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Abstract
Conventional slot waveguides (CSWs) consisting of an isotropic low-index material sandwiched by two high-index silicon wires have been extensively used in functional photonic devices, including chemical sensing, optical modulating, and all-optical signal processing, due to its significantly enhanced electric field perpendicular to the interfaces in the slot layer. However, there are two drawbacks to be improved if the CSWs are used for signal transmission in photonic integrated circuits, including the crosstalk between waveguides and direct butting mode conversion efficiency (MCE) to a silicon (Si)-strip waveguide. In this study, we propose an anisotropic SW with bulk transition metal dichalcogenide (ASWTMD) to relieve the two shortcomings by replacing the isotropic low-index slot layer with a bulk molybdenum disulfide layer having a high refractive index and giant optical anisotropy. We demonstrated the crosstalk reduction (CR) of the proposed ASWTMD by analyzing the mode profile, power confinement, and coupling strength. We also investigated the MCE by examining the mode overlap ratio and power evolution. The proposed ASWTMD shows significant CR and superior MCE for the transverse electric and transverse magnetic modes compared to those of a CSW with a SiO2-slot layer. The present design paves the possible extensibility to other transition metal dichalcogenides (TMDs) for designing state-of-the-art TMD-based photonic devices exploiting their extraordinary optical properties.
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1. Introduction
Slot waveguides (SWs) [1,2], consisting of an isotropic low-index slot layer (silicon dioxide (SiO2) or air) sandwiched by two high-index silicon (Si) wires, have attracted significant attention because of their strong electric field confined in a nanometer-wide slot much smaller than the mode decay length inside the slot. The strong electric field relying on the continuity of the normal component of the electric displacement field at high-index-contrast interfaces can significantly enhance light-matter interactions, benefiting SWs to apply in various functional devices, such as chemical or biochemical sensing [3–6], optical modulating [7–9], and all-optical signal processing [10–14]. Although the electric field is significantly enhanced in the slot region, the leakage of evanescent wave spreading out of SWs is to be reckoned with, leading to relatively strong crosstalk between adjacent devices. To reduce waveguide crosstalk, some authors [15–18] surrounded Si strips with subwavelength wavelength gratings (SWGs), controlling the optical momentum of evanescent waves to suppress the crosstalk of transverse electric (TE) field, called relaxed total internal reflection (TIR) [18]. However, the SWGs enhance the coupling effect of the transverse magnetic (TM) field, increasing the crosstalk of TM mode. This idea [18] is beneficial to building an ultra-short and high extinction ratio polarization beam splitter based on Si strips [16–18] or CSWs [19]; however, it is not easy to realize polarization-insensitive photonic components with low-crosstalk.
Another crucial problem of using conventional SWs (CSWs) is its mode conversion to a Si-strip waveguide, an elementary component in photonic integrated circuits (PICs). Generally, the CSWs suffer from relatively higher propagation loss (∼10 dB/cm) than that (∼1.5 dB/cm) of Si-strip waveguides [20–23], leading to severe limitations for realizing low-loss PICs. Therefore, a large number of slot-strip converters [24–30] have been reported to improve the mode conversion efficiency (MCE), such as adiabatic tapers [24], sharp tips [25], multi-mode interference (MMI) coupler [26], direct butting coupler [27], balanced power splitter, and tunable phase-matched taper [28], three parts, including a strip taper, 2 × 2 MMI region, and slot taper [29], and a non-uniform strip with sinusoidal profiles [30]. Among them [24–30], a direct butting coupler [27] has a major advantage on the ultra-short device length but accompanies by a cost of lower MCE than those of other structures [24–26,28–30] due to the mode mismatching between a CSW with non-Gaussian-like profile and a Si-strip waveguide with Gaussian-like one.
