Fig. 1. Concept of the Vernier spectral interferometry. (a) Optical pulse sampling in conventional dual-comb method. The relative timing between the local (black) and signal (red or blue) pulses proceeds in increments of ΔTr with each sequential pulse. The ΔTr should satisfy the Nyquist sampling theorem, otherwise resulting in under-sampling. (b) After the dispersive time-stretch, the broadened pulses of the local (black) and signal (blue) interact, leading to frequency-varied fringes. The dual-comb interferometry converts into spectral interferometry. (c) Circular mapping of sample distance (the plane spacing marked by the yellow and red circles) to the measured results (d1, d2). Measured results do not exceed the NAR of the two bands.

Fig. 2. Schematic of the experimental setup. Pulse trains generated from dual-comb sources are launched into a dispersion module and divided into sample and local sources by WDMC. The dual-wavelength bands are coupled into the sample and reference arms and separated to interact with local sources. The dual-wavelength-band results are captured by an acquisition module consisting of two photodiodes and one oscilloscope.

Fig. 3. (a) Schematic of mock-locked fiber laser (MLFL) and DPLL. The comb 1 possesses the same configuration as comb 2. The phase locking system is based on a two-channel FPGA platform, and it consists of electrical components, such as mixers, low-pass filters (LPFs), and electrical amplifiers (EAs). (b) The dual-comb spectra are divided into two distinct wavelength bands, λ-band 1 (red shaded area) and λ-band 2 (blue shaded area). (c) Stability of the repetition rate difference: reference clock (red) and dual-comb system (black).

Fig. 4. Signal properties of the Vernier spectral interferometry at different locations. (a)–(c) Signals at distances of 0.960 mm, 100.12 mm, and 308.047 m, respectively. (a1), (a2) Normalized digitized signals of band 1 (red) and band 2 (blue). (a3), (a4) Enlarged views of peaks in (a1) and (a2). (a5), (a6) Fourier transform spectra of the fringes. The gray region denotes the acquisition bandwidth limitation. (b1)–(b10) Signal properties at a distance of 100.12 mm. (c1)–(c10) Signal properties at a distance of 308.047 m.

Fig. 5. Results of measuring displacement. (a), (b) Results of band 1 (marked in red) and band 2 (marked in blue). The insets show the zoom-in views of the results at a distance of 107.048 mm. (c) Averaging results and fitting lines. Results of band 1 are marked by the red triangle, while band 2 is characterized by the blue inverted triangle. (d) Residual error of the measured results and fitting line.

Fig. 6. Results of the stationary sample. (a) Measured results of λ-band 1 (red) and λ-band 2 (blue). (b) The estimated value of p is derived from Eq. (6). (c) Distance measurement results recovered from the two measured results in (a). (d) Allan deviation varies with averaging time.

Fig. 7. Dynamic distance results. A sinusoidal voltage-driven fiber stretcher accomplishes distance variation. (a) Measured results (wrapped) of band 1 (red) and band 2 (blue). (b) Estimated value of p. (c) Retrieved traces of the distance variation. (d) Residuals of the measured results and a standard sinusoidal function.