# Far Off-Resonance Laser Frequency Stabilization Technology
## 2 Far Off-Resonance Laser Frequency Stabilization by Pound-Drever-Hall Technique
at National Physical Laboratory (NPL) placed an optical reference cavity on an active vibration isolation platform to reduce external vibration noise [29]. The laser locked by the PDH technique achieved sub-hertz linewidth and relative frequency stability of \(10^{-15}\) level at 5 s. Since then, commercially passive or active vibration isolation platforms have been adopted for vibration reduction [30; 31; 32]. [PERSON] et al. introduced the first vibration-insensitive optical cavity at the Joint Institute for Laboratory Astrophysics (JILA) in 2005 [33]. Figure 2 shows that it is a vertically-oriented low-expansion glass cavity, and the cavity support points are arranged to be in the symmetry plane between the two mirrors on the cavity ends. Noteworthy, this design takes the advantage of potential constant length distance between the two mirrors by setting the optical axis along the direction of gravity. The acceleration along the direction of gravity causes the compression of the upper part accompanied by stretch of the lower part, which results in an unchanged length of the optical cavity. The associated reduction of the vibration sensitivity of the effective cavity length resulted in a simple and compact reference cavity system for laser stabilization at the level of 1 Hz linewidth. [PERSON] et al. at JILA locked a diode laser to the thermal-noise-limited passive optical cavity in vertical mounting configuration by PDH technique [31]. When the resolution band width (RBW) was reduced to 150 mHz, linewidths of 220 mHz could be observed with non-negligible power in low-frequency noise sidebands. Based on the design concept of the vibration-insensitive optical cavity, horizontally-oriented vibration-insensitive  
Figure 1: Schematic illustration of the PDH technique. PDH: Pound–Drever–Hall; EOM: electro–optic modulator; PBS: polarizing beam splitter; \(\lambda/4\): quarter wave plate; and F–P: Fabry–Pérot. optical cavities were also designed similarly through numerically simulations by finite-element analysis [34; 35; 36]. High frequency-stabilized, narrow-linewidth lasers have been constructed by PDH technique to lock the laser frequencies to the resonance of reference cavities. The frequency stability of a laser which is locked to a reference cavity is dominated by the effective length stability of the reference cavity. To maintain the stability of the cavity length, glass or single crystal silicon with ultra-low thermal expansion coefficient (ULE), vacuum chambers with high temperature stability, special geometry, and mounting configurations for reference cavities were developed for less sensitivity to vibration and temperature fluctuation [37; 38; 39; 40; 32]. Thermal noise arises from the Brownian motion, which can be naturally reduced by cooling optical cavities to cryogenic temperature. An optical cavity supported in a vibration-insensitive configuration was designed by Physikalisch-Technische Bundesanstalt (PTB) and JILA. The cavity spacer and mirror substrates made of single-crystal silicon were cooled to 124 K [41]. The thermal-noise-limited frequency instability was estimated to be around \(7\times 10^{-17}\). With vibration-insensitive configuration for the optical cavity, a laser at \(1.5\,\mu m\) was locked to the optical cavity, resulting in a linewidth of less than 40 mHz and a fractional frequency instability of \(1\times 10^{-16}\) at short timescales in 2012. In 2017, the individual laser linewidth obtained from the phase noise spectrum or the direct beat note between the two lasers was as small as 5 mHz at 194 THz [42]. In PDH technology, a laser is usually locked to the F-P cavity. With a very narrow cavity mode, the high-finesse cavity can reduce the linewidth, suppress the phase noise, and improve the short-term stability of the laser. Narrow-linewidth lasers achieved by the PDH locking technique are central to precision spectroscopy, optical atomic clocks, gravitational wave detection, and tests involving fundamental physics.