Compact 459 nm Cs cell optical frequency standard with $2.1\times{10}^{-13}/\sqrt{\tau}$ short-term stability

We achieve a compact optical frequency standard with an extended cavity diode laser locked to the 459 nm 6S$_{1/2}$ - 7P$_{1/2}$ transition of thermal $^{133}$Cs atoms in a $\phi$ 10 mm $\times$ 50 mm glass cell, using modulation transfer spectroscopy (MTS). The self-estimated frequency stability of this laser is $1.4\times{10}^{-14}/\sqrt{\tau}$. With heterodyne measurement, we verify the linewidth-narrowing effect of MTS locking and measure the frequency stability of the locked laser. The linewidth of each laser is reduced from the free-running 69.6 kHz to 10.3 kHz after MTS stabilization, by a factor of 6.75. The Allan deviation measured via beat detection is $2.1\times{10}^{-13}/\sqrt{\tau}$ for each MTS-stabilized laser. In addition, we measure the hyperfine structure of the 7P$_{1/2}$ energy level based on the heterodyne measurements, and calculate the magnetic dipole constant $A$ of the Cs 7P$_{1/2}$ level to be 94.38(6) MHz, which agrees well with previous measurements. This compact optical frequency standard can also be used in other applications that require high-stability lasers, such as laser interferometry, laser cooling, geodesy, and so on.