Recently, a family of two-dimensional (2D) semiconductor transition metal dichalcogenides (TMDs) [31] with unique electric and optical properties, such as MoS2, MoSe2, WS2, and WSe2, have been extensively applied in a large number of optoelectronic and photonic devices, including photodetectors [32,33], sensors [34,35], optical modulators [36,37], and lasers [38,39], due to their extraordinary electrical and optical properties, demonstrating even better performances than that of the graphene-based devices [40]. Optically, the layered TMDs perform giant optical anisotropy [41] in the visible and near-infrared (near-IR) spectral intervals because of the significant difference between layered in-plane covalent bonding and inter-layer van der Waals interaction. Among the TMDs mentioned above, the bulk MoS2 exhibits the smallest optical loss with almost zero absorption and the highest refractive index exceeding that of the traditional semiconductor materials, e.g., Si, Germanium, and gallium antimonide, in the telecommunication wavelength of 1,550 nm [42]. Therefore, the bulk MoS2 serves as an ideal material to form lossless photonic devices with tighter mode confinement than those of the traditional semiconductors. In this study, we propose an anisotropic SW with TMD (ASWTMD) by replacing the isotropic slot of a CSW with a bulk MoS2 to accomplish the polarization-independent crosstalk reduction (CR) and significantly improve the direct butting MCEs. The mode profiles, power concentrations, and coupling lengths of TE and TM modes of the present ASWTMD are analyzed to demonstrate its superiority in CR by comparing it with that of the CSWs. We also considered the power evolutions from CSWs to Si-strip and investigated the MCE as functions of the geometry parameter and working wavelength.
2. CR of the proposed ASWTMD
In the geometrical arrangement, there are vertical (supporting quasi-TE modes) [1,2] and horizontal (supporting quasi-TM modes) [43–49] SWs. Meanwhile, in the fabrication aspect, a horizontal SW can be fabricated easily and precisely due to avoiding the high aspect ratio of a vertical one that causes significant sidewall roughness in modern fabrication techniques. Generally, an isotropic material is used in the slot layer for the previously reported slot-based waveguides or devices [43–49]. In this study, we propose an ASWTMD to significantly reduce the crosstalk of the TE and TM modes in photonic devices based on the CSWs. Another benefit of the proposed structure is to improve the MCE between CSWs and Si-strip waveguides based on a direct butting approach (discussed in the next section).
Figure 1(a) shows a 3D schematic of the proposed ASWTMD with thickness tslot sandwiched by two Si wires with thickness tSi and width wSi on a SiO2 substrate. Here, the TMD chosen is MoS2 with an anisotropic refractive index [nxx, nyy, nzz], where nxx (= nzz) a + b/λ2 is the in-plane refractive index, and nyy = c + d/λ2 is the refractive index along the crystal axis (y-axis) of bulk MoS2; a = 3.84, b = 0.44 µm2, c = 2.44, and d = 0.17 µm2 are the Cauchy parameters [50], and λ is the working wavelength in free space. At the optical communication wavelength λ = 1,550 nm, we obtain the values nxx = nzz = 4.023 and nyy = 2.511 that show a giant refractive index difference Δn ≈ 1.5 [Fig. 1(b)]. The upper Si wire was drawn using a translucent color for a clear view of the interior structure. The numerical results were analyzed using the COMSOL Multiphysics based on the rigorous finite element method.
Fig. 1. (a) 3D schematic diagram of the proposed ASWTMD consisting of a bulk MoS2 slot layer with thickness tslot sandwiched by two Si wires with width wSi and thickness tSi on a SiO2 substrate. (b) The dispersion relation of bulk MoS2, where nxx (= nzz) and nyy are the in-plane and out-of-plane components, respectively, and Δn is the refractive index difference.
Before the analyses of the proposed ASWTMD, we describe the currently promising approaches to prepare the bulk MoS2 films. The practical fabrications of MoS2 films can be divided into top-down and bottom-up approaches. In the top-down approach, exfoliation methods [51,52] yield materials with different shapes, sizes and layer numbers. By peeling off some parts of bulk MoS2 crystal with the tape and transferring into the substrate. Although the produced flakes are typically <10 µm in their lateral size, the dimensions are sufficient area to be useful for the proposed waveguide devices with several µm2. The main advantage of this approach is offering the highest quality films, but it is difficult to precisely control the numbers (thickness) of layers. In contract, the popular bottom-up approach is chemical vapor deposition (CVD) [53–55] compatible to the modern semiconductor technology. The CVD promises several advantages including large-scale growth, highly uniform, precise control of thickness of MoS2 films along with batch fabrication for nanoscale electronic and photonic circuitry.
