Nonlinear high-harmonic generation in micro-resonators using a pump at telecom wavelengths is a popular technique used to extend the operating range of applicationfs such as f-2f and 2f-3f self-referencing systems and coherent communications in the visible region^1-16. Recent demonstration of visible emission generation in microring resonators (MRRs) via third-harmonic generation (THG) in a number of nonlinear integrated optical circuit platforms, such as silicon nitride^7-9, composite AlN/Si₃N₄^10-11, lithium niobite (LN) on insulator¹², and highly doped silica glass (HDSG)¹³, promises the potential of on-chip integration of a visible reference source in the advanced integrated circuits using these platforms. This is especially important in the development of compact and practical self-referencing and sensing systems.

The critical criteria for effective THG of visible emission in wavelength selective MRRs lies is the phase matching between the resonances of the pump mode and the third harmonic (TH) modes. Similar to that of the Kerr frequency comb generation^17-26, understanding and controlling the thermal behaviors of these modes remain a challenge in maximizing their THG efficiency²⁷, as both the linear and nonlinear thermo-optial (TO) effects impart refractive index variations of the waveguide^{19-26, 28-42}. For platforms with a positive linear TO coefficients, the linear TO effect gives rise of tens of pm/℃ temperature dependent wavelength shifts (TDWS) to the cold-cavity resonances of MRRs^17-19. It is possible to compensate the intrinsic linear TDWS by adding a cladding having a negative TO coefficient, such as polymer^31-36 or TiO₂^37-38 and liquid crystal³⁹ to offset the thermal dependency³⁰. These types of structures can achieve nearly athermal operation over temperature ranges of tens of degrees. However, the THG process must also account for the nonlinear TO effects as the localized heat generated by the high intra-cavity pump power can also introduce a redshift to the resonance^{18, 28-29}. While this nonlinear TO effect can be compensated by the photorefractive effect in LN waveguide devices^23-24 or by changing the structure of MRRs⁴¹ or by adopting the electrical or optical cooling approaches^{26, 42}, these compensation methods are either unavailable or add complexity to the design in the complementary metal oxide semiconductor (CMOS) compatible MRRs devices.

Therefore, it is necessary to explore other approaches in the realization of temperature insensitive operation that will also address effects such as Kerr nonlinearity induced phase shift in the determination of the phase matching condition.

This work presents the development of a dynamic model to investigate the thermal dependency of the THG phase matching condition, which is affected by the linear and nonlinear TO effects induced phase mismatch. By using this model, we indirectly measure the thermal coefficients as well as Q-factor of the visible mode. We also perform the steady state analysis to investigate the TDWS of the TH modes as well as the existence of athermal THG operation on the thermal mismatch between the pump mode and the TH modes. We are able to verify the model's predictions experimentally in a series of silicon rich HDSG add-drop MRR filters. We further classify the different types of visible TH modes from their THG efficiencies as well as TDWS. By precisely pairing the pump and the TH modes at different thermal mismatches, it is now possible to configure effective TDWS of the TH modes for a given MRR leading to athermal THG at different wavelengths.

In this work, we consider both linear and nonlinear TO effects of THG in MRRs. The discussion requires steady and uniform intra-cavity power distribution for both the pump and the TH waves, in which both optical power and net heat are assumed to be evenly distributed over the ring cavity, so that the averaged local thermal nonlinearity still correlates with the Kerr nonlinearity in the moving reference frame. Experimentally, this uniform intra-cavity power distribution is achieved by injecting continuous wave (CW) laser into the resonance with large normal dispersion. Different from generating Kerr frequency combs^17-18, the sweep speed of the pump wavelength has to be slow so that the cavity can be thermally self-stabilized.

When intra-cavity power of the pump I_p is much larger than I_t, the intra-cavity power of TH emission, it is reasonable to neglect the self-phase modulation (SPM), cross-phase modulation (XPM), and the pump depletion induced by the TH emission. By using the steady-state assumption for I_p and I_t, from the coupled thermal dynamics model for THG in MRRs (See Supplementary Discussion Ⅲ), we have¹³

$ $$\begin{array}{l}\left(-\mathrm{i}{\mathit{\Omega }}_{\mathrm{p}}-\mathrm{i}{\xi }_{\mathrm{p}}{\omega }_{\mathrm{p}0}\delta T-\frac{{\kappa }_{\mathrm{p}}}{2}\right)a_{\mathrm{p}}\\ -\mathrm{i}\left({\mathit{\Theta }}_{\mathrm{p}}+g_{\mathrm{p}\mathrm{p}}\right)I_{\mathrm{p}}{a}_{\mathrm{p}}-\mathrm{i}\sqrt{{\kappa }_{\mathrm{p}\mathrm{e}}}{p}_{\mathrm{i}}=0,\end{array}$$ $  (1)

