Diffraction engineering for silicon waveguide grating antenna by harnessing bound state in the continuum

1 Introduction

A silicon (Si) waveguide grating antenna (SWGA) is an Si-on-insulator (SOI) waveguide with periodic grating structures that diffract the guided mode into the radiation mode. SWGAs have been widely applied in various photonic integrated systems, e.g. light detection and ranging (LiDAR) [1], [2], [3], high-resolution microscopy [4], biophotonic probing [5], and low-cost biosensors [6], [7]. The spatial resolution is critical for all these applications, so the small far-field divergence is desired for SWGA. However, the input light can only propagate over a short distance (less than a few hundred micrometers) in conventional SWGAs due to the significant diffraction strength, leading to the relatively large far-field divergence. In SWGA, the conventional diffraction formula can be written as [8]

where κ is the coupling coefficient, Δn_eff is the index perturbation caused by the grating structures, n_eff is the effective index of the guided mode, and Λ_grt is the grating pitch. Thus, one has to reduce the index perturbation to achieve weak diffraction. One possible solution is to reduce the corrugation of the grating structures, e.g. SWGA with a 10 nm feature size can provide 1 mm propagation length [9]. However, it is quite challenging to realize such a small feature size considering the current state of fabrication technology. An alternative is to combine the high-index Si waveguide with the low-index Si-nitride overlay [8], [10], [11]. However, such Si/Si-nitride hybrid platform requires additional deposition and etching steps, leading to the complex fabrication process. Furthermore, for SWGA-based systems, the beam steering process is commonly required to scan the surrounding environment and locate the receivers [3], [12]. The SWGA diffraction angle can be steered by tuning the working wavelength. However, the diffraction strength is wavelength sensitive for conventional SWGAs, so the output power and far-field divergence will deviate in the beam steering process, which makes the scanning operation unstable.

The photonic bound state in the continuum (BIC) is a lossless nonradiating state localized within the continuum spectrum of the radiation mode [13], [14]. In practice, the ideal BIC will transform into quasi-BIC due to material absorption, surface roughness, fabrication imperfection, substrate leakage, or other perturbations [15]. Quasi-BICs can be realized on various platforms, e.g. photonic crystals [16], asymmetric metasurfaces [17], dual-grating membranes [18], fiber Bragg gratings [19], and low-contrast waveguides [20], [21]. In this paper, the novel concept of BIC is introduced to engineer the diffraction in SWGA. A new degree of freedom is attained in diffraction engineering by introducing the “modified” diffraction formula. The proposed structure is an Si waveguide with periodic nanogrooves on the side walls. The diffraction strength can be dramatically depressed even with a large feature size by tailoring the intrawaveguide interference. Such interference effect also induces a flattened dispersion curve for the diffraction strength. Moreover, the proposed BIC-based SWGA is designed on the all-Si platform without heteromaterial integrations, leading to an easy fabrication process. One might find that some similar SWGAs with side-wall corrugations have been demonstrated in LiDAR systems [1], [9]. However, these reported structures still suffer from the significant diffraction strength and short propagation length, as they are all designed based on the conventional diffraction formula and the BIC nature has not been revealed. Our proposed design is the first one that introduces the BIC effect into SWGA optimization, paving the way for precise diffraction engineering and high-performance integrated optical antennas.

2 Design and principle

Figure 1A and B shows the schematics for the proposed SWGA, which is an Si waveguide with periodic nanogrooves on the side walls. The device is designed based on the SOI platform with an Si core layer (n_Si=3.46, h_Si=250 nm) and an SiO₂ buffer layer (n_SiO2=1.45, h_SiO2= 3 μm). The cladding is chosen to be a PMMA polymer layer (n_PMMA=1.50, h_PMMA=500 nm) [22], [23], [24], [25], [26]. In this paper, we only consider the transverse-electric (TE) polarization. For such a structure, the bound state is the fundamental TE mode (TE₀) that propagates along the Si waveguide (toward the +x-direction), whereas the continuum is the radiation mode that propagates in free space. In SWGA, the incident TE₀ mode can be coupled into the radiation mode [E_L(−) and E_R(+)] by nanogrooves. It should be noted that nanogrooves also excite the scattering modes [E_L(+) and E_R(−)] confined within the Si core region (see also Supplementary Material, Section 1). Hence, the side-wall emission can be dramatically depressed by building a destructive intrawaveguide interference between E_L(±) and E_R(±), leading to diffraction-limited quasi-BIC [20], as shown in Figure 1C. It should be noted that such phenomenon is induced by the “accidental” BIC effect [14], which emerges by introducing the interference between different diffraction channels while preserving the structural symmetries. A “modified” diffraction formula is obtained to describe the BIC effect by inserting an interference term into Eq. (1) [20], [21]:

Figure 1:

Working principle for BIC-based SWGA.

