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Abstract
A silicon photonics platform with a reduced silicon layer thickness, which is suitable for hybrid thin-film lithium niobate traveling-wave electro-optic modulators, is used to design low loss waveguides, precise directional couplers, high-quality-factor silicon microring resonators and broad-top coupled microring filters. These designs are verified with experimental measurements and show a way to include such components without requiring a second layer of crystalline silicon of different thickness for this purpose.
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1. Introduction
Recent advancements in thin-film lithium niobate (TFLN) photonics have improved the state-of-the-art in high-bandwidth and low-voltage integrated electro-optic modulator (EOM) devices [1–8]. Currently, there is interest in the combination of TFLN EOM’s and other types of photonic components which need not, or cannot, be fully realized using TFLN alone. These useful components can perhaps be realized and iteratively improved at a lower cost and more easily using conventional integrated photonics than in novel thin-film electro-optic materials. In particular, silicon (Si) photonics already has a library of integrated device components which can be fabricated in large volumes using cost-effective wafer-scale processing without having to develop a custom procedure. Our approach adds TFLN EOM functionality selectively where needed as part of a larger lightwave circuit [9–11]. Among the passive components that can be added are directional couplers, microring resonators and coupled resonator optical waveguides (CROWs) with either low-quality-factor (Q) or high-Q characteristics. When integrated with TFLN EOM technology in a way that minimizes impairments, this combination of passives in crystalline Si photonics and the EOM devices using TFLN can provide a low-cost, yet high-performance integrated photonics platform.
Our TFLN EOM devices are based on hybrid optical modes, in which light is partially contained in the TFLN layer and partially in an underlying rib waveguide formed in Si [9–11]. By controlling the dimensions of the Si waveguide over a certain range, the mode fraction contained in TFLN can be varied over a wide range without having to etch or pattern TFLN. A fabrication process has been described elsewhere for a back-end integration process based on low-temperature bonding of TFLN dies [12]. An example is shown in Fig. 1, for a chip labeled “Microchip #2” which consists of TFLN over oxide and a handle material, bonded over a selected portion of a larger Si photonics die, labeled “Microchip #1”. The cross-sectional schematic is shown in Fig. 1(a) and typically uses a Si layer thickness of about 150 nm, which is thinner than the thickness of about 230 nm used in a conventional Si photonics process at 1550 nm [13]. The standard thickness is not suitable for EOM modes as discussed in Ref. [9–11]. With the introduction of a different Si layer thickness, the question arises if the original designs for the passive components should be retained, and a bi-layer process be implemented with additional fabrication complexity. Alternatively, can a single Si layer achieve good component performance with a simple re-design in the thin-Si layer?
Fig. 1. (a) Cross-section of thin-Si platform using an unetched TFLN layer in a portion of the hybrid chip. (b) Photograph of a bonded chip. Microchip #1 is a Si photonic chip on which all the waveguiding features are defined in a layer of crystalline Si that is part of the SOI (silicon on insulator) wafer. Microchip #2 contains the unetched TFLN layer and is bonded upside-down before the handle is removed for electrode fabrication. (c) Optical modes in the three cross-sections labeled “A”, “B”, and “C” in panel (a). These images show the magnitude of the simulated Poynting vector of the TE-polarized fundamental mode.
In this report, we discuss some of the outcomes of such an integration approach and present a strategy for the passive components designed in the thin-Si layer outside the bonded region. This complements our earlier study of passive components formed using hybrid modes within the region contained under the bonded TFLN layer [9]. This work also complements another earlier study in which high-Q silicon microring resonators and low-Q coupled-microring filters were integrated with an all-Si (carrier depletion) EOM [14]. In Section 2, we briefly review the thin-Si platform for hybrid devices in a historical context. In Sections 3-5, we investigate dispersion, directional couplers, microring resonators and coupled-microring resonator filters in this platform. In Section 6, we confirm experimentally that this platform also supports RF-optical index matching out to very high RF frequencies, beyond 100 GHz. Therefore, this platform is, in principle, for multi-functional integrated photonics which combines TFLN-based EOM and crystalline Si-photonics-based optical processing.
