## Abstract

We present an interferometric method to analyze transparent samples using complex fringes generated by a parallel phase shifting radial shear interferometer using two coupled interferometers. Parallel interferograms are generated using two interferometers: the first one generates the polarized base pattern, and the second system is used to generate parallel interferograms allowing the adjustment of the $x - y$ positions of the parallel interferograms. To obtain the optical phase map, parallel phase shift is generated by collocating polarizing filters at the output of the system; the polarizers are placed at arbitrary angles since they do not require adjustment because of the phase-recovery algorithm. The optical phase was processed using a two-step algorithm based on a modified Gram–Schmidt orthogonalization method. Such an algorithm has the advantage of not being iterative and is robust to amplitude modulation. The proposed method reduces the number of captures needed in phase-shifting interferometry. We applied the developed system to examine static and dynamics phase objects.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Radial shearing interferometry is one of the most powerful techniques in many domains in optical testing. This method was first proposed by Brown [1] and Hariharan and Sen [2,3]. It has been used in various applications such as spherical surface testing [4–7], wavefront sensing [8–11], and characterization of the emerging beam of a laser [12,13]. Classically, a radial shearing interferometer (RSI) can be implemented with conventional optical components conveniently using lenses and the refraction law [14,15]. In such a way, the incident wavefront in an RSI is expanded, or compressed, in a radial direction. Other different proposals include the use of zone plates [16,17], cube-type beam splitters (BSs) [18], and adaptive optics [19–21]. These interferometers are convenient in applications where the samples of interest have a spherical wavefront or radial symmetry.

On the other hand, this interferometer does not require a plane reference wavefront [6,7,22–25]; nevertheless, one of the disadvantages of the RSI is the difficulty in measuring dynamic wavefronts, since the solution of closed fringes with one interferogram presents a sign error when recovering the optical phase [8,26,27]. One solution to this problem is to use a fast camera that allows the capture of consecutive frames, but only if the variation of the wavefront is slow. In our proposal, we can obtain the advantages of simultaneous phase shift interferometry to improve the accuracy of dynamic measurements and perform real-time analysis [28–32]. Nowadays, several optical systems have been developed to retrieve the optical phase data in a single capture employing polarization, using micropolarizing arrays [30,33], grating interferometers [34], and liquid-crystal spatial modulators [35], among others. These systems have been employed in several fields of application, as in optical metrology, holography, and electronic speckle pattern interferometry (ESPI) [10–14,36,37]. The principal objective of these systems is to recover the optical phase to be able to calculate characteristics of the incident wavefront or physical properties of transparent samples. The main purpose of this work is to measure variations of phase maps of transparent objects using two interference patterns with an unknown phase shift in one shot. The configuration is based on a radial shear-polarized Mach–Zehnder interferometer (MZI) that generates a base pattern with known polarization properties [30,31] and a cyclic path interferometer for replication purposes. Experimental results of different static transparent samples, such as an oil drop and a lens, are obtained. For the study of dynamic events, experimental results of temporal variations of a water drop placed on a slide moving because of gravity are presented.

## 2. INTERFEROMETRIC SYSTEM DESIGN AND BASIC PRINCIPLE

Figure 1 shows the diagram of the polarized parallel phase-shifting interferometer. A spatially filtered polarized plane wavefront linearly polarized at 45 deg is coming from a laser operating at 532 nm. This wavefront is incident at the polarizing beam splitter (PBS) of an MZI. In each arm of the interferometer, a Galilean telescope (${{\rm Gt}_1}$ and ${{\rm Gt}_2}$) is placed. In one arm, the telescope is inverted with respect to the other [38] to reduce the size of the incident beam. The expanded and contracted polarized wavefronts recombine at the BS to give radially sheared wavefronts with orthogonal circular polarizations [pattern base, ${P_b}(x,y)$]. The ${P_b}(x,y)$ enters into the cyclic trajectory system where the incident pattern is divided in amplitude: (a) a pattern is reflected and follows the ${{\rm M}_3} - {{\rm M}_4} - {\rm BS}$ trajectory and emerges from the system, and (b) the other pattern is transmitted following the ${{\rm M}_4} - {{\rm M}_3} - {\rm BS}$ trajectory and emerges from the system. Both patterns enter the polarizer array (PA) to generate patterns with visible fringes and arbitrary phase shifts.

