Temporal phase shifting in automatic interferogram analysis offers very high accuracy of phase retrieval providing that several experimental conditions are met. The paper is focused on the calibration error of unequal phase changes across the interferogram field, i.e., tilt-shift error. For its detection the lattice-site representation of phase shift angles is proposed. The error can be readily discerned using (N+1) algorithms with the last frame overlapping the first one. Four and five frame algorithms are considered. The influence of experimental parameters on the error detection sensitivity is discussed. Numerical studies are complemented by experimental results.
©2006 Optical Society of America
Among automated interferogram analysis methods, temporal phase shifting (TPS) technique is recognized as the one providing most accurate wave-front extraction [1–4]. The measurement accuracy of the TPS depends on the phase and intensity type errors [5, 6] caused by several practical limitations. The first group contains all errors related to the phase shift fidelity, including the phase shifter performance and measurement environment instabilities (e.g., mechanical vibrations, air turbulence, etc.). Intensity errors include detector nonlinearity, detector noise and average intensity discrepancy between TPS frames. This statement concerns both the interferogram phase [1–6] and contrast or intensity modulation [7,8] determination. Accurate phase shift calibration procedures or modern self-calibrating algorithms are applied to minimize measurement errors.
This paper is devoted to tilt-shift errors caused by tilting or twisting of the PZT system during translation of the reference mirror in an interferometer. Commercial systems have automatic calibration procedures to eliminate or minimize this error and provide a sequence of evenly spaced reference phase shifts . Although no further details are usually disclosed those procedures are most probably based on modern self-calibrating techniques similar, for example, to the ones published by Chen et al  and by Dobroiu et al . Both techniques assume a linear dependence of phase shifts with respect to pixel position. A first-order Taylor series expansion of the phase-shift errors (including both translational and tilt-shift errors) was used in  for defining this departure and subsequent error compensation. The second approach  is based on blockwise processing of interferograms divided into small regions with uniform phase steps. Using calculated contrast maps the algorithm globally adjusts the phase-shift planes and compensates both translational- and tilt-shift errors.
In our paper we present a simple method for detecting the presence of tilt-shift error and its semi-quantitative estimation. It does not require extensive iterative calculations of the phase shift maps for the component TPS interferograms. It represents a competitive solution to the recently published improved max-min scanning (IMMS) method  which requires plotting light intensity values versus driving voltage, low-pass filtering and least-squares curve-fitting to obtain a smooth intensity variation curve, finding maximum and minimum intensity values by gradient computation and phase unwrapping. Our technique is based on an elegant approach of Gutmann and Weber for phase shifter calibration and error detection . The authors introduced the so-called lattice-site representation of the phase shift angle in detector pixels. It represents a modification of the shift angle histogram. In the lattice-site representation, a population of points is plotted in the Cartesian coordinate system with their coordinates corresponding to the numerator and the denominator in the phase shift angle equation assuming digitization of the measured intensities into 256 gray levels. Gutmann and Weber considered a popular five frame algorithm [14, 15]. Straight lines passing through the origin represent lattice-sites with equal shift angles. Authors proved the superiority of the lattice-site representation over the conventional histograms, both for finding the mean phase step value between the interferograms and identifying some error sources (signal-to-noise ratio, unequal phase steps, intensity quantization, presence of higher harmonics). In recent works further extensions of the lattice-site representation of the phase shift angle distribution to detection of different average intensities of the interferograms  and nonlinear detection of a two-beam interferogram  have been reported.