$ $$\begin{array}{l}\left(-\mathrm{i}{\mathit{\Omega }}_{\mathrm{t}}-\mathrm{i}{\xi }_{\mathrm{t}}{\omega }_{\mathrm{t}0}\delta T-\frac{{\kappa }_{\mathrm{t}}}{2}\right)a_{\mathrm{t}}\\ -\mathrm{i}\left({\mathit{\Theta }}_{\mathrm{t}}+g_{\mathrm{t}\mathrm{p}}\right)I_{\mathrm{p}}{a}_{\mathrm{t}}=\mathrm{i}{g}_{\mathrm{T}\mathrm{H}}\hslash {\omega }_{\mathrm{p}}{a}_{\mathrm{p}}^3,\end{array}$$ $  (2)

where, |a_p|² = I_p/ħω_p and |a_t|² = I_t/3ħω_p corresponds to the photon numbers of the pump and TH emission, respectively, and ω_p is the angular frequency of input pump. Ω_p = ω_p-ω_p0 and Ω_t = 3ω_p-ω_t0 are the cold-cavity resonance detuning of the pump and TH wave, respectively, where ω_p0 and ω_t0 are the cold-cavity resonance frequencies of pump and TH modes at temperature T₀, respectively. Here, δT = T - T₀ is the difference between temperature T and T₀. ξ_p and ξ_t are the linear TO coefficients of the pump and TH modes, respectively²⁸, κ_p and κ_t are the overall losses of the pump and TH modes, respectively. Here, we define Θ_p and Θ_t as the nonlinear TO shift rate of the pump and the TH emission in rad/J, (See Supplementary Discussion Ⅲ), respectively. g_pp is the nonlinear factors of the pump and g_tp is the XPM factors of TH emission and g_TH is the growth rate of the THG. In Eq. (1), κ_pe is the external coupling rate of the pump and |p_i|² = P_i/ħω_p refers to the photon number of P_i. In Eq. (2), XPM effect dominates the Kerr nonlinearity.

The overall THG efficiency depends strongly on the phase matching condition between the pump and the TH modes. It is more convenient to express the phase matching condition in terms of the phase mismatch Δβ_total between these modes, which can be approximately expressed as

$ \Delta {\beta _{{\rm{total}}}} \approx \frac{{\Delta {n_{\rm{g}}}}}{c}\left[ {{\mathit{\Omega }_{\rm{t}}} + {\mathit{\Omega }_{{\rm{TL}}}}(\delta T) + {\mathit{\Omega }_{{\rm{TNL}}}}(\Delta T) + {\mathit{\Omega }_{{\rm{KNL}}}}} \right], $  (3)

where, the terms Ω_TL = ξ_tω_t0δT, Ω_TNL = Θ_tI_p, and Ω_KNL = g_tpI_p represent the linear TO phase mismatch, the nonlinear TO phase mismatch, and the Kerr nonlinear phase mismatch in THG, respectively. Here, Δn_g is the difference of group velocity and the c is the speed of light in vacuum. From Eq. (2), we can analytically write the dependence of I_t of the TH mode as

$ {I_{\rm{t}}} = \frac{4}{3} \cdot \frac{{g_{{\rm{TH}}}^2I_{\rm{p}}^3}}{{\mathit{\kappa }_{\rm{t}}^2 + 4{{\left[ {{\mathit{\Omega }_{\rm{t}}} + {\mathit{\Omega }_{{\rm{TL}}}} + {\mathit{\Omega }_{{\rm{TNL}}}} + {\mathit{\Omega }_{{\rm{KNL}}}}} \right]}^2}}}. $  (4)

As shown in Eq. (4), the maximum efficiency occurs at its minimum mismatch. We also notice that, due to the nonlinear phase mismatches, the intensity of THG signal is proportional to the intra-cavity energy instead of the input pump power. Therefore, we should define$\eta = 4g_{{\rm{TH}}}^2/\left( {3\kappa _{\rm{t}}^2} \right)$as the overall THG efficiency in J^-2. For HDSG MRRs, since the calculated Kerr nonlinear coefficient of the TH emission γ_tt =1.35 W^-1·m^-1 @ 517 nm, we can neglect the Ω_KNL term with a factor g_tp < 6.24×10⁷ pJ^-1 in Eq. (2) in the analysis which is two orders of magnitude smaller than Θ_t as shown in Table 1⁴³.

