Fig. 1. Schematic diagram of fiber-optic Michelson microprobe and displacement interference-sensing method. SMF, single-mode fiber; GRIN, gradient refractive index lens; NPBS, non-polarizing cube beam splitter; w0, beam waist radius of light source from SMF; wt, beam waist radius of fiber Michelson microprobe; l, beam waist position of microprobe. The upper-right diagram displays the transmission state of the Gaussian beam across the microprobe sensor and spot intensity distributions at SMF end face, GRIN output, and beam waist position.

Fig. 2. Coordinate system for (a) measurement and (b) reference optical field transmission of fiber-optic Michelson microprobe with misaligned GRIN. M1, target mirror; M2, reference reflective surface of NPBS inner surface; L0, distance between SMF and GRIN; Lg, GRIN length; p, distance between GRIN and NPBS; dt, size of NPBS; Zwd, working distance of fiber microprobe; d, d1, radial displacement of GRIN concerning SMF optical axis; O, O1, centers of incident and emergent planes of the GRIN; θ, tilt angle of GRIN relative to SMF end face. θ, d directions shown are positive directions. Blue arrows represent reference planes on misaligned optical axis (blue dashed lines), and red lines represent reference planes corresponding to centers of mirror-symmetric expansion.

Fig. 3. Effect of (a) radial displacement and (b) tilt angle of GRIN concerning SMF end face on spatial-fiber coupling efficiency (Zwd=l=60  mm). (c) Measurement light-coupling efficiencies as a function of working distance at different misalignment amounts. (d) Schematic diagram for effect of spatial-fiber coupling efficiency on interference contrast under microprobe element misalignment. Red beam represents measurement light; blue beam represents reference light. Effect of (e) radial displacement and (f) tilt angle of GRIN concerning SMF end face on interference contrast (Zwd=l=60  mm).

Fig. 4. (a) Nonlinear displacement errors (Zwd=l=60  mm) with conventional PGC and proposed PGC demodulation method under interference light intensity variation. Inset: nonlinear error amplification curve with the proposed PGC demodulation method. (b) Microprobe-sensing signal demodulated phase SINAD under different alignment deviations between GRIN and SMF. Displacement nonlinear errors of microprobes (c) with conventional PGC and (d) proposed PGC demodulation method under different alignment deviations of GRIN. Inset: residual errors amplified curve with ideal alignment.

Fig. 5. Schematic diagram of experimental verification setup for microprobe interference-sensing model and nonlinear errors correction in microprobe interferometric sensor. FC, fiber-optic circulator; PIN, InGaAs photodetector; ADC, analog-to-digital converter; DDS, direct digital frequency synthesizer; DAC, digital-to-analog converter. Inset: schematic diagram and actual photo of the fiber-optic microprobe sensor.

Fig. 6. Experimental and simulation results of spatial-fiber coupling efficiencies and contrast at (a), (b) different radial displacements and (c), (d) different tilt angles of GRIN. Blue curves represent the measurement light-coupling efficiency, and red ones represent the reference light-coupling efficiency.

Fig. 7. (a) Coupling efficiency and interference contrast of assembled microprobe sensor. (b) Demodulation phase SINAD of microprobes with optimized assembly and alignment deviation at different target displacements. (c) Residual nonlinear errors of microprobe sensing under GRIN alignment deviation after the conventional PGC demodulation, proposed PGC demodulation, and combination with the interference-sensing model. (d) Enlarged view of the gray shaded area of (c).