Thermo-refractive noise in silicon nitride microresonators

TL;DR

This paper characterizes thermo-refractive noise in silicon nitride microresonators, demonstrating it as the dominant thermal noise source and providing quantitative bounds to inform future photonic device design.

Contribution

It provides the first detailed measurement and theoretical validation of thermo-refractive noise in silicon nitride microresonators, highlighting its significance.

Findings

Thermo-refractive noise dominates thermal noise in the platform.

Measurements agree with theoretical models and simulations.

Quantitative bounds on thermal noise scaling and magnitude.

Abstract

Thermodynamic noise places a fundamental limit on the frequency stability of dielectric optical resonators. Here, we present the characterization of thermo-refractive noise in photonic-chip-based silicon nitride microresonators and show that thermo-refractive noise is the dominant thermal noise source in the platform. We employed balanced homodyne detection to measure the thermo-refractive 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 thermo-refractive noise can prove valuable in the design considerations and performance limitations of future photonic integrated devices.

Figures

Tables

Table 1. Table 1: Physical properties used for both the theoretical models and FEM simulations of the thermo-refractive noise of Si 3 ​ N 4 subscript Si 3 subscript N 4 \text{Si}_{3}\text{N}_{4} micoresonators

Physical properties	Values
Density ($\rho$)	$3.29\text{×}{10}^3\mathrm{kg}{m}^{-3}$
Refractive index ($n_0$)	1.996
Thermo-optic coefficient ($dn/dT$)	$2.45\text{×}{10}^{-5}{K}^{-1}$
Thermal conductivity ($\kappa$)	$30W{m}^{-1}{K}^{-1}$
Specific heat capacity ($C$)	$800J{\mathrm{kg}}^{-1}{K}^{-1}$

Equations

$$⟨\delta T^2⟩=\frac{k_B{T}^2}{\rho CV}$$

$$S_{\delta T}(\omega )=\frac{k_B{T}^2}{\sqrt{{\pi }^3\kappa \rho C\omega }}\sqrt{\frac{1}{2p+1}}\frac{1}{R\sqrt{d_z^2-d_r^2}}\frac{1}{(1+(\omega {\tau }_d{)}^{3/4}{)}^2}$$

$$S_{\delta T}(\omega )=\frac{k_B{T}^2{R}^2}{12\kappa V_{\mathrm{eff}}}{\left(1+{\left(\frac{R^2\rho C\omega }{3^{5/3}\kappa }\right)}^{3/2}+\frac{1}{6}{\left(\frac{R^2\rho C\omega }{8l^{1/3}\kappa }\right)}^2\right)}^{-1}$$

Full text

I Introduction

Optical microresonators are used in numerous practical applications, in particular integrated, silicon-based lasers Liang and Bowers (2010), narrow linewidth lasers Gundavarapu et al. (2019) and photonic microwave oscillators Capmany and Novak (2007). The small mode volume of microresonators enables strong optical nonlinearities harnessed in emerging technologies such as microcombs Kippenberg et al. (2018). The small mode volume also enhances sensing capabilities used in fundamental research, e.g. cavity quantum optomechanics Aspelmeyer et al. (2014). However, the small mode volume comes at a cost: limitations on the microresonator’s frequency stability arise from thermal fluctuations such as thermo-refractive (TRN) and thermoelastic noise. These fluctuations were first theoretically described in the context of Laser Interferometer Gravitational-Wave Detectors Braginsky et al. (2000), and place limits on the frequency stability of an interferometer. Thermal fluctuations are particularly strong in small mode volume optical resonators and place fundamental limits on applications that require high frequency stability, e.g. optical sensing Foreman et al. (2015), optomechanical displacement sensing Arcizet et al. (2009), dissipative Kerr soliton microcomb generation Kippenberg et al. (2018), electro-optical modulators Pavlov et al. (2015), opto-electronic oscillators Liang et al. (2015) and Kerr squeezing Hoff et al. (2015). Therefore, different kinds of thermal noises have been extensively studied Braginsky et al. (2000); Gorodetsky and Grudinin (2004); Matsko et al. (2007); Weng et al. (2018), but theoretical analyses show inconsistencies between different platforms Kondratiev and Gorodetsky (2018) and rely on auxiliary measurements of material parameters that are not always well-known. Experimental characterization is therefore essential to understand the limitations of a specific system.

