Observation of Aulter–Townes splitting in subwavelength grating metamaterial ring resonators

Coherent interactions of light and multilevel atoms can produce many interesting and useful phenomena, such as electromagnetically induced transparency (EIT)^1,2 and Autler–Townes splitting (ATS).³ ATS can create a transparency window via strong-coupling-field-induced splitting of energy levels,^3,4 which has been observed and studied in many physics systems, including atomic systems,⁵ molecular systems,⁶ and whispering-gallery-mode micro-resonator (WGMR) systems.⁷ WGMR systems, such as directly coupled silica microspheres⁷ and microtoroids,⁸ have been proven to be a promising platform for classical and all-optical analogs to ATS. In these systems, ATS originates from the intra-resonator mutual coupling of the degenerate clock-wise (CW) and counter-clock-wise (CCW) modes induced by scattering,^9,10 leading to the breaking of degeneracy and resonance splitting. This phenomenon has been observed in ring resonators with rough sidewalls¹¹ and grating structures.^12,13

ATS is of special interest to sensing applications because it is intrinsically immune to noise. Nanophotonic sensors are typically implemented by monitoring the resonance wavelength shift caused by ambient refractive index variation. These highly sensitive sensors are prone to external environment instabilities, such as temperature fluctuation and mechanical disturbance. In ATS, as the splitting modes experience the same noise sources, the system noise can be significantly suppressed with optical differential detection.¹⁴ ATS-based sensors have been proven to be capable of detecting feeble refractive index disturbances, such as single nanoparticles on the order of tens to hundreds of nanometers.^15,16 Therefore, there is substantial interest in implementing ATS in on-chip sensors to reduce the noise level.

In this letter, we experimentally prove that ATS can be generated in subwavelength grating metamaterial waveguide ring resonators (SGMRRs). Conventionally, SGM is treated as a homogeneous anisotropic material without internal reflection, which is often a convenient and valid simplification, especially in non-resonant structures.^17,18 In resonant structures, however, the internal reflection in SGM must be considered because of the enhancement effect of these structures.¹⁹ Figure 1(a) shows the schematic of a typical all-pass SGMRR, which is formed by a subwavelength grating metamaterial ring side-coupled to an SGM bus waveguide. The SGMRR is constructed from a circular array of rectangular silicon pillars. Adiabatic tapers are placed on both ends of the SGM bus waveguide to facilitate the mode transition between the strip and SGM waveguide and reduce the reflection at the interfaces. The SGM waveguide is defined by the period (Λ), duty cycle (DC), length (l), waveguide width (w), and height (h) of silicon pillars. DC is defined as the proportion of the silicon pillar in the period (DC = l/Λ). The radius of SGMRR is 10 μm. The gap (g) between the SGM waveguide and SGMRR is tuned to meet the critical coupling condition. The period Λ is selected to be 300 nm, which is adequately small to suppress the diffraction (Λ < λ₀/2n_eff). The strength of the split, determined by the internal reflection and Q factor,  can be controlled by adjusting the structural parameters of the SGM waveguide.

The field distribution of the transverse-electric (TE) mode of the SGM waveguide is simulated via Finite-Division Time-Domain (FDTD) simulation. As shown in the insets of Fig. 1(b), the difference between the electric field distributions in and between silicon segments is evident, which triggers weak reflection. According to Ref. 20, when the bending radius of an SGM is larger than 5 μm, the bending-induced mode delocalization leads to a <1% mode mismatch with a straight SGM waveguide. Thus, the internal reflection in the SGMRR can be approximated by a straight SGM waveguide with the same number of periods. The reflection characteristic of the SGM waveguide can be modeled by a multilayer stack composed of silicon (segment 1) and cladding (segment 2), as illustrated by the inset at the bottom of Fig. 1(b). The cladding is set to the refractive index of deionized (DI) water as it was used in the experimental demonstration. Since the mode is less confined to the core region, the asymmetricity of the field distribution along the z axis is less significant compared to a conventional strip waveguide. The optical characteristics of the SGM waveguide can be described with the transfer matrix method. The refractive indices of segments are approximated by the effective refractive indices²¹ estimated by electric field weighted average refractive index.

