1. INTRODUCTION

Optical measurement methods have been widely accepted in scientific and industrial research due to their high sensitivity and lack of contact [1–4]. In this manner, the ESPI technique is commonly employed as an accurate optical method for full-field three-dimensional (3D) displacement assessment [3–8], for surface deformation distributions tracking [5–8], for non-destructive testing [9–11], for experimentally verification with models of applied mechanics, and for shape measurement [10–13], as well as in several other fields of science and engineering [14–17]. In ESPI, object displacement is associated to the phase alteration of the interferograms that are obtained through subtraction between two speckle patterns that represent an initial and a deformed state [18,19]. In dynamic events, the use of an electro-mechanic system to apply phase-shifting requires time to synchronize the image recording and the phase shift, making it sensitive to the presence of vibration during the measurement. These difficulties affect the application of ESPI measurements to various fields, such as vibrational analysis in mechanics [15–17,20], study of cardiac hypertrophy [21], material studies in fabrication of cardiac stem cells [22], and eye diseases [23,24]. In order to minimize environmental effects, simultaneous phase-shifting techniques based on polarization have been presented in systems such as a Mach–Zender configuration [25] and a Michelson interferometer [26] (both with polarization modulation and a 4-$\textit{f}$ system to replicate three interferograms), a polarized Radial Shear interferometer that replicates over three patterns [27], and a Speckle interferometer [28] and a Twyman-Green interferometer [29] (both with a Phase-Cam as a four-interferogram replicating system). The results reported in Refs. [28,29] show that only the phase map was retrieved. Polarizing phase-shifting techniques have the advantages of allowing rapid measurement, achieving best results with low contrast fringes, and being able to vary the sensitivity by varying the number of fringes. An advantage of the proposed system, over those reported by other authors, is that it does not require the use of two or more cameras to capture simultaneous patterns and special elements such as pixelated masks or synchronizing pulsed lasers [30–38]; besides, the optical phase is processed by using a known two-step algorithm based on the Gram-Schmidt method [39–42]. However, the use of polarization to generate the shifts limits the application when the materials under test alter the degree of polarization of the system [36–38]. On the other hand, some authors have proposed the use of three-beam systems with adjustable aperture multiplexing to introduce multiple tunable carrier frequencies into a three-beam electronic speckle pattern interferometer, which has the advantage of being able to measure the out-of-plane displacement and its first-order derivative simultaneously [43]. Under the same concept of three beams, others have developed a dual-directional shearography method for simultaneously measuring the surface strain in two directions [44]. Recently, other researchers have developed a shearography system capable of simultaneous and dynamic in-plane and out-of-plane strain measurements for measuring crack-tip deformation fields [45]. Some of these systems are simpler than the one we have developed, however, they have the disadvantage that they still use Fourier techniques for processing the optical phase, which requires higher resolution acquisition devices, since they are used with higher frequency fringes. Thus, one of the advantages of the presented system consists in the phase-step method. The phase-step method allows the use of low frequency fringes, which also allows the use of more accessible cameras of standard resolution. As a proposal, we present a dynamic parallel phase-shifting electronic speckle pattern interferometer DPP-ESPI), that uses two coupled systems: a polarized out-of-pane ESPI system to generate the speckle pattern, and a system based on a Michelson configuration to replicate the original pattern. This second system generates two simultaneous parallel patterns with relative phase shifts, which can be changed independently by operating linear polarizers. Through the proposed system we have shown, simultaneous n-interferograms can be used to process speckle patterns using a coupled interferometer as a replicator. The system has the advantage of not using diffractive elements such as pixelated masks, and it allows measuring dynamic events in real time [46,47]. The significant advance of the presented system is that it does not use special components to generate the replicates of the interferograms, such as diffraction gratings or micropolarizer arrays, and the position x–y of the speckle patterns can be adjusted by moving and tilting the mirrors. Additionally, it simplifies the number of phase steps needed to process the optical phase.

2. FRINGE FORMATION IN ESPI

Fringes as seen on the computer monitor are merely manifestations of an array of numbers in memory that represent irradiance changes caused by phase changes in the pixels. They are not even “fringes” in the strict interferometry sense, but only loci of constant irradiance change. The displayed fringes are a measure of the amount of surface displacement between two deformed (stressed) states, or between a stressed state and an unstressed state. The mathematical descriptions of these processes are given as

The resulting difference recorded will be

The square-root in Eq. (3) describes the background illumination. The first sine-term gives the stochastic speckle noise which varies randomly from pixel to pixel. This noise is modulated by the sine of the half phase difference induced by the deformation. This low frequency modulation of the high frequency speckle noise is recognized as an interference pattern. The interferogram’s phase $\Delta \phi$ associated to the deformation is determined by the scalar product of the displacement vector, $\vec d$, and the sensitivity vector $\vec e$. Then, a displacement $\vec d(p)$ of a displaced point $p$ on the object surface can be measured by means of the following expression [12,18]:

