Fig. 1. (a) Noise source for real-time DSI. Phase noise from the laser source brings the variation in the filter window of the Fourier domain. Intensity noise is added to the amplitude of spectral fringes via the photodetector and analog–digital converter. (b) Theoretical resolution calculated by the model, which evolves with data volume and SNR. The lower evolution result accounts solely for the impact of intensity noise, whereas the upper one incorporates the effects of phase noise. (c) Principle of phase noise elimination. When the orthogonally polarized target and reference pulses are incident at a 45 deg angle to the PBS, they decompose along its fast and slow axes, forming a set of IGMs with a π phase difference that are used to extract and eliminate the phase noise in an envelope. (d) Theoretical resolution of real-time OPSI varies with measurement frame rate and detection bandwidth.

Fig. 2. (a) Data processing procedure for real-time OPSI. The time series data recorded by the oscilloscope are interpolated and segmented according to roundtrip time (panels I and II). Then, the time-domain interferograms are mapped to the frequency domain and are de-enveloped using two interferograms with orthogonal polarization (panels III and IV). The phase results are extracted either by performing a cosine fitting on the de-enveloped IGMs or by applying a linear fit to the Fourier-filtered phase curve, both of which are fundamentally equivalent (panel V). (b) Numerical simulations performed across different levels of phase noise, with intensity noise fixed at −40dB. (c) Numerical simulations conducted with varying intensity noise levels, keeping phase noise constant at −75dB. The orange and red curves in panels (b) and (c) illustrate the noise-limit resolution before and after correction. The green and purple circles represent the phase resolution results for DSI and OPSI, respectively.

Fig. 3. Schematic diagram of experimental setup. OSC, oscilloscope; PD, photodetector; PBS, polarization beam splitter; OC, optical coupler; PM, phase modulator; Cir, circulator; Col, collimator; Mir, mirror; PM ISO, polarization-maintaining isolator. PC, polarization controller; WDM, wavelength division multiplexer; DCF, dispersion compensation fiber; ISO, isolator; Insets: time series signals detected by two PDs.

Fig. 4. (a) Spectral evolution along with time before de-enveloping. Left: single-shot interference fringe in panel (a). Right: optical spectra acquired simultaneously by an optical spectrum analyzer. Below: normalized phase for data in panel (a) retrieved using the CPFT method. (b) Spectral evolution along with time after the de-envelope. Left: single-shot interference fringe in panel (b) [Interference data (light red circle) and the fitting results of IGM after de-envelope with φ (green curve) and without φ (black curve)]. Below: normalized phase for data in panel (b) retrieved using the cosine fitting method. (C) De-enveloped spectra with fitting lines under three different detection bandwidths. (d) Normalized phase evolutions of real-time OPSI under three different bandwidths (Blue circle: 4 GHz, red square: 1 GHz, and green triangle: 500 MHz).

Fig. 5. (a) Spectral evolution with a 200 kHz phase modulation. (b) Phase evolutions for real-time DSI (gray triangles and cyan squares represent solutions derived by the FT and CPFT methods, respectively) and real-time OPSI (purple circles) under driving voltage amplitudes of 1 V and 5 mV. (c) Zoom-in phase evolution under 5 mV and phase evolution with an update rate of 2 MHz. (d) Phase evolutions under the modulation frequencies of 2 and 5 MHz.

Fig. 6. (a) Improvement of resolution with decreasing frame rate. (b) Normalized phase with frame rates of 22.2 MHz (green circle) and 40 kHz (red circle) over a time duration of 500μs. (c) A comparison of the real-time OPSI and a commercial picometer interferometer for measuring a damped oscillation cycle of ∼125Hz with an amplitude of ∼150nm. (d) A zoom-in of two regions (a) and (b) from panel (c) highlights the real-time OPSI’s ability to maintain the same resolution as the picometer interferometer while achieving 10 times the temporal resolution.