In this paper popular four and five frame algorithms are considered. It is shown that lattice-site representations of the phase shift angle distribution provide easy to interpret information on the occurrence of the tilt-shift error under the following experimental conditions: a) equal phase shifts between data frames; b) the first and last frames should be captured with the same relative phase between the object and reference beams, c) linear interferogram detection, d) low noise level, and e) sufficient interferogram intensity modulation. It is interesting to note that the first and second condition is met in the fringe-locking calibration procedure described by Cheng and Wyant , and is characteristic to N+1 algorithms [4, 18]. Our last and first frame capturing condition has nothing to do, however, with fringe tracking. Detector linearity is a common requirement in automatic interferogram processing techniques. Since the magnitude of tilt-shift error is much smaller than translational error of the PZT driven reference mirror, it is clear that other error sources must also be small as required by d) and e). Our quantitative estimations of visual detectability of a double-cone-like lattice-site patterns characteristic to the tilt-shift errors call for the signal-to-noise ratio of at least 16 dB and the interferogram intensity modulation of min. 50 gray levels (in case of 8-bit digitization of the interferograms). These two requirements are readily satisfied by the interferograms commonly met in optical shop testing. Noisy interferograms, however, encountered in experimental mechanics studies, can be treated after applying very effective spin-filtering of component TPS frames [19, 8]. Numerical investigations of main characteristics of the method, including practical departures from the requirements a), b) and c) mentioned above, are corroborated by experiments. Further quantitative studies are out of scope of this paper because of its extent.
2. Phase shift angle histogram and its lattice-site representation
In experimental realization, due to unavoidable disturbances, phase shift values α(x,y) between frames will be different for different CCD pixels [1–4]. By analyzing populations of pixels with the same shift angle we can seek the information on individual experimental errors. Phase shift angle distributions (histograms) for algorithms with an unknown but constant shift angle α between frames can be calculated using:
1. for four frame algorithms — the equation quoted by Kreis :
2. for five frame algorithms - the equation [14,15]:
where I1, I2, I3, I4 and I5 denote intensity distributions of component interferograms (for notation brevity their dependence on spatial coordinates x,y has been omitted).
Our calculations have shown that histograms calculated by Carré equation (3) and Eq. (1) give very close results. The form of Eq. (1) enables straightforward formation of the lattice-site representation for the four frame algorithm.
In all our lattice-site representations we display each combination of the numerator and denominator values, Eqs. (1) and (2), by a black dot in a two-dimensional diagram with the nominator value as the ordinate and the denominator value as the abscissa. Gutmann and Weber  displayed the population of each numerator and denominator combination by a corresponding gray level. Since the asymmetries and spreads of lattice-site representations as compared to the error-free case are of main interest and importance, they are better displayed using our binary representation than the gray level one. It combines two highly populated cores and tails  into one structure which becomes a double-cone-like spread pattern in case of the tilt-shift error presence.
3. Tilt-shift error detection-numerical simulations
In the ideal TPS experiment the histogram and lattice-site representations show a narrow symmetric peak and straight line passing through the origin, respectively. High frequency additive intensity noise, phase shifter errors, quantization of interferogram intensity values, higher harmonics in the fringe intensity profile, different average intensities of component frames and nonlinear recording of a two-beam interferogram cause irregular shape and spread of both the histogram and lattice-site patterns [13,8,16].
The tilt-shift error under consideration corresponds to a nonuniform phase shift across the reference (phase shifted) beam. First, let us explain the principle of our approach using Eq. (2). It can be readily seen that when the first and last frames are captured with the same relative phase between the object and reference beams and random errors are not present (including additive intensity noise mentioned above), the numerator in Eq. (2) vanishes. In case of tilt-shift and/or twist-shift errors, however, this is no longer true because these two data frames have different fringe distributions. For example, if the first frame I1(x,y) contains straight fringes and the tilt axis is parallel to the direction of fringes, the last frame I5(x,y) contains a fringe pattern with slightly larger or smaller period, depending on the beam tilt direction. On the other hand, when the tilt axis is perpendicular to the fringe direction, the fringes in the frame I5(x,y) are slightly rotated and of different period with respect to the ones in frame I1(x,y).
Simulated two-beam interferogram intensity distributions had approx. three spatial periods. The bias (DC) and modulation (AC) values were assumed to be equal to 128 and 80 intensity levels, respectively (arbitrary units). The additive, high frequency intensity noise (gaussian distribution with zero mean) of +/- 6 intensity levels was assumed.