When the wave travels in segment m (m = 1, 2), the phase changes by φ_m = (n_eff,mk − iα/2)d_m. Here, k is the wave number in a vacuum. n_eff,m and d_m represent the effective refractive index and length of the medium, respectively. α is the propagation loss of each segment. The transfer matrix is

The interface matrix S describes transmittance and reflectance at a boundary, where r and t are the reflection and transmission coefficients based on the Fresnel theory. S can be written as²¹ 

The transfer matrix M₀ of one section of the SGM waveguide is modeled by^22,23

Here, S represents the transfer matrix at the interface between two segments. The subscripts of S describe the propagation direction. The transfer matrix of N sections of the SGM waveguide can be expressed as $M=M0N$⁠. Then, the reflection R of the SGM waveguide can be readily calculated by $R=M21/M222$⁠. Figure 1(c) shows the reflection spectra of the SGM waveguide with different DCs ranging from 0.5 to 0.8, which show a periodic fluctuation with the highest reflection as large as 0.126. The upper limit of the DC range is chosen to ensure the period of the subwavelength structure satisfies the subwavelength condition. When DC is larger than 0.8, the operating wavelength is close to the Bragg wavelength, resulting in prominent reflections. Thus, in this region, the periodic structure can no longer be considered a subwavelength structure. The spectra in Fig. 1(c) indicate that an SGM waveguide with a larger DC can generate a higher reflection. When the DC of an SGM increases, the optical field becomes more confined inside the SGM waveguide, leading to an increase in the n_eff of the SGM waveguide, as shown in the inset of Fig. 1(c).¹⁷ As a result, the n_eff,m and, consequently, the reflection coefficients r in matrix S increase. This property can be leveraged to tune the reflection strength. Similarly, the optical field confinement is also wavelength-dependent due to the waveguide dispersion. The mode field extends more into the cladding and substrate at longer wavelengths, resulting in a decrease in reflection.

The transmission characteristics of the SGMRR can be analyzed by slightly modifying the simplified model in Ref. 24, in which the reflection of the SGMRR is equivalent to a reflector in a microring resonator.²⁴ However, the point-coupling approximation, which is widely adopted in conventional ring resonator analysis,²⁵ is no longer valid due to the large mode field of the SGM waveguide. Thus, the phase shift φ_c induced by the coupling region can no longer be ignored in the model²⁶ as a modified transfer matrix M = φ_cMφ_c. As will be shown later, the existing of φ_c is deterministic to the features of the transmission spectrum of the SGMRR. The spectra transmission T of the proposed device can be obtained by Ref. 22,

Here, κ is the coupling coefficient between the SGMRR and the SGM waveguide.

To unveil the characteristics of the ATS in SGMRR, the transmission spectrum of an SGMRR with DC = 0.7 is analyzed with the theoretical model introduced above. The corresponding reflection spectrum of an SGM with the same number of periods as the SGMRR is also simulated. Both spectra are plotted in Fig. 2(a). Since the Δλ_s = λ₊ − λ₋ of the splitting resonant modes depends on the strength of the internal reflection within the SGMRR, the periodic reflection characteristic of the SGM makes Δλ_s wavelength dependent. The resonance near 1.555 μm does not split because of the near-zero reflection at the wavelength. The Δλ_s are 211, 114, and 277 pm for resonances A, B, and C, respectively, as shown in the insets of Fig. 2(a). Due to the reflection-induced wavelength-dependent loss, the split modes have different Q factors.¹⁴ To confirm the occurrence of the ATS, the field profiles (Hz) of λ₋ and λ₊ are simulated by FDTD and plotted in Figs. 2(b) and 2(c), respectively. The redistribution of the optical field is eminent, which leads to the deviation of the effective indices of the splitting modes based on mutual-coupled CW and CCW modes.¹⁴ It is worth noting that φ_c introduces a phase shift between the optical mode and periodical modulation, leading to the wavelength-dependent characteristic of Δλ_s.²⁷ Without φ_c, only resonances in phase with the periodical modulation will split.

To verify the prediction of the model, SGMRRs with different DCs (DC = 0.7, 0.8) and gaps (g = 0.4, 0.6 μm) are fabricated and measured. The devices are fabricated on SOI with a 220 nm thick single-crystal silicon device layer and a 2 μm thick buried oxide (BOX) layer. Diluted ZEP520A resist is used for E-beam lithography (NanoBeam nB5) to assure the successful fabrication of the nanostructures. Then the designs are transferred into the single-crystal device layer by inductively coupled plasma (ICP) etching. The designs are pre-compensated to minimize proximity effect-induced fabrication errors. Figure 3 is the scanning electron microscopy (SEM) image of a 10 μm radius SGMRR with DC = 0.7 and g = 0.6 μm. Due to the nature of the lithography and etching processes, the fabricated device always deviates from the design.

