Thermorefractive noise in silicon-nitride microresonators

Thermodynamic noise places a fundamental limit on the frequency stability of dielectric optical resonators. Here, we present the characterization of thermorefractive noise in photonic-chip-based silicon-nitride (${\text{Si}}_3{\text{N}}_4$) microresonators and show that thermorefractive noise is the dominant thermal noise source in the platform. We employed balanced homodyne detection to measure the thermorefractive noise spectrum of microresonators of different diameters. The measurements are in good agreement with theoretical models and finite element method simulations. Our characterization sets quantitative bounds on the scaling and absolute magnitude of thermal noise in photonic-chip-based microresonators. An improved understanding of thermorefractive noise can prove valuable in the design considerations and performance limitations of future photonic integrated devices.

Figure 1

Comparison of FEM simulation results of thermorefractive noise with the theoretical predictions (infinite heat bath and thermal decomposition). The graphs show the cavity frequency noise spectral density of a ${\text{Si}}_3{\text{N}}_4$ microresonator with a free spectral range (FSR) of 1 THz, 99G Hz, and 20G Hz. At high frequency, the simulation curves match better with the infinite heat bath assumption. At low frequency, the simulation curves experience a cutoff because of the finite size of the modeled chip, which behaves more similar to the thermal decomposition method.

Figure 2

Balanced homodyne setup for measuring thermorefractive noise of the ${\text{Si}}_3{\text{N}}_4$ microresonator. The electro-optic modulator (EOM) phase modulates the light at 140 MHz to generate the PDH error signal for locking the laser to the cavity mode. The AOM is modulated with an rf tone with a central frequency of 80 MHz. The rf tone is then phase modulated at 50 kHz with a 1.14 rad modulation depth, which provides an absolute calibration peak for the noise measurements. A piezo mirror is used in the local oscillator path to lock the homodyne at the phase quadrature. The measurement bandwidth is limited by the detector bandwidth of 80 MHz.

Figure 3

Calibrated thermorefractive frequency noise spectra measured using different probing power. A ${\text{Si}}_3{\text{N}}_4$ microresonator with free spectral range of 1 THz was probed with 1 to 120 $\mu \mathrm{W}$ of optical power. The bottom line shows the shot noise level with $1\mu \mathrm{W}$ of input optical power. The normalized units utilized in the inset figure are obtained by integrating over 200 kHz to 2 MHz and dividing by the average of the integrated values. It indicates that the probe is weak enough to avoid other laser-induced thermal effects (e.g., photothermal noise), and importantly reveals the power independence expected for thermorefractive noise.

Figure 4

Verification of the size dependence of thermorefractive noise in integrated ${\text{Si}}_3{\text{N}}_4$ microresonators. (a) Scanning electron microscopy (SEM) image of a 1 THz-FSR ${\text{Si}}_3{\text{N}}_4$ microresonator ring (dark blue). (b) SEM image of the cross section of the waveguide. The cross-section image is color shaded to help identify different regions. (c) Thermorefractive noise measured in integrated ${\text{Si}}_3{\text{N}}_4$ microresonators with free spectral ranges of 1 THz, 200 GHz, 100 GHz, and 88 GHz. The 88 GHz-FSR sample has a bigger width 2.0 $\mu \mathrm{m}$ compared to other samples 1.5 $\mu \mathrm{m}$) and there is some run to run variation in sidewall angles. A 30-point moving average was applied to the data. The dotted lines show the FEM simulation results and the dashed lines show the theoretical predictions from Eq. (2). Data below $10\mathrm{kHz}$ were truncated due to the excess locking noises, and also data above 10 MHz due to the detector's nonlinear response. Additional data can be found in the Supplemental Material [34].