Optical Kerr nonlinearity underlies many important nonlinear optical phenomena [1,2], such as self-/cross-phase modulation (SPM/XPM), soliton formation, and supercontinuum generation, that have found broad applications ranging from laser mode locking, frequency metrology, to quantum information processing. In combination with high-Q micro-/nano-cavities, the optical Kerr effect enables intriguing functionalites such as optical switching [3] and frequency comb generation [4], which have attracted significant interest in recent years. To date, a variety of materials have been explored for this purpose, from dielectrics such as silica [4], Si₃N₄ [5], CaF₂ [6], and MgF₂ [7], to semiconductors such as silicon [8], chalcogenide glass [9], III–V materials [10,11], and diamond [12]. However, optical materials generally exhibit a fairly weak third-order nonlinearity and thus a substantial optical intensity is required for exciting the nonlinear optical processes, which in turn is usually accompanied with other detrimental effects such as two-photon absorption, thermal perturbation, and/or mechanical instability. Therefore, a nonlinear optical medium combining a wide bandgap with excellent optical, thermal, and mechanical properties is essential for integrated nonlinear photonic application. Silicon carbide (SiC) is a wide-bandgap semiconductor with superior material properties that has attracted considerable interest very recently to develop various SiC micro-/nanophotonic devices based upon the polytypes of 3C, 4H, and 6H [13–20]. Its significant thermal conductivity, mechanical rigidity, and hardness make it particularly suitable for nonlinear photonic applications that require high optical intensity while maintaining thermal and mechanical resilience. In particular, its large refractive index (∼2.6) infers [1, 21] that SiC would exhibit a strong Kerr nonlinearity. To date, however, very little information is known, except a few characterizations available for 6H-SiC at wavelengths of 694 nm [22], 780 nm [23], and 800 nm [24], and for nanocrystalline SiC at a wavelength of 1064 nm [25].

Here we demonstrate a high-Q amorphous SiC (a-SiC) microresonator, with an optical Q of 1.3 × 10⁵, which is about one order of magnitude larger than other SiC devices [13–18, 20] and more than twice of what we demonstrated on the 3C-SiC platform recently [19]. In particular, with such a high optical Q, we are able to characterize the Kerr nonlinearity of a-SiC, with n₂ = (5.9 ± 0.7) × 10⁻¹⁵ cm²/W in the important telecom band around 1550 nm. This value is significantly larger than Si₃N₄ [26], AlN [11], diamond [12], silica [2], CaF₂ [21], and MgF₂ [21] that are currently used for frequency comb generation [4]. Compared with other material forms of SiC, a-SiC can be easily deposited by various approaches with high quality [27] and is thus very convenient for making micro-/nano-photonic devices. The one we employed was deposited by plasma enhanced chemical vapor deposition, with a gas mixture of SiH₄, CH₄ and Ar at 350°C, on a silicon substrate. The devices were fabricated on the as-grown material via electron-beam lithography and plasma etching, similar to the process we used previously [19]. Figure 1(a) and (b) show a fabricated microdisk with a fairly smooth sidewall, which implies a high optical quality. The cavity transmission spectrum of the device is shown in Fig. 2(a) for a quasi-transverse magnetic (quasi-TM) polarization with the electric field dominantly perpendicular to the device plane. The mode families are identified by the approach in Ref. [19]. The device exhibits high optical Q for the TM3, TM4, and TM5 mode families, with intrinsic optical Q typically above 1 × 10⁵. Two detailed examples are shown in Fig. 2(c) and (d) for TM5 modes located at 1498 and 1545 nm, respectively. Both modes are doublets with an intrinsic optical Q as high as 1.3 × 10⁵, and mode splitting 2β/(2π) of 0.95 and 1.1 GHz, respectively (see the Appendix).

 

Fig. 1 (a) Scanning electron microscopic (SEM) image of a fabricated microdisk, with a thickness of 570 nm and a radius of 6 μm. (b) SEM image of the device sidewall.

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Fig. 2 (a) Cavity transmission of an a-SiC microresonator for the quasi-TM polarization. Cavity modes are labeled as (n,m) where n and m stand for the radial and azimuthal mode numbers, respectively. Green and blue curves denote the transmission spectra scanned by different tunable lasers. (b) Optical mode profiles simulated by the finite element method. (c) and (d) show the transmission spectra of the cavity modes located at 1498 nm and 1545 nm, respectively, with experimental data in blue and theoretical fitting in red.