2. Thin-Si waveguide platform for hybrid TFLN integration
Over the last decade, many ways of incorporating high-bandwidth TFLN EOMs with Si photonics have been reported. For example, in Ref. [15], partially-etched TFLN was adhered to a silicon microchip using a low-refractive-index epoxy. In Ref. [6], partially-etched TFLN was bonded using benzocyclobutene polymer over etched silicon waveguides, which requires precision aligned bonding. In Ref. [3], a TFLN slab was rib-loaded with silicon nitride which was deposited using plasma-enhanced chemical vapor deposition (PECVD) which can be achieved at relatively low temperatures (about 300°C), followed by patterning the silicon nitride layer. In place of silicon nitride, deposited tantalum oxide can be used for a similar purpose of confining the optical mode [16], or amorphous silicon [17,18]. In our approach, a layer of crystalline Si, part of a silicon-on-insulator (SOI) wafer, is processed using a wafer-scale photonics foundry process, then the wafer is segmented into individual dies for bonding to TFLN using plasma-activated oxide bonding [9,10]. The TFLN layer is not etched and precision alignment is not required during bonding. No polymers are used in the bonded stack. Electrodes can be formed on top of TFLN after handle removal [10] as indicated in Fig. 1(a), or buried in the SOI stack as part of the SOI foundry process [19].
Previously, bonding of TFLN to patterned Si waveguides was used to demonstrate low-speed Mach-Zehnder interferometer structures [20] and microring resonators with a compact footprint [21,22] which do not require large-area bonding. Using an Si layer of conventional (230 nm) thickness, the majority of the optical mode is contained in the silicon region, and only the evanescent tail is left to interact with the TFLN film. This substantially decreases the efficiency of the EOM and increases the modulation voltage. We narrow down the Si features in both height (across the entire wafer) and width (using a lithographic mask), which allows us to decrease or increase the mode fraction in TFLN with only a small margin on either side (to cutoff and multi-mode behavior respectively [9]). We use the precision lithography of modern wafer-scale Si microfabrication to reliably maintain these dimensions over long (millimeter-scale) distances to precisely define hybrid modes whose optical and radio-frequency (RF) indices of refraction can be matched [23].
As shown schematically in Fig. 1, our platform consists of two crystalline materials, Si and (x-cut) TFLN, whose refractive index is higher than that of the dielectric cladding materials SiO2 (for conventional components) and air (for the EOM section). We use hybrid modes, in which light is distributed between TFLN and Si (and the cladding) and the mode fraction in TFLN can be controlled by the size of the features in Si. This is shown in Fig. 1(c). We have shown less than 20% or more than 80% of the light (Poynting vector magnitude of the TE-polarized fundamental waveguide mode) at 1550 nm can be contained in the TFLN layer without etching it. The mode labeled “B” in Fig. 1(c) uses wider Si rib features and is useful for input/output transitions from the feeder waveguide mode which is labeled mode “A”. Mode “C” in which about 80% of the light is in LN uses a narrow rib width for the electro-optic phase shifter section. In some approaches, passive components can be designed using Mode “C”; however, we prefer to use mode “A” which is unlikely to have DC or low-frequency drift problems. All three modes use the same Si layer thickness so that only one lithography step is needed, which reduces fabrication cost and complexity. The single-height Si waveguide continuously connects the EOM device to the other devices such as splitters and filters without breaks and interruptions. This reduces back-reflections and phase jumps (in transmission) that can be found at abrupt, butt-coupled interfaces and also does not require three-dimensional tapers for matching Si waveguides of different heights [24]. A study of the tapers and transitions was previously carried out: the transition between modes A and B can have a low calculated reflection coefficient of about 0.1 dB - 0.2 dB over a wide range of wavelengths between 1260 nm and 1650 nm, and the transition between modes B and C is adiabatic and has very low loss [23].
3. High-Q microring resonator design and measurement
The micro-resonator in the racetrack configuration [25] (simply called a microring for simplicity) consists of a bus waveguide, couplers based on the directional coupling principle [26], and the resonator itself, and we will discuss each aspect in turn.
The microring is formed using mode A, in which the Si thickness is 150 nm and the width is 650 nm. The mode has an effective index neff = 2.28 and group index ng = 3.85 and small mode area Aeff = 0.21 µm2 at 1550 nm wavelength. (For a similar waveguide with slightly narrower width of 580 nm, neff = 2.27 and ng = 3.77; the latter is experimentally verified in Section 4). The silicon photonic fabrication was performed at Sandia National Laboratories. Optical transmission through a diced-out test chip was measured by coupling in light (transverse electric, TE, polarization in the plane of the chip) from a tunable-wavelength laser using multi-axis micropositioner stages and lensed tapered fibers, and measured on a photodetector. A propagation loss coefficient of 0.8 dB.cm-1 was measured at 1550 nm by comparing the transmission through waveguides of different length that were fabricated on the same chip, and is reported in Fig. 2(a). This mode is well confined in the vicinity of the Si rib, and can support a bending radius of less than 5 µm with low optical loss [9]. Conservatively, a radius of 10 µm is used in the microring design of Section 4. Figure 2(b) shows the group-velocity dispersion parameter (symbol: D, units: ps.nm-1.km-1) based on a fourth-order centered differentiation of the calculated modal effective refractive index versus wavelength around 1.55 µm. The values of D are about a factor of five lower than typical values of D ∼ + 3,000 ps.nm-1.km-1 for a fully-etched Si waveguide of 525 nm width and 226 nm thickness [27]. The reduced values of D are comparable to the material dispersion of silicon DSi ∼ -900 ps.nm-1.km-1 indicating that the geometry does not influence the overall dispersion as strongly as it does in conventional Si waveguides. The sign of D is negative (i.e., normal dispersion), rather than positive (anomalous dispersion). The normal dispersion regime in the feeder waveguides can help suppress nonlinear optical processes such as four-wave mixing, self-frequency shifting and pulse self-compression which are not desired in the feeder bus or transition regions [28,29].
