1. Introduction

Visible emission from micro-resonators by third-harmonic generation (THG) processes pumped at telecom wavelengths have been demonstrated in a variety of platforms, such as silica [1], silicon [2], LiNbO₃ [3], silicon nitride [4], and composite AlN/Si₃N₄ [3,5], etc. Silica is among the first of these platforms that have demonstrated THG emission due to the ability of making very high Q-factor fused silica cavities. THG and sum-frequency generation (SFG) have been demonstrated in silica microtoroids [1], microspheres [6] and microbottle [7]. These works lay the foundation of silicon microphotonic emitters with emission expanded to the visible band while keeping the pump power at a manageable level. While these cavities can achieve extremely high Q-factors of up to a billion, their intrinsic nonlinearity which is orders of magnitude lower than other nonlinear materials such as LiNbO₃, Si₃N₄ and Si, limits their emission conversion efficiencies. On the other hand, silicon rich complementary metal–oxide–semiconductor (CMOS) compatible platforms such as silicon (Si) and silicon nitride (SiN) have much higher nonlinearity and they have been used to demonstrate THG emission in resonance structures such as micro-ring resonator (MRR) [4,8] and photonic crystals [2] with much higher efficiency than the silica cavities. For example, utilizing THG and SFG, Wang et. al. [8] have demonstrated a visible frequency comb based on the SiN MRR that phase matched the pump mode with multiple visible higher order modes to obtain more efficient phase matching than with a single visible higher order mode.

One nonlinear optics platform that has gained attention in recent years is the highly-doped silica glass [9,10]. The CMOS-compatible highly-doped glass has an adjustable refractive index between 1.45 and 1.9 in the C-band [11]. Although its nonlinearity is an order of magnitude lower than silicon, it has negligible nonlinear loss in the telecom wavelengths leading to a very high nonlinear figure-of-merit, with Kerr nonlinearity that is approximately 5 times that of silica [10]. Moreover, its mature fabrication process and precise dispersion controllability make this platform promising and robust.

In this work, THG in highly-doped silica glass MRRs is demonstrated for the first time. In the dynamic phase-matching process, one can achieve momentum conservation for THG by carefully adjusting the dispersion as well as utilizing the thermal detuning. Since the redshift introduced to the cold resonance from the thermal effect of the pump can affect the accuracy of the measurement, monitoring the exact in-cavity power in the experiment is vital in the account of the actual power consumption. In the experiment, a thermoelectric cooler (TEC) is used to control the global temperature of the device, by tuning the global temperature and sweeping the pump wavelength with a continuous-wave (CW) laser, the THG power dependence of the in-cavity power is thoroughly investigated. Maximum conversion efficiency is achieved when TH mode overlaps the pump cavity mode, along with the THG wavelength aligns with the pump wavelength simultaneously [12]. Moreover, a thermal nonlinear model is developed to calibrate the TH resonance wavelength by subtracting the pump intensity induced thermal nonlinear shift. This work enables new capability of frequency converter for tunable on-chip visible laser sources and provides additional insights into the mechanism of the THG emissions.

2. Device description

A series of three highly-doped silica glass MRRs were fabricated to investigate the THG dependency with Q-factors. The devices under investigation consist of identical rectangular core dimension of 2 µm by 1 µm, core index of 1.70, and ring radius of 135 µm surrounded by silica. The waveguides were fabricated using the CMOS-compatible process described in [9–11]. We varied the Q-factors of the MRRs by adjusting the gap separation between the bus and the ring. Figures 1(a)-1(c) show the measured responses of the fabricated devices. The measured coupling losses for all three devices are less than 1 dB per facet where the extracted loaded Q-factors are 5.9×10⁵, 8.5×10⁵ and 1.4×10⁶ for MRR with gap separations of 0.8 µm, 1.0 µm and 1.2 µm, respectively. It is worth noting that the MRR with gap separation of 0.8 µm is near critically coupled while the other two are under coupled. Figure 1(e) shows the calculated dispersions for the TE and TM modes indicating both modes are in normal dispersion at around 1550 nm. This allows us to simplify the interpretation of the results as there is only a one to one correspondence between the pump and the emission wavelengths. We also measured the nonlinear coefficient of the waveguide with a one meter long waveguide having the same waveguide geometry as the MRR and from the same wafer. Using the four-wave mixing (FWM) method ascribed in [13], the measured FWM conversion efficiency times nonlinear coefficient Re{γ} is plotted as a function of the wavelength detuning between pump and signal, shown as in Fig. 1(f). By fitting the curve to the theoretical formula, Re{γ} is obtained to be 310 W⁻¹km⁻¹.