The transmission spectra of the fabricated devices are measured by exciting the SGMRR with an amplified spontaneous emission (ASE) source and measuring the output with an optical spectrum analyzer (OSA). Subwavelength grating couplers are leveraged to couple light in and out.²⁸ The peak coupling efficiency of the subwavelength grating coupler is −6.24 dB at 1568 nm, with a 3 dB bandwidth of 45 nm. The devices are immersed in DI water during the testing. Figure 4(a) depicts the measured spectrum of the SGMRR at DC = 0.7 and g = 0.6 μm, and the corresponding theoretical reflection. Δλ_s of the split resonances is plotted in the same figure as pentagrams. It can be clearly observed that Δλ_s is wavelength dependent. Figure 4(b) shows the details of all resonance peaks in Fig. 4(a). Four splitting resonances are observed. The maximum splitting of 0.798 nm occurs at resonance D. Δλ_s of splitting resonances varies periodically with wavelength, which closely matches the theoretical reflection curve. Resonances C and F are at wavelengths with near-zero reflection. Thus, the two resonances do not split. Resonance A shows a mode broadening phenomenon, which is because the linewidth is not narrow enough (low Q) to resolve a doublet in the splitting spectrum.

To further investigate the effects of structural parameters on the ATS phenomenon, SGMRRs with different gaps between the bus waveguide and the ring and different DCs are fabricated and measured. Varying the gap not only alters the coupling strength (κ) and consequently the Q factor, but also alters the additional phase φ_c introduced in the coupling region. Two representative resonances with non-zero reflection [Fig. 5(a)] and near-zero reflection [Fig. 5(b)] are chosen and plotted as the gap changes. When the gap changes from 0.6 to 0.4 μm, the Δλ_s of the resonance at 1.538 μm decreases from 0.583 to 0.216 nm, as shown in Fig. 5(a). Since the internal reflection strength is independent of the coupling strength between the resonator and the bus waveguide, the change in Δλ_s is most likely caused by φ_c. The resonance in Fig. 5(b) remains non-split when the gap changes. In the meantime, the Q factor of the resonance in Fig. 5(b) increases from 4541 to 7237 when the gap increases from 0.4 to 0.6 μm. It is due to the near-zero reflection that no matter how Q increases, the splitting does not exist. Since the critical coupling condition is broken by varying the gap, the extinction ratio (ER) of the resonance is compromised. Therefore, a trade-off has to be made between Δλ_s and ER.

The transmission, reflection spectra, and the Δλ_s of the resonances in SGMRR with DC = 0.8, and g = 0.4 μm are shown in Fig. 5(c). Figure 5(d) shows the resonance peaks in Fig. 5(c). The SGMRR with DC = 0.8 shows a completely different transmission spectrum compared to the SGMRR with DC = 0.7. The structural parameters of the SGM waveguide are essential to its internal reflection characteristics, thus leading to different Δλ_s. Four splitting resonances are observed, with the maximum splitting of 0.318 nm at resonance C. Three resonances do not split, and the corresponding reflection of resonances E and G are close to 0, except for resonance B. The discrepancy is potentially caused by a manufacturing error. Due to the waveguide dispersion, the reflection variation is slower at shorter wavelengths. As a result, the fabrication errors induced reflection variation is wavelength dependent. It is possible that the discrepancy at a certain wavelength is so large that the resonance does not split at all while the reflection spectrum shows a peak at the wavelength. Resonance E has the largest Q factor (∼12 253). For other resonances, the experimental results show a good correlation between Δλ_s and the reflection. The fabrication errors could change the shape of the silicon pillars and the gap between the bus waveguide and the ring resonator. As a result, the reflection strength, the intrinsic loss, and the coupling coefficient would be affected at the resonance of interest, leading to variation in the transmission spectrum. Although the variation of transmission spectra can be observed even for devices with identical designs, the ATS phenomenon always exists in devices with high Q, as predicted by the theoretical model.

In conclusion, this paper theoretically predicts and experimentally demonstrates ATS in SGMRRs. In previous studies, the SGM waveguide has been considered a homogeneous structure without reflection. Here, we show that the reflection in SGM can enable the mutual coupling between CW and CCW modes, leading to ATS. The strength of the split can be adjusted by the structural parameter of the SGM and the Q factor of the cavity. Leveraging the ATS phenomenon, SGMRRs can potentially be used in ultra-sensitive biological particle sensing as the split modes based differential detection method is naturally noise immune. Those characteristics make the ATS of SGMRRs a promising approach for sensing applications.

This research is supported by the Science, Technology, and Innovation Commission of Shenzhen Municipality (Grant No. JCYJ20210324131614040); Basic and Applied Basic Research Foundation of Guangdong Province (Grant No. 2020B1515130006).