1. Introduction

Mid-infrared (mid-IR) silicon (Si) photonics, spanning wavelengths from 2 to 20 µm [1], has attracted much interest in recent years due to its potential application in various fields such as absorption spectroscopy, chemical and biological sensing, environmental monitoring, and free-space communication [1–4]. Initial applications have targeted photonic integrated circuits and optical communications around the near-IR wavelength of 1.55 µm. However, driven by the demand for high capacity data transmission in modern networking, the near-IR regime has begun to face the so-called “capacity crunch.” To address this pressing issue, the scope of Si photonics has been expanded to include the short mid-IR range [2,5,6]. In recent years, many Si photonics components, including waveguides [7], electro-optic modulators [8], directional couplers [9], multimode interferometers [10], ring resonators [11], grating couplers [12–14], and polarizing beam splitters [15,16], have been proposed and demonstrated. Compared with near-IR photonics using silicon-on-insulator (SOI) structures based on mature complementary metal-oxide semiconductor compatible technology, mid-IR band Si photonics has thus far received much less attention. This is because the silicon dioxide (SiO₂) in the SOI platform has strong absorption loss in the mid-IR wavelength range of λ = 2.6–2.9 µm and around λ > 3.6 µm [17,18]. Hence, choosing an appropriate platform for operating in the mid-IR band is essential to building low-loss mid-IR photonic components. At present, several different transparent platforms, including silicon-on-sapphire, silicon-on-nitride, silicon-on-calcium-fluoride (SOCF), and silicon-on-lithium-niobate, have been used in the mid-IR spectrum; the design and fabrication of each platform have been reviewed and discussed previously in [1–4]. Additionally, the Germanium-based integrated photonics has also reported for the extension of the transparent window up to 15 µm [19]. In addition to extending operating wavelengths, advanced multiplexing technologies [20–24] can further increase data traffic volumes for optical information processing and sensing. Among them, mode-division multiplexing (MDM) [21–24], which encodes each optical signal on the different spatial modes of multimode waveguides, allows significant transmission capacity scaling. The key component to realizing MDM technology is a mode converter able to transform given spatial modes into desired modes. There are three approaches to mode conversion: Phase matching [25–34], beam shaping [35,36], and coherent interference [37–39]. The most popular phase-matching technique is to provide the phase difference and field overlap between the given and target modes by perturbing refractive-index profiles along the transverse and propagation directions.

Ding et al. [25] adopted a tapered directional coupler-based TE₀-to-TE₁ converter with relatively long device length of 50 µm for the operating wavelength of 1550 nm on a SOI platform, and the measured insertion loss (IL) and inter-modal crosstalk (CT) are 0.3 and <−16 dB, respectively. In [26,28], continuous graded and rectangular periodic indices were introduced to a Si waveguide along the transverse and propagation directions, respectively. A transverse electric (TE) mode converter between the TE₀ and TE₁ modes, with device length of around 23 µm, conversion efficiency of around 96.5%, and power transmissions of around 92%, were numerically and experimentally demonstrated. Chen et al. [27] designed various mode-order converters based on asymmetric tapered structures with device length of around 20 µm. Their simulation results show high conversion efficiency (CE) of around 0.1 dB over a wavelength of 1520 to 1580 nm. In [29], using deposited phase arrays of Si nanorods on lithium niobate waveguides, mode converter device lengths between TE₀ and TE₁ modes (with power transmission 92.9%) and between TE₀ and TE₂ modes (with power transmission 82.9%) were 5.4 and 8.4 µm, respectively. However, no clear design principle was presented in this work, and the fabrication precision of the phase arrays can be difficult to control due to size variation between the Si nanorods. A tilted subwavelength periodic slot engraved on a Si waveguide was proposed for converting TE₀ to TE_j modes, with j in the range 1 to 4 [31]. Device lengths were about 5 to 10 µm, with conversion loss around 1.2 dB and inter-modal crosstalk (CT) around −10 dB. However, the smallest feature size of about 50 nm is difficult to fabricate precisely, requiring moderate relaxation of geometry conditions. In [32], the researchers proposed an on-chip mode converter, achieving forward conversion using two cascaded Bragg reflection processes instead of a general backward conversion process, with the bandwidth and central wavelength adjustable according to their theoretical analysis. However, the power transmission spectrum of the converter between TE₀ and TE₂ modes were only around 20 nm. TE-polarized mode converters based on deeply etched polygonal slots [33] have been reported based on the theoretical analysis of transformation optics [40,41]. The device length between TE₀ and TE₁ modes was around 24 µm, with a conversion efficiency of around 97.6% and crosstalk of <−20 dB over the wavelength range of around 100 nm. However, different polygonal slots are required for scaling the design to different mode-order converters. For beam shaping [35,36] and coherent interference [34–36] techniques, the main challenges are the large device footprint and limited operating bandwidth. Nevertheless, the mode converters mentioned above [25–39] all operate at the telecommunication wavelength of 1.55 µm. To the best of our knowledge, mode converters operating in the mid-IR range have yet to be studied.