Recently, Huang et al. [56] proposed an atmospheric pressure CVD (APCVD) through a reaction between MoCl5 precursor and H2S reactive gas at ambient temperature, followed by a two-step annealing process, to deposit MoS2 thin films on a variety of substrates. Comparing these measurements with that of mechanical exfoliation, the thick MoS2 films fabricated by the APCVD are close to single crystal structures and maintain the required optical properties, revealing great promise for nanoscale photonic devices. In 2021, Timpel et al. [57] proposed a versatile, safe, and low-temperature process, ionized jet deposition (IJD) method, to successfully grow large-area MoS2 films with thicknesses up to ∼200 nm on Au/Si3N4 substrate remarkably possess electronic and optical properties as encapsulated 2D MoS2. They also fabricated MoS2-based resistive random memory (ReRAM) devices (Fig. 5(c) in [57]) comprised a thick MoS2 film sandwiched vertically by two Au layers on Si3N4/quartz substrate to demonstrate the electronics properties. Clearly, the structure of the ReRAM is similar to our waveguide device if replacing Au layers by Si ones. Although the exfoliation methods are difficult to precisely control the thickness of MoS2 film, the authors [58–66] adopted mechanically exfoliated approach via repeated peeling due to the simplicity and obtaining high-quality films.
The fabrication flow of the present ASWTMD are schematically shown in Fig. 2 including the steps: (1) depositing a Si (green) and a photoresist (PR) (cyan) films on a SiO2 substrate (brown); (2) defining the lower Si layer with a hard mask followed by PR exposure, development and etching; (3) coating a PR film followed by moving out of PR film on the Si layer by e-beam lithography; (4) depositing a thick MoS2 film using the APCVD or IJD approaches that can precisely control film thickness [56,57] followed by PR lift out; (5) repeating step 3; (6) depositing the upper Si layer and then lifting PR out.
At the parameters, tslot = 60 nm, tSi = 150 nm, wSi = 400 nm, nSiO2 = 1.444 [67], and nSi = 3.480 [67] operating at λ = 1,550 nm, we show the mode profiles of the TE (major component of electric field Ex experiencing nxx = 4.023) and TM (major component of electric field Ey experiencing nyy = 2.511) modes of the proposed ASWTMD in Figs. 3(a) and 3(b), respectively, and those of a CSW with a popular SiO2 slot layer (CSWSiO2) in Figs. 3(c) and 3(d), respectively, for comparison.
Fig. 3. Electric field profiles of (a) TE - and (b) TM modes of the proposed ASWTMD and those of (c) TE- and (d) TM modes of the CSWSiO2, at tslot = 60 nm, wSi = 400 nm, and tSi = 150 nm. The normalized fields of Ex of (a) and (c) are shown in (e) along the red lines of insets; the normalized fields of Ey of (b) and (d) are shown in (f).
For the TE mode, the proposed ASWTMD exhibits tighter mode confinement in the x-direction, mainly affecting the waveguide crosstalk, than that of the CSWSiO2 because nxx > nSiO2 in the slot layer. Moreover, the effective indices of the TE modes are ne = 2.687 and ne = 2.138 for the ASWTMD and CSWSiO2, respectively. Figure 3(e) shows the normalized fields of Ex of the ASWTMD (labeled MoS2) and CSWSiO2 (labeled SiO2) at the central line of the slot layer (the red line of the inset). For the TM mode, we observe looser mode confinement of the proposed ASWTMD in the y-direction than that of the CSWSiO2 because of a lower index contrast between Si and MoS2 (nyy = 2.511) than that between Si and SiO2. Therefore, the mode distribution of the proposed ASWTMD results in better mode confinement in the x-direction than that of the CSWSiO2 (demonstrated later). The effective indices of the TM modes are ne = 2.137 and 1.617 for the ASWTMD and CSWSiO2, respectively. Figure 3(f) shows the normalized fields of Ey in the y-direction (see the red line of the inset).
To quantitatively assess the mode confinement, we investigate the power fractions in the entire waveguide (including the slot and two Si wires) and slot region versus tslot, as shown in Fig. 4(a), where the legends XY-w and XY-sl denote the power fractions of the X (TE or TM) mode of the structures Y (Mo: ASWTMD or SiO2: CSWSiO2) inside an entire waveguide (w) and slot (sl) layer, respectively. At tslot = 60 nm, the TE and TM power fractions inside waveguide of the ASWTMD (solid lines) are TEMo-w = 94.1% and TMMo-w = 86.6%, respectively; those of the CSWSiO2 (dotted lines) are TESiO2-w = 84.9% and TMSiO2-w = 65.3%, respectively. The results demonstrate that the proposed structure achieves significantly stronger mode confinements than those of the CSWSiO2.
