical switching power of the switch, defined as the power required to switch from one state to another, as a function of $W_{strip}$ of the used SWG-SW phase shifter is plotted Fig. 9(a). As can be seen, the switching power for all of the TE0-TE2 modes show a decrease trend with the increase of $W_{strip}$. The normalized $\mathrm {d}n_{\mathrm {eff}} /dT$ as a function of $W_{strip}$ then can be translated from the data in Fig. 9(a) and is plotted in Fig. 9(b). The theoretical results, obtained using the phase analysis method, are also included in this figure for comparison. It can be seen that the change trends of $\mathrm {d}n_{\mathrm {eff}} /dT$ with $W_{strip}$ for the different modes are in rough agreement with the simulated data. The deviations between the experimental and theoretical results and the fluctuations of the measured data are attributed to fabrication imperfections, manufacturing non-uniformity, etc.
Fig. 9. (a) Experimental electrical switching power of the multimode switch as a function of $W_{strip}$ of the SWG-SW phase shifter. (b) Normalized $\mathrm {d}n_{\mathrm {eff}} /dT$ with respect to $W_{strip}$ of the SWG-SW, translated from the data in (a).
From Figs. 9(a) and 9(b), we can also see that the switching power and $\mathrm {d}n_{\mathrm {eff}} /dT$ of the TE0-TE2 modes are very close to each other when the $W_{strip}$ is 0.66 um. The difference in $\mathrm {d}n_{\mathrm {eff}} /dT$ between the TE0-TE2 modes at this point is calculated to be less than $\pm 1.5$%. Thus, this switch can provide a mode-insensitive switching operation. The experimental $W_{strip}$ enabling this mode-independent behavior of 0.66 um is slightly smaller than 0.69 um predicted by the simulations [shown in Fig. 5], which is attributed to fabrication variations.
Then, to further characterize the switching behavior of this mode-insensitive switch, we measured the optical powers at the bar and cross ports at the wavelength 1550 nm as a function of electrical power for each of the TE0-TE2 modes. The electrical power was swept from 0 mw to 130 mw. with a step of 0.5 mw. The results for the TE0-TE2 modes are plotted in Figs. 10(a)–10(c), respectively, where the coupling losses due to the grating couplers have been calibrated out. The switching powers for the the TE0-TE2 modes are highly close to each other, which are measured to be 28.736 mw, 28.653 mw and 29.56 mw (obtained from the fitted curves of optical power versus electrical power), respectively, corresponding to a difference of <$\pm 1.5$%, confirming the mode-insensitive characteristic of the switch. The highest extinction ratios (ERs) for the TE0-TE2 modes are measured to be 38.2 dB (31 dB), 37.9 dB (37 dB), and 31.9 dB (20.5 dB) for the cross (bar) state, respectively. The small difference between the heating powers of the different modes should mainly come from the difference in waveguide width or/and thickness between the two arms of the balanced MZI structure caused by fabrication non-uniformity, which can result in a mode-dependent phase difference between them. This can be alleviated by improving the fabrication uniformity and arranging the two arms to be closer to each other. To perform the mode-insensitive switching in the presence of a small difference in heating power between the modes, one can use proper common operation powers for all modes. The common powers can be carefully chosen to be close to the heating powers of all modes to achieve the best compromise between the extinction ratios of the different modes.
Fig. 10. Measured optical power at the bar and cross ports versus the applied electrical power of the mode-insensitive switch for the (a) TE0 (b) TE1 and (c) TE2 modes.
Figures 11(a)–11(c) show the experimental transmission spectra of the mode-insensitive switch at the bar and cross states for the input modes of TE0-TE2, respectively, after calibrating out the coupling losses of the grating couplers. The measured wavelength range is 1520-1560 nm. As can be seen, the mode crosstalk for the TE0-TE2 modes at the bar (cross) states are <-14.8 dB (<-24.6 dB), <-13.4 dB (<-9.3 dB), and <-15.2 dB (<-9.2 dB), respectively, over the entire measurement wavelength range. The insertion loss at the wavelength of 1550 nm for the TE0-TE2 modes at the bar (cross) states are about 7.2 dB (3.4 dB), 5.6 dB (3 dB), and 2.6 dB (3.5 dB), respectively, The loss of the TE0 mode at the bar state is larger than those of the other cases, which could be attributed to a fabrication non-uniformity-induced calibration issue. The losses at the wavelength of 1520 nm for the TE0-TE2 modes at the bar (cross) states are also estimated to be 3.9 dB (4 dB), 4 dB (4 dB), and 5.2 dB (4.8 dB), respectively. The experimental performances of the mode-insensitive optical switch are also summarized in Table 2.