#### A. Radial Shear Phase-Shifting Interferometry

In the first stage of the implemented system, the PBS transmits the horizontally polarized beam and reflects the vertically polarized beam. The separated beams recombine at the output and pass through the quarter-wave plate (QWP) placed at 45 deg with respect to the axial axis. The radially sheared wavefronts have opposite circular polarization and their beam sections are $O(x,y) = {\rm circ}[ {\rho /{M_a}} ] \cdot \exp \,\{ { {\,i\phi ( {x/{M_a},y/{M_a}} )\,} \}} $, and $R(x,y) = {\rm circ}[ \rho ] \cdot \exp \,\{ { {\,i\phi ( {x,y} )\,} \}} $ [39,40]. The amplitude ${P_b}(x,y)$ of the output of the MZI is given by where $\rho = \sqrt {{x^2} + {y^2}} $, ${M_a} = 1/R$ denotes the relative magnification of the pupils, and ${J_L}$ and ${J_R}$ the matrix of circular polarizations to the left and right, respectively. Figure 2 shows the stages of generation of parallel interferograms where at the first stage, the system does not generate an interference pattern; this is observed in Fig. 2(a). In order to observe an interferogram, it is necessary to place a linear polarizer to verify that the two beams interfere; see Fig. 2(b). This is because the beams have cross-circular polarizations, which, when interfered, generate a constant intensity field that does not have a fringe pattern [30,31]; this can be seen in Eq. (1), ${| {{P_b}(x,y)} |^2} = cte$. However, this pattern maintains the polarization properties required to generate phase shifts by placing a linear polarizer. Thus, by placing the auxiliary polarizer and rotating it at any angle $\psi $, an interference pattern can be observed that maintains a constant amplitude modulation [31]. That is, when each field is observed through a linear polarizing filter whose transmission axis is at angle $\psi $, the new polarization states are## 3. TWO-STEP FRINGE PATTERNS BASED ON GRAM–SCHMIDT ALGORITHM

To recover the phase map of the interferograms presented in Eq. (5), we implemented the two-step Gram–Schmidt (GS) algorithm proposed in Refs. [39,40]. The advantage of using a two-step system is that we can avoid the sign problem in closed fringes in one shot without losing spatial resolution. Typically, Eq. (5) can be expressed as

According to Eq. (4), ${A_k}$ represents the background illumination, ${B_k}$ is the amplitude modulation, $\Delta \phi (x,y)$ is the phase map to be recovered, ${\sigma _k}$ is the phase step generated from the polarization states, and ${\eta _k}$ is the speckle noise distribution. Given that we are capturing two steps in one shot, the background illumination as well as the amplitude modulation present variations between the two interferograms that can be interpreted as a spatial–temporal dependency. In order to normalize the fringe patterns, we implemented a prefiltering process, as proposed in Ref. [39], by using isotropic pattern normalization [40]. Other normalization methods can be implemented, such as Gabor filters bank [41], Hilbert–Huang transform [42], or even deep neural networks [43], just to mention some of them. All the prefiltering process does is lead to a propensity for the loss of information, depending on the frequency it is tuned to filter out and the pattern that is to be analyzed. Given the results of the prefiltering process (where ${A_{k}} = \;0$ and ${B_{k}} = \;1$) and considering ${\sigma _1} = 0$ and ${\sigma _2} = \sigma $, we can reformulate Eq. (6) for the two interferograms as

Since we do not know the phase shift $\sigma $ because of the unknown rotation angle $\psi $, we calculate the phase distribution by orthonormalizing the patterns as proposed by Vargas *et al.* in Ref. [39]. Nevertheless, we also are interested in calculating the phase shift $\sigma $ and the rotation angle $\psi $ of the polarizer, in which case we implement the step calculation algorithm of Flores and Rivera in Ref. [44].