Fig. 4. Normalized power as functions of the (a) tslot and (b) working wavelength λ for ASWTMD and CSWSiO2, where TEMo(SiO2)-w denotes the power fraction of the TE mode in the waveguide for the ASWTMD (CSWSiO2), and TMMo(SiO2)-sl denotes the power fraction of the TM mode inside the slot.
As tslot increases, the TEMo-w and TMMo-w moderately increase, whereas that of the TMSiO2-w significantly decreases. Moreover, the difference in power fractions between ASWTMD and CSWSiO2 increases. The results show that the mode confinements of the proposed structure are nearly independent of tslot. For the power fractions inside the slot regions, the TE and TM power fractions of the proposed ASWTMD are TEMo-sl = 25.7% and TMMo-sl = 36.2%, respectively; those of the CSWSiO2 are TESiO2-sl = 14.8% and TMSiO2-sl = 38.8%, respectively. For the proposed design, the TEMo-sl and TMMo-sl increase as tslot increases. In contrast, the TESiO2-sl and TMSiO2-sl saturate while tslot exceeds 50 nm. For the TE mode based on typical TIR mainly confining light in the high-index materials, the field peak of the proposed ASWTMD locates in the slot [Fig. 3(a)] because nxx > nSi. Meanwhile, that of the CSWSiO2 locates in the lower Si wire [Fig. 3(c)] because nSi > nSiO2. For TM mode relying on fulfilling the continuity of the normal component of the electric displacement field, more power of the CSWSiO2 is confined in the slot than [Fig. 3(d)] that of the proposed ASWTMD [Fig. 3(b)] because nyy > nSiO2. Note that more power of the proposed ASWTMD is pushed toward the Si wires [Fig. 3(f)]. Figure 4(b) shows the power fractions as a function of wavelength. Apparently, except that the TMSiO2-w shows a significant variation as λ varies, others are almost invariant in a broad bandwidth from λ = 1,400 to 1,650 nm.
Figures 5(a) and (b) show the field overlaps of the TE and TM modes considering the crosstalk between adjacent waveguides, respectively, for a coupled ASWTMD (labeled as MoS2 in the inset) and CSWSiO2 (labeled as SiO2) systems at an edge-to-edge separation s = 600 nm. The proposed structure significantly reduces the crosstalk of the TE and TM modes. Note that the CR of the TM mode is relatively small compared to that of the TE mode due to the smaller difference between nyy and nSiO2 than that between nxx and nSiO2.
Fig. 5. Field profiles of (a) TE- and (b) TM-symmetrical modes of a coupled ASWTMD and CSWSiO2 systems for tslot = 60 nm, wSi = 400 nm, tSi = 150 nm, and s = 600 nm along the x-direction at the central line of the slot layer [similar to the red line of the inset of Fig. 3(e)].
According to the coupled-mode theory (CMT) [68], the coupling length Lc = λ/[2(ne − no)] is used to evaluate the degree of crosstalk of a coupled waveguide system, where ne and no are the effective refractive indices of the even and odd modes, respectively. The calculated values of Lc of the TE mode are Lc = 78.9 and 21.82 µm for ASWTMD and CSWSiO2, respectively, at tslot = 60 nm, wSi = 400 nm, tSi = 150 nm, and s = 200 nm. Figures 6(a) and (b) show the TE power evolutions (Poynting vector flows) through a distance of L = 10 µm of the ASWTMD (PTE–MoS2) and CSWSiO2 (PTE–SiO2), respectively, which exhibit the 3D field propagations.
Fig. 6. Poynting vector flows of the TE mode for the (a) proposed ASWTMD (PTE-MoS2) and (b) CSWSiO2 (PTE-MoS2) at s = 200 nm; those of the TM modes for the (c) proposed ASWTMD (PTM-MoS2) and (d) CSWSiO2 (PTM-SiO2) at s = 400 nm. Here, L = 10 µm denotes the power propagation distance.
