$1.5\mu \mathrm{m}$ Lasers with Sub-10 mHz Linewidth

We report on two ultrastable lasers each stabilized to independent silicon Fabry-Pérot cavities operated at 124 K. The fractional frequency instability of each laser is completely determined by the fundamental thermal Brownian noise of the mirror coatings with a flicker noise floor of $4\times {10}^{-17}$ for integration times between 0.8 s and a few tens of seconds. We rigorously treat the notorious divergences encountered with the associated flicker frequency noise and derive methods to relate this noise to observable and practically relevant linewidths and coherence times. The individual laser linewidth obtained from the phase noise spectrum or the direct beat note between the two lasers can be as small as 5 mHz at 194 THz. From the measured phase evolution between the two laser fields we derive usable phase coherence times for different applications of 11 to 55 s.

Figure 1

Modified Allan deviation for Si2 (squares), Si3 (circles), and the ULE-cavity laser (diamonds) derived from three-cornered hat estimations. We used a 3.4 h data set for $10\mathrm{ms}\le \tau \le 4\mathrm{s}$ and a 24.2 h data set for $8\mathrm{s}\le \tau \le 8192\mathrm{s}$, recorded in the same day. The green line represents the expected thermal noise of the silicon cavities. The dashed line illustrates the instability where the rms phase fluctuations are 1 rad for a given $\tau$ [34]. The intersections with the instability curves of the Si lasers result in coherence times of around 11 s. Linear frequency drifts in each data set were subtracted. The inset shows a schematic of the measurement setup.

Figure 2

PSD of phase fluctuations of a Si stabilized laser, obtained as one half of the PSD of the Si3—Si2 beat. The red line shows the expected flicker frequency noise corresponding to the thermal noise at $T=124\mathrm{K}$. The inset shows the rms phase noise integrated down from 10 MHz. A value of $1{\mathrm{rad}}^2$ is obtained after integrating down to 6.8 mHz (blue markers) leading to a FWHM linewidth of 13.6 mHz.

Figure 3

FFT spectrum of the beat note between lasers Si2 and Si3 (Hanning window, frequency resolution 7.2 mHz).

Figure 4

The evolution of the phase difference between the two Si lasers. The first 4 s segment $T_0$ is used to estimate the average frequency $\overline{\nu }$ at $t=0\mathrm{s}$. For $t=0–12\mathrm{s}$, the phase deviation from the expected $2\pi \overline{\nu }t$ is calculated. 100 consecutive curves are shown with thin gray lines. The red lines indicate the $\pm \mathrm{\Delta }{\varphi }_{\mathrm{rms}}$ range, evaluated statistically from 20 750 curves.