Silicon nitride ($\text{Si}_{3}\text{N}_{4}$) is a space compatible Brasch et al. (2014), CMOS-compatible material Moss et al. (2013) with a large Kerr nonlinear coefficient, an absence of two photon absorption in the telecommunication window, ultralow losses Pfeiffer et al. (2016); Ji et al. (2017), and a wide transparency window from visible to mid-infrared. These properties have in particular been exploited for chip-scale frequency combs Kippenberg et al. (2011, 2018), as well as coherent low-pulse-energy supercontinuum generation in the near- Johnson et al. (2015) and mid-infrared Guo et al. (2018). Recent advances Pfeiffer et al. (2018); Liu et al. (2018); Ji et al. (2017) in fabrication of integrated $\text{Si}_{3}\text{N}_{4}$ microresonators have enabled optical quality factors $Q>10^{7}$, for which the fundamental limits imposed by TRN will be relevant for the frequency stability, and can be a limiting factor in future applications such as microwave generation Capmany and Novak (2007). Although recent measurements of the carrier-envelope-offset frequency noise of microcombs Drake et al. (2018) have already shown clear evidence that thermal noise is limiting the performance of certain applications, TRN has so far only been investigated in simple geometries (notably silica microspheres and microtoroids Arcizet et al. (2009); Gorodetsky and Grudinin (2004)), and has not yet been measured in photonic integrated microresonators.

Here, we present the first characterization of TRN in $\text{Si}_{3}\text{N}_{4}$ microresonators, and compare FEM simulations with measurements of TRN using balanced homodyne detection. The results show that TRN is the dominant thermal noise source over frequencies ranging from $10\text{\,}\mathrm{kHz}$ to $10\text{\,}\mathrm{MHz}$. Refined theoretical and modeling approaches are therefore required to analyze the TRN in $\text{Si}_{3}\text{N}_{4}$ microresonators.

II Thermo-Refractive Noise

In an optical resonator, thermo-refractive noise leads to fluctuations of the resonance frequency due to fluctuations of refractive index, $n$, caused by thermodynamic fluctuations of temperature, $\delta T$, whose variance is:

[TABLE]

where $T$ is the temperature of the heat bath, $k_{B}$ the Boltzmann constant, $\rho$ the density, $C$ the specific heat, and $V$ the volume. Using the material parameters of $\text{Si}_{3}\text{N}_{4}$ and the optical mode volume of a typical $1\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$ free-spectral-range (FSR) microresonator, we can obtain a value of the standard deviation of the heat bath temperature as $\sqrt{\langle\delta T^{2}\rangle}\sim$60\text{,}\mathrm{\SIUnitSymbolMicro}\mathrm{K}. Combined with the measured thermo-optic coefficient, $\mathrm{d}n/\mathrm{d}T=$2.45\text{\times}{10}^{-5}\text{\,}\mathrm{K}^{-1}, of $\text{Si}_{3}\text{N}_{4}$ Arbabi and Goddard (2013), the fractional frequency fluctuation can be estimated asGorodetsky and Grudinin (2004) $\sqrt{\langle\delta f^{2}\rangle}/f\sim 7\times 10^{-10}$, and the absolute frequency fluctuation $\sqrt{\langle\delta f^{2}\rangle}$ is around $150\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$, which is above $1\%$ of the cavity linewidth in $\text{Si}_{3}\text{N}_{4}$ microresonators Liu et al. (2018).

The thermo-refractive noise and the thermo-elastic noise, which are both consequences of thermodynamic temperature fluctuations, were previously experimentally observed in silica microspheres Gorodetsky and Grudinin (2004) and theoretically analyzed (but not observed) in crystalline resonators Matsko et al. (2007). In most cases, thermo-refractive noise is the largest among the thermal noises. For chip-based $\text{Si}_{3}\text{N}_{4}$ microresonators, TRN is also expected to be the largest noise source, as the thermo-optic coefficient ($\mathrm{d}n/\mathrm{d}T$) is larger than the thermal expansion coefficient. However, the different modelling approaches taken in the analysis of the two previously mentioned platforms result in different predictions for the geometric dependency of TRN as well as its magnitude at low offset frequencies Kondratiev and Gorodetsky (2018).

We now describe the two approaches taken for modelling TRN in microresonators: the first model assumes a homogeneous microresonator in an infinite heat bath, yielding the following expression for the effective temperature fluctuations: Kondratiev and Gorodetsky (2018):

[TABLE]

where $R$ is the ring radius of the microcavity (geometry shown in Fig. 4(a)), $d_{z}$ and $d_{r}$ are halfwidths of the fundamental mode with orbital number $l$, azimuthal number $m$, meridional mode number $p=l-m$, $\tau_{d}=\frac{\pi^{1/3}}{4^{1/3}}\frac{\rho C}{\kappa}d_{r}^{2}$, and the definitions of the other parameters can be found in Table 1. The key features are the $\omega^{-1/2}$ dependence at low frequency, the $\omega^{-2}$ dependence at high frequency, and the $R^{-1}$ overall scaling. This model gave satisfactory agreements with experimental measurements in microspheres Gorodetsky and Grudinin (2004).