(A) 3D view of SWGA. (B) Enlarged top view of SWGA with some key parameters labeled. (C) Schematics for the BIC effect in SWGA. The bound state is the TE₀ mode that propagates along the Si waveguide (toward the +x-direction), whereas the continuum is the radiation mode that propagates in free space. In SWGA, the nanogrooves diffract the confined TE₀ mode into the radiation mode [E_L(−) and E_R(+)]. One should note that nanogrooves also excite the scattering modes [E_L(+) and E_R(−)] confined within the Si core region, so the side-wall emission in SWGA can be dramatically depressed by building a destructive intrawaveguide interference between E_L(±) and E_R(±). Thus, only a small part of light can emit from the top and bottom surfaces (E_T and E_B), leading to limited diffraction with an ultrasmall far-field divergence.

where κ is the “modified” coupling coefficient, n_s is the effective index of E_L(+) and E_R(−), w_wg is the waveguide width, r_grt is the nanogroove radius, λ₀ is the working wavelength, and φ₀ is the intrinsic phase difference between E_L(±) and E_R(±). For the conventional diffraction formula [see Eq. (1)], the diffraction strength is solely determined by the index perturbation. For the “modified” diffraction formula [see Eq. (2)], a new degree of freedom is introduced to engineer the diffraction by exploiting the intrawaveguide interference. From Eq. (2), the diffraction-limited quasi-BICs can be built at some critical waveguide widths w_c:

Such interference effect has also been observed in other BIC-based platforms such as dual-grating membranes [18] and low-contrast waveguides [20], [21]. However, a small part of light can still emit from the top and bottom surfaces (E_T and E_B) even at critical waveguide width w_wg=w_c, as the top/bottom emission is not forbidden (see Figure 1C). Thus, the rigorous diffraction formula can be written as

where Γ is a parameter quantifying the incomplete diffraction elimination. The intrinsic phase difference is φ=π for SWGA with a symmetric geometry [18], [21]. As an initial setting, the nanogroove radius is chosen to be r_grt=60 nm. The working wavelength is set to be λ₀=1.55 μm. It should be noted that E_L(+) and E_R(−) can be roughly approximated as the slab modes with confinement only in the z-dimension (see also Supplementary Material, Section 1). The effective index is then calculated to be n_s=2.95 for E_L(+) and E_R(−) using the finite element method. According to Eq. (3), the critical waveguide width is calculated to be w_c,m=0.64 + 0.52m μm for the mth-order quasi-BIC.

The diffraction angle θ_d for SWGA can be determined by the following equation:

where n_eff is the effective index of the TE₀ mode, λ₀ is the working wavelength, and Λ_grt is the grating pitch. The grating pitch is then chosen to be Λ_grt=700 nm with the corresponding θ_d≈25° at the 1.55 μm wavelength. To verify the BIC effect, we then calculate the diffraction strength for SWGA with varied w_wg, as shown in Figure 2A. Here, the diffraction strength α is defined as