3.1. The case of exact 90 deg phase shifts, αi=90 deg
3.1.1. Linear detection, γ=1
Tilt axis parallel to rectilinear fringes. Figures 1 and 2 show calculated phase shift angle histograms and their lattice-sites using Eqs. (1) and (2) for equal translational phase steps of 90 deg and tilt values of 1.0 and 10 µrad between adjacent TPS frames. In general, the ratio of the subsequent phase shifts is constant at each pixel across the field of view (this remark is also valid for Section 3.2). The tilt axis was assumed to be parallel to fringe direction and located at the left edge of the interferogram. Linear recording, γ=1, has been assumed. Under the simulation conditions the presence of the tilt-shift error (obtained by detecting the characteristic double-cone-like shape of a lattice-site pattern) could be established for the tilt values of approx. 1 µrad. The detectability is influenced by the signal-to-noise ratio and the interferogram intensity modulation. Our quantitative estimations call for the signal-to-noise ratio of min. 16 dB and intensity modulation of min. 50 gray levels (256 gray level quantization). Please note that no information on the tilt-shift error can be obtained from the standard histograms.
Tilt axis perpendicular to rectilinear fringes. Figure 3 shows simulation results for the orthogonal orientation between the direction of interference fringes and the tilt axis of the reference beam. Because of a great similarity of the results for the tilt value of 1.0 µrad, only the case of 10 µrad tilt is presented. The tilt axis was assumed to coincide with the bottom edge of the interferogram.
The difference between Figs. 2 and 3 is that when the axis of tilt is parallel to the direction of fringes, the histograms and lattice-site representations have a discrete or quasi-discrete character (because of discrete phase jumps in pixels). On the other hand, for orthogonal orientation of both directions the histograms and lattice-site patterns have more continuous distribution. It is a consequence of simultaneous rotation and period change of fringes in adjacent TPS frames (in the former case only the period change is encountered). In the two special cases shown, the divergence angle of the characteristic double-cone-like lattice-site pattern is proportional to the tilt-shift error magnitude. The cone-like shape itself is characteristic to tilt-shift error and manifests its presence.
It is to be noted that in the case of equal phase steps and the four frame algorithm, phase shift angle representations are the same for the first (1 to 4) and second (2 to 5) groups of frames. This is true for Figs. 1–3 and Figs. 5–7, related to special cases of tilt axes set parallel and perpendicular to simulated straight fringes.
In Figs. 1–3 histograms and lattice-site representations of the phase shift angle for four and five frame algorithms with 90 deg phase steps are given. In case of the latter algorithm we have a perfect coincidence of interferograms in the last and the first frames as mentioned above. This is however not true for the four frame algorithm, but because of the coincidences between odd and even numbered interferograms in this case, the numerator in Eq. (1) (used in this case) vanishes as required. The assumption about (N+1) character of the four frame algorithm calls for the phase steps of 120 deg. This case was verified numerically and experimentally and will be illustrated at the end of the paper. For the sake of conciseness, however, we will focus on a popular self-calibrating five frame algorithm with nominal 90 deg phase steps. Accompanying four frame histograms and lattice-site representations serve the purpose of showing the presence and influence of other experimental errors on the tilt-shift error detection, i.e., nonlinear recording of interferograms and unequal phase steps.
Arbitrary orientation of tilt axis with respect to straight fringes. This most general case includes a huge number of combinations of tilt-shift error values for simultaneous tilts about x and y axes, including their signs (tilts of the same and/or opposite directions). We have performed numerical simulations for several combinations and a general conclusion following is that the double-cone-like envelope still manifests presence the tilt-shift error in all those cases. Its quantitative estimation, however, is not straightforward.
To illustrate this complex issue Fig. 4 shows histograms and lattice-sites for five frame algorithm for two cases of simultaneous tilt-shift errors about x and y axes: -5 µrad and 10 µrad (left), and -10 µrad and 10 µrad (right), respectively.
By comparing Figs. 3 and 4 the changes of the shape and location of histograms along the horizontal (angular) axis can be readily identified. Likewise, lattice-site patterns have a different angular divergence and orientation when compared to the case of pure y axis tilt, see Fig. 3. Possible nonlinear detection introduces characteristic elongations and deformations of the patterns as described below for special cases of tilt axes set parallel and perpendicular to fringe direction (Figs. 5–10). Detailed studies aiming at establishing quantitative relationships between the lattice-site divergences and tilt-shift errors are out of scope of this paper and will be given in a separate publication.