The second model Matsko et al. (2007) uses the thermal decomposition method which does not take into account the interaction with the environment. As a consequence, there is a low frequency cut-off due to the finite dimension of the resonator, which results in the following (single-sided) temperature noise spectral density:

[TABLE]

where $V_{\mathrm{eff}}$ is the effective mode volume. Here the expression still has a $\omega^{-2}$ dependence at high frequency, but features an overall scaling with $R$ between $R^{-3}$ and $R^{-4}$ depending on how $V_{\mathrm{eff}}$ scales with radius.

The temperature fluctuations can be converted into a frequency noise spectral density through $S_{\delta f}=(f_{0}\frac{1}{n_{0}}\frac{dn}{dT})^{2}S_{\delta T}$, where $f_{0}$ is the resonance frequency. However, Eq. (2) and Eq. (3) are both idealized cases assuming homogeneous materials and either the infinite heat bath or isolated model. Clearly, these assumptions do not match the geometry of integrated $\text{Si}_{3}\text{N}_{4}$ resonators consisting of complex waveguide structures comprising different materials, which invalidates the assumption of homogeneity. We therefore performed an FEM simulation based on the fluctuation-dissipation theorem Kondratiev and Gorodetsky (2018); Weng et al. (2018), where we simulate the thermo-refractive noise of a $\text{Si}_{3}\text{N}_{4}$ microresonator embedded in a $\text{Si}\text{O}_{2}$ substrate, using the actual geometry.

We can now compare the FEM simulation results with the theoretical expressions (see Fig. 1). Because the thermal properties of $\text{Si}_{3}\text{N}_{4}$ depend on the material characteristics and the fabrication process, they can exhibit significant variation. We use the median values of the physical properties reported in the literature for both the theoretical predictions and the FEM simulations (see Table 1). The optical mode parameters used in Eq. (2) and Eq. (3) are retrieved from FEM simulations of the $\text{Si}_{3}\text{N}_{4}$ microresonators. For the scaling with ring radius, the simulation curves match the infinite heat bath theory Eq. (2) well, while the deviation from the thermal decomposition method becomes larger as the radius increases. It indicates that, according to FEM simulations, $\text{Si}_{3}\text{N}_{4}$ microresonators experience thermo-refractive noise more similar to microspheres at high frequency, while at low frequency, the noise is reduced due to the cladding region ($\text{Si}\text{O}_{2}$) and the chip geometry. The measurement results, the simulation curves and the infinite heat bath curves are compared in Fig. 4.

Due to the $R^{-1}$ scaling and the strong light confinement the $\text{Si}_{3}\text{N}_{4}$ waveguide offers, the computed spectral density of thermo-refractive frequency noise for resonators with FSR ranging from 20 GHz and 1 THz is sufficiently high to be probed without an extremely pure and stable laser. In crystalline microresonators, the much larger mode volume and smaller thermo-optic coefficient ($\mathrm{d}n/\mathrm{d}T$) make it much more challenging to measure TRN Lim et al. (2017), and correspondingly it is typically not a practical limit.

III Measurement scheme and results

The measurement scheme in this study employs a balanced homodyne setup to measure the phase fluctuation of the transmitted laser caused by the cavity frequency noise. The cavity frequency noise is calibrated using a calibration tone. The laser (external cavity diode laser at $1550\text{\,}\mathrm{n}\mathrm{m}$, linewidth $\sim 30$ kHz) is locked to the cavity resonance via the Pound-Drever-Hall (PDH) locking method Drever et al. (1983) with a feedback bandwidth of a few kilohertz. The power level is sufficiently low to avoid any unwanted thermal effects. The measurement setup is illustrated in Fig. 2.

In order to provide an absolute calibration of our measurement, we use a calibration tone generated by phase modulating the RF tone applied to the acousto-optic modulator (AOM). A modulation depth of $1.14\text{\,}\mathrm{r}\mathrm{a}\mathrm{d}$ is calibrated by sideband fitting using heterodyne detection. The calibration tone induces an extra phase fluctuation transduced by the cavity, and results in a narrow peak in the frequency noise spectrum. Because the frequency modulation depth is known for the calibration peak, we can use it to calibrate the absolute magnitude of the corresponding homodyne signal. The modulation frequency and the modulation depth are chosen at $50\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ and $1.14\text{\,}\mathrm{r}\mathrm{a}\mathrm{d}$ to be outside the PDH locking bandwidth and to keep the frequency modulation depth smaller than the cavity linewidth.