Fig. 11. Measured transmission spectra of the mode-insensitive switch for the input modes of (a) TE0 (b) TE1 (c) TE2.
Table 2. Experimental results of the mode-insensitive switch within the wavelength range from 1520 nm to 1560 nm; ER: extinction ratio; CT: crosstalk; IL: insertion loss; BW: bandwidth.
A performance comparison between the present switch and other recently reported SOI-based mode-insensitive switches [27,45] is given in Table 3. Note that the 3-port switch described in [45] is chosen for the comparison. As can be seen, the current switch exhibits the smallest power consumption. This could be mainly because the switches proposed in [27,45] used overly wide waveguides (with widths of 4 um) to realize mode-independent thermo-optic phase shifters, which can reduce the heating efficiency. The extinction ratios of our switch are also significantly larger than those of the others. An additional advantage of the present switch (not shown in Table 3) is the smaller theoretical difference between $\mathrm {d}n_{\mathrm {eff}} /dT$ of the different modes for the used thermo-optic phase shifter compared with those in [27,45] ($\pm 0.5 {\% }$ versus $\pm 1 {\% }$). This difference, owning to the large design flexibility of SWG structures in controlling $\mathrm {d}n_{\mathrm {eff}} /dT$, can potentially be further decreased by using, for example, the artificial neural network [41,42] to further optimize the phase shifter. The crosstalks of the demonstrated switch are comparable with those of the others. The loss for the TE0 mode at the wavelength of 1550 nm of our switch is larger than those of the other works, which could be attributed to the fabrication non-uniformity-induced calibration issue as discussed before. The overall size of the present switch is slightly larger than that in [27], but is much smaller than that in [45]. The device size can be significantly reduced by, for example, using thermal isolation trenches between the arms of the MZI [10]. This can largely decrease the thermal crosstalk between the two arms, thus allowing them to be located much closer to each other to reduce the vertical dimension of the switch.
Table 3. Performance comparison of reported SOI-based mode-insensitive switches; P$_{\pi }$: electrical power required for realizing a $\pi$ phase shift; BW: bandwidth.
5. Conclusion
To summarize, the possibility of exploiting sub-wavelength structures to engineer the mode TO coefficients ($\mathrm {d}n_{\mathrm {eff}} /dT$) of individual modes or potentially specific polarization states for integrated-optics devices has been suggested. The relationships between $\mathrm {d}n_{\mathrm {eff}} /dT$ of the TE0-TE2 modes and the different parameters of a SWG-assisted strip waveguide (SWG-SW) have been studied using the phase analysis method and the photonic band-structure approach, and a part of the results have been verified through experiments. The results have shown that $\mathrm {d}n_{\mathrm {eff}} /dT$ of different modes of a SWG-SW are closely related to the various waveguide parameters with different relationships. Further, a mode-insensitive switch for the TE0-TE2 modes on the SOI platform using a $\mathrm {d}n_{\mathrm {eff}} /dT$ engineered SWG-SW as a mode-independent thermo-optic phase shifter has been designed and experimentally demonstrated. The experimental switching powers for the TE0-TE2 modes are only $\sim$29 mW. The measured maximum extinction ratios (ERs) for the cross (bar) states at the wavelength 1550 nm are 38.2 dB (31 dB), 37.9 dB (37 dB), and 31.9 dB (20.5 dB) for the TE0-TE2 modes, respectively. The experimental mode crosstalk at the bar (cross) states for the TE0-TE2 modes are <-14.8 dB (<-24.6 dB), <-13.4 dB (<-9.3 dB) and <-15.2 dB (<-9.2 dB), respectively, over the wavelength range of 1520-1560 nm.
The current study is expected to offer new design possibilities for a variety of integrated photonic devices. It provides a new route for achieving mode- or polarization-insensitive thermo-optic phase shifters, which are key elements for various mode- or polarization-independent tunable devices (e.g., switches and variable attenuators) for mode- or polarization-division multiplexing systems. Other potential applications of this work in integrated optics include designing athermal devices, optimizing the thermal sensitivity of specific modes or polarization states for diverse tunable devices, minimizing the power consumption of mode or polarization switches, etc. The proposed concept can potentially be used on other SWG structures to further increase the degrees of freedom in $\mathrm {d}n_{\mathrm {eff}} /dT$ engineering, such as multibox SWGs, multiple strip waveguide-assisted SWGs, etc.
Funding
National Natural Science Foundation of China (62105089, 62105381); National Key Research and Development Program of China (2021YFB2801500).
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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