In this article, we propose a mode converter operating at the mid-IR wavelength of 3.4 µm, comprising a single etched parallelogram slot filled with silicon nitride (Si₃N₄) on a Si waveguide. To avoid the SiO₂ substrate’s high absorption loss [17], we use a crystalline calcium fluoride (CaF₂) substrate because of its transparency in the spectral range from visible to mid-IR wavelengths of 8 µm, and its low refractive index of around 1.4 at a wavelength of 3.4 µm [42]. Additionally, Si photonic devices have been experimentally integrated on a CaF₂ substrate [43]. Compared with other mid-IR photonic platforms, the SOCF platform possesses the higher refractive-index contrast, making the proposed device more compact. In addition to using the conventional solid materials, a suspended waveguide platform was reported to build the mid-IR photonic devices operating at longer wavelengths for sensing applications in the molecular fingerprint [44]. Notably, the suspended waveguide can offer an even higher refractive-index contrast [45]. Differing from published reports relying on massive parameter sweeps [29–34], we provide analytical formulas to determine the proposed structure’s geometry parameters by considering the phase-matching condition (PMC) and the profiles of the coupling coefficient in coupled-mode theory (CMT) [46]. From a practical perspective, the relatively simple geometry of the proposed design poses easy fabrication challenges, making the fabrication processes feasible and easy. Numerical examples are provided to demonstrate high performance in terms of conversion efficiency, power transmission, and inter-modal crosstalk. The spectral response within the 3.34 to 3.46 µm range and fabrication tolerances are also analyzed. Notably, the numerical results of the present device operating at the wavelength of 1.55 µm are also offered to validate the feasibility.

2. Design principles of the proposed mode converter

Figure 1 shows a 3D schematic diagram and top view of the proposed mode converter, comprising an etched parallelogram slot filled with Si₃N₄ on a Si waveguide, deposited on a CaF₂ substrate. The operating wavelength is at the mid-IR wavelength of λ = 3.4 µm. Instead of using SiO₂ in the slot and substrate, as widely used in near-IR systems, we used Si₃N₄ and CaF₂ for the slot and substrate, respectively, because of their transparency at the present operating wavelength [1,4,18]. Along the transverse (x-direction) and propagation (z-direction) directions, the tilted slot with the angle θ and depth h_et induces dielectric perturbations in the Si waveguide, achieving mode conversion through inter-modal coupling. Before addressing the theoretical aspect of the proposed mode converter, we outline the fabrication steps. A patterned hard mask of the present Si waveguide of width W_Si is prepared using electron beam lithography in advance. (1) Depositing a positive photoresist (PR) thin film of height h_Si on a CaF₂ substrate to define the Si waveguide by the preceding mask, a PR exposure with ultraviolet light, development, and an etching process. (2) Adopting focused ion beam (FIB) milling [47] with high beam currents to directly write the tilted slot region of depth h_et. (3) A Si₃N₄ layer of height h_et is deposited using chemical vapor deposition in the slot. Subsequently, a chemical mechanical polishing is used to flatten the plane of the slot. In the step 2, differing from the optical lithography with larger spot size of few tens of nanometers, a maskless FIB with beam size of <5 nm in diameter can relieve the fabrication tolerance of acute angles A to D of parallelogram region, effectively reducing the length of transverse direction of the acute angles. Compared with the FIB technique, the scanning probe lithography has emerged as an alternative type of lithography with sub-10-nm resolution despite the slow writing speed [48,49]. The coupling mechanism of the proposed structure can be approximately described by the CMT dealing with an electromagnetic wave propagating along a perturbed dielectric structure.

