Open Access
Add to BibSonomy
Abstract
A new method, to the best of our knowledge, in digital speckle pattern interferometry is introduced. It depends on extending the applicability of using LiNbO3 crystal as a holographic recording medium for the evaluation of the difference in displacement between two similar objects (master and test), and displaying the comparison result in the form of an interference pattern. The method is a two-stage process. In the first stage, two states (un-displaced and displaced states) of the master object are recorded in a LiNbO3 crystal using the usual holographic arrangement. In the second stage, various phase stepping algorithms are applied, using the reconstructed wavefronts of the master object as a holographic optical element to produce a reference wave field in the comparison process. Recording and analysis of the difference correlation fringes are performed using the phase wrapping algorithm and real experiment.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The invention of phase-shifting interferometry (PSI) was a major breakthrough in the field of interferometry, providing a method to measure the optical phase to unprecedented accuracy [1–4]. PSI has been performed with almost all types of interferometric imaging systems through use of different algorithms to extract a phase map from several intensity fringe patterns [1–4]. For all algorithms, a discrete or continuous temporal phase shift is introduced. By measuring the intensity as the phase is shifted, the phase can be obtained. In the so-called phase-stepping technique, the phase is stepped by a known amount between each intensity measurement, whereas in the so called integrating-bucket technique the intensity is integrated while the phase is being shifted. A large number of phase-stepping algorithms have been proposed [5–14].
The PSI allows measurements of wavefront deviations of the order of λ/200 with repeatability close to λ/1000. Thus, the PSI has been widely used not only the surface profile measurement of diffuse object [15], highly reflective object [16], analysis of thickness or thickness variation of transparent plates [17–19], displacement measurement [20] in the industrial field, but also ranging to the areas of comparative measurement in speckle interferometry [21]. While the PSI is widely used, it is susceptible to the instability of the laser source and the disturbing environment, which can lead to measurement errors that have not been completely solved. Specifically, phase-shift errors due to external vibration [22], the nonlinearity of piezoelectric transducer (PZT) and drift [23], the changes in the background and amplitude intensities due to fluctuations in light source output and ambient light, random noise and nonlinearity of the camera, and speckle noise are arisen. For instance, Schwider et al studied that the phase shift errors in the N-step phase-shifting method (PSM) generated systematic phase error with a double frequency of the interferometric fringes [24]. Hariharan et al devised a five step PSM to reduce constant phase-shift errors [6]. Creath reported that increasing the number of phase shift could be robust to phase-shifting error [1]. Besides, there are many previous studies on phase errors due to variations in background [25] and amplitude intensities in recorded interference fringes, especially for interferometry with a tunable laser diode (LD) [26,27]. There have been some pertinent prior studies proposed on deformation testing and measurement using phase and phase derivative extraction from digital holography fringes [28–31]. We recently developed an accurate phase analysis technique that is a modified version of the phase-shifting differential digital speckle pattern interferometry (PSDDSPI) [32], where the phase-shifting and 2D Itoh phase wrapping algorithms [33–35] are applied to ensure the highest accuracy for the measurements.
Instead of employing traditional holographic and speckle interferometry, which do not allow for a direct interferometric comparison of the two nominal specimens subjected to similar stress levels, the novel method relies on the use of the LiNbO3 crystal as a recording medium. Each displacement field must be calculated independently in order to ascertain the difference in displacement between the two object surfaces. Only then can the numerical results be compared. However, in practice, even for small stresses, holographic and speckle interferometry cause fringe overcrowding. Without strong magnification, the fringe system becomes too dense to observe, making the evaluation laborious and occasionally even impossible. The new approach have been designed to reduce or solve these challenges. It also extends the application of the double reference beam technique to address the problem of comparing displacement fields of two identical samples. The difference correlation fringes can be obtained and analyzed using both the wrapped phase maps to extract the difference fringe map which are corresponding to the difference correlation fringes to be able to measure the difference in displacement accurately with high performance. The presented method also studied the phase- stepper errors, and the effect of increasing the number of phase shift on the phase-shifting error. It also studied the dependence of the error due to phase stepper (radian) on the wavelength of the used light, for three, four and five phase stepping algorithms. A comparative analysis of the most effective phase stepping methods is provided in order to get more precise extracted phase result. The provided results demonstrate that the five-phase stepping algorithm yields more accurate results than the three- and four-phase stepping algorithms when it comes to the wrapped phase map of the difference in displacement of the two stressed objects.