 

Fig. 1. Characteristic of MRRs with cross-section geometry of 2 µm × 1µm. (a-c) Filter response (Black-dotted: TM drop; Magenta-dotted: TM through; Red-solid: drop fitting; Blue-solid: through fitting) characteristic of MRR with gap value of 0.8 µm (a), 1.0 µm (b) and 1.2 µm (c), respectively. (d) Dependence of loaded quality factor on the gap. (e) Calculated dispersion of the fundamental quasi-TE and quasi-TM mode. Inset: Fundamental quasi-TM mode profile. (f) Nonlinearity coefficient Re(γ) times the normalized four-wave mixing efficiency η, versus wavelength detuning Δλ.

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3. C + L bands TH mode measurement

3.1 Simulation and experimental setup

The effective indices of the fundamental TE and TM pump modes, along with the TH modes are simulated using FemSIM. Figure 2(a) shows the simulated mode spectra for both the pump and TH modes and reveals that suitable phase-matching conditions for THG and pump modes occur between the 15^th and 19^th TH higher order modes, shown in Fig. 2(b), in the 180 to 200 THz range, covering the C + L bands.

 

Fig. 2. (a) Effective refractive index for the fundamental pump mode and higher order TH modes of the 200 GHz MRR. (b) Mode filed distribution for the higher order modes. (c) Schematic experimental setup for the THG. A collimator is putted on top of the ring resonator to collect scattered light. (TLs: tunable laser source; OBF: optical bandpass filter; PM: power meter.) (d) 1548.85 nm pump and corresponding 516.15 nm THG spectra. Inset: Photograph of green emission from THG under pump power of ∼200 mW.

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Figure 2(c) shows the experimental setup where a tunable single-frequency CW laser (HP 81680A) with linewidth ∼100 kHz amplified by a C and L bands high power erbium-doped fiber amplifier (EDFA) (Amonics, AEDFA-CL-30-R-FA). A bandpass filter and a polarization controller between the EDFA output and the input of the MRR are used to filter out the unwanted ASE and for the selection of the desired polarization. Since there is negligible coupling between the bus waveguides and the ring resonator at the THG wavelength for the selected gap separations, the resonator is effectively isolated from the buses at the TH frequency with no THG signal coupled to the output waveguide. Thus, it is necessary to collect the THG emission from the top of the MRR with a collimator (Thorlabs, F671SMA). The collected emission is then transferred to the optical spectrometer (Ocean Optics, USB2000+, 0.38 nm stepsize) by an optical cable for the determination of both the intensity and frequency of the signal. The emission collection system is calibrated using a reference silicon detector (Thorlabs, DET100A, 400−1100 nm). The large area detector is first placed less than 5 mm directly above of the ring resonator and the detected photocurrents and powers at different pump powers are mapped to the photon counts obtained by the emission collection system. Tap couplers at the input and output of the MRR are to monitor the input and output powers, respectively, while the optical spectrum analyzer at the output is to measure the pump spectrum. Figure 2(d) shows the measured pump and visible spectra at 1548.85 nm and 516.15 nm, respectively, along with an image of the emission at pump power of approximately 200 mW.

