## 1 Introduction  
Laser interferometers now play a significant role in high-precision displacement measurements. However, the intensity and frequency instability of the light source dominates the measurement uncertainty of displacement-measuring interferometers. Normally, thermally-stabilized He-Ne lasers, of single or dual frequency, are used [1; 2; 3; 4]. However, a frequency stability in the order of \(10^{-6}\)-\(10^{-8}\)and an intensity stability of 0.1% are insufficient for measurements at a nanometer order of accuracy. Moreover, He-Ne laser sources are voluminous and have a short lifetime, low output power, and high operating voltage. Compared with He-Ne lasers, laser diodes (LDs) have some advantageous features, such as compactness, a long operating time, a high output power, and a wide range of wavelengths. However, the frequency of LDs changes rapidly due to temperature fluctuations or impure injection currents. Current efforts aim to stabilize the frequency of LDs by locking to the reference frequency of atoms or molecules [5; 6; 7]. In high-precision displacement measurements, some well-known types of interferometer have been developed: homodyne, heterodyne, and frequency modulation (FM) interferometers. The homodyne interferometer can achieve a measurement resolution at a picometer level in some special conditions [8; 9]. Unfortunately, the measurement uncertainty of homodyne interferometers is dominated by both the frequency and intensity instability of the light source, environmental disturbances, and the misalignment of optics. Moreover, the short measurement range and ambiguity of direction also limit the application of homodyne interferometers in industry. A heterodyne interferometer is less sensitive to environmental changes, but the maximum measurable speed of a heterodyne interferometer is limited by the heterodyne frequency [10; 11; 12]. The high cost and complicated system are other disadvantages of heterodyne interferometers. Among these interferometers, the FM interferometer offers some advantages. The time-dependent interference signal of an FM interferometer is a series of harmonics of the modulation frequency. Two consecutive harmonics in the interference signal can be detected synchronously by lock-in amplifiers (LIAs) to determine the phase shift and then to calculate the displacement. The LIA detects the specific harmonic in which the frequency synchronizes with the reference signal and removes anything unsynchronized with the reference signal. Therefore, the LIA can detect a weak signal buried in a strong noise, and hence, the effect of the imperfect inference signal caused by environmental disturbance or light source instability can be reduced. However, the measurement speed of LIAs is limited by the cutoff frequencies of the lowpass filters, and the cutoff frequencies must be less than the modulation frequency. This means that a higher modulation frequency for LDs can achieve a higher measurement speed of the FM interferometer. Another drawback of the FM is that the harmonic intensities depend on the modulation index of the LD via the presence of the Bessel functions. The modulation index is a function of the modulation excursion [13; 14]. In previous works, the high-frequency stability of LD sources by locking them to I\({}_{2}\) hyperfine transitions was proposed [15; 16; 17; 18; 19]. To lock the frequency of LDs, their frequencies must be modulated and turned around the frequency transition of I\({}_{2}\). LDs offer direct frequency modulation by modulating the injection current and turn over long ranges without mode-hopping. Therefore, electro-optic modulators (EOMs) and frequency shifters are unnecessary. The systems are stable and compact and do not suffer from the residual amplitude modulation effect of EOMs [20; 21; 22; 23]. The relative frequency stability of locked LDs can reach 10\({}^{-9}\) or 10\({}^{-11}\), and they can improve the measurement accuracy of laser interferometers.