2. Optical setup
Basic elements of the optical arrangement (Fig. 1) are the master (MO) and test (TO) objects; reference waves (pump beams) with average intensity I2 and I3, which can be switched on and off separately ; LiNbO3 with 5 mm x 5 mm x 5 mm size. The light source is a coherent beam of low-power laser diodes with different wavelengths (0.6700 nm, 0.6328 nm, 0.5780 nm, 0.5640 nm, 0.4360 nm, and 0.3600 nm). CCD camera with a pixel resolution of 1280 × 1024, and pixel size 4.65 × 4.65 µm/pixel. CCD camera has no function during the recording of the master object. It is used only in the second step, i.e., during the comparison process. Piezoelectric transducer (PZT) is used for getting the phase shifts required for automatic fringe analysis. The objects to be compared were two identical bars. The bars available to be displaced axially or vertically or rotated along the vertical axes.
Fig. 1. Optical set-up for comparative measurement in speckle interferometry using a double reference beam technique with LiNbO3 Crystal as a recording medium: OL: objective lens, L1, and L2: lenses, D1 and D2: pupil apertures, BS1 and BS2: beam splitters, L3 : imaging lens, Crystal LiNbO3 (CCD): observation plane, M1, M2 and M3: mirrors, I1: average intensity of the object beam, I2 and I3: average intensity of the pump beams (reference beams). MO, and TO: maser and test objects.
3. Analysis of the interference pattern
Depending on the experimental arrangement which is shown in Fig. 1. The experimental part is divided into two steps process. In the first step, holograms of the master object in displaced and non-displaced states are recorded in the LiNbO3 crystal. In the second step, phase-shifting algorithms are applied, and the interferometric speckle patterns of the test object in displaced and non-displaced states are recorded by CCD camera. In this step, the holographically reconstructed master object wavefronts are used as references in the comparison process.
The hologram of the non-displaced master object is recorded by the reference beam I2, and at this time, the reference beam I3 is stopped. Then the master object is displaced and the second hologram is recorded at the same plate (LiNbO3 crystal). This second exposure is made by reference beam I3, and at this step, the reference beam I2 is stopped. The average intensity I1, representing the object beam (master or test object). Because the holograms of the master object are recorded by two reference beams, the two states of the master object can be reconstructed independently. The experimental arrangement used in the second step is the same as that in the first step, with the main difference that now the CCD camera is used. Also in this step, the master object is replaced by the test object [36,37]. The phase step technique is used only in the second step, for example by monitoring the object beam mirror of the interferometer on a PZT. Accurate calibration is then very important to obtain the desired phase shifts between data frames [24,25].
The simple processing of the operation can be explained as: ${A_m}$ and $A_m^{\prime}$ are the complex amplitudes of the retrieved master object, before and after displacement, respectively,
where $\varphi _m^{\prime}$ is the phase term introduced due the displacement of the master object.Let ${A_t}$ be the complex amplitude of the light scattered by the test object in its initial state:
Similar to the previous analysis, let $A_t^{\prime}$ be the complex amplitude of the scattered light of the test object in its final (displaced) state:
where $\varphi _t^{\prime}$ is the phase term introduced due the displacement of the test object.The complex amplitude generated by the retrieved master object interferes with the wave field scattered by the test object in their non-displaced states is:
The intensity distribution on the front panel of the CCD camera is:
where $\Delta \varphi = ({\varphi _t} - {\varphi _m})$, is the phase difference corresponding to the difference in non-displaced states of the test and master objects.Also, the complex amplitude generated by the retrieved master object interferes with the wave field scattered by the test object in their displaced states is:
The presented treatment proposed that the zero-order light transmitted through the crystal does not reach the image sensor due to the off-axis optics and is ignored [38].
As mentioned above, also, the CCD camera receives the scattered light by the test object and master objects in their displaced states. . The intensity distribution on the front panel of the CCD camera is:
As shown in Eq. (10), the value $\sin (\frac{{\Delta {\varphi ^{\prime}}}}{2})$, which representing the difference in displacements between the two objects modulated with a high-frequency random speckle noise. As a final result of the analysis, difference correlation fringes which is modulated with the noise term displayed on the monitor of the computer.
3.1. Phase-stepping techniques
As already mentioned, the experimental arrangement used in the second step is the same as that in the first step, with the main difference that the phase-stepping technique is used only in the second part, by surveillance one of the mirror of the interferometer on a PZT. The desired phase shifts between data frames need an accurate calibration to be obtained [39].