 

Fig. 1. (a) 3D schematic diagram and (b) top view of the proposed mode converter.

Download Full Size | PDF

For a given dielectric perturbation of Δɛ (x, y, z) of a waveguide, the amplitude conversion between the input mode p and output (converted) mode q for the condition of co-directional coupling is expressed by the following [46]:

3. Numerical demonstrations of performance and fabrication tolerance

The relative permittivities of Si, Si₃N₄, and CaF₂ at a wavelength of λ = 3.4 µm are n_Si = 3.4293, n_Si3N4 = 1.9304, and n_CaF2 = 1.4148 [42], respectively. The thickness of the Si waveguide and depth of the etched parallelogram Si₃N₄ slot chosen were h_Si = 220 nm and h_et = 100 nm, respectively, as shown in Fig. 1(a). In this work, the numerical results were obtained using COMSOL Multiphysics based on a rigorous finite-element method. The computational window was enclosed by perfectly matched layers that effectively absorb the outgoing power. The first step in the design process is to select a suitable width for the Si waveguide, able to support the converted guided modes. For the TE₀-to-TE₁ mode converter of width W_Si = 3.25 µm, the effective refractive indices of TE₀ and TE₁ modes are n_{eff_0} = 2.0234 and n_{eff_1} = 1.8216, respectively. According to the PMC modified of Eq. (4) with condition q = 1, the total length of the TE₀₁ (TE₀-to-TE₁) mode converter is L = 16.84 µm. The next step is to find the tilted angle θ and width W_slot of the slot. As shown in Fig. 1(b), the W_slot and θ are related by the trigonometry of tanθ = W_Si/(L – W_slot) = t_slot/W_slot. Once t_slot is obtained, the θ and W_slot can be determined to completely define geometry parameters for the proposed structure. To maintain a non-zero κ_0q(z) through the entire conversion process, the parallelogram geometry must be fulfilled by the condition W_slot < L/2. Therefore, the maxima of W_slot and t_slot must be smaller than L/2 and W_Si, respectively. Before deriving the parameters, we selected the transverse position x as the independent variable of κ_0q because the actual propagation distance z depends on θ. To investigate the effect of κ_0q on mode conversion, we examined profiles of κ_0q(x) with different values of W_Si (3.25, 4.25, 5.25, and 6.25 µm for TE₀₁, TE₀₂, TE₀₃, and TE₀₄, respectively), where q = 1, 2, 3, and 4, along the x-direction from x = 0 (point A in Fig. 1(b)) to x = W_Si (point B in Fig. 1(b)), as shown in Fig. 2(a). The choice of W_Si is to enable waveguiding for the required modes and to avoid waveguiding for high-order modes. Also, the coupling coefficient profile of κ_0q(z) is calculated based on different cross section along propagation distance z from the positions of z = A to z = C with effective-index perturbation for different q’s. In the parallelogram, t_slot = W_Si is chosen for the TE₀₁ converter, and the last-half triangle ΔBCD accumulates the remaining half-period phase, as per Fig. 2(b), which shows the profiles of κ_0q(z) from z = 0 to z = L. Thus, we obtain θ = 21.1° and W_slot = L/2 = 8.42 µm for the TE₀₁ converter.

 

Fig. 2. Coupling coefficient profiles of (a) κ_0q(x) and (b) κ_0q(z) versus the transverse position x and propagation distance z of the proposed structure, respectively, at a wavelength of λ = 3.4 µm, with parameters h_Si = 220 nm, h_et = 100 nm, and a fixed W_Si for a specific q.

Download Full Size | PDF

For the TE₀₂ converter with W_Si = 4.25 µm, the device length is L = 21.06 µm according to the PMC where q = 2. Observing the profile κ₀₂(x) from t_slot = W_Si/2 (the second point of κ₀₂= 0) to W_Si (the third point of κ₀₂= 0), maximum coupling strength occurs at t_slot = 2W_Si/3, as depicted by a red arrow in the profile of κ₀₂(x). This is because the maximum power exchange occurs at Δβ = 0 and |κ_0q|L = π/2 in the CMT. The relationship of |κ_0q|L = π/2 can be calculated by

 

Fig. 3. The obtained geometry relations of the proposed structure with the width W_Si of Si waveguide, where q is the mode number and α is shrinking factor.

