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Low-noise Kerr frequency comb generation with low temperature deuterated silicon nitride waveguides

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Abstract

We report very low-loss deuterated silicon nitride (SiNx:D) micro-ring resonators fabricated by back-end CMOS compatible low-temperature plasma-enhanced chemical vapor deposition (PECVD) without annealing. Strong confinement micro-ring resonators with a quality factor of > 2 million are achieved, corresponding to a propagation loss in the 1460-1610 nm wavelength range of ∼ 0.17 dB/cm. We further report the generation of low-noise coherent Kerr microcomb states including different perfect soliton crystals (PSC) in PECVD SiNx:D micro-ring resonators. These results manifest the promising potential of the back-end CMOS compatible SiNx:D platform for linear and nonlinear photonic circuits that can be co-integrated with electronics.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Silicon nitride (SiNx) as an important alternative optical waveguide material to silicon and silicon dioxide has attracted much attention due to its combination of wide transparency window, low intrinsic loss, fabrication flexibility, strong optical confinement and high-power handling capacity. The development of low-loss silicon nitride waveguide technology has boosted a wide range of applications in nonlinear photonics [1], sensing [2], LiDAR [3] and quantum optics [4]. Particularly, silicon nitride has become one of the most mature material platforms in the area of Kerr nonlinear photonics. As an important breakthrough in integrated photonics, high quality factor (Q) SiNx-based resonators have enabled coherent soliton microcombs generation with repetition rates from microwave to terahertz frequencies [5,6], which opens up numerous opportunities for on-chip spectroscopy [7], miniature optical atom clock [8], microwave photonics [5] and optical neuromorphic computing [9]. The realization of SiNx soliton microcombs requires very low-loss waveguides with tailored dispersion profile to achieve the double balance of the dispersion and the nonlinearity as well as the cavity loss and the parametric gain [10].

To obtain high-quality silicon nitride films for low-loss high-confinement waveguides, several deposition methods including low-pressure chemical vapor deposition (LPCVD), plasma enhanced chemical vapor deposition (PECVD), sputtering and atomic layer deposition (ALD) have been developed [1113]. The LPCVD process directly uses heat to initiate chemical reaction of precursor gases, which requires a high temperature (>800 °C) but provides high purity and uniformity. The as-deposited stoichiometric Si3N4 film usually has high tensile stress and special annealing treatment is required to avoid cracking when depositing thick films (>400 nm). PECVD techniques utilize energetic plasma to assisted deposition reaction, which provides a much lower deposition temperature (below 400 °C) with higher deposition speed than LPCVD process. Film stress can be well controlled, allowing large film thickness (>1 μm) in a single run [14]. The low thermal budget involved in PECVD is back-end compatible with CMOS processing, offering the attractive potential of integrating SiNx photonic layers on top of prefabricated electronic or active optoelectronic layers.

SiNx films formed by both LPCVD and PECVD methods using silane (SiH4) as precursor gas (SiNx:H) contain a certain amount of nitrogen-hydrogen (N-H) bond, which causes absorption loss in the 1500-1550 nm wavelength range. The N-H bond residue levels in low-temperature deposited SiNx films is much higher than LPCVD films, hence introducing a higher material loss and limiting the application for efficient Kerr comb generation. To date, all SiNx-based soliton combs using ultralow loss (<0.1 dB/cm @1550 nm) waveguides are fabricated using LPCVD technique combined with high-temperature thermal annealing (above 1200 °C) to remove N-H bonds [1518]. However, the high deposition temperature and high thermal annealing temperature are not compatible with CMOS electronic circuits, which makes LPCVD silicon nitride a front-end-of-line (FEOL) platform. The N-H bond residue in PECVD SiNx film can also be minimized by high-temperature annealing [19,20] but the advantage of back-end CMOS compatibility will be lost.

To provide low material loss in the telecommunications band with low-temperature, low-stress PECVD technology, an alternative method is to use deuterated silane (SiD4) instead of traditional hydrogenated silane (SiH4) as the precursor gas to form deuterated silicon nitride (SiNx:D) films [2123]. The N-H bond is replaced by nitrogen-deuterium (N-D) bond, which has no absorption loss in the telecom band [24,25]. Integrated SiNx:D photonic circuits for polarization-insensitive arrayed-waveguide grating [21], microwave photonic filters [26] and direct hybrid integration [27,28] have been demonstrated and exhibit good performance. Deuterated silicon oxide has also been investigated as low-loss cladding for weak-confinement LPCVD SiNx waveguide [29]. In the field of integrated nonlinear photonics, only modulational instability (MI) frequency combs have been previously reported in PECVD SiNx:D waveguides [22,25]. The inability to achieve soliton comb generation is believe to be associated with the relative large absorption loss in PECVD SiNx:D material.