3. DYNAMIC PARALLEL PHASE-SHIFTING ESPI

Figure 1 shows the diagrams that compose the DPP-ESPI system. Figure 1(a) shows the polarized out-of-plane system, while Fig. 1(b) shows the Michelson configuration as the replication system (RS). In Fig. 1(a), a wavefront directed towards a beam splitter (${{\rm{BS}}_1}$) is linearly polarized at 45° by using a polarizer (${P_{45^\circ}}$). On each arm of the system, a microscope objective (MO) is placed to expand the object beam (A) and the reference beam (B). On the object arm, right circular polarization is generated through the ${{\rm{PH}}_{\rm{A}}} - {{\rm{Q}}_{\rm{A}}}(45^\circ)$ system. On the reference arm, left circular polarization is generated using the ${{\rm{PV}}_{\rm{B}}} - {{\rm{Q}}_{\rm{B}}}(- 45^\circ)$ system. In the reference arm, the scattering plate (SP) was selected according to maintain statistical properties of the scattered beam. In addition, we select the proper exit pupil size of the optical system in order to acquire completely and uniformly the scattered light from the SP [48]. Each resolution element in the scattering plate (SP) contains many scatterers. Thus, the exit pupil of the optical system is completely and uniformly filled with the light scattered by the SP [48].

 

Fig. 1. (a) Out-of-plane scheme with polarization components and the Michelson replicating system (MRS). (b) Replication system based on a Michelson configuration.

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The polarized wavefronts with orthogonal circular polarizations are combined by beam splitter ${{\rm{BS}}_2}$. In the Michelson replication system (MRS), the beams are divided by beam splitter ${{\rm{BS}}_3}$, which belongs to the Michelson configuration that operates as a replicator system for the incoming pattern. Inside the RS, mirrors ${{\rm{M}}_{\rm{R}}}$ and ${{\rm{M}}_{\rm{L}}}$ are adjusted in order to have laterally displaced equidistant replicas of the input. Every beam travels on its respective axis, right ($R$) or left ($L$), towards the observer (a CMOS camera with zoom lens) going through a linear polarizer array, PA (${P_{{{{\Psi}}_1}}}$ or ${P_{{{{\Psi}}_2}}}$). Every polarizer makes it possible for the combined pattern in a beam to interfere.

A. Polarizing and Replicated Shifted Patterns

In the general case, following the out-of-plane system shown in Fig. 1(a), and using the Jones matrix approach for polarization, the object and reference beam sections of the wavefronts with opposite circular polarization are

Considering of that the out-of-plane system and the Michelson system are coupled, beam ${\vec J_O}$ is replicated. Then, when each replicated field is observed through a linear polarizing filter, whose transmission axis is at an angle ${{\Psi}}$, the new polarization state of every beam is

The irradiance distribution of the obtained pattern can be expressed as [29,37]

B. Michelson Replication System

The speckle patterns generated by the out-of-plane interferometer enters into an MRS that does not operate as an interferometer. Instead, it is used to generate two replicas from the incident speckle patterns. In order to observe the two speckle interferograms, it is necessary to place a polarizer array (PA) shown in Fig. 1. The MRS system allows us to separate the replicas at a distance ($x_0$) by moving the mirror distances and the tilt of the mirrors ${M_L}$ and ${M_R}$. This method has the advantage of generating two replicas without minimum changes, as it would be the case for diffractive elements or holographic masks [29,36]. In the replication stage, the two generated patterns by the MRS pass through the polarizer array (PA), which is formed by linear polarizers placed at known angles. This allows us to generate parallel speckle patterns with known phase shifts that will be used to recover the optical phase. Considering Eq. (4) and the polarizer array (PA), the two patterns generated by the replication system can be expressed by Eqs. (11) and (12).

C. Some Considerations of Polarization

By the experience obtained in previous implementations [36,37], specific steps must be considered to introduce phase changes by polarization: (1) the angular aperture is controlled with a varying diaphragm from a commercial zoom lens (ZL) (f-number of 16) to register fully polarized speckle patterns and to prevent depolarization by the sample; (2) several aluminum plates should be tested, varying their surface roughness [32,33] and checking the variation of the output polarized light with an analyzer. If small changes are observed in the polarization states, models considering amplitude modulation variation can be implemented as in Refs. [36,37]. Experimentally, small changes were observed in the polarization states, so in this case the polarization states are not totally circular and can be elliptical according to the following equation:

However, the analysis of the polarization states is outside the scope of this investigation, and due to the fact that the output polarization is elliptical, the shifting can be considered constant only for some regions of the ellipse. For this reason, we only have a margin of 15° of phase-shift, for which the amplitude of the patterns is kept constant, and the linear phase-shift can be applied according to Eq. (11) and Eq. (12) [36,37].