Fig. 2. (a) Measured optical propagation loss of thin-Si waveguides at 1550 nm. (b) Calculated group-velocity dispersion coefficient. (c) Length-integrated directional coupling coefficients for different values of the gap between the two waveguides as indicated in the legend. Two target designs (0.005 and 0.01) are indicated by the dashed horizontal lines. (d) Mode field (|Ex|) profile of the symmetric mode in the directional coupler composed of single waveguides with Mode “A” in Fig. 1(c). (e) Mode field (|Ex|) profile of the anti-symmetric mode. (f) Group refractive indices of the symmetric mode (blue), and anti-symmetric mode (red), for the thin-Si directional coupler (solid lines) compared to conventional 230 nm height silicon waveguides (dashed lines).
Figure 2(c) shows the calculated coupling coefficient |κ|2 of a directional coupler between a straight waveguide and a second waveguide which comprises two back-to-back quarter-bend waveguides [this is indicated in Fig. 3(c)]. In this calculation, the Si height is 150 nm, width is 650 nm and the gap between the waveguides at the narrowest point varies between 300 nm and 440 nm as indicated in the figure legend. The coupling coefficient κ is defined as the length-integrated amplitude coupling coefficient following the convention described in Ref. [30]. The overall coupler length is about twice the quarter-bend radius and is about 18.0 µm, with straight sections of the racetrack about 9.0 µm long. For a high-Q micro-resonator, we seek a small value of |κ|2, typically around 0.005 to 0.01 which is indicated by the dashed lines. A flat-top filter would use a larger value of |κ|2 and a coupled-resonator optical waveguide (CROW) requires a graded sequence of |κ|2 [31], varying from 440 nm to 300 nm in gap width. Figure 2(b) shows that the wavelength dependency (i.e., dispersion) of |κ|2 is lower for wider gaps which have a weaker coupling. For a 400 nm gap, d|κ|2/dλ = 0.1 µm-1 which is about one order of magnitude lower than d|κ|2/dλ = 8 µm-1 of fully-etched Si waveguide couplers in the conventional platform [32].
Fig. 3. (a) Schematic drawings of two types of Bézier bends for use in a waveguide-coupled microring resonator. (b) Calculated (loaded) quality factor (Q) versus the magnitude of the inter-waveguide coupling coefficient. (c) Schematic of a microring resonator using the Type II design. (d) Measured transmission over 1540 nm to 1575 nm, (e) Transmission measured with a high resolution near 1542.5 nm, showing a loaded Q of 1.2 × 105, close to the design target shown by dashed lines in panel (b).
The reason for the lower coupling dispersion is the smaller difference between the indices of the symmetric and anti-symmetric supermodes of the coupler, shown in Fig. 2(d) and 2(e), respectively. The group index of the supermodes is plotted in Fig. 2(f), as well as the difference between the group indices, which determines the coupling dispersion [32]. In this calculation, a slightly larger gap (by about 40 nm) was used for the 230 nm thick silicon coupler to achieve the same coupler crossover length. Compared to the thin Si directional coupler, the conventional directional coupler’s index difference is about twice as large.
Two types of Bézier bends are shown in Fig. 3(a), labeled Type I and Type II. The initial and final widths are 0.65 µm and in the Type I design, the most aggressive Bézier bend design is used, in which the two control points of a quarter bend coincide. In the Type II design, a less aggressive bend is used which is in-between the conventional (circular) ring and the Type I design [33,34]. The inset to Fig. 3(b) shows the equation used to calculate Q, based on the round-trip circumference L, (single-pass) round trip loss a = exp(-αL), based on α=1 dB/cm passive optical loss coefficient (assumed for simplicity), the resonance wavelength λres = 1550 nm, and the through coupling coefficient of the coupler, t = (1-|κ|2)1/2. The Type II design has a slightly better performance, but the difference should be too small to be significant in most practical cases. Using a dashed line in Fig. 3(b), we indicate the target value of the (loaded) quality factor Q = 1 × 105 which we have used elsewhere for efficient four-wave mixing and spontaneous four-wave mixing using microrings with the conventional Si height [35].