In this work, we demonstrate low-loss, high-Q micro-ring resonators for coherent Kerr comb generation on a low-temperature PECVD SiNx:D platform. Propagation loss as low as 0.17 dB/cm throughout the 1460-1610 nm wavelength range is achieved in the strong confinement waveguide. A further detailed study on thermal-optic bistability reveals the relative contribution of material absorption loss to the total loss, offering guidelines for further reduction of the waveguide loss. Finally, we present the results of generating low-noise Kerr microcomb states including different perfect soliton crystals (PSC), which to the best of our knowledge is the first demonstration in PECVD SiNx:D micro-ring resonators.

2. SiNx:D film preparation and device fabrication

Based on our previously reported deposition technique on an inductively coupled plasma enhanced chemical vapor deposition (Oxford PlasmaPro100 ICP-PECVD) platform [30], we start the SiNx deposition process optimization at temperatures of below 300℃. Deuterated silane (SiD4) gas and pure Nitrogen (N2) are used as the precursor gases for Si and N, and their flow rates are set so that their partial pressure value are the same as SiH4 and N2 in the optimized conventional silane recipe. Keeping the gas flow rates constant, total deposition chamber pressure, ICP power and lower electrode RF power values are tuned to optimize the film quality, which is characterized in terms of the film deposition rate, stress and wet etching rate in 10% hydrofluoric acid.

The impact of the isotopic substitution of deuterium for hydrogen is investigated by Fourier-transform infrared (FT-IR) spectroscopy [22]. As shown in Fig. 1(a), the fundamental absorption peak is shifted from 3.05 μm to 4.03 μm, the corresponding first overtone of this absorption shifts from 1.52 μm to 2.01 μm, thus the material loss in the S and C telecommunications bands is reduced. Figure 1(b) shows an early comparison results between SiNx:D and hydrogenated SiNx:H micro-ring resonators (MRRs) with same waveguide geometry. It is clear that most of the resonances exhibits a uniform low propagation loss of below 1 dB/cm within the 1525-1575 nm wavelength in SiNx:D resonators, while SiNx:H waveguide features a dramatic N-H bond-induced loss peak in the 1520-1550 nm range. Further optimized results are reported later in this paper.

 figure: Fig. 1.

Fig. 1. (a) FTIR spectra for PECVD-deposited SiNx:H and SiNx:D films. (b) Propagation loss within 1525-1575 nm wavelength for SiNx:D and SiNx:H MRR with similar size. (c) Microscope image of an 80-μm-radius SiNx:D MRR. (d) Simulated dispersion D2 of the SiNx:D waveguide with a cross-section of 2 μm × 880 nm.

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To realize anomalous dispersion for Kerr nonlinear photonics at the pump wavelength of 1550 nm (∼193.4 THz optical frequency), the cross-section of the waveguide in this work is design to be 2 μm × 880 nm, which results in an optical confinement factor of > 0.92 for both fundamental transverse electric (TE0) and fundamental transverse magnetic (TM0) modes. The simulated dispersion profile in Fig. 1(d) suggests that both TE0 and TM0 modes satisfy the anomalous dispersion condition for achieving phase matching in microcomb generation.

To fabricate the devices, firstly we deposited 880 nm SiNx:D films in one continuous run at a deposition temperature of 270 °C on a Si wafer with a 5 μm thermally-grown SiO2 layer. The refractive index of the SiNx:D film is ∼1.99 at 1550 nm as measured by ellipsometry. No chemical-mechanical polishing (CMP) is applied to the as-grown film. Micro-ring resonators with different radii ranging from 50 to 300 μm are defined using a Vistec EBPG5000+ electron beam lithography (EBL) system at 100kV in AR-P 6200 resist with a thickness of 800 nm. The pattern is then transferred to the silicon nitride film by reactive ion etching (Oxford Instrument Plasmalab System 100 RIE180) with a CHF3:O2 chemistry. After photoresist stripping and piranha cleaning, the devices are fully cladded with a 3-μm ICP-PECVD SiOx:H layer using conventional SiH4 precursor. The microscope image of an 80-μm-radius MRR with a straight bus waveguide is shown in Fig. 1(c). The bus waveguide is slightly narrower than the ring waveguide to achieve selective excitation of fundamental modes with high coupling ideality [31,32]. The gap between ring waveguide and bus waveguide is 250 nm.