In the present measurements, a phase shift ${\raise0.7ex\hbox{$\pi $} \!\mathord{/ {\vphantom {\pi 2}}}\!\lower0.7ex\hbox{$2$}}$ per exposure is applied for the three, four, and five phase step algorithms. By using the phase-shifting device, three, four and five phase shifted intensity patterns are produced [40].
The intensity distribution of the correlograms can be expressed as [28]:
For three phase stepping algorithm, ${\alpha _1} = {\raise0.7ex\hbox{$\pi $} \!\mathord{/ {\vphantom {\pi 4}} }\!\lower0.7ex\hbox{$4$}}$, ${\alpha _2} = {\raise0.7ex\hbox{${3\pi }$} \!\mathord{/ {\vphantom {{3\pi } 4}} }\!\lower0.7ex\hbox{$4$}}$, and ${\alpha _3} = {\raise0.7ex\hbox{${5\pi }$} \!\mathord{/ {\vphantom {{5\pi } 4}} }\!\lower0.7ex\hbox{$4$}}$, for four phase stepping algorithm, ${\alpha _1} = 0$, ${\alpha _2} = {\raise0.7ex\hbox{$\pi $} \!\mathord{/ {\vphantom {\pi 2}} }\!\lower0.7ex\hbox{$2$}}$, ${\alpha _3} = \pi $, and ${\alpha _4} = {\raise0.7ex\hbox{${3\pi }$} \!\mathord{/ {\vphantom {{3\pi } 2}} }\!\lower0.7ex\hbox{$2$}}$, and for five phase stepping algorithm, ${\alpha _1} = 0$, ${\alpha _2} = {\raise0.7ex\hbox{$\pi $} \!\mathord{/ {\vphantom {\pi 2}} }\!\lower0.7ex\hbox{$2$}}$, ${\alpha _3} = \pi $, and ${\alpha _4} = {\raise0.7ex\hbox{${3\pi }$} \!\mathord{/ {\vphantom {{3\pi } 2}} }\!\lower0.7ex\hbox{$2$}}$, and ${\alpha _5} = 2\pi $. The second sine terms of Eqs. (11) to (15) are modulated by a sine terms denoting high frequency random speckle noise.
The phase difference $\Delta {\varphi ^{\prime}}$ can be obtained by using the three, four, and five phase step algorithms from the intensity distributions of Eqs. (11) to (15), and is given by
For three phase stepping algorithm,
For four phase stepping algorithm,
For five phase stepping algorithm,
3.2 Error due to phase stepper error
The phase-stepper error, is the main error affecting most phase shifting interferometric (PSI) techniques [41]. For the 3, 4 and 5 position techniques, phase step size is ${\raise0.7ex\hbox{$\pi $} \!\mathord{/ {\vphantom {\pi 2}} }\!\lower0.7ex\hbox{$2$}}$, which is assumed to be fixed and known for the techniques of phase shifting interferometry (PSI). The phase may be estimated with error, due to the non-linearity in the motion of a PZT performing the phase-stepping, or a miscalibration of phase step size. In the presented method, the general equation for the error in the phase map due to the phase step error can be derived as follow:
The following analysis is assumed to apply to every point (x,y) in the interferogram.
Assuming an error ${\varepsilon _r}$ in the size of the phase step, i.e.
where $\alpha _r^{\prime}$ is the achieved phase step, and ${\alpha _r}$ is the correct phase step. Using as an equation for the intensity at a point for the phase angle ${\alpha _r}$,Substituting this into the general equation of phase stepping,
The error in the calculated value of $\Delta {\varphi ^{\prime}}$ will be,
Using the orthogonality relations of the sine and cosine functions and assuming that ${\varepsilon _r}$ is small, we can put $\cos {\varepsilon _r} = 1$ and $\sin {\varepsilon _r} = {\varepsilon _r}$, then the following expression of the error due phase stepper error [42] can be derived:
From Eq. (25), and after the simplification, the upper and lower terms will be:
Upper term =
Lower term =
So,
4. Experimental results and discussions
Figure 1 shows the optical set-up for comparative measurement in speckle interferometry using a double reference beam technique with LiNbO3 Crystal as a recording medium: Nearly half portion of the transmitted beam (object beam) illuminates the master (MO) and test (TO) objects. Remaining portion of the laser beam (pump beam or reference beam) enters the crystal directly. The pump beam given from the same laser source is set at an angle of 45° (external) with respect to the object beam. As a result, and inside the crystal, there is interference between the pump beam (reference beam) and the object beam, which results in the formation of the specklegram. According to the theoretical analysis described above, and as a final result of the analysis, difference correlation fringes which is modulated with the noise term displayed on the monitor of the computer, representing the difference of the displacements between the master and test objects. The result of the measurement made when the two objects master and test were displaced axially, as illustrated in Fig. 2(a)–2(c), which show the difference correlation fringes when the test and master objects are axially shifted by 2 µm and 3 µm respectively for three phase stepping $\pi /4$, $3\pi /4$ and $5\pi /4$.