where S is the radiation power and l_grt is the propagation length. Here, the SiO₂ buffer layer thickness h_SiO2 is assumed to be infinite. The numerical calculation is carried out using the 3D finite-difference time-domain method. More simulation details can be found in Supplementary Material, Section 2. According to the conventional diffraction formula [see Eq. (1)], α should monotonously decrease with the increasing w_wg, as the index perturbation is weaker with a larger w_wg. However, it can be seen in Figure 2A that α periodically reaches a minimum value, which indicates the diffraction depression induced by the BIC effect. Similar phenomena have also been observed in other BIC-based photonic systems [18], [20], [21]. Critical waveguide widths are calculated to be w_c,1=600 nm, w_c,2=1.14 μm, and w_c,3=1.68 μm for the first three quasi-BICs (qBIC₁, qBIC₂, and qBIC₃), which show great agreement with the analytical model [see Eq. (3)]. Here, qBIC_m denotes the mth-order quasi-BIC. Figure 2B shows the calculated α using a logarithmic scale. It can be observed in Figure 2B that the diffraction is not fully eliminated for quasi-BICs, which is mainly due to the top/bottom emission. The diffraction strength is calculated to be α₁=2.0×10⁻³ dB/μm, α₂=6.0×10⁻⁴ dB/μm, and α₃=1.7×10⁻⁴ dB/μm for qBIC₁, qBIC₂, and qBIC₃, respectively. One can find that α can be reduced further for higher-order quasi-BICs. However, the relatively small w_wg is usually required to obtain the relatively high group index n_g and ensure an efficient beam steering (see Supplementary Material, Section 3). Thus, the waveguide width is chosen to be w_wg=600 nm to ensure a weak diffraction as well as a high beam steering efficiency.

Figure 2:

Diffraction properties for BIC-based SWGA.

Calculated (A) linear-scale and (B) logarithmic-scale diffraction strength α with varied waveguide width w_wg at the 1.55 μm wavelength. The diffraction strength periodically reaches a minimum value at some critical waveguide widths w_c, which is mainly due to the BIC effect. The first critical waveguide width is calculated to be w_c,1=600 nm. (C) Calculated α spectra with varied w_wg. The dot markers denote the zero-dispersion wavelengths. For the BIC-based SWGA with w_wg=w_c,1, the interference-induced zero dispersion can be obtained at the central wavelength, leading to a flattened dispersion curve. (D) Calculated α spectra with varied nanogroove radius r_grt.

We then calculate the α dispersion curves for BIC-based SWGAs, as shown in Figure 2C. For SWGA with w_wg=600 nm=w_c,1, the zero dispersion (∂α/∂λ=0) can be obtained at the central wavelength, leading to a flattened dispersion curve with Δα<1.0×10⁻³ dB/μm over the 100 nm wavelength range from 1.5 to 1.6 μm. We also calculate the α dispersion curves for SWGAs with w_wg=580 nm<w_c,1 and w_wg=620 nm>w_c,1, as shown in Figure 2C. One can find that the dispersion flattening can still be achieved, but the zero-dispersion wavelength is slightly shifted. Such dispersion flattening is mainly induced by the interference effect. The destructive interference condition can be perfectly satisfied only at the central wavelength, so the wavelength deviation will introduce the phase mismatch between E_L(±) and E_R(±), leading to the negative α dispersion (∂α/∂λ<0) when λ<λ₀ and the positive α dispersion (∂α/∂λ>0) when λ>λ₀. As a comparison, we also calculate the α dispersion curve for the conventional SWGA with shallowly etched nanoholes on the top surface (see Supplementary Material, Section 4). The diffraction strength is calculated to be α=7.8×10⁻² dB/μm for conventional SWGAs, which is much higher than the calculation result for the BIC-based SWGA. One can also find that the α dispersion is always positive for conventional SWGAs, which is mainly due to the weaker light confinement and the stronger disturbance at the longer wavelength. We then calculate the α dispersion curves with varied r_grt, as shown in Figure 2D. It can be found that α can be further reduced by choosing an even smaller r_grt. The diffraction strength is calculated to be α=4.3×10⁻⁴ dB/μm for the BIC-based SWGA with r_grt=40 nm. Moreover, dispersion flattening can still be maintained with varied r_grt. However, the smaller feature size will give rise to fabrication difficulties. Thus, the nanogroove radius is finally chosen to be r_grt=60 nm to obtain a weak diffraction as well as an easy fabrication process.