3. Statistical analysis of propagation loss

The transmission spectra of the MRRs are collected using conventional wavelength scanning method with wavelength sweeping from 1460 to 1610nm using a Keysight 8164B Lightwave Measurement System. Light from the narrow-linewidth tunable laser source (Keysight 81606A TLS) is butt-coupled to tapered waveguides with a tip width of 150 nm through a single-mode lensed fiber. The fiber-chip coupling loss is about 2.6 dB/facet. The measured resonances are fitted with Lorentzian curve to extract the loaded quality factor (QL). The intrinsic Q (Qi) and propagation loss α can be retrieved by calculating equation (1):

$${Q_i} = \frac{{2{Q_L}}}{{1 \pm \sqrt {T{}_{\min }} }}\textrm{ = }\frac{{\textrm{2}\pi {n_g}}}{{\alpha {\lambda _0}}}$$
where Tmin refers to the normalized transmission at resonance, ng is the group index, and λ0 is the resonant wavelength. The positive and negative sign in the denominator correspond to under- and over-coupled condition [33].

We measured 22 MRRs with different radii on the same chip, which contains more than a total of 5000 resonances for both TE0 and TM0 modes. Resonances with low fitting accuracy (R2<0.9) are filtered out to avoid under- or over-estimating. For resonators with bending radii larger than 80 μm, bending loss is negligible for TE0 mode (Fig. 2(a)) while TM0 mode (Fig. 2(d)) is more sensitive to waveguide bending and exhibits a minimum acceptable bending radius of >100 μm. The histogram of propagation loss from MRRs with radius >80 μm indicates a most probable value of 0.17 dB/cm for TE0 mode (Fig. 2(b)) and 0.23 dB/cm for TM0 mode (Fig. 2(e)). The inset of Fig. 2(b) shows that highest QL of 2.29 million is achieved in a 300-μm-radius MRR, corresponding to a propagation loss of 0.11 dB/cm. The propagation loss is generally uniform in the 1460-1610 nm wavelength range as shown in Fig. 2(c) and (f), except a slight larger loss appears in 1485-1535 nm wavelength for TE0 mode. This could be brought by the hydrogen bond in the PECVD SiO2 up cladding [29]. This trend is not clearly seen in the TM0 mode due to a larger statistical dispersion. The loss value achieved in this work is lower than annealed PECVD SiNx:H waveguides [20] and sputtered hydrogen-free SiNx waveguides [12], indicating a better inherent optical property for the SiNx:D material.

 figure: Fig. 2.

Fig. 2. Statistical analysis of propagation loss for TE0 mode (left) and TM0 mode (right) based on 22 resonators, including (a) (d) box charts of propagation loss for different radius, (b) (e) histogram of >2500 resonances for estimating the most probable value, (c) (f) box charts of propagation loss in 1460-1610 wavelength range. The black solid lines show the median value and the shaded regions show the median absolute deviation.

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4. Thermo-optic bistability measurement and material absorption loss analysis

To probe the loss origins and estimate the intrinsic material loss of SiNx:D waveguide around 1550 nm quantitatively, we perform a thermo-optic bistability measurement and analysis [34]. The thermal bistability originates from power coupled to the resonant mode being partially absorbed by the SiNx material and the cladding SiO2 material, which results in local heating that shifts the resonance frequency. To extract the absorbed fraction ζ of the coupled power, a series of power dependent experiments is conducted firstly. With increasing power, the resonance of TE0 mode gradually undergoes linewidth broadening and exhibits a skewed Lorentzian line shape (Fig. 3(a)). The asymmetric curve can be fitted by Eq. (2):

$$\textrm{T} = 1 - \frac{{{\gamma _0}{\gamma _{ex}}}}{{[{\Delta \omega - 2\pi {f_D}(1 - T)} ]^{2} + ({\gamma _0} + {\gamma _{ex}})^{2}/4}}$$
where γ0 and γex refer to intrinsic linewidth and external linewidth respectively, and fD refers to the resonance drag [35]. The slope of the linear relationship between fD and input power Pin is the thermal susceptibility χth, which is 837.33 MHz/mW for the selected resonance (Fig. 3(b)). The relationship between χth and ζ is describe as Eq. (3):
$$\zeta = {\chi _{_{th}}}{K_{_C}}{(\delta {f_{res}}/\delta T)^{ - 1}}$$

 figure: Fig. 3.