3.2 THG conversion efficiency

Figure 2(d) indicates that the simulated mode spectra in Fig. 2(a) have correctly predicted the observation of THG emission having the suitable phase matched condition when pumped in the C band. However, the measured TH wavelength of 516.15 nm is not exactly one-third of the pump wavelength of 1548.85 nm, rather it is slightly blue-shifted due to the thermally induced resonance shift from the pump power. To obtain a more accurate THG conversion efficiency measurement, the pump laser is then swept across a wider range from 1534 nm to 1584 nm in small steps to monitor the behavior of the THG emission with respect to the in-cavity pump power. Figure 3(a) shows the detected pump power at the drop port across the swept wavelength range. At each resonance, the drop power increases with the swept wavelength due to the thermal induced resonance shift. This build-up of in-cavity power reaches a maximum before dropping to zero when the pump wavelength finally walks off from the resonance. It is interesting to note that the THG emission spectrum in Fig. 3(b) shows that TH modes can be observed at almost every resonance, with some resonances having multiple peaks correspond to phase-matched to multiple TH modes. Maximum THG emission occurs when the phase-matching condition is perfectly satisfied, where the pump wavelength, pump cavity mode, THG wavelength, and TH mode overlap completely. There are now three types of detuning: the detuning of pump wavelength and pump cavity mode, the detuning of THG wavelength and TH mode, and detuning of pump cavity mode and TH mode, all of which contribute to the intensity of the THG emission.

 

Fig. 3. C + L bands phase-matching condition and measured THG power as a function of detuning. Drop-port power (a) and THG intensity (b) dependence on pump wavelength at swept step of 2 pm. Measured cold resonances at 25 °C are plotted in gray line for quasi-TM polarization. Note that the discontinuity of x-axis between 1562 nm and 1566 nm is due to the unstable pump power in that range probably caused by mode hopping.

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Although the relationship between the maximum THG efficiency and the in-cavity pump power is cubic, large nonlinear coefficient γ can introduce strong nonlinear phase shift to the in-cavity pump energy and the in-cavity pump power may not be proportional to the input pump power [14]. Here, the drop power is used to monitor the in-cavity pump energy as they have a more straightforward relationship [15]. When sweeping the pump across a resonance, the drop response exhibits thermal broadening as shown in Fig. 4(a) where the extent of the broadening depends directly on the input power. In principle, phase matching between the pump and the TH modes can occur anywhere within the response and the TH emission power varies depends on where the phase matching occurs within the response. Subsequently, the emission power depends on the separation between the phase matching location and the cold resonance. Weaker emission will be generated if the location is closer to the cold resonance, while stronger emission will be produced when the location is closer to the apex of the broadened response. In order to locate the perfect phase matching condition, a thermal electric controller is employed to control the global temperature of the device and the cold resonance location of the resonator. It is important to note that a change of the global temperature will shift the resonances of both the pump and TH modes, although these modes have slightly different thermal coefficients. Figure 4(b) shows the measured THG emission responses across the same TM pump resonance at around 1548.4 nm at global temperatures between 19 °C and 43 °C showing the Lorentzian responses with their peak emissions decrease with increase global temperatures.

 

Fig. 4. Experimental C-band phase-matching condition and measured TH power as a function of detuning. (a) Normalized cavity thermal shift under pump power of 219.1 mW. Cold resonance is marked as blue line. Power response recorded from drop port is denoted as maroon dots and solid line. (b) Pump wavelength sweeping under different global temperatures, and its corresponding THG power. (c) THG power dependence of in-cavity power for MRRs with different gap values. (d) Microscope photo of the green emission from the three MRRs simultaneously. Gap sizes are marked below each MRR.