Fig. 2. (a)–(c) Difference correlation fringes when the test and master objects are axially shifted by 2 µm and 3 µm respectively for three phase stepping $\pi /4$, $3\pi /4$, and $5\pi /4$, (d) wrapped image of the difference correlation fringes, (e) row 310 of the wrapped phase image. The used wavelength was 0.6700 µm.
Figure 2(d) shows the wrapped image of the difference correlation fringes. Figure 2(e) shows row 310 of the wrapped phase image. The used wavelength was 0.6700 µm.
Figure 3(a)–3(d) show the difference correlation fringes when the test and master objects are axially shifted by 2 µm and 3 µm respectively for four phase stepping 0, $\pi /2$, $\pi $, and $3\pi /2$. Figure 3(e) shows the wrapped image of the difference correlation fringes. Figure 3(f) shows row 310 of the wrapped phase image. The used wavelength was 0.4360 µm.
Fig. 3. (a)–(d) Difference correlation fringes when the test and master objects are axially shifted by 2 µm and 3 µm respectively for four phase stepping 0, $\pi /2$, $\pi $, and $3\pi /4$, (e) wrapped image of the difference correlation fringes, (f) row 310 of the wrapped phase image. The used wavelength was 0.4360 µm.
Figure 4(a)–4(e) show the difference correlation fringes when the test and master objects are axially shifted by 2 µm and 3 µm respectively for five phase stepping 0, $\pi /2$, $\pi $, $3\pi /2$, $2\pi $. Figures 4(f) and 4(g) show the wrapped image of the difference correlation fringes, and row 310 of the wrapped phase image in Fig. 4(f), respectively. The used wavelength was 0.3600 µm.
Fig. 4. (a)–(e) Difference correlation fringes when the test and master objects are axially shifted by 2 µm and 3 µm respectively for five phase stepping 0, $\pi /2$, $\pi $, $3\pi /2$, and $2\pi $, and (f) wrapped image of the difference correlation fringes, (g) row 310 of the wrapped phase image. The used wavelength was 0.3600 µm.
Due to the phase wrapping effect, which corresponds to three and four phase stepping, respectively, there is a discontinuity in the center in Figs. 2(d) and 3(e). Using the five phase stepping improves it, as Fig. 4(f) illustrates. Figures 5(a) and 5(b) show the wrapped image of the error due phase stepper using three phase stepping algorithm, and the row 310 of the wrapped image 5(a), respectively. The used wavelength was 0.6700 µm. Figures 5(c) and 5(d) show the wrapped image of the error due phase stepper using four phase stepping algorithm, and the row 310 of the wrapped image 5(c) respectively. The used wavelength was 0.4360 µm. Figure 5(e) and 5(f) show the wrapped image of the error due phase stepper using five phase stepping algorithm, and the row 310 of the wrapped image in Figs. 5(e), the used wavelength was 0.3600 µm. Depending on Eq. (22), Figs. 5(b), 5(d), 5(f), and Fig. 6 can be shown the dependence of the error due to phase stepper (radian) on the wavelength of the used light, for different phase stepping algorithm. As we can see from this figure, the phase error decrease with increasing the used wavelength, and also with increasing the number of phase stepping. This means, by using the five phase stepping, the more accurate results of the extracted phase can be obtained.
Fig. 5. (a) Wrapped image of the error due phase stepper using three phase stepping algorithm, (b) row 310 of the wrapped image (a), the used wavelength was 0.6700 µm. (c) Wrapped image of the error due phase stepper using four phase stepping algorithm, (d) row 310 of the wrapped image (c), the used wavelength was 0.4360 µm. (e) Wrapped image of the error due phase stepper using five phase stepping algorithm, (f) row 310 of the wrapped image (e), the used wavelength was 0.3600 µm.