Fig. 3. Absorption loss measurement via thermal-optic bistability analysis. (a) Skewed Lorentzian fitting to extract resonance shift. (b) Linear correlation between extracted resonance shift and on-chip power, revealing a thermal susceptibility χth of 837.33 MHz/mW. (c) Numerical simulation of steady-state temperature distribution of an 80 μm radius silicon nitride ring waveguide with absorbed optical power of 40.5 mW. (d) Simulated transient effective temperature Teff rise with time. The curve is fitted with exponential model to calculate thermal decay rate γT and thermal conductance Kc of the resonator. (e) Absorbance of thermal grown SiO2 and PECVD grown SiOx:H thin films by FT-IR spectroscopy.

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KC is the thermal conductance of the waveguide and the temperature-induced frequency shift (δfres/δT) is measured using a temperature-controlled sample stage. Thermal conductance KC is obtained by a finite-element (FEM) simulation with transient heat transfer model [36,37]. The thermal dynamic response in a micro-resonator is described by Eq. (4):

$$\frac{{\textrm{d}{T_{eff}}}}{{dt}} ={-} {\gamma _T}({T_{eff}} - \frac{{{P_{abs}}}}{{{K_C}}})$$
where γT is the thermal decay rate, Pabs is the absorbed power, Teff is the effective temperature calculated by Eq. (5):
$${T_{eff}}(t) = \frac{{\int {T(\vec{r},t)} {n^2}(\vec{r}){{|{E(\vec{r})} |}^2}{d^3}\vec{r}}}{{\int {{n^2}(\vec{r}){{|{E(\vec{r})} |}^2}} {d^3}\vec{r}}}$$

By assuming a heat source with same intensity profile as optical intensity distribution, the time response of temperature is recorded and fitted with exponential curve as shown in Fig. 3(c) and (d). Combined the simulated parameter KC=0.0017 W/K with the measured (δfres/δT) = -2.915 GHz/°C and χth = 837.33 MHz/mW, the absorbed fraction ζ is estimated to 0.49, indicating that almost half of the propagation loss (i.e., ∼ 0.08 dB/cm or 8 dB/m) is accounted for by the absorption loss. This absorption loss value is still larger than that of an ultra-smooth annealed LPCVD Si3N4 waveguide (∼1.5 dB/m) measured using the same methodology.

The slight increase of ∼ 0.07 dB/cm in the waveguide loss at around 1515 nm (N-H absorption peak wavelength) compared to the average values in the 1560-1600 nm range implies that the H-content in the SiOx:H cladding plays a non-negligible role despite the high optical power confinement factor of 0.92 in the SiNx:D core. This is corroborated by the existence of the hydrogen-induced absorption peaks in the FT-IR spectrum of the PECVD grown top SiOx:H which are missing from that of the thermally grown bottom SiO2 cladding in Fig. 3(e). Assuming the increased waveguide loss is entirely due to the cladding absorption, it converts to an absorption loss of 1.95 dB/cm at 1515 nm in the SiOx:H upper cladding material where 3.59% of the total optical power exists. At 1550 nm where the thermo-optical bistability measurement is carried out, the waveguide loss is ∼ 0.02 dB/cm higher than the 1560-1600 nm average, suggesting the SiNx:D absorption loss contribution is ∼ 0.06 dB/cm or 6 dB/m. Replacing the SiOx:H cladding with hydrogen-free SiOx:D material therefore can be an effective solution for further decreasing the absorption loss [29].

Other potential absorption loss mechanisms include impurity-related absorption [34] or the silicon nano-crystals in the low-temperature deposited film with non-stoichiometry, which remains to be identified with total reflection X-ray fluorescence (TXRF) measurement [16] or high-resolution transmission electron microscope (HR-TEM) observation [38]. The rest 50% of propagation loss from scattering can be further reduced with future improvements on waveguide fabrication including chemical-mechanical polishing (CMP) and photoresist reflow [18,39].

