Temporal phase unwrapping for under-sampled spectral signal with tilted interference

Fourier transform Speckle profilometry can provide profiling capabilities, eradicating 2𝜋 phase ambiguities through spectral signals obtained using wavelength tuning. Limitations in the wavelength tuning steps and range restrict the range and resolution of height differences that could be measured. In addition, in the alignment of the interferometric setup, the absolute path delay of the sample with respect to the reference arm needs to be set within the range to provide the carrier frequency in the interference signal for subsequent analysis through temporal Fourier Transform. On the other hand, the removal of phase ambiguities through an unambiguous synthetic phase map offers profiling capabilities where the resolution of the height measurement can be improved through the subsequent reduction of synthetic wavelengths generated from a set of wavelength-tuned interferograms. We explore temporal phase unwrapping for an under-sampled spectral signal resulting from an arbitrary long path delay between the sample and reference arm. Tilted interference is employed to access the phase of individual wavelength-tuned interferograms. The spatial carrier frequency in the interference pattern allows us to determine the phase through spatial Fourier Transform. We demonstrate the profiling capability using a tunable laser light source having a wavelength tuning range of 36nm ( 900nm to 936nm) that can be scanned in steps of 0.05nm. The relaxation in the path delay between the sample and reference arm, unrestricted by the minimum scanning step of wavelength, can extend the measurement capabilities.