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The TH emission scans at various global temperatures were performed for all three MRRs at resonances around 1584 nm. Note that all the thermal behaviors studied here are in steady-state [16]. Figure 4(c) shows the measured THG power as a function of the in-cavity power for the three MRRs. All three curves show the expected cubic relationship in the third-harmonics process and with the detected TH emission power increases with Q values. Here, the in-cavity pump power ${P_c}$ is estimated from dividing the measured drop port power ${P_d}$ by its coupling coefficient [15]. The pump powers in the bus waveguide P_i are 219.1 mW, 273.6 mW and 389.6 mW, corresponding to the extracted maximum on-chip conversion efficiencies, $\eta = {P_{TH}}/P_i^3$, of 5.3×10⁻⁶ W⁻², 8.9×10⁻⁶ W⁻², 2.7×10⁻⁵ W⁻², for gap separations of 0.8 µm, 1.0 µm, 1.2 µm, respectively. With the converted TH power is proportional to ${(\textrm{Q}/V)^3}$ where V is the optical mode volume [1], the expected conversion efficiency ratio of the three MRRs should be 1.0: 3.0: 13.4, instead the extracted ratio from the experiment of 1.0: 4.7: 132.2, where the highest Q value MRR does not agree well with the expected trend. It is believed that the difference is due to the underestimation of the high Q of the MRR in the linear measurement. In comparison, the highest measured conversion efficiency of the MRRs with loaded Q-factor of 10⁶ is three orders of magnitude lower than that of the silica microsphere with Q-factor of 10⁷ [6], five orders of magnitude higher than that in Si₃N₄ MRR with imperfect phase-matching condition [4], and two orders of magnitude lower than the results in [8] which demonstrates frequency comb in the green using SiN miroresonators. A table of comparison among the different platforms is presented in Table 1.

Table 1. Quality factor and conversion efficiency for optical platforms demonstrating harmonic generations.

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The fact that the experimental setup in Fig. 2(c) does not collect the TH emission directly from the waveguide output might not capture all the TH emission and have greatly underestimated the conversion efficiency. Here, we introduce a new parameter $\Re $ which is the ratio of the detected power to the overall in-cavity TH power to account for the underestimation. The maximum conversion efficiency $\eta $ can then be expressed as ${P^{\prime}_{TH}} = \Re {P_{TH}} = \Re \eta P_i^3$, where ${P^{\prime}_{TH}}$ and ${P_i}$ are the measured scattered TH emission power and input pump power, respectively. The obtained maximum conversion efficiency including $\Re $ is listed in Table 2. Note that Q_IP and Q_LP represent intrinsic and loaded quality factor at the pump wavelength, respectively.

Table 2. Conversion efficiency of 200 GHz MRRs with different gap separations.

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4. Calibrated THG resonance shift using pump induced phase mismatch

Figure 5 shows the measured thermal nonlinear coefficients Θ as functions of the pump wavelength over C + L bands for the three MMRs. As shown in Fig. 5, Θ is inversely proportional to the wavelength of cold resonance, i.e. proportional to ${\omega _0}$, which is consistent with Eq. (8) in the appendix.

 

Fig. 5. Measured thermal nonlinear coefficient Θ versus TM pump cold resonance at 25 °C.

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Moreover, the uncertainty of the measured visible wavelength is limited by the resolution of the spectrometer of 0.38 nm and is included in the error bar in Fig. 6(a). As discussed in the Appendix, thermal nonlinear and Kerr nonlinear induced phase mismatch ${\Omega _M}$ caused TH maximum blue-shift was shown in Fig. 6(a). If we assume the pump and TH wave have the same thermal nonlinear coefficient, the phase matching condition of THG can be approximately written as

 

Fig. 6. (a) Measured and calibrated THG wavelength versus pump wavelength. (b) The corresponding in-cavity pump energy at the THG wavelength of every THG emission band maximum.

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5. Conclusions

Visible green emission through nonlinear process of THG when pumped at telecom wavelengths is demonstrated in a series of highly-doped MRRs having various coupling coefficients. The behavior of THG emitted power as a function of the in-cavity pump power is investigated through the dynamic thermal model to obtain the maximum TH emission powers and verified from their cubic relationship with the in-cavity pump power. Thermal nonlinear coefficients are experimentally derived across the C + L bands, which are used to calibrate the cold-cavity THG resonance shift by subtracting the phase mismatch caused by thermal nonlinear effect from the measured TH wavelength, returning the ratio of the pump and the calibrated TH wavelengths to the three-fold relationship. Although the devices presented in this work cannot couple the THG output to the waveguide, with proper waveguide dispersion engineering, the emission from the top of the ring can be used as a high quality illumination backlight 2f-3f referencing frequency comb source with applications in microscopy and bio-imaging. Ongoing work addresses the THG coupling of ring directly to waveguide.