Although it might be intuitively obvious, it is worthwhile to note that in the case of presence of simultaneous tilts about x and y axes of the phase shifted reference mirror, histograms and lattice-sites for four frame groups (1 to 4 and 2 to 5) are no longer identical as in the special cases discussed above. This issue is not discussed because the difference between histogram and lattice-site distributions for the two four frame groups is mainly dictated by phase step inequalities (see Section 3.3) encountered in practice.
3.1.2. Nonlinear detection, γ≠1
The analyses of the influence of the phase shift error presented below should include the influence of the second main experimental error — detector nonlinearity [1–8, 21]. Nonlinear recording has been simulated according to Eq. (4)
For simplicity we have chosen coefficient b to be equal to 4×10-6 with other coefficients set to zero. I denotes sinusoidal pattern intensity distribution.
In the case of the ideal step calibration αi=90 deg and for a very small tilt-shift error of 1.0 µrad the results are almost independent of the orientation of the tilt axis with respect to the direction of simulated straight fringes. Figure 5 shows computation results.
For such a small tilt-shift error, the nonlinear detection has almost negligible influence on the five frame algorithm, except for some magnification-like effect, i.e., the lattice-site pattern elongation. On the other hand the results for the four frame algorithm are dramatically different, showing characteristic bow-tie or eight-number-like shapes that were discussed in . Unfortunately these results are rather useless for the tilt-shift error detection but are helpful for identifying nonlinear registration. This is a consequence of relatively high sensitivity of four-frame algorithms with 90 deg phase steps [1–6] (including Carré’s one ), to camera nonlinearities.
Figures 6 and 7 show the phase shift angle representations for large tilt-shift error value of 10 µrad and γ≠1. Considerable changes of all histogram distributions and four frame lattice-site patterns are readily seen when compared with Figs. 2 and 3. On the other hand five frame algorithm lattice-site patterns, especially their double-cone-like shape envelope divergence, are only slightly influenced by detection nonlinearity. This feature is characteristic to the absence of the translational phase step error, i.e. when αi=90 deg and relative insensitivity of that algorithm to detector nonlinearities.
3.2. The case of linear translational phase shift error, αi=const≠90 deg
Computer simulations were performed for several equal phase step values different from 90 deg in the range from 70 to 110 deg and equal tilt errors between the frames. The conclusions can be summarized as follows.
3.2.1. Linear detection, γ=1
1. For very small values of tilt-shift error, e.g., 1 µrad, the information on its presence can be discerned from the lattice-site representation patterns in the form of a double-cone-like pattern for the whole range of αi=70 - 110 deg in case of the five frame algorithm. On the other hand, very narrow conventional histograms and shape of four frame algorithm lattice-site patterns do not allow for straightforward identification of the presence of the tilt-shift errors. This fact is in agreement with the statement at the end of point b), Section 3.1.1.
2. For larger tilt-shift error values the lattice-site patterns characteristic to mutual orientation between the reference element tilt axis and fringe direction (see Figs. 2, 3, 6, and 7) preserve their shape. They are proportionally rotated in the lattice-site coordinate system with respect to the case of αi=90 deg. As for the most important parameter of the tilt-shift error pattern, i.e., the divergence of the double-cone-like envelope, it remains proportional to the phase shift value in a nonlinear angular scale of the lattice-site coordinate system. This is why it might be perceived as approximately the same for αi in the range 70–90 deg. It changes, however, for αi>90 deg and becomes smaller with αi increasing.
3.2.2. Nonlinear detection, γ≠1
As in the case of αi=90 deg, the error of nonlinear detection of the component TPS frames introduces deformations to histograms and lattice-site patterns. As a result the information on tilt-shift error is obscured. In the case of exact translational tilts, αi=90 deg, the deformation of the five frame algorithm pattern was rather small, see Fig. 6 and Fig. 7. Now for αi=const≠90 deg, the deformations are considerable. Figure 8 shows lattice-site patterns for the phase steps of 70 and 110 deg and tilt-shift error of 1 µrad. For this small error value the patterns are, as before, almost the same for the tilt axes parallel and perpendicular to fringe direction.
