Results and Discussions In the experiment, the piezoelectric ceramics were loaded with signals of 600, 700, 800, and 900 kHz, and the influence of low-frequency noise on displacement measurements was explored (Fig. 4). The results show that the noise in the measured noise power spectra in this frequency band remain higher than the shot noise reference, although no signal less than 600 kHz is loaded on the piezoelectric ceramics. This is because a variety of noises in the experimental environment, such as those from mechanical vibration, beam pointing, and laser intensity, affect the measurement results. In the experiment, the signal changes under different driving voltages were measured at a center frequency of 2 MHz. Upon adjusting the driving voltage, the noise in the power spectrum of the signal light at 2 MHz is 3 dB to 7 dB higher than the shot noise baseline (Fig. 7). At 2 MHz, the signal-to-noise ratio can be increased by increasing the optical power of the signal (Fig. 9). In the experiment, by changing the driving voltage amplitudes under different signal powers, the noise in the signal light is set 3 dB higher than the shot noise. The measurement results show that the minimum measurable displacement of the system decreases with an increase in the optical power of the signal. When the signal light power is 90 μW, the minimum measurable displacement is 0.176 nm.

Measurement has always been important in science; the improvement of precision measurement technology can facilitate the exploration of the micro world and has great significance in daily life and production activities. Atomic clocks, for example, can accurately measure time and frequency, which is crucial for coordinating information systems. Similarly, the accurate measurement of displacement is important. For example, the measurement of beam lateral displacement can be used in many fields, such as in atomic force microscopy, optical imaging, space satellite position stability, optical tweezers, and biological measurements. In displacement measurement, noise interference from mechanical vibration, beam pointing, laser intensity, and other factors is inevitable. These noises affect measurement results and increase the difficulty of measurement. However, most of the noise is concentrated at low frequencies. To improve the signal-to-noise ratio of beam lateral displacement measurement, we test the influence of signal optical power on beam lateral displacement measurement. We prove that increasing the power of the light signal can improve the signal-to-noise ratio of beam lateral displacement measurement and reduce the minimum measurable displacement.

Figure 3 shows the experimental devices. An all-solid-state single-frequency laser outputs infrared light with a wavelength of 1064 nm through a Faraday optical isolator and phase modulator. A beam splitter then divides the infrared light into two parts. One part of the laser generates a TEM₀₀ mode signal through mode conversion cavity 2 (MC2). The other part uses mode conversion cavity 1 (MC1) to generate the TEM₀₀ mode required for interference adjustment or TEM₁₀ mode as background light. MC1 and MC2 are locked by two sets of electronic servo systems. Before the displacement measurement, the TEM₀₀ mode is outputted by MC1, and its power is adjusted to the same level as that of the signal light. Simultaneously, a high-voltage amplifier is used to drive a piezoelectric ceramic 1 (PZT1) to control the local light phase of a balanced homodyne detection system, and the interference visibility of the two beams is then adjusted to ensure high coincidence of the two beams after passing through the beam splitter. Following the interference visibility adjustment, the TEM₁₀ local light generated by MC1 is locked with the power of 1 mW, and the TEM₀₀ signal light generated by MC2 is locked with the power of 50 μW. PZT2 is driven by a sinusoidal signal outputted by an arbitrary waveform generator to produce a small optical transverse displacement. The noise power spectra of signal light under different measurement conditions can be obtained by changing the signal optical power or adjusting the frequency and intensity of the signal source.

In this study, the minimum measurable displacements at different frequencies from 0 Hz to 3 MHz are tested, and the signal-to-noise ratios obtained at 2 MHz for different driving voltages are measured. The changes in the signal-to-noise ratio and minimum measurable displacement of the signal light for different powers are investigated. This study will promote research on methods and techniques for improving the extent of beam lateral displacement measurement and provide a theoretical and experimental basis for the application of beam transverse displacement measurement.