Analytical procedure for determining lengths from fractional fringes

Abstract

The development of stabilized multifrequency lasers makes fractional fringes an increasingly attractive technique for length measurement. Determination of an unknown length from the measured fractional fringes is aided by the development of analytical equations for the length and its uncertainty, and criteria are given for selecting the wavelengths.

Synthetic adjacent pulse repetition interval length method to solve integer ambiguity problem: theoretical analysis

“…The PU problem, which has been extensively studied previously [19]- [21], is only briefly reviewed here to introduce basic concepts. Assuming h to be the distance between two points, the PU problem can be expressed as…”

“…As described in previous studies [19,20], when we know that h < Λ 12 in advance, we can estimate the integral part ϕ N12 based on the two excess fractional parts. Then, the true value of the distance h can be unambiguously determined.…”

Algebraic solution for phase unwrapping problems in multiwavelength interferometry

“…An interesting example that highlights this difference is depicted in Fig. 3 of reference [6], where the strategy in [2,3] is limited to n × Δ2 < Δ1 with n = 1, 2, 3, etc.. Other phase retrieval techniques as EF [6,[10][11][12], CRT based techniques [7][8][9], and other methods [4][5][6]13], reduce the ambiguity in M01 and thereby increase the UMR by utilizing the information in the remaining factional fringe orders. Here, this is accomplished by calculating the value of μ12 using (M01+E01)Λ01 = (M03+E03)Λ03, …”

“…Extended beat-wavelength approaches [2][3][4] tried to overcome this problem, by extending the measurement range beyond the largest beat wavelength. Such systems extend the UMR, or in other words, relax the requirements on the wavelength separations for a given UMR to be achieved.…”

“…Multi-wavelength interferometry (MWI) has been developed for a number of applications in the field of optical metrology to overcome this problem. Conventional [1] or extended [2][3][4] beat wavelength approaches increase the unambiguous measurement range (UMR) for a wide range of measurement wavelengths, however, these procedures do not always make use of the entire information contained in the measured phase [5,6]. Other methods such as the Chinese Remainder Theorem (CRT) [5,[7][8][9], are restricted to particular sets of measurement wavelengths [5] and require in the presence of noise a partial least squares approach [6,9].…”