# A New Method to Verify the Measurement Speed and Accuracy of Frequency Modulated Interferometers
## 1 Introduction  
High-precision technology is increasingly critical for industrial applications. The demand for high-speed and precise processing has been on the rise in various fields. To meet these requirements, different types of displacement measuring sensors are available, including capacitive sensors, linear encoders, and laser interferometers. In a short measurement range, capacitive sensors can obtain sub-nanometer resolution [1, 2]. There are some disadvantages of capacitive sensors, such as sensitivity to temperature and humidity, short stand-off distance, and relatively low bandwidth [3, 4]. Linear encoders, which can measure both distance and displacement at nanometer resolution over a long measurement range, are widely used as machine tools [5, 6]. However, the complex structure and large volume limits their application in ultra-precision measurements. Among these sensors, displacement measuring interferometers have increasingly been adopted because of their high level of accuracy and traceability to the definition of the meter. Numerous displacement measuring interferometers have been developed, such as homodynes [7, 8, 9], heterodynes [10, 11, 12], and frequency/phase modulation interferometers [13, 14, 15, 16]. In open-air environments, the measurement range and accuracy of the homodyne interferometer are reduced due to the refractive index fluctuation [17]. The heterodyne frequency is less than 20 MHz, and, hence, the measurement speed is approximately 5 m/s [18, 19]. For a frequency modulation interferometer, the measurement speed is limited by the modulation frequency of the laser source. Laser diodes (LDs) are widely used as the laser source in interferometers dueto their advantageous features, such as high power, compactness, and long lifespan,. In particular, the high modulation frequency of LDs can be obtained by applying a current modulation at the GHz level [20,21]. Hence, the measurement speed of the frequency modulation interferometer can theoretically reach a level of 10 m/s. Therefore, a considerable challenge for the interferometer is the measurement speed verification at nanoscale uncertainty. Normally, a piezo-electric (PZT) actuator uses the motion of the mirrors for interferometry because of its high motion resolution. However, the travelling speed of the PZT stage is limited to several kHz. In addition, a mechanical displacement PZT system typically shows hysteresis, which is a type of cyclic error [22, 23, 24]. Hysteresis introduces uncertainties into measurements, which should be suppressed or removed. Voice coil actuators are also used for many high-precision motion applications [25,26]. The actuator can achieve a resolution of 2 nm over a range of 1 mm [27]. Two major disadvantages of voice coil actuators are heat output and disturbance from moving wires [28]. Hence, voice coil actuators are not suitable for wide-range nano-positioning at high speed. The measurement speed and measurement precision of displacement measuring interferometers are key parameters. For interferometers, the measurement speed or rate implies the time required to take one displacement measurement in the unit of seconds or Hz. In our previous works, a high modulation frequency of 3 MHz was successfully applied to the LD to improve the measurement speed of the frequency modulated interferometer [13,14,29]. Moreover, a high-precision phase meter was developed to measure the phase change in the interferometer [12]. To verify the measurement accuracy of our proposed interferometers, the displacement of the target mirror was measured using a capacitive sensor (D100, Physik Instrumente) integrated into the PZT stage. However, the bandwidth of the capacitive sensor was less than 3 kHz, and the resolution was 1 nm over a displacement range of 2 \(\mu\)m [13,14]. Therefore, it remains a significant challenge to verify both parameters using a nanometer resolution and high-speed displacement actuator. In this paper, a high-speed virtual displacement without hysteresis is proposed and validated.