Fundamental noise dynamics in cascaded-order Brillouin lasers

The dynamics of cascaded-order Brillouin lasers make them ideal for applications such as rotation sensing, highly coherent optical communications, and low-noise microwave signal synthesis. Remarkably, when implemented at the chip scale, recent experimental studies have revealed that Brillouin lasers can operate in the fundamental linewidth regime where optomechanical and quantum noise sources dominate. To explore new opportunities for enhanced performance, we formulate a simple model to describe the physics of cascaded Brillouin lasers based on the coupled mode dynamics governed by electrostriction and the fluctuation-dissipation theorem. From this model, we obtain analytical formulas describing the steady-state power evolution and accompanying noise properties, including expressions for phase noise, relative-intensity noise, and power spectra for beat notes of cascaded laser orders. Our analysis reveals that cascading can enhance laser noise, resulting in a broader emission linewidth and larger intensity fluctuations with increased power. Consequently, higher-coherence laser emission can be achieved if indefinite cascading can be prevented. In addition, we derive a simple analytical expression that enables the Stokes linewidth to be obtained from spectra of beat notes between distinct cascaded laser orders and their relative powers. We validate our results using stochastic numerical simulations of the cascaded laser dynamics.

Figure 1

Fundamentals of Brillouin lasing. (a) Energy conservation and (b) wave vector phase matching requirements. Brillouin coupling mediated (c) optical amplifier and (d) energy transfer. (e) Ring-resonator-based Brillouin laser. Here, spatial growth and depletion of the optical fields around the resonator are exaggerated for illustrative purposes; this spatial growth must be small for the mean-field approximation to remain valid.

Figure 2

Illustration of a cascaded Brillouin laser. A laser of frequency ${\omega }_{\mathrm{pump}}$ and linewidth $\mathrm{\Delta }{\nu }_{\mathrm{pump}}$ pumps an optical resonator. Light in the ${\omega }_0$ mode (blue) can scatter to ${\omega }_1$ (red) by emitting a phonon. When lasing, the ${\omega }_1$ optical mode can act as a pump for higher Stokes orders. Here, $g_m$ and ${\tilde{\gamma }}_m$ respectively quantify the Brillouin coupling rate between the $m$ and the $(m+1)\mathrm{th}$ modes and the optical decay rate of the $m\mathrm{th}$ mode.

Figure 3

Illustration of noise dynamics in cascaded Brillouin lasers. Tiles represent optical and acoustic modes. The mixer symbol represents the nonlinear optomechanical coupling between two optical modes and one acoustic mode. (a) Below threshold for cascaded lasing, optomechanical coupling enables noise transfer between the $m=0$ and the $m=1$ through spontaneous Brillouin scattering from the phonon mode $b_0$. (b) Above threshold for cascaded lasing, noise can be injected into the $m=1$ mode from spontaneous scattering from thermal phonons in the $b_0$ and $b_1$ modes.

Figure 4

Steady-state laser power for the laser parameters given in Table 1. Solid lines represent theoretical predictions for the steady-state powers given by Eqs. (36) and (37), and solid points denote the steady-state powers obtained from stochastic simulations of Eqs. (2) and (3). From top to bottom, the power curves are $P_0$, $P_1$, $P_2$, $P_3$, $P_4$, and $P_5$.

Figure 5

Simulated pump phase ${\phi }_0$ and external pump phase ${\phi }_{\mathrm{pump}}$ as a function of time for $P_{\mathrm{pump}}=50$ mW. Simulation parameters given in Table 1.

Figure 6

Relative-intensity noise of the first Stokes order (a) prior to cascaded lasing [point (a) of inset], (b) cascaded lasing to two Stokes orders [point (b) of inset], and (c) cascaded lasing to three Stokes orders [point (c) of inset]. Gray dashed line (c) is the theory curve from (a), included for comparison.

Figure 7

Comparison of the phase noise of the first Stokes order, above and below threshold for cascaded lasing. The solid lines are calculated using Eq. (58), and the points represent simulated phase noise, open circles for point B of the inset and red points for point A. The emitted laser power is $P_1=5.2\mathrm{mW}$ for both curves, whereas the power supplied to the laser is respectively $197\mathrm{mW}$ and $369\mathrm{mW}$.

Figure 8

Phase noise of beat note of the first and third Stokes laser orders for parameters given in Table 1. The power spectrum given by Eq. (64) is represented as the solid line. The gray dots represent the simulated beat note phase noise power spectrum obtained by numerically solving Eq. (10). The on-chip pump power is 756 mW.

Figure 9

Power dynamics of the first Stokes (red, left), third Stokes (green, bottom right), and first-third beat note (gray, top right) linewidth.