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We employ these two high-Q modes to characterize the Kerr nonlinearity, with a pump-probe XPM scheme [28]. An intense wave launched into the mode at 1545 nm and a weak wave into the mode at 1498 nm function as the pump and the probe, respectively. A sinusoidal temporal modulation of the pump power would result in a modulation of the refractive index of the device through the optical Kerr effect, which is in turn experienced by the probe beam and then transduced into its transmission out of the cavity. For a modulation of the intracavity pump energy of δU_p(Ω) at a modulation frequency of Ω, the modulated probe power, δP_s(Ω), detected at the cavity transmission is given by (see Appendix)

The optical Kerr effect responds instantaneously to the pump modulation while the photothermal effect responds only within a certain bandwidth. As a result, the probe modulation will be dominantly contributed by the Kerr effect in the high-frequency region. Moreover, Eq. (1) shows that the magnitude of probe modulation depends on δU_p and H_s(Ω), both of which depends critically on the cavity Q (see the Appendix). A sufficiently high optical Q will provide high transduction gain to boost the probe modulation well above the detector noise background, thus allowing us to measure the Kerr nonlinear susceptibility.

To measure the n₂ value, we carried out the pump-probe experiment (Fig. 3). The continuous-wave (CW) pump power was modulated sinusoidally and then launched right near the center of the cavity resonance (Δ_p = 0), with a dropped optical power of 28 μW and a modulation power of 228 nW. The CW probe wave was launched into the cavity with a laser-cavity detuning around Δ_s = Γ_ts/2 to maximize the cavity transduction of the probe signal. The dropped probe power was 3.1 μW, which was carefully checked to produce negligible nonlinear effect while maintaining good signal-to-noise ratio for the detected signal. The transmitted probe wave was detected by a high-speed detector and recorded by a network analyzer.

 

Fig. 3 Schematic of the experimental setup. The pump wave is modulated in amplitude by a lithium niobate modulator which is driven by a network analyzer. The high-speed detector 1 is used to detect the modulated probe signal which is recorded by the network analyzer. VOA: variable optical attenuator. MUX/DEMUX: optical multiplexer and de-multiplexer. The microscopic image shows an a-SiC microdisk with a delivery tapered fiber.

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Figure 4 shows the recorded modulation spectrum of the probe wave. The probe modulation is dominated by the photothermal effect at the low frequency, whose amplitude decreases with increasing modulation frequency (Fig. 4, inset) until ∼100 MHz, beyond which the optical Kerr effect starts to dominate and the probe modulation reaches a plateau. Fitting the experimental data with the theory, we obtain a Kerr nonlinear coefficient n₂ of (5.9 ± 0.7) × 10⁻¹⁵ cm²/W. Table 1 shows clearly that a-SiC has a larger refractive index and Kerr nonlinearity than other materials currently used for Kerr frequency comb generation [2, 4, 6, 7, 11, 12, 26].

 

Fig. 4 Modulation spectrum of the probe wave, |δP_s(Ω)|/P_0s, with experimental data shown in blue and theoretical fitting shown in red. The dashed and dotted curves show the individual contributions of the Kerr effect and thermal response, respectively. The gray curve shows the detector noise background. The inset shows the detailed thermal response.

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Table 1. Comparison of a-SiC with other materials used for frequency comb generation

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In summary, we have demonstrated a high-Q amorphous SiC microresonator and measured its Kerr nonlinearity. The strong Kerr nonlinearity and the high optical Q, together with other excellent material properties of SiC, render a-SiC devices promising for integrated nonlinear photonic applications such as optical parametric oscillation and frequency comb generation.