5. Low-noise Kerr microcomb generation

The experimental setup of Kerr comb generation and characterization is shown below in Fig. 4(a). The CW light from a Keysight 81606A TLS is amplified by an erbium-doped fiber amplifier (EDFA) and launched into the chip through a lensed fiber. The transmission spectrum of the selected 80-μm-radius MRR in Fig. 4(b) shows an efficient excitation of TE0 mode with higher-order mode suppression. We choose a TE0 resonance with QL of 0.95 million and Qi of 1.54 million for pumping, as shown in Fig. 4(c). With on-chip power of 13.5 mW, optical parametric oscillation (OPO) with the first pairs of sidebands can be observed in the optical spectrum analyzer (OSA) as shown in Fig. 4(d). By slowly increasing frequency detuning under a larger pumping power, low-noise Kerr microcombs with double-FSR-spacing (comb 1) and single-FSR-spacing (comb 2) can be occasionally obtained (Fig. 4(e) and (f)).

 figure: Fig. 4.

Fig. 4. (a) Experimental setup diagram for frequency comb generation and measurement. TLS: tunable laser source. FPC: fiber polarization controller. TEC: temperature controller. DUT: device under test. OSA: optical spectrum analyzer. ESA: electrical spectrum analyzer. PD: photodetector. (b) Transmission spectrum in 1460-1610 nm wavelength range of the 80-μm-radius MRR. (c) Transmission of the selected pumping resonance around 1550 nm. (d) Initial state of parametric oscillation measured with on-chip pump power of 13.5 mW. (e) Spectra of the generated frequency combs and (f) the corresponding intensity noise.

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To further investigate the comb dynamics, the pump transmission during the scan is detected by a photodetector (PD) and recorded by an oscilloscope, as shown in Fig. 5(a). Here the on-chip pumping power is ∼100 mW and the sweeping rate is 20 nm/s. With the pump frequency scanning over the resonance, the output spectra shown in Fig. 5(b) exhibit several transitions including (I) Turing pattern, (II, IV, V) modulational instability (MI) comb and (III) solitonic crystal state (Fig. 5(d)). The comb state can evolve repeatedly from MI comb to the perfect soliton crystal (PSC) state indicated by low RF intensity noise in Fig. 5(c) and an upward staircase in the transmission trace. With frequency detuning further increases, the PSC state melts down and the incoherent noisy MI comb forms again. The phenomenon can be stably accessed in various devices while it is not in consistence with the traditional soliton crystal dynamics [40,41]. The underlying switching mechanisms are still under investigation by considering other competing nonlinear optical processes. Figure 5(e) and (f) show the PSC spectra with different numbers obtained in other devices. The demonstrated energy-efficient 9-PSC and 7-PSC state with large frequency spacing can be useful for terahertz applications [6].

 figure: Fig. 5.

Fig. 5. (a) Oscilloscope trace of pump transmission signal in a 100-μm-radius MRR with on-chip power of ∼100 mW. (b) Spectra of the generated frequency combs and (c) corresponding intensity noise, I-V refers to different pump detuning position in (a). (d) The enlarged solitonic crystal state (III) with ASE noise filtered out before light coupling into the waveguide. (e) 9-PSC and (f) 7-PSC generation in the SiNx:D MRR.

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6. Conclusions

To conclude, we have successfully optimized low temperature (270 °C) ICP-PECVD-deposited deuterated silicon nitride material that supports high-confinement optical waveguide with low loss of 0.17 dB/cm throughout the 1460–1610 nm wavelength range by eliminating the N-H bond absorption in the S and C telecommunication bands that is the limiting factor for conventional silane-based SiNx:H. By means of thermo-optic bistability analysis, the material absorption loss of the SiNx:D waveguide is found to be ∼ 8 dB/m. In consideration of the slightly higher total propagation loss in the 1500-1530 nm wavelength range and the hydrogen-related absorption peaks in the PECVD SiO2 cladding revealed by FT-IR spectra, we believe the waveguide absorption loss can be further reduced with deuterated oxide cladding in future.