4. TWO-STEP ALGORITHM

To retrieve the phase map from two interferograms it is possible to apply a phase step estimator [37,49,50]. We considered the phase estimator reported by Vargas et. al. [39], in which the phase is retrieved by means of

In order to unwrap the phase, a recursive approach [51] was implemented. The expression to unwrap the phase is given by

From Eq. (4), since the contribution of the transverse sensitivity vector components ${e_x}$ and ${e_y}$ is approximately zero in an out-of-plane system, the out-of-plane displacement $w$ of a point $P$ can be evaluated through the following [12,18,52–54]:

5. EXPERIMENTAL RESULTS

Figure 2 shows the DPP-ESPI setup composed of the out-of-plane system and the MRS, which are coupled. Laser light ($\lambda = 632\;{\rm{nm}}$) with vertical and horizontal polarization components is divided into illumination (A) and reference (B) beams by beam splitter ${{\rm{BS}}_1}$. Following the polarization modulation described in Section 3.A, the beams are circularly polarized with opposite directions. They are combined by means of ${{\rm{BS}}_2}$ and considered as the incoming beam for the MRS.

 

Fig. 2. (a) Aluminum plate clamped along its shorter edges. (0, 0, 0): origin of the reference system that coincides with the optical axis. (b) Experimental setup—out-of-plane interferometer. ${\boldsymbol{\lambda}}$, 632 nm; ${{\textbf{P}}_{45^\circ}}$, ${{\textbf{PH}}_{\textbf{A}}}$, ${{\textbf{PV}}_{\textbf{B}}}$, polarizers; ${{\textbf{BS}}_{\textbf{1}}}$, ${{\textbf{BS}}_{\textbf{2}}}$, beam splitters; ${{\textbf{M}}_{\textbf{A}}}$, ${{\textbf{M}}_{{\textbf{B1}}}}$, ${{\textbf{M}}_{{\textbf{B2}}}}$, mirrors; ${{\textbf{Q}}_{\textbf{A}}}$, ${{\textbf{Q}}_{\textbf{B}}}$, quarter wave plates; ${\textbf{M}}{{\textbf{O}}_{\textbf{A}}}$, ${\textbf{M}}{{\textbf{O}}_{\textbf{B}}}$, microscope objectives; ${\textbf{O}}$, object; ${\textbf{SP}}$, scattering plate. Michelson replicating system (MRS) to generate two interferograms: ${{\textbf{BS}}_{\textbf{3}}}$, beam splitters; ${{\textbf{M}}_{\textbf{R}}}$, mirror to direct a replicated pattern towards $\boldsymbol P(\textbf{0}^\circ)$; ${{\textbf{M}}_{\textbf{L}}}$, mirror to direct a replicated pattern towards $\boldsymbol P(- \textbf{5}^\circ)$; $\boldsymbol P(\textbf{0}^\circ)$ and $\boldsymbol P(- \textbf{5}^\circ)$, linear polarizers.

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As we are having the same number of reflections in mirrors and beam splitters for both beams, the sign of both circular polarization states change equal to maintaining its orthogonality. After going through the polarizer array (PA), every replicated beam goes through its respective linear polarizer, which causes the object and reference beams to interfere. To avoid errors in the calculation of the phase due to the contrast changes and modulation of the fringes by errors in the angle (${{\Psi}}$) of the polarizers, the two polarizers of the PA were cut with a laser cutting machine (LX-6090 elite), and for their placement the intensities were measured with a calibrated photodetector to ensure that the polarizers were placed at the correct angles (the polarizing film that was used is commercially available, Edmund Optics, TechSpecs high contrast linear polarizing film). Then, the interfered beams are imaged onto the CMOS sensor, which is on $z$ axis and located at a linear distance of 180 cm from the object $O$, yielding an image size of ${{2056}} \times {{1542}}$ pixels, (pixel size, ${3.45}\;\unicode{x00B5}{\rm m} \times {3.45}\;\unicode{x00B5}{\rm m}$ -EO-32122M camera). The object we used was an aluminum plate clamped on its sides [Fig. 1(a)], of $40\;{\rm{mm}} \times 20\;{\rm{mm}}$ in size and 3 mm in thickness, and placed on the $xy$ plane at $z = 0$. The sample was subjected to mechanical inflection when point $({0,0,0})$ was loaded from the back by a screw placed on a micrometer translation stage. According to the reference system, the illumination source ${{\rm{MO}}_A}$ was located at (17.78 cm, 0 cm, 25.39 cm), corresponding to an angle of incidence of 35° measured with respect to axis $z$and to $\overline {{{\rm{MO}}_{\rm{A}}} - {\rm{O}}}$ distance of 31 cm. One of the disadvantages of generating several patterns in a single image, is to cut the same region for all the patterns, since incorrect cropping induces errors in the calculation of the optical phase. In order to register and process two interferograms on a single image, we placed an aperture at the entrance of the system, calculated the centroid of each replica, and generated a rectangular mask around each centroid for the two interferograms; additionally, the calibration is verified using a letters pattern (see Fig. 3), and verifying that the replicas of the letters were correctly aligned. Because of this, some trade-offs appear while placing several images over the same detector field, but for low frequency interferograms with respect to the inverse of the pixel spacing, the influence of these factors seems to be rather small [37]. As it was already mentioned, spatial resolution is compromised, and this can be noted by employing high frequency fringes. This can be resolvable using an aperture at the entrance of the system to identify a common point on the system and employ image registration techniques. Under our working conditions, we did not require any numerical compensation. Due to using MI as a replication system our alignment procedure is coming from adjusting the mirror distances and tilt, ensuring that our beams remain at the same distance from the optical axis when the incident occurred in the PA.