Appendix

Here we provide the details of the theory describing the pump-probe modulation under the optical Kerr effect and the photothermal effect. Our situation is quite different from [28] since the optical modes involved in the pump-probe modulation are both doublets, each of which consists of two counter-propagating modes (named as the forward and backward modes) coupled with each other, as shown in Fig. 5. In this case, the pump and probe fields inside the cavity satisfy the following equations

(2)$$\frac{d{a}_{\text{pf}}}{dt}=\left(i{\mathrm{\Delta }}_p-{\mathrm{\Gamma }}_{\text{tp}}/2\right)a_{\text{pf}}+i{\beta }_p{a}_{\text{pb}}-i{g}_T\delta T{a}_{\text{pf}}+i{\gamma }_p\left(U_{\text{pf}}+2U_{\text{pb}}\right)a_{\text{pf}}+i\sqrt{{\mathrm{\Gamma }}_{\text{ep}}}{A}_p,$$

(3)$$\frac{d{a}_{\text{pb}}}{dt}=\left(i{\mathrm{\Delta }}_p-{\mathrm{\Gamma }}_{\text{tp}}/2\right)a_{\text{pb}}+i{\beta }_p{a}_{\text{pf}}-i{g}_T\delta T{a}_{\text{pb}}+i{\gamma }_p\left(U_{\text{pb}}+2U_{\text{pf}}\right)a_{\text{pb}},$$

(4)$$\frac{d{a}_{\text{sf}}}{dt}=\left(i{\mathrm{\Delta }}_s-{\mathrm{\Gamma }}_{\text{ts}}/2\right)a_{\text{sf}}+i{\beta }_s{a}_{\text{sb}}-i{g}_T\delta T{a}_{\text{sf}}+2i{\gamma }_s{U}_p{a}_{\text{sf}}+i\sqrt{{\mathrm{\Gamma }}_{\text{es}}}{A}_s,$$

(5)$$\frac{d{a}_{\text{sb}}}{dt}=\left(i{\mathrm{\Delta }}_s-{\mathrm{\Gamma }}_{\text{ts}}/2\right)a_{\text{sb}}+i{\beta }_s{a}_{\text{sf}}-i{g}_T\delta T{a}_{\text{sb}}+2i{\gamma }_s{U}_p{a}_{\text{sb}},$$

where a_jf and a_jb (j = p, s stands for the pump and probe, respectively) are the intracavity fields of the forward and backward propagating modes, respectively. A_j is the input field which is launched into the cavity only along the forward direction. The field amplitudes are normalized such that U_jf = |a_jf|² and U_jb = |a_jb|² represent the intracity energies of the forward and backward modes, and P_0j = |A_j|² is the input optical power. U_p = U_pf + U_pb is the total pump energy inside the cavity. β_j is the coupling coefficient between the forward and backward propagating modes. Δ_j = ω_j − ω_0j is the detuning of laser frequency ω_j from the cavity resonance ω_0j. Γ_0j and Γ_tj are the photon decay rate of the intrinsic and loaded cavity, respectively, and Γ_ej describes the photon external coupling rate, which are related to the optical Q as Γ_ij = ω_0j/Q_ij (i = 0, e, t). The cavity transmitted power is given by $P_j={\left|A_j+i\sqrt{{\mathrm{\Gamma }}_{\text{ej}}}{a}_{\text{jf}}\right|}^2$.

 

Fig. 5 Schematic of forward and backward propagating modes inside the cavity.

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In Eqs. (2)–(5), we have neglected the SPM and XPM from the probe wave since it is much weaker than the pump. ${\gamma }_j=\frac{c{\omega }_{0\text{j}}{n}_2}{n_0^2{V}_{\text{eff}}}$, where $V_{\text{eff}}=\sqrt{V_p{V}_s}$ and V_p and V_s are the effective mode volume of the pump and probe mode, respectively. The finite-element simulations show that V_s = 30.1 μm³ and V_p = 30.4 μm³ for the two cavity modes at 1498 and 1545 nm, respectively. The third terms on the right-hand side of Eqs. (2)–(5) describe the photothermal effect [29], where $g_T=\frac{d{\omega }_0}{dT}=-\frac{{\omega }_0}{n}\frac{dn}{dT}$ stands for the photothermal coupling coefficient. The associated temperature modulation δT(t) is governed by

(6)$$\frac{d\delta T}{dt}=-{\mathrm{\Gamma }}_T\delta T+{\eta }_T{U}_p,$$

where the first term describes the thermal relaxation induced by the heat dissipation to surrounding environment and the second term describes the thermal heating induced by the linear absorption from the pump field inside the cavity. The thermal relaxation rate Γ_T and the thermal heating coefficient η_T both depend on the device material and geometry [29].