Figure 3(e) shows the measurement of a high-Q microring using this platform. Our design for the racetrack used four Type II Bézier bends with a nominal radius of 10 µm, a straight section of 8.95 µm and a gap of 0.38 µm. The measured free-spectral range was 7.6 nm near 1550 nm, which was close to the design target of 7.5 nm. The resonance that is shown in Fig. 3(e) is near 1542.5 nm, which is one FSR away from the resonance closest to 1550 nm. (In a typical photon pair-generation experiment, the pump would be positioned on this resonance in order to generate a photon at 1550 nm through spontaneous four-wave mixing [35].) The measured contrast in transmission is about 20 dB which suggests that the device is close to critical coupling. The measured loaded quality factor is Q = 1.2 × 105, which is slightly better than the design target of 1.0 × 105. This suggests that the experimental optical propagation loss is slightly less than 1 dB/cm which was assumed in the model and agrees with the separate loss measurement of 0.8 dB/cm reported in Fig. 2. The low propagation loss is consistent with earlier calculations which predicted a reduced optical propagation loss for thin-and-wide Si waveguides compared to the conventional (500 nm x 230 nm, height x width) design [36]. In fact, the measured Q’s are comparable to those of the best microrings using a conventional Si layer height which we have reported as being useful for bright and efficient entangled photon-pair generation [14,35]. Therefore, it is likely that we can achieve similar pair-generation performance at 1550 nm in the thin-Si platform in the future, which would eliminate the difficulty of including a second layer of crystalline silicon of different thickness for this purpose.
4. Low-Q coupled-microring filter design and measurement
Figure 4(a) shows the schematic of a coupled-microring optical filter, which is a simple prototype of a more general class of optical filters based on coupled-resonator optical waveguides (CROWs) that can be used as integrated optical filters with a lower back-reflection than in-line waveguide gratings. Such filters have been combined with nonlinear optical microrings performing frequency conversion and filtering without requiring a separate chip for this purpose [37]. We have previously shown that rather than cascade a large number of resonators in one continuous segment, improved performance can be achieved by concatenating multiple shorter segments [31]. Therefore, the performance of elementary coupled-ring filters (as short as two rings in this case) is of interest. Here, we use the thin-Si platform and relatively strongly coupled resonators, with each racetrack using the same four Type II Bezier bends with a nominal radius of 10 µm. A slightly narrower width of 580 nm is used, with a longer straight section of 20 µm in the coupler region, and gap of 0.3 µm to achieve stronger coupling than in the high-Q ring design. Simulations predict a relatively flat-top response with a full-width at half-maximum (3 dB) bandwidth of greater than 100 GHz at 1550 nm.
Fig. 4. (a) Schematic drawing of a coupled-microring filter. (b) The drop-port transmission is measured and compared with the transmission through a conventional waveguide on the same chip. (c) A magnification of the transmission around 1550 nm. (d) Free Spectral Range (FSR) versus wavelength with an average value of 0.77 THz.
The experimental measurement is shown in Fig. 4(b) for a wide range of wavelengths measured using a swept-wavelength laser, and in the vicinity of 1550 nm in Fig. 4(c). The grey line shows the transmission through a test waveguide in the vicinity of the coupled-microring filter, so that the insertion loss of the filter can be obtained by comparison. The measurement of the reference waveguide shows some ripple from Fabry-Perot effects from the polished edges of the test chip; these are extraneous to the design. This ripple is also superimposed on the filter transmission and has not been factored out. Therefore, the actual transmission response near the peak of 1551 nm is likely to be slightly flatter than shown in this raw data.
Within the range of wavelengths from 1520 nm to 1620 nm (which span the C and L bands), the band-center insertion loss (IL) ranges between 0.57 dB to 1.15 dB, with an average of 0.9 dB. This is similar to the insertion loss we reported earlier for coupled silicon microrings in the conventional silicon platform [37]. The free spectral range (FSR) is about 0.76 THz (6.07 nm) at 1550 nm with a bandwidth of about 125 GHz (1 nm), for an effective finesse (F) of about 6. A low value of F is important in a silicon-based optical filter to prevent a buildup of optical power in the rings which would cause nonlinear loss or phase shift, which is undesirable and would degrade the performance. The measured extinction ratio exceeds 30 dB at the lowest points of the stopband around 1550 nm as shown in Fig. 4(c) and is likely to be higher as it may be limited by the detector noise. There are also some ripples in the stop-band of the coupled-resonator filter; these are usually the result of disorder and have been seen in other CROW structures in the conventional Si platform as well [38,39]. When multiple stages of such filters are cascaded, these interference effects should be mitigated or eliminated. Consistent with the observation of low dispersion discussed in Section 3, there is relatively little change in the FSR as shown in Fig. 4(d). Using the formula ng = c/(L x FSR) which relates the FSR to the group index (ng), circumference (L) and speed of light (c) [40], the fitted ng = 3.75 ± 0.056 (one standard deviation), compared to the simulated ng = 3.77 for mode A (580 nm width, 150 nm thickness) at 1550 nm.