Appendix A: Thermal nonlinear model of the pump

As mentioned in Section 3.2, the observed TH resonances blue-shift is caused by the thermal nonlinearity introduced phase mismatch. For a better understanding of this phenomenon, we developed a thermal nonlinear model of the pump to describe the phase mismatching caused by the surrounding temperature, in-cavity pump energy induced thermal nonlinear effect and Kerr nonlinear effect.

When considering the thermal effects, according to [16], the resonance wavelength ${\lambda _r}(T + \Delta T) \cong {\lambda _0}[1 + \xi (T - {T_0} + \Delta T)]$ is a function of the temperature difference ΔT and the thermal coefficient ξ, where ${\lambda _0}$ is pump cold resonance wavelength at ${T_0}$. While the thermal coefficient is affected by thermal expansion ɛ and thermal index change, and can be expressed as $\xi = \varepsilon + \frac{1}{{{n_0}}}\frac{{dn}}{{dT}}$. Thermo-optic coefficient $dn/dT$ is measured to be 1.74×10⁻⁵ (1/°C) [18]. Here, $\Delta T$ corresponds to high pump power introduced on cavity net heat, which is determined by the generated heat ${q_{gen}}$ as well as diffused heat ${q_{dif}}$ and can be described by a rate equation [16,19]:

(2)$$C\frac{{d\Delta T}}{{dt}} = \frac{{d{q_{gen}}}}{{dt}} - \frac{{d{q_{dif}}}}{{dt}} = {P_c}\frac{{{\textrm{Q}_P}}}{{{\textrm{Q}_{abs}}}} - U\Delta T(t)$$

where, C denotes the heat capacity in J/°C and ${P_c}$ is the in-cavity pump power. ${\textrm{Q}_{abs}}$ represents the Q-factor relating to the pump absorption. Here, $U = K{S_{eff}}/{L_{eff}}$ indicates the effective thermal conductivity between the cavity mode volume and the surrounding, in unit of W/m·°C. While K is material thermal conductivity and ${S_{eff}}$ denotes the area of effective surface of the in-cavity pump mode and ${L_{eff}}$ is the effective length from the cavity mode volume to the surrounding. From Eq. (2), we can get the rate equation of ΔT as a function of in-cavity pump power and heat diffusing rate:

(3)$$\frac{{d\Delta T}}{{dt}} = \frac{{{\textrm{Q}_P}}}{{C{\textrm{Q}_{abs}}}}{P_c} - \frac{U}{C}\Delta T(t)$$

If we define the pump frequency detuning from ${\omega _0} \equiv 2\pi c/{\lambda _0}$ as $\Omega (\Delta T) = {\omega _P} - {\omega _0}$, where, ${\omega _P}$ denotes pump angular frequency. The thermal resonance shift can be written as

(4)$${\omega _r}(\Delta T) - {\omega _0} = 2\pi c\left( {\frac{1}{{{\lambda_r}}} - \frac{1}{{{\lambda_0}}}} \right) = \frac{{ - \xi (T - {T_0} + \Delta T)}}{{1 + \xi (T - {T_0} + \Delta T)}}{\omega _0}$$

where, ${\omega _r} \equiv 2\pi c/{\lambda _r}$ is the resonance peak angular frequencies at $T + \Delta T$. Note that since $\xi (T - {T_0} + \Delta T) \ll 1$, we thus yield ${\omega _r} - {\omega _0} \approx{-} \xi (T - {T_0} + \Delta T){\omega _0}$.

According to [5,19], if applied to the above dynamic thermal model, we can write a time-domain rate equation describing the dynamics of pump in MRRs as

(5)$$\frac{{d\varphi }}{{dt}} = [ - i\Omega - i\xi (T - {T_0} + \Delta T){\omega _0} - \frac{{\delta {\omega _0}}}{2}]\varphi - i\hbar {\omega _P}g{|\varphi |^2}\varphi - i\sqrt \kappa {p_i}$$

where, ${|\varphi |^2} = 2\pi R{P_c}/(\hbar {\omega _p}{\nu _g})$ corresponds to the photon numbers of pump wave and R is the radius of MRR. ${\nu _g}$ is the group velocity. In Eq. (5), the fourth term on the right-hand side of equal sign corresponds to self-phase modulation (SPM) effect. Meanwhile, the drop port output can be written as ${p_d} ={-} i\sqrt \kappa \varphi $, where, ${|{{p_x}} |^2} = {P_x}/\hbar {\omega _p},x = i,\textrm{ }d,$ are the amplitudes of the input pump and the drop port pump, respectively, while $g = \gamma \nu _g^2/2\pi R$ represents the nonlinear parameter, $\delta {\omega _0}$ and $\kappa $ denote the overall loss and the externally coupling rate, respectively.