We observe that most PTE–MoS2 remains in the through (upper) port; however, a considerable amount of the PTE–SiO2 couples to the cross (lower) port, revealing the significant CR of the proposed structure. For the TM mode, the calculated Lc’s of the ASWTMD and CSWSiO2 are Lc = 51.2 and 15.2 µm, respectively, at s = 400 nm. Figures 6(c) and 6(d) show the TM power evolutions of the ASWTMD (PTM–MoS2) and CSWSiO2 (PTM–SiO2), respectively. The results confirm that the proposed ASWTMD shows significant CR for the TE and TM modes.
At the end of the section, we investigate the Lc dependence on s and λ, as shown in Figs. 7(a) and (b), respectively. It can be seen that Lcs of the TE and TM modes of the proposed structure (solid lines) are longer than those of the CSWSiO2 (dotted lines). For instance, the Lcs of the TE (TM) modes are 1,761.4 (114.8) and 229.7 (25.2) µm for the ASWTMD and CSWSiO2, respectively, while setting the distance s = 500 nm. These results also correspond to the previous analyses of the mode distribution, power fraction, and field overlapping. Figure 7(b) shows the Lc’s versus λ to understand the spectral response. The results demonstrate that the slope rates of Lc with respect to λ are analogous to the two structures.
Fig. 7. Coupling length Lcs of the TE and TM modes as functions of the (a) separation of adjacent waveguides s and (b) working wavelength λ of the proposed ASWTMD (solid lines) and the CSWSiO2 (dotted lines).
3. Mode conversion between the proposed ASWTMD and a typical Si-strip waveguide
In this section, we investigate the MCE of the proposed ASWTMD to the Si-strip waveguide. Figures 8(a) and (b) show the TE and TM mode profiles of a typical Si-strip waveguide with width wSi = 400 nm and height hSi = 360 nm, respectively. Note that the mode profiles of the proposed ASWTMD and CSWSiO2 have been illustrated in Fig. 3. To compare the mode matching degrees between the Si-strip waveguide and the proposed structure with the same parameters as that in Fig. 3, we plot the normalized fields of Ex along the x-direction and those of the Ey along the y-direction, as shown in Figs. 8(c) and (d), respectively. Obviously, the TE mode profiles of the Si-strip and the proposed structure are highly matching. Although their TM mode profiles show moderate mismatching, the deviation degree is smaller (some fraction of the mode field spreads out of the slot region to the Si wires) than that between the Si-strip and CSWSiO2 [Fig. 3(f)]. According to the CMT [68], the mode overlap ratio (Γ) derived from the orthonormal eigenmode relations is used to quantitatively estimate the field overlap or power coupling efficiency between two modes of a waveguide, as shown below.
Fig. 8. Field profiles of (a) TE- and (b) TM modes of the Si-strip waveguide at tslot = 60 nm, WSi = 400 nm, and hSi = 150 nm. The normalized fields of (a) Ex and (d) Ey of the Si-strip waveguide and the proposed structure.
We show the MCEs of the ASWTMD (labeled by TE(TM)-MoS2) and CSWSiO2 (TE(TM)- SiO2) as a function of tslot at WSi = 400 nm and hSi = 150 nm in Fig. 9(a). For the proposed ASWTMD, the MCEs of the TE and TM modes are greater than 99%, ranging from tslot = 20 to 90 nm. Although the TE mode of the CSWSiO2 achieve MCE of greater than 90%, the TM mode varies from MCE = 94.1% at tslot = 20 nm to 74.8% at tslot = 100 nm. The results demonstrate that the proposed structure can significantly improve more than the CSWSiO2. Figure 9(b) shows the MCE dependence on λ at tslot = 60 nm. The proposed ASWTMD shows a performance of MCE of greater than 99.5% in an ultrabroad band of 250 nm, whereas the MCE of TM mode of the CSWSiO2 is between 86.1% and 79.5%.
Fig. 9. Mode conversion efficiencies (MCEs) of ASWTMD and CSWSiO2 as functions of (a) tslot and (b) λ.