## 4. EXPERIMENTAL RESULTS

For the processing of obtaining the interferograms in a single image, a closed diaphragm is placed to generate well-defined pupils in order to detect the centroid of patterns and easily locate them. The complementary metal oxide semiconductor (CMOS) camera used has a resolution of ${2048}\;{\rm pixels}\; \times \;{1536}\;{\rm pixels}$ (3.1 MP). The individual patterns have a spatial resolution of ${800}\;{\rm pixels}\; \times \;{800}\;{\rm pixels}$ and are treated as separate images by the proposed algorithm. For this purpose, we capture two images over the same detector field. Since we have low-frequency interferograms with respect to the inverse of the pixel spacing, the influence of errors in the capture seems to be rather small if noticeable; this is an advantage of the phase-shift technique. The beam size of the collimated incident wavefront is 2 cm. To obtain the results here presented, first, we took a reference phase, which is the phase of the coverslip without the sample; then, the samples were placed on a slide. Finally, we obtain the phases of the reference and the sample separately. The phases shown in the results present the subtraction of the reference phase, generated by the empty coverslip, and the phase of the sample is put on the coverslip. Because the interferometric system generates phase shifts through the modulation of polarization, this method is not suitable for birefringent samples. Other limitations of the arrangement presented are the size of the samples, because they are limited by the size of the optical components used. Previous reports have shown that by analyzing 100 frames each, showing the temporal variation obtained by each pixel, the standard deviation in time of each pixel of the demodulated phase maintained is in a 0.2 rad range [45]. Among other advantages of this system, compared to previously published ones [46], are that the system reduces the number of phase steps required to process the optical phase and obtain parallel phase shift by collocating polarizing filters at arbitrary angles, given that they do not require adjustment because of the phase-recovery algorithm, and our configuration allows dynamic measurements and does not require vibration isolation [46,47].

Given that the GS algorithm orthonormalizes the fringe patterns, the phase step between them is always 90 deg; nevertheless, by using the estimation proposed in Ref. [44], it is possible to estimate the real phase step between the two interferograms. Such results are presented in Table 1, corresponding to each of the results presented in the upcoming figures. Figure 3 presents typical interferograms obtained with the interferometer. Figure 3(a) shows the two parallel patterns obtained. In Fig. 3(b) we present the phase map. Figure 4 shows the results obtained when a water drop is placed on a coverslip; for this case, it was necessary to wait for the flow to stabilize. The parallel patterns show the deformation generated on the original pattern [see Fig. 4(a)]. In Fig. 4(b), the unwrapped phase is shown. Figure 5 shows the case of an oil drop placed by the same procedure on a coverslip. Figure 5(a) shows the parallel patterns obtained in a single capture of the camera, and in Fig. 5(b), we present the phase map; in this case, it can be seen that the oil introduces a larger phase difference because its refractive index is greater than that of the water. For this reason, the frequency of the fringes is greater. In these cases, the relative phase shift calculated by the algorithm is 75.31 deg. The patterns with this symmetry allow the study of spherical surfaces and lenses. To validate it, a lens with a focal distance of $f\; = \;{30}\;{\rm mm}$ was placed in one of the MZI arms, and its corresponding adjustments were made [22–24,32]. The results obtained are shown in Fig. 6. Due to the change in the size of the patterns, small adjustments were made in the zoom focus of the CMOS camera and the angles of the polarizers. Figure 6(a) shows the obtained parallel patterns; in Fig. 6(b) we present the recovered phase map. It can be seen that the phase introduced by the sample has small irregularities due the low quality of the tested lens.

