# Far Off-Resonance Laser Frequency Stabilization Technology
## 2 Far Off-Resonance Laser Frequency Stabilization by Pound-Drever-Hall Technique  
In 1946, [PERSON] first reported that microwave oscillators successfully derived stabilization of their frequency from a resonant cavity [23]. In 1983, after [PERSON]'s studies on radio frequency microwave stabilizer, [PERSON] et al. first locked a dye laser and gas laser onto a stable F-P reference interferometer, and jet stream dye lasers could achieve sub-100 Hz linewidth [24]. Subsequently this technique was called Pound-Drever-Hall (PDH) technique. The PDH technique is based on signals reflected from a stable F-P cavity. A schematic illustration of the PDH technique is presented in Figure 1. A laser beam, the electric field of which is given by \(E=E_{0}e^{i\omega t}\), is phase-modulated using an electro-optic modulator (EOM) with a phase modulation frequency of \(\Omega\) at local oscillator (LO) range and a depth of \(\beta\). After the beam passes through this EOM, its electric field is given by \(E^{\prime}=E_{0}e^{i[\omega t+\beta\sin{(\Omega t)}]}\). When \(\beta<1\), the electric field can be simplified as \(E^{\prime}=E_{0}[J_{0}(\beta)e^{i\omega t}+J_{1}(\beta)e^{i(\omega+\Omega)t} -J_{1}(\beta)e^{i(\omega-\Omega)t}]\) with Bessel functions [15; 25]. This indicates the existence of actually three different beams being incident on the cavity: a carrier, with frequency \(\omega\), and two sidebands with frequencies \(\omega\pm\Omega\). Further, the modulated light is incident into an F-P cavity, and the reflected beam from the F-P cavity is sent into a photodetector using a polarizing beam splitter (PBS) and a quarter wave plate (\(\lambda/4\)). Output of photodetector is compared with the signal of LO via a mixer. The mixer's output contains signals at both direct current (or very low frequency) and twice the modulation frequency. After passing through a low-pass filter, the frequency discrimination signal (the PDH signal) is obtained as \(\varepsilon=2\sqrt{P_{c}P_{s}}Im[F(\omega)F^{*}(\omega+\Omega)-F^{*}(\omega)F( \omega-\Omega)]\)[15; 25], which then passes through a servo and into the tuning port on the laser, thus locking the laser to the cavity. In 1984, [PERSON] et al. mounted the cavity in a cradle on a 5-wire pendulum suspension to minimize vibrational disturbances and locked a ring dye laser to this independent cavity [26]. By heterodyning the light from two such independent systems and studying the fluctuations in the resulting beat frequency spectrum, single laser linewidths of the order of 750 Hz were successfully achieved and this corresponds to a standard Allan variance of about \(2.2\times 10^{-13}\) with integration time of 1 s. Furthermore, they also investigated a number of other sources of noise such as residual optical feedback from laser light scattered or reflected back into the cavities using external optical components. In 1992, [PERSON] et al. locked two-diode laser-pumped Nd:YAG lasers to an F-P interferometer with a finesse of 27,500, yielding a robust and easily repeatable beat note linewidth of 700 mHz [27]. In 1999, [PERSON] et al. at the National Institute of Standards and Technology (NIST) narrowed the linewidth of a 563 nm laser beam to 0.6 Hz by the PDH technique. The fractional frequency instability for the light locked to the cavity was found to be \(3\times 10^{-16}\) at 1 s [28]. The performance of the previous study on narrow-linewidth lasers was limited by length instability of optical reference cavities due to imperfect vibration isolation. [PERSON] et al. isolated the high-finesse (\(\mathcal{F}>150,000\)) F-P cavity, where the spacer between the cavity mirrors was composed of a low-thermal-expansion material. The cavity was protected from vibrational noise by mounting it on an aluminum v-block inside an evacuated chamber. The vacuum chamber was mounted on a passively isolated optical table which was suspended by vertical strands of surgical tubing stretched to \(\sim\)3 m. The method for vibration reduction was complex and thus hard to realize. In 2004, [PERSON] et al.