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Abstract
A temporal electronic speckle pattern interferometry (ESPI) system is proposed for in-plane rotation measurement. The relationship between the rotation angle and the phase change distribution is established and the rotation direction is indicated by the sign of the partial differential of the phase change distribution. Temporal phase modulation is applied in the proposed symmetric illumination ESPI system. The phase is recovered by the temporal intensity analysis method which uses the temporal evolution history of the light intensity. The system can perform dynamic measurements and provide results in off-line real-time. Preliminary experiments were carried out with a continuously rotating target to show the feasibility and the dynamic feature of the temporal ESPI system. At present, the mean absolute error of the experiment is 0.39 arcsec.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Electronic speckle pattern interferometry (ESPI) plays a vital role in the field of optical metrology and is featured by the ability of carrying out full-field non-destructive testings with high accuracy in real-time. Various ESPI systems have been reported for different scenarios, e.g., out-of-plane/in-plane deformation measurement [1–3], shape/profile modeling [4], and surface roughness evaluation [5,6]. Regarding to in-plane rotation measurement, one of the challenges for ESPI system is to establish a relationship in mathematics between the phase change and the in-plane rotation angle. Another crucial issue is to determine the sign of the in-plane rotation.
In 1999, A. K. Nassim et al. proposed a dual-wavelength ESPI system to analyze the in-plane rotation [7]. By illuminating the target with two wavelengths successively, the rotation angle and sign are determined by the evaluation of fringe tilt. Two years later, B. Ráczkevi et al. described a system that only requires a single wavelength by introducing two movable mirrors [8]. The application of either dual-wavelength or movable mirrors provides a method to retrieve the phase in corresponding ESPI system, which, however, limits the time resolution of the measurement system. K. M. Abedin et al. established a fringe counting ESPI system [9] which is free of the problems above, but the accuracy provided by fringe counting technique is just passable and the rotation direction cannot be determined. Besides, the in-plane rotation can be converted into out-of-plane rotation by switching the observation direction. This conversion method falls generally outside the scope of this paper, so we retain the details in [10]. Recently, we proposed a spatial carrier ESPI system to determine the in-plane rotation angle and the direction [11]. The spatial carrier is generated by asymmetric illumination coupled with two apertures, and the phase is recovered from the carrier. However, during our further studies, we found that the asymmetric illumination brings in a systematic error resulting from the shearing item.
In this paper, we propose a temporal ESPI system to eliminate the systematic error in our previous work. The accuracy is improved over ten times than before. Temporal phase modulation is applied to enable the phase retrieval in symmetric illumination ESPI system, so that the shearing item is removed and the relationship between the phase and the rotation angle is revised accordingly. The phase is evaluated from the temporal carrier which is introduced by our newly developed phase modulator. The rotation direction is discriminated directly by the sign of the partial differential of the phase change distribution. In contrast to conventional techniques that the interferograms are captured before and after rotation, the proposed method captures the interferograms continuously while the target rotates, so there is no reduction of time resolution. As a result, the proposed system can perform dynamic measurements and provide results in off-line real-time.
2. Principle
2.1. Phase change distribution and in-plane rotation
The optical configuration of the proposed temporal ESPI system is illustrated in Fig. 1. A linearly polarized laser beam is directed into our home-made modulator and the modulated beam (by polarization control) is then expanded and collimated. The two components of the modulated beam are separated by a polarizing beam splitter (PBS) according to their polarization directions. Then, the target is illuminated by the two beam components symmetrically and simultaneously. A 45° oriented polarizer helps to make the two orthogonal components interfere with each other and a CCD camera is placed in front of the target to capture the interferograms.
Fig. 1 System configuration. The precise and controllable in-plane rotation of the target is provided by a micromechanical rotation stage. The lithium niobate crystal is driven by an external power supply. Note that the power supply are omitted for clarity. LN: lithium niobate crystal; Q: quarter-wave plate; L: lens; AP: aperture; PBS: polarizing beam splitter; M: mirror; P: polarizer; k : wave vector.