5. RF optical index matching for high-bandwidth modulators
The measurements of the single high-Q microring and coupled low-Q microrings in Sections 3 and 4, respectively, were performed using diced-out segments of a single larger chip which also contained a hybrid TFLN-Si Mach-Zehnder modulator (MZM).
In a traditional (diffused-waveguide) LN modulator, the microwave index is about 4.2 [41] which is close to the group index (ng(optical) ∼ 4.25) that can be achieved when using the conventional 230 nm thickness Si photonics platform. However, the optical mode is highly confined to the Si region and does not overlap well with the RF field which is primarily in the LN region. When using a thin Si platform, the optical mode can be pushed out to overlap better with TFLN [see mode “C” in Fig. 1(c)] but because of the reduced dispersion, this also results in a significantly lower ng(optical) than 4.25, and therefore, the target n(RF) must be much lower than its value for a traditional LN EOM device to achieve RF-optical index matching.
The optical index is sensitive to the dimensions of the Si waveguide, the thickness of the TFLN slab, and the thickness of the thin oxide layer between the TFLN and Si layers whose critical dimensions are shown in Fig. 1(a). Typical values of ng at 1550 nm are 2.38 for 300 nm Si rib width and 2.31 for 275 nm Si rib width with a 150 nm Si layer thickness. The RF phase refractive index, n(RF), is mainly controlled by the geometry of the traveling-wave electrodes (metal thickness, electrode spacings, materials used in the cladding and substrate etc.) [42,43]. A segment of our periodically-loaded traveling wave electrode design is shown in a scanning electron microscope (SEM) image in the inset to Fig. 5(a). The period is 25 µm, the slots are of depth 4 µm, with a length of 5 µm. The gap between the “S” and “G” electrodes is 8 µm and the total length of the electrodes is 1 cm. The electrodes are made of gold with a thickness of 0.9 µm and a thin TiN adhesion layer as described elsewhere [12].
Fig. 5. (a) Microscope image of a hybrid Si/TFLN chip using the cross-section shown in Fig. 1. Insets show the traveling-wave electrode structures for a Mach-Zehnder modulator. (b) Measured S parameters up to 118 GHz. (c) Inferred microwave refractive index (solid blue line) compared to the optical group index (dashed black line).
A measurement of the electrode structures was performed using landed RF GSG probes and an electrical Vector Network Analyzer (VNA). Figure 5(b) shows the measured S parameters. The dashed line in Fig. 5(c) shows the optical group index of mode “C” in the thin-Si platform at 1550 nm; this value is essentially independent of the RF frequency. The RF index was calculated from the measured S parameters and is shown in Fig. 5(c) over the range of 100 MHz to 118 GHz. Outside of the usual low-frequency behavior which raises nRF below 1 GHz [44,45], the RF and optical indices are well matched to within 1%, which is at least as good as traditional LN MZM devices [46]. Electro-optic measurements of the MZM devices are not the subject of this report and will be reported elsewhere.
6. Conclusion
We have shown that a hybrid TFLN-Si photonics platform can not only achieve RF-optical index matching, but also provide a useful platform for high-Q Si microring resonators and low-Q coupled-microring Si optical filters. This does not require a second level of crystalline waveguides with a different thickness to accommodate the silicon photonics components and is therefore a simpler design and fabrication process. The lower waveguide dispersion and lower coupling dispersion may be helpful to design passive devices such as rings and filters over a wide bandwidth. As we continue to improve and validate the hybrid EOM designs, the power of the already-existing Si foundry process can also be exercised to incorporate increasingly more complex and multi-functional components with TFLN. In our approach, the TFLN layer is not etched or patterned, and is bonded as a back-end step after the Si fabrication has been completed. The overall performance is expected to exceed that of an all-Si device in which we have integrated a carrier-depletion EOM with a high-Q microring and coupled-microring filter [14]. We did not have to design the high-Q ring and microring filter in the TFLN platform. A mature Si fabrication process leads to good control over both layer thicknesses and etched feature dimensions, which in turn, leads to good performance of high-Q resonators, optical filters, and traveling-wave electro-optic modulators. The crystalline nature of the Si layer in this platform can be leveraged to incorporate dopants, diodes and detectors. Taken together, these positive attributes of the design and fabrication process should benefit many applications in classical and quantum photonics such as light routing, optical filtering, wavelength conversion, frequency comb generation and entangled photon-pair creation that are of current interest.
