To obtain an understanding of phase modulation in heterogeneous integrated systems, it is important to understand the interaction between TMDs and an MRR, as the spectral distance of each TMD’s exciton relative to the waveguide probing wavelength (here, *λ*_{op}=1550 nm) is noncoincidental; thus, the real versus imaginary part index decay away from the exciton resonance has a different impact on the telecom-operating photonic structures. Unlike other larger-bandgap TMDs (for example, MoS_{2}, WS_{2}, MoSe_{2}, and WSe_{2}), the optical bandgap of a few-layered MoTe_{2} (~0.9 eV) is smaller than the bandgap of Si (~1.1 eV) and allows a facile integration with an Si photonic platform. Here, the aim is to improve the waveguide bus-to-ring coupling efficiency by shifting the phase through the introduction of a few-layered MoTe_{2} flakes on top of the resonator (Figure 1A–C). By varying the MoTe_{2} coverage length, we observe a tunable coupling response. To achieve TMD loading of the rings, we use a precise and cross-contamination-free transfer using the 2D printer method recently developed [23]. The transmission spectra before and after MoTe_{2} transfer reveal a definitive improvement of coupling in terms of visibility defined as the transmission amplitude ratio (*T*_{max}/*T*_{min}) upon TMD loading (Figure 1D). This hybrid device shifts toward the critical coupling regime compared to before transfer where it was overcoupled.

Figure 1: TMD-loaded MRR.

(A) Schematic representation and (B) optical microscopy image of a taped-out Si photonic chip with grating coupler. Optical micrograph of a grating coupler designed for TM mode at 1550 nm in the field of a view of 100× objective (inset). (C) Zoomed image of an MRR (*R*=40 μm and *W*=500 nm) covered by an MoTe_{2} flake of length (*l*) precisely transferred using our developed 2D printer technique. (D) Transmission output before and after the transfer of MoTe_{2} showing an improvement of coupling efficiency as it brings the device close to the critical coupled regime after the transfer of the TMD layer.

We fabricated a set of ring resonators with different percentages of MoTe_{2} coverage between 0% and 30% (Figure 2), which allowed us to extract different parameters to understand coupling physics. The cavity quality (*Q*) factor is found to decrease from 1600 to 900 as the ring coverage is raised from 0 to 27% (Figure 2A). We attribute this as a gradual increase of loss arising from both MoTe_{2} absorption near its band edge corresponding to indirect bandgap of 0.88 eV [24] and the small impedance mismatch between bare and TMD-covered sections of the rings. In contrast to the monotonic behavior of the quality factor, the minimum transmission (*T*_{min}) at resonant wavelength initially decreases until 10% coverage is reached and then increases for higher coverage (Figure 2B). The visibility (*T*_{max}/*T*_{min}) shows the opposite trend being maximal near 10% of TMD coverage (Figure 2C). In combination, these findings indicate the tunability of the coupling condition by means of varying TMD coverage.

Figure 2: TMD-loaded tunable coupling effect.

Variation of (A) *Q* factor, (B) minimum transmission (*T*_{min}), (C) visibility (*T*_{max}/*T*_{min}), (D) difference between the self-coupling coefficient and the roundtrip transmission coefficient |*a* – *r*| as a function of MoTe_{2} coverage. The monotonic decrease of * Q* factor suggests increasing loss for higher coverage, i.e. more transferred TMD. Tunability of coupling effect, i.e. transition from the overcoupled to the undercoupled regime via critical coupled condition (dashed vertical line), is evident from the variation of *T*_{min}, *T*_{max}/*T*_{min}, and |*a* – *r*|. (E) Corresponding figures for different coverage (bare ring, 6% and 22%) showing the advantages of precise transfer by 2D transfer methods.

