Laser frequency stabilization based on steady-state spectral-hole burning in Eu${}^{3+}$:Y${}_2$SiO${}_5$

We present and analyze a method of laser frequency stabilization via steady-state patterns of spectral holes in Eu${}^{3+}$:Y${}_2$SiO${}_5$. Three regions of spectral holes are created, spaced in frequency by the ground state hyperfine splittings of ${}^{151}$Eu${}^{3+}$. The absorption pattern is shown not to degrade after days of laser frequency stabilization. An optical frequency comparison of a laser locked to such a steady-state spectral-hole pattern with an independent cavity-stabilized laser and a Yb optical lattice clock demonstrates a spectral-hole fractional frequency instability of $1.0\times {10}^{-15}{\tau }^{-\frac{1}{2}}$ that averages to $8.5_{-1.8}^{+4.8}\times {10}^{-17}$ at $\tau =73$ s. Residual amplitude modulation at the frequency of the RF drive applied to the fiber-coupled electro-optic modulator is reduced to less than $1\times {10}^{-6}$ fractional amplitude modulation at $\tau >$ 1 s by an active servo. The contribution of residual amplitude modulation to the laser frequency instability is further reduced by digital division of the transmission and incident photodetector signals to less than $1\times {10}^{-16}$ at $\tau >$ 1 s.