A Cartesian coordinate system is established with respect to the target. Assuming the z-axis is along the observation direction and all the laser beams propagate in the xz-plane, the sensitivity vector of the system is denoted as [12,13]
where , , and are the unit vectors along the positive x-, y-, and z-axis, respectively, being and the two wave vectors of the illuminating beams, is the wave vector of the object beam, λ is the operating wavelength, and θ is the incidence angle respected to the object beam. Note that in the proposed configuration the incidence angles for both illuminating beams are the same. For a general case, the expression for the phase change distribution, ∆φ (x, y), is proportional to the projection of the displacement vector, , onto the sensitivity vector, , denoted as where ∆lx, ∆ly, and ∆lz are the x, y, and z components of the displacement vector, respectively. The system is sensitive to the x-direction, as illustrated in Eq. (2).The relationship between the phase change distribution and the rotation angle is then established geometrically. As depicted in Fig. 2, the rotation is assumed to proceed in counterclockwise direction. An arbitrary point, P (x1, y1), on the target surface is selected for the following discussion. After introducing a rotation with an angle of (0, 0, Ω), the selected point goes to P′(x2, y2). Herein we define the range of the rotation angle, Ω, is [0, 90°]. Another point, Q(x1, y3), is considered, which locates at the vertical direction of P(x1, y1) along the y-axis, where y1 > y3, and the corresponding displaced point is Q′(x4, y4). The lateral displacements of both points along the sensitive direction are represented by
where the minus-plus sign is applied to the corresponding rotation direction, respectively, i.e., minus for counterclockwise and plus for clockwise. Since the rotation angle over the entire target surface is identical, the partial differential of the lateral displacement distribution, ∆lx (x, y), can be calculated by using Eq. (3) along the direction which is perpendicular to the sensitivity vector, namely the y-axis, denoted as Finally, from Eqs. (2) and (4), the relationship between the phase change distribution and the rotation angle is established as where ∆φ(x, y) is obtained by the phase retrieval algorithm described in Section 2.3. The rotation direction is indicated by the sign of the partial differential in Eq. (5). In the current coordinate system, positive values represent counterclockwise rotations and negative values represent clockwise rotations. Considering the range of the rotation angle, Ω, the domain of the arcsin function is [0, 1). If the partial differential is evaluated to be negative which indicates a clockwise rotation, then a minus sign is applied in the function to correct the domain. The minus-plus sign in Eq. (4) has the same meaning as it does have in Eq. (5).Fig. 2 Mathematical model of the in-plane rotation counterclockwise. P(x1, y1) is an arbitary point on the target surface. Points P and Q share the same abscissa. O, center of rotation; Ω, rotation angle; ∆lx, lateral displacement along the sensitive direction.
2.2. Polarization-controlled phase modulator
In the present method for in-plane rotation measurement, the desired phase is carried by a temporal carrier, so that the phase can be recovered by symmetric illumination which eliminates the systematic error resulting from asymmetric illumination. The temporal carrier is introduced by a polarization-controlled phase modulator. Unlike conventional phase modulators, our newly developed phase modulator is immune to the orientation errors and capable to be used with commercial cameras. It consists of a bulk of z -cut lithium niobate crystal (LiNbO3) and a stationary quarter-wave plate (see Fig. 3). The three-fold rotation axis of the crystal is along <001> direction and the dimension is d-width, d-height, and L-length. The longitudinal (L ) direction is along the three-fold axis. Under an application of a rotating electric field, the crystal operates in transverse mode. In [001] plane (square cross section in the crystal), the electric field is generated and rotates about the three-fold axis. For clarity we establish another coordinate system (x′y′z′) in Fig. 3, then the electric field is denoted as
where Ex′, Ey′, and Ez′ are the electric field components parallel to the corresponding axis, respectively, E0 is the amplitude, ω is the angular velocity, and t denotes time parameter.Fig. 3 The phase modulator consists of a powered-up lithium niobate crystal (LiNbO3, LN) and a stationary quarter-wave plate. The four rectangular surfaces are covered with electrodes, respectively. ω is the angular frequency of the driving electric field and t denotes the time parameter. Greek letters in each dash line square represent corresponding angles which are marked by short arc. The azimuth (β) of the u v-axis is time-varying. The double-arrow red lines stand for polarizations (input, Vy″, and Vx″) and F is the fast axis of the quarter-wave plate. The coordinate transformations and detailed interpretations regarding to all parameters can be found in Section 2.2.
