We propose the use of the sinusoidal frequency modulation technique to improve both the frequency stability of an external cavity laser diode (ECLD) and the measurement accuracy and range of a displacement-measuring interferometer. The frequency of the ECLD was modulated at 300 kHz by modulating the injection current, and it was locked to the b ₂₁ hyperfine component of the transition 6-3, P(33), ¹²⁷ I ₂ (633 nm) by the null method. A relative frequency stability of 6.5 × 10 ⁻¹¹ was achieved at 100 s sampling time. The stabilized ECLD was then utilized as a light source for an unbalanced Michelson interferometer. In the interferometer, the displacement and direction of the target mirror can be determined using a Lissajous diagram based on two consecutive and quadrant-phase harmonics of the interference signal. Generally, the measurement range of the interferometer by the proposed method is limited by the modulation index and the signal-to-noise ratio of the harmonics. To overcome this drawback, suitable consecutive harmonic pairs were selected for the specific measurement ranges to measure the displacement. The displacements determined in the specific ranges by the proposed method were compared with those observed by a commercial capacitive sensor. From the comparison, the proposed method has high precision to determine the displacement. The measurement range was also extended up to 10 m by selecting a suitable modulation index and suitable consecutive pairs of harmonics.

In this paper, we present 10-pm-order mechanical displacement measurements using heterodyne interferometry. The measuring system includes a single-path heterodyne interferometer and a phase meter based on a phase-locked loop (PLL). It is not easy to measure a mechanical displacement of 10 pm or less owing to electronics and environmental noises in the interferometer. To solve this problem, the improvement of the noise floor is required for the phase meter. A PLL algorithm, which is programmed on a field-programmable gate array module, is used for efficient noise reduction of the phase meter. The interferometer combined with a stiff piezoelectric flexure stage is placed in a vacuum chamber. The measurement comparisons and the noise floor evaluations are performed between air and vacuum to evaluate effects from their environments. The interferometer has two spatially separated beams with different frequencies and two balanced optical arms. The measurement results demonstrate that the system combined with the above components is capable of measuring mechanical displacements of 11 pm in air and vacuum. A noise floor of 0.2 pm/ H z between 50 Hz and 100 Hz can be obtained in vacuum. In this paper, the setup of the interferometer, the signal processing of the PLL, experiments, and results are discussed.

We propose a sinusoidal phase modulation method to achieve both the frequency stabilization of an external-cavity laser diode (ECLD) to an ¹²⁷ I ₂ saturated absorption transition near 633 nm and displacement measurement using a Mach–Zehnder interferometer. First, the frequency of the ECLD is stabilized to the b ₂₁ hyperfine component of the P(33) 6-3 transition of ¹²⁷ I ₂ by combining sinusoidal phase modulation by an electro-optic modulator and frequency modulation spectroscopy by chopping the pump beam using an acousto-optic modulator. Even though a small modulation index of m = 3.768 rad is utilized, a relative frequency stability of 10 ⁻¹¹ order is obtained over a sampling time of 400 s. Secondly, the frequency-stabilized ECLD is applied as a light source to a Mach–Zehnder interferometer. From the two consecutive modulation harmonics (second and third orders) involved in the interferometer signal, the displacement of the moving mirror is determined for four optical path differences ( L ₀ = 100, 200, 500, and 1000 mm). The measured modulation indexes for the four optical path differences coincide with the designated value (3.768 rad) within 0.5%. Compared with the sinusoidal frequency modulation Michelson interferometer (Vu et al 2016 Meas. Sci. Technol . 27 105201) which was demonstrated by some of the same authors of this paper, the phase modulation Mach–Zhender interferometer could fix the modulation index to a constant value for the four optical path differences. In this report, we discuss the measurement principle, experimental system, and results.

This paper shows how to suppress residual amplitude modulation (RAM) appeared in a commercial electro-optic modulator (EOM) to improve an iodine-frequency-stabilized laser diode (LD) using frequency modulation spectroscopy (FMS). Since the RAM of an EOM is affected by the temperature of the EOM, the DC-offset applied to the EOM and the polarization of the incident beam, it should easily fluctuate with drifts in temperature, applied voltage, and polarization. The fluctuation of the RAM of the EOM, which might decrease the frequency stability of the iodine-frequency-stabilized LD, must be eliminated by some means. In this paper, we present a theoretical analysis and experimental results to express the RAM error from the EOM in a LD frequency stabilization system near 633 nm using iodine-saturated absorption. A RAM compensation system, which actuates the temperature of the EOM and the DC-offset applied to the EOM, is constructed for the commercial EOM. Using the RAM compensation system, the RAM is reduced to 10^− 4 order, and frequency stability of ~ 3.67 × 10^− 11 for 1000 s is achieved for the LD. The proposed method can be applied for normal frequency stabilizing LDs, which use a commercial EOM and a low modulation index with FMS.