Funding
U.S. Government; National Aeronautics and Space Administration (80NSSC17K0166); Office of Naval Research (N00014-21-1-2805); U.S. Department of Defense (HR001120S0008).
Acknowledgments
The authors thank: C. Ma, M. Ruesing, C. Valdez, P. O. Weigel and J. Zhao (formerly of UC San Diego) for earlier contributions and discussions on this topic; A. Lentine, N. Boynton, T. A. Friedman, S. Arterburn, C. Dallo, A. T. Pomerene, A. L. Starbuck, and D. C. Trotter (Sandia National Laboratories, Applied Microphotonic Systems) for discussions and fabrication assistance; C. Coleman, R. Scott and G. Lee (Keysight Technologies) for discussions and measurement assistance. Part of this work was performed at the San Diego Nanotechnology Infrastructure (SDNI) of UCSD, a member of the National Nanotechnology Coordinated Infrastructure, which is supported by the National Science Foundation (Grant ECCS-2025752). This research was developed in part with funding from the Defense Advanced Research Projects Agency (DARPA) and the U.S. Government. This paper describes objective technical results and analysis. The views, opinions and/or findings expressed are those of the authors alone and should not be interpreted as representing the official views or policies of the Department of Defense or the U.S. Government.
Disclosures
The authors declare 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.
References
1. W. Sohler, H. Hu, R. Ricken, V. Quiring, C. Vannahme, H. Herrmann, D. Büchter, S. Reza, W. Grundkötter, S. Orlov, H. Suche, R. Nouroozi, and Y. Min, “Integrated optical devices in lithium niobate,” Opt. Photonics News 19(1), 24 (2008). [CrossRef]
2. L. Chen, Q. Xu, M. G. Wood, and R. M. Reano, “Hybrid silicon and lithium niobate electro-optical ring modulator,” Optica 1(2), 112–118 (2014). [CrossRef]
3. A. Rao, A. Patil, P. Rabiei, A. Honardoost, R. DeSalvo, A. Paolella, and S. Fathpour, “High-performance and linear thin-film lithium niobate Mach–Zehnder modulators on silicon up to 50 GHz,” Opt. Lett. 41(24), 5700–5703 (2016). [CrossRef]
4. A. Boes, B. Corcoran, L. Chang, J. Bowers, and A. Mitchell, “Status and potential of lithium niobate on insulator (LNOI) for photonic integrated circuits,” Laser Photonics Rev. 12(4), 1700256 (2018). [CrossRef]
5. C. Wang, M. Zhang, X. Chen, M. Bertrand, A. Shams-Ansari, S. Chandrasekhar, P. Winzer, and M. Lončar, “Integrated lithium niobate electro-optic modulators operating at CMOS-compatible voltages,” Nature 562(7725), 101–104 (2018). [CrossRef]
6. M. He, M. Xu, Y. Ren, J. Jian, Z. Ruan, Y. Xu, S. Gao, S. Sun, X. Wen, L. Zhou, L. Liu, C. Guo, H. Chen, S. Yu, L. Liu, and X. Cai, “High-performance hybrid silicon and lithium niobate Mach–Zehnder modulators for 100 Gbits-1 and beyond,” Nat. Photonics 13(5), 359–364 (2019). [CrossRef]
7. D. Zhu, L. Shao, M. Yu, R. Cheng, B. Desiatov, C. J. Xin, Y. Hu, J. Holzgrafe, S. Ghosh, A. Shams-Ansari, E. Puma, N. Sinclair, C. Reimer, M. Zhang, and M. Lončar, “Integrated photonics on thin-film lithium niobate,” Adv. Opt. Photonics 13(2), 242 (2021). [CrossRef]
8. J. Lin, F. Bo, Y. Cheng, and J. Xu, “Advances in on-chip photonic devices based on lithium niobate on insulator,” Photonics Res. 8(12), 1910 (2020). [CrossRef]
9. P. O. Weigel, M. Savanier, C. T. DeRose, A. T. Pomerene, A. L. Starbuck, A. L. Lentine, V. Stenger, and S. Mookherjea, “Lightwave circuits in lithium niobate through hybrid waveguides with silicon photonics,” Sci. Rep. 6(1), 22301 (2016). [CrossRef]
10. P. O. Weigel, J. Zhao, K. Fang, H. Al-Rubaye, D. Trotter, D. Hood, J. Mudrick, C. Dallo, A. T. Pomerene, A. L. Starbuck, C. T. DeRose, A. L. Lentine, G. Rebeiz, and S. Mookherjea, “Bonded thin film lithium niobate modulator on a silicon photonics platform exceeding 100 GHz 3-dB electrical modulation bandwidth,” Opt. Express 26(18), 23728 (2018). [CrossRef]
11. X. Wang, P. O. Weigel, J. Zhao, M. Ruesing, and S. Mookherjea, “Achieving beyond-100-GHz large-signal modulation bandwidth in hybrid silicon photonics Mach Zehnder modulators using thin film lithium niobate,” APL Photonics 4(9), 096101 (2019). [CrossRef]
12. V. Mere, F. Valdez, X. Wang, and S. Mookherjea, “A modular fabrication process for thin-film lithium niobate modulators with silicon photonics,” JPhys Photonics 4(2), 024001 (2022). [CrossRef]
13. A. L. Lentine, C. T. DeRose, P. S. Davids, N. J. D. Martinez, W. A. Zortman, J. A. Cox, A. Jones, D. C. Trotter, A. T. Pomerene, A. L. Starbuck, D. J. Savignon, T. Bauer, M. Wiwi, and P. B. Chu, “Silicon photonics platform for national security applications,” in 2015 IEEE Aerospace Conference (IEEE, 2015), pp. 1–9.
