Ambiguity on the phase sign

When extracting the phase   from an interferogram using an analytic inversion of the fringe pattern, we have to solve the problem of the phase sign, which appears because the cosine function is not bijective but is even and periodic. Indeed, a phase determined using a single fringe pattern remains undefined with respect to the sign and to an additive constant, that is, an integer multiple of . Each phase demodulation method includes an inverse trigonometric function. Each inverse trigonometric function can be expressed using the arctangent function, for example as .

The arctangent function gives a result included in the interval .

For most phase demodulation techniques, the argument of the arctangent function is a ratio for which the numerator characterizes the phase sine and the denominator the phase cosine. It is thus interesting to consider numerator and denominator independently to obtain a value in the interval by considering the sign of each quantity. The four possible situations as for the sign of the sine, cosine and tangent are indicated on figure 1.

However, the previous step allows a determination on the interval and is not sufficient to suppress the ambiguity about the phase constant (i.e. modulo ) and the sign.

A convenient technique to extract the sign is to take into account some information about the experimental conditions generating the optical phase, and the signal distribution within the fringe pattern. For example, the direction in which the studied structure moves can be known « a priori », which will determine the sign of the phase variation.