Still, the absorption loss is sufficiently low for integrated nonlinear photonics. A parametric oscillation threshold as low as 13.5 mW was observed in the dispersion-engineered high-Q micro-ring resonators on this material platform. It can be further reduced by replacing the SiOx:H cladding with hydrogen-free SiOx:D material and optimizing fabrication including CMP and photoresist reflow. The generation of low-noise frequency comb with different perfect soliton crystal (PSC) states has been successfully demonstrated, which to our best knowledge is the first report of PSC generation on a PECVD SiNx platform. The low loss, low stress, and fabrication flexibility derived from back-end CMOS compatibility makes the PECVD SiNx:D an appealing material platform for heterogeneous integration with active electronic and photonic components and other temperature-sensitive materials in future 3D hybrid integrations.

Funding

National Natural Science Foundation of China (11774437, 61975243, U1701661); Guangdong Basic and Applied Basic Research Foundation (2019A1515010858, 2021B1515020093); National Key Research and Development Program of China (2018YFB1801800, 2019YFA0706302); Guangzhou Municipal Science and Technology Project (202103030001); Science and Technology Planning Project of Guangdong Province (2018B010114002); Local Innovative and Research Teams Project of Guangdong Provincial Pearl River Talents Program (2017BT01X121).

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.

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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.

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Figures (5)

Fig. 1.
Fig. 1. (a) FTIR spectra for PECVD-deposited SiNx:H and SiNx:D films. (b) Propagation loss within 1525-1575 nm wavelength for SiNx:D and SiNx:H MRR with similar size. (c) Microscope image of an 80-μm-radius SiNx:D MRR. (d) Simulated dispersion D2 of the SiNx:D waveguide with a cross-section of 2 μm × 880 nm.
Fig. 2.
Fig. 2. Statistical analysis of propagation loss for TE0 mode (left) and TM0 mode (right) based on 22 resonators, including (a) (d) box charts of propagation loss for different radius, (b) (e) histogram of >2500 resonances for estimating the most probable value, (c) (f) box charts of propagation loss in 1460-1610 wavelength range. The black solid lines show the median value and the shaded regions show the median absolute deviation.
Fig. 3.
Fig. 3. Absorption loss measurement via thermal-optic bistability analysis. (a) Skewed Lorentzian fitting to extract resonance shift. (b) Linear correlation between extracted resonance shift and on-chip power, revealing a thermal susceptibility χth of 837.33 MHz/mW. (c) Numerical simulation of steady-state temperature distribution of an 80 μm radius silicon nitride ring waveguide with absorbed optical power of 40.5 mW. (d) Simulated transient effective temperature Teff rise with time. The curve is fitted with exponential model to calculate thermal decay rate γT and thermal conductance Kc of the resonator. (e) Absorbance of thermal grown SiO2 and PECVD grown SiOx:H thin films by FT-IR spectroscopy.
Fig. 4.
Fig. 4. (a) Experimental setup diagram for frequency comb generation and measurement. TLS: tunable laser source. FPC: fiber polarization controller. TEC: temperature controller. DUT: device under test. OSA: optical spectrum analyzer. ESA: electrical spectrum analyzer. PD: photodetector. (b) Transmission spectrum in 1460-1610 nm wavelength range of the 80-μm-radius MRR. (c) Transmission of the selected pumping resonance around 1550 nm. (d) Initial state of parametric oscillation measured with on-chip pump power of 13.5 mW. (e) Spectra of the generated frequency combs and (f) the corresponding intensity noise.
Fig. 5.
Fig. 5. (a) Oscilloscope trace of pump transmission signal in a 100-μm-radius MRR with on-chip power of ∼100 mW. (b) Spectra of the generated frequency combs and (c) corresponding intensity noise, I-V refers to different pump detuning position in (a). (d) The enlarged solitonic crystal state (III) with ASE noise filtered out before light coupling into the waveguide. (e) 9-PSC and (f) 7-PSC generation in the SiNx:D MRR.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

Q i = 2 Q L 1 ± T min  =  2 π n g α λ 0
T = 1 γ 0 γ e x [ Δ ω 2 π f D ( 1 T ) ] 2 + ( γ 0 + γ e x ) 2 / 4
ζ = χ t h K C ( δ f r e s / δ T ) 1
d T e f f d t = γ T ( T e f f P a b s K C )
T e f f ( t ) = T ( r , t ) n 2 ( r ) | E ( r ) | 2 d 3 r n 2 ( r ) | E ( r ) | 2 d 3 r
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