Lattice-site patterns for tilt-shift error of 10 µrad are given in Fig. 9 and Fig. 10. A quick look at the results proves undesired and harmful influence of recording nonlinearities on the tilt-shift error detection. At the same time the fact of nonlinear detection can be readily inferred from calculated lattice-site patterns.
3.3. The case of unequal translational phase shifts, αi≠const.
This most general case of phase shift errors (including translational and tilt ones) contains a huge number of combinations of phase step values. Now the phase shifts at each interferogram point are no longer the same. From heuristic considerations and numerical simulations it follows that the inequality of phase step values decreases resolution and ability to detect the tilt-shift error. Although in some cases including small tilt-shift error values the detection is still feasible, it is of qualitative character only. The double cone-like envelope of the lattice-site representation pattern is no longer sharp at the origin of the lattice-site coordinate system. The “waist” becomes broader because unequal values of the phase steps in the experiment without tilt-shift error give narrow elliptic lattice-site patterns [13, 16]. Correspondingly, the final shape of the pattern depends on relative magnitudes of both errors, the shape of interference fringes and the localization of tilt axis. All these remarks concern the case of linear detection. Detection nonlinearities make the situation progressively more complex.
Taking into consideration the extent of the paper we will illustrate intuitively most favorable case in which the fifth frame coincides with the first one (translational phase step values). Figures 11, 12, and 13 illustrate exemplary results for phase displacements of the 2nd, 3rd, 4th and 5th frames with respect to the 1st one equal to 79, 202, 270 and 360 deg (instead of 90, 180, 270 and 360 deg for the ideal calibration case, Section 3.1). The results for the first and second group of four frames (4f1 and 4f2) are included to show the influence of different phase step values. Figure 11 shows the results for 1 µrad tilt-shift error; they are common for the reference beam tilt axis running parallel and perpendicular to fringes. Figures 12 and 13 illustrate calculation results for tilt-shift error of 10 µrad and the tilt axis parallel and perpendicular to fringe direction, respectively.
At the end of this Section the most general case of unequal phase steps between the frames (including translational and tilt-shift errors) with simultaneous tilts about x and y axes should be commented. It combines all considerations presented separately up to now. All observations established enable the detection and identification of component experimental errors as will be shown below.
4. Experimental work
Experimental work was conducted using two interferometer types — Fizeau and Twyman-Green. In both cases the reference element was shifted by three piezoceramic transducers placed along the optical element circumference and angularly separated by 120 deg. PZTs driving voltage was controlled by computer.
4.1. Experimental results using Fizeau interferometer
The interferometer was mounted with vertical orientation of optical axis. A glass plate with nearly flat front surface and ground back was used as a test specimen. In null-fringe adjustment mode two closed fringes of irregular shape were observed. The reference flat was axially displaced by the 3PZT placed along a circumference of diameter of 50 mm.
Histogram and lattice-site phase shift angle representations calculated from selected interferogram recordings are presented in Figs. 14–17. Figure 14 shows the results for a null-fringe adjustment of a phase-step calibrated instrument (on average αi=90 deg).
Using the findings presented in  and Section 3 of this paper the following measurement deficiencies can be readily identified based on Fig. 14: nonlinear detection (γ≠1), slight phase step variations and tilt-shift error. First two errors are easily detected and identified from four frame lattice-sites (asymmetrical quasi horizontal “eights”) and histogram widths and asymmetries. The information on tilt-shift error is well displayed in the five frame lattice-sites. Note its finite width at the origin caused by the interferogram noise and slightly unequal phase steps. Quasi-horizontal direction of the double-cone-like lattice-site is a cumulative result of the average value of translation displacement and tilt-shift error. From numerical simulations we can estimate equal absolute values of tilts about x and y axes with the first one having positive and the second one negative sign. Figure 5 shows the similar case, however without nonlinearities in recording clearly present in the experimental data. Additionally, we have also calculated the interferogram intensity modulation distribution using the same data frames. Uniform distribution grey-level maps without parasitic fringes were obtained for both four and five frame algorithm data.