Figure 1(a) illustrates the individual contributions to the cavity resonance tuning due to the linear and nonlinear TO effects. It shows that while the Ω_TNL can only produce a redshift, Ω_TL can produce a red or blue shift depending on whether the temperature is ramped up or down^{10, 19-22}. From the nearly phase matching assumption, i.e. Δβ_total ≈ 0, we have Ω_TL + Ω_TNL = ξ_tω_t0δT + Θ_tI_p ≈ - Ω_t from Eq. (3) which implies a decrease of δT resulting in a negative Ω_TL that can be used to compensate the increase of Ω_TNL, as shown in Fig. 1(b). A better way to understand the interplay between the thermal behaviors of the pump and TH modes is to consider the thermal dynamic of these modes separately. We can denote τ_i = ξ_iω_i0/Θ_i, i = p, t, as the ratio between the linear TO compensation rates of decreasing δT and the nonlinear TO shift rate induced by I_p for the pump and TH modes, respectively, as the measure of the effective TDWS of the modes. It is now possible to map the δT-λ_p relationship and plot the TDWS of TH (red dot lines) and the pump mode (blue lines) separately as shown in Fig. 1(b), where λ_p is the pump wavelength to obtain the resulting effective TDWS of the TH mode (red solid lines).

Figures 1(c)-1(e) show three scenarios of the effective TDWS of the TH mode in the δT-λ_p diagrams at different values of τ_t. As long as the TH mode is thermally matched with the pump mode, i.e. the thermal mismatch Δτ = τ_p - τ_t ≈ 0, their resonances can be kept correlated while maintaining the necessary phase matching condition, without the need for any external compensation, as shown in Fig. 1(d). In contrast, the thermal mismatch between the pump and TH modes will cause misalignment between their resonances and affect the THG efficiency. Figures 1(c) and 1(e) show when it is under- and over-compensated, respectively.

In MRRs, the dependence of I_p to the input pump power P_i and the pump detuning Ω_p is a nonlinear relationship instead of the linear one^44-45. In the HDSG MRRs, the dominant nonlinear TO effect further reduces the nonlinear threshold by two orders of magnitude. In this case, only configuring a fixed Δτ may not keep the thermal matching condition over a broad range of temperature. From Eq. (2), the existence of stable athermal modes can be found by equating ∂Ω_t/∂T = 0. Applying this athermal condition into Eq. (1) leads to the following normalized equation which describes the dependence of Ω_p on P_i and I_p in nearly athermal THG via steady state analysis, (see Supplementary information Eq. S5.3).

$ {{F}^{2}}=\left[ 1+{{\left( \rho -\alpha \right)}^{2}} \right]\rho , $  (5)

where, α = 2Ω_p/κ_p is the normalized pump detuning which is proportional to Ω_p, ρ = -2Θ_pΔτδT/κ_p is the normalized intra-cavity pump power, and F² = 8Θ_pκ_peΔτP_i/(κ_p³τ_t) is the normalized intra-cavity pump power.

In Eq. (5), only when $\left| \alpha \right|>\sqrt{3}$, can there be three equilibria for one value of F²⁴⁵, which is the criteria for the existence of stable athermal TH modes. Figure 2(a) plots the nonlinear relationship between α and ρ using Eq. (5) with different F², in which the real roots of α are plotted as dashed lines. In Fig. 2(a), the green and blue solid lines show that α remains constant within a certain temperature range of δT, which implies the existence of stable athermal TH modes built up by fixed input pump powers. Moreover, different from the case of Kerr nonlinearity⁴⁵, in athermal THG, the opposite sign of α implies that the dominating nonlinear TO effect can generate stable athermal modes on the blue edge of the pump resonance that are accessible by the sweeping of the pump signal.

