In many metrological applications the data being measured is associated to the phase difference codified in two fringe patterns. This phase difference can be recovered directly with what are called Differential Phase Shifting Algorithms (DPSAs) by using a combination of irradiance values from both patterns in the arctangent argument. Use of such algorithms requires characterisation mechanisms to inform of their sensitivity to the various random and systematic error sources, which is the same as for well-studied Phase Shifting Algorithms (PSAs). Thus, we present a new analysis of error propagation for DPSAs taking into account the frequency shifting property of the employed arctangent function. The general analysis is verified for significant specific cases associated to large errors that appear during phase difference evaluation using the Monte Carlo method, which provides a characterisation of a DPSA’s sensitivity; this is an alternative to spatial and temporal techniques but has wholly coinciding results. Monte Carlo simulation opens up the possibilities for the analysis of other error types for any DPSA.
©2010 Optical Society of America
1. Differential evaluation
Phase shifting algorithms (PSAs) [1,2] are the most widely used and studied evaluation tool as they provide optical phase results with high precision in a greater measurement range and little influence from stationary noise.
It is well known that PSAs use at least M=3 irradiance values sm(ϕ,αm) shifted uniformly in phase by an amount of αm and with a harmonic contribution given by the ak(r) coefficients3]4] for both patterns ϕ, Eq. (2), and ϕ+Δϕ, Eq. (4).
However, Δϕ can be obtained directly using Differential Phase Shifting Algorithms (DPSAs) – with a very similar design to the PSAs from which they inherit their characteristic use of an inverse trigonometric function, generally arctan – and this avoids the duplication of the evaluation processes demanded by calculation using PSAs, which can mean losses in exactness and precision. Furthermore, when Δϕ does not complete a whole period, its continuous values are recovered directly with no need for unwrapping. References can be found in the literature to DPSAs constructed without a specific design method [5–7], and whose sensitivity has hardly been studied, and DPSAs designed with expressions [8–10] that suitably combine two known PSA patterns to recover Δϕ directly. The former are called Specific DPSAs (SDPSAs) and the latter Generic DPSAs (GDPSAs). Worth noting among the generic type are Asymmetric GDPSAs (AGDPSAs), which are obtained by means of a least square fit similar to that used in PSAs , so called because they do not need to use the same PSA in the original pattern and in the modified oneEq. (6)a), and SGDPSA2, Eq. (6)b).
There are currently no generalised studies on the sensitivity of the various DPSAs to different error sources [11–13] that would allow them to be chosen to guarantee the highest exactness, resolution and repeatability or derive compensation or cancellation mechanisms like PSAs can [14–18]. In both cases the most significant systematic errors are provoked by values of the phase shifts (α,β) that are distinct from the nominal and by fringe patterns with undesired harmonics, k>1 in Eq. (1) and g>1 in Eq. (3). Previous works provide either a qualitative characterization of these errors with a Fourier description of the differential phase-shifting evaluation [13, 19] or a quantitative analysis by linearization of error [11, 12]. Now to analyze these two GDPSA error sources we present a quantitative estimate based on the frequency shifting property of the arctan function from which all GDPSAs are derived (Eq. (5) and (6)), thus revealing their propagation mechanisms. The expressions obtained are verified with the Monte Carlo Method (MCM) contrasted with the quantitative analysis, to open up the possibilities for analyzing other error types.
2. Differential error analysis
The generic functional dependence of the error in the phase calculation for a PSA obtained from the arctangent function can be obtained by attending to its frequency shifting property [20, 21] in such a way arguments with specific frequency components give phase errors at harmonics of these frequencies. A similar procedure can be carried out for GDPSAs. Thus for AGDPSAs, starting in the same way from the identity of the tangent of a remainder for the phase difference Δϕ with and without error, expressing each one by the least squares fit of the AGDPSAs (Eq. (5))Eq. (7) and first order information selected, the following is obtainedEq. (6) for SGDPSAs, and C 1=C 2 and μ=ν is considered, for they must use the same PSA precursor in both patterns and have the same harmonic contribution, then the generic expression for the error is obtained for the SGDPSAs1Equation (10) does not have a trigonometric function in the denominator as Eq. (11),12) have that can derive in the presence of inconvenient discontinuities in the error.