 

Fig. 3. Calibration pattern.

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As it was already mentioned, spatial resolution is compromised, and this can be noted by employing high frequency fringes. This can be resolvable using an aperture at the entrance of the system to identify a common point on the system, using low frequency fringes, and employing image registration techniques. After acquiring the simultaneous speckle pattern that represented the initial state, an applied load induces a displacement in the plate in the z-axis which increases every 1 µm. This displacement was registered by taking 290 speckle images, with a camera acquisition configuration of 30 images per second. The 290 speckle patterns were correlated with the reference pattern to obtain the same number of interferograms. Figure 4 shows simultaneous double interferograms extracted from the 290 frames (Visualization 1). Due to the superposition of two patterns in the middle zone of every interferogram shown in Fig. 4, the ${I_1}$ and ${I_2}$ interferograms were selected according to the selected square region in each frame. Therefore, each interferogram corresponds to an evaluated area of $25\;{\rm{mm}} \times 14\;{\rm{mm}}$. Since an estimator assumes the amplitude and the background intensity of the interferograms to be constant [55], the patterns were pre-processed by using a mean filter (window size: ${{8}} \times {{8}}$, loop number: 23). After computing the wrapped phase and unwrapping phase ${{\Delta}}\phi$ for every pair of pre-processed interferograms through Eqs (11) and (12), the displacement maps were computed through Eq. (18). The sensitivity vector component ${e_z}$ was calculated by using the ${{\rm{MO}}_{\rm{A}}}$ and CMOS coordinates. Wrapped phases ${{\Delta}}\phi$ and displacement maps for the four states are shown in Fig. 5. Temporal evolution of displacement fields are shown in Visualization 2. The maximum displacement achieved in this experiment was 0.89 µm.

 

Fig. 4. (a)–(d) Replicated patterns in one single shot for four states. (Visualization 1: interferograms evolution video).

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Fig. 5. (a)–(d) Left: wrapped phase. Right: four states of the displacement field evolution (Visualization 2: evolution maps video).

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6. CONCLUSIONS

We have presented a dynamic ESPI system based on a polarized out-of-plane configuration and a replication system, which simultaneously obtain two parallel speckle patterns with known phase shifts for displacement measurements. Although a dynamic event requires introducing the use of fast detectors, pulsed laser, and robust optical arrangements, the technique we have developed allows one to both reduce the phase steps and, in conjunction with the algorithm presented, obtain the phase in a single capture of the camera by using an easy implementation system. DPP-ESPI allows capturing of phase-shifted interferograms that evolve in time, and which are adequate to compute the phase associated with the displacement.

Previously, the experimental viability of combining polarization with ESPI was verified, in order to establish the experimental conditions that would allow measurements using the phase shift by polarization method [56]. However, the system already reported has the limitation of generating phase shifts in stages, thereby the system is only able to analyze static events or of very slow, almost stationary temporal variation. In the presented system, this technical limitation was solved by coupling a Michelson interferometer that operates as a replicating system of the speckle pattern.

The proposed system could be applied to the characterization of micromechanical devices, the calculation of deformation in dynamic structures, the detection of cracks in metal plates and photovoltaic cells, the study of the movements of biological objects of the human body in vitro or in vivo, or characterizing the mechanical properties of thin films among others. The advancements in dynamic ESPI have allowed for the development of dynamic methods known as 4D measurements, which can operate in the presence of environmental disturbances while preserving measurement accuracy.