Download Full Size | PDF

Detailed design parameters for the proposed converters are listed in Table 1. It is clear that device length increases as the converted mode-order increases. Note that a smaller shrinking factor α is chosen for the higher-order mode converter. As stated above, the value of α = 0.97 is sufficient to achieve the deviation from the PMC between the TE₀ and TE₁ modes, making a significant reduction of inter-modal CT of the target TE₂ and undesired TE₁ modes for a TE₀₂ converter. As q increases, the number of undesired TE_q−1 mode coupling increases because of more guided modes supported in a wider waveguide. To reduce the maximum undesired TE_q−1 mode coupling, we need to choose a smaller α (< 0.97) for a TE_0q converter with q > 2. This is because the effective-index contrast between TE₀ and TE_q−1 modes increases as q increases, and therefore, a smaller α is enough to deviate their PMC, reducing the inter-modal CT between the target TE_q and the undesired highest-order TE_q−1 modes.

Table 1. Geometry parameters of the TE₀ → TE_q mode converters operating at a wavelength of λ = 3.4 µm, with parameters h_Si = 220 nm and h_et = 100 nm.

View Table | View all tables in this article

In terms of coupling strength, the calculated Κ_0q’s of the TE₀₁ and TE₀₂ converters are close to the theoretical value of π/2, ensuring the proposed design can achieve extremely high-power conversion. However, Κ₀₃ and Κ₀₄ show larger deviations from π/2, resulting in lower power conversion between the input and target modes. The profiles of κ₀₃(z) and κ₀₄(z) are also shown in Fig. 2(b). It can be seen that the profiles of κ_0q(z) are symmetrical to the propagation distance of L_opt/2. According to the results in Table 1, we show the simulated electric field (E_x) evolutions of the four converters in Fig. 4 along the propagation distance. The numerical algorithm to simulate the evolutions of electric field (E_x) is based on the COMSOL build-in direct solver of multifrontal massively parallel sparse. It can be clearly observed that the input TE₀ modes for different Si waveguide widths launched from the left sides are converted to high-order modes of TE_q after the propagating slots. The field profiles of the TE₀ mode are first squeezed into the sharp corners because of the lower index of the slot and then scattered across the slot region to form the desired high-order modes. Note that the squeeze order becomes increasingly strong as the converted mode order increases. The numerical results demonstrate that the proposed etched parallelogram slot can effectively accomplish arbitrary mode-order conversion. Notably, considerable feature sizes can be precisely fabricated with modern technologies, and the relatively simple fabrication process ensures the proposed structure is feasible.

 

Fig. 4. Electric field (E_x) profiles of the (a) TE₀₁, (b) TE₀₂, (c) TE₀₃, and (d) TE₀₄ mode converters at the plane of y = 100 nm, where y = 0 nm is the bottom of the Si waveguide, at a wavelength of λ = 3.4 µm, with parameters h_Si = 220 nm and h_et = 100 nm.

Download Full Size | PDF

To quantify mode converter performance, we assessed three crucial indices, including: (1) Conversion efficiency (CE) - defined as the power ratio between desired output mode and input mode; (2) Inter-modal crosstalk (CT) - defined as the power ratio between output undesired and desired modes (namely to evaluate the mode purity); and (3) Transmission (T) - defined as the total transmitted power. The three indices are formulated by the following:

Next, we analyzed the spectral response of the present structure over the wavelength band of 3.34–3.46 µm, and the calculated CE, CT, and T, taking the material dispersion [39] into account, as shown in Fig. 5. Over the broadband range of 100 nm from the wavelengths of 3.35–3.45 µm, the CEs of the TE₀₁ and TE₀₂ converters were >90%, TE₀₃ was >80%, and TE₀₄ was >70%, as shown in Fig. 5(a). We observed transmissions of >90% to be insensitive over the band of 120 nm, as shown in Fig. 5(b). In terms of target mode purity, the CT of the TE₀₁ converter was <−20 dB, as shown in Fig. 5(c), and the maximum CTs for all converters were generally <−15 dB, as shown in Figs. 5(d)–5(e), except one TE₃-to-TE₄ case with CT of <−10 dB, as shown in Fig. 5(f). These results validate the proposed structure’s ability to support broadband operation of 120 nm with sufficiently high performance.