The normalized error values for each particular case can be obtained by considering different algorithms and errors in an individualised way. Thus the above general expressions can be particularised by applying the same linear approximation in each case. For instance, possibly the main source of systematic error affecting differential evaluation is the discrepancy in the nominal value of the additional relative phases of the original (αm,βp) and modified tp(ϕ+Δϕ,βp) pattern. The generic expression of the calibration error in linear approximation  for a DPSA of M shifts in the original pattern and P shifts in the modified one isEq. (10), (11) or (12) in each case. As an example we analyse the sensitivity of the AGDPSA (17a), the SGDPSA1 (17b) and SGDPSA2 (17c) from Schwider-Hariharan [23, 24], to these two error sources.Table 1 ) fits Eq. (10) with (μ,ν)=(1,1) in the same way that the SGDPSAs fit Eqs. (11) and (12). Table 2 shows in this case that the presence of third order harmonics corresponds with one factor (μ,ν)=(3.3). In all cases the GDPSAs are observed to inherit insensitivity to detuning of their precursor PSA and the presence of second order harmonics. So, it can be appreciated that the sensibilities of the Schwider-Hariharan PSA are the same as for the Schwider-Hariharan GDPSAs so in this latter case compensation capabilities are inherited from their precursors.
3. Monte Carlo simulation method
The previous analytical relationships can be contrasted with protocols from statistical theory and provide characterisation of DPSA sensitivities. This is an alternative to spatial and temporal techniques [10–14] with wholly coinciding results. Thus MCMs are a suitable tool if completion or verification of deviation calculations is required  as they do not present limitations with regards to the possible non-linear nature of the equations to be used and can handle dependent input variables. Moreover, they are especially suitable when specific errors of the differential evaluation, such as variations in the original and modified local background irradiance a 0(r)≠b 0(r) and modulation amplitude a 1(r)/a 0(r)≠b 1(r)/b 0(r), are present. In this way the error in the phase difference Δϕ can be obtained indirectly from the irradiance values of the original (1) and modified (3) patterns Δϕ=Δϕ(s 1,s 2,…,sM,t 1,t 2,…,tP) attributing to each point a continuous Probability Distribution Function (PDF) . MCMs use pseudorandom numbers distributed statistically  to guarantee generation of a discrete representation of the output quantities. When these PDFs are combined in a DPSA a sufficiently large number of times, I, a PDF of the phase difference ΔϕI is obtained at each point characterised by its expectation average value and standard deviation that they can be directly associated with the error in the measurement of EΔϕ . In this way a complete characterisation is gained for the propagation of errors of the patterns sm(ϕ,αm) and tp(ϕ+Δϕ,βp) to the phase difference Δϕ.
To illustrate the response of these algorithms to the MCM method above described, the Schwider-Hariharan GDPSAs (Eq. (17)a-17c)) sensitivity to the two main systematic error sources Eqs. (13-16) is analyzed as an example. Their reliability is contrasted with the analytical results provided in Tables 1 and 2 where, in order to obtain comparable results in both cases, an error of 10% is assumed in the additional phases and in the undesired harmonics amplitudes.
The success of the MCM lies mainly in the correct choice of PDF associated to each error type . A uniform or rectangular PDF is usually associated if the corresponding variation limits are known and the probability to obtain a value between them is the same. In the case of the error in the additional phases (αm,βp) there is no reason to think that one value is more likely than another, except for technical specifications from the manufacturer of the phase shift element. Then a rectangular PDF can be chosen, centred at αm and βp in each case and whose width will be determined by the non void q th and r th addends in the Eqs. (14). The simulation was done in a MATLAB computer routine where a one-dimensional calculation was carried out on 300 values considering I=104 iterations with an equivalent sampling rate.
Figure 1 shows the width of any of the coefficients (εq,χr) from which the cases q=r=1 and q=r=2 are considered separately below. On introducing these perturbations in the original (1) and modified (3) patterns rectangular PDFs are equally obtained at both first (Fig. 2 ) and second (Fig. 3 ) order. The next step consists of combining these patterns in a DPSA. As systematic error is being dealt with, the M irradiance values that codify the phase ϕΙ in each iteration I, and the P values that codify ϕΙ+ΔϕI at each point must be affected by the same random value in each iteration I.
On combining the patterns affected by detuning error, the obtained PDF, which is not now represented, is very narrow and the error is null in all cases in the same way as it is for the precursor PSA [23,24]. Figure 4 shows the second order error EΔϕ where it can be seen that the best behaved is the AGDPSA, as the SGDPSAs show inconvenient discontinuities. The value obtained with the MCM is compared with the analytical result provided by Table 1, where great coincidence can be seen between both results.
The same reasons that justify using a rectangular PDF for phase shift error also point to choosing it for the possible appearance of undesired higher order harmonics.
Following a similar procedure (Figs. 5 -7 ) curves are obtained (Fig. 8 ) that account for the sensitivity to the presence of third order harmonics of the DPSAs. They are all also insensitive to the second order just like their PSA precursor, where it can be seen that their amplitude is very great and even greater than that for the error in the phase shift (Fig. 4). The curves obtained with the MCM in Fig. 8 fit the analytical results in Table 2 better than those in Fig. 4 fit the results in Table 1. It can be seen that the PDFs in the first case are not as uniform as in the second one.