The characterized samples are integrated $\text{Si}_{3}\text{N}_{4}$ microresonators with radius $R\sim 23-$1200\text{,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$, and FSR ranging from $1\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$ to $20\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$. The Q-factors of these microresonators fabricated by the photonic Damascene reflow process Liu et al. (2018); Pfeiffer et al. (2018) are typically $Q>10^{7}$. The measured noise spectrum is thus filtered by the cavity resonance at high offset frequencies. The resonance linewidth of each microresonator and the response function of the bias tee before the spectrum analyzer are measured and compensated for through data post-processing.

We first verified the power-independent nature of thermo-refractive noise (as expected from Eqs.1-3) by performing an input power sweep (shown in Fig. 3). The frequency noise level remains the same when varying the laser power of the probe signal by more than two orders of magnitude (from $1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{W}$ to $120\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{W}$), showing that photothermal noise is not making a significant contribution. We next investigated the dependence on optical mode volume. Fig. 4 (c) presents the measurement results for four different cavity radii, together with the corresponding theoretical curves and FEM simulation curves. The observable background noise sources arise from local oscillator shot noise, several technical spikes, and the calibration peak at $50\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$. The calibration peaks in the off-resonance noise spectrum and the LO shot noise spectrum from Fig. 4 (c) are the result of residual amplitude modulation from the AOM. However, by utilizing phase modulation of the AOM RF tone, a signal-to-noise ratio of $20\text{\,}\mathrm{d}\mathrm{B}$ can be obtained for the calibration peak.

Good agreement of the measured spectra with the simulation curves for both frequency dependence ($\propto\omega^{-1/2}$) and radius dependence ($\propto R^{-1}$) is clearly observed, which also confirms the validity of Eq. (2) as a theoretical prediction of TRN in the $\text{Si}_{3}\text{N}_{4}$ microresonator platform. By assuming that the spectrum matches the FEM simulation in the low frequency range, the total frequency fluctuations due to TRN could be retrieved through integration over the high frequency data and the low frequency FEM curves, e.g. the $1\text{\,}\mathrm{T}\mathrm{H}\mathrm{z}$ microresonator has a resonance frequency instability of around $240\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$, which agrees well with our previous estimate of $150\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$. The agreement further indicates that the heat exchange with the surrounding environment is largely responsible for the generation of thermo-refractive noise in this system. However, in the low frequency range, the approximation of a homogeneous medium in Eq. (2) will break down due to heat exchange with the media outside the waveguide, as is indicated by the multiple saturation steps of the FEM simulation curves at low frequency shown in Fig. 1. Though the heat exchange with the outer layer makes it difficult to derive a simple expression for the thermo-refractive noise in $\text{Si}_{3}\text{N}_{4}$ microresonators, it offers a possibility to bypass the thermal limit Kimble et al. (2008). The design of such a thermal-noise-reduced photonic microresonator will be more important when more applications truly reach the thermal limit of their performance, e.g. to realize integrated ultra narrow linewidth lasers.

IV Conclusion

We present the first characterization of thermo-refractive noise in the integrated $\text{Si}_{3}\text{N}_{4}$ microresonator platform. The presence of thermo-refractive noise in photonic integrated resonators could limit the performance for many applications. Therefore, we measured the thermo-refractive noises of $\text{Si}_{3}\text{N}_{4}$ microresonators with a wide range of FSR. The measurement results are in good agreement with both the FEM simulation and the theoretical prediction for both frequency dependence ($\propto\omega^{-1/2}$) and radius dependence ($\propto R^{-1}$). The characterization of thermo-refractive noise in $\text{Si}_{3}\text{N}_{4}$ microresonator systems can serve as a standard for comparison of the noise features observed in associated applications such as microwave generation. Finally, these results might provide new insights for the silicon photonics community, enabling a better understanding of the formation of thermo-refractive noise as well as a path towards systems which bypass thermal noise limitations.

Funding Information

This publication was supported by Contract HR0011-15-C-0055 and HR0011181003 from the Defense Advanced Research Projects Agency (DARPA), Defense Sciences Office (DSO), by the Swiss National Science Foundation under grant agreement No.163864, and the Russian Foundation for Basic Research project 17-02-00522.

Acknowledgments

All samples were fabricated at the Center for MicroNanoTechnology (CMi) at EPFL. We acknowledge Rui Ning Wang for the assistance in sample fabrication.