Figure 10 shows the power evolutions from the input ASWTMD or CSWSiO2 waveguide (left side) with a length L1 = 1 µm through the connecting interface at z = 1 µm directly butting to the output Si-strip waveguide with a length L2 = 2 µm. Figures 10 (a) and (c) show that the TE- and TM power evolutions of the proposed ASWTMD exhibit extremely low reflections at the connecting interfaces at z = 1 µm, respectively. The low reflection of the TE mode of the CSWSiO2 is also observed in Fig. 10 (b). Obviously, the input TM mode of the CSWSiO2 not only suffers a back reflection from the connecting interface but excites the first-order TE mode supported by the Si strip waveguide, resulting in a strong non-uniform field profile as shown in Fig. 10(d).
Fig. 10. Power evolutions of the TE modes for (a) ASWTMD and (b) CSWSiO2; those of TM modes for (c) ASWTMD and (d) CSWSiO2, at tslot = 60 nm, hSi = 150 nm, and WSi = 400 nm. Here, L1 = 1 µm and L2 = 2 µm denote the waveguide lengths of the ASWTMD (or CSWSiO2) and Si-strip waveguide, respectively.
Figure 11 shows the power distributions at some positions for clear observation of the mode conversion.
Fig. 11. Power distributions of the (a) TE and (b) TM modes of the proposed ASWTMD and those of the (c) TE and (d) TM modes of the CSWSiO2 at some positions along the propagation direction (z-direction).
Around the interface, the variation of power distribution of the proposed ASWTMD is slight [Fig. 11(a)] for the TE mode because of high mode matching. For the TM mode of the proposed ASWTMD, the variation of power distribution reflects the moderate mode mismatching; however, it shortly transforms to the desired mode at only 100 nm from the interface [Fig. 11(b)]. For the CSWSiO2, significant mode mismatching increases the propagation distances to achieve complete mode conversions [Figs. 11(c) and (d)]. Clearly, stable TE and TM modes of the CSWSiO2 (ASWTMD) are not reached until propagating distances of 200 (10) and 400 (100) nm, respectively, from the interface, responding to the better mode matching between the proposed ASWTMD and a Si-strip waveguide.
Lastly, we consider the extension of the proposed ASWTMD to the O-band spanning wavelength range from 1260 nm to 1360 nm. The O-band is considerably used for high-speed Ethernet network due to its low dispersion and loss. The commercial applications of O-band aim at data center interconnect, passive optical network, and metro/long haul datacom [69]. By the chromatic dispersions as shown in Fig. 1(b), the refractive indices of bulk MoS2 film along different orientations show slight variations from O- to C-bands. Although the spectral responses of Lc [Fig. 7(b)] and MCE [Fig. 9(b)] are calculated at the range of λ = 1400 nm to 1650 nm in the present work, the dependences of Lc and MCE on λ can be well predicted by the trends in Figs. 7(b) and 9(b). Therefore, the proposed idea might be extended to the O-band, significantly increasing the network transmission volume.
4. Summary
This paper proposed ASWTMD to reduce the waveguide crosstalk and improve MCE between SW and Si-strip waveguide modes compared to those of the CSWSiO2, where one of the TMDs, MoS2, is chosen as the slot layer because of its smallest optical loss and highest refractive index. For the TE mode, MoS2 shows a higher in-plane refractive index than that of the Si wires, making the tighter mode confinement achieve better CR. Moreover, the mode confinement of the TM mode is improved because of the larger refractive index along the crystal axis of MoS2 than that of the SiO2 layer. However, locating subwavelength gratings in between the Si-strip waveguides or CSWs only reduces the crosstalk of the TE mode but enhances the coupling of the TM mode. Therefore, the mode profile, power confinement, and coupling strength of the proposed ASWTMD confirm the better CR for the TE and TM modes than those of the CSWSiO2. The other advantage of the proposed structure is that the mode conversions to the Si-strip waveguide show much higher mode overlaps than those of the CSWSiO2, ranging from wavelengths 1,400 to 1,650 nm. Additionally, the power evolutions demonstrate the low power reflections at a direct butting interface. We expect that the giant optical anisotropy and high refractive index of the TMDs might have more applications in diverse PICs.
Funding
Ministry of Science and Technology, Taiwan (111-2112-M-005-012).
Acknowledgments
The authors would like to thank Enago (www.enago.tw) for the English language review.
Disclosures
The author declares 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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