Phonon-Limited-Linewidth of Brillouin Lasers at Cryogenic Temperatures

Laser linewidth is of central importance in spectroscopy, frequency metrology, and all applications of lasers requiring high coherence. It is also of fundamental importance, because the Schawlow-Townes laser linewidth limit is of quantum origin. Recently, a theory of stimulated Brillouin laser (SBL) linewidth has been reported. While the SBL linewidth formula exhibits power and optical $Q$ factor dependences that are identical to the Schawlow-Townes formula, a source of noise not present in conventional lasers, phonon occupancy of the Brillouin mechanical mode is predicted to be the dominant SBL linewidth contribution. Moreover, the quantum limit of the SBL linewidth is predicted to be twice the Schawlow-Townes limit on account of phonon participation. To help confirm this theory the SBL fundamental linewidth is measured at cryogenic temperatures in a silica microresonator. Its temperature dependence and the SBL linewidth theory are combined to predict the number of thermomechanical quanta at three temperatures. The result agrees with the Bose-Einstein phonon occupancy of the microwave-rate Brillouin mode in support of the SBL linewidth theory prediction.

Figure 1

Experimental setup and Brillouin laser action. (a) Experimental setup showing external cavity diode laser (ECDL) pump, erbium-doped fiber amplifier (EDFA), polarization control (PC), and circulator coupling to the cryostat. Green lines indicate optical fiber. A fiber taper is used to couple to the microresonator. Pump and even-ordered stimulated Brillouin laser (SBL) waves propagate in the forward direction while odd-ordered SBL waves propagate in the backward direction and are coupled using the circulator. Photodetectors (PD) and an oscilloscope monitor the waves propagating in both directions. A fast photodetector measures the first and third beatnote which is measured using an electrical spectrum analyzer (ESA) and phase noise [L(f)] analyzer. An optical spectrum analyzer (OSA) also measures the backward propagating waves. The pump laser is locked to the microresonator optical resonance using a Pound-Drever-Hall (PDH) feedback loop. (b) Schematic of the optical fiber taper coupling setup inside the cryostat. Optical fiber (red) is glued to an aluminum holder which is fixed on a 3-axis piezoelectric stage. The microresonator is mounted on a copper plate. (c) Top view of the 6 mm wedge disk resonator. (d) Illustration of cascaded Brillouin laser action. Pump and even Stokes orders propagate in the forward direction while odd orders propagate in the backward direction. Green curves represent the Brillouin gain spectra. Brillouin shift frequency ($\mathrm{\Omega }$) and free-spectral range (FSR) are indicated. (e) Optical spectrum measured using the OSA and showing cascaded Brillouin laser action to fifth order. Inset: typical electrical beatnote spectrum produced by the first- and third-order SBL signals.

Figure 2

Aligning the third-Stokes wave to the Brillouin gain spectrum maximum. (a) Illustration showing spectral placement of Stokes wave with respect to Brillouin gain spectrum maximum. Variation of the pumping wavelength causes the Brillouin shift frequency ($\mathrm{\Omega }$) to vary and thereby scans the Stokes wave across the Brillouin gain peak. (b) Measured spectra of $P_{\text{clamp}}$, the clamped third-order Stokes wave power (black circle). The three panels show measurements performed at $T=300$, 77, and 8 K. Each color corresponds to a distinct pumping wavelength. The first-order Stokes wave also appears in the spectral map as the stronger peak near the third-order Stokes wave. The pump wave is not observable in the linear-scale spectrum as it propagates in the direction opposite to the first-order and third-order Stokes waves. $P_{\text{clamp}}^{\min }$ is determined from the fitted red curve as the minimum power point.

Figure 3

(a) Phase noise spectra of the first- and third-order SBL beatnotes at three temperatures (300, 77, and 8 K). The Eq. (7) fit within the 30–50 kHz band at each temperature is the black dotted line. The blue bands give the 1.5 dB variation in this fit over multiple traces. (b) Red points are minimum SBL linewidths ($\mathrm{\Delta }{\nu }_{\min }$) resulting from the fittings in (a). (c) Thermal phonon occupancies ($n_T$) calculated from the measured $\mathrm{\Delta }{\nu }_{\min }$ and $g_0$ (see Table 1) are plotted (red points) versus temperature. $n_T$ from the Bose-Einstein occupancy is given as the black line. Triangular point: $n_T$-based temperature calibration to 22 K using Bose-Einstein result.