Figure 15 shows the calculation results for the null-fringe mode detection and step error introduced deliberately to one of the PZTs. Note the changes in all patterns, especially in the five frame lattice-site representation. The double-cone-like pattern has an increased divergence angle and is rotated clockwise. As in previous case, we calculated the interferogram modulation distributions for four and five frame algorithms. Note that the visibility of parasitic fringes is proportional to the phase step error magnitude. They are not presented here for all cases because the resulting modulation maps are too dark to be well reproduced. Weak parasitic fringes (of double frequency with respect to interference fringes) can be observed in grey-level maps for both cases of four frame calculations in the vicinity of the PZT shifter that was translated by erroneous control signal.
Figure 16 shows the case of a similar error as in Fig. 15 but obtained for interferograms with eight interference fringes (finite-fringe detection). A smoothing effect in all histograms and lattice-site patterns is observed because of an increased number of fringes. As before, information about variations of phase steps, detection nonlinearities and tilt-shift error is discernable. In the calculated intensity modulation map double frequency parasitic fringes are observable in the vicinity of erroneously controlled PZT. Their detection is easier now because of an increased number of interferogram fringes. The parasitic modulation fringes appear, as before, for the four frame calculations only.
Figure 17 illustrates the case of a deliberately produced erroneous phase step by another PZT. The error introduced here had a larger value and different sign than the case illustrated in Fig. 16. The character of histograms and lattice-sites is in agreement with numerical simulations, see Fig. 12. Although Fig. 12 corresponds to linear detection case, the nonlinearity in experimental data mainly changes the scale of the patterns, preserving their character, see Section 3. Note an increased divergence angle of the double-cone-like lattice-site (five frame algorithm), its counterclockwise rotation, and short “neck” at the origin due to phase step variations and detection nonlinearities. Also high frequency noise present in interferogram makes the patterns appear more “fuzzy” than computer simulations. Calculated modulation maps support, as before, the lattice-site pattern analyses.
4.2. Experimental results using Twyman-Green interferometer
One of two Twyman-Green interferometer mirrors was actuated by three PZTs placed along a circumference of diameter of 15 mm. From a very extensive set of experiments we will present here only one illustrating the applicability of our approach to N+1 type algorithms. Specifically, the case of 3+1 frame algorithm with three steps of 120 deg and the last frame coinciding with the first one will be shown. The same set of data will be used to discuss applicability of our calculations to noisy interferograms.
Figure 18 shows one of the interferograms, its horizontal cross-section and calculated modulation maps  for two sets of four frames. The modulation grey-level map for a five frame algorithm is not presented as it does not contain any parasitic modulation fringes. Note rather noisy character of the interferogram. It contains dust, scratches and dense spurious fringes from a double reflection inside the cube beam-splitter used. Those fringes do not move upon phase shifting. To improve the histogram and lattice-site calculation results spin filtering of component TPS interferograms was applied [19, 8], see Figs. 19 and 20.
The patterns in Fig. 20 for two four frame sets indicate very clearly the case of erroneous reference mirror tilt about the axis almost parallel to the interferogram fringes, see Fig. 2. The tilt between subsequent frames was approx. equal to 15 µrad. Linear recording of interferograms and some departure from 120 deg steps can be readily concluded as well. The latter follows from presence of narrow ellipses in the five frame lattice-sites (see also five frame results shown in Figs. 11 and 12). Asymmetric number of “whiskers” in Fig. 20 in four frame lattice-sites can be attributed to slightly different phase steps and interferogram noise.
The technique proposed belongs to testing methods, i.e., it mainly detects and identifies the presence of errors. Obtaining detailed quantitative information on the tilt axis location, tilt plane and tilt amount is not as straightforward and should be investigated. Great simplicity and sensitivity of the method should be emphasized together with simultaneous detection and identification of most important TPS errors such as phase stepper miscalibration and nonlinear recording. Because of these features our approach using five frame lattice-sites aided by four frame ones should find interest in automated procedures of interferometric optical shop testing and instrument calibration. The interferometer construction and experimental process reliability can be also comprehensively assessed.
This work was supported by KBN (Polish Committee for Scientific Research) Grant No. 4T10C/001/27 and, in part, by statutory activity funds.
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