The optimized parameters are summarized as follows: w_wg=600 nm, Λ_grt=700 nm, r_grt=60 nm, and h_Si=250 nm. The light propagation profiles are then calculated for the optimized SWGA, as shown in Figure 3A and B. It can be observed that the incident TE₀ mode can be coupled into the radiation mode by nanogrooves (see Figure 3A). Moreover, for the BIC-based SWGA, the side-wall emission is depressed, whereas the top/bottom emission is dominant (see Figure 3B). As a comparison, we also calculate the light propagation profiles with w_wg=800 nm>w_c,1, as shown in Figure 3C and D. One can observe that the light emits from both side walls and top/bottom surfaces, leading to a much stronger diffraction. We also calculate the element factor for the BIC-based SWGA [27], as shown in Supplementary Material, Section 5. It can be found that the far-field pattern is azimuthally uniform for SWGA with w_wg=600 nm=w_c,1 at different wavelengths. For the above analysis, the SiO₂ buffer layer thickness h_SiO2 is assumed to be infinite. However, there is usually an Si substrate buried under the SiO₂ buffer layer [28], which will reflect a small part of the radiation mode and slightly affect the diffraction (see Supplementary Material, Section 6). The diffraction strength is calculated to be α=2.1×10⁻³ dB/μm for the optimized SWGA on the commercial SOI platform with h_SiO2=3 μm, which is quite close to the calculation result for SWGA with infinite h_SiO2, indicating a weak bottom reflection.

Figure 3:

Calculated light propagation profiles (|E_y| field) for BIC-based SWGAs with (A and B) w_wg=600 nm=w_c,1 and (C and D) w_wg=800 nm>w_c,1.

The logarithmic scale is used to enhance the contrast. For SWGA with w_wg=w_c,1, the side-wall emission is depressed, whereas the top/bottom emission is dominant. For SWGA with w_wg>w_c,1, the light emits from both side walls and top/bottom surfaces.

3 Fabrication and characterization

To verify the above analysis, we fabricated the BIC-based SWGAs on the 250 nm SOI platform (SOITEC, Inc.). The SOI chip was first spin-coated with the positive-tone electron-beam resist (PMMA) and prebaked at 180°C for 10 min. An electron beam lithography (Raith 150-II) process with the exposure dose of 300 μC/cm² was performed to define the pattern. The Si core layer was then fully etched by implementing an inductively coupled plasma reactive ion etching process with a gas mixture of C₄F₈ and SF₆. Another overlay exposure with PMMA resist followed by a 50 nm shallow etching was performed to fabricate the TE-type grating couplers for the fiber-chip coupling and polarization selectivity [28]. Finally, the whole chip was spin coated with a 500-nm-thick PMMA layer as the cladding.

To observe the BIC effect, one possible solution is to characterize the far-field pattern. However, for the optimized SWGA, the centimeter-scale propagation length is required to obtain >99% power efficiency (l_{20 dB}=20 dB/α≈1 cm), leading to an extremely small far-field divergence <0.02°. Although the Fourier imaging system with a high resolution <0.01° has been developed in Ref. [3] to characterize the optical antenna, it is still quite challenging to measure the far-field pattern with such a small divergence, as the main lobe can only occupy a few pixels [11]. To directly demonstrate the BIC effect, we measured the diffraction strength for fabricated SWGAs. A series of SWGAs with varied l_grt (from 400 to 2000 μm) were fabricated to evaluate α, as shown in Figure 4A. The Si waveguides without nanogrooves were also fabricated on the same chip for normalization. Figure 4B and C shows the scanning electron microscopy (SEM) image for fabricated SWGA and Si waveguide. The tunable laser (Agilent 81600B) was used to excite the TE₀ mode at the input ports (i₁, i₂, …, i₁₀). The power meter (Aglient 81635) was used to detect the transmission responses at the output ports (o₁, o₂, …, o₁₀). Figure 5A shows the measured excess losses for fabricated SWGAs with w_wg=600 nm and varied l_grt at the 1.55 μm wavelength. Here, the excess loss is defined as

Figure 4:

The fabricated SWGAs.

(A) Optical microscopy image for fabricated Si waveguides and SWGAs with varied propagation length l_grt. SEM images for fabricated (B) Si waveguide and (C) SWGA. (D) Enlarged optical microscopy image for the fabricated shallow-ridge taper.

Figure 5:

Measurement results for fabricated SWGAs.