In the absence of the optical Kerr effect and the photothermal effect, Eqs. (2)–(5) can be solved easily for a continuous-wave input, with a cavity transmission given by

(7)$$T_j\equiv \frac{P_j}{P_{0\text{j}}}={\left|1-\frac{{\mathrm{\Gamma }}_{\text{ej}}}{2}\left(L_{\text{jc}0}+L_{\text{js}0}\right)\right|}^2,$$

where L_jc0 ≡ 1/(Γ_tj/2 − iΔ_jc) and L_js0 ≡ 1/(Γ_tj/2 − iΔ_js), with Δ_jc = Δ_j + β_j and Δ_js = Δ_j − β_j (j = p, s). P_0j is the input power. Physically, the coupling between the forward and backward propagating modes lifts their frequency degeneracy and results in two standing waves inside the cavity with a frequency separation 2β_j. As a result, the cavity transmission manifests as the well-known doublet. This is evident in the transmission spectra shown in Fig. 2(c) and (d), which deviate slightly from a single-Lorentzian shape. Equation (7) was used to fit the transmission spectra of the passive cavity to obtain the cavity Q, external coupling rate, and mode splitting.

For an input pump field modulation of δA_p(Ω) at frequency Ω, Eqs. (2)–(6) lead to a transmitted probe modulation given in Eq. (1), where δU_p(Ω) and H_s(Δ_s) are given by

(8)$$\delta U_p(\mathrm{\Omega })=\frac{{\mathrm{\Gamma }}_{\text{ep}}}{2}{A}_p\delta A_p^{*}(-\mathrm{\Omega })\left(L_{\text{pc}0}{L}_{\text{pc}}^{-}+L_{\text{ps}0}{L}_{\text{ps}}^{-}\right)-\frac{{\mathrm{\Gamma }}_{\text{ep}}}{2}{A}_p^{*}\delta A_p(\mathrm{\Omega })\left(L_{\text{pc}0}^{*}{L}_{\text{pc}}^{+}+L_{\text{ps}0}^{*}{L}_{\text{ps}}^{+}\right),$$

(9)$$\begin{array}{ll}{H}_s({\mathrm{\Delta }}_s)=\hfill & {\mathrm{\Gamma }}_{\text{es}}\left(i\mathrm{\Omega }-{\mathrm{\Gamma }}_{\text{ts}}+\frac{{\mathrm{\Gamma }}_{\text{es}}}{2}\right)\left({\mathrm{\Delta }}_{\text{sc}}{\left|L_{\text{sc}0}\right|}^2{L}_{\text{sc}}^{+}{L}_{sc}^{-}+{\mathrm{\Delta }}_{\text{ss}}{\left|L_{\text{ss}0}\right|}^2{L}_{\text{ss}}^{+}{L}_{\text{ss}}^{-}\right)\hfill \\ \hfill & +\frac{{\mathrm{\Gamma }}_{\text{es}}^2}{2}{\mathrm{\Delta }}_s\left(L_{\text{sc}0}^{*}{L}_{\text{ss}0}{L}_{\text{ss}}^{+}{L}_{\text{sc}}^{-}+L_{\text{sc}0}{L}_{\text{ss}0}^{*}{L}_{\text{ss}}^{-}{L}_{\text{sc}}^{+}\right),\hfill \end{array}$$

where $L_{\text{jc}}^{\pm }\equiv 1/(\left[i\left({\mathrm{\Delta }}_{\text{jc}}\pm \mathrm{\Omega }\right)\mp {\mathrm{\Gamma }}_{\text{tj}}/2\right]$ and $L_{\text{js}}^{\pm }\equiv 1/\left[i\left({\mathrm{\Delta }}_{\text{js}}\pm \mathrm{\Omega }\right)\mp {\mathrm{\Gamma }}_{\text{tj}}/2\right]$ (j = p, s). Equations (1), (8), and (9) completely describe the pump-probe modulation under the optical Kerr effect and photothermal effect. They were employed to fit the experimental data (Fig. 4) with three parameters, γ_s, Γ_T, and η_T. The Kerr nonlinear coefficient n₂ is thus obtained from the fitted γ_s, which is given in the main text. The thermal relaxation rate Γ_T/2π is fitted to be 0.46 MHz, which corresponds to a thermal relaxation time around 0.35 μs.

Acknowledgments

This work was supported by National Science Foundation under grant ECCS-1408517. It was performed in part at the Cornell NanoScale Science & Technology Facility (CNF), a member of the National Nanotechnology Infrastructure Network.

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