Assuming that the laser propagates along the z′-axis (three-fold axis), the central section normal to the laser beam is selected to describe the refractive index distribution and the indicatrix degenerates into an ellipse with z′ = 0, denoted as
where no is the intrinsic refractive index of the ordinary wave; γ is the electro-optic tensor. To write Eq. (7) in principal axis system, a principal axis transformation is applied to rotate the original principal axis system about the origin with an angle of β (we would like to point out that this rotation is carried out mathematically and it should be distinguished with the target rotation which is denoted as Ω in Section 2.1). The relationship between the original and the new principal axis systems is defined as where u and v are the new axes, is the rotation matrix. In the new principal axis system, the refractive index distribution is denoted asObviously, in the principal axis system, the coefficient of the cross term is zero, such that the rotation angle, β, can be described by the rotating electric field as
where ω is the angular velocity of the applied electric field. Equations (9) and (10) show that the rotation of the refractive index ellipse is time-related and the ellipse rotates in the opposite sense to that of the applied electric field with an angular velocity of ω/2. Inside the ellipse, the major and minor axes are the allowed polarization directions which also rotate along with the ellipse rotation. The lengths of the semi-major axis and the semi-minor axis numerically equal to the refractive indices (nu and nv) for the corresponding polarization directions.Assuming that the length of the crystal along the z′-axis is L, the phase difference between the two allowed polarizations is
where λ is the operating wavelength and ξ is Equations (11) and (12) show that if E0 is fixed, the phase difference between the two allowed polarizations is constant and independent of time.By far, we can reach two consensus: 1. inside the powered-up crystal, the two allowed polarization directions are rotating and always maintain the orthogonality; 2. by applying a half-wave voltage, the phase delay between the two rotating orthogonal polarization directions can be set to π. Assuming that the input laser is linearly polarized with an arbitrary azimuth and propagates along the three-fold axis in the powered-up crystal, the function of the crystal is described in the x′y′-plane which is perpendicular to the incident laser beam, denoted as
where Vcrystal represents the laser beam leaving the powered-up crystal, the item marked as “phase delay π” is the phase delay provided by the powered-up crystal when the driver is operating in half-wave voltage mode, and the items marked as “input laser” describe a linearly polarized laser beam which has an arbitrary polarization azimuth (denoted as α with respect to the x′-axis). The rotation matrix and the inverse rotation matrix are applied for rotating the coordinate system.Then the modulated beam, Vcrystal, passes through a quarter-wave plate (QWP) whose fast axis is oriented at an arbitrary angle, η, with respect to the x′-axis. The output can be calculated as
where the x′y′ coordinate is rotated with an angle of η − π/4, represented by , Vx″ and Vy″ are the two components along the axes in the x″y″ coordinate, respectively. Note that the coordinate transformation does not influence the physical features of the modulation, but changes the mathematical representation into an easy-reading form. In front of the image plane, a polarizer is placed with an orientation of ψ to make the two orthogonally polarized beams interfere with each other and also a coordinate rotation with an angle of ψ is applied to give an easy-reading result, denoted as where Vimg is the complex amplitude of the laser beam at the image plane, is the rotation matrix, the second matrix is the Jones matrix of the polarizer, and ψ is the arbitrary azimuth of the polarizer. At last, the interferometric intensity is described as where (·)† denotes the conjugate transpose operation. Euler’s formula is applied for the simplification and only the real part is meaningful and shown in the equation. From Eqs. (13)–(16), the orientations of the input polarization (α), the quarter-wave plate (η), and the polarizer (ψ) are all arbitrary. Once an interferometric system is built up, the parameters, α, η, ψ, will be fixed and the phase item, 2η + 2α, is regarded as the initial phase. The time-varying phase shift is only related to the modulation. The azimuth of the polarizer, ψ, just determines the visibility of the interferometric pattern. Therefore, the proposed phase shifter is immune to the orientation errors of the optical components and insensitive to the input polarization direction.2.3. Phase retrieval in temporal ESPI system
During the measurement of the in-plane rotation, the interferograms are recorded continuously in time sequence. Here, we assume that the recording rate satisfies the Nyquist-Shannon sampling theorem regarding to the carrier frequency. The temporal intensity analysis method is applied to recover the phase change. In this case, a time parameter is taken into account and considering the sensitivity vector of the system, the phase change is then represented by ∆φ(x, y, t). The time-related intensity distribution at the image plane is denoted as
where Euler’s formula is applied, I0(x, y) is the background intensity, Iv is the visibility, ω is the angular velocity of the crystal driver, and φ0 is the initial phase where 2η and 2α are included (see Eq. (16)). The temporal intensity evolution history of a single pixel, located at (x1, y1), is extracted for further illustration. 1D Fourier transform is applied to convert the temporal intensity evolution history, I(x1, y1, t), into Fourier domain, denoted as [14] where the upper-case letters represent the corresponding Fourier spectra referring to each item in Eq. (17), the asterisk denotes the complex conjugate function, ft is the transform variable, and f0 = ω/π is the temporal frequency which is introduced by the phase modulator. A filter is used to isolate the sideband, C(ft − f0), in Fourier domain and the inverse Fourier transform is applied to the isolated sideband to obtain the complex function, c(x1, y1, t). Then the phase change, Δφ(x1, y1, t) is recovered by where Im and Re represent the real and imaginary parts, respectively.By far, the phase evolution history of the selected pixed is obtained. In the measurement, the temporal intensity analysis method is applied to all the pixels over the entire surface. For each captured frame, there is a corresponding phase map and the in-plane rotation can be measured by Eq. (5) for each sampling moment.