 

Fig. 5. (a) Conversion efficiency (CE) and (b) transmission (T) of the proposed mode converter, and the inter-modal crosstalk (CT) of the (c) TE₀₁, (d) TE₀₂, (e) TE₀₃, and (f) TE₀₄ converters versus the wavelength, with conditions of h_Si = 220 nm and h_et = 100 nm.

Download Full Size | PDF

Table 2. Performance of the TE₀ → TE_q mode converters operating at a wavelength of λ = 3.4 µm, with parameters h_Si = 220 nm and h_et = 100 nm.

View Table | View all tables in this article

To evaluate device feasibility, we investigated fabrication tolerance by examining the effects of geometry deviation on performance. We first analyzed CE, CT, and T versus variation in angle Δθ of the etched parallelogram slot, as shown in Fig. 6. Within Δθ < ± 2°, the CEs of the converters TE₀₁ to TE₀₃ were >75%, as shown in Fig. 6(a), and all converters achieved T > 80%, as shown in Fig. 6(b).

 

Fig. 6. (a) Conversion efficiency (CE) and (b) transmission (T) of the proposed mode converter and the inter-modal crosstalk (CT) of the (c) TE₀₁, (d) TE₀₂, (e) TE₀₃, and (f) TE₀₄ converters, versus the variation of slot angle Δθ under conditions of h_Si = 220 nm and h_et = 100 nm.

Download Full Size | PDF

Decreasing the slot angle appears to reduce insertion loss in total transmitted power. The maximum CTs of all converters remained <−10 dB, except for the CT between TE₃ and TE₄ modes with Δθ < 0. This is because decreasing θ generally reduces the input-to-target mode coupling strength. Nevertheless, applying the condition of Δθ > 0 can effectively improve the CT between TE₃ and TE₄ modes. The other critical parameter is the etched depth of the slot. The relationship between performance and etched thickness Δh_et of the slot and the calculated results are shown in Fig. 7. Decreasing the etched depth appears to moderately increase the CEs and Ts, as shown in Figs. 7(a) and 7(b), respectively, but significantly reduce the modal purity. The increase of transmission as h_et decreases is also due to the smaller index perturbation having the same effect induced by decreasing the parallelogram angle. In contrast, shallow depth leads to reduced coupling strength, resulting in a higher CT. Within the range Δh_et < ± 10 nm, the CEs and Ts of all converters were >80%, as shown in Figs. 7(a) and 7(b), respectively, and the maximum CTs of all converters remained <−10 dB, as shown in Figs. 7(c) to 7(f), respectively. Extending the range to Δh_et > ±10, only the CT between the TE₃ and TE₄ modes exceeded −10 dB. Fortunately, by selecting a deeper etched thickness of Δh_et > 0, CT between the TE₃ and TE₄ modes remain <−10 dB. Overall, we consider the fabrication tolerances from the above results to be satisfactory, making the proposed structure practically feasible using the proposed mathematical formulas for obtaining geometry parameters.

 

Fig. 7. (a) Conversion efficiency (CE) and (b) transmission (T) of the proposed mode converter and inter-modal crosstalk (CT) of the (c) TE₀₁, (d) TE₀₂, (e) TE₀₃, and (f) TE₀₄ converters, versus the variation in etched thickness Δh_et of the slot under conditions of h_Si = 220 nm and h_et = 100 nm.

