A phase recovery procedure using interferograms acquired in highly noisy environments as severe vibrations is described. This method may be implemented when disturbances do not allow obtaining equidistant phase shifts between consecutive interferograms due to tilt-shift and nonlinearity errors introduced by the vibrating conditions. If the amount of the tilt-shift is greater than π radians, it will lead a sign change in the phase estimation. This situation cannot be handled correctly by algorithms that consider small errors or non-equidistant phase shifts during the phase shifting process under moderate disturbances. In experimental applications, it is observed that the tilt-shift is often the most dominant error in phase differences that one must deal with. In this work, a Fourier technique is used for the processing and recovering of the cosine of the phase differences. Once the phase differences are obtained, the phase encoded in the interferograms is determined. The proposed algorithm is tested in two sets of interferograms obtained from the analysis of an optical component, finding an rms error in the phase reconstructions of 0.1388 rad.
© 2014 Optical Society of America
Phase determination procedures from interferograms are required to decode the object information under analysis. It is widely known that phase shifting techniques are very accurate procedures if experimental conditions are free of air turbulences, vibrations and when the object under study remains stable while at least three images are taken [1,2]. If these conditions are not fully met errors arise in the phase, but usually a good estimation can be obtained provided that the sources of disturbances are moderate . The mechanical vibrations problems have been extensively studied because it is often the most limiting factor for an accurate interferometric measurement. The simplest effect of a vibrating environment in an interferometric setup is the introduction of an erroneous phase shift. In order to deal with such situation several procedures that estimate the actual phase shift have been developed [4–7]. A more general effect is that the phase difference among interferograms is non constant but it may contain a small tilt. Under these conditions robust solutions that often work recursively have been developed to obtain the phase accurately [8–11]. Other approaches that involve special hardware requirements have also been employed to reduce or compensate mechanical vibrations in interferometric measurements [12–14]. In this work the phase recovery from a set of three fringe patterns acquired in highly vibrating environments is considered. The requirement of additional devices like a high speed camera or a phase shifter element is avoided and only a basic interferometric setup is required to obtain the fringe patterns to be analyzed. It is observed that under high-amplitude vibrations the main effect in an interferometric system is a misalignment. The phase difference among recorded interferograms in general, may contain piston, tilt and defocus errors, being tilt the most predominant term. In such manner the wavefront differences among the intensity patterns may change the sign across the interferogram field. Under this condition the referenced algorithms for phase recovery will fail. An exception is the algorithm given in  which relies in methods for phase recovery from a single interferogam with closed fringes which makes of this procedure a complex and time consuming task. Here, a simpler but effective approach is developed. The cosines of the phase differences are calculated and processed with a Fourier technique to recover the wavefront error. Once the phase differences are estimated, they are used to recover the phase information from the set of real interferograms in spite of the mechanical vibrations involved.
2. Estimation of the cosines of phase differences among interferograms
A set of three interferograms acquired in a vibrating environment may be mathematically modeled as:
Where, Iα(x,y) is the background illumination, Iβ(x,y) is the modulation intensity, ϕ(x,y) is the phase to be recovered, and φ1(x,y) and φ2(x,y) are the phase terms introduced due to vibrations. A set of real interferograms that corresponds to Eqs. (1), (2) and (3) can be observed in Figs. 1(a), 1(b) and, 1(c), respectively.
In order to eliminate the background and modulation terms an interferogram normalization process is applied, resulting in :
Where, ψ(x,y) = ϕ(x,y) + φ1(x,y), and ξ(x,y) = ϕ(x,y) + φ2(x,y). Two wrapped phases with sign changes are obtained for each interferogram applying two orthogonally orientated filters of one half of the spectrum in the Fourier domain as described by Kreiss . The two procedures mentioned above, fringe pattern normalization and the wrapped phase calculations are closely related techniques that are addressed by band pass filtering in the frequency domain, as shown in Figs. 2(a)-2(i). The normalized interferograms, In(x,y), In1(x,y), and In2(x,y) can be observed in Figs. 2(a), 2(d), and 2(g), respectively. Let us denote the wrapped phases with sign changes in the x direction seen in Figs. 2(b), 2(e), and 2(h) as ϕwx(x,y), ψwx(x,y), and ξwx(x,y), respectively. Similarly, we denote the wrapped phases with sign changes in the y direction seen in Figs. 2(c), 2(f), and 2(i) as ϕwy(x,y), ψwy(x,y), and ξwy(x,y), respectively.
The following relations hold for the wrapped phases and the actual phase encoded in the interferograms:
The cosine of the phase differences between interferograms I(x,y) and I1(x,y), are calculated as follows:
Similarly, the cosine of the phase differences between interferograms I(x,y) and I2(x,y) are obtained as:
The calculated intensities, Ic1(x,y) and Ic2(x,y) delimited by a pupil function, can be observed in Figs. 3(a) and 3(b), respectively. They correspond to the cosine of the phase differences φ1(x,y) and φ2(x,y) introduced due to the vibrating environment,
It may be noticed that the phase differences are not constant since the intensities Ic1(x,y) and Ic2(x,y) range from [-1,1]. This behavior indicates that φ1(x,y) and φ2(x,y) are mainly described by a non-constant function:
3. Phase differences estimation
The phase differences nearly correspond to a linear function. Thus, the interferograms shown in Figs. 3(a) and 3(b) could be, in theory, analyzed with the Fourier procedure. However, for the example presented here the fringe density is not high enough to separate the lobules in the frequency domain as is required with the Takeda’s method . In order to obtain the phase differences φ1(x,y) and φ2(x,y) we calculate two new interferograms with twice the frequency content,
The interferograms of twice the phase differences, Ĩc1(x,y) and Ĩc2(x,y), are shown in Figs. 4(a) and 4(b) respectively. In these interferograms it is seen clearly the claim done, earlier in this work, of a linear behavior of the phase differences in a high amplitude vibrating environment. A slight curvature of the fringes can be appreciated in Fig. 4(a). This effect is related with a small defocus term.