With the purpose of showing the capability of the optical system to process dynamic events, the dynamic phase evolution of water moving for gravity on a microscope slide is shown in Fig. 7. It is important to clarify that the results are obtained using an optical table without pneumatic suspension, and the polarized array used is formed by recycled polarized film placed at arbitrary angles, which presents the advantage of not using a micropolarized array. These results show that dynamic phase objects can be analyzed with the proposed optical system. Figure 7(a) shows a representative frame of the temporal evolution of parallel interferograms (Visualization 1). In Visualization 2, it is possible to see the temporal evolution of the phase map; Fig. 7(b) shows a representative frame.

#### A. Spiral and Fork Fringes

In previous reports [48], it has been shown that patterns can be generated with spiral symmetries without using a Bessel beam. In the results presented in Fig. 8, the spirals that are generated when a phase step is introduced obstructs half of the beam in one of the arms of the MZI. The differences between Figs. 8(a) and 8(b) are due to the inclination of the phase step with respect to the optical axis. These experimental results show that, when tilting the phase step, spirals within the two arms are generated, as shown in Fig. 8(a), and with one arm, as shown in Fig. 8(b). Figure 8(a) shows the case of a topological charge of 2, and Fig. 8(b) can be associated with a topological charge of 1. In both cases, it is shown that they also have fork fringes, which is useful if they are to be used as optical traps because these fork fringes represent an optical vortex in the optical phase of the wavefront.

The interferograms with radial symmetry allow one to easily generate this type of spiral fringe, which can be applied in other areas such as optical traps. With these results, it is useful to know the sign of the phase [48,49].

#### B. Optical Alignment Test

Figure 9 shows how the fringe pattern moves if the object is not aligned perpendicularly with the optical axis; as the sample progressively inclines, the center of the pattern and the fringes move in the same direction, so this technique could be used to align the test object. Figures 9(a)–9(c) show the displacement of the center of the interference patterns; in Figs. 9(b)–9(d), the associated inclination of the test object is presented. The shift generated in the fringes is a linear function with the angle of inclination of the sample. This is ${\rm shift}(\theta )\; = m \cdot \theta + b$, where $m$ is the slope of the line adjusted to the data obtained, $\theta $ is the angle of inclination of the sample, and $b$ is the point where it intercepts the axis. The parameters calculated using linear regression are the following: $m = 2.39$ and $b = 0.0064$, and the correlation factor is 0.99. For this case, the individual patterns have a spatial resolution of ${500}\;{\rm pixels}\; \times \;{500}\;{\rm pixels}$.

## 5. CONCLUSIONS

In conclusion, an adaptation of a radial shear interferometer with a coupled system to simultaneously achieve two parallel interferograms with unknown phase shift for phase measurements has been presented. This technique allows one to reduce the phase steps, and, combined with the proposed algorithm, allows one to obtain the phase in a single shot of the camera. One of the advantages of the proposed system is that it can be easily adapted to generate spirals that indicate the sign of the phase or to generate optical vortices with possible applications as optical traps. The symmetry of the interferograms allowed us to adjust the position of the test object so that it is perpendicular to the optical axis. This interferometric system allows capture of phase-shifted interferograms evolving in time to be used for phase extraction. The phase-shifting techniques have the advantages that they permit rapid measurement, can get best results with low-contrast fringes, and can vary the sensitivity by varying the amount of radial shear. This technique could be applied for the study of transparent biological tissue, calculation of deformation spherical structures, and ophthalmological lens characterization. Knowing the mean refractive index, we can calculate the mean thickness of thin films, which is very convenient in materials science.

## Funding

Consejo Nacional de Ciencia y Tecnología (A1-S-20925).

## Acknowledgment

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper. Author N. I. Toto-Arellano acknowledges the support provided by the National Council of Science and Technology, “Fondo Sectorial de Investigación para la Educación.” Author V. H. Flores thanks CONACYT for the postdoctoral grant provided.

## Disclosures

The authors declare no conflicts of interest.

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