A series of HDSG four-port single ring MRR add-drop filters are fabricated for the investigation in which a ring resonator with radius of R = 135 μm is side coupled to an input and output bus waveguide as shown in Fig. 2(b). The cross-section of both the rings and the bus waveguides are 2 μm × 1 μm, which consist of highly doped glass of refractive index 1.70 and cladded with silica. The Q-factors of the MRRs are varied by using a gap separation of 0.8 μm (R-1, R-3) and 1.0 μm (R-2), respectively. The corresponding transmission spectrum and the Q factors of the devices are shown in Figs. 1(a) and (b) as well as table 2 in ref.¹³, respectively. The advantage of the device having a drop port is to allow the direct monitoring of I_p. All the devices are pigtailed to fiber arrays where the coupling loss at the pump wavelength is between 0.75 and 1 dB/facet. It is important to note that the coupling coefficient between the ring and the bus is between 2% to 7% at the pump wavelength but there is virtually no coupling at the TH wavelength, as can be seen from the TH emission photos in Fig. 2(c), where only the ring is lit up during THG. Additional details on the fabrication of the MRR devices can be found in refs.^46-48.

The schematic of the experimental setup for monitoring the THG emission is shown in Fig. 2(d). A CW pump from the TLS, (Agilent 8164A) is amplified by an EDFA (Amonics, AEDFA-CL-30-R-FA). An OBF is used to reject the amplified spontaneous emission (ASE) of the amplified pump before it is launched into the input port of the MRR add-drop filter in the experiment. Since the THG emission cannot couple to the bus waveguide, it is necessary to collect the emission from the top of the ring by a collimator (Thorlabs F671SMA), then the collected signal is transmitted to the spectrometer (Ocean Optics, 0.38 nm resolution) via an optical fiber for the spectral analysis. A large area silicon detector (Thorlabs DET100A) is used to calibrate the emission collection system, where the detector is placed directly on the emitting ring with the collected power as the reference. Next, the power is compared to the photon counts from the spectrometer for the calibration. A TEC is used to control the on-chip temperature T of MRR device in the various stages of the experiment.

By using the experimental setup shown in Fig. 2(d), the response of the TH mode emission at various temperatures T are obtained by scanning the pump signal across the free spectral range (FSR) of the MRR while maintaining the device temperature constant with the TEC¹². The measured wavelength of maximum TH emission at various T of the different TM TH modes at two adjacent resonances of the MRR near 1550 nm is shown in Fig. 3(a). Fig. 3(b) shows the measured maximum intensities of the TH modes of Fig. 3(a) as a function of I_p. The result shows that it is feasible to classify these modes according to their THG efficiencies, with values in the order of 1.0 nJ^-2, 0.05 nJ^-2, 0.001 nJ^-2, as type Ⅰ, type Ⅱ, and type Ⅲ, respectively. Here, we calculated the THG efficiencies of type Ⅰ TH modes by using ${I'_{{\rm{tmax}}}} = {\mathcal{R}}{I_{{\rm{tmax}}}} = 4{\mathcal{R}}g_{{\rm{TH}}}^2I_{{\rm{pmax}}}^3/\left( {3\kappa _t^2} \right) = {\mathcal{R}}\eta I_{{\rm{pmax}}}^3$, with η as the THG efficiency. Since we can only measure the scattered signal of the TH emission instead of directly measuring the intra-cavity TH emission energy, η cannot be fully determined due to the unknown overall photodiode responsivity of the collimator$\mathcal{R}$. The measured THG efficiencies ${\cal R}\eta $ are 0.937 nJ^-2 in R-1 and 0.454 nJ^-2 in R-2.

Besides the THG efficiencies, each type has its own unique values of ξ_t and Θ_t that can be extracted from the effective TDWS measurement at two or more resonance locations of the modes. The calculated ξ_t and ξ_t of the type Ⅰ and type Ⅱ TH modes in table 1 show that the linear TO coefficient ξ_t of these TH modes have similar values as the pump mode ξ_p where all four TM-TM THG cases having negative thermal matchings, with Δτ < 0. This is caused by their linear TO phase mismatches over-equalizing the nonlinear TO phase mismatches. For these TH modes, their TDWS rates d_t are proportional to the thermal mismatch -Δτ.

For HDSG MRRs, the measured nonlinear TO shift rate Θ_p of the TM pump mode decreases more rapidly than the TE mode across the C- and L-bands, (see Supplementary Fig. S2(b). The result indicates different polarization combinations of the pump mode and the generated TH mode give rise to Δτ with different signs, which can lead to a positive, negative, or even zero TDWS. In MRRs R-1 and R-2, with 200-300 mW TM pumps, TE athermal TH modes with TDWS of |3d_t| ≤ 0.62 pm/℃ can be achieved as shown in Figs. 4(a) and 4(b). Figure 4(c) compares the measured photon count on the peak of TH modes as a function of I_p. As shown in Fig. 4(c), in device R-2 the generated TE athermal TH mode has a THG efficiency of ${\cal R}\eta $= 0.0226 nJ^-2, which is one order of magnitude smaller than that of type Ⅰ TM-TM THG combination. We also notice that the measured THG efficiency of the TE athermal modes is proportional to the Q-factor of MRR. The device R-2 with higher Q-factor can achieve THG efficiency one order of magnitude greater than that in R-1.