The interferograms that correspond to Eqs. (16) and (17) are processed in the Fourier domain and the unwrapped results are shown in Figs. 5(a) and 5(b) for the phase encoded in Ĩc1(x,y) and Ĩc2(x,y), respectively. These phases are divided by two and are used to obtain the wrapped phase differences, φ1w(x,y) and φ2w(x,y), that are encoded in interferograms Ic1(x,y) and Ic2(x,y) as can be appreciated in Figs. 5(c) and 5(d), respectively. A dynamic range was found for φ1(x,y) and φ2(x,y) of 4.6995 and 4.9888 rad., respectively.
4. Phase recovery procedure
The phase recovery process can now be achieved by using the information of the phase differences already calculated. Ideally, we would need only two normalized interferograms and its phase difference. The wrapped phase, ϕw(x,y), can be found with the following formula:
Where the atan2[∙] is the arctangent function that accepts two arguments corresponding to the sine and cosine functions and returns the result modulo 2π. In the above equation and in the subsequent ones the spatial dependence (x,y) has been omitted for simplicity. The wrapped phase found with the above formula is observed in Fig. 6(a). Similarly the wrapped phase found with the information of In(x,y), In2(x,y), Ic2(x,y), and φ2w(x,y) is shown in Fig. 6(b). Both wrapped phases exhibit some regions with high amount of noise. This occurs in those places where the sine of φ1w(x,y) or φ2w(x,y) is zero or nearly zero. The application of an unwrapping phase algorithm to these wrapped phases would lead to obtain a continuous phase with inconsistencies. This situation may be overcome by using the information of both phase differences and the normalized interferograms, according to the following conditional:
The above conditional compares the magnitude between the sum and the difference of the sine of the phase differences to ensure a non-zero or nearly zero division which would make the process very sensitive to noise. The wrapped phase calculated with Eq. (19) is seen in Fig. 7(a). It is observed a great reduction in noise as compared with the wrapped phases shown in Figs. 6(a) and 6(b).
In order to test the reliability of the proposed method, a second set of interferograms is used. The phase estimation was carried out in the same form. The first interferogram showed in Fig. 1(a) is used again as I(x,y). The second and third fringe patterns, I1(x,y) and I2(x,y) are shown in Figs. 8(a) and 8(b), respectively. The cosine of the phase differences are observed in Figs. 8(c) and 8(d) and the wrapped and unwrapped phase results are seen in Figs. 8(e) and 8(f), respectively. Both unwrapped solutions appear identical with a maximum and minimum error of 2.8 and -2.3 rads. An rms error (Erms) of 0.1388 rad. was found with the following formula:
For further investigation of the consistency of the obtained results we have applied a three step phase shifting interferometry (PSI) algorithm. One of the used interferograms has been already shown in Fig. 2(b) the other two interferograms are seen in Figs. 9(a) and 9(b). The estimated wrapped and unwrapped phases are showed in Figs. 9(c) and 9(d). The same results obtained for the wrapped and unwrapped phases with the developed procedure can be appreciated in Figs. 9(e) and 9(f), respectively. The recovered phases with our method and the PSI technique show an rms error (Erms) of 0.3160 rad. The error increases but a good concordance is still maintained considering that the measured dynamical range of the phase is around 133 rad., it represents a percentage error of 0.24%. This demonstrates also a good repeatability in the obtained measurements. The difference in results between the PSI and our method are mainly due to the normalization process. We found that the background and modulation terms among interferograms with a high non constant phase difference may differ, which avoids the cancelation of these terms as occurs with classical PSI techniques. Due to this situation the normalization procedure was required for the method here presented.
A phase recovery process from three interferograms acquired in high amplitude vibrating environment has been demonstrated. In such conditions the phase differences among interferograms may contain a tilt term that previous algorithms cannot handle correctly. This occurs because the phase differences may change its sign throughout the interferogram field infringing the main assumption of a nearly constant phase shift. The interferograms are first normalized and the cosines of the phase differences are then calculated with the method of Kreis. After that, the phase differences are estimated with a Fourier method given that the wave front error mainly contains a tilt term. In the given examples it was necessary to calculate two additional interferograms of twice the phase differences in order to apply the Takeda’s method. These results were used to calculate the wrapped phase differences between the first interferogram with the second and the third ones. Once the phase differences are available, they are used along with the normalized interferograms in a conditional procedure to assess the wrapped phase. Two sets of interferograms were analyzed with the proposed technique with great agreement between both results. A third set of interferograms were processed with a PSI technique that confirms the validity of the developed algorithm. All the interferograms showed here were obtained by using a commercial Fizeau interferometer under uncontrolled high vibrating conditions. Since the Fizeau is a common path interferometer that is less sensitive to mechanical vibrations as compared with a two arm interferometer or a Mach-Zhender configuration the developed method may be used for in situ measurements when the obtained data cannot be analyzed with other procedures. Further investigation is in progress to obtain the phase from interferograms with an arbitrary phase difference. Due to the random nature of the vibrations, the global sign of the phase remains undetermined. It could be estimated by observing the shift of the fringes as a result of introducing some tilt or defocus before the interferograms acquisition.
References and links
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