Download Full Size | PDF

To validate feasibility of the proposed formulations to the optical communication wavelength of λ = 1.55 µm, we investigate the performances of the proposed mode converter on a SOI platform. Therefore, we replaced the CaF₂ substrate used at the mid-IR wavelength of λ = 3.4 µm with the SiO₂ substrate at the near-IR wavelength of λ = 1.55 µm. The relative permittivities of Si, Si₃N₄, and SiO₂ at a wavelength of λ = 1.55 µm are n_Si = 3.480, n_Si3N4 = 1.996, and n_SiO2 = 1.444 [42], respectively. The thickness of the Si waveguide and depth of the etched parallelogram Si₃N₄ slot chosen were h_Si = 220 nm and h_et = 100 nm, respectively, the same as those of above example. Detailed design parameters for the converters are listed in Table 3. Note that the device lengths of the TE₀₁ with W_Si= 1.10 µm and TE₀₂ with W_Si= 1.50 µm are only 5.47 and 7.23 µm, respectively.

Table 3. Geometry parameters of the TE₀ → TE_q mode converters operating at a wavelength of λ = 1.55 µm, with parameters h_Si = 220 nm and h_et = 100 nm.

View Table | View all tables in this article

According to the results in Table 3, the simulated electric field (E_x) evolutions of the two converters are in Fig. 8.

 

Fig. 8. Electric field (E_x) profiles of the (a) TE₀₁and (b) TE₀₂ mode converters at the plane of y = 100 nm, where y = 0 nm is the bottom of the Si waveguide, at a wavelength of λ = 1.55 µm, with parameters h_Si = 220 nm and h_et = 100 nm.

Download Full Size | PDF

Based on the definitions in Eq. (6), the calculated performances of the proposed converter operating at the wavelength of λ = 1.55 µm are listed in Table 4. We observe that the CEs of the TE₀₁ and TE₀₂ converters are greater than 94% and 82%, respectively; the maximum CTs of TE₀₁ and TE₀₂ are less than −18 and −11 dB, respectively; and the T values for the two converters are >96%. Also, the T of the TE₀₁ converter with a slightly larger angle of θ = 21.9° is slightly lower than that of other TE₀₂ one with θ = 19.1° due to increased energy scattering.

Table 4. Performance of the TE₀ → TE_q mode converters operating at a wavelength of λ = 1.55 µm, with parameters h_Si = 220 nm and h_et = 100 nm.

View Table | View all tables in this article

The obtained results demonstrate that our proposed idea can be accurately extended to other bands. Comparing our device with reported mode converters only using dielectric materials, the device length and performances are listed in Table 5. Considering both the device length and performances, it can be seen that the proposed device shows superior figure of merit (i.e., [CE or CT] divided by device length) than the previously reported designs. Importantly, the proposed mode converter accomplishes arbitrary mode-order conversion operating at various wavelength bands using a single tilted slot structure.

Table 5. Comparison of various mode converters operating at the wavelength of λ = 1.55 µm.

View Table | View all tables in this article

4. Summary

We proposed a compact mode converter comprising an etched parallelogram slot filled with Si₃N₄ on a Si waveguide on a CaF₂ substrate, able to achieve any waveguide mode conversion at the mid-IR wavelength of 3.4 µm. We constructed analytical formulas for the design parameters by considering the PMC and maximum coupling strength of CMT. Numerical examples were provided, demonstrating that the TE₀-to-TE₁, TE₀-to-TE₂, TE₀-to-TE₃, and TE₀-to-TE₄ converters were able to achieve conversion efficiencies of >92.7%, >91.7%, >88.2%, and >75.8%, respectively, with power transmission and inter-modal CT values of >93% and <−10 dB, respectively, for the four converters. Device lengths of the shortest TE₀-to-TE₁ and longest TE₀-to-TE₄ converters were just 16.84 µm and 24.61 µm, respectively. Over a broad wavelength range of 100 nm, the CEs of the first three converters were >80% and >70% for the TE₀-to-TE₄ converter. Analysis of fabrication tolerances showed that the conditions of Δθ > 0° and Δh_et > 0 nm achieved a high T of >80% and low CT of <−10 dB. The proposed idea can be scaled not only for arbitrary-order mode conversion but also for other operating bands. In addition, the simulation results of our device operating at the wavelength of 1550 nm show that the device lengths of the TE₀-to-TE₁ (with the CE 94%) and TE₀-to-TE₂ (with the CE 82%) converters can be shrunk to 5.47 and 7.23 µm, respectively, demonstrating the feasibility of the proposed idea to other bands. Overall, this work provides a universal model for effectively designing mode converters and other photonic components.