Design strategies and quantitative characterisation for the three GDPSA families constructed from PSAs show that PSAs transfer their compensatory capabilities to them in such a way that the exhaustive knowledge held nowadays about PSAs enables GDPSA design with these same compensatory capabilities. We have deduced a general expression for all systematic error sources in GDPSAs by means of an approximation of the error using the arctangent function from which all GDPSAs derive (Eqs. (10), (11) and (12)). A complex harmonic dependence has been observed in several combinations of the original ϕ and modified ϕ+Δϕ phase that induce us to seek optimisation techniques for DPSA behaviour with a view to, without cancelling, avoid discontinuities and minimize error amplitudes. As an example, we analyzed Schwider-Hariharan GDPSAs and we can observe discontinuities in phase shift error and a great amplitude error in the third harmonic for the Schwider-Hariharan SGDPSAs, on the other hand, Schwider-Hariharan AGDPSA shows a good behaviour in both cases. MCM shows itself to be a technique that is comparable to traditional spatial and temporal methods, which completes those already existing in the literature and obtains identical results to these. Its use can also be extended to the calculation of random errors (also called environmental or stochastic errors) using, in this case, a Gaussian PDF or even carrying out complex error combinations in order to analyze their joint contribution.
The authors are grateful for funding received from Xunta de Galicia (07DPI002CT) and Ministerio de Ciencia e Innovación (DPI2008-06818-C2-01/DPI).
References and links
1. B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10(3), 33–55 (1999). [CrossRef]
2. D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (Taylor&Francis Group 2005).
3. J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
4. D. C. Ghiglia, and M. D. Pritt, Two-dimensional phase unwrapping, Wiley, New York, 1998.
5. K. A. Stetson, “Theory and applications of electronic holography,” in SEM Conference on Hologram Interferometry and Speckle Metrology: 294–300 (1990).
10. J. Burke and H. Helmers, “Complex division as a common basis for calculating phase differences in electronic speckle pattern interferometry in one step,” Appl. Opt. 37(13), 2589–2590 (1998). [CrossRef]
11. M. Miranda and B. V. Dorrío, “Error-phase compensation properties of differential phase-shifting algorithms for Fizeau fringe patterns ,” in RIAO/OPTILAS 2007 AIP Conf. Proc. 992, 993–998 (2007). [CrossRef]
12. M. Miranda, and B. V. Dorrío, “Error behaviour in Differential Phase-Shifting Algorithms,” Proc. SPIE 7102, 71021B–1 - 71021B–9 (2008).
13. M. Miranda, and B. V. Dorrío, “Design and assessment of Differential Phase-Shifting Algorithms by means of their Fourier representation,” in Fringe 2009 (Elsevier, Stuttgart, 2009) 153–159.
16. J. F. Mosiño, D. M. Doblado, and D. M. Hernández, “Calculus of exact detuning phase shift error in temporal phase shifting algorithms,” Opt. Express 17(18), 15766–15771 (2009). [CrossRef] [PubMed]
19. M. Miranda and B.V. Dorrío, “Fourier analysis of two-stage phase-shifting algorithms,” J. Opt. Soc. Am. A, doc. ID 115412 (posted 1 December 2009, in press).
20. K. G. Larkin and B. F. Oreb, “Propagation of errors in different phase-shifting algorithms: a special property of the arctangent function,” Proc. SPIE 1775, 219–227 (1992).
21. J. M. Huntley, “Random phase measurements errors in digital speckle pattern interferometry,” Opt. Lasers Eng. 26(2-3), 131–150 (1997). [CrossRef]
23. 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]
25. M. G. Cox and P. Harris, “An outline of Supplement 1 to the Guide to the Expression of Uncertainty in Measurement on numerical methods for the propagation of distributions,” Meas. Tech. 48(4), 336–345 (2005). [CrossRef]
26. M. A. Herrador, A. G. Asuero, and G. González, “Estimation of the uncertainty of indirect measurements from the propagation of distributions by using the Monte-Carlo method: An overview,” Chemom. Intell. Lab. Syst. 79(1-2), 115–122 (2005). [CrossRef]
27. M. G. Cox and B. R. L. Siebert, “The use of a Monte Carlo method for evaluating uncertainty and expanded uncertainty,” Metrologia 43(4), S178–S188 (2006). [CrossRef]
28. V. Álvarez-Valado, H. González-Jorge, B. V. Dorrío, M. Miranda, F. Rodríguez, J. L. Valencia, F. J. Yebra, J. Rodriguez, and J. Rodríguez J, “Testing phase-shifting algorithms for uncertainty evaluation in interferometric gauge block calibration,” Metrologia 46(6), 637–645 (2009). [CrossRef]
29. R. R. Cordero, J. Molimard, A. Martínez, and F. Labbe, “Uncertainty analysis of temporal phase-stepping algorithms for interferometry,” Opt. Commun. 275(1), 144–155 (2007). [CrossRef]