14. C. Ma, X. Wang, and S. Mookherjea, “Progress towards a widely usable integrated silicon photonic photon-pair source,” OSA Continuum 3(6), 1398 (2020). [CrossRef]
15. A. J. Mercante, P. Yao, S. Shi, G. Schneider, J. Murakowski, and D. W. Prather, “110 GHz CMOS compatible thin film LiNbO3 modulator on silicon,” Opt. Express 24(14), 15590–15595 (2016). [CrossRef]
16. P. Rabiei, J. Ma, S. Khan, J. Chiles, and S. Fathpour, “Heterogeneous lithium niobate photonics on silicon substrates,” Opt. Express 21(21), 25573–25581 (2013). [CrossRef]
17. L. Cao, A. Aboketaf, Z. Wang, and S. Preble, “Hybrid amorphous silicon (a-Si:H)–LiNbO3 electro-optic modulator,” Opt. Commun. 330, 40–44 (2014). [CrossRef]
18. Y. Wang, Z. Chen, L. Cai, Y. Jiang, H. Zhu, and H. Hu, “Amorphous silicon-lithium niobate thin film strip-loaded waveguides,” Opt. Mater. Express 7(11), 4018 (2017). [CrossRef]
19. N. Boynton, H. Cai, M. Gehl, S. Arterburn, C. Dallo, A. Pomerene, A. Starbuck, D. Hood, D. C. Trotter, T. Friedmann, C. T. DeRose, and A. Lentine, “A heterogeneously integrated silicon photonic/lithium niobate travelling wave electro-optic modulator,” Opt. Express 28(2), 1868 (2020). [CrossRef]
20. I. Bakish, R. Califa, T. Ilovitsh, V. Artel, G. Winzer, K. Voigt, L. Zimmermann, E. Shekel, C. Sukenik, and A. Zadok, “Voltage-induced phase shift in a hybrid LiNbO3-on-silicon Mach-Zehnder interferometer,” in Advanced Photonics 2013 (OSA, 2013), p. IW4A.2.
21. Y. S. Lee, G.-D. Kim, W.-J. Kim, S.-S. Lee, W.-G. Lee, and W. H. Steier, “Hybrid Si-LiNbO3 microring electro-optically tunable resonators for active photonic devices,” Opt. Lett. 36(7), 1119–1121 (2011). [CrossRef]
22. L. Chen and R. M. Reano, “Compact electric field sensors based on indirect bonding of lithium niobate to silicon microrings,” Opt. Express 20(4), 4032 (2012). [CrossRef]
23. P. O. Weigel, F. Valdez, J. Zhao, H. Li, and S. Mookherjea, “Design of high-bandwidth, low-voltage and low-loss hybrid lithium niobate electro-optic modulators,” JPhys Photonics 3(1), 012001 (2021). [CrossRef]
24. S. Abe, H. Hara, S. Masuda, and H. Yamada, “Fabrication of Vertical-Taper Structures for Silicon Photonic Devices by Local-Thickness-Thinning Process,” in 2021 26th Microoptics Conference (MOC) (IEEE, 2021), pp. 1–2.