The performance of a ring resonator is determined by two coefficients: the self-coupling coefficient (*r*), which specifies the fraction of the light transmitted on each pass through the coupler, and the roundtrip transmission coefficient (*a*), which specifies the fraction of the light transmitted per pass around the ring. For the critical coupling condition, i.e. when the coupled power is equal to the power loss in the ring 1 – *a*^{2}=*k*^{2} or *a*=*r* (*k*=cross-coupling coefficient), the transmission at resonance becomes zero. At this point, the difference (|*a* – *r*|) is found to be minimum at ~10% of the coverage (Figure 2D), suggesting close to critical coverage as |*a* – *r*| is inversely proportional to the square root of the visibility term [25]. Thus, the coupling condition is tunable from the overcoupled regime (*r*<*a*) to the undercoupled regime (*r*>*a*) via the critical coupled regime (*r*=*a*) as a function of TMD coverage. Being able to operate at critical coupling is important for active device functionality; for instance, the extinction ratio of an MRR-based electro-optic modulator is maximized at critical coupling [26], [27].

To understand the coupling mechanism of a ring resonator, it is important to extract and distinguish coupling coefficients (*a* and *r*), as they are governed by different factors in design and fabrication. However, it is not possible to decouple both the coefficients without additional information, as *a* and *r* can be interchanged [Eq. (1)]. The transmission from an all-pass MRR (Figure 3A) is given by

Figure 3: Coupling coefficients for TMD integrated hybrid Si MRR.

(A) Schematic representation of MRR showing self-coupling coefficient (*r*), cross-coupling coefficient (*k*), and roundtrip transmission coefficient (*a*), where the ring is partially covered by a MoTe_{2} flake. (B) Propagation loss (*α*_{TMD-Si}) for TMD-covered portion of the ring found to be 0.4 dB/μm using cutback measurement. (C) Tunability of roundtrip transmission coefficient explains the exponential decrease as a function of flake coverage. (D) Relationship between critical coverage (%) and power coupling coefficient (*k*^{2}) assuming lossless coupling (*r*^{2}+*k*^{2}=1).

$${T}_{n}=\frac{{a}^{2}+{r}^{2}-2ar\hspace{0.17em}\text{cos}\phi}{1+{r}^{2}{a}^{2}-2ar\hspace{0.17em}\text{cos}\phi}$$(1)

where *φ* is the roundtrip phase shift and *a* is the roundtrip transmission coefficient related to the power attenuation coefficients by

$${a}^{2}=\mathrm{exp}\mathrm{(}-{\alpha}_{\text{Si}}\mathrm{(}2\pi R-l\mathrm{)}\mathrm{)}*\text{exp(}-{\alpha}_{\text{TMD-Si}}*l\text{)}$$(2)

where *l* is the TMD coverage length and *α*_{Si} and *α*_{TMD-Si} are the linear propagation losses for Si waveguide and TMD-transferred portion of the Si waveguide in the ring, respectively. We find the propagation loss for Si and TMD-Si to be 9.7 dB/cm (Figure SI3) and 0.4 dB/μm (Figure 3B), respectively, via the cutback method at 1550 nm.

Inserting these values into Eq. (2), we find the roundtrip transmission coefficient (*a*) to be tuned as a function of TMD coverage (Figure 3C). The variation of *a* from 0.97 to 0.01 as a function of coverage confirms the transition from the overcoupled to the undercoupled regime as *a*=1 suggests that there is no loss in the ring. Hence, the loss tunability in MRRs can be manipulated accordingly by controlling the coverage length [28]. The MRR transmission at resonance leads to the form: ${T}_{\text{n,res}}={\mathrm{(}\frac{a-r}{1-ar}\mathrm{)}}^{2};$ therefore, the critical coverage (*a*=*r*) anticipates zero-output transmittance. Hence, for a given MRR (fixed *k*^{2}), the critical coverage value could be determined assuming lossless coupling (*r*^{2}+*k*^{2}=1; Figure 3D).

