Spatio-spectral phase unwrapping for multi-wavelength heterodyne interferometry

Optical path difference (OPD) measurements typically are characterized by 2 pi phase discontinuities, since optical interference can only determine the wavefront module the wavelength. Obtaining an estimate of the actual phase profile from OPD measurements necessitates removing the discontinuities, or 'unwrapping' the phase. Monochrome phase unwrapping algorithms have been developed over the years in two broad categories - local and global. Local algorithms generally work by starting at some initial point within the OPD map, assigning an arbitrary surface profile value at that point, and working outward from that location, detecting and removing phase jumps pixel-by-pixel. Global algorithms seek simultaneous solutions over the entire map, or on extended regions within it, that minimize overall error according to some criterion. For macroscopic surface feature measurements, digital differencing techniques can be used to extend the measurement range from optical wavelengths to synthetic wavelengths on the order of centimeters or more, at the expense of accentuated noise effects. Surface estimates obtained by conventional unwrapping techniques applied at different synthetic wavelengths may not agree closely, especially if the physical surface has discontinuities. This paper presents a new algorithm which uses phase information at each of a number of synthetic wavelengths to produce a composite surface estimate that has the macroscopic surface feature capture capabilities associated with long synthetic wavelengths, yet also has the low-noise behavior associated with shorter synthetic wavelengths. The algorithm has been tested using both modeled surface data and laboratory heterodyne measurements of actual calibrated surfaces.