3. Experimental results and discussions
In this section, we first demonstrate the feasibility of the proposed temporal ESPI system without the rotation of the target and then an in-plane rotation measurement was carried out to show the accuracy and the dynamic feature of the experimental system. Three issues are discussed at last, which are the quality of the phase modulation, the measuring range, and the upper limit of the angular velocity of the in-plane rotation.
3.1. Experiments
The system arrangement is shown in Fig. 1. A He-Ne laser (Carl Zeiss Jena HNA 188-2) was used as light source with a wavelength of 632.8 nm and a power of 50 mW. The laser was linearly polarized with an extinction ratio of 1000:1. The dimension of the LiNbO3 crystal was 5 (d) × 5 (d) × 30 (L) mm3 and the laser beam propagated along the 30 mm direction (three-fold axis). The half-wave voltage of the crystal was uπ = d · E0 = 349 V. Here, we would like to point out that when calculating the half-wave voltage, a revised electro-optic coefficient, γ22 = 7 × 10−12 mV−1, for extremely low frequencies was taken into account to make the phase modulator operate in a desired frequency band [15, 16]. The angular velocity of the applied electric field was set to ω = 34π s −1. The fast axis of the quarter-wave plate is oriented at 45° with respect to either of the allowed polarization directions in the PBS. The laser beam is expanded and collimated by two lenses with focus lengths of −6 mm (L1) and 75 mm (L2) and an aperture was used to eliminate the undesired beam part. Then the two components of the modulated beam were separated by a PBS according to their polarization directions. Two mirrors (M1 and M2) directed the two laser beams onto the target surface with an identical incident angle of θ = 45° (symmetric illumination). A charge-coupled device (CCD) camera (Basler, piA640-210gm) with a zoom lens was placed in the front of the target to capture the interferometric images. The region of interest was 160 × 300 pixels and the frame rate was 272 f ps. A polarizer was inserted between the target and the CCD camera to make the two orthogonal linearly polarized components interfere with each other. The target was an aluminum alloy plate which was fixed on a micromechanical rotation stage.
The first experiment was performed without the rotation of the target. The acquisition time was 2 s and 544 frames were captured during the experiment. The 80th horizontal slice of each frame was extracted and stacked together in time sequence to illustrate the temporal intensity evolution history. As shown in Fig. 4(a), from the top to the bottom, periodic intensity evolution is observed. Then the temporal intensity evolution history of the pixel located at (100,150) of each frame was selected to have a detailed insight of its Fourier spectrum. As shown in Fig. 5, the DC component is the spectrum of the background intensity and the temporal frequency is exactly 34 Hz according to the system configuration (see Eq. (17) with ω = 34π s −1).
Fig. 4 (a) illustrates the normalized intensity evolution history. (b), (c), and (d) shows three captured frames, where the 80th horizontal slices are marked with red line. For all of the captured 544 frames, the 80th horizontal slices are extracted and stacked together in time sequences to generate (a).
Fig. 5 The Fourier spectrum of the temporal intensity evolution history of the pixel located at (100,150) of each frame. DC component is the spectrum of the background intensity. 34 Hz is the temporal frequency which is introduced by the phase modulator.