To verify the results, a second device R-3 is measured, having the same design as R-1. As shown in Fig. 4(c), the two measured TE athermal TH modes in R-3 have similar THG efficiencies as R-1, which indicates that they belong to one TH mode family. The measured spectra of these two athermal TH modes in MRR R-3 generated by TM-TE combination were shown in Fig. 5(c) and Fig. 5(f). In the same device, Fig. 5(a) shows that, with the TM pump mode, the generated TM TH visible mode yields d_t = 2.35 pm/℃ which is close to that of R-2 shown in Table 1. In this case, a negative Δτ leads to a positive TDWS. For the TE-TE combination in Fig. 5(b), it gives rise to a negative TDWS of d_t = -2.84 pm/℃ due to the under-compensation of the linear TO effect. The results indicate that the TE-TE and TM-TM combination having opposite TDWS trends. Athermal operation with perfect thermal matching Δτ ≈ 0 can be achieved with the TM-TE combination in R-1 (Fig. 4(a)), R-2 (Fig. 4(b)), and R-3 (Figs. 5(c) and 5(f)). Figure 5(c) show an TDWS at the TH wavelength of only |d_t| ≤ 0.09 pm/℃ can be achieved in R-3, which is a clear demostration of the use of linear TO phase mismatch to equalize the nonlinear TO phase mismatch when thermal matching occurs. This also shows that by selecting different combinations of the pump and TH modes, one can control the direction of the TDWS via configuring the thermal mismatch Δτ in these HDSG MRRs.

For athermal TH modes, since the measured pump wavelength slope is less than the resolution of the OSA of 1 pm, it is not possible to calculate ξ_t and Θ_t accurately as in Fig. 3(a). From Eq. (2), the nearly phase matching assumption of the athermal modes can be used to determine the pump detuning Θ_p and T₀. Figures 5(c) and 5(f) indicates the two TE TH modes belong to the same type of TH mode in MRR R-3 and have similar ξ_t in the L-band. By assuming that the ratio of ξ_t to Θ_t in R-3 is the same as that in R-1 as shown in table 1, we get a trial value of ξ_t = 9.0×10^-6 /℃. From the data of Figs. 5(c) and 5(f), we can obtain Δτ = 0.36 pJ/℃ and Θ_t = 2.3 ×10⁹ rad/pJ for Fig. 5(c) and Δτ = 0.29 pJ/℃ and Θ_t = 1.9 ×10⁹ rad/pJ for Fig. 5(f) by using the estimated ξ_t.

One can also estimate Q_TH of the TE TH mode by using the extracted Θ_t and ξ_t along with the assumption of Δτ ≈ 0. Figs. 5(d) and 5(g) show the measured spectrum and fitted curve using Eq. (4) and the estimated values of Δτ and Θ_t. As shown in Figs. 5(d) and 5(g), both of the athermal TH modes have a Q_TH of 3.5 ×10⁵, which is close to the intrinsic Q of the TH mode and smaller than Q_p = 4.32 ×10⁵ of the pump mode.

For the HDSG MRR, the measured Θ_p is approximately two orders of magnitude higher than the shift rate from the Kerr effect. Such difference is about 10 times greater than that in the Si₃N₄ MRR¹⁷. Meanwhile, both of these platforms have similar linear TDWS at around 20 pm/℃^13-18. It indicates that in the HDSG MRRs, the nonlinear TO effect can redshift the resonance of pump mode over a much broader bandwidth of about 10 times compared to the Si₃N₄ MRR. This covers more TH mode resonances and makes thermal matching easier to be achieved by finding a close to zero Δτ in the HDSG MRRs. We notice that MRRs with different gap separations between the bus and ring, coupling rates, and Q-factors, also lead to different wavelength dependent values of Θ_p¹⁷.