Fig. 6. Dependence of the error due to phase stepper (radian) on the wavelength of the used light, for different phase stepping algorithm.
5. Conclusion
Accurate phase analysis technique is developed, which is a modified version of the phase-shifting differential digital speckle pattern interferometry, where the phase-shifting and phase-wrapping algorithms are applied to ensure the highest accuracy for the measurements. The new method depends on using the LiNbO3 crystal as a recording medium instead of using the conventional method. It also extends the application of the double reference beam technique to address the problem of comparing displacement fields of two identical samples.
New theoretical formulae discovered to gave a high documentation for the experimental work. The more accurate measurement of the extracted phase can be obtained by using the phase wrapping algorithm applied at 0.6700 µm, accordingly the phase error decrease with increasing the used wavelength, also with increasing the number of phase stepping. This means, by using the five phase stepping, the more accurate results of the extracted phase can be obtained. A good approval was reached between the theoretical formulae and the resultant experimental results.
Funding
National Science Technology and Innovation Plan (NSTP) (12-NAN2287-10); King Abdul-Aziz City for Science and Technology (MAARIFH).
Acknowledgement
National Science Technology and Innovation Plan (NSTP) (grant No.12-NAN2287-10), the King Abdul-Aziz City for Science and Technology (MAARIFH), Kingdom of Saudi Arabia.
Disclosures
The author declare no conflict of interest.
Data availability
Data supporting the results presented in this paper can be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
References
1. K. Creath, “Phase-measurements interferometry techniques,” in Progress in Optics, E. Wolf, eds. (Elsevier Science, 1988), Vol. 26, pp. 349–393.
2. J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, eds. (Elsevier Science, 1990), Vol. 28, pp. 271–359.
3. J. E. Greivenkamp and J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, D. Malacara, eds. (Wiley, 1991), Chap. 14, pp. 501–598.
4. R. S. Sirohi and M. P. Kothiyal, “Heterodyne and phase shifting interferometry,” in Optical Components, Systems, and Measurement Techniques, R. S. Sirohi, eds. (Marcel Dekker, 1991), pp. 219–246.
5. D. W. Phillion, “General methods for generating phase- shifting interferometry algorithms,” Appl. Opt. 36(31), 8098–8115 (1997). [CrossRef]
6. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error compensating phase calculating algorithm,” Appl. Opt. 26(13), 2504–2506 (1987). [CrossRef]
7. K. G. Larkin and B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9(10), 1740–1748 (1992). [CrossRef]
8. Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32(19), 3598–3600 (1993). [CrossRef]
9. J. Schwider, O. Falkenstorfer, H. Schreiber, et al., “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993). [CrossRef]
10. B. Zhao and Y. Surrel, “Phase shifting: six-sample self calibrating algorithm insensitive to the second harmonic in the fringe signal,” Opt. Eng. 34(9), 2821–2822 (1995). [CrossRef]
11. P. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34(22), 4723–4730 (1995). [CrossRef]
12. J. Schmit and K. Creath, “Extended averaging technique for derivation of error compensating algorithms in phase shifting interferometry,” Appl. Opt. 34(19), 3610–3619 (1995). [CrossRef]
13. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996). [CrossRef]
14. K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” J. Opt. Soc. Am. A 13(4), 832–843 (1996). [CrossRef]
15. V. Srinivasan, H Liu, and M Halioua, “Automated phase-measuring profilometry of 3-D diffuse objects,” Appl. Opt. 23(18), 3105 (1984). [CrossRef]
16. Y Kim, K Hibino, M Sugita, et al., “Surface profile measurement of a highly reflective silicon wafer by phase-shifting interferometry,” Appl. Opt. 54(13), 4207 (2015). [CrossRef]
17. P. de Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt. 39(16), 2658 (2000). [CrossRef]
18. Y. Zhao, R Yang, and C Yang, “An orthogonal phase-shifting interferometry and its application to the measurement of optical plate,” Optik 127(20), 8841–8846 (2016). [CrossRef]
19. Y. Kim, K. Hibino, N Sugita, et al., “Measurement of optical thickness variation of BK7 plate by wavelength tuning interferometry,” Opt. Express 23(17), 22928 (2015). [CrossRef]
20. J. Lee and G. Jiang, “Displacement measurement using a wavelength-phase-shifting grating interferometer,” Opt. Express 21(21), 25553 (2013). [CrossRef]
21. N. A. Moustafa, J. Kornis, and Z. Füzessy, “Active phase-shifting comparative speckle pattern interferometry,” Int. Symp. on Applied Optical Metrology, Balaton füred, 1998.