In the second experiment, the same system configuration was used. The in-plane rotation was introduced by the micromechanical rotation stage which is driven by a piezoelectric transducer (PZT) to achieve enough accuracy. The resolution provided by the PZT-driven micromechanical rotation stage is 0.1 arcsec. The direction is counterclockwise with respect to the camera plane. The acquisition time was 4 s and 1088 frames were captured during the experiment. The phase is recovered by the method illustrated in Section 2.3. The sine-cosine filter [17, 18] was applied to remove the random noise. For the recovered phase maps, Visualization 1 is provided as supplementary material to illustrate the dynamic feature of the proposed system (the frame rate of the video playback is 30 f ps ). The recovered phase is time-related which results from the in-plane rotation of the illuminated target. Nine phase maps are selected from the supplementary video and shown in Fig. 6 whose corresponding frame index runs from 250 to 1050 with a step length of 100. In both the supplementary video and Fig. 6, the wrapped phase maps describe the phase change distributions on the image plane. The total phase is then obtained by phase unwrapping technique [19]. To evaluate the angle and the direction of the in-plane rotation, the gradient (the partial differential item in Eq. (5)) of each phase map is calculated along the direction which is perpendicular to the fringe in Fig. 6. The values of the partial differentials were all positive, which indicates the in-plane rotation is counterclockwise. Finally, the measured angles are shown in Fig. 7. The mean absolute error (MAE) between the measured results and the theoretical values is 0.39 arcsec.
Fig. 6 The recovered phase maps. From (a) to (i), the corresponding frame index runs from 250 to 1050 with a step length of 100. The equivalent time interval is 100 (f rames )/272 (f ps ).
Fig. 7 Dynamic measurement results of a continuously rotating target. (a) shows the measured results and the theoretical values. (b) shows the absolute errors of the measurement.
3.2. Discussions
The first issue we would like to discuss is the quality of the phase modulation. In both of the experiments, the laser beam is expected to pass through the center of the square cross section of the powered-up crystal. Inside the crystal, the electric field is uniform along the center line in the three-fold axis direction, whereas in the off-axis area, due to the cross talk between the x′- and y′- components of the electric field, there will be certain distortion of the synthetic field and the distortion reaches the maximum at the edge of the crystal. In the presented experiments, the aperture acted as a spatial filter to help eliminating the undesirable beam to guarantee the quality of the phase modulation.
Another issue is the measuring range of the proposed in-plane rotation measurement system. Based on the analysis in Section 2.1, the rotation angle, Ω, is limited by [0, 90°]. If the rotation angle exceeds the upper limit, the arcsin function in Eq. (5) becomes an multivalued function which leads to an ambiguity in the angle evaluation. Hence, theoretically, the measuring range is [−90°, 90°], if both directions are considered. However, in practice, the “fringe” density of the wrapped phase change distribution (see Fig. 6) must be taken into consideration. The partial differential in Eq. (5) fails when the fringes become too dense. Here, we provide a potential solution that when dense fringes occur, the current frame is selected as a new start point to reset the wrapped phase change distribution. In other words, the captured frame set can be divided into several subsets in time domain. Within each single frame subset, the fringe density is suitable to carry out the measurement. Then for a single moment, the angle is the summation of the angle within the current subset and all the calculated angles in previous subsets. But note that the errors accumulate from each summation operation, which results in a trade-off between the accuracy and the range of the measurement.
The last issue is the upper limit of the angular velocity of the in-plane rotation, which can be measured by the proposed temporal ESPI system. To implement a feasible temporal intensity analysis (see Section 2.3), the sidebands in Fourier domain should be separated from the background frequency (DC component), which means the equivalent frequency of the phase change is required to be lower than the temporal frequency (f0 in Eq. (18)), that is
where the partial differential indicates the instant phase change rate which is further related to the angular velocity of the in-plane rotation (see Eq. (5)). When the requirement in Eq. (20) is not satisfied, the sidebands can be separated again by increasing the temporal frequency. On the other hand, the sampling frequency of the camera should be increased accordingly to meet the Nyquist-Shannon sampling theorem.4. Conclusion
This paper presents a temporal electronic speckle pattern interferometry system for in-plane rotation measurement. Symmetric illumination is applied to remove the systematic errors which result from asymmetric illumination. The accuracy is improved over ten times than before. Both of the angle and the direction of the rotation can be determined simultaneously. The relationship between the phase change distribution and the rotation angle has been established. The phase is carried by a temporal frequency which is introduced by our newly developed phase modulator. Temporal intensity analysis method is used to recover the phase from the carrier. In preliminary experiments, a continuously rotating target was measured and the results are provided in off-line real-time, which show the feasibility and the dynamic feature of the proposed system. The mean absolute error of the presented in-plane rotation measurement is 0.39 arcsec.