25. L. Caruso and I. Montrosset, “Analysis of a racetrack microring resonator with MMI coupler,” J. Lightwave Technol. 21(1), 206–210 (2003). [CrossRef]
26. E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48(7), 2071–2102 (1969). [CrossRef]
27. E. Dulkeith, F. Xia, L. Schares, W. M. J. Green, and Y. A. Vlasov, “Group index and group velocity dispersion in silicon-on-insulator photonic wires,” Opt. Express 14(9), 3853–3863 (2006). [CrossRef]
28. A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, “Tailored anomalous group-velocity dispersion in silicon channel waveguides,” Opt. Express 14(10), 4357 (2006). [CrossRef]
29. B. A. Daniel and G. P. Agrawal, “Dependence of dispersive and birefringence properties of silicon nanowires on waveguide dimensions,” Opt. Lett. 35(2), 190 (2010). [CrossRef]
30. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36(4), 321–322 (2000). [CrossRef]
31. J. R. Ong, R. Kumar, and S. Mookherjea, “Ultra-high-contrast and tunable-bandwidth filter using cascaded high-order silicon microring filters,” IEEE Photonics Technol. Lett. 25(16), 1543–1546 (2013). [CrossRef]
32. R. Aguinaldo, Yiran Shen, and S. Mookherjea, “Large dispersion of silicon directional couplers obtained via wideband microring parametric characterization,” IEEE Photonics Technol. Lett. 24(14), 1242–1244 (2012). [CrossRef]
33. H. P. Bazargani, J. Flueckiger, L. Chrostowski, and J. Azaña, “Microring resonator design with improved quality factors using quarter Bezier curves,” in CLEO: 2015 (OSA, 2015), p. JTu5A.58.
34. T. Y. L. Ang, J. R. Ong, S. T. Lim, C. E. Png, X. Guo, and H. Wang, “Versatile bezier bends for silicon photonics,” in 2017 Conference on Lasers and Electro-Optics Pacific Rim, 2017 Conference on Lasers and Electro-Optics Pacific Rim (Optica Publishing Group, 2017), p. s1869.
35. C. Ma, X. Wang, V. Anant, A. D. Beyer, M. D. Shaw, and S. Mookherjea, “Silicon photonic entangled photon-pair and heralded single photon generation with CAR > 12,000 and $\textrm{g}\,\hat{}\,(2)(0)$ < 0006,” Opt. Express 25(26), 32995 (2017). [CrossRef]
36. T. Barwicz and H. A. Haus, “Three-dimensional analysis of scattering losses due to sidewall roughness in microphotonic waveguides,” J. Lightwave Technol. 23(9), 2719–2732 (2005). [CrossRef]
37. J. R. Ong, R. Kumar, and S. Mookherjea, “Silicon microring-based wavelength converter with integrated pump and signal suppression,” Opt. Lett. 39(15), 4439 (2014). [CrossRef]
38. M. L. Cooper and S. Mookherjea, “Modeling of multiband transmission in long silicon coupled-resonator optical waveguides,” IEEE Photonics Technol. Lett. 23(13), 872–874 (2011). [CrossRef]
39. S. Mookherjea and M. A. Schneider, “Avoiding bandwidth collapse in long chains of coupled optical microresonators,” Opt. Lett. 36(23), 4557 (2011). [CrossRef]
40. M. L. Cooper, G. Gupta, M. A. Schneider, W. M. J. Green, S. Assefa, F. Xia, D. K. Gifford, and S. Mookherjea, “Waveguide dispersion effects in silicon-on-insulator coupled-resonator optical waveguides,” Opt. Lett. 35(18), 3030 (2010). [CrossRef]
41. M. M. Howerton and W. K. Burns, Broadband Traveling Wave Modulators in LiNbO3 (Cambridge University, 2002).
42. X. Liu, B. Xiong, C. Sun, J. Wang, Z. Hao, L. Wang, Y. Han, H. Li, and Y. Luo, “Sub-terahertz bandwidth capactively-loaded thin-film lithium niobate electro-optic modulators based on an undercut structure,” Opt. Express 29(25), 41798 (2021). [CrossRef]
43. P. Kharel, C. Reimer, K. Luke, L. He, and M. Zhang, “Breaking voltage–bandwidth limits in integrated lithium niobate modulators using micro-structured electrodes,” Optica 8(3), 357 (2021). [CrossRef]
44. G. Ghione, M. Goano, G. L. Madonna, G. Omegna, M. Pirola, S. Bosso, D. Frassati, and A. Perasso, “Microwave modeling and characterization of thick coplanar waveguides on oxide-coated lithium niobate substrates for electrooptical applications,” IEEE Trans. Microwave Theory Tech. 47(12), 2287–2293 (1999). [CrossRef]
45. P. Singh, R. Bansal, and A. K. Verma, “Dispersion analysis of coplanar waveguide for low frequency,” in V. Janyani, M. Salim, and K. K. Sharma, eds. (2013), p. 87600I.
46. G. K. Gopalakrishnan, W. K. Burns, R. W. McElhanon, C. H. Bulmer, and A. S. Greenblatt, “Performance and modeling of broadband LiNbO/sub 3/ traveling wave optical intensity modulators,” J. Lightwave Technol. 12(10), 1807–1819 (1994). [CrossRef]