Appendix B: Stability analysis

When using a CW pump which makes the pump power uniformly distributed in the cavity, a thermally self-stabilized system will be built up to maintain the equilibrium between the in-cavity power induced absorbed-heat and the dissipated-heat, in which we can assume $\frac{{d\Delta T}}{{dt}} = \frac{{d\varphi }}{{dt}} = 0$ and then we get the steady-state solutions of Eqs. (3) and (5) which looks like

(6)$$[ - i\Omega - i\xi (T - {T_0} + \Delta T){\omega _0} - \frac{{\delta {\omega _0}}}{2}]\varphi - i\hbar {\omega _p}(\Theta + g)\varphi = i\sqrt \kappa {p_i}$$

(7)$$\frac{{{\textrm{Q}_P}}}{{{\textrm{Q}_{abs}}}}{P_c} = U\Delta T(t) = \frac{{{\textrm{Q}_P}}}{{{\textrm{Q}_{abs}}}}\frac{{{\nu _g}{I_P}}}{{2\pi R}}$$

Where ${I_P} = {|\varphi |^2}\hbar {\omega _P}$ is in-cavity pump energy and

(8)$$\Theta = \frac{{{\textrm{Q}_P}\xi {\omega _0}}}{{U{\textrm{Q}_{abs}}}}\frac{{{\nu _g}}}{{2\pi R}}$$

is the thermal nonlinear coefficient of the pump in 1/J/s. From Eq. (6), the dependence of $\Omega $ to the surrounding temperature T and the in-cavity energy of the MRR can be written as

(9)$$\Omega ={-} \xi {\omega _0}({T - {T_0}} )- ({\Theta + g} ){I_p} \pm \sqrt {\kappa \frac{{{P_i}}}{{{I_p}}} - \frac{{\delta \omega _0^2}}{4}} $$

In Eq. (9), the first term on the right side is T induced phase mismatch ${\Omega _T} = \xi {\omega _0}(T - {T_0})$. The second term is in-cavity pump induced thermal nonlinear and Kerr nonlinear phase mismatch ${\Omega _M} = (\Theta + g){I_P}$ to ${\omega _0}$. Note that when $\Theta > > g$, the SPM terms can be neglected. Figure 7 plots in-cavity energy I_P as a function of Ω at different T using Eq. (9). The measured data are shown as different symbols which are well matched by the plotting curves of Eq. (9). Here, measured parameters were used for the plotting where $\xi = 1.28 \times {10^{ - 5}}\textrm{ (1/}^\circ \textrm{C)}$, $\Theta = 3.15 \times {10^9}\textrm{ (1/pJ)}$, and $\delta {\omega _0} = {\omega _0}/{\textrm{Q}_P}$. The results indicate that the measured pump resonance red-shifting is caused by the phase mismatches introduced by the TEC changing as well as the thermal nonlinear effect.

 

Fig. 7. In-cavity energy dependence of pump cold resonance detuning under different temperatures of TEC. Here, $\xi = 1.28 \times {10^{ - 5}}\textrm{ (1/}^\circ \textrm{C)}$, $\Theta = 3.15 \times {10^9}\textrm{ (1/pJ)}$ and $\delta {\omega _0} = {\omega _0}/{\textrm{Q}_P}$.

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Funding

National Natural Science Foundation of China (R-IND12101, 61675231); Natural Science Foundation of Fujian Province (2017J01756); Chinese Academy of Sciences (XDB24030300).

Disclosures

The authors declare no conflicts of interest.

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