The model in Eq. (5) also provides insights to maximize the Θ_p value for broadening the temperature range of stable atheraml THG. Since the real roots of α in Eq. (5) only exist when |F²| ≥ |ρ| and the smaller roots cannot lead to an athermal mode, from $\alpha = \rho \pm \sqrt {{F^2}/\rho - 1} $, we have

$ {\mathit{\Omega }_{\rm{p}}} = {\mathit{\Theta }_{\rm{p}}}\Delta \tau \left| {\delta T} \right| + \sqrt {\frac{{{\kappa _{{\rm{pe}}}}{P_{\rm{i}}}}}{{{\tau _{\rm{t}}}\left| {\delta T} \right|}} - \frac{{\kappa _{\rm{p}}^2}}{4}} . $  (6)

Equation (6) determines the dependence of athermal TH resonance frequency on temperature fluctuation δT, in which the first and second terms on the right-hand side of Eq. (6) represents the linear TO effect and the nonlinear TO effect, respectively. The increase of |δT| can simultaneously enhance the first term and reduce the second, and vice versa, which gives a similar form as the bandgap reference in electrical circuit design⁴⁹. With a non-zero and positive Δτ, the linear TO effect can equalizes the nonlinear TO effect over a certain temperature range and provide temperature independent frequency reference when the pump frequency is fixed. In contrast to the external linear feedback control scheme^{17, 42}, the thermal matching can automatically retain the phase matching of THG in the athermal mode.

Equation (6) also shows the operating temperature range of athermal THG can be determined by setting the calculated ∂Ω_p/∂(δT) to zero, which gives an approximately inversed cubic relationship between δT and τ_t. Since the values of ξ_t are on the same order of ξ_p, a close to zero τ_t can easily be obtained with a large Θ_p value, which is in the same order as Θ_t for broadening the operating temperature range. Contrarily, although the athermal THG still exists at small Θ_p values, it shrinks the operating temperature range to less than one degree and severely limits its effectiveness.

Using the parameters extracted from the measured athermal TH modes, the calculated α as functions of ρ at a fixed Δτ are plotted in Figs. 5(e) and 5(h) using Eq. (5). The gray squares in Figs. 5(e) and 5(h) are from the measured result of Figs. 5(c) and 5(f), respectively. It shows a non-zero and positive Δτ (0.3 pJ/℃) can achieve perfect power insensitive athermal THG with less than hundreds of MHz variation within 12 ℃ and 6 ℃ ranges, respectively. This range is comparable to the reported passive athermal TDWS approaches for MRRs^{34, 38} as shown in table 2. In Figs. 5(e) and 5(h), note that the variation of their wavelengths is only a few picometers, which is close to the accuracy limit of TLS and OSA.

In conclusion, we present a comprehensive study of the thermal behavior of the visible modes in micro-resonators generated by the THG process pumped with telecom wavelengths. A dynamic model that includes both the linear and nonlinear TO effects is used to explain the pump and third harmonic resonance shifts for a given input pump power in the MRRs. Using this model, we predict and experimentally demonstrate visible third harmonics modes having TDWS between -2.84 pm/℃ and 2.35 pm/℃ when pumped at the L-band. By precisely matching the pump and the TH modes at different thermal mismatches, it is now possible to configure effective TDWS for a given MRR leading to athermal THG at different wavelengths. We have also identified orthogonally pumped athermal visible TH modes with a TDWS of 0.05 pm/℃ over a temperature range of 12 ℃. In table 2, we compared the proposed deterministic athermal THG scheme with other athermal approaches which use a negative TO coefficient overlay to compensate linear TO effect or equalize the nonlinear TO effect by optimizing the structure of MRRs. Unlike the previous works, our proposed scheme uses only one CW laser pump with fixed wavelength and power to directly generate athermal TH modes in a silicon rich CMOS-compatible MRRs without any external compensation scheme. This configurable THG approach can generate athermal visible emission mode with high Q-factor, approximately-zero TDWS, tens of degree temperature range, and no operation bandwidth limitation. This finding further promises the realization of temperature insensitive highly efficient THG for potential 2f-3f self-referencing in metrology, biological and chemical sensing applications.

We are grateful for financial supports from the Natural Science Foundation of Fujian Province (Grant No. 2017J01756); National Natural Science Foundation of China (Grant No. R-IND12101, No. 61675231); Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB24030300).

S. H. W. developed the original concept. B. E. L., R. R. D. and X. W. designed and fabricated the integrated devices. Y. L. performed the experiments. Y. L. and S.T.C. contributed to the development of the experiment. S. H. W., S. T. C., and B. E. L. contributed to the writing of the manuscript. L. W. and S. T. C. supervised the research.

The authors declare no competing financial interests

Supplementary information for this paper is available at https://doi.org/10.29026/oea.2020.200028.