The detuning phase shift error is a common systematic error observed in temporal phase shifting (TPS) algorithms. This error, generally due to miscalibration of the phase shifter, is solved by using a quadrature filter insensitive to this detuning error. To compare algorithms, this error is frequently analyzed numerically. However, in this work we present an exact and analytical expression to calculate such error which is applicable to any kind of filters with real or complex frequency response. Finally, a table with the detuning error for several algorithms is reported.
© 2009 OSA
Temporal Phase Shifting (TPS) techniques are used widely for wave front extraction [1–8]. However, the accuracy of these measurements is limited by the presence of several systematic errors. One important error to estimate is the detuning phase shift error, which is due to a miscalibration of the phase shifter and it is introduced by the data gathering process. To minimize the error due to detuning, the temporal signal’s carrier must have exactly the carrier frequency assumed in the TPS algorithm; otherwise, an erroneous phase is estimated. In a previous work , we obtained an exact expression by using a phasorial method to analytically calculate the detuning phase shift error of the TPS algorithm. However, this demonstration is only valid for symmetrical TPS algorithms having a real frequency response. Although, in ref  the author show how a non symmetrical filter can be transformed into a symmetric filter, the main purpose of this paper is to find a general useful expression to calculate the detuning phase shift error in terms of the frequency response of any TPS algorithm, symmetrical or not without the use of any transformation for the filter. This paper is organized as follows: in Section 2 we discuss the fact that a detuning phase shift error is a very common systematic error, and we obtain a generalized expression to calculate this error. In Section 3, some particular cases of non symmetrical TPS algorithms are considered, as three and four frames cases, and a table of analytical expressions of detuning errors for several known algorithms is reported. Finally in Section 4, the conclusions are discussed.
2. Error Detuning in Phase-Shifting Interferometry (PSI)Eq. (3), we have , and it can be expressed asEq. (5), we have three components, and to obtain a quadrature filter we only have two possible options, as long as . The first one is the case for a filter tuned onto the right side of the frequency axis, which for frequencies and , meets the condition , while . The second option is for a filter tuned onto the left side of frequency axis, and the quadrature conditions for the frequencies and are with . That is, the quadrature conditions are given only by the magnitude of the filter. Then, there are two possible solutions to recover the desired phase. Therefore, for a quadrature filter tuned onto the right side case, the output of the TPS algorithm, becomesFig. 1 and are given by,Eq. (8) and rearranging we haveEq. (10) isEq. (11) has been reported previously in [7,8], where, σ is the correlation factor. In the same way, r is the ratio of detuning, and the relationship between both values is given by1–8]. Substituting Eq. (11) into the definition of detuning error, we find that this error is expressed as,7,8]. the authors find the same equation, but they are only able to calculate an approximation for it. In a previous work , by using a phasorial method, we find an expression for the detuning error applicable to symmetrical filters as a function of ratio r. Here, we expand our previous work to both symmetrical and non-symmetrical quadrature filters, and additionally, we establish the relationship between both formalisms.
To solve Eq. (13), we take the tangent for both terms, and we have,Eq. (11) and after some trigonometric substitutions, Eq. (14) becomes,8], the ratio reported in this work is more general than the previously reported ratio, in spite of both being referred to as r. That is, from Eq. (15) and the ratio r we can obtain the exact detuning error for any TPS algorithm. Then, from Eq. (15) we can observe that if , then and , no detuning error is present, and the erroneous phase becomes the desired phase . On the other hand, assuming a small detuning error, we have that and . Hence, Eq. (16) is reduced to,7,8]. Notice that this result is almost the same result presented here; however, we consider the expression here reported to be more practical. To compare against the results reported in literature, we maximize our exact result in Eq. (17) with respect to φ , obtaining the following expression,Eq. (18) for the maximum detuning error is exact when tuned onto the left side; in this fashion, it coincides exactly with the detuning error that was evaluated numerically [1–7]. We can repeat all the steps described above for a quadrature filter tuned onto the right side of the frequency axis, or for sign minus in Eq. (8) and we obtain the following result, which is equivalent to having changed the sign of , then, we have9]. This expression is a very versatile way to evaluate the detuning phase shift error analytically or numerically, instead of the approximation reported in literature [7,8].
3. Some Examples of Error Detuning in Phase-Shifting Interferometry
In this Section we analyze some popular TPS algorithms, such as the three and four frame cases.
3.1 Three frame algorithm case
One three frame non symmetric TPS algorithm is given byEq. (18), the exact detuning error for is
3.2 Four frame algorithm
The four non symmetric frame TPS in cross algorithm is given byEq. (19) we obtain,
3.2 Other algorithms
In Table 1 , the value r, for some detuning phase shift errors for several TPS algorithms are presented. We notice that many of them have the form tann(Δ/2) for n integer.
An exact and analytical algorithm to evaluate the detuning error in phase shifting algorithms was obtained from algebraic methods. The expression is applicable to any kind of (PSI) algorithms, symmetrical or not. The derived expression was compared with other well known approximations. Finally, this expression was successfully applied to evaluate and obtain the detuning error for some well known quadrature filters.
This work was partially supported by CONACyT under grant No. 42771.
References and links
1. J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf ed., (North Holland, Amsterdam, Oxford, New York, Tokyo, 1990).
2. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometry for testing optical surfaces and lenses,” Appl. Opt. 13(11), 2693–2703 (1974). [CrossRef] [PubMed]
3. H. Schreiber, J. H. Brunning, and J. E. Greivenkamp, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara ed., (John Wiley & Sons, Inc., Hoboken, New Jersey 2007).
4. M. Servin, and M. Kujawinska, “Modern fringe pattern analysis in Interferometry,” in Handbook of Optical Engineering, D. Malacara and B. J. Thompson eds., (Marcel Dekker, 2001).
5. D. Malacara, M. Servin, and Z. Malacara, “Phase Detection Algorithms,” in Interferogram Analysis for Optical Testing, D. Malacara ed., (Taylor & Francis Group, 2005).
6. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22(21), 3421–3432 (1983). [CrossRef] [PubMed]
7. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase measuring interferometry,” J. Opt. Soc. Am . A 7(4), 542–551 (1990). [CrossRef]
8. J. E. Hernández and D. Malacara, “Exact linear detuning error in phase shifting algorithms,” Opt. comm. 180, 9–14 (2000).