Funding
China Scholarship Council (CSC) (201507090066, 201408080029).
References and links
1. V. D. Madjarova, H. Kadono, and S. Toyooka, “Dynamic electronic speckle pattern interferometry (DESPI) phase analyses with temporal hilbert transform,” Opt. Express 11, 617–623 (2003). [CrossRef] [PubMed]
2. M. Lu, S. Wang, L. Aulbach, and A. W. Koch, “Simultaneous displacement and slope measurement in electronic speckle pattern interferometry using adjustable aperture multiplexing,” Appl. Opt. 55, 5868–5875 (2016). [CrossRef] [PubMed]
3. F. M. Santoyo, M. C. Shellabear, and J. R. Tyrer, “Whole field in-plane vibration analysis using pulsed phase-stepped espi,” Appl. Opt. 30, 717–721 (1991). [CrossRef] [PubMed]
4. L.-S. Wang and S. Krishnaswamy, “Shape measurement using additive-subtractive phase shifting speckle interferometry,” Meas. Sci. Technol. 7, 1748–1754 (1996). [CrossRef]
5. P. de Groot, “Interferometric laser profilometer for rough surfaces,” Opt. Lett. 16, 357–359 (1991). [CrossRef] [PubMed]
6. N. Werth, F. Salazar-Bloise, and A. Koch, “Influence of roughness in the phase-shifting speckle method: An experimental study with applications,” Rev. Sci. Instruments 85, 015114 (2014). [CrossRef]
7. A.-K. Nassim, L. Joannes, and A. Cornet, “In-plane rotation analysis by two-wavelength electronic speckle interferometry,” Appl. Opt. 38, 2467–2470 (1999). [CrossRef]
8. B. Ráczkevi, F. Gyímesi, and S. Mike, “One-wavelength in-plane rotation analysis in electronic speckle pattern interferometry,” Opt. Lasers Eng. 35, 33–40 (2001). [CrossRef]
9. K. Abedin, M. Wahadoszamen, and A. Haider, “Measurement of in-plane motions and rotations using a simple electronic speckle pattern interferometer,” Opt. & Laser Technol. 34, 293–298 (2002). [CrossRef]
10. Y. P. Kumar, S. Chatterjee, and S. S. Negi, “Small roll angle measurement using lateral shearing cyclic path polarization interferometry,” Appl. Opt. 55, 979–983 (2016). [CrossRef]
11. M. Lu, S. Wang, L. Aulbach, M. Jakobi, and A. W. Koch, “Non-phase unwrapping interferometric approach for a real-time in-plane rotation measurement,” Opt. Lett. 42, 1986–1989 (2017). [CrossRef] [PubMed]
12. A. Martínez, J. A. Rayas, R. Rodríguez-Vera, and H. J. Puga, “Three-dimensional deformation measurement from the combination of in-plane and out-of-plane electronic speckle pattern interferometers,” Appl. Opt. 43, 4652–4658 (2004). [CrossRef] [PubMed]
13. A. Koch, M. Ruprecht, O. Toedter, and G. Häusler, “Optische messtechnik an technischen oberflächen,” Expert. Renningen-Malmsheim, Ger. (1998).
14. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982). [CrossRef]
15. J. F. Nye, Physical properties of crystals: their representation by tensors and matrices (Oxford University, 1985).
16. C. C. Davis, Lasers and electro-optics: fundamentals and engineering (Cambridge University, 2014).
17. P. Jacquot and J.-M. Fournier, Interferometry in speckle light: Theory and applications (Springer Science & Business Media, 2012).
18. H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999). [CrossRef]
19. J. C. Estrada, M. Servin, and J. A. Quiroga, “Noise robust linear dynamic system for phase unwrapping and smoothing,” Opt. Express 19, 5126–5133 (2011). [CrossRef] [PubMed]
