The   discontinuity of the extracted phase appears when an extreme value,   or  , is reached; the phase then jumps to the other end of the interval,  or, even though physically the optical phase is continuous and relatively softly increasing or decreasing. As an example, let us consider a continuous phase proportional to a physical quantity (displacement, thickness, velocity...) varying linearly along a pixel row of the detector. This variation is illustrated on figure 2.

Figure 2 : continuous variation of the optical phase  

When demodulating the interferometric signal encoding for this variation, the arctangent function gives a wrapped phase variation, with phase jumps. This property is illustrated on figure 3.

Figure 3 : wrapped optical phase  

Reconstructing the physically continuous phase variation can be done by adding or subtracting multiples of , and thereby suppressing the phase jumps. The operation consisting in determining the order of each fringe and therefore restoring the physical continuity of the phase map is called phase unwrapping [3]. The process usually begins at an arbitrary point of the phase map for which the order is equal to 0. After unwrapping, the phase can be considered to be known within an additional value. This value stays undetermined, unless it is possible to know a point in the phase map where the phase is indeed equal to zero. The problem vanishes for a light source of large spectrum (white light, for example), since the interference fringes are then located around a white fringe corresponding to a zero optical path difference.

Figure 4 represents a line of an experimental phase map obtained after demodulation. The phase jumps are clearly visible.

Figure 4 :  wrapped phase (from an experiment).

Figure 5 shows the result obtained after unwrapping this phase. The starting point of the algorithm is located at pixel 300.

Figure 5 : unwrapped phase  

The zero of this phase variation cannot be easily determined.

This result represents an introduction to phase unwrapping, and the various algorithms will not be detailed. Indeed, phase unwrapping techniques have become more and more sophisticated in the past 10 years with the appearance of powerful algorithms that are sometimes quite difficult to implement. For a complete description, the reader can use reference [3].  

The unwrapping operation is relatively easy to implement when the number of pixels per fringe is large, and when the signal to noise ratio of the interferogram is high. In this case, unwrapping the phase consists in adding/subtracting each time a discontinuity is detected in   modulo . The phase map can be scanned vertically or horizontally. In practice, this ideal case is not very common because the noise level can be quite high, in particular for speckle interferometry methods. Phase unwrapping then becomes extremely complicated.

Algorithms can be classified into two categories:

• those independent from the path followed on the phase map, and called “global” algorithms; they operate in a single step by identifying, isolating and excluding the locations of   that could lead to an error when identifying discontinuities; the unwrapping process is done by following an arbitrary path on the phase map.

• those dependent on the followed path, which are called “local”algorithms; they give a continuous phase, point by point, by following a given path.

Recently developed phase unwrapping algorithms can be efficient even in the difficult situations where the noise level is comparable to the phase value, where interferograms show a small modulation, where the phase exhibits a steep phase jump due to a discontinuity of the measured physical value, or where the pixel density per fringe is small.

Algorithms will be distinguished according to their computation time, their sensitivity to error propagations during unwrapping, and their robustness to previously mentioned elements. Despite the large progress made in the past years, we have to mention that no algorithm can behave positively when many sources of problems add up. Each of them is only efficient to solve one particular problem, and requires additional information to cover all cases. In that sense, up to now, no fully automated algorithm has been developed.

This strategy is therefore mostly insensitive to noise, but on the other hand is very sensitive to fringe openings (loss of continuity of the contour lines  ) and will propagate errors on the whole phase map. Nevertheless, the experimentalist will always  have the possibility to work again on the problematic regions to improve the accuracy of the algorithm.