22. P. Groot, “Vibration in phase-shifting interferometry,” J. Opt. Soc. Am. A 12(2), 354 (1995). [CrossRef]
23. C. Ai and J. Wyant, “Effect of piezoelectric transducer nonlinearity on phase shift interferometry,” Appl. Opt. 26(6), 1112 (1987). [CrossRef]
24. J. Schwider, R. Burow, K. Elssner, et al., “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22(21), 3421 (1983). [CrossRef]
25. Y. Surrel, “Design of phase-detection algorithms insensitive to bias modulation,” Appl. Opt. 36(4), 805 (1997). [CrossRef]
26. Y. Ishii, J. Chen, and K. Murata, “Digital phase-measuring interferometry. with a tunable laser diode,” Opt. Lett. 12(4), 233 (1987). [CrossRef]
27. P. Hariharan, “Phase-stepping interferometry with laser diodes: effect of changes in laser power with output wavelength,” Appl. Opt. 28(1), 27–29 (1989). [CrossRef]
28. J. Ramaiah and R. Gannavarpu, “Dynamic displacement measurement in digital holographic interferometry using eigenspace analysis,” Appl. Opt. 60(33), 10468–10476 (2021). [CrossRef]
29. A. V. Satya Vithin, A. Vishnoi, and R. Gannavarpu, “Phase derivative estimation in digital holographic interferometry using a deep learning approach,” Appl. Opt. 61(11), 3061–3069 (2022). [CrossRef]
30. G. Rajshekhar and R. Pramod, “Fringe demodulation using the two-dimensional phase differencing operator,” Opt. Lett. 37(20), 4278–4280 (2012). [CrossRef]
31. G. Rajshekhar and R. Pramod, “Application of complex-lag distributions for estimation of arbitrary order phase derivatives in digital holographic interferometry,” Opt. Lett. 36(19), 3738–3740 (2011). [CrossRef]
32. N. A. Moustafa, J. Kornis, and Z. Füzessy, “Comparative measurement in speckle interferometry using holographically generated reference wave by single reference beam technique,” Opt. Commun. 172(1-6), 9–16 (1999). [CrossRef]
33. N. A. Moustafa, J. Kornis, and Z. Füzessy, “Phase-shifting comparative speckle pattern interferometry,” Proc. of Int. Symp. on Automotive Technology and Automation, Düsseldorf 7, 119–123 (1998).
34. K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt. 21(14), 2470 (1982). [CrossRef]
35. D. C. Ghiglia and M. D. Pritt, “Two-dimensional phase unwrapping: theory, algorithms, and software,” May 1998.
36. P. K. Rastogi, “Comparative holographic Moiré interferometer in real time,” Appl. Opt. 23(6), 924–927 (1984). [CrossRef]
37. Z. Füzessy and F. Gyimesi, “Difference holographic interferometry: technique for optical comparison,” Opt. Eng. 32(10), 2548–2556 (1993). [CrossRef]
38. Y. Zhang, Q. Lu, and B. Ge, “Elimination of zero-order diffraction in digital off-axis holography,” Opt. Commun. 240(4-6), 261–267 (2004). [CrossRef]
39. J. Schwiegeling, Optical Specification,Fabrication, and Testing, SPIE (28 October 2014),V. PM252, pp. 191-195.
40. N. A. Moustafa, J. Kornis, and Z. Füzessy, “Comparative measurement by phase-shifting digital speckle interferometry using holographically generated reference wave,” Opt. Eng. 38(7), 1241–1245 (1999). [CrossRef]
41. T. Van Wingerden, J. H. Frankena, and C. Smorenburg, “Linear approximation for measurement errors in phase shifting interferometry,” Appl. Opt. 30(19), 2718–2729 (1991). [CrossRef]
42. A. R. Nicolaus, “Evaluation of Fizeau interferences a comparison of phase-stepping algorithms,” Proc. SPIE 1319, 237–238 (1990). [CrossRef]
43. M. Servin, J. Antonio Quiroga, and M. Padilla, Fringe Pattern Analysis for Optical Metrology: Theory, Algorithms, and Applications, (John Wiley & Sons, 2014).