Si-based MRRs provide a compact and ultra-sensitive platform as refractive index sensor to find an unknown index of the 2D materials at telecom wavelength (1.55 μm). In essence, the shift in resonant wavelength can be used to sense the optical properties affecting entities on the Si core or the cladding [29]. We observe a gradual resonance red shift from bare to increased coverage of MoTe_{2} of 4.5%, 10%, and 17%, respectively (Figure 4A). The change in resonant wavelength (*λ*_{res}) of an MRR independent on group index, which takes into account the dispersion of the waveguide, can be defined as ${n}_{g}={n}_{\text{eff}}-{\lambda}_{0}\frac{d{n}_{\text{eff}}}{d\lambda}$ for our case, as the resonance shift occurs over a narrow wavelength range, where *dn*_{eff}/*dλ*<<1, considering first-order dispersion term, we approximate *n*_{g} as *n*_{eff} [30], [31]. Therefore, the change in effective mode index (Δ*n*_{eff}) is related to change in resonance (Δ*λ*) following $\Delta {n}_{\text{eff}}=\frac{\Delta \lambda}{{\lambda}_{\text{res}}}*{n}_{\text{eff,control}},$ where *n*_{eff,control} is the effective mode index for Si MRR (i.e. without any TMD flakes transferred). The effective index for the control sample (Si ring+SiO_{2} cladding) can be found from FEM eigenmode analysis choosing the TM-like mode in correspondence with our TM grating designs used in measurements (Figure SI4). We map out the resonance shift (Δ*λ*) as a function of MoTe_{2} coverage (Figure 4B, i) for needed calibration to determine the unknown index using a semiempirical approach. The positive change in the effective index (Δ*n*_{eff}) as a function of MoTe_{2} coverage relates to an increased effective mode index with TMD transfer (Figure 4B, ii) and the corresponding red shifts thereof.

Figure 4: Ring resonator as refractive index sensor.

(A) Normalized transmission spectra for different coverage showing gradual red shift. (B) Variation of resonance shift (Δ*λ*) and effective index change (Δ*n*_{eff}) extracted from (A) as a function of MoTe_{2} coverage length. The mapped-out resonance shift as a function of coverage length provides the calibration curve to determine the unknown index of the TMDs. (C) Mode profile (|*E*|) for the portion of the ring with MoTe_{2} transferred flakes from eigenmode analysis. (D) FEM results for effective index *n*_{eff} versus the MoTe_{2} index *n*_{TMD} to extract the material index from experimental results. The green dashed line exhibits the obtained MoTe_{2} index from our results as 4.36.

Now, we obtain a resonance shift of 1 nm for 4% coverage, which gives a corresponding change in effective index of 0.001 (Figure 4B). At this point, it is important to establish the relation between the effective mode index and the refractive index of the unknown TMDs. As, in our case, the MRR is partially covered by MoTe_{2} flakes and the resonance shift arises from a change in effective mode index due to the change in coverage length, the effective refractive index of the ring can be formulated as an effective length-fraction index via

$${n}_{\text{eff},\text{ring}}=\frac{\mathrm{(}2\pi R-l\mathrm{)}*{n}_{\text{eff,control}}+l*{n}_{\text{eff}}}{2\pi R}$$(3)

where *R* is the radius of the ring and *l* is the MoTe_{2} coverage length. Using Eq. (3), it is straightforward to find *n*_{eff} after TMD transfer.

Once *n*_{eff} is known, we can find the effective index of the heterogeneous optical mode [Si-on-insulator (SOI) plus TMD] through eigenmode analysis for device geometry and provide experimentally measured thickness values of the flake (Figure SI5). By sweeping the value of the TMD (*n*_{TMD} range=[3.5, 5]) material index in cross-sectional eigenmode analysis of the waveguide structure in the MoTe_{2} transferred section of the ring (Figure 4C), we can match the numerically obtained effective index with that found from aforementioned experimental results. We find the index of the bulk MoTe_{2} material to be 4.36 (Figure 4D). This index value is closely aligned with reported values in previous studies for bulk MoTe_{2} [32]. We also find the imaginary part of the material index from the cutback method varying flake sizes to be ~0.011. This semiempirical approach has several advantages over conventional ellipsometric technique to determine the refractive index of the unknown materials; for instance, in ellipsometry, large uniform flakes (few millimeters to centimeters) are needed as the beam spot size is generally large, which are still challenging to obtain with uniform properties across such scales. However, for the method presented here, small TMD flakes (~500 nm) can be measured. In fact, the limit of this technique is not bounded by the MRR waveguide width, as a partial coverage of the top waveguide is also possible to measure but with linear phase shift scaling with respect to the area covered. Besides this, to determine the refractive index of the materials from the ellipsometry data, the experimental data need to be fitted by an exact physical model that bears ambiguity and thus introduces additional inaccuracy.

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