(A) Measured excess losses (ELs) for fabricated SWGAs with w_wg=600 nm and varied l_grt at the 1.55 μm wavelength. The red dashed line shows the linear fitting. The diffraction strength was measured to be α≈3.3×10⁻³ dB/μm. (B) Measured α for fabricated SWGAs with varied w_wg at the 1.55 μm wavelength. The qBIC₁ can be found at w_wg≈600 nm. (C) Measured α for fabricated SWGAs with w_wg=600 nm at different wavelengths. The error bars refer to the standard errors of the linear fittings. The gray dashed lines show the simulation results. (D) Measured transmittances at the o′i ports for fabricated SWGAs with w_wg=600 nm=w_c,1 and w_wg=800 nm>w_c,1. The gray dashed line shows the detection limit (DL). The red and blue dashed lines show the linear fittings.

where EL is the excess loss, T_SWGA is the measured transmittance for SWGA, and T_wg is the transmittance for the Si waveguide. By linearly fitting the measured excess losses, the diffraction strength was characterized to be α≈3.3×10⁻³ dB/μm, which agrees well with the simulation result. SWGAs with varied w_wg were also fabricated on the same chip to demonstrate the BIC effect. Figure 5B shows the measured α with varied w_wg at the 1.55 μm wavelength. It can be seen that the measured α reaches a minimum value at critical waveguide width w_c≈600 nm, indicating the diffraction-limited qBIC₁. The α dispersion was also measured for fabricated SWGA with w_wg=600 nm to verify the dispersion flattening effect, as shown in Figure 6C. From the measurement results, one can observe a flattened dispersion curve with zero dispersion around the 1.55 μm wavelength. To further demonstrate the BIC effect, we also fabricated some shallow-ridge tapers to characterize the side-wall emission of fabricated SWGAs (see Supplementary Material, Section 7), as shown in Figure 4D. Each shallow-ridge taper can collect the side-wall emission over a 50 μm propagation length. The spacing between the adjacent shallow-ridge taper is 200 μm. The TE₀ mode was launched at the i₀ input port using the tunable laser, whereas the collected side-wall emission was detected at the output ports (o′1, o′2, …, o′10) using the power meter, as shown in Figure 4A. Figure 5D shows the measured transmittances at the o′i ports for fabricated SWGAs with w_wg=600 nm and w_wg=800 nm. For fabricated SWGA with w_wg=600 nm=w_c,1, the side-wall emission is both weak and uniform, indicating the diffraction depression effect induced by the quasi-BIC. In contrast, for fabricated SWGA with w_wg=800 nm>w_c,1, the side-wall emission decays rapidly over a short distance, indicating the significant diffraction strength. To further evaluate the performance of fabricated SWGAs, we also predict the divergence angles δθ_{3 dB} by calculating the far-field intensities according to the measured α, as shown in Figure 6. The calculation details for the near-to-far transform can be found in Ref. [11]. For SWGA with w_wg=600 nm=w_c,1, the divergence angle decreases as the truncated propagation length varies from l_grt=0.5 to 10 mm. An extremely small divergence angle δθ_{3 dB}≈0.027° can be predicted for SWGA with l_grt=10 mm, indicating ultrauniform diffraction. In contrast, for SWGA with w_wg=800 nm>w_c,1, the divergence angle keeps nearly constant as the truncated propagation length varies from l_grt=0.5 to 10 mm, indicating that most of the input power has been diffracted over a short propagation length. Such rapid power decay also leads to a relatively large divergence angle δθ_{3 dB}≈0.25°.

Figure 6:

Predicted far-field intensities for fabricated SWGAs with varied w_wg and l_grt according to the measured α at the 1.55 μm wavelength.

4 Conclusion and discussion

In conclusion, we propose and demonstrate a novel approach to engineer the diffraction in SWGAs by exploiting the BIC effect. The diffraction strength can be tailored using intrawaveguide interference. The side-wall emission can be dramatically depressed by building quasi-BICs at some critical waveguide widths, leading to the weak and uniform diffraction process. The extremely weak diffraction strength (~3.3×10⁻³ dB/μm) has been experimentally demonstrated for the fabricated device with a large feature size (~60 nm). An ultralong propagation length l_grt>1 cm and an ultrasmall far-field divergence ~0.027° can be predicted from the measurement results. Moreover, the diffraction strength dispersion can be flattened for SWGA at critical waveguide width, which is also induced by the BIC effect. Zero dispersion has been experimentally observed for the fabricated BIC-based SGWA. The BIC-based SWGA is designed based on the all-Si platform, so the device can be easily realized using the CMOS-compatible fabrication technology. Moreover, such novel BIC effect can also be used to engineer the diffraction strength for the silicon-nitride antenna at visible wavelengths [29], as shown in Supplementary Material, Section 8. Our proposed design is the first one that introduces the BIC effect into SWGA optimization, paving the